Evaluating Transient Power Losses in Renewable Energy Integrated Power Networks
EMMA SJ ¨ ODIN
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
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Abstract
This thesis investigates the resistive losses incurred in returning a power network to a synchronous state following a transient stability event, or in main- taining this state in the presence of persistent stochastic disturbances. We quantify these transient power losses, the so-called “Price of Synchrony”, using the squared H2norm 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 cou- pled 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 first- and 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 H2norms for both system models are shown to be a function of trans- mission 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 syn- chronous 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 demon- strate 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.
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 stör- ning. Dessa transientförluster, som vi benämner ”synkronismens pris”, utvär- deras med hjälp av H2 -normen för ett linjärt dynamiskt system. I ett första steg modellerar vi ett stort nätverk av synkrongeneratorer och erhåller ett sy- stem med kopplade svängningsekvationer av andra ordningen. Sedan utvidgas denna modell för att även omfatta dynamiska laster och asynkrongenerato- rer, 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 bå- de nätverksfel och fluktuationer i elförsörjningen orsakade exempelvis av vind- eller solkraft.
För att utvädera transientförslusterna används först en typ av reducera- de, eller ”jordade”, laplacianer för att beskriva ett reducerat system som är asymptotiskt stabilt. Vi visar sedan att H2 -normen för det ursprungliga sy- stemet 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 ordning- ens, andra ordningens eller kombinerad dynamik skiljer sig kraftigt åt, så kan deras H2-normer vara lika för nätverk av samma storlek med lika dämpnings- koefficienter. Dessa resultat indikerar att nätanslutna förnybara energikällor inte kommer att öka transientförlusterna om deras regulatorer kan bli anpas- sade till dämpningen hos befintliga synkrongeneratorer. De visar dock också att en ökad utbredning av distribuerad generation, särskilt i mellan- och låg- spänningsnät, kommer att öka transientförlusterna eftersom de växer linjärt med antalet generatorer, samt att denna effekt inte kan mildras genom att öka antalet anslutningar.
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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 vi
1 Introduction 1
1.1 Scope . . . 3
1.2 Related Work . . . 4
2 Preliminaries 7 2.1 Power System Dynamics . . . 7
2.1.1 Classification of Power System Stability . . . 8
2.1.2 The Swing Equation . . . 8
2.2 Network Descriptions and Graph Laplacians . . . 10
2.2.1 The Admittance Matrix . . . 10
2.2.2 Consensus Dynamics and Graph Laplacians . . . 11
2.2.3 Properties of Graph Laplacians . . . 12
2.3 The H2 Norm . . . 13
2.4 Renewable Power Generation . . . 14
2.4.1 Synchronous vs. Asynchronous Generators . . . 15
2.4.2 Wind Power . . . 16
2.4.3 Other Sources . . . 17
3 Resistive Losses in Synchronizing Power Networks 19 3.1 Problem Formulation . . . 20
3.1.1 System Dynamics . . . 20
3.1.2 Performance Metrics . . . 21
3.2 Evaluation of Losses . . . 23
3.2.1 System Reduction . . . 23
3.2.2 H2 Norm Calculation . . . 23
3.2.3 Special Case: Equal Line Ratios . . . 26
3.2.4 H2 Norm Interpretations for Swing Dynamics . . . 27
3.3 Generalizations and Bounds . . . 29
3.3.1 Network-Characteristic Bounds on Losses . . . 29
3.3.2 Generator Parameter Dependence . . . 30
3.4 Numerical Examples . . . 31 vi
CONTENTS vii
3.4.1 Line Ratio Variance . . . 31
3.4.2 Increased Network Size . . . 32
3.4.3 Marginal Losses for Added Lines . . . 33
3.4.4 Effects of Generator Placement . . . 35
3.5 Discussion . . . 36
4 Losses in Renewable Energy Integrated Systems 39 4.1 Problem Formulation . . . 40
4.1.1 Network Model . . . 41
4.1.2 Model of Asynchronous Machines . . . 41
4.1.3 System Dynamics . . . 43
4.1.4 System Inputs . . . 45
4.1.5 Performance Metric . . . 45
4.2 Input-Output Analysis . . . 46
4.2.1 Stability . . . 46
4.2.2 H2 Norm Calculations . . . 47
4.2.3 Properties of the Augmented Network Laplacians . . . 48
4.2.4 Relation to Previous Results . . . 49
4.3 Case Studies . . . 51
4.3.1 Increased Synchronous Damping . . . 51
4.3.2 Effects of Generator Placement . . . 51
4.4 Discussion . . . 53
5 Conclusions and Directions for Future Work 55 A Appendices to Chapter 3 57 A.1 Proof of Lemma 3.2 . . . 57
A.2 Proof of Lemma 3.3 . . . 59
A.3 H2 Norm With Simultaneously Diagonalizable Laplacians . . . 59
A.4 Proof of Corollary 3.6 . . . 61
B Appendices to Chapter 4 63 B.1 Proof of Theorem 4.1 . . . 63
List of Figures 65
Bibliography 67
Chapter 1
Introduction
The electric power system is undergoing large and rapid changes, primarily due to the growing interest in replacing fossil fuel-based power generation with renewable energy sources. Factors driving this replacement are growing concerns about cli- mate change and global warming, diminishing supplies of fossil fuels causing price increases [7] and government mandates world-wide [31]. The Nordic countries, in- cluding 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 en- ergy. 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) pre- scribes 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 men- tioning. 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 over- all 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
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 the- sis however, the concept of synchronization in renewable energy integrated power networks is studied in a different context. We assume that the network will re- turn 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 to- tal 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 3
1.1 Scope
In this thesis, the transient resistive power loss – the price of synchrony – is eval- uated for large power networks, for which we formulate the dynamics as a 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 es- sential 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.
1.2 Related Work
A special case of the problem of rotor-angular or synchronous stability is the tran- sient stability problem, which is associated with large angular disturbances due to e.g. generator or line failures, or the intermittency of the power sources in a renew- able energy integrated system. There is a large body of transient stability 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 equiva- lence 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 damp- ing, 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 op- eration 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
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 use- ful 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 generation- load 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 sys- tem 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 Êiand 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 transmis- sion systems [34], assumes:
i. a flat voltage profile; Vi= V0, ’i = 1, ..., N,
ii. that the line resistance is negligible compared to the reactance in all lines, and iii. that the voltage angle 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
Pe,i¥ÿ
j≥i
bij(◊i≠ ◊j). (2.3)
Substituting this into (2.1) leads to Mi¨◊i+ —i˙◊i = ≠ÿ
j≥i
bij[◊i≠ ◊j] + Pm,i. (2.4) This is the linear version of the swing equation in the classical machine model, 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 angle ◊i and a speed Êi = ˙◊. Any deviations from a steady state will propagate 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].
b12
b23 b13
◊1 Ê1 Ê2
Ê3
Figure 2.2: Mechanical analogy to the power system dynamics and the swing equa- tion (2.4). A deviation of one oscillator’s phase angle ◊iand/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 admit- tance matrix is a type of graph Laplacian or Laplacian matrix; a matrix represen- tation of a network or graph.
Graph Laplacians arise naturally in state space formulations of so-called consen- sus problems, in which a system of agents cooperate with a certain control objective.
Since the coupled oscillator dynamics (2.4) are a type of such consensus dynamics, this type of problem will also briefly be reviewed at this stage, along with 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:
yij = 1
zij = gij ≠ jbij, where gij = r2rij
ij+x2ij and bij = r2xij
ij+x2ij are respectively the conductance and suscep-
2.2. NETWORK DESCRIPTIONS AND GRAPH LAPLACIANS 11 tance of the line. Furthermore, each node i œ N may have a shunt conductance gi
which is the conductance of the node’s connection to ground.
Now we can define the admittance matrix Y by:
Yij :=
Y_ __ ] __ _[
¯gi+ÿ
k≥i
(gik≠ jbik), if i = j,
≠(gij ≠ jbij), if i ”= j and j ≥ i,
0 otherwise.
(2.5)
where j ≥ i denotes a line between nodes i and j. The diagonal elements Yiiof 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+ ¯G) ≠ jLB, (2.6)
where LG is called the conductance and LBthe susceptance matrix. ¯Gis 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: ˙xi = ui, i = 1, . . . , n where the control objective is for all agents to eventually reach the same state x1(t) = x2(t) = · · · = ¯x(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
Lij :=
Y_ __ ] __ _[
ÿ
kœNi
aik, if i = j,
≠aij if i ”= j and j œ Ni,
0 otherwise,
(2.7)
where aij are positive weights of the graph which describes how the agents (nodes) are connected. The elements on the diagonal 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.
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:
˙x = ≠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 ¯x 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.
ii. Zero row/column sums. Since Lii = ≠qj”=iLij, 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., UúU = I), such that L = Uú U, where = diag{⁄1, ⁄2, . . . , ⁄n} is a diagonal matrix of L’s eigenvalues 0 = ⁄1 Æ ⁄2Æ . . . Æ ⁄n.
2.3 The H
2Norm
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
˙x(t) = Ax(t) + Bw(t) (2.9)
y(t) = Cx(t),
where x œ Rn, w œ Rm and y œ Rp. Its p ◊ m-dimensional transfer matrix is given by G(s) = C(sI ≠ A)≠1B. If the system is asymptotically stable, we can define its H2 norm by
||G||2H2 = 1 2fi
⁄ Œ
≠Œ||G(jÊ)||2FdÊ, (2.10) 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(BúXB), (2.11)
where X is the observability Gramian given by
AúX+ XA = ≠CúC.3 (2.12)
The matrix equation (2.12) is referred to as the Lyapunov equation.
In this thesis, we will use the H2 norm to evaluate the resistive losses in sys- tems of oscillating generators during the synchronization transient. This usage is supported by some of the H2 norm’s standard interpretations, of which three are
2The Frobenius norm is defined as the sum of the absolute values of all entries in a matrix:
||A||2F =qn i=1qm
j=1|aij|2 = tr(AúA).
3||H||H2 can also be calculated using the controllability Gramian XC; ||H||2H2 = tr(CXCCú), with AXC+ XCAú= ≠BBú. This formulation will however not be used in this thesis.
presented below. The physical meaning of these interpretations for our 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
tæŒE{yú(t)y(t)}.
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{x0xú0} = BBú, then the H2 norm (squared) is the time integral
||H||2H2 =⁄ Œ
0 E{yú(t)y(t)}dt
of the resulting transient response. This interpretation is closely related to inter- pretation (iii):
iii. Sum of impulse responses. If H were a single-input-single-output (SISO) system, the 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 =ÿm
i=1
⁄ Œ
0 yiú(t)yi(t) dt.
2.4 Renewable Power Generation
A large scale introduction of renewable energy sources to the power grid is, as men- tioned in Chapter 1, apart from introducing high levels of disturbances, likely 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 iner- tia. 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 integra- tion 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 alterna- tors. 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;
s= Ê0≠ Ê 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.
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 tur- bines 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 stand- alone systems, are connected to the grid via a synchronous generator. The
2.4. RENEWABLE POWER GENERATION 17
(a) Fixed-speed (b) DFIG
Figure 2.5: Principles of fixed-speed wind turbines with squirrel cage induction generators (a) and doubly-fed induction generators (DFIGs). Since induction gen- erators 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 world- wide 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
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 Sec- tion 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 en- countering 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 previ- ously 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
r12+ jx12 r23+ jx23
r24+jx24
r34+jx34
r45+jx45 r46+jx46
r56+ jx56 r67+ jx67
G1
{V1, ✓1}
G2
{V2, ✓2}
G3
{V3, ✓3}
G4
{V4, ✓4}
G5
{V5, ✓5}
G6
{V6, ✓6}
G7
{V7, ✓7}
Figure 3.1: An example of a network with N = 7 generator nodes. Each line has 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 as- sumptions, see e.g. [34], to represent the interactions between the generators through
3.1. PROBLEM FORMULATION 21 the network of impedances. The dynamics of each generator i œ {1, . . . , N} are then given by (2.4). By also making use of the susceptance matrix LB, defined by Equa- tions (2.5)-(2.6), we can write rewrite the differential equation (2.4) in state space form as:
d dt
C◊ Ê D
=
C 0 I
≠M≠1LB ≠M≠1B D C◊
Ê D
+ C 0
M≠1 D
w (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 gen- erator 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 formu- lation (3.1), which allows a uniformly sized disturbance to have a larger influence on small-inertia generators. Depending on the character of disturbances, both defi- nitions 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 defini- tion, 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)
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 = ◊úLG◊, where ◊ is the state vector introduced in (3.1). We therefore choose to define an output of (3.1) as
y = CÂ =: ËC1 0È C◊
Ê D
, (3.5)
where Ploss= yúy. Since LG is positive semidefinite, see Section 2.2.3, we can take C1 as the unique positive semidefinite matrix square root L1/2G , which is what we assume from now on.
Equations (3.1) and (3.5) can then be rewritten as
d dt
C◊ Ê D
=
C 0 I
≠M≠1LB ≠M≠1B D C◊
Ê D
+ C 0
M≠1 D
w (3.6a)
y=ËL
12
G 0È C◊
Ê D
(3.6b)
We denote the input-output mapping of (3.6) by H.
The total real power losses incurred in returning this system to a synchronous state after a disturbance can be quantified using the input-output H2 norm, 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 non- zero 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 xúLQx, 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 xúLQx 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 23
3.2 Evaluation of Losses
In order to compute the input-output response of (3.6), we first define a reduced system ˜H 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 = MI 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 kthrows and columns of LGand LBrespectively, by ˜LG
and ˜LB. The states of the reduced system ˜◊ and ˜Ê are then obtained by discarding the kth 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 kth 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 ˜H:
d dt
C˜◊
˜Ê D
=
C 0 I
≠M1 ˜LB ≠M—I D C˜◊
˜Ê D
+ C 0
M1 I D
˜w (3.7a)
=: A˜„ + B ˜w;
˜y =˘LG12 0È C˜◊
˜Ê D
=: C ˜„. (3.7b)
By the assumption of a network where the underlying graph is connected, the grounded Laplacians ˜LG and ˜LB are positive definite Hermitian matrices (see e.g.
[40]). All of the poles of system ˜H are thus strictly in the left half plane and the input-output transfer function from ˜w to ˜y has a finite H2 norm.
3.2.2 H2 Norm Calculation
The (squared) H2 norm of the system ˜H is given by Equations (2.11) - (2.12).
We call the obsevability Gramian ˜X and partition it into four submatrices. The
