System Normapproaches for Power ... - Research Collection

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Doctoral Thesis

System Norm Approaches for Power System Stability Analysisand Control

Author(s): Poolla, Bala Kameshwar

Publication Date: 2019

Permanent Link: https://doi.org/10.3929/ethz-b-000340384

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bala kameshwar poolla

S Y S T E M N O R M A P P R O A C H E S F O R P O W E RS Y S T E M S TA B I L I T Y A N A LY S I S A N D C O N T R O L

diss . eth no. 25823

S Y S T E M N O R M A P P R O A C H E S F O R P O W E RS Y S T E M S TA B I L I T Y A N A LY S I S A N D C O N T R O L

A dissertation submitted to attain the degree of

doctor of sciences of eth zurich

(Dr. sc. ETH Zurich)

presented by

B A L A K A M E S H WA R P O O L L AM. Tech., Control System Engineering, IIT Kharagpur

B. Tech. (H), Electrical Engineering, IIT Kharagpur

citizen of India

accepted on the recommendation of

Prof. Dr. Florian A. Dörfler, examinerProf. Dr. Mihailo R. Jovanovic, co-examiner

Prof. Dr. John W. Simpson-Porco, co-examiner

2019

ETH ZurichIfA – Automatic Control LaboratoryPhysikstrasse 3

8092 Zurich, Switzerland

doi: 10.3929/ethz-b-000340384

isbn: 978-3-906916-58-3

©Bala Kameshwar Poolla, 2019

To Mom and Dad!"# "$% &'( ) !*+,*

“But, after all, who knows, and who can say,whence it all came, and how creation happened?the gods themselves are later than creation,so who knows truly whence it has arisen?

Whence all creation had its origin,the creator, whether she/he fashioned it or whether she/he did not,the creator, who surveys it all from highest heaven,she/he knows– or maybe even she/he does not know.”

–Nasadiya SuktaRig Veda, 10:129-6, 7

Translation: AL Basham

A C K N O W L E D G E M E N T S

Firstly, I would like to thank my advisor Prof. Florian Dörfler for givingme the opportunity to work at ETH Zürich under his guidance. Florianhas been a fantastic mentor and a powerhouse of inspiration who has lefta lasting impact with his zeal, enthusiasm, and passion for research. Hehas always pushed me to strive for “perfection in action”, while providingample freedom and encouragement to explore topics which were of interestto me. It is indeed a privilege to be Florian’s first doctoral student.

I would also like to thank my committee members Prof. Mihailo Jovanovicand Prof. John W. Simpson-Porco for their insightful comments not only onmy thesis but throughout my doctoral study. A lot of work in this thesis isinspired by their previous research and the long exchanges Florian and Ihave had with them over the years. I am also very grateful to Prof. Li Nafor the brief but valuable time I spent with her research group at Harvard.

I would like to thank Prof. Amit Patra, Prof. Santanu Kapat, Prof. Au-robinda Routray, and Prof. Sourav Patra who deeply influenced me duringmy final years at IIT Kharagpur. They were instrumental in developing andnurturing my interest in research and for introducing me to the enrichingworld of control.

The Automatic Control Laboratory (IfA) at ETH offered a fantastic re-search environment. I would like to thank Prof. Manfred Morari, Prof.John Lygeros, Prof. Roy Smith, and Prof. Maryam Kamgarpour for theirefforts in ensuring a lively atmosphere at the lab. I was fortunate to assistthem in teaching some of the courses offered by the laboratory, which wasa very rewarding experience. I have also immensely benefitted from myconversations with the numerous visiting professors and researchers at IfA.

I would like to acknowledge two of my most significant collaborators,colleagues, and lunch-mates, Saverio Bolognani for helping me stay afloat,especially in my initial days at IfA, for his incredible patience and ex-ceptional teaching skills; Dominic Groß for his penchant for rigour andexactness, apart from a shared sense of humour. It was a pleasure to spendso much time interacting, bouncing off ideas, engaging in research withthese two extremely knowledgeable and affable people.

Many thanks to my other collaborators– Theodor Borsche for the longdiscussions on low-inertia systems and for sharing the many power systemstest cases that he developed; Nima Monshizadeh for his technical correct-

vii

ness and imaginative approach to problem-solving; and the industriousmaster students whom I co-supervised over the course of my doctoralstudies.

I had the good fortune of sharing the stunning views of the city andthe lake from my lovely office ETL K 26 with Christian Conte, MarkoTanaskovic, Stefan Bötschi, Marcello Colombino, and Ashish Cherukuri. Iwould like to thank them for indulging in intellectually stimulating dis-cussions, though they often led to no noteworthy conclusions; Tony Wood,Paul Beuchat, Francesca Parise, and Yvonne Stürz for the shared travails oforganizing the annual IfA Open House; Ashish Hota for the long discus-sions on cricket and Indian politics; all members of Florian’s group– CatalinArghir, Adrian Hauswirth, Nicolò Pagan, Taouba Jouni, Jeremy Coulson,Liviu Aolaritei, Lukas Ortmann, Miguel Picallo Cruz, Irina Subotic, WenjunMei, and Robin Delabays for the engrossing weekly meetings and the fun-filled yearly retreats; other lab members– Basilio Gentile, Sandro Merkli,Tyler Summers, Nikolaos Kariotoglou, Joe Warrington, Sergio Grammatico,Xiaojing Zhang, Angelos Georghiou, Claudia Fischer, Andreas Hempel,Bart van Parys, Dario Paccagnan, Ben Flamm, Marius Schmitt, Alex Liniger,Damian Frick, Felix Rey, Annika Eichler, Chitrupa Ramesh, Giampaolo Tor-risi, Tobias Sutter, Orçun Karaca, Luca Furieri, Ilnura Usmanova, GeorgiosDarivianakis for making for making IfA enjoyable and productive; the IfAsecretariat– Martine Wassmer, Sabrina Baumann, Tanja Turner, Alain Bolle,and Markus Thierer for always having a solution to administration relatedqueries and for keeping the lab running.

It would be amiss to not mention the lovely country of Switzerlandfor Roger Federer, chocolates, and extremely generous tax-payers fundingpublic education; my illustrious academic forefathers (including the likesof Joseph Fourier, Joseph-Louis Lagrange, Leonhard Euler, Pierre-SimonLaplace, Carl Gauß) for passing down their knowledge to subsequentgenerations; the anonymous reviewers who painstakingly reviewed myresearch and provided critical feedback; friends from IIT Kharagpur for themaintaining the “high tempo”; Yagnesh Ramalingam for his friendship andall the conversations- from the mundane to the sublime over the years.

Finally, I would like to thank my parents, who have been an inexhaustiblesource of motivation for their wholehearted support in all my endeavours.I remain eternally indebted to them.

viii

A B S T R A C T

This thesis fundamentally concerns with stability analysis of power systemsfrom a system norm viewpoint. A major part of this thesis is devotedto frequency stability problems in low-inertia power grids. Subsequently,the viability of system norms as an effective control-theoretic tool forperformance analysis of primal-dual saddle-point algorithms and variationsthereof is elaborated upon.

While the share of renewable-based distributed generation has been onthe rise, there has also been a decline in the conventional synchronous-basedgeneration. The renewable-based power generation interfaced with the gridvia power electronic converters, however, does not provide rotational inertia,an inherent feature of synchronous machines. This absence of inertia hasbeen highlighted as the prime source for the increasing frequency violationswhich have severely impacted grid stability. As a countermeasure, virtualor synthetic inertia, fast frequency response, emulated by advanced controltechniques have been proposed. We discuss in depth, the appositeness,implementation, and optimal tuning of such devices. To factor in for theeconomics of the provision of such virtual inertia devices, we also constructa market mechanism inspired by the ancillary service markets, pivotedaround social welfare optimization and the VCG payment rule. The re-sulting mechanism ensures truthful bidding to be the dominant biddingstrategy and guarantees non-negative payoffs for agents providing virtualinertia while maximizing social welfare.

Conventionally, time-domain and spectral metrics have been engagedto gauge the resilience of power systems. In this work, we focus on per-formance metrics based on the notion of system norms, accounting fornetwork coherency as well as efficient use of control energy. We illustratehow such an approach constitutes a compelling tool and results in tractablereformulations of power system stability problems.

We motivate the analysis by considering small-scale, linear multi-machine/multi-inverter power system models and develop computational approachesfor solving the non-convex optimal inertia placement problem. Next, wedevelop explicit models of two particular implementations of virtual inertiaand fast frequency response, the grid-following and grid-forming schemes.These implementations, while conclusively capturing the key dynamic char-acteristics of virtual inertia emulation also remain amenable for integration

ix

with large-scale, non-linear, high-fidelity power system models. The pa-rameters of these devices are tuned and the specific location in the powersystem optimized, in order to improve resilience. A comprehensive casestudy based on the model of the South-East Australian system is used toillustrate the effectiveness of such devices.

The system norm approach developed in this thesis can also be used tostudy optimal frequency regulation problems in power systems. These prob-lems can be solved via saddle-point methods, which have recently attractedrenewed interest as a systematic technique to design distributed algorithmsfor solving convex optimization problems. However, when implementedonline as dynamic feedback controllers, these algorithms are often subjectto disturbances and noise. As the conventional standalone convergence rateevaluation fails to provide the complete picture of performance, quantify-ing the input-output performance becomes more meaningful, thereby, alsounderscoring the wide-applicability of system norms. Lastly, we extend ouranalysis to a more generic resource allocation problem and compare theinput-output performance of various centralized and distributed saddle-point implementations.

x

Z U S A M M E N FA S S U N G

Diese Arbeit behandelt die Stabilitätsanalyse von Stromnetzen mittels Sys-temnormen. Der Großteil dieser Arbeit beschäftigt sich mit dem Problemder Frequenzstabilität in trägheitsarmen Stromnetzen. Des Weiteren wirddie Verwendung von Systemnormen als effektives regelungstheoretischesWerkzeug zur Analyse der Güte von primal-dualen Algorithmen für Sattel-punktprobleme untersucht.

Während der Anteil der regenerativen dezentralen Stromerzeugungsteigt, ist ein Rückgang der konventionellen synchronen Stromerzeugungzu verzeichnen. Im Gegensatz zu Synchronmaschinen verfügt die über Fre-quenzumrichter an das Energienetz angeschlossene regenerative Stromer-zeugung jedoch über keine rotierende Masse und Trägheit.

Diese fehlende Trägheit ist eine der Hauptquelle für die zunehmendenFrequenzabweichtungen, die die Netzstabilität stark beeinträchtigen. AlsGegenmaßnahme werden in der Literatur fortschrittliche Regelungsverfah-ren wie virtuelle oder synthetische Trägheit und schnelle Primäregelungvorgeschlagen.

In dieser Arbeit wird ausführlich die Eignung, Implementierung undoptimale Regelerauslegung für solche Systemdienstleistungen betrachtet.Um die Wirtschaftlichkeit der Bereitstellung solcher virtueller Trägheitsein-richtungen zu berücksichtigen, wird ein Marktmechanismus untersucht, dersich an den Märkten für Systemdienstleistungen orientiert und mittels desVickrey-Clarke-Groves-Mechanismus eine Wohlfahrtsfunktion optimiert.Der resultierende Mechanismus stellt sicher, dass wahrheitsgetreues Bie-ten die dominante Strategie ist, garantiert nicht-negative Auszahlungenfür die Agenten die virtuelle Trägheit bereitstellen, und maximiert dieWohlfahrtsfunktion.

Die Stabiltät und Robustheit von Stromnetzen wird in der Regel mittelsEigenwertanalyse oder Metriken im Zeitbereich (d. h. mittels Simulationen)untersucht. In dieser Arbeit konzentrieren wir uns hingegen auf Metri-ken, die auf dem Begriff der Systemnormen basieren, und unter anderemFrequenstabilität sowie der effizienten Nutzung von Regelenergie berück-sichtigen. Wir veranschaulichen, dass ein solcher Ansatz ein wertvollesWerkzeug zur Analyse von Stabilitätsproblemen in Stromnetzen darstelltund in Optimierungsproblemen mündet, die effizient lösbar sind.

xi

Wir motivieren die theoretische Analyse, indem wir lineare modellevon Übertragungsnetzen mit mehreren Maschinen und Umrichtern undgeringer Komplexität betrachten und entwickeln numerische Verfahrenum das nicht konvexen Problem der Berechnung einer optimalen Träg-heitsverteilung zu lösen. In einem weiteren Schritt entwickeln wir expliziteModelle von zwei spezifischen Implementierungen (grid-following undgrid-forming) von virtueller Trägheit und schneller Primärregelung.

Diese Implementierungen bilden die wichtigsten dynamischen Eigen-schaften der Trägheitsemulation ab und eignen sich für die Integration mitnichtlinearen hochgenauen Simulationsmodellen von großen Stromnetzen.Im weiteren werden sowohl die Parameter als auch der Einsatzort im Strom-nnetz optimiert, um die Belastbarkeit des Netzes zu erhöhen. Anhand einerumfassenden Fallstudie, die auf dem Modell des südostaustralischen Über-tragungsnetzes basiert, wird die effektivät der vorgeschlagenen Methodeveranschaulicht.

Der in dieser Arbeit entwickelte Ansatz auf Basis von Systemnnormenkann auch zur Untersuchung von Problemen der optimalen Frequenzre-gelung in Stromnetzen verwendet werden. Diese Probleme können mitHilfe von Sattelpunkt-Methoden gelöst werden, die in jüngster Zeit alssystematische Methode zur Entwicklung verteilter Algorithmen zur Lösungkonvexer Optimierungsprobleme auf Interesse gestoßen sind.

Bei der Online Implementierung als dynamische Regler werden die-se Algorithmen Störungen und Rauschen ausgesetzt. Da eine herköm-liche Betrachtung von Konvergenzraten keine vollständigen Bewertungder Robustheit oder Güte erlaubt werden in dieser Arbeit Systemnormenverwendet um die Ein-/Ausgangsgüte zu charakterisieren. Dies unter-streicht die breite Anwendbarkeit von Systemnormen. Zuletzt erweiternwir unsere Analyse auf ein allgemeineres Problem der Ressourcenzuwei-sung und vergleichen die Güte verschiedener zentralisierter und verteilterSattelstützpunkt-Implementierungen.

xii

C O N T E N T S

List of Figures xviList of Tables xix1 introduction 1

1.1 Motivation 1

1.2 Outline 4

1.3 Publications 5

2 preliminaries 7

2.1 Signal and System Norms 7

2.2 Swing Equation 9

2.2.1 Inertia Constant 9

2.3 Frequency Stability 10

3 optimal placement of virtual inertia 13

3.1 Related Works 13

3.2 Contributions 13

3.3 Problem Formulation 15

3.3.1 System Model 15

3.3.2 Coherency Performance Metric 16

3.4 Optimal Inertia Allocation 19

3.4.1 Performance Bounds 19

3.4.2 Noteworthy Cases 21

3.4.3 Explicit Results for a Two-Area Network 24

3.4.4 A Computational Method for the General Case 26

3.4.5 Economic Allocation of Resources 29

3.4.6 The min-max Problem: Optimal Robust Allocation 29

3.5 Case Study: 12-Bus-Three-Region System 31

3.6 Conclusions 37

4 implementation of virtual inertia and fast frequency

response 39



4.3 System Model 41

4.3.1 Modeling of Virtual Inertia Devices 41

4.3.2 Disturbance Model 45

4.4 Performance Metrics and Design Constraints 45

4.4.1 Performance Metrics 45

xiii

xiv contents

4.4.2 Design Constraints 47

4.5 Closed-loop System and H2 Optimization 48

4.5.1 Closed-loop System Model and Linearization 48

4.5.2 Virtual Inertia as Output Feedback 50

4.5.3 H2 norm Optimization 51

4.5.4 Complexity of the Gradient Computation 52

4.6 Test Case Description 53

4.7 Simulation Results 53

4.7.1 Validity of the Linearized Model 55

4.7.2 Optimal Tuning and Placement of VI Devices 55

4.7.3 Contrasting Allocations for VI Implementations 56

4.7.4 Impact of VI Devices on Frequency Stability 56

4.7.5 Time-domain Responses 59

4.8 Summary and Conclusions 61

5 a market mechanism for virtual inertia 63



5.3 Problem Background 65

5.3.1 System Model 65

5.3.2 Virtual Inertia and Post-fault Behavior 67

5.4 Centralized Planning Problem for Inertia 70

5.5 A Market Mechanism for Inertia 72

5.5.1 Overview and Preliminaries: Mechanism Design 72

5.5.2 VCG Market Mechanism 74

5.6 Numerical Case Study 79

5.6.1 Simulation Setup 79

5.6.2 Simulation Results 81

5.7 Conclusions 82

6 performance of linear-quadratic saddle-point algo-rithms 85



6.3 Saddle-Point Methods and H2 Performance 89

6.4 H2 Performance of Saddle-Point Methods 91

6.4.1 Regularized Saddle-Point Methods 94

6.4.2 Augmented Saddle-Point Methods 97

6.5 Dual and Distributed Dual Methods 102

6.5.1 Centralized Dual Ascent 102

6.5.2 Distributed Dual Augmented Lagrangian 103

contents xv

6.6 Application to Optimal Frequency Regulation 107

6.6.1 Power Network Model and Frequency Regulation 107

6.6.2 Performance of Primal-Dual Frequency Controllers 109

6.7 Application to Resource Allocation Problems 112

6.8 Conclusions 120

7 conclusions and outlook 121

7.1 Stability Analysis and Frequency Control in Future Low-Inertia Power Systems 121

7.2 Future Outlook 123

a appendix 125

a.1 Expression for the H2 Norm 125

a.2 Proof of Observability Gramian Lemma 127

a.3 Gradient Computation for H2 Norms via Perturbation 128

a.4 Gradient Computation for Output Feedback 129

a.5 Gradient-based Optimization of H2 Norms 130

a.6 Proof of Global Convergence to Optimizer Lemma 131

bibliography 133

L I S T O F F I G U R E S

Figure 1.1 World energy consumption cumulative growth %since 1990. Data source: Financial Times/BP. 2

Figure 1.2 World energy consumption in billion tonnes of oilequivalent. Data source: Financial Times/BP. 2

Figure 1.3 Schematic representation of the paradigm shift inthe nature of power generation from a larger shareof conventional sources (left) to that of power elec-tronics interfaced renewable generation. 3

Figure 2.1 Schematic representation of power system frequencyresponse to large disturbances, such as generatorfaults/ network splits (upper panel) and to smallerpersistent disturbances, such as fluctuating powergeneration from renewables (lower panel). 11

Figure 2.2 Typical time scales of frequency-related dynamics inconventional power systems as well as typical timescales of frequency control. 12

Figure 3.1 Cost function profiles for identical and weakly dis-similar πi/di ratios for the two-area network. 25

Figure 3.2 Optimal inertia allocation for a two-area systemwith non-identical damping coefficients di, and dis-turbances inputs varying from (π1, π2) = (0, 1) to(π1, π2) = (1, 0). We choose d1 = 1 < d2 = 2,mbdg = 25, and a12 = 1 as the system parame-ters. 26

Figure 3.3 A 12 bus three-region system test case. Grid parame-ters: Transformer reactance 0.15 p.u., line impedance(0.0001 + 0.001j) p.u./km. 31

Figure 3.4 Optimal inertia allocations, performance compari-son for the test case in Figure 3.3, under differentscenarios- 1. 33

Figure 3.5 Optimal inertia allocations, performance compari-son for the test case in Figure 3.3, under differentscenarios- 2. 34

xvi

list of figures xvii

Figure 3.6 Relative performance loss (%) as a function of penaltyγ with capacity constraints. 0%, 100% correspondto the optimal allocation, no additional allocationrespectively. 35

Figure 3.7 Time-domain plots for angle differences, frequencies,and control effort m4 · θ4 for a localized disturbanceat node 4. 36

Figure 3.8 The eigenvalue spectrum of the state matrix A fordifferent inertia profiles, where m? has been opti-mized for a disturbance at node 4. 37

Figure 4.1 A schematic representation of a grid-following vir-tual inertia device. 42

Figure 4.2 Interconnection of a single grid-following virtualinertia device with power set-points according to(4.3). 43

Figure 4.3 A schematic representation of a grid-forming virtualinertia device. 44

Figure 4.4 Interconnection of a single grid-forming VI devicemodeled via (4.4). 44

Figure 4.5 Closed-loop system interconnection for the grid-forming VI with tuning parameters Kform, whereVSC is the voltage source converter. 49

Figure 4.6 Closed-loop system for the grid-following VI withtuning parameters Kfoll, where CPS is the control-lable power source. 49

Figure 4.7 South-East Australian Power System line diagram.The crossed out generators are replaced by con-stant power sources to mimic a low-inertia scenario,whereas the red lightning symbols are the locationswhere disturbances are injected. The circles withVI inscribed within indicate the virtual inertia anddamping devices distributed across the power sys-tem. 54

Figure 4.8 Distribution of relative linearization errors for loadsteps ranging from −250 MW to +250 MW at thenodes indicated in Figure 4.7 for both grid-forming,grid-following configurations. 55

xviii list of figures

Figure 4.9 Optimal inertia and damping allocations for theAustralian system for the grid-forming and grid-following configurations. 57

Figure 4.10 Distribution of generator frequency nadirs, maxi-mum generator RoCoF, and VI power injections forload steps ranging from −350 MW to −150 MW atthe nodes indicated in Figure 4.7 for different con-verter configurations. 58

Figure 4.11 Time-domain plots for generator frequencies, gener-ator RoCoF, generator power injections, and powerinjections of the VI devices for the original system,grid-following, and grid-forming configurations fora step disturbance of 200 MW at node 508. 60

Figure 5.1 A schematic representing the operator-agent inter-action for the centralized and market-based mech-anism design and highlighting the richness of theproblem under consideration. We depict differentnodes Ai through rectangular boxes and their non-homogeneity in the value of inertia at these nodesthrough different colors. The number of circles ineach rectangle indicate the number of agents; thecolor, their dissimilarities; and the radii of the cir-cles, their inertia capacities. 71

Figure 5.2 Schematic representing the operator-agent mecha-nism through a regulator proposed mechanism de-sign. 76

Figure 5.3 A 12 bus three-region system test case with gridparameters as in Figure 3.3 and virtual inertia agentsat buses 2, 4, 8, 12. 80

Figure 5.4 Inertia profiles for the grid with a primary controleffort penalty and the associated worst-case perfor-mance. 82

Figure 5.5 Virtual inertia contributions from agents and the to-tal monetary cost incurred under regulatory, market-based setups. 83

Figure 5.6 Average payment and average cost profiles for agentsparticipating in the market-based auction for procur-ing virtual inertia. 84

Figure 6.1 System norm for regularized dynamics as a functionof ε, for parameters τx = 1, τν = 1, tc = 1, tb = 3,Q = 3I5, and S = [0.82 0.90 0.13 0.91 0.63]. 98

Figure 6.2 System norm for regularized dynamics as a functionof ε, for parameters τx = 1, τν = 1, tc = 1, tb = 1,Q = 0.05I5, and S = [0.82 0.90 0.13 0.91 0.63]. 98

Figure 6.3 Steady-state variance for the unaugmented case, forparameters n = 2, Q = diag(4, 25), τx = 1, τξ =

1, τν = 1, τµ = 1, E = [1 − 1]>; the remainingparameters do not influence the results. 117

Figure 6.4 Steady-state variance for the augmented case, forparameters n = 2, ρ = 100, Q = diag(4, 25), τx = 1,τξ = 1, τν = 1, τµ = 1, E = [1− 1]>; the remainingparameters do not influence the results. 118

Figure 6.5 RAdualdist (ρ) for Q = diag(4, 25, 16, 49). 119

Figure 6.6 RAcent, RAdist, RAdualdist for Q = diag(4, 4, 4, 9) and

line graph. 120

L I S T O F TA B L E S

Table 2.1 H constants for various modes of generation. 10

Table 4.1 Performance metrics for a load increase of 200 MWat node 508. 62

Table 6.1 Comparison of squaredH2 norm expressions. 115

Table 7.1 Summary of results on the basis of the optimizercharacteristics. 123

Table 7.2 Summary of results on the basis of the performancebounds. 123

xix

1I N T R O D U C T I O N

1.1 motivation

The past decade has seen a concerted focus on alternate sources of en-ergy to replace conventional synchronous machine-based generation asindicated in Figure 1.1 and Figure 1.2. A majority of the concerns forcingsuch a shift– namely greenhouse emissions, safety of nuclear generationand waste disposal, etc., are effectively addressed by cleaner alternatives,primarily-wind and photovoltaics. These sources are interfaced by meansof power electronic converters (see Figure 1.3). Their large-scale integration,however, has raised concerns about system stability (especially frequencystability) [1]–[3] and has been recognized as one of the prime concernsby transmission system operators [4]–[8]. In the event of faults such asloss of generators, sudden fluctuation in power injections due to variablerenewable sources, tie line faults, system splits, loss of loads, etc., leadingto frequency deviations, the inherent rotational inertia [7], [9], [10] of thesynchronous machines acts as a first response by providing (or absorbing)kinetic energy to (or from) the system. This coupled with the damping pro-vided by governors assures system stability. In contrast, converter-interfacedgeneration fundamentally offers neither of these services, thus, makingthe system prone to instability. This thesis is primarily concerned with thestability analysis and frequency control in such future low-inertia powersystems.

Not only are low levels of inertia troublesome, but particularly spatiallyheterogeneous and time-varying inertia profiles can lead to destabilizingeffects as shown in an interesting two-area case study [11]. It is, there-fore, not surprising that rotational inertia has been recognized as a keyancillary service for power system stability and a plethora of mechanismshave been proposed for inertia emulation (also known as virtual or syn-thetic inertia) [12]–[15] through a variety of devices, ranging from windturbine control [16] over flywheels to batteries [17]. Inertia monitoring [18],virtual inertia provision by transmission operators, and introducing market-sourcing mechanisms to this end have also been suggested [19], [20]. Thisalso, therefore, invites questions related to the provision and integrationof inertia-based ancillary services into the existing architecture. However,

1

2 introduction

Wind, Solar Hydro Gas Coal Oil Nuclear

1226

82 78 71 37 29

Figure 1.1: World energy consumption cumulative growth % since 1990. Datasource: Financial Times/BP.

Oil 4.3

Coal 3.8

Gas 3.1

Hydro 0.9Nuclear 0.6Wind, Solar 0.40.03

0.5

1.82.2

3.2

1990 2015

Figure 1.2: World energy consumption in billion tonnes of oil equivalent. Datasource: Financial Times/BP.

1.1 motivation 3

as this is an emerging research area, there is limited literature on marketdesign mechanisms for virtual inertia provision.

Figure 1.3: Schematic representation of the paradigm shift in the nature of powergeneration from a larger share of conventional sources (left) to that ofpower electronics interfaced renewable generation.

Conventionally, the total inertia and primary frequency control in thesystem were the main metrics utilized for system resilience analysis [6].Other commonly used performance metrics to quantify power systemrobustness include time-domain and spectral metrics such as frequencynadir, RoCoF (rate of change of frequency), and power system dampingratio [11]. In [21] the power system is reduced to a single swing equationwith first-order turbine dynamics and the damping ratio, peak overshootare considered as measures of system robustness. These approaches andmetrics are, however, not immediately scalable for analysis of large-scalepower systems. In addition, most of the numerical methods often result inproblems involving significant computational ability.

In this thesis, we address these issues of system stability and frequencycontrol by utilizing concepts from control theory and propose tractablereformulations of the problem. Among the repertoire of available mathe-matical tools, system norms present themselves as a powerful yet promisingapproach for such analyses. As we show in detail in the subsequent chap-ters, not only do these effectively capture information relating to system

4 introduction

performance contained in the spectral and time-domain metrics, numericalmethods involving system norms significantly improve tractability. Theseadvantages of system norms also find favour while dealing with problemsrelated to optimization, thereby enabling us to advance our approaches toother applications involving analyses of input-output maps. In addition, sys-tem norms can be used to study a very general set of algorithms. The familyof such algorithms also encompasses, in particular, the optimal frequencyregulation problem. As shown in [22] the standard primary/secondarycontrol dynamics of a power network can themselves be interpreted as aprimal-dual algorithm for solving an optimal frequency regulation problem.The resulting dynamics can be interpreted as a concatenation of power sys-tem dynamics along with a real-time distributed control layer for the powergrid, ostensibly replacing the secondary/tertiary centralized control layers.Extensions of this framework to load-side optimal frequency regulation andmixed generator-side/load-side optimal frequency regulation can be foundin [23], [24] and in [25] for a port-Hamiltonian perspective.

1.2 outline

This thesis is organized as follows. The Chapters 3–5, primarily deal withsystem norm analysis of low-inertia power grids, whereas Chapter 6 con-cerns with the input-output performance analysis of saddle-point algo-rithms via the prism of system norms. In especially considers applicationsrelating to optimal frequency regulation and resource allocation. A briefoverview of the chapters is presented below.

In Chapter 2, we revisit some of the preliminary concepts from con-trol and power systems which form the building blocks of this work. Inparticular, we investigate signal and system norms, swing equations, andfrequency stability with a heightened focus on the role of rotational inertiain conventional power systems.

In Chapter 3, we pursue the questions raised in [11] regarding the detri-mental effects of spatially heterogeneous inertia profiles and how they canbe alleviated by inertia emulation throughout the grid. In particular, weaddress the allocation problem of “where to optimally place inertia ?”. Weconsider a linear, network-reduced power system model along with anH2 performance metric accounting for the network coherency. A set ofclosed-form global optimality results for particular problem instances aswell as a computational approach resulting in locally optimal solutions arepresented. Further, the robust inertia allocation problem, wherein the opti-

1.3 publications 5

mization is carried out accounting for the worst-case disturbance locationis also considered. As an illustration of our results, a three-region powergrid case study is investigated with different placement heuristics in termsof different performance metrics.

In Chapter 4, explicit models of grid-following and grid-forming virtualinertia (VI) devices are developed for inertia and fast frequency emulationin low-inertia systems. We propose a computationally efficient H2 norm-based algorithm to optimally tune the parameters and the placement of theVI devices in order improve the resilience of low-inertia power systems byexploiting the interpretation of VI devices as feedback controllers. A casestudy based on a high-fidelity model of the South-East Australian systemis used to illustrate the effectiveness of such devices.

In Chapter 5, we propose a market mechanism inspired by the ancillaryservice markets in power systems. We consider a linear, network-reducedpower system model along with a robust H2 performance metric penalizingthe worst-case primary control effort. With social welfare maximization forthe system operator as a benchmark, we construct a market mechanism inwhich bids are invited from agents providing virtual inertia based on theclassical Vickrey-Clarke-Groves (VCG) payment rule. A three-region casestudy is considered for simulations and a comparison with a regulatoryapproach to the same problem is presented.

In Chapter 6, we use the H2 system norms to quantify how effectivelyprimal-dual distributed algorithms reject external disturbances. For thelinear primal-dual algorithms arising from quadratic programs, we providean explicit expression for the H2 norm and quantify the performance im-provements achieved by other variations (such as regularized, augmented)of the algorithm. We contrast the input-output performance of variouscentralized and distributed saddle-point implementations by consideringtwo particular applications: power system optimal frequency regulation bymeans of distributed primal-dual controllers and the more generic resourceallocation problems.

1.3 publications

The body of work presented in the following chapters borrows from previ-ously published and/or submitted articles. The articles corresponding toeach chapter are listed below.

Chapter 2

6 introduction

• B. K. Poolla, D. Groß, T. S. Borsche, S. Bolognani, and F. Dörfler,“Virtual inertia placement in electric power grids", in Energy Marketsand Responsive Grids, S. Meyn, T. Samad, I. Hiskens, and J. Stoustroup,Eds., vol. 162, Springer, 2018, pp. 281–305.

Chapter 3

• B. K. Poolla, S. Bolognani, and F. Dörfler, “Optimal placement ofvirtual inertia in power grids", IEEE Transactions on Automatic Control,vol. 62, no. 12, pp. 6209–6220, 2017.

• B. K. Poolla, S. Bolognani, and F. Dörfler, “Placing rotational inertia inpower grids", in Proc. American Control Conference, 2016, pp. 2314–2320.

Chapter 4

• B. K. Poolla, D. Groß, and F. Dörfler, “Placement and Implementa-tion of Grid-Forming and Grid-Following Virtual Inertia and FastFrequency Response", IEEE Transactions on Power Systems. In Press.

• D. Groß, S. Bolognani, B. K. Poolla, and F. Dörfler, “Increasing theresilience of low-inertia power systems by virtual inertia and damp-ing", in Proc. IREP Bulk Power System Dynamics and Control Symposium,2017.

Chapter 5

• B. K. Poolla, S. Bolognani, L. Na, and F. Dörfler, “A market mechanismfor virtual inertia", Available at https://arxiv.org/abs/1711.04874.

Chapter 6

• J. W. Simpson-Porco, B. K. Poolla, N. Monshizadeh, and F. Dörfler,“Input-output performance of linear-quadratic saddle-point algorithmswith application to distributed resource allocation problems", IEEETransactions on Automatic Control, To appear.

• J. W. Simpson-Porco, B. K. Poolla, N. Monshizadeh, and F. Dörfler,“Quadratic performance of primal-dual methods with application tosecondary frequency control of power systems", in Proc. IEEE Controland Decision Conference, 2016, pp. 1840–1845.

2P R E L I M I N A R I E S

In this chapter, we introduce some concepts from control theory and powersystems which form the basis for the analysis in the subsequent chap-ters. Furthermore, for self-containment, each chapter separately includesnotation relevant to its contents.

2.1 signal and system norms

•L2 normThe squared L2 norm of square-integrable signal u(t) given by

‖u‖22 :=

∫ ∞

0u(t)>u(t)dt,

represents the total energy contained in the signal. From the Parseval’stheorem, the frequency domain representation is

‖u‖22 :=

12π

∫ ∞

0U(jω)>U(jω)dt,

where U(jω) is the Fourier transform of u(t).

•L∞ normThe L∞ norm of an amplitude bounded signal w(t) is given by

‖w‖∞ := supt≥0|w(t)|,

and represents the maximum value the signal can take.

•H2 NormConsider the linear time-invariant system

x = Ax + Bη,

z = Cx ,(2.1)

7

8 preliminaries

where η is the disturbance input signal and z is the performance output.With x(0) = 0, we denote the linear operator from η to z by G. If (2.1) isinput-output stable, its H2 norm ‖G‖H2 is defined as

‖G‖2H2

:=1

2π

∫ ∞

−∞Tr(G(−jω)>G(jω))dω,

where G(jω) = C(jωIn − A)−1B is the frequency response of (2.1) for anidentity matrix In of appropriate dimension and Tr(·) is the trace operator.Another interpretation of ‖G‖2

H2is the steady-state variance of the output,

i.e.,‖G‖2

H2:= lim

t→∞E[z(t)> z(t)], (2.2)

when each component of η(t) is stochastic white noise with unit covariance(E[η(t) η(t′)>] = δ(t− t′)In), where E is the expectation operator and δ(t)is the Dirac delta function. Therefore, ‖G‖H2 measures how much theoutput varies in steady-state under stochastic disturbances.

If the state matrix A is Hurwitz (in this thesis, we also explore a fewextensions for state matrices with a zero eigenvalue), then the H2 norm isfinite, and can be computed (see Chapter 4 of [26] and Appendix A.1) as

‖G‖2H2

= Tr(B>PB), (2.3)

where the observability Gramian P = P> � 0 is the unique solution of theLyapunov equation

PA + A>P + C>C = 0. (2.4)

If the pair (C, A) is observable, then P is positive definite.

•H∞ Norm (Stable systems)For the system considered in (2.1), its H∞ norm ‖G‖H∞ is defined as thegreatest possible ratio of the L2 norm of the output signal to the L2 normof square integrable input signals η, i.e.,

‖G‖H∞ := sup‖η‖2 6=0

‖Gη‖2

‖η‖2.

In the frequency domain, the H∞ norm is defined as the supremum of thelargest singular value of the transfer function over the imaginary axis

‖G‖H∞ := supω

σmax(G(jω)).

2.2 swing equation 9

2.2 swing equation

The swing equation reasonably approximates synchronous machine behav-ior around the steady-state of operation and is usually employed for systemanalysis. This equation can be derived by applying Newtonian mechanicsto rotating bodies. Let Tm be the mechanical torque corrected for lossesand Te be the electrical torque. The net torque dictates the change in themechanical angular velocity ωm of the machine, i.e.,

Jddt

ωm = Tm − Te, (2.5)

where J is the effective moment of inertia of the machine.

2.2.1 Inertia Constant

The inertia constant H is defined as the kinetic energy in watt-seconds atthe rated speed divided by the rated power of the machine (VA base) [27].For a rated angular velocity ω0m, the inertia constant is

H :=12

Jω20m

VAbase. (2.6)

On rearranging (2.5) and substituting the above expression, we have

2Hddt

(ωm

ω0m) =

Tm − Te

VAbase/ω0m. (2.7)

As the electrical angular velocity ωr and the mechanical angular ro-tor velocity ωm are a factor of the number of poles p f of the machineapart (ωr = ωm p f ), the electrical angular position δθ with respect to asynchronously rotating reference for δθ(0) = δθ

0 is δθ = ωrt−ω0t + δθ0 . The

term ω0 = ω0m p f is the rated electrical angular velocity.In terms of the electrical angular position and the inertia constant, the

swing equation (2.7) can be expressed as

2Hω0

d2

dt2 δθ =Tm − Te

VAbase/ω0m.

The typical values of H constants are enumerated in Table 2.1 [28].Though most power system analyses rely on swing equations, we be-

lieve that such models may become questionable while analyzing futurepower systems. We shall investigate converter models, applicable for powersystems with significant renewable generation in Chapter 4.

10 preliminaries

Generation H constant

Turbine generator condensing

–1800 rpm 9− 6 seconds


Turbine non-condensing


Water wheel generator

–slow speed (<200 rpm) 3− 2 seconds

–high speed (>200 rpm) 4− 2 seconds

Synchronous condenser

–large 1.25 seconds

–small 1 second

Synchronous motor with varying load 2 seconds

Table 2.1: H constants for various modes of generation.

2.3 frequency stability

The amount of inertia available in the grid directly affects the rate ofchange of frequency (RoCoF) in the instants that immediately follow a largedisturbance, as depicted in upper panel of Figure 2.1 (see for example thenumerical investigation in [29] for a quantification of this relation). Basedon this understanding, RoCoF is typically adopted as the main metric toevaluate the robustness of the system in terms of frequency stability, for thefollowing reasons:

(i) Steep changes in grid frequency (i.e., large RoCoF) are poorly toleratedby the prime movers of power plants, leading to a higher probability offurther disconnections, and ultimately to cascading events.

(ii) Immediately after a fault, the grid frequency will be different at differentbuses. Therefore, larger RoCoF translates into potentially larger voltageangle differences across power lines, and therefore, a higher probability ofprotection tripping and even network splitting.

2.3 frequency stability 11

(iii) Moreover, as the governor response of the generators (primary frequencyregulation) does not act until seconds after an incident, the RoCoF directlyaffects the lowest frequency reached by the grid (the frequency nadir,in Figure 2.1). A low-frequency peak can lead to the disconnection ofgenerators, load shedding intervention, and is therefore dangerous forsystem stability.

ωnominal frequency

RoCoF (max rate of change of frequency)

frequency nadir

energy unbalance

restoration time

ωnominal frequency

secondary control

ωnominal frequency

RoCoF (max rate of change of frequency)

frequency nadir

energy imbalance

restoration time

ωnominal frequency

secondary controlenergy imbalance

Figure 2.1: Schematic representation of power system frequency response tolarge disturbances, such as generator faults/ network splits (upperpanel) and to smaller persistent disturbances, such as fluctuatingpower generation from renewables (lower panel).

Based on historical observations of exceptional events in the continentalEuropean grid, operators have derived recommendations regarding themaximum allowed values for the RoCoF (typically 500 mHz/s–1 Hz/s).

The robustness of the system frequency against power imbalance dis-turbances can be also assessed and quantified via different metrics. Forexample, if faster primary control mechanisms are deployed (by exploitingthe flexibility of the power converters, or the smart loads available in thegrid, or battery storage as in [17]) then the time scale separation betweenprimary control and inertial response of the grid may become less sharp.(Refer to the time scales of frequency control in Figure 2.2 from [10]). Insuch a scenario, the frequency nadir should be explicitly evaluated (and notindirectly, via the RoCoF, which falls short in describing the entire responsecurve).

Another approach for the assessment of frequency stability consists ofevaluating a signal norm for the post-fault frequency response. As depicted

12 preliminaries

15 min 75 min5 s 30 s

Inertial Response

Primary Control

Secondary Control - AGC

Tertiary Control

Generator ReschedulingConverter-Interfaced Generation

Figure 2.2: Typical time scales of frequency-related dynamics in conventionalpower systems as well as typical time scales of frequency control.

in Figure 2.1, the total area between the frequency evolution and the steady-state post-disturbance frequency can be interpreted as the energy imbalancecaused by the disturbance. It is, therefore, a meaningful metric to describehow promptly (and efficiently) the system is restored to its nominal operat-ing condition after a large scale disturbance. Interestingly, this signal normis also informative in another scenario. When a large number of renewablesources are connected to the grid, power imbalance will not only be causedby large (although rare) events, such as the loss of a generator, but also bysudden unpredicted fluctuations of uncontrollable sources such as windand solar. The size of this latter class of events is expected to be smaller, buttheir occurrence is quite more frequent. For analysis purposes, one shouldconsider a persistent disturbance on the power in-feed of the buses whererenewable sources are connected. In this case, the aforementioned signalnorm would describe the amplification gain between these disturbancesand the resulting fluctuations in the grid frequencies at all the buses, asdepicted in the bottom panel of Figure 2.1 for a generic grid bus i.

3O P T I M A L P L A C E M E N T O F V I RT UA L I N E RT I A

3.1 related works

The problem of inertia allocation has been reported before [11], but we areaware only of the study [30] explicitly addressing the problem. In [30], thegrid is modeled by the linearized swing equations and eigenvalue dampingratios as well as transient overshoots (estimated from the system modes)are chosen as optimization criteria for placing virtual inertia and damping.The resulting problem is non-convex, but a sequence of approximationsleads to some insightful results.

In comparison to [30], we focus on network coherency as an alternativeperformance metric, that is, the amplification of stochastic or impulsivedisturbances via a quadratic performance index measured by the H2 norm[31]. As performance index, we choose a classic coherency criterion penaliz-ing angular differences and frequency excursions, which has recently beenpopularized for consensus and synchronization studies [32]–[37] as well asin power system analysis and control [38]–[40]. We demonstrate that thisH2 performance metric is not only more tractable than spectral metrics, butit is also very meaningful for the problem at hand, i.e., it measures the effectof stochastic fluctuations (caused by loads and/or variable renewable gener-ation) as well as impulsive events (such as faults or deterministic frequencyerrors caused by markets) and quantifies their amplification by a coherencyindex directly related to frequency volatility. Finally, in comparison to [30],the damping or droop coefficients are not decision variables in our problemsetup in this chapter, since these are often determined by the system physics(in case of damping), the outcome of primary reserve markets (in case ofprimary control), or scheduled according to cost coefficients, ratings, orgrid-code requirements [41]. We, however, consider an extended analysiswith droop coefficients in Chapter 4.

3.2 contributions

The contributions of this chapter are as follows. We provide a comprehen-sive modeling and analysis framework for the inertia placement problem in

13

14 optimal placement of virtual inertia

power grids to optimize an H2 coherency index subject to capacity and bud-get constraints. The optimal inertia placement problem is characteristicallynon-convex, yet we are able to provide explicit upper and lower boundson the performance index. Additionally, we show that the problem admitsan elegant and strictly convex reformulation for a performance index re-flecting the effort of primary control which is often advocated as a remedyto low-inertia stability issues. In this case, the optimal inertia placementproblem reduces to a standard resource allocation problem, where the costof each resource is proportional to the ratio of expected disturbance overinertia. A similar simplification of the problem is obtained under some rea-sonable assumptions on the ratio between the disturbance and the dampingcoefficient at every node. For the case of a two-area network, a closed-formanalysis is possible, and a series of observations are discussed.

Furthermore, we develop a computational approach based on a gradientformula that allows us to find a locally optimal solution for large networksand arbitrary parameters. We show how the combinatorial problem of allo-cating a limited number of inertia-emulating units can be also incorporatedinto this numerical method via a sparsity-promoting approach. Finally, anysystem norm such as H2 assumes that the location of the disturbance (ora distribution thereof) is known. While empirical fault distributions areusually known based on historical data, the truly problematic faults inpower grids are rare events that are poorly captured by any disturbancedistribution. To safeguard against such faults, we also present a robustformulation of the inertia allocation problem in which we optimize the H2norm with respect to the worst possible disturbance.

A detailed three-region network has been adopted as a case study for thepresentation of the proposed method. The numerical results are also illus-trated via time-domain simulations, that demonstrate how an optimization-based allocation exhibits superior performance (in different performancemetrics) compared to heuristic placements and, perhaps surprisingly, theoptimal allocation also uses less effort to emulate inertia.

From the methodological point of view, this chapter extends the H2performance analysis of second-order consensus systems to non-uniformdamping, inertia, and input matrices (disturbance location). This technicalcontribution is essential for the application that we are considering, as theseparameters dictate the optimal inertia allocation in an intertwined way.

In Section 3.3, we motivate our system model and the coherency perfor-mance index. Section 3.4 presents numerical inertia allocation algorithmsfor general networks and provides explicit results for certain instances of

3.3 problem formulation 15

cost functions and problem scenarios. Section 3.5 presents a case study ona three-region network accompanied with time-domain simulations and aspectral analysis. Finally, Section 3.6 concludes the chapter.

notation We denote the n-dimensional vectors of all ones and zerosby 1n and 0n, whereas 0 is a matrix of zeros of appropriate dimension.The n × n identity matrix is In. For A ∈ Rn×n, A> is its transpose andTr(A) = ∑n

i=1 Aii is its trace. For a positive semi-definite matrix Q � 0, Q12

is its square root. Given an index set I with cardinality |I| and a real-valuedarray {x1, . . . , x|I|}, we denote by x ∈ R|I| the vector obtained by stackingthe scalars xi and by diag{xi} the associated diagonal matrix. The vector eiis the i-th vector of the canonical basis for Rn.

3.3 problem formulation

3.3.1 System Model

Consider a power network modeled by a graph with nodes (buses) V ={1, . . . , n} and edges (transmission lines) E ⊆ V × V . We consider a small-signal version of a network-reduced power system model [27], [42], wherepassive loads are eliminated via Kron reduction [43], and the network isreduced to active buses i with linearized dynamics

mi θi + di θi = pin,i − pe,i, i ∈ {1, . . . , n}, (3.1)

where pin,i and pe,i refer to the power input and electrical power output,respectively. If bus i is a synchronous machine, then (3.1) describes theelectromechanical swing dynamics for the generator rotor angle θi [27],[42], mi > 0 is the generator’s rotational inertia1, and di > 0 accounts forfrequency damping or primary speed droop control (neglecting rampinglimits). If bus i connects to a renewable or battery source interfaced with apower electronics inverter operated in grid-forming mode [44], [45] (see alsoChapter 4), then θi is the voltage phase angle, di > 0 is the droop controlcoefficient, and mi > 0 accounts for power measurement time constant[46], a control gain [47], or arises from virtual inertia emulation through adedicated controlled device [12]–[14]. Finally, the dynamics (3.1) may alsoarise from frequency-dependent or actively controlled frequency-responsiveloads [27]. In general, each bus i will host an ensemble of these devices, and

1 We can relate the coefficient m and inertia constant H in Section 2.2.1 by H = mω0 (p.u.).


the quantities mi and di are lumped parametrizations of their aggregatebehavior.

Under the assumptions of identical unit voltage magnitudes, purelyinductive lines, and a small-signal approximation, the electrical poweroutput at the terminals is given by [27]

pe,i =n

∑j=1

bij(θi − θj), i ∈ {1, . . . , n}, (3.2)

where bij ≥ 0 is the inverse of the reactance between nodes {i, j} ∈ E .The state-space representation of the system (3.1)–(3.2) is then[

θ

ω

]=

[0 In

−M−1L −M−1D

] [θ

ω

]+

[0

M−1

]pin, (3.3)

where M = diag{mi} and D = diag{di} are the diagonal matrices ofinertia and damping/droop coefficients, and L = L> ∈ Rn×n is the networkLaplacian matrix with off-diagonal elements lij = −bij and diagonals lii =∑n

j=1,j 6=i bij. The states (θ, ω) ∈ R2n are the stacked vectors of angles andfrequencies and pin ∈ Rn is the net power input– all of which are deviationvariables from nominal values.

3.3.2 Coherency Performance Metric

We consider the linear power system model (3.3) driven by the inputs pin,iaccounting either for faults or non-zero initial values (modeled as impulses)or for random fluctuations in renewables and loads. We are interested in theenergy expended in returning to the steady-state configuration, expressedas a quadratic cost of the angle differences and frequency displacements∫ ∞

0

{12

n

∑i,j=1

aij(θi(t)− θj(t))2 +n

∑i=1

si ω2i (t)

}dt. (3.4)

Here, si are positive scalars and we assume that the nonnegative scalars aij =aji ≥ 0 induce a connected graph– not necessarily identical to the power griditself. We denote by S the matrix diag{si}, and by N the Laplacian matrixof the graph induced by the aij. In this compact notation, N = L would bean example of local error penalization [32], [33], while N = In − 1n1

>n /n

penalizes global errors.

3.3 problem formulation 17

Aside from consensus and synchronization studies [32]–[37] the co-herency metric (3.4) has recently also been also used in power systemanalysis and control [38]–[40].

The above metric (3.4) represents a generalized energy2 in synchronousmachines. Indeed, for aij = bij (where bij is the inverse of the power linereactance) and si = mi, the metric (3.4) accounts for the potential andkinetic energy in swing mode oscillations. Following the interpretationproposed in [38], for aij = gij (where gij are the power line conductances),the metric (3.4) accounts for the transient resistive losses (i2R losses) inthe grid lines when linearized around the no-load profile. This can bevisualized by considering unit magnitude node voltages (in the per unitframe), i.e., Vi(t) = 1∠θi(t). The current between two nodes i, j under theeffect of disturbance is, therefore, gij (Vi(t)−Vj(t)) = gij (θi(t)− θj(t)), andthe expression for heat losses follows. To be mathematically more precise,gij(θi(t)− θj(t))2 is the second-order series expansion of the full AC lossesaround the baseline solution Vi = 1∠θi(t) used also for the DC power flow.

Next, consider the case si = di where di is a damping coefficient. Thus, thesecond term of the cost function reduces to the sum of di ω2

i – correspondingto the power losses in the machine. Overall, the integrand is in units ofpower and the integral thereof (our cost function) is in units of energy.

We also recall from Section 2.2, that the kinetic energy of the generatoris 0.5 Ji ω2

i , where Ji is the rotational inertia. For the case where si = mi,we get an equivalent second term of the above integrand representingthe mechanical energy at time t. For the integral to be in units of energyan appropriate (per second) normalization has to be used. The requirednormalization3 is an artefact of the per unit system in power engineering.

Adopting the state representation in (3.3), the performance metric (3.4)can be rewritten as the integral

∫ ∞0 yp(t)>yp(t)dt of the performance output

yp =

[N

12 0

0 S12

]︸︷︷︸

= C

[θ

ω

]. (3.5)

2 In functional analysis any L2 norm of the form∫ ∞

0 ‖x(t)‖22dt is colloquially referred to as

energy, independent of the units.3 In general, coefficients aij and si are used for our optimization formulation as design parame-

ters. If the cost function is to be interpreted in terms of energy, then appropriate normalizationsare needed.


In order to model the localization of the disturbances in the grid, weparametrize the input pin as

pin = Π12 η, Π = diag{πi}, πi ≥ 0,

where Π is assumed to be known from historical data among other sources.We, therefore, obtain the state-space model[

θ

ω

]=

[0 In

−M−1L −M−1D

]︸︷︷︸

= A

[θ

ω

]+

[0

M−1Π1/2

]︸︷︷︸

= B

η. (3.6)

In the following, we refer to the input-output map (3.5)–(3.6) as G =(A, B, C). If the inputs ηi are Dirac impulses, then (3.4) is the squared H2norm ‖G‖2

H2of the system [31] and can be interpreted as either of the

following:

(i) measuring the energy amplification of the outputs yj(t), for unit im-pulses (modeling faults or initial conditions) at all inputs ηi(t)=δ(t) withstrengths π1/2

i > 0 for each node i ∈ {1, . . . , n}.

(ii) quantifying the steady-state total variance of the output for a systemsubjected to unit variance stochastic white noise inputs ηi(t). These inputscan model stochastic fluctuations of renewable generation or loads. Thematrix Π1/2 = diag{π1/2

i } quantifies the probability of occurrence of suchfluctuations at each node i.

In general, the H2 norm of a linear system can be calculated efficientlyby solving a linear Lyapunov equation. In our case an additional linearconstraint is needed to account for the marginally stable and undetectablemode z0 = [1>n 0>n ]

> corresponding to an absolute angle reference.

Lemma 1 (H2 norm via observability Gramian) For the state-space system(A, B, C) defined above, we have that

‖G‖2H2

= Tr(B>PB), (3.7)

where the observability Gramian P ∈ R2n×2n is uniquely defined by the followingLyapunov equation and an additional constraint via z0 = [1>n 0>n ]

>:

PA + A>P + C>C = 0, (3.8)

Pz0 = 02n. (3.9)

3.4 optimal inertia allocation 19

3.4 optimal inertia allocation

We assume that each node i ∈ {1, . . . , n} has a nonzero4 inertia coefficientmi > 0 and we are interested in optimally allocating additional virtualinertia in order to minimize the H2 norm (3.4), subject to upper boundsmi at each bus (accounting for the available capacity or installation space)and a total budget constraint mbdg (accounting for the total cost of theinertia-emulating devices). This problem statement is summarized as

minP, mi

‖G‖2H2

= Tr(B>PB) (3.10a)

subject to 1>n m ≤ mbdg, (3.10b)

mi ∈ [mi, mi], i ∈ {1, . . . , n} (3.10c)

PA + A>P + C>C = 0, Pz0 = 02n, (3.10d)

where (A, B, C) are the matrices of the input-output system (3.5)–(3.6).Observe the bilinear nature of the Lyapunov constraint (3.10d) featuringproducts of A and P, and recall from (3.6) that the decision variables mi alsoappear as m−1

i in A. Hence, the problem (3.10) is non-convex and typicallyalso large-scale. We, however, can provide general lower and upper bounds.

3.4.1 Performance Bounds

Theorem 2 (Performance bounds) Consider the power system model (3.5)–(3.6), the squared H2 norm (3.7), and the optimal inertia allocation problem (3.10).Then, the objective (3.10a) satisfies

π

2d

Tr(NL†) +n

∑i=1

simi

≤ ‖G‖2H2≤ π

2d

Tr(NL†) +n

∑i=1

simi

, (3.11)

where π = mini{πi}, π = maxi{πi}, d = mini{di}, and d = maxi{di}.

Proof of Theorem 2: Let the observability Gramian P be the block matrix

P =

[X1 X0

X>0 X2

].

4 Observe that the case mi = 0 leads to an ill-posed model (3.1) whose number of algebraic anddynamic states depend on the system parameters.


With this notation, the squared H2 norm (3.7), ‖G‖2H2

reads as

Tr(B>PB) = Tr(ΠM−2X2) =n

∑i=1

πiX2,ii

m2i

, (3.12)

where we use the ring commutativity of the trace and the fact that Π1/2

and M−1 are diagonal and therefore commute. The constraint (3.10d) canbe expanded as[

X1 X0

X0> X2

]A + A>

[X1 X0

X0> X2

]+

[N 0

0 S

]= 0. (3.13)

By right-multiplying the equation (1,1) of (3.13) by the Moore-Penrosepseudoinverse L† of the Laplacian L, we obtain

−X0M−1LL† − LM−1X>0 L† = −NL†.

By the constraint (3.9) we have that[1>n 0>n

]P =

[0>n 0>n

]which implies

1>n X0 = 0>n . This together with the identity LL† = (In − 1n1>n /n), implies

that LL†X0 = X0. Then, using the ring commutativity of the trace, and itsinvariance with respect to transposition of the argument, we obtain

2 Tr(M−1X0) = Tr(NL†). (3.14)

On the other hand, equation (2,2) of (3.13) implies that

X>0 + X0 = X2M−1D + DM−1X2 − S.

Similarly as before we left-multiply by M−1, use trace properties and thecommutativity of matrices M−1, D to obtain

2 Tr(M−1X0 − DM−2X2) = −Tr(M−1S). (3.15)

Thus, (3.14) and (3.15) together deliver

Tr(DM−2X2) =12

Tr(M−1S + NL†). (3.16)

From (3.12) we obtain the relations

π

n

∑i=1

X2,ii

m2i≤ ‖G‖2

H2≤ π

n

∑i=1

X2,ii

m2i

,


which can be further bounded as

π

d

n

∑i=1

diX2,ii

m2i≤ ‖G‖2

H2≤ π

d

n

∑i=1

diX2,ii

m2i

. (3.17)

The structural similarity of (3.16) and (3.17) allows us to state upper andlower bounds by rewriting (3.17) as in (3.11).

Notice that in the bounds proposed in Theorem 2, the network topol-ogy described by the Laplacian L enters only as a constant factor, and isdecoupled from the decision variables mi. Moreover, in the case N = L(short-range error penalty on angles differences), this offset term becomesjust a function of the grid size: Tr(NL†) = Tr(LL†) = n− 1.

Theorem 2 (and its proof) sheds some light on the nature of the opti-mization problem that we are considering, and in particular on the roleplayed by the mutual relation between disturbance strengths πi, dampingcoefficients di, their ratios πi/di, frequency penalty weights si, and thedecision variables mi. These insights are further developed hereafter.

3.4.2 Noteworthy Cases

In this section, we consider some special choices of the performance metricand some assumptions on the system parameters, which are practicallyrelevant and yield simplified versions of the general optimization problem(3.10), enabling in most cases the derivation of closed-form solutions.

We first consider the performance index (3.4) corresponding to the effortof primary control. As a remedy to mitigate low-inertia frequency stabilityissues, additional fast-ramping primary control is often put forward [11].The primary control effort can be accounted for by the integral

YD(m) =

∫ ∞

0θ(t)>Dθ(t)dt. (3.18)

Hence, the effort of primary control (3.18) mimics the H2 performancewhere the performance matrices in (3.5) are chosen as S = D and N =0. This intuitive cost functions allows an insightful simplification of theoptimization problem (3.10).

Theorem 3 (Primary control effort minimization) Consider the power sys-tem model (3.5)–(3.6), the squaredH2 norm (3.7), and the optimal inertia allocationproblem (3.10). For a performance output characterizing the effort of primary con-trol (3.18), i.e., S = D and N = 0, the optimization problem (3.10) can beequivalently restated as the convex problem


minmi

n

∑i=1

πimi

(3.19a)

subject to (3.10b)− (3.10c), (3.19b)

where we recall, πi describes the strength of the disturbance at node i.

Proof of Theorem 3: With S = D and N = 0, the Lyapunov equation (3.13)together with the constraint (3.9) is solved explicitly by

P =

[X1 X0

X>0 X2

]=

12

[L 0

0 M

].

The performance metric as derived in (3.12) therefore becomes

‖G‖2H2

=n

∑i=1

πiX2,ii

m2i

=12

n

∑i=1

πimi

.

This concludes the proof.

The equivalent convex formulation (3.19) yields the following importantinsights. First and foremost, the optimal solution of (3.19) is unique (as longas at least one πi is greater than zero) and also independent of the networktopology and the line susceptances. It depends solely on the location andstrength of the disturbance as encoded in the coefficients πi. For example,if the disturbance is concentrated at a particular node i, that is, πi 6= 0 andπj = 0 ∀ j 6= i, then the optimal solution is to allocate the maximal inertia atnode i, i.e., mi = min{mbdg, mi}. If the capacity constraint (3.10c) is relaxed,the optimal inertia allocation is proportional to the disturbance πi

1/2.We now consider a different assumption that also allows a similar simpli-

fied analysis in other notable cases.

Assumption 1 (Uniform disturbance-damping ratio) The ratio $ = πi/diis constant for all i ∈ {1, . . . , n}. �

Notice that the droop coefficients di are often scheduled proportion-ally to the rating of a power source to guarantee fair power sharing [41].Meanwhile, it is reasonable to expect that the disturbances due to variablerenewable fluctuations scale proportionally to the size of the renewable


power source. Hence, Assumption 1 can be justified in many practicalcases, including of course the case where both damping coefficients anddisturbances are uniform across the grid. Aside from that, Assumption1 may be of general interest since it is common in many studies with aspatially invariant setting [33], [37], [38]. Under this assumption, we have thefollowing result.

Theorem 4 (Optimal allocation with uniform disturbance-damping ratio)Consider the power system model (3.5)–(3.6), the squared H2 norm (3.7), and the

inertia allocation problem (3.10). Let Assumption 1 hold. Then, the optimizationproblem (3.10) can be equivalently restated as the convex problem

minmi

n

∑i=1

simi

(3.20a)

subject to (3.10b)− (3.10c), (3.20b)

where we recall that si is the penalty coefficient for the frequency deviation at i.

Proof of Theorem 4: From Assumption 1, let $ = πi/di > 0 be constant forall i ∈ {1, . . . , n}. Then we can rewrite (3.12) as

‖G‖2H2

=n

∑i=1

viX2,ii

m2i

= $

n

∑i=1

diX2,ii

m2i

.

This is equal, up to the scaling factor $, to the left hand side of (3.16). We,therefore, have

‖G‖2H2

=$

2Tr(M−1S + NL†), (3.21)

which is equivalent, up to multiplicative factors and constant offsets, to thecost of the optimization problem (3.20a).

Again, as in Theorem 3, Theorem 4 reduces the original optimizationproblem to a simple convex problem for which the optimal inertia allocationis independent of the network topology. Indeed, the physical intuition ofAssumption 1 is that the disturbance is dissipated at every node in thesame proportion, and thus network effects are negligible.

This setting also allows us to highlight that the cost function for inertiaallocation needs to be chosen insightfully. For example, consider a frequency


penalty S proportional to the inertia coefficients, S = cM for some c ≥ 0(including c = 0):∫ ∞

0

{12

n

∑i,j=1

aij(θi(t)− θj(t))2 + cn

∑i=1

mi ω2i (t)

}dt.

This choice penalizes the variation in kinetic energy as it decays to zero– a standard penalty in power systems. The subsequent corollary showsthat this cost function is independent of (and thus not meaningful for) theinertia allocation.

Corollary 5 (Kinetic energy penalization with uniform disturbance to damp-ing ratio) Let Assumption 1 hold, and let the penalty on the frequency deviationsbe proportional to the allocated inertia, that is, S = cM. Then, the performancemetric ‖G‖2

H2is independent of the inertia allocation, and assumes the form

‖G‖2H2

=$

2

(c n + Tr(NL†)

),

where $ = πi/di > 0, for all i ∈ {1, . . . , n}, is the disturbance-damping ratio.

3.4.3 Explicit Results for a Two-Area Network

In this section, we focus on a two-area power grid as in [11] to obtain someinsight on the nature of this optimization problem. We also highlight theprominent role of the ratios πi/di in Assumption 1 and bounds (3.11).

In the case of a two-area system, it is possible to derive an analyticalsolution P(m) of the Lyapunov equation (3.10d), as a closed-form functionof the vector of inertia allocations mi. We thus obtain an explicit expressionfor the cost (3.10a) as

‖G‖2H2

= f (m) := Tr(B(m)>P(m)B(m)), (3.22)

where in the two-area case f (m) reduces to a rational function of polynomi-als of orders 4 in the numerator and the denominator, in terms of inertiacoefficients mi.

As the explicit expression is more convoluted than insightful, we will notshow it here but only report the following observations:

(i) The problem (3.10) admits a unique minimizer.

(ii) For sufficiently large bounds mi, the budget constraint (3.10b) becomesactive, that is, the optimizers satisfy m?

1 + m?2 = mbdg. In this case, m2 =

mbdg−m1 can be eliminated, and (3.10) can be reduced to a scalar problem.


2 4 6 8 10

1

2

3

4

5

6

m1

f(m

1)

π1/d1 = π2/d2π1/d1 6= π2/d2

Figure 3.1: Cost function profiles for identical and weakly dissimilar πi/di ratiosfor the two-area network.

(iii) Identical πi/di ratios and frequency penalties si result in identical opti-mal allocations m?

1 = m?2 (as predicted by Theorem 4), if capacity constraints

are absent. If πi/di > πj/dj, then m?i > m?

j (see the example in Figure 3.1,where we eliminated m?

2 = mbdg −m?1).

(iv) For sufficiently uniform πi/di ratios, the problem (3.10) is stronglyconvex. We observe that the cost function f (m) is fairly flat over the feasibleset (see Figure 3.1).

(v) For strongly dissimilar πi/di ratios, we observe a less flat cost function.If disturbance affects only one node, for example, π1 = 1 and π2 = 0, thenstrong convexity is lost.

From the above facts, we conclude that the input scaling factors πi play afundamental role in the determination of the optimal inertia allocation. Toobtain a more complete picture, we linearly vary the disturbance input ma-trices from (π1, π2) = (0, 1) to (π1, π2) = (1, 0), that is, from a disturbancelocalized at node 2 to a disturbance localized at node 1. For each value of(π1, π2), we compute the optimal inertia allocation for the cost functionwith N = L and S = I2.

The resulting optimizers are displayed in Figure 3.2 showing that inertiais allocated dominantly at the site of the disturbance, which is in linewith previous case studies [11], [30]. Notice also that depending on thevalue of the budget mbdg, the capacity constraints mi, and the πi/di ratios,the budget constraint may be active or not. Thus, perhaps surprisingly,


0.2 0.4 0.6 0.8 10

5

10

15

20

25

π1 = 1− π2

Opt

imal

iner

tia

allo

cati

on m?1

m?2

mbdgm?

1 + m?2

Figure 3.2: Optimal inertia allocation for a two-area system with non-identicaldamping coefficients di, and disturbances inputs varying from(π1, π2) = (0, 1) to (π1, π2) = (1, 0). We choose d1 = 1 < d2 = 2,mbdg = 25, and a12 = 1 as the system parameters.

sometimes not all inertia resources are allocated. Overall, the two-area casepaints a surprisingly complex picture.

3.4.4 A Computational Method for the General Case

In Section 3.4.2 and Section 3.4.3, we considered a subset of scenarios andcost functions that allowed the derivation of tractable reformulations andsolutions of the inertia allocation problem (3.10). In this section, we considerthe optimization problem in its full generality. As in Section 3.4.3, we denoteby P(m) the solution of the Lyapunov equation (3.10d), and express thecost ‖G‖2

H2as a function f (m) of the vector of inertia allocations mi. In

the following, we derive an efficient algorithm for the computation of theexplicit gradient ∇ f (m) of f (m) in (3.22).

In general, most computational approaches can be sped up tremendouslyif an explicit gradient is available. In our case, an additional significantbenefit of having a gradient ∇ f (m) of f (m) is that the large-scale set ofnonlinear (in the decision variables) Lyapunov equations (3.10d) can beeliminated and included into the gradient information. In the following,we provide an algorithm that achieves so, using the routine Lyap(A, Q),which returns the matrix P that solves PA + A>P + Q = 0 together withPz0 = 02n, where Q = C>C.


Theorem 6 (Gradient computation) Consider the objective function (3.22),where P(m) is a function of m via the Lyapunov equation (3.10d). The objectivefunction is differentiable for m ∈ Rn

>0, its gradient at m is given by AlgorithmA.3.

The proof of Theorem 6 is partially inspired by [35], [48] and relies ona perturbation analysis of the Lyapunov equation (3.10d) combined withTaylor and power series expansions.

Proof of Theorem 6: In order to compute the gradient of (3.22) at m ∈ Rn>0,

we make use of the relation

∇µ f (m) = ∇ f (m)>µ, (3.23)

where ∇µ f (m) is the directional derivative of f in the direction µ ∈ Rn,defined as

∇µ f (m) = limε→0

f (m + εµ)− f (m)

ε, (3.24)

whenever this limit exists. From (3.22) we have that

f (m + εµ) = Tr(B(m + εµ)>PB(m + εµ)), (3.25)

where P is a solution of the Lyapunov equation

PA(m + εµ) + A(m + εµ)>P + C>C = 0, (3.26)

and where by A(m + εµ) we denote the system matrix defined in (3.6),evaluated at m + εµ. The matrices A(m + εµ) and B(m + εµ) viewed asfunctions of scalar ε can be expanded in a Taylor series around ε = 0 as

A(m + εµ) = A(0)(m,µ) + A(1)

(m,µ)ε +O(ε2),

B(m + εµ) = B(0)(m,µ) + B(1)

(m,µ)ε +O(ε2),

(3.27)

with coefficients A(i)(m,µ) and B(i)

(m,µ), i ∈ {0, 1}. To compute the coefficients

of the Taylor expansion in (3.27), we recall the scalar series expansion of1/(mi + εµi) around ε = 0:

1(mi + εµi)

=1

mi− εµi

m2i+O(ε2).


Using the shorthand Φ = diag(µi), we therefore have

A(0)(m,µ) =

[0 In

−M−1L −M−1D

], B(0)

(m,µ) =

[0

M−1Π12

],

A(1)(m,µ) =

[0 0

ΦM−2L ΦM−2D

], B(1)

(m,µ) =

[0

−ΦM−2Π12

].

Accordingly, the solution of the Lyapunov equation (3.26) can be ex-panded in a power series as

P = P(m + εµ) = P(0)(m,µ) + P(1)

(m,µ)ε +O(ε2), (3.28)

and therefore the Lyapunov equation (3.26) becomes

(P(0) + εP(1) +O(ε2))(A(0) + εA(1) +O(ε2))+

(A(0) + εA(1) +O(ε2))>(P(0) + εP(1) +O(ε2)) + C>C = 0,

where we dropped the subscript (m, µ) for readability. By collecting termsassociated with powers of ε, we obtain two Lyapunov equations determiningP(0) and P(1):

P(0)A(0) + A(0)>P(0) + C>C = 0, (3.29)

P(1)A(0) + A(0)>P(1) + (P(0)A(1) + A(1)>P(0)) = 0. (3.30)

By the same reasoning as used for equation (3.8), the first Lyapunovequation (3.29) is feasible with a positive semi-definite P(0) satisfyingP(0)z0 = 02n. The second Lyapunov equation (3.30) is feasible by anal-ogous arguments. Finally, by using (3.25) together with (3.27) and (3.28),we obtain

f (m + εµ) = f (0)(m,µ) + f (1)

(m,µ)ε +O(ε2),

where f (0)(m,µ) = f (m) and

f (1)(m,µ) = Tr

(2 B(1)

(m,µ)

>P(0)(m,µ)B

(0)(m,µ) + B(0)

(m,µ)

>P(1)(m,µ)B

(0)(m,µ)

). (3.31)

From (3.24), it follows that ∇µ f (m) = f (1)(m,µ) as defined in (3.31), thereby

implicitly establishing differentiability of f (m).This concludes the proof, as the algorithm computes each component of

the gradient ∇ f (m) by using the relation (3.23) with the special choice ofµ = ei for i ∈ {1, . . . , n}.


3.4.5 Economic Allocation of Resources

In this section, we focus on the planning problem of optimally allocatingvirtual inertia when economic reasons suggest that only a limited numberof virtual inertia devices should be deployed (rather than at every gridbus). Since this problem is generally combinatorial, we solve a modifiedoptimal allocation problem, where an additional `1-regularization penaltyis imposed, in order to promote a sparse solution [49]–[51].

The regularized optimal inertia allocation problem is then

minP, mi

Jγ(m, P) = ‖G‖2H2

+ γ‖m−m‖1 (3.32a)

subject to (3.10b)− (3.10d), (3.32b)

where γ ≥ 0 trades off the sparsity penalty and the original objectivefunction. As in (3.10c) the allocations mi are lower bounded by a positivemi, the objective (3.32a) can be rewritten as

Jγ(m, P) = Tr(B>PB) + γ1>n (m−m). (3.33)

Observe that the regularization term in the cost (3.33) is linear and differen-tiable. Thus, problem (3.33) fits well into our gradient computation algo-rithm and a solution can be determined within the fold of Algorithm A.3by incorporating the penalty term. Likewise, our analytic results in Sec-tion 3.4.2 can be re-derived for the cost function (3.33). We highlight theutility of the performance-sparsity trade-off (3.33) in Section 3.5.

3.4.6 The min-max Problem: Optimal Robust Allocation

Thus far we have assumed knowledge of the disturbance strengths encodedin the matrix Π. While empirical disturbance distributions from historicaldata are generally available to system operators, the truly problematicand devastating faults in power systems are rare events that are poorlypredicted by any (empirical) distribution. Given this inherent uncertainty,it is desirable to obtain an inertia allocation profile which is optimal forthe most detrimental disturbance. This problem belongs to the domain ofrobust optimization and can be interpreted as a zero-sum game between


the power system operator and the adversarial disturbance. The robustinertia allocation problem can then be formulated as the min-max problem

minmi

maxπi

f (m, π) (3.34a)

subject to π ∈ P , (3.34b)

(3.10b)− (3.10c), (3.34c)

where f (m, π) = Tr(B(m, π)>P(m)B(m, π)) and P is the set of possibledisturbances with a non-empty interior. As a special instance, consider

P ={

π ∈ Rn : 1>n π ≤ πbdg, 0 ≤ πi

}5, (3.35)

where we normalized the disturbances by πbdg > 0.Recall from (3.6)–(3.7) that the objective f (m, π) is linear in π. Therefore,

we can express it as f (m, π) = π>g(m), where gi(m) = P(m)2,2i/m2i is only

a function of m, for i ∈ {1, . . . , n}. Hence, by strong duality, we can rewritethe inner maximization problem as the equivalent dual minimization prob-lem

minχ, µi

πbdg χ (3.36a)

subject to χ ≥ 0, µi ≥ 0, ∀i (3.36b)

gi(m) + µi = χ , ∀i, (3.36c)

where χ and µi are the dual variables associated with the constraints (3.35).The min-max problem (3.34) is then equivalent to:

minmi , χ, µi

πbdg χ (3.37a)

subject to (3.10b)− (3.10c), (3.37b)

(3.36b)− (3.36c). (3.37c)

The minimization problem (3.37) has a convex objective and constraints,barring (3.36c). However, we already have the gradient of the individualelements, dgi(m)/dm which can be computed from Algorithm A.3 asd f (m, π)/dm by substituting πi = 1 and πj = 0 ∀ j 6= i. The availabilityof the gradient of this set of non-linear equality constraints considerablyspeeds up the computation of the minimizer.

5 The choice of such a set is justified as it bounds the energy or power of the disturbance signal(as is the case in a linear-quadratic setting) and can be evaluated as the squared 2 norm orπ1/2

i × π1/2i = πi .

3.5 case study : 12-bus-three-region system 31

25km10km

25km10km

25km

110km

110km 110km

1

1

2

3 4

5

6

78

910 11

12

1570MW

1000MW100Mvar

567MW100Mvar

400MW 490MW

611MW164Mvar

1050MW284Mvar

719MW133Mvar

350MW

700MW208Mvar

700MW293Mvar

200Mvar

350Mvar

69Mvar

Figure 3.3: A 12 bus three-region system test case. Grid parameters: Transformerreactance 0.15 p.u., line impedance (0.0001 + 0.001j) p.u./km.

By direct inspection or computation (see Section 3.5), we observe that therobust optimal allocation profile tends to make the cost (3.34a) indifferentwith respect to the location of the disturbance, as is customary for similarclasses of min-max (adversarial) optimization problems. For the special caseof primary control effort minimization as in Theorem 3, the min-max problem(3.34) simplifies to

minmi , χ, µi

πbdg χ (3.38a)

subject to χ ≥ 0, µi ≥ 0, ∀i (3.38b)

(3.10b)− (3.10c), (3.38c)1

mi+ µi = χ, ∀i. (3.38d)

In this case, the robust optimal allocation profile tends to make the inertiaallocations mi equal for all i, inducing a valley-filling strategy that allocatesthe entire inertia budget and prioritizes buses with lowest inertia first.

3.5 case study : 12-bus-three-region system

In this section, we investigate a 12-bus case study illustrated in Figure 3.3.The system parameters are based on a modified two-area system fromExample 12.6 of [27] with an additional third area, as introduced in [30].In this system, three buses are not available for inertia allocation and aretherefore eliminated via Kron reduction, resulting in a 9-bus equivalent.

We investigate this example computationally, using Algorithm A.3 todrive standard gradient-based optimization routines, while highlighting


parallels to our analytic results. We analyze different parametric scenariosand compare the inertia allocation and the performance of the proposednumerical optimization (a local optimum) with two plausible heuristics: thefirst one can be deduced from the conclusions in [11], [30] and consists inallocating the available budget uniformly across the grid, in the absenceof capacity constraints, that is, mi = muni = mbdg/n; the second followsfrom the intuition developed in Theorem 3 and consists in allocating themaximum inertia allowed by the bus capacity, in the absence of a budgetconstraint, that is, mi = mi (which we set as mi = 4mi). We consider twodisturbance scenarios: a uniform disturbance affecting all nodes identically,and a localized disturbance at node 4 alone. For the performance metric,we choose N = L and S = I9. The following conclusions can be drawnfrom the above test cases– some of which are perhaps surprising andcounterintuitive:

(i) Our locally optimal solution achieves the best performance among thedifferent heuristics in all scenarios; see Figure 3.4 and Figure 3.5.

(ii) As depicted in Figure 3.4(a), for uniform disturbances with capacity con-straints on the individual buses, the optimal solution does not correspondto allocating the maximum possible inertia at every bus. In Figure 3.4(b),we note that when only a total budget constraint is present, the optimalsolution is remarkably different from the uniform allocation of inertia atthe different nodes. For both scenarios, the performance improvementwith respect to the initial allocation and the different heuristics is modestand confirms the intuition (Section 3.4.3) regarding the flatness of the costfunction.

(iii) In stark contrast, a localized disturbance results in adding inertia dom-inantly to the disturbed node (node 4) as an optimum choice. The latteris also in line with the closed-form results in Theorem 3. Furthermore,an additional inertia allocation to all other (undisturbed) nodes may bedetrimental for the performance, as shown in Figure 3.4(c).

(iv) The robust allocation approach proposed in Section 3.4.6 is investigatedin Figure 3.5(e) and Figure 3.5(f). The ensuing inertia profiles in additionto being robust to disturbance location, also result in a significantly lowerworst-case cost compared to the heuristic allocations. We also observea generally flatter optimal inertia profile, in tune with the conclusionsfollowing the reformulation (3.38) developed for another cost function.


1 2 4 5 6 8 9 10 120

50

100

150

node

iner

tia

(a) uniform disturbance subject to capacity constraints (3.10c)

mm?

m

0

0.05

0.1

0.15

cost

1 2 4 5 6 8 9 10 120

30

60

90

node

iner

tia

(b) uniform disturbance subject to budget constraint (3.10b)

mm?

m

0

0.05

0.1

0.15

cost

1 2 4 5 6 8 9 10 120

50

100

150

node

iner

tia

(c) localized disturbance at node 4 subject to capacity constraints (3.10c)

mm?

m

0

0.09

0.18

0.27

cost

Figure 3.4: Optimal inertia allocations, performance comparison for the test casein Figure 3.3, under different scenarios- 1.


1 2 4 5 6 8 9 10 120

45

90

135

180

node

iner

tia

(d) localized disturbance at node 4 subject to budget constraint (3.10b)

mm?

muni

0

0.09

0.18

0.27

cost

1 2 4 5 6 8 9 10 120

50

100

150

node

iner

tia

(e) robust inertia allocation subject to capacity constraints (3.10c)

mm?

robm

0

0.09

0.18

0.27

cost

1 2 4 5 6 8 9 10 120

30

60

90

node

iner

tia

(f) localized disturbance at node 4 subject to budget constraint (3.10b)

mm?

robmuni

0

0.09

0.18

0.27

cost

Figure 3.5: Optimal inertia allocations, performance comparison for the test casein Figure 3.3, under different scenarios- 2.


0

2

4

6

8

10

Car

dina

lity

0 0.5 1 1.5 2 2.5 3

0

20

40

60

80

100

γ (10−4)

Rel

ativ

ePe

rfor

man

ceLo

ss[%

]

localized, carduniform, cardlocalized, perfuniform, perf

Figure 3.6: Relative performance loss (%) as a function of penalty γ with capac-ity constraints. 0%, 100% correspond to the optimal allocation, noadditional allocation respectively.

(v) The sparsity-promoting approach proposed in Section 3.4.5 is examinedin Figure 3.6. For a uniform disturbance without a sparsity penalty, inertiais allocated at all nine buses of the network. A modest penalty of γ =6e−5, however, yields an allocation at only seven buses with a mere 1.3%degradation in performance. For sparser allocations, the performance lossbecomes more significant. The optimal allocation is inherently sparser inthe case of localized disturbances. Even without a sparsity penalty, virtualinertia would be assigned to only buses 4 and 6, when the disturbance is atnode 4. An allocation exclusively at bus 4 (economically preferable), withnegligible performance loss can be arrived at, with a penalty of γ > 2e−4.

(vi) Figure 3.7 shows the time-domain responses to a localized impulseat node 4, modeling a post-fault condition. Figure 3.7(a) (respectively,(b)) shows that the optimal inertia allocation according to the proposedH2 performance criteria is also superior in terms of frequency overshootand angle differences (respectively, frequencies). Figure 3.7(c) displays thefrequency response at node 5 of the system. Note from the scale of thisplot that the deviations are insignificant. Similar comments also apply toall other signals which are not displayed here. Finally, Figure 3.7(d) showsthe control effort6 m4 · θ4 expended by the virtual inertia emulation at the

6 In our notion of ‘control effort’, i.e., m · θ, we have m = m0 +madd, where m0 is the initial inertiapresent, madd is the additional virtual inertia that we propose to add at the correspondingbus, which is an aggregate model accounting for the contribution of one or more synchronous


0 10 20 30 40 50 60

−4

−2

0

2

4

·10−2

t [s]

∆θ 4−

∆θ 1

0 10 20 30 40 50 60

−0.1

0

0.1

t [s]

∆ω

4

0 10 20 30 40 50 60−2

0

2

4

·10−3

t [s]

∆ω

5

0 10 20 30 40 50 60

−2

0

2

t [s]

Con

trol

effo

rt

mmunim?

Figure 3.7: Time-domain plots for angle differences, frequencies, and controleffort m4 · θ4 for a localized disturbance at node 4.

3.6 conclusions 37

−0.18 −0.15 −0.12 −0.09 −0.06 −0.03 0

−3

−1.5

0

1.5

3

Real Axis

Imag

inar

yA

xis

conemmunim?

Figure 3.8: The eigenvalue spectrum of the state matrix A for different inertiaprofiles, where m? has been optimized for a disturbance at node 4.

disturbed bus. Perhaps surprisingly, we observe that the optimal allocationm = m? requires the least control effort.

(vii) Figure 3.8 plots the eigenvalue spectrum for different inertia profiles.The initial inertia profile m, marginally outperforms other allocations withrespect to both the best damping asymptote (most damped nonzero eigen-values) as well as the best damping ratio (narrowest cone). As is apparentfrom the plots in Figure 3.7, this case also leads to inferior time-domainperformance compared to the optimal allocation m?, which has slightlypoorer damping asymptote and ratio. These observations reveal that thespectrum holds only partial information and advocate the use of the H2norm as opposed to spectral performance metrics (as in [30]).

3.6 conclusions

In this chapter we considered the problem of placing virtual inertia inpower grids based on an H2 norm performance metric reflecting networkcoherency. This formulation gave rise to a large-scale and non-convex op-timization program. For certain cost functions, problem instances, and inthe low-dimensional two-area case, we could derive closed-form solutionsyielding some, possibly surprising insights. Next, we developed a computa-

generators, together with inertia-emulating power converters. For the synchronous machines,the analogous ‘control effort’ would be given by m0 · θ, which is its natural rate of change ofstored kinetic energy.


tional approach based on an explicit gradient formulation and validatedour results on a three-region network. Suitable time-domain simulationsdemonstrated the efficacy of our locally optimal inertia allocations overintuitive heuristics. We also examined the problem of allocating a finitenumber of virtual inertia units via a sparsity-promoting regularization.

Our computational and analytic results are well aligned and suggestinsightful strategies for the optimal allocation of virtual inertia. Contrary topopular belief, it is the location of disturbance and the placement of inertiain the grid, rather than the total inertia in a power system that dictates itsresilience. In the next chapter we explore more detailed system models andextend the analysis to fast frequency response.

4I M P L E M E N TAT I O N O F V I RT UA L I N E RT I A A N D FA S TF R E Q U E N C Y R E S P O N S E

4.1 related works

Several studies have been carried out to propose control techniques tomitigate the loss of rotational inertia and damping. One extensively studiedtechnique relates to using power electronic converters to mimic synchronousmachine behavior [13], [52]–[54]. Other studies develop methods which relyon concepts ranging from simple proportional-derivative to more complexcontrols, e.g., under the name of Virtual Synchronous Generators. All thesestrategies depend on some form of energy storage such as batteries, super-capacitors, flywheels, or the residual kinetic energy of wind turbines [55],which acts as a substitute for the kinetic energy of machines.

These investigations have also established the efficacy of virtual inertia(VI) and fast frequency response (FFR), i.e., primary frequency controlwithout turbine delay, as a short-term replacement for machine inertiain low-inertia power systems. As power converters operate at a muchfaster time scales compared to conventional generation, it is plausible toforesee future power systems based on predominantly converter-interfacedgeneration, without a major distinction between different time-scale controlssuch as inertia and fast frequency response, and primary frequency controlprovided by synchronous machines [10], [56], [57]. In this chapter, weexclusively focus on power systems with reduced inertia due to loss ofsynchronous machines and utilize virtual inertia, fast primary frequencycontrol as a remedy.

As shown in the previous chapter, not only is virtual inertia and primaryfrequency control vital, but its location in the power system is equallycritical. Further, there can be a degradation in the performance due toill-conceived spatial inertia distributions [58], even if the total virtual inertiaadded to the power system is identical. The problem of optimally tuningand placing the virtual inertia and primary frequency controllers basedon system norms (see [59]–[62]) has as been explored for small-scale testcases with linear models [58], [60], [61]. In [21], [30], [63] time-domain andspectral metrics such as RoCoF, nadir, and damping ratios are considered. In[30], [63] a sequential linear programming approach is used to optimize the

39

40 implementation of virtual inertia and fast frequency response

allocation of grid-following virtual inertia and primary frequency control.This method directly optimizes the frequency nadir and RoCoF.

4.2 contributions

As contributions, this chapter develops explicit models of converter-basedvirtual inertia devices that capture the key dynamic characteristics of phase-locked loops (PLLs) [64] used in grid-following virtual inertia devices andof grid-forming controls such as virtual synchronous machines [14], [52],droop control [10], [65], and machine matching control [53], [66], [67], thatcan be used to provide virtual inertia. In addition, these models are suitablefor integration with large-scale, non-linear power system models, thusallowing for parameter tuning through tractable optimization problems.

Moreover, the applicability of system norms as a performance metricfor power system analysis is established beyond the prototypical swingequation, by considering detailed models. To this end, we propose a compu-tationally efficient H2 norm-based algorithm to optimally tune the parame-ters and the placement of the VI devices in order improve the resilience oflow-inertia power systems. The key idea of this algorithm is to exploit theinterpretation of VI devices as feedback controllers. Though the algorithmis applicable for a broader class of services offered by power electronicdevices, we concentrate on virtual inertia and fast frequency response.

Finally, a high-fidelity model of the South-East Australian power systemis modified to replicate a low-inertia scenario and used for an extensivecase study. This modified system is augmented with VI devices to studytheir impact on system stability and validate the optimal tuning obtained byapplying the proposed optimization algorithm. Moreover, through extensivesimulations, we validate the linearized models used in the H2 optimizationalgorithm and study the impact of both grid-forming and grid-followingvirtual inertia on the disturbance responses of the non-linear power system.Lastly, time-domain simulations are presented to study system stability andto compare the response of grid-following and grid-forming VI in detail.

In Section 4.3, the power system model, converter models in both grid-following and grid-forming implementations are presented. The key perfor-mance metrics for grid stability and design constraints are identified andsuitably defined in Section 4.4. In Section 4.5, a computational approachto identify the location of the VI devices to improve post-fault response oflow-inertia power systems is proposed. In Section 4.6, the low-inertia modelbased on the South-East Australian system is presented. A case study based

4.3 system model 41

on the two implementations of virtual inertia is presented in Section 4.7, andsuitable metrics are investigated to quantify the improvements in systemstability. Finally, Section 4.8 concludes the chapter.

4.3 system model

We consider a high-fidelity, non-linear power system model, consistingof synchronous machines with governors, automatic voltage regulators(AVRs), power system stabilizers (PSSs), constant impedance and constantpower loads, and renewable generation that is abstracted by constant powersources on the time scales of interest. The dynamical model of the powersystem is given by a differential-algebraic equation [68]

xs = fs(xs, zs), (4.1a)

0 = gs(xs, zs, i), (4.1b)

where 0 is a vector of zeros, xs ∈ Rnxs is the state vector and contains (but isnot limited to) the mechanical states of the generators, their controllers andthe states of other devices (i.e., non-linear dynamic loads and renewablein-feed). The three-phase transmission network is modeled by the algebraicequation (4.1b) in current-balance form (see Section 7.3.2 of [42]). The vectorzs ∈ Rnzs comprises the AC signals of the transmission network, such astransmission line currents and bus voltages vk ∈ R3. Moreover, the vectori ∈ Rni contains the three-phase currents ik ∈ R3 injected at each busk ∈ {1, . . . , nb}. In this section, we will use the current i to incorporateexplicit models of converter-based virtual inertia devices and disturbancesinto the power system model (4.1). We note that, zs can be expressed asa function of the states xs and the current injections i. Moreover, aftercombining the power system model (4.1) with suitable models of powerconverters, we shall use the control inputs of the power converters toprovide virtual inertia and fast frequency response. We now elaborate onthe dynamics of the VI devices and the disturbance model.

4.3.1 Modeling of Virtual Inertia Devices

The VI devices are power electronic devices that mimic the inertial re-sponse of synchronous generators. In the following we consider the twomost common implementations: grid-following and grid-forming [57]. Agrid-following virtual inertia device is controlled to inject active power


proportional to the frequency deviation and rate of change of frequency es-timated by a phase-locked loop. In contrast, the grid-forming virtual inertiadevice is a voltage source that responds to power imbalances by changingthe frequency of its voltage. In this chapter, we model both types of VIdevices as local dynamic feedback controllers. Even though we focus on twoprototypical implementations of virtual inertia, the approach proposed inthis chapter can be used for any other arbitrary controller transfer function.

vk

ik

to grid

Figure 4.1: A schematic representation of a grid-following virtual inertia device.

grid-following The grid-following VI device is controlled to injectactive power proportional to the frequency deviation and RoCoF of theAC voltage at the bus where it is connected. To this end, each such virtualinertia device uses a frequency estimator, i.e., a phase-locked loop, thatsynchronizes to the bus voltage vk to obtain an estimate θk of the bus voltagephase angle ∠vk, the frequency estimate ωk, and the RoCoF estimate ˙ωk.We model such a synchronization device as

˙θk = ωk, (4.2a)

τk ˙ωk = −ωk − KP,kvq,k − KI,k

∫vq,k, (4.2b)

where vq,k is the q-axis component of the bus voltage vk in a dq-framewith angle θk and vq,k ≈ θk −∠vk for small angle differences. Moreover, τk,KP,k, and KI,k are the filter time constant, proportional gain, and integralsynchronization gain. With τk = 0 the model (4.2) reduces to the standardsynchronous reference frame phase-locked loop (SRF-PLL) with a PI loopfilter (i.e., ˙θk = −KP,kvq,k − KI,k

∫vq,k) commonly used in control of power

converters [64], [69]. However, the SRF-PLL with a PI loop filter does notprovide an explicit RoCoF estimate. In contrast, incorporating a filter with

4.3 system model 43

time constant τk into the loop filter of the SRF-PLL allows us to obtain anexplicit RoCoF estimate.

Remark 1 Extensive simulations indicate that using a standard SRF-PLLin combination with a realizable differentiator results in a worse controlperformance than integrating the RoCoF estimation into the PLL. Further-more, the input into the PLL (i.e., vk) is often subject to pre-filtering and τkcan also be interpreted as moving the pre-filter into the loop filter (see thediscussion in [69]). �

At the nominal steady-state, we have θk → ∠vk and ωk → 0. Thisis because, we consider a reference frame rotating at the nominal gridfrequency.

With the frequency, RoCoF estimates in (4.2), the VI device is modeled as

P?VI,k = Kfoll,k [ωk ˙ωk]

>, Q?VI,k = 0, (4.3)

where Kfoll,k = [dk mk] are the control gains and P?VI,k and Q?

VI,k are theset-points for the power injection of the grid-following VI device. Theelements mk ≥ 0 are referred to as virtual inertia (reacts proportional to thederivative of the measured frequency), and dk ≥ 0 as the virtual damping(reacts proportional to the measured frequency itself).

xs = fs(xs, zs), 0 = gs(xs, zs, i)

KfollPLL(4.2)

PowerSource

ik

P?VI

[ωk ˙ωk]>

vk

Figure 4.2: Interconnection of a single grid-following virtual inertia device withpower set-points according to (4.3).

The VI device utilizes a current source that injects the three-phase currentik at node k (see Figure 4.1) and tracks the power references P?

VI,k, Q?VI,k with

time constant τfoll = 100 ms. Figure 4.2 shows the overall control strategy.

grid-forming The grid-forming VI device uses a voltage source con-nected to the grid via an LC filter with parasitic losses (see Figure 4.3) that


generates a voltage vVI,k with an angle θVI,k = ∠vVI,k that is a function ofthe power in-feed of the VI devices. The device is modeled via

θVI,k = ωVI,k, (4.4a)

mkωVI,k = −dkωVI,k − PVI,k, (4.4b)

where θVI,k, ωVI,k are the angle and frequency of voltage generated by the

vVI,k

ik

vkto grid

Figure 4.3: A schematic representation of a grid-forming virtual inertia device.

grid-forming VI device, PVI,k is the active power from the grid-forming VIdevice into the grid, mk > 0 is the virtual inertia constant, and dk ≥ 0 thevirtual damping constant. The amplitude of the voltage is regulated at the

xs = fs(xs, zs), 0 = gs(xs, zs, i)

RLCfilter

vkVI

(4.4)

ik

vVI,k

PVI,k

Figure 4.4: Interconnection of a single grid-forming VI device modeled via (4.4).

nominal operating voltage of the bus to which the device is connected1. Theoverall signal flow for the grid-forming VI device is shown in Figure 4.4.The second-order active power-frequency droop characteristics (4.4) are thecore operating principle of a wide range of grid-forming control algorithmsfor power converters. For instance, under the assumption that the controlledinternal dynamics of the converter are sufficiently fast, droop control witha low pass filter in the power controller (see Figure 5 of [70]) is equivalent

1 We assume that the voltage is regulated, however, voltage support can also be incorporatedby extending the model of the power electronic devices by some form of reactive powercompensation or voltage regulation.

4.4 performance metrics and design constraints 45

to (4.4) (see Lemma 4.1 of [71]). Similarly, (4.4) can be explicitly recoveredfor a wide range of grid-forming controls by applying model reductiontechniques that eliminate the fast controlled internal dynamics of powerconverters [72]. This includes virtual synchronous machines that directlyenforce a second-order frequency droop behavior (see Section II-A of [14]and Section II-B of [52]) as well as controls based on matching the dynamicsof power converters to that of a synchronous machine [53], [66], [67], [72](see Remark 2 and Section 3.3 of [53]).

4.3.2 Disturbance Model

We consider a general class of disturbance signals ηk(t) that act at thevoltage buses of the power system (4.1) through the current injection ik. Thisapproach can be used to model a wide range of faults such as load steps,fluctuations in renewable generation, or generator outages (i.e., by cancelingthe current injection of a generator). For brevity of presentation, we focus onfaults that map changes in active power injection at every bus (i.e., changesin demand or generation) to current injections ik at every voltage bus k. Wedenote by η = (η1, . . . , ηnd) the disturbance vector that corresponds to, e.g.,changes in load or generation (fluctuations of renewables).

4.4 performance metrics and design constraints

In this section, we discuss several performance metrics typically utilized instability analysis of power systems. As an alternative to these conventionalmetrics, we propose system norms (as in Chapter 3) as a tool to assesstransient stability and for optimizing the allocation of VI devices. Further,we discuss design constraints on the inertia, damping gains arising fromlimits on the maximum power output of the VI devices and specificationsimposed by grid-codes.

4.4.1 Performance Metrics

Based on the model presented in Section 4.3, we now formally define a setof performance metrics that we shall use to assess the frequency stability ofthe grid, when subjected to disturbances. Using the response of the systemfollowing a disturbance input η(t), several time-domain metrics can bedefined. In particular, given a negative step disturbance, e.g., a sudden loadincrease or loss of generation, at time t = 0, we define the following indices


on the time-domain evolution of the frequency vector ω = (ω1, . . . , ωn) thatcollects the frequencies at different buses. Let the frequency nadir |ωk|∞,maximum RoCoF2 |ωk|∞ at each bus k be given by

|ωk|∞ := maxt≥0|ωk(t)| , (4.5)

|ωk|∞ :=∣∣∣∣min

t≥0ωk(t)

∣∣∣∣ . (4.6)

Further, let ωG = (ωG,1, ωG,2, . . .)= [ω>G,1, ω>G,2, . . .]>, ωG = (ωG,1, ωG,2, . . .),PG = (PG,1, PG,2, . . .), and PVI = (PVI,1, PVI,2, . . .) collect the generator fre-quencies, RoCoF, the governor power injections, and the active powerinjections of VI devices. For the same step disturbance as considered above,we also define the peak power injection by the virtual inertia devices as wellas the peak power injection due to the governor response of synchronousmachines ∣∣PVI,k

∣∣∞ := max

t≥0

∣∣PVI,k(t)∣∣ ,

∣∣PG,k∣∣∞ := max

t≥0

∣∣PG,k(t)∣∣ . (4.7)

Next, for any disturbance input η (not necessarily a step) we definethe metrics quantifying the energy imbalance and control effort on a timehorizon τ. The integrals

Eτ,η(t),ω :=∫ τ

0ω>G ωG dt, Eτ,η(t),ω :=

∫ τ

0ω>G ωG dt, (4.8)

capture the frequency and RoCoF imbalance post-fault, and

Eτ,η(t),PVI:=∫ τ

0P>VIPVI dt, Eτ,η(t),PG

:=∫ τ

0P>G PG dt, (4.9)

encode the virtual inertia and damping effort of the converters, and themachine governor efforts.

Consider a weighted sum of the metrics in (4.8)–(4.9), i.e.,

Jτ,η(t) :=∫ τ

0rGP>G PG+ rVIP>VIPVI + rωω>G ωG + rω ω>G ωG dt,

with non-negative scalars rG, rω, rω, and rVI trading off the relative efforts.In the context of transient frequency stability analysis of power systems,

faults are typically modeled as steps capturing, e.g., an increase in load or

2 The RoCoF for the generators is computed by filtering the frequency derivative through alow-pass filter [73].

4.4 performance metrics and design constraints 47

loss of generation. The main purpose of virtual inertia and fast frequencyresponse is to improve the transient behavior of the power system imme-diately after such a fault. However, unless the time horizon τ is chosencarefully, the quadratic cost Jτ,η(t) for a step disturbance is a questionablemetric for optimizing virtual inertia and fast frequency response because itis dominated by the post-fault steady-state deviation. To avoid this prob-lem, we consider the H2 norm that can be interpreted as the energy of thesystem response to impulse disturbance inputs. Specifically, the H2 normcan be obtained by perturbing the system with a unit impulse δ(t) at everydisturbance input ηk individually and summing over the resulting infinitehorizon costs J∞,δk(t). To clarify the interpretation in the context of powersystems, note that the impulse response of a linear system is equal to thetime derivative of its step response. In other words, as time goes to infinitythe step response of a stable system will settle to constant values, but thesignals in the integral of the cost J∞,δk(t) tend to zero. Therefore, the H2norm predominantly captures the initial transient. This is in line with theuse of virtual inertia and fast frequency response to stabilize the frequencybefore slower controls and ancillary services act to control the long-termpost-fault steady-state behavior of the system.

As the H2 norm measures short-term energy imbalance, it is a suitableproxy for transient power system stability. Concurrently, H2 norms result intractable, well understood design and optimization problems that apply toa broader class of disturbances than the classical power system metrics andin special cases, allow to solve min-max problems arising by maximizingover the disturbance vector η while minimizing over inertia coefficients asin Section 3.4.6. In the remainder, we will use H2 norms for control designand tuning. However, for evaluation purposes, we also consider the metrics(4.5)–(4.7) commonly used in power system analysis.

4.4.2 Design Constraints

In addition to the performance metrics presented in the previous section,the virtual inertia devices are also subject to constraints on their powerinjection. Moreover, the net damping is constrained by grid-codes andprimary control reserve markets3. To account for constraints on the netdamping, we impose an upper bound on the sum of damping gains ofthe VI devices, i.e., ∑k dk ≤ dsum. This ensures realistic results in line with

3 As per ENTSO-E specifications, 70MW at 200mHz is an approximate requirement for Switzer-land [74].


the power system operation. Further, we use constraints on the individualinertia and damping gains to account for the maximum power rating of thepower converters. Notably, it has been observed from empirical data, thatthe maximum frequency deviation and the maximum RoCoF do not occurat the same time (see Section III-C of [58], see also the scatter plot in Figure3 of [75]). Therefore, the inertia response and the damping response donot attain their peak values simultaneously. Based on this observation, themaximum power injection constraint can be approximated by the constraints

mk ≤Pmax,k

|ω|max, dk ≤

Pmax,k

|ω|max,

where Pmax,k is the power rating of the k-th converter and |ω|max and |ω|maxare a priori estimates of the maximum RoCoF and frequency deviation. Welimit the individual damping and inertia gains to be non-negative.

4.5 closed-loop system and H2 optimization

In this section we present a computational approach to answer the questionof “how and where to optimally use virtual inertia and damping?” viaappropriate tuning of the gain matrices Kfoll and Kform in order to improvethe post-fault response of a low-inertia power system.

4.5.1 Closed-loop System Model and Linearization

As in Chapter 3, the placement and tuning of VI devices can be recastas a system norm (input-output gain) minimization problem for a linearsystem. To this end, we combine the power system model (4.1) with thedisturbance model presented in Section 4.3.2 and either grid-following(4.3) or grid-forming (4.4) virtual inertia device models. Next, we defineinputs and outputs of the system interconnected with the grid-formingand grid-following devices that allow us to optimize the virtual inertia anddamping gains, mk and dk respectively.

For each of the grid-forming devices, we define yform,k = (ωVI,k, PVI,k) asthe output, collecting its internal frequency variable and its active powerinjection; and uform,k = ωVI,k as the control input. On choosing Kform,k =

−[dk m−1k m−1

k ], (4.4b) can be re-expressed as

ωVI,k = uform,k = Kform,k yform,k. (4.10)

4.5 closed-loop system and H2 optimization 49

The resulting overall system with input uform, output yform, and gain matrixKform is shown in Figure 4.5.

PowerSystem

VSC

Kform

Distη

i

+

uform yform

ωG

ωG

PG

PVI

︸︷︷︸

yp

Figure 4.5: Closed-loop system interconnection for the grid-forming VI withtuning parameters Kform, where VSC is the voltage source converter.

Similarly, for each grid-following device we define yfoll,k = (ωk, ˙ωk) asthe output, collecting the frequency and RoCoF estimates from the PLL;and the active power set-point as the control input ufoll,k = P?

VI,k. WithKfoll,k = [dk mk], (4.3) can be re-expressed as

P?VI,k = ufoll,k = Kfoll,k yfoll,k. (4.11)

The resulting overall system with input ufoll, output yfoll, and gain matrixKfoll is shown in Figure 4.6.

PowerSystem

Kfoll

PLLCPS

Distη

i

+

yfollufoll

ωG

ωG

PG

PVI

︸︷︷︸

yp

Figure 4.6: Closed-loop system for the grid-following VI with tuning parametersKfoll, where CPS is the controllable power source.

Next, let x and u denote the states of the power system equipped withVI devices and the control inputs of these VI devices. Additionally, let yand yp denote the outputs and the performance outputs corresponding to


the performance metrics discussed in Section 4.4.1. The overall dynamicalmodel translates to

x = f (x, z, u, η), (4.12a)

0 = g(x, z), (4.12b)

(y, yp) = (h(x), hp(x)). (4.12c)

Next, we linearize these dynamics around a nominal operating point. Inthis process, the algebraic equation (4.12b) can be eliminated by exploitingthe fact that its Jacobian with respect to x has full rank at operating pointsthat do not correspond to voltage collapse. Likewise, we can also removethe unobservable mode (with zero eigenvalue) corresponding to absoluteangles to obtain a linearization

∆x = A∆x + B∆u + Bgη, (4.13a)

∆y = C∆x, ∆yp = Cp∆x, (4.13b)

where ∆x, ∆y, ∆yp, ∆u are the resulting deviation states, measurementoutputs, performance outputs, control inputs; and Bg = BΠ1/2 is thedisturbance gain matrix which encodes (via Π = diag{π1, π2, . . .}) thelocation and (relative) strengths of the disturbances η. The states x andoutputs y are different for both VI implementations. For grid-followingimplementation, these correspond to x = (xs, xPLL), yfoll = (ω, ˙ω) whereas,x = (xs, xVI), yform = (ωVI, PVI) refer to grid-forming implementation.

4.5.2 Virtual Inertia as Output Feedback

The control inputs are constructed via static output feedback (4.13), i.e.,

∆ufoll =

d1 m1 . . . 0 0...

.... . .

......

0 0 . . . dnc mnc

︸︷︷︸

Kfoll

∆ω1

∆ ˙ω1...

∆ωnc

∆ ˙ωnc

︸︷︷︸

∆yfoll

, (4.14)

4.5 closed-loop system and H2 optimization 51

∆uform =

α1 β1 . . . 0 0...

.... . .

......

0 0 . . . αnc βnc

︸︷︷︸

Kform

∆ωVI,1

∆PVI,1...

∆ωVI,nc

∆PVI,nc

︸︷︷︸

∆yform

, (4.15)

are the control inputs for the grid-following, grid-forming implementationsrespectively, with feedback matrices Kfoll, Kform, and where αk = −dk m−1

k ,βk = −m−1

k , nc is the number of virtual inertia devices.For our analysis, the performance output is selected as

∆yp = Cp ∆x =(

r12ω∆ωG, r

12ω ∆ωG, r

12G∆PG, r

12VI∆PVI

),

where the states x depend on the VI implementation. For impulse distur-bances, the infinite horizon integral of quadratic penalties on frequencydeviations, RoCoF, as well as power injections from VI devices and genera-tors is given by

J∞,δ(t) =

∫ ∞

0∆y>p ∆yp dt, (4.16)

the system energy imbalance. With Acl = A + BKC, and combining (4.13)and (4.14) (respectively, (4.15)) the resulting dynamical system G is givenby

∆x = Acl∆x + Bgη, ∆yp = Cp ∆x . (4.17)

4.5.3 H2 norm Optimization

To compute the H2 norm between the disturbance input η and the perfor-mance output yp of the system (4.17), let the so-called observability GramianPK denote the positive definite solution P of the Lyapunov equation

PAcl + A>cl P + C>p Cp = 0, (4.18)

parameterized in K for the given system matrices A, B, C, and Cp. Basedon the observability Gramian PK, the H2 norm ‖G‖2

H2is given by [31]

J∞,δ(t) = ‖G‖2H2

= Tr(Bg>PKBg). (4.19)


Thus, the optimization problem to compute the optimal allocation withrespect to the H2 norm ‖G‖2

H2is

min J∞,δ(t) (4.20a)

subject to K ∈ SC . (4.20b)

The set SC = S ∩ C, where S encodes the structural constraint on K, i.e.,the purely local feedback structure of the virtual inertia control in (4.14) and(4.15). Hereafter, we consider C to be the set of constraints on the controlgains discussed in Section 4.4.2. The optimization problem (4.20) can tunethe gain of any VI device in the system. Sparse allocations (i.e., with fewVI devices with significant contribution) can be obtained by including an`1-regularization penalty in the optimization as in Section 3.4.5.

Note that evaluating the cost function requires solving the Lyapunovequation (4.18), which is non-linear in P and K. In general, the optimizationproblem (4.20) is non-convex and may be of very large-scale. However, byexploiting the feedback structure of the problem, the gradient of ‖G‖2

H2with respect to K can be computed efficiently (see Appendix A.4 andAppendix A.5) and can be directly used to solve (4.20) via scalable first-ordermethods (e.g., projected gradient) or to speed up higher-order methods.

4.5.4 Complexity of the Gradient Computation

In Chapter 3, gradient-based optimization methods are used to directlyoptimize the inertia constants of a linearized networked swing equationmodel to minimize the H2 norm of a power system. For a system with nbuses, the gradient computation in Theorem 6 requires the solution of n− 1Lyapunov equations of dimension 2n, resulting in a complexity of O(n4).In contrast (4.20) includes more realistic models of virtual inertia devicesand the gradient of J∞,δ(t) can be computed by solving two Lyapunovequations of a higher dimension, thereby reducing the complexity to O(n3).In [63] a sequential linear programming approach is used to optimizethe allocation of grid-following virtual inertia and damping. This methoddirectly optimizes the frequency nadir and RoCoF. However, every iterationof the optimization algorithm in [63] requires computing the eigenvalues ofthe linearized system which has complexity O(n3), as well as time-domainsimulations and the solution of a linear program, resulting in far highercomputation complexity than the proposed method.

4.6 test case description 53

4.6 test case description

To illustrate our algorithms for optimal inertia and damping tuning, we usea test case based on the 14-generator, 59-bus South-East Australian system[3], [76] shown in Figure 4.7. It is equipped with higher-order models forturbines, governors, power system stabilizers, and voltage regulators. Thissystem has several interesting features, for instance its string topologyand weak coupling between South Australia (area 5) and the rest of thesystem. The SIMULINK version [77] of the model [76] was developed forthe light loading scenario. Variations of this model have also been studiedas low-inertia test cases in [63], [78].

For this chapter the model presented in [77] was modified to obtain a low-inertia case study by replacing synchronous machines located at the buseslabeled 101, 402, 403, and 502 with constant power sources4 that inject thesame active and reactive power as the original generators. This modelingchoice is based on the high penetration of renewable generation in thereal-world power system (particularly in area 5) [78] that does not providefrequency support. The model was augmented with 15 VI devices acrossthe system (see Figure 4.7). For brevity of the presentation we consider twoscenarios, in the first scenario the VI devices are all grid-forming, in thesecond scenario they are all grid-following (see Section 4.3). In the casestudy in [63] motor loads with non-negligible inertia are used to ensure thatthe notion of a frequency signal (as input the VI devices) is well defined. Inthis work, we do not require this assumption. Finally, we use constant powerinjections at six locations (indicated by a red bolt) to simulate disturbances.The SIMULINK model of the benchmark system including virtual inertiadevices is available online [79].

4.7 simulation results

In this section we compare the performance of the original system with theclosed-loop system equipped with virtual inertia and damping devices. Weconsider both the grid-following, grid-forming modes of implementationand mainly focus on the performance metrics defined in Section 4.4.1.

4 In other words, sources that maintain constant active and reactive power injections regardlessof the frequency or voltage at their point of connection.


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Figure 4.7: South-East Australian Power System line diagram. The crossed outgenerators are replaced by constant power sources to mimic a low-inertia scenario, whereas the red lightning symbols are the locationswhere disturbances are injected. The circles with VI inscribed withinindicate the virtual inertia and damping devices distributed acrossthe power system.

4.7 simulation results 55

4.7.1 Validity of the Linearized Model

As discussed in (4.13), we optimize the virtual inertia and damping gainsusing a linearization of the system at the nominal operating point. Tovalidate the linearized model we compare it to the non-linear model forstep disturbances at the six locations shown in Figure 4.7 ranging from−250 MW to +250 MW. In Figure 4.8, the relative linearization errors fordifferent performance metrics are plotted- both for the grid-following andthe grid-forming virtual inertia and damping implementations. The plotsreveal a concentration of data points in the −10% to +10% band. Thisindicates that the linear approximation of the model closely resembles thenon-linear model and justifies the effectiveness of our approach.

−50 −25 0 25 500

10

20

30

∆∣∣ωG,k

∣∣∞ [%]

sam

ples

[%]

−50 −25 0 25 500

10

20

30

∆∣∣ωG,k

∣∣∞ [%]

sam

ples

[%]

−50 −25 0 25 500

10

20

30

∆∣∣PVI,k

∣∣∞ [%]

sam

ples

[%]

−50 −25 0 25 500

10

20

30

∆∣∣PG,k

∣∣∞ [%]

sam

ples

[%]

Figure 4.8: Distribution of relative linearization errors for load steps rangingfrom −250 MW to +250 MW at the nodes indicated in Figure 4.7 forboth grid-forming, grid-following configurations.

4.7.2 Optimal Tuning and Placement of VI Devices

The optimal inertia and damping profiles for the system are computedusing the optimization problem (4.20). We consider the same weighted per-formance outputs (4.5.2) for both grid-forming, grid-following VI and set


the penalties to rω = 0.1, rω =0.2, rG=0.2, and rVI=0.2, thereby identicallypenalizing the power injections from the VI devices and the synchronousmachines. Further, the disturbance gain matrix Π, in Section 4.5.1 is chosensuch that πi = 1. In other words each node is subjected to equally sizeddisturbances. Finally, using the approach outlined in Section 4.4.2 we choosethe constraints such that ∑k dk ≤ 420 MWs/rad, dk ≤ 40 MWs/rad, andmk ≤ 18.5 MWs2/rad. These constraints ensure that the total damping doesnot exceed realistic values, and that the power output of the converters isroughly limited to 40 MW for frequency deviations in the normal operatingregime. The resulting inertia and damping allocations for the above param-eters and constraints are depicted in Figure 4.9 (a) and Figure 4.9(b) for thegrid-forming and grid-following implementations respectively. Finally, weobserve that no significant performance gains can be achieved by optimiz-ing the PLL gains beyond applying standard tuning techniques (see [64],[69]).

4.7.3 Contrasting Allocations for VI Implementations

The two allocations highlight some interesting features. Note that the op-timized allocations are not uniform across the system. In fact, the virtualinertia for both implementations is predominantly allocated in area 5. Inci-dentally, the blackout in South-Australia in 2016 was also in this area [5].Moreover, uniform allocations, chosen as the initial guess for the optimiza-tion, are typically not optimal (see also Section 3.5). Another facet of theallocations is that the gains for the grid-following virtual inertia devicesare limited by the constraints imposed in the optimization. This may beprimarily attributed to time delays (RoCoF estimation, response time τfoll ofthe power source, etc.) encountered for the inertial response. To compensatefor these delays, the allocation for the grid-following VI relies on largerinertia and damping gains (see Figure 4.9) at some nodes as well as largertotal damping and inertia (refer Table 4.1). While significant inertia anddamping is allocated at all nodes in the case of grid-forming virtual inertiadevices, the grid-following implementation results in negligible allocationsfor some nodes outside of area 5.

4.7.4 Impact of VI Devices on Frequency Stability

To investigate the effect of the VI devices, we consider the non-linear modelof the South-East Australian grid with the optimal VI device parameter


102 208 212 215 216 308 309 312 314 403 405 410 502 504 5080

10

20

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50

node

(a) Grid-forming

inertia [MW s2/rad]damping [MW s/rad]

102 208 212 215 216 308 309 312 314 403 405 410 502 504 5080

10

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50

node

(b) Grid-following

inertia [MW s2/rad]damping [MW s/rad]

Figure 4.9: Optimal inertia and damping allocations for the Australian systemfor the grid-forming and grid-following configurations.

values from Figure 4.9. Next, individual step disturbances at the six lo-cations, as shown in Figure 4.7 are considered. These disturbances rangefrom −375 MW to −150 MW and capture a load increase (or equivalently aloss of generation). In Figure 4.10, the resulting distribution of post-faultfrequency nadirs

∣∣ωG,k∣∣∞, maximum RoCoF

∣∣ωG,k∣∣∞ of all generators for

the system without VI devices, the system with grid-following VI devices,and the system with grid-forming VI devices are represented along with thepeak power injections

∣∣PVI,k∣∣∞ from the VI devices. We make the following

observations:


0 0.05 0.1 0.150

5

10

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25

|ωG,k|∞ [Hz]

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ples

[%]

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sam

ples

[%]

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50

|ωG,k|∞ [Hz/s]

sam

ples

[%] low-inertia

grid-followinggrid-forming

0 0.2 0.4 0.60

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ples

[%]

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ples

[%]

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sam

ples

[%]

Figure 4.10: Distribution of generator frequency nadirs, maximum generator Ro-CoF, and VI power injections for load steps ranging from −350 MWto −150 MW at the nodes indicated in Figure 4.7 for different con-verter configurations.


(i) The mean and variance of the distribution of frequency nadirs is smallerfor the system equipped with the VI devices as compared to system withoutVI devices.

(ii) The grid-following implementation has a smaller mean frequency nadir,but larger variance and longer tail in comparison to grid-forming coun-terpart. However, this comes at the expense of a much larger peak powerinjection for certain disturbances. Note that, the maximum injection for thegrid-following VI, for the same set of disturbances is roughly three timesthat of the grid-forming VI implementation.

(iii) While we observe a smaller mean and a shorter tail of the distributionof the maximum generator RoCoF for grid-following virtual inertia, thehistograms suggest that the impact of the virtual inertia devices on the max-imum RoCoF is modest. However, it is noteworthy that these histogramsonly depict the maximum RoCoF

∣∣ωG,k∣∣∞ at generator buses. Moreover, the

maximum is typically attained during the first swing of the system aftera fault. In contrast, the time-domain simulations depicted in Figure 4.11

show that virtual inertia devices can have significant impact on the RoCoFωG after the first swing. These differences are not captured when usingthe maximum RoCoF

∣∣ωG,k∣∣∞ as performance metric, but are accurately

captured by the H2 norm.

We conclude that the VI devices have the expected positive impact onfrequency stability. Moreover, the differences between the two VI implemen-tations appear to be mostly related to differences in the maximum powerinjection. In the next section we will investigate the time-domain responseof the system with and without VI devices in more detail.

4.7.5 Time-domain Responses

We now simulate a load increase of 200 MW at node 508, this represents arealistic contingency in the system (see Section II of [3]). Broadly speaking,this disturbance could also represent a loss of 200 MW renewable generationin area 5 and is of the type considered in the H2 optimization. Moreover,due to the low levels of rotational inertia in area 5, the placement of this faultcorresponds to the worst-case location. The system responses are illustratedin Figure 4.11 and underscore the efficacy of virtual inertia and dampingdevices in a low-inertia power system. The grid-following and grid-formingVI implementation with the optimal allocations from Figure 4.9 are sim-ulated and compared with the response of the original system. Table 4.1


0 2 4 6 8 10 12 14−150

−100

−50

0

50

t [s]

ωG

[mH

z]low-inertiagrid-followinggrid-forming

0 2 4 6 8 10 12 14

−0.2

0

0.2

t [s]

ωG

[Hz/

s]

0 2 4 6 8 10 12 14

0

5

10

15

t [s]

P G[M

W]

0 2 4 6 8 10 12 14−10

0

10

20

t [s]

P VI

[MW

]

Figure 4.11: Time-domain plots for generator frequencies, generator RoCoF, gen-erator power injections, and power injections of the VI devices forthe original system, grid-following, and grid-forming configurationsfor a step disturbance of 200 MW at node 508.

4.8 summary and conclusions 61

shows the key performance indicators discussed in Section 4.4.1. The toppanels of the time-domain plots in Figure 4.11 illustrate the frequenciesand the RoCoF of the 10 generators in the system for the two different VIimplementations and the original system. The power injections from thegenerators and the 15 virtual inertia and damping devices across the powersystem are plotted in the bottom two panels of the figure. The key insightsdrawn from a closer analysis of these plots and the table are summarizedbelow:

(i) While both VI implementations improve the frequency nadir and maxi-mum RoCoF, the grid-forming VI implementation performs better in termsof the absolute values. Further, the total inertia and damping is also less.

(ii) The maximum active power maxk∣∣PVI,k

∣∣, injected by a single virtualinertia device as well as the maximum power maxt≥0

∣∣∑k PVI,k(t)∣∣ injected

by all the virtual inertia devices combined is smaller for the grid-formingvirtual inertia devices. Thus, grid-forming virtual inertia achieves a betterperformance with a lesser control effort in comparison to the grid-followingconverters.

(iii) A drop in the maximum governor response maxk∣∣PG,k

∣∣, of a singlesynchronous machine compared to the original system is observed due tothe active power injections from the virtual inertia devices.

(iv) A decrease in the H2 norm is observed for both VI implementations, i.e.,the H2 norm is an effective proxy for other time-domain metrics relevantfor analysis [58].

(v) Another difference in the implementations pertains to the computationtimes for solving the optimization problem. Using MATLAB on a Corei7-6600U CPU, the optimization for grid-forming VI takes around 60s incomparison to 160s for the grid-following VI for identical penalties.

4.8 summary and conclusions

In this chapter we considered the problem of low-inertia power systemsequipped with grid-following or grid-forming VI implementation usingpower electronics interfaced renewable energy sources. We modeled thesetwo implementations as dynamic feedback control loops that provide virtualinertia and damping. A system norm-based optimization approach wasused to study the problem of optimal placement and tuning of these devices.


Metric Original Grid-Following Grid-Forming

∑i mi [MW s2/rad] - 111.8 99.9

∑i di [MW s/rad] - 420 375.9

maxk∣∣ωG,k

∣∣∞ [Hz/s] 0.34 0.31 0.27

maxk∣∣ωG,k

∣∣∞ [mHz] 128.6 112.1 104.3

maxk∣∣PVI,k

∣∣∞ [MW] - 21.98 15.62

maxt≥0

∣∣∑k PVI,k(t)∣∣ [MW] - 68.1 41.1

maxt≥0

∣∣∑k PG,k(t)∣∣ [MW] 50.5 39.9 45.9

H2 norm 11.72 9.92 9.66

Table 4.1: Performance metrics for a load increase of 200 MW at node 508.

Our proposed tuning algorithm was far more scalable and computationallyefficient in comparison to some of the other existing approaches in theliterature and the one discussed in Chapter 3. Further, we showcasedthe capabilities of such VI devices on a high-fidelity, non-linear modelof the South-East Australian power system and illustrated their efficacy.For a range of disturbances both types of virtual inertia implementationsimproved the system resilience compared to the system without virtualinertia. The results once again show that the system robustness does notonly depend on the amount of virtual inertia used but can also dependon the specific implementation and location of virtual inertia. However,this fact is in contrast to typical paradigms of ancillary service marketsthat value energy or the total amount of damping and inertia but notlocation or specifics of the implementation and therefore do not capture thisaspect. Some results on a market mechanism that considers the location ofvirtual inertia devices shall be discussed in the next chapter. Given that ourproposed tuning algorithm is computationally efficient, our approach canalso be used to optimize a virtual inertia allocation with respect to multiplelinearized models, each modeling different dispatch points and changes inmodel structure (e.g., system splits, tripping of generators, etc.).

5A M A R K E T M E C H A N I S M F O R V I RT UA L I N E RT I A

5.1 related works

In this chapter, we consider the provisioning of virtual inertia units andthe affiliated payment architecture. The problem is analyzed through theframework of ancillary service markets and auction theory and hinges oninvestigating potential conflicts as a result of the interaction between multi-ple non-cooperative stakeholders (e.g., end-consumers, transmission systemoperators, renewable energy source providers, etc.) and proposing a plausi-ble solution which safeguards individual interests. Pricing as an instrumentto assist frequency regulation is not new, having being discussed in [80].Numerous works [81]–[83] have weighed-in on markets for regulation inpower systems. Some variants of the problem of mechanism design forelectricity markets have been considered in [84], [85], while [86] discusses agame-theoretic characterization of market power in energy markets. Amongthe several market mechanisms, energy markets based on the celebratedVickrey-Clarke-Groves (VCG) mechanism [87]–[89] have been previouslyexplored in [90]–[92]. Such a mechanism maximizes a pre-defined globalcost (e.g., social welfare of all stakeholders), while allowing individualagents to operate as per their private interests. However, to the best of ourknowledge, market design involving virtual inertia provision to ensure therobust system performance is fairly unexplored.

5.2 contributions

The contributions of this chapter are as follows. We construct and proposea framework wherein the procurement and pricing of virtual inertia acrossthe power network coupled with their effect on system performance isreformulated as an H2 system norm optimization problem. To this end,we present two approaches: the centralized problem formulation– which isconsidered as an efficiency benchmark and a market-based approach. Boththese constructions cater to long-term planning as well as the day-ahead orthe real-time (e.g., 5 minute) procurement scenarios. Though the centralizedapproach seems appealing due to its efficiency, it is quite idealistic, as all

63

64 a market mechanism for virtual inertia

participating agents need to report their true cost curves to the systemoperator. Such a scenario would result in agents reporting inflated costcurves for maximizing profit. To overcome this shortcoming, we adoptthe VCG framework from mechanism design as the foundation of ourproposed market mechanism. This mechanism incentivizes the agents toparticipate (by guaranteeing non-negative payoffs) and to report their truecost curves–as the optimal bidding strategies. We also recover the efficiencyof our benchmark solution, as the two procurement problems coincide,maximizing a social welfare objective. Though VCG has been well-studied,our mechanism is rendered novel due to the nature of the problem weaddress. Our min-max problem setup with infinite-dimensional couplingconstraint (an LTI system) is a non-trivial example for applying VCG. Thesocial welfare objective considered in this chapter is a linear combinationof costs accounting for the worst-case post-fault dynamic response of thepower network and the costs of procuring virtual inertia. Moreover, becauseof the special structure of the problem, the VCG mechanism applied to ourproblem is proved to be individual rational i.e., every market participantis guaranteed to receive non-negative net revenue. Our analytic resultsrest on a deliberately stylized power system model and an H2 frequencyperformance criterion as in Chapter 3, though they can be extended (nu-merically) to more detailed models and other performance indices as longas the underlying optimization problems can be solved efficiently. Finally,as the value of inertia and procurement thereof, is location-dependent asdiscussed in previous chapters, we show that this non-homogeneity pre-vents the existence of a global price for virtual inertia. Moreover, even witha large number of agents, it becomes critical to have them reveal the truecost and they do not possess enough market power unless they collude.

The remainder of this section introduces some notation. Section 5.3provides a background on the problem by revisiting some previously estab-lished results on H2 norm-based coherency metrics. Section 5.4 introducesthe different costs associated with virtual inertia and provides a benchmarkprocurement problem. Such a problem admits a solution which maximizesa measure of social welfare. Section 5.5 presents an efficient market-basedsetup for inertia procurement. Section 5.6 presents a case study and simula-tions on a three-region network, where we compare the market-based andclassic regulatory approaches. Finally, Section 5.7 concludes the chapter.

notation We denote the n-dimensional vectors of all ones and zerosby 1n and 0n, whereas 0 is a matrix of zeros of appropriate dimension.

5.3 problem background 65

The n × n identity matrix is In. For A ∈ Rn×n, A> is its transpose andTr(A) = ∑n

i=1 Aii is its trace. For a positive semi-definite matrix Q � 0,Q

12 is its square root. Given an index set I with cardinality |I| and a real-

valued array {x1, . . . , x|I|}, we denote by x ∈ R|I| the vector obtained bystacking the scalars xi, by diag{xi} the associated diagonal matrix and byblkdiag(a1, a2, . . . , an) a block-diagonal matrix with the elements {ai} onthe diagonals. The vector ei is the i-th vector of the canonical basis for Rn.

5.3 problem background

In this section, we present our power system model and integral-quadraticH2 performance criterion for frequency stability. We consider a powersystem model consisting of network-reduced swing equations accountingfor the dynamics of synchronous machines and virtual inertia devices as in[11], [30], [60], [93] and Chapter 3. This coarse-grain model for the powersystem frequency dynamics together with the H2 metric allows us to deriveour market mechanism in an analytic and insightful fashion. We remark thatthe results in this chapter hold equally true when considering higher-ordermodels of synchronous machines including governor dynamics, inertia-emulating power converters as in Chapter 4, and also other performancemetrics [30], [63], [93] as long as the underlying optimization problems canbe solved efficiently.

5.3.1 System Model

We consider the system model considered in Chapter 3. The power networkis modeled by an underlying graph with nodes (buses) V = {1, . . . , n} andedges (transmission lines) E ⊆ V × V . Next, a small-signal version of anetwork-reduced power system model [42] is considered, where passiveloads are modeled as constant impedances and eliminated via Kron re-duction [43], and the network is reduced to the sources i ∈ {1,. . . , n} withlinearized dynamics. The assumptions of identical unit voltage magnitudes,purely inductive lines, and a small-signal approximation [42] result in

mi θi + di θi = pin,i − ∑{i,j}∈E

bij(θi − θj), ∀ i, (5.1)

where pin,i is the net (i.e., electrical and mechanical) nodal power input,bij ≥ 0 is the inverse of the reactance between nodes {i, j} ∈ E . If bus i


only hosts a single synchronous machine, then (5.1) describes the electrome-chanical swing dynamics for the generator rotor angle θi [42], mi > 0 is thegenerator’s rotational inertia, and di > 0 accounts for frequency dampingor primary speed droop control (neglecting ramping limits).

The renewable energy sources interfaced via power electronic inverters[44] are also compatible with this setup. For such an interconnection, θi isthe voltage phase angle, di > 0 is the droop control coefficient, and mi > 0either accounts for power measurement time constant [46], or arises fromvirtual inertia emulation through a dedicated controlled device [12]–[14], oris simply a control gain [47]. Finally, the dynamics (5.1) may also arise fromfrequency-dependent or actively controlled frequency-responsive loads [27].In general, each bus i may host an ensemble of these devices, and thequantities mi, di reflect their aggregate behavior.

We wish to characterize the response of the interconnected system todisturbances in the nodal power injections, possibly representing faults,disconnection of loads or generators, intermittent demand, or fluctuatingpower generation from renewable sources as in [11], [30], [58], [60], [63],[93], [94]. To do so, we consider the linear power system model (5.1) drivenby the inputs pin,i parametrized as pin,i = πi

12 ηi. In this parametrization,

ηi is a normalized disturbance, and the diagonal matrix Π = diag{πi}encodes the information about the strength of the disturbance at differentbuses, and therefore, describes also the location of the disturbance. We canobtain the state-space model for this linearized system as[

θ

ω

]=

[0 In

−M−1L −M−1D

]︸︷︷︸

= A

[θ

ω

]+

[0

M−1Π1/2

]︸︷︷︸

= B

η,

where M = diag{mi} and D = diag{di} are the diagonal matrices of inertiaand damping/droop coefficients, and L ∈ Rn×n is the symmetric networkLaplacian matrix. The states x = (θ, ω) ∈ R2n are the stacked vectors ofangles and frequencies (deviations from the nominal values). Note thatA[1>n 0>n ]

> = 02n and thus, [1>n 0>n ]> is the right eigenvector corresponding

to the eigenvalue zero or the absolute angles. To gauge the robustness of apower system to disturbances, we study generalized energy functions asmetrics [58], e.g., a quadratic function expressed as the time-integral (3.4)∫ ∞

0

{12

n

∑i,j=1

aij(θi(t)− θj(t))2 +n

∑i=1

si ω2i (t)

}dt. (5.2)


This generalized energy can be rewritten as the integral∫ ∞

0 yp(t)>yp(t)dt,where yp(t) = C x(t) is the corresponding performance output for an inputu(t). Note that by construction, the matrix C is such that C[1>n 0>n ]

> = 02n1.

The above performance metric can be interpreted as the energy amplifi-cation of the output y (resp., its steady-state total variance) for impulsivedisturbances η (resp., unit variance white noise in a stochastic setting) andis commonly referred to as the squared H2 norm as discussed in Chapter 3.The suitability of this metric to describe the stability and robustness of thesystem post fault is discussed in Chapter 3 and Chapter 4.

Here, we recall a tractable approach for computing the energy met-ric presented in Lemma 1. For the state-space system G(A, B, C), withA[1>n 0>n ]

> = C[1>n 0>n ]> = 02n. The H2 norm, Y(m, π) is given by

‖G‖2H2

= Y(m, π) = Tr(B>PB), (5.3)

where P ∈ R2n×2n is the observability Gramian, a solution to (3.8)–(3.9).It is known that the existence of the solution of (3.8) depends on the

positive semi-definiteness of C>C. Furthermore, from [60], we conclude thatthe solution P is unique under the constraint (3.9). This result generalizesthe uniqueness of P, to positive semi-definite matrices C>C under certainassumptions. Lemma 1 allows us to analyze the generalized energy function(5.2) via an elegant H2 norm optimization problem.

5.3.2 Virtual Inertia and Post-fault Behavior

The minimization of (5.3) through suitable choice of inertia coefficients miattenuates the energy amplifications due to disturbances. We stress herethat as the H2 norm is a function of both the spatial distribution of theinertia m in the grid and the location and strengths of the disturbancesπ (through the input matrix B and the cost function C), the effect of thelocation also impacts the market mechanism.

We briefly revisit some of the key results from Chapter 3 which form thebuilding blocks for our analysis:

(i) Generalized energy function– The performance metric (5.2) computedas in (3.7)–(3.9) is generally non-convex in the inertia parameter m. This istroublesome as it significantly complicates the analysis and study of theconsequences of loss in synchronous inertia in large-scale power systems.

1 As the system matrix is not Hurwitz, this choice ensures that the uncontrollable mode of A isalso unobservable from C.


A meaningful metric which singles out the effect of inertia on the post-fault frequency response is desirable in order to quantify and mitigate anyobstacles arising due to the loss in synchronous inertia. To tackle these twoconcerns simultaneously, we consider the output y(t) = D1/2ω(t) as thecost in (5.2). This yields

YD(m, π) = ∑i

∫ ∞

0[ω(t)>D ω(t)]i dt, (5.4)

which penalizes the frequency excursions at each node via the primarycontrol effort in (5.1) for disturbances at all inputs ui(t). This cost functionis also justified from a system operator’s viewpoint as significant resourcesare employed to contain frequency violations through droop control torestore frequency stability, which is effectively captured here. We note thata penalty on primary control effort is also (and especially) meaningfulwhen considering more detailed higher-order power system models [58]where the optimal control strategy trades-off already existing droop controland additional costly inertia emulation.

(ii) Primary control effort– The performance metric YD(m, π) in (5.4), fur-ther, as a special case, admits a closed-form expression of the squared H2norm via Lemma 1 which is convex in the inertia variable m, i.e.,

‖G‖2H2

= YD(m, π) = ∑i

πimi

, (5.5)

where πi, mi are the disturbance strength, inertia at node i respectively.Note that the performance metric is linear in the disturbance strengths πi.

Furthermore, as discussed in Section 5.3.1, Π encodes the specific locationof a disturbance besides the strength. As grid specifications necessitateperformance guarantees against all possible contingencies, it is appropriateto consider a robust performance metric accounting for the worst-casedisturbance. We recall from Section 3.4.6, that such worst-case requirementscan be incorporated by exploiting the linearity of the performance metric(5.5) in π. We denote by P , the set collecting all the possibly occurringdisturbances (by historical data, or forecasts). As a representative, we re-consider the following normalized set for the rest of the chapter

P :{

π ∈ Rn+, 1>n π ≤ πtot

}. (5.6)


The set P is a proxy for a set of bounded energy disturbances. Let Y(m)be the worst-case performance, over the set of normalized-disturbances inP , i.e.,

Y(m) = maxπ∈P

Y(m, π). (5.7)

As Y(m, π) is linear in π, the maximization problem (5.7) can be reformu-lated as a tractable minimization problem, which for the primary controleffort (5.5) is given by

Y(m) =

minρ≥0

πtot ρ

subject to m−1i − ρ ≤ 0, ∀ i,

(5.8)

where ρ is the dual multiplier associated with the inequality budget con-straint (5.6) on the disturbances.

(iii) Convex upper bounds for generalized energy functions– As we observedin (5.5), the objective is convex in the inertia variable m, but this is nottrue for any general output matrix C. However, it is possible to obtain anupper-bound on the generalized energy function which is convex in m, asshown below.

Remark 2 (Convex upper bound) For a generic H2 performance metric(5.2) with an output matrix C, a convex (in m) upper bound is

Y(m, π) ≤ π · 12d

Tr(L† Q1) +n

∑i=1

(Q2)imi

︸︷︷︸= Ub(m)

,

where π = maxi{πi}, d = mini{di}, L† is the pseudoinverse of the networkLaplacian L, and Q = C>C = blkdiag(Q1, Q2). For the disturbance set Pin (5.6), the worst-case upper bound Ub(m) on the performance Y(m, π) interms of the inertia variable m is given by

U (m) = maxπ∈P

π Ub(m) = πtot Ub(m).

Alternatively, this bound can be expressed as

U (m) =

minρ≥0

πtot ρ

subject to Ub(m)− ρ ≤ 0.

Note that the structures of U (m) and (5.8) are strikingly similar. �


In the subsequent sections we shall focus on the primary control effortfor our analyses. However, we remark that this choice is not particularlyrestrictive, as investigating a convex upper bound (as in Remark 2) forgeneric cost functions (5.2) leads to comparable analyses and analogousresults– due to convexity in the inertia variable m.

5.4 centralized planning problem for inertia

In this section, we motivate the inertia planning problem from the systemoperator’s context. Consider a setting, wherein the system operator acquiresvirtual inertia from multiple agents, with the provision of more than oneagent per bus. A possible regulatory solution involves sourcing inertia fromeach agent proportional to its capacity. This may, however, be uneconomicalas virtual inertia units may be dissimilar based on the underlying technol-ogy and may not have identical costs. In order to make a more informedchoice, it is, therefore, necessary to consider the costs entailed for virtualinertia provision by these units.

The grid operator needs safeguarding against the worst-case limits [29]on ||ω(t)||∞. Thus, a significant virtual inertia and fast frequency responsesupport in terms of large power injections for short durations of time post-fault is necessary to offset large frequency violations and arrest the rateof change of frequency ω. We recall from (5.1) that the power injected byvirtual inertia devices during the initial post-fault transient is proportionalto ||m ω(t)||∞. As the service of providing such high peak-power is notusually provided by the standard converters, it may require a dedicatedoversized power converter interface.

A previous study [95] considers the problem of economic evaluationalong with the design aspects of virtual inertia providing devices. Thecost for provision of peak power is revealed as the overriding factor indetermining the overall cost for such devices. Furthermore, the normalizedcost of physical inertia is proposed as a trading unit, rendering all themonetary costs proportional to m. Apart from this, agents would also incursome costs towards the maintenance of these resources over their lifetime.Together, all these contribute to the net cost of providing virtual inertia as aservice. We consider c(m), a convex non-decreasing function to account forthe lumped cost of m.

The system operator, equipped with the true cost information of all agents,determines the virtual inertia allocation by optimizing system performance

5.4 centralized planning problem for inertia 71

System Operator

nodes

bids bk(µk)costs ck(µk)capacity µk

A1 An

k

Figure 5.1: A schematic representing the operator-agent interaction for the cen-tralized and market-based mechanism design and highlighting therichness of the problem under consideration. We depict differentnodes Ai through rectangular boxes and their non-homogeneity inthe value of inertia at these nodes through different colors. The num-ber of circles in each rectangle indicate the number of agents; thecolor, their dissimilarities; and the radii of the circles, their inertiacapacities.

in an economical manner. Mathematically, this translates to solving thefollowing optimization problem

minµ

υY(m(µ)) + ∑k∈A

ck(µk) (5.9a)

subject to 0 ≤ µk ≤ µk, ∀ k, (5.9b)

where Y(m) is the worst-case performance metric as in (5.8), A = A1 ∪. . . ∪An is the total number of agents, Ai is the total number of agents atnode i, µk is the virtual inertia obtained from agent k, and υ > 0 trades-offthe coherency metric and the costs. We note that m(µ) can be expressed as

mi = m0i + ∑

k∈Ai

µk, ∀ i, (5.10)

where m0 is the stacked vector of existing/residual (after the synchronousmachines are replaced) inertia at the nodes and µk is the procured virtualinertia from each agent k. We model the capacity constraints of each agentvia (5.9b), where µk is the maximum inertia procurement capacity fromagent k. This set up is schematically represented in Figure 5.12. In thefollowing, we use µ ∈ M as a shorthand for the constraints in (5.9b).

2 For the centralized problem which is considered a benchmark, the agents only provide thecosts c(µk) and not the bids b(µk).


The solution of (5.9) is the allocation that maximizes the social welfare,by virtue of minimizing the worst-case coherency metric (5.8) and the costof virtual inertia. Such an allocation is inherently efficient and constitutesthe benchmark solution of the virtual inertia procurement problem. Thecentralized problem (5.9) and the variations thereof have been studied in[58], [60], [93]. However, in liberalized energy markets, the cost functionsck(µk) are private information of each agent and therefore the systemoperator can not directly solve (5.9). We consider this framework in the nextsection.

Remark 3 (Planning and day-ahead scenarios) Both the centralized iner-tia procurement problem (5.9) and the market-based approach (that wepropose in Section 5.5) can be considered on two operational time-scales.If analyzed in a long-term planning framework, virtual inertia is predom-inantly employed to counteract the ill-effects of integrating renewables,coupled with a planned phasing-out of synchronous machines. As the costsare biased towards high peak-power devices, a market is needed to incen-tivize investment and installation of such devices. On the other hand, in aday-ahead scenario, virtual inertia is deployed for instantaneous frequencysupport in case of time-varying inertia profiles [11], incidental re-dispatches,and anticipated fluctuations. This market can be employed at different timescales, including real-time (e.g., 5 minutes). �

5.5 a market mechanism for inertia

In this section, we consider a market-based procurement approach, inspiredby ancillary service markets. In such a setting, the power system regulatordevises a market mechanism– where the system operator invites bidsfor virtual inertia in lieu of a fair compensation to the agents providingthe service. The net additional burden is eventually borne by the endconsumers. The objectives that govern the design of such a mechanismare: safeguard against the agents benefiting from reporting inflated bids;ensure that the resulting payments incentivize agents to participate in theauction by assuring non-negative returns; and guarantee the efficiency ofthe benchmark centralized planning problem discussed in Section 5.4.

5.5.1 Overview and Preliminaries: Mechanism Design

We consider a mechanism design approach to design an auction for thesystem operator to procure virtual inertia at the buses. Specifically, each

5.5 a market mechanism for inertia 73

service provider with its private cost function ck(·) is one agent. The systemoperator:

(i) collects a bid bk(·) from each agent k,

(ii) determines an allocation/procurement µk(b), and

(iii) computes the payment pk(b) for each agent k via a mechanism.

Here, b = (b1, . . . , bn) is the vector of collected bids and we alternativelyuse (bk, b−k) to denote b. Each agent k aims to choose its bidding curvebk(µk) optimally, in order to maximize individual utility, which is a measureof profit. The utility function uk, is computed as the difference between thepayment and the investment costs, i.e., the utility evaluated when µk inertiaunits are provided by agent k at bid bk is given by

uk(bk, b−k) = pk(bk, b−k)− ck(µk(bk, b−k)), (5.11)

and depends also on all other agents’ bids b−k.Different mechanism design approaches differ on the types of bids, al-

location rule, and payment rule. Due to the Revelation Principle, in thischapter, we only consider direct mechanism design, meaning that the bidsbk are the same type of cost functions ck. In the following, we formalize afew of these notions via game-theoretic definitions [96] before proceedingto the main results.

Nash equilibrium: In a non-cooperative game-theoretic setting, a Nash equi-librium is a set of strategies for the players, such that a unilateral deviationby any participant (strategies of other players unchanged) does not resultin any gain in the payoff. For the game under consideration, this translatesto a set of bids (b?1 , . . . , b?n) being a Nash equilibrium, if for all agents k, theutility uk(bk, b−k) satisfies

uk(b?k , b?−k) ≥ uk(bk, b?−k), ∀ bk,

where bk, b−k are the bids of agent k and agents excluding k, respectively.

Dominant strategy: A strategy for any player is said to be dominant, ifirrespective of the strategies of other players, the payoff is larger than anyother of its strategies. This is a stronger result than a Nash equilibriumstrategy, as it is alien to the strategies of other players.


For the game considered above, this translates to a bid b?k being a dominantstrategy for agent k, if the utility is maximized, i.e.,

uk(b?k , b−k) ≥ uk(bk, b−k), ∀ b−k, bk.

Incentive-compatibility: A mechanism is said to be incentive-compatible ifevery player maximizes its payoff by bidding true costs bk = ck, i.e.,

uk(ck, b−k) ≥ uk(bk, b−k), ∀ b−k, bk.

Observe that an incentive-compatible mechanism results in a dominantstrategy (thus, also a Nash equilibrium) with truthful bids b?k = ck.

Mechanism Structure– From a mechanism design viewpoint, the keycharacteristics of the proposed market mechanism are as follows. The “typespace” corresponds to the set of convex, non-decreasing cost curves ck(µk)which is the private information of each agent k. The “message space” isthe set of non-decreasing, convex bidding curves bk(µk) such that bk(0) = 0for each agent k. The mechanism thus designed is a direct mechanism fromthe Revelation Principle.

5.5.2 VCG Market Mechanism

The auction theory literature is rich with mechanisms proposed to in-centivize the participation of agents while prohibiting inflated bidding.A particularly popular one, which respects both the requirements, is theVickrey-Clarke-Groves (VCG) mechanism [87]–[89]. In this chapter, weassume that the regulator adopts this mechanism as a possible approach.

The system operator sets up an auction to procure virtual inertia forbuses i ∈ V . We denote by k ∈ A, the different agents who participate inthe auction (refer to Figure 5.1). Each agent simultaneously submits non-decreasing and convex bidding curves bk(µk) such that bk(0) = 0, togetherwith the margins on the maximum (µk) virtual inertia that can be provided.

The system operator instead of (5.9), determines the allocation vectorµVCG for individual bids bk(µk) for each service provider k, by solving theoptimization problem

µVCG(b) := arg minµ∈M

υY(m(µ)) + ∑k∈A

bk(µk)︸︷︷︸:= B(µ, b)

. (5.12)


We recall from (5.8) that Y(m(µ)) is convex in m (therefore, also convex inµ). The objective B(µ, b) being convex in the µ, admits an optimal allocationfrom (5.12) which is unique and that concurrently minimizes the worst-casecoherency metric (5.8), while choosing the most economical bids.

The agents k ∈ A receive a commensurate compensation for the virtualinertia µk they provide, according to the VCG payment rule (5.14). Theknowledge of the underlying mechanism is assumed to be known a priorito both the operator and agents. The payment made to agent k translates toits externality, i.e., the difference of the system costs when agent k is absentfrom the auction and the cost when agent k’s contribution is excluded.

Let µVCG-k be the vector of optimal allocations of agents, when agent kabstains from the auction. This corresponds to the solution of the same op-timization problem (5.12) with the same bids, however, with the constraintmodified as the setM−k :=M∩ (µk = 0), accounting for the absence of k,i.e.,

µVCG-k(b) := arg minµ∈M−k

B(µ, b). (5.13)

The payment pVCGk to agent k is computed as

pVCGk =

{B(µVCG-k, b)

}−{B(µVCG, b)− bk(µ

VCGk )

}, (5.14)

where we recall that µVCG and µVCG-k are defined in (5.12) and (5.13)respectively.

The market mechanism is completely defined by the map that dictatesthe allocations µk and the payments pk, given the bids bk collected fromthe agents. Figure 5.2 illustrates the operator-agent mechanism, where wemake the allocations µ(b) and the payments p(b) explicit functions of thebids.

For the market setup in Figure 5.2, we investigate suitable bidding strate-gies for the agents to maximize their utilities, in the theorem below.

Theorem 7 (VCG-based inertia auction) Consider the market setup describedin Section 5.5.2, defined by the allocation function µVCG in terms of the bids (5.12)and the payment function pVCG in (5.14). Then for every agent k:

(i) the VCG mechanism is incentive-compatible,

(ii) the utilities of and payments to the agents are non-negative3, and

3 A mechanism additionally ensuring non-negative utilities is said to be individually rational.


System Operator

minµ∈M

maxπ∈P

υ Y(m(µ), π) + ∑k∈A

bk(µk)

↑ bidbk(µk)

←→mechanism

design(Regulator)

↓ payment,allocation

(pVCGk , µVCG

k )

Individual Agent

maxbk

pVCGk (bk, b−k)− ck(µ

VCGk )︸︷︷︸

= uk(bk, b−k)

(5.15)

Figure 5.2: Schematic representing the operator-agent mechanism through aregulator proposed mechanism design.

(iii) the allocation µVCG(c) is efficient, i.e., solves the social welfare maximizationproblem (5.9).

Proof of Theorem 7: In the following we use µ? as a shorthand for µVCG, andµ as a shorthand for µVCG-k. We have from (5.12),

µ?(b) = arg minµ∈M

B(µ, b), (5.16)

where µ?(b)=µVCG(b) is a function of the bids b = (bk, b−k).As before, let µVCG-k be the vector of optimal allocations of agents, when

agent k abstains from the auction. We have,

µ(b) = arg minµ∈M−k

B(µ, b), (5.17)

where µ(b)=µVCG-k(b) is a function of the bids (b−k). The individual pay-ment to agent k as per the VCG payment rule (5.14) is

pVCGk =

{B(µ(b), b)

}−{B(µ?(b), b)− bk(µ

?k (b))

}.


Each agent maximizes individual utility with the local optimizationproblem for agent k as in (5.15). At the optimal allocation (µVCG, pVCG),with bids (bk, b−k), the utility of agent k is

uk(bk, b−k) = pVCGk (bk, b−k)− ck(µ

VCGk (bk, b−k)) (5.18)

= B(µ(b), b)−υY(m(µ?(b)))−∑j 6=k

bj(µ?j (b))− ck(µ

?k (b))︸︷︷︸

= −B(µ?(b), (ck, b−k))

. (5.19)

We recall from (5.10) that m is a function of µ. Hence, we can expressm(µ?(b)) as a function of µ?(b). Furthermore, we know from (5.15) that theagents desire to maximize individual utilities through an optimal biddingstrategy bk, preferably, independent of b−k.

Note that the first term in (5.19), B(µ(b), b) is the objective evaluated atthe optimizer obtained in (5.17). Via the constraint µk = 0 and bid char-acteristic bk(0) = 0, we conclude that the term B(µ(b), b) is independentof (bk, µk). The utility uk in (5.18) is therefore maximized when the secondterm B(µ?(b), (ck, b−k)) attains a minimum. Let B?(µ?, (b†

k , b−k)) be theminimum of B(µ?(b), (ck, b−k)), i.e.,

B? = minbk

υY(m(µ?(b))) + ∑j 6=k

bj(µ?j (b)) + ck(µ

?k (b)).

From (5.16), note that

B(µ?(bk, b−k), (bk, b−k)) = minµ∈M

B(µ, (bk, b−k)). (5.20)

Therefore, by changing arguments from bk to ck we get

µ?(ck, b−k) = arg minµ∈M

B(µ, (ck, b−k)), (5.21)

B(µ?(ck, b−k), (ck, b−k)) = minµ∈M

B(µ, (ck, b−k)). (5.22)

From equations (5.20), (5.21), (5.22), and for the truthful bid bk(µk) =ck(µk), the set µ?(b) = µ?(bk, b−k) is the minimizer of B(µ?(b), (ck, b−k)),with the minimum value of B?(µ?, (b†

k , b−k)) = B(µ?(ck, b−k), (ck, b−k)).The above argument applies to each agent k. Hence, bidding the true

cost bk = ck is a dominant strategy, and thus also a Nash equilibrium.Furthermore, as this bid bk = ck, is independent of the choice of the bids ofother agents b−k, such a mechanism is incentive-compatible.


The utility of agent k when all agents bid true costs, u?k (ck, c−k), is

u?k (ck, c−k) = B(µ(c), c)−B(µ?(c), c),

where c = (ck, c−k). As the first term B(µ(c), c) is evaluated over a smallerconstraint set M−k, it is always larger than or equal to the second termB(µ?(c), c). Consequently, it follows that bidding one’s true cost also resultsin non-negative utilities u?

k ≥ 0 and thus also non-negative paymentspVCG

k (c) ≥ 0 from (5.15). As each agent bids the true cost, the problems(5.12) and (5.9) coincide. Hence, the optimal allocation µVCG(c) minimizesthe social cost in the centralized problem (5.9) and maximizes social welfare.This completes the proof.

Theorem 7 establishes that under the framework proposed in Figure 5.2,the agents maximize their respective utilities by bidding their true costfunctions. Such a characteristic is desirable in order to prevent agents fromaccumulating profits by reporting inflated bids. We wish to highlight thatthis theorem adapts and extends the original VCG results to more complexobjectives with dynamic coupling, e.g., an H2 norm setting, to arrive at apayment scheme for virtual inertia.

In addition, the proof also helps us to reflect on the assumptions made toarrive at the result. We highlight that primarily, we relied on the convexityof the performance metric (5.5) and the convexity of the bids (Section 5.5.1)in the inertia variable m(µ). On inspection, the proof relies on the ability tocompute a global optimum of the combined cost function B(µ, b). Whenconsidering large-scale, non-linear power system models, computing thelocal minima for such a large dimensional problem is fairly easy, however,it is very hard to compute the global minimum (even if it exists). As theVCG-based payments rely on recomputing the objective without the agents’contribution, it can be visualized that the numerically computed optimizercould converge to any of the multiple local minima (depending on thesolver, initialization, etc.) and therefore, could have varied objective valuesat these minimizers. Under such scenarios, non-negativity of the utilityfunction cannot be assumed or proved. These assumptions can, however,be relaxed if global minimum can be computed explicitly or numerically.

Remark 4 (Alternate problem formulation) The planning problem (5.9)can alternatively be posed as minimizing costs subject to meeting certain

5.6 numerical case study 79

performance guarantees. In such a framework, the optimization problemtranslates to

minµ∈M ∑

k∈Ack(µk) (5.23a)

subject to Y(m(µ)) ≤ Y , (5.23b)

where Y is a pre-specified performance guarantee which the system opera-tor desires to meet. Observe that upon dualizing the constraint (5.23b) witha Lagrange multiplier υ, we obtain

maxυ≥0

−υY + minµ∈M ∑

k∈Ack(µk) + υY(m(µ))

. (5.24)

The inner minimization problem corresponds to the original problem for-mulation (5.9). The outer maximization problem, on the other hand, isa tractable scalar optimization problem that can be solved via standarditerative procedures. As the optimal bidding strategy has been shown to beindependent of υ in Theorem 7, the same bids can be used in all iterationsof the outer maximization problem. �

Remark 5 (Generality of the result) Although we discuss a particular in-stance based on a coarse-grain swing equation model and an H2 perfor-mance metric here, Theorem 7 also holds true for more general systemmodels such as the non-linear, high-fidelity South-East Australian systemdiscussed in Chapter 4 and other performance metrics Y(m(µ)), as long asthe arg min of the expressions (5.12), (5.13) can be evaluated efficiently. �

5.6 numerical case study

In this section, we present a few illustrations which support the precedingdiscussions on the centralized planning problem and the market-basedauction mechanism. A 12-bus case study depicted in Figure 5.3 is con-sidered. The system parameters are identical to the three-region systemfrom Section 3.5. Further, we assume that each node receives virtual inertiacontributions from a number of dissimilar agents.

5.6.1 Simulation Setup

We examine the alternate formulation presented in Remark 4 for the casestudy in Figure 5.3. This setup enables us to impose guarantees on the


25km10km

25km10km

25km

110km

110k

m 110km

1

1

2

3 4

5

6

78

910 11

12

1570MW

1000MW100Mvar

567MW100Mvar

400MW 490MW

611MW164Mvar

1050MW284Mvar

719MW133Mvar

350MW

700MW208Mvar

700MW293Mvar

200Mv

ar

350Mv

ar 69Mvar

4a ... 4g2a

2c

12a 12b

8a

8c

Figure 5.3: A 12 bus three-region system test case with grid parameters as inFigure 3.3 and virtual inertia agents at buses 2, 4, 8, 12.

worst-case performance via (5.23b). Further, we note that the constraint(5.23b) requires solving a maximization problem over the disturbance set(5.6). This can, however, be circumvented by rewriting the maximization asa set of inequality constraints. We express (5.23b) from (5.5), (5.7) asmax

π ∑i

πimi

≤ Y (5.25a)

subject to πi ≥ 0, ∀ i (5.25b)

1>n π ≤ πtot. (5.25c)

The linearity of (5.25a) in π dictates that the maximum can be obtainedby evaluating the objective at the vertices of the polytope given by (5.25b)–(5.25c), i.e., (5.25) is equivalent to the set of inequality constraints given by

πtot

mi≤ Y , ∀ i. (5.26)

The resulting set of constraints from (5.26) is linear in the decision variablesµ and can be directly incorporated in the optimization problem (5.24). Forthe purpose of simulations, we choose a non-trivial performance guaranteeY = 0.29. This choice of Y requires an additional virtual inertia allocation atthe buses labeled 2, 4, 8, 12 only, as the other buses by virtue of m0 alreadyhave sufficient inertia capability based on (5.26) to meet this constraintspecification.

5.6 numerical case study 81

5.6.2 Simulation Results

The design of the market facilitates virtual inertia at each bus to be con-tributed by a number of agents. In our case study, we consider 3, 7, 3, 2

agents connected to buses numbered 2, 4, 8, 12 respectively, and labeledas (2a,. . . , 2c), (4a, . . . , 4g), (8a,. . . , 8c), (12a, 12b). Further, we assumethat the maximum inertia that each agent can contribute is given by thestacked vector µ = [20, 40, 60, 20, 40, 20, 40, 20, 40, 20, 20, 40, 60, 20, 40]. Notethat the dissimilarity in the maximum inertia support can arise due tonature underlying technology of the power electronics interface, the sizeof the energy device, maximum power rating among others. We recallfrom Section 5.5.2 that the bids and costs are expressed as non-decreasingconvex curves. Here, we consider a special subset of linear functions, i.e.,ck(µk) = ck µk for the costs. As the costs incurred for provision of virtualinertia depend on the operating technology [95], we divide the costs intothree monetary categories4 for a qualitative analysis: low– (c = 1), medium–(c = 5), high– (c = 10). The true cost vector for the agents is chosen ascsim = [1, 5, 1, 5, 5, 1, 5, 10, 5, 5, 5, 5, 10, 1, 5], where the csim

k element corre-sponds to the cost of procuring unit inertia from agent k. We summarizeour observations below:

(i) In Figure 5.4, we plot the inertia profiles and the corresponding worst-case performance for the three-region case study. The robust optimal alloca-tion problem yields a valley-filling profile as in Section 3.5 which rendersall buses identical with respect to the expected disturbance.

(ii) In Figure 5.5, we compare the allocations and total procurement costsfor three mechanisms: a possible regulatory allocation (where inertia is allo-cated proportionally to the capacity of the virtual inertia devices µ, in orderto meet the specifications regardless of cost), market-based (Figure 5.2),and the centralized (5.9) mechanism. Each allocation is such that the per-formance requirements in (5.25) are met. The optimal allocations and thecosts for the centralized and market-based mechanisms coincide. As theregulatory strategy does not factor for the cost-curves of the agents, itresults in a higher overall cost.

(iii) Figure 5.6 plots the average VCG-based payments (payment received perunit of virtual inertia) and average costs (cost per unit of virtual inertia) foreach agent. Note that the agents 2b, 4e, 8c, 12b do not provide any virtual

4 The costs are indicative of the trend and do not necessarily reflect the exact cost of procurement.


1 2 4 5 6 8 9 10 120

10

20

30

40

50

node

iner

tia

mm?

0

0.3

0.6

0.9

1.2

1.5

wor

st-c

ase

perf

orm

ance

Figure 5.4: Inertia profiles for the grid with a primary control effort penalty andthe associated worst-case performance.

inertia due to higher per-unit cost of inertia. The average payment for eachagent providing virtual inertia support and connected to the same node isidentical. However, the cheapest providers are preferred– this is reflectedin payments larger than their costs (which is also their bid). Ultimately, theutility (the difference between payment and cost incurred) of each agentdepends on the cost curves of all the agents that are co-located at the samebus.

(iv) Another interesting observation is that the number of agents do notnecessarily determine the extent of influence on pricing dynamics or marketpower, e.g., the agents 2a, 2c, 4c, 12a have the lowest cost of the serviceand yet they are not utilized to their capacity µ. This is due to the fact thatsystem performance YD(m, π) is both inertia and location dependent. Thistherefore, also rules out the concept of a single-clearing price based on thecost and capacity.

5.7 conclusions

We motivated the need for virtual inertia markets and considered the prob-lem of economic, incentivized procurement of virtual inertia in low-inertiapower grids. Two schemes for virtual inertia procurement were introducedand analyzed via an objective embedding a robust performance metricaccounting for grid stability and the cost of virtual inertia procurement. Incontrast to a regulatory approach, the proposed market-based decentralized

5.7 conclusions 83

2a 2b 2c 4a 4b 4c 4d 4e 4 f 4g 8a 8b 8c 12a 12b0

5

10

15

20

agents

iner

tia

regulatorymarket-basedcentralized

0 50 100 150 200 250 300 350 400 450

total cost (units)

Figure 5.5: Virtual inertia contributions from agents and the total monetary costincurred under regulatory, market-based setups.

mechanism while respecting performance constraints, adequately compen-sated the agents for their contribution. Such a decentralized approach alsoresulted in a virtual inertia allocation which maximized the social welfareof the centralized planning problem. This mechanism was partially inspiredby the ancillary service markets and the payments to agents were based onthe VCG rule. Such a mechanism enforced truthful bidding by the agents,achieved incentive compatibility, and non-negative utilities. We finally pre-sented a few illustrations for a three-region case study underscoring thebenefits of such a mechanism, while highlighting the peculiarities such asthe absence of a single-clearing price, due to the heavy location-dependenceof inertia as a service.

With more renewable integration in the power grids, virtual inertia ispoised for a greater role, and the issues of economical procurement andmarkets for this service will gain further prominence. This chapter is anattempt in this direction and we believe that such decentralized auctionmechanisms for virtual inertia procurement will be integrated within thestructure of existing power markets. In addition, we wish to point out thatthe decentralized mechanism presented here is model-agnostic. Though weillustrate our approach through a stylistic swing equation-based model, it


2a 2b 2c 4a 4b 4c 4d 4e 4 f 4g 8a 8b 8c 12a 12b0

1.5

3

4.5

6

7.5

agents

mon

etar

yun

its

marginal paymentmarginal cost

Figure 5.6: Average payment and average cost profiles for agents participating inthe market-based auction for procuring virtual inertia.

can be extended to more complex system dynamics as long as convexity ofthe objective is established. Open problems not considered here pertain torobust auction mechanisms which counteract shill-bidding and collusionamong various agents.

6P E R F O R M A N C E O F L I N E A R - Q UA D R AT I CS A D D L E - P O I N T A L G O R I T H M S

6.1 related works

In the previous chapters, we have utilized system norms for stability anal-ysis of low-inertia power systems. These can also be used to study theinput-output performance of generic sets of algorithms, a subset of whichincludes primal-dual/ saddle-point methods. These methods belong to theclass of continuous-time gradient-based algorithms for solving constrainedconvex optimization problems. Introduced in the early 1950s [97], [98],these algorithms are designed to seek the saddle points of the optimizationproblem’s Lagrangian function. These saddle points are in one-to-one corre-spondence with the solutions of the first-order optimality (KKT) conditions,and the algorithm, therefore, drives its internal state towards the globaloptimizer of the convex program; see [99]–[101] for convergence results.

A key motivation for studying primal-dual methods stems from their re-cent application relating to optimal frequency regulation in power networks.These networks are designed to operate around a nominal frequency (e.g.,50 Hz or 60 Hz), and deviations from this nominal value indicate the globalimbalance of power supply and demand. The so-called primary control is aproportional control layer implemented at sources (see Chapter 9 of [102])or loads [103] which attempts to balance supply and demand over fasttime-scales, stabilizing the grid to an off-nominal frequency. Higher-levelcentralized control layers termed secondary and tertiary control are thentasked with regulating the grid frequency to its nominal value, meetingoperational constraints, and optimizing the grid by minimizing the cost ofgeneration. This combined control/optimization problem is referred to asthe optimal frequency regulation (OFR) problem. The rise of distributedgeneration allows us to leverage the many new controllable power electronicdevices within the grid that will enable frequency control to be distributedacross both the generation and load side, thereby decreasing total costwhile enhancing system robustness. Due to sensing and communicationconstraints, these devices should act based on minimal information andideally without detailed model information or precise knowledge of systemparameters and generation/load forecast.

85

86 performance of linear-quadratic saddle-point algorithms

Apart from control applications in power systems [22], [23], [25], [41],[104], [105], these algorithms have also attracted renewed attention fore.g., in the context of machine learning [106], for solving convex resourceallocation problems [107], in the control literature for solving distributedconvex optimization problems [108] where agents cooperate through acommunication network to solve an optimization problem with minimal orno centralized coordination, and non-smooth convex optimization problems[109]. Applications of distributed optimization include utility maximization[99], congestion management in communication networks [110] amongothers. While most standard optimization algorithms require centralizedinformation to compute the optimizer, saddle-point algorithms often yielddistributed strategies in which agents perform state updates using onlylocally measured information and communication with some subset ofother agents. The control-theoretic interpretations of these algorithms areavailable in [100], [111]–[116].

Rather than solve the optimization problem offline, it is sometimes desir-able to run these distributed algorithms online as controllers, in feedbackwith system and/or disturbance measurements, to provide references sothat the optimizer can be tracked in real-time as operating conditionschange. Such algorithms offer promise for online optimization, especiallyin scenarios with streaming data. However, when saddle-point methodsare implemented online as controllers, they become subject to disturbancesarising from fluctuating parameters and noise (the precise nature of thesedisturbances being application dependent). In the power system context,disturbances take the form of varying power injections, arising from fluc-tuating load demands or from uncontrolled renewable generation. Thesechanging power injections tug on the frequency of the network, and it is un-clear how well primal-dual controllers can accommodate these disturbanceswhile regulating the frequency and maintaining optimality. The standardmethod for assessing optimization algorithms– namely, convergence rateanalysis, is also insufficient to capture the performance of the algorithm.Indeed, an algorithm with a fast convergence rate would be inappropri-ate for control applications if it responded poorly to disturbances duringtransients, or if it greatly amplifies measurement noise in steady-state.

The appropriate tool for measuring dynamic algorithm performanceis instead the system norm [117], as commonly used in feedback systemanalysis to capture system response to exogenous disturbances. Recentwork in this direction includes input-to-state-stability results [118], [119],

6.2 contributions 87

finite L2 gain analysis [120], and the robust control framework proposed in[121], [122].

The purpose of this chapter is to continue this line of investigation. Inparticular, the case of saddle-point algorithms, applied to optimizationproblems with quadratic objective functions and linear equality constraintsleads to a very tractable instance of this analysis problem where many basicquestions can be asked and accurately answered. These include:

(i) how do saddle-point algorithms amplify disturbances which may enterthe objective function and/or equality constraints?

(ii) how does performance in the presence of disturbances change whenthe initial optimization problem is reformulated (e.g., dual or distributedformulations)?

(iii) how do standard modifications to optimization algorithms, such asregularization and Lagrangian augmentation, affect these results?

We are specifically interested to answer the above questions in the con-text of studying the quadratic performance of primal-dual methods forfrequency control in power systems. To this end, we shall first present someinteresting and relevant generic results and later consider the implicationsfor the frequency regulation problem. Another exciting application relatesto the problem of resource allocation which shall also be considered indetail.

6.2 contributions

The three main contributions of this chapter are as follows. First, in Section6.4 we consider the effect of disturbances on the saddle-point dynamicsarising from linearly constrained, convex quadratic optimization problems.We quantify the input-output performance of the method via the H2 systemnorm and for a relevant input-output configuration– derive an explicitexpression for the norm as a function of the algorithm parameters. We find(Theorem 9) that the squared H2 norm scales linearly with the number ofdisturbances to both the primal and dual variable dynamics.

Second, we study two common modifications to the Lagrangian opti-mization paradigm: regularization and augmentation. We show that regu-larization strictly improves the transient H2 performance of saddle-pointalgorithms (Theorem 10). However, this improvement in performance is notusually monotone in the regularization parameter and the system norm


may achieve its global minimum at some finite regularization parametervalue. For augmented Lagrangian saddle-point methods, we derive anexplicit expression for the H2 norm (Theorem 12); the results show thataugmentation may either improve or deteriorate the H2 performance. Forcases when standard augmentation deteriorates performance, we proposeaugmented dual distributed saddle-point algorithm which strictly improvesperformance (Section 6.5.2).

Third and finally, in Section 6.6 and Section 6.7 we apply our results tooptimal frequency regulation and resource allocation problems respectively.We compare and contrast the input-output performance of the differentalgorithms we have considered. The results show that distributed imple-mentations can perform equally well as centralized implementations, butthat significant performance differences can appear between the algorithmsonce augmentation is considered.

Taken together, these results provide fairly complete answers to the ques-tions (i)–(iii) outlined in Section 6.1, for the class of problems considered.A similar study in an H∞ or L2 gain framework requires a significantlydifferent analysis and is not presented here. As background, in Section 6.3we review saddle-point algorithms for the relevant class of optimizationproblems.

notation The n× n identity matrix is In, 0 is a matrix of zeros of appro-priate dimension, while 1n (resp. 0n) are n-vectors of all ones (resp. zeros).If f : Rn → R is differentiable, then ∂ f

∂x : Rn → Rn is its gradient. ForA ∈ Rn×n, A> is its transpose and Tr(A) = ∑n

i=1 Aii is its trace. If S ∈ Rr×n

has full row-rank, then SS† = Ir where S† = S>(SS>)−1 is the Moore-Penrose pseudoinverse of S. For a positive semi-definite matrix Q � 0, Q

12

is its square root. The symbol ⊗ denotes the Kronecker product. Given ele-ments {ai}n

i=1 (scalars, vectors, or matrices), col(a1, . . . , an) = (a>1 , . . . , a>n )>

denotes the vertically concatenation of the elements (assuming compatibledimensions), and blkdiag(a1, a2, . . . , an) is a block-diagonal matrix with theelements {ai} on the diagonals.

graphs and graph matrices A graph is a pair G = (V , Eu), where Vis the set of vertices (nodes) and Eu is the set of undirected edges (unorderedpairs of nodes). The set of neighbours of node i ∈ V are denoted by N (i).If a label e ∈ {1, . . . , |Eu|} and an arbitrary orientation is assigned to eachedge, we can define a corresponding directed edge set E ⊂ V × V withelements e ∼ (i, j) ∈ E . The node-edge incidence matrix E ∈ R|V|×|E| is

6.3 saddle-point methods and H2 performance 89

defined component-wise as Eke = 1 if node k is the source node of edge eand as Eke = −1 if node k is the sink node of edge e, with all other elementsbeing zero. If the graph is connected, then ker(E>) = Im(1|V|). A graph isa tree (or acyclic) if it contains no cycles, and in this case ker(E) = {0|E |}.

6.3 saddle-point methods and H2 performance

We consider the constrained quadratic optimization problem

minx∈Rnx

J(x) :=12

x>Qx + x>c (6.1a)

subject to Sx = Wbb, (6.1b)

where x ∈ Rnx , Q = Q> � 0 is positive definite, c ∈ Rnx and b ∈ Rnb

are parameter vectors, and S ∈ Rnr×nx , Wb ∈ Rnr×nb with nr < nx. Wemake the blanket assumption that S and Wb have full row rank, whichsimply means that the constraints Sx = Wbb are not redundant. While theright-hand side Wbb of the constraints is apparently over-parameterized,this formulation is natural when considering particular problem instances.In the resource allocation problem of Section 6.7, Wb = [1 1 · · · 1] and b isa vector of demands; the product Wbb is simply the total demand.

The problem (6.1) describes only a subclass of the optimization problemsto which saddle-point algorithms are applicable; more generally one canalso consider strictly convex costs and convex inequality constraints aswell. We restrict our attention to (6.1), as this case will allow LTI systemanalysis techniques to be applied, and represents a large enough class ofproblems to yield some general insights. Intuitively, the performance of thesaddle-point algorithm on (6.1) should indicate a “best” case performancefor the general case, as the objective J(x) is smooth and strongly convex,and (6.1) is free of hard inequality constraints. Under these assumptions,the convex problem (6.1) has a finite optimum, the equality constraints arestrictly feasible, and (6.1) may be equivalently studied through its Lagrangedual with zero duality gap [123]. The Lagrangian Lϑ : Rnx ×Rnr → R of(6.1) is

Lϑ(x, ν) =12

x>Qx + c>x + ν>(Sx−Wbb), (6.2)


where ν ∈ Rnr is a vector of Lagrange multipliers. By strong duality, theKKT conditions

∂Lϑ

∂x(x, ν) = 0nx ⇐⇒ 0nx = Qx + S>ν + c,

∂Lϑ

∂ν(x, ν) = 0nr ⇐⇒ 0nr = Sx−Wbb,

(6.3)

are necessary and sufficient for optimality. From these linear equations onecan quickly compute the unique global optimizer (x?, ν?) to be[

x?

ν?

]=

[−Q−1(S>ν? + c)

−(SQ−1S>)−1(Wbb + SQ−1c)

]. (6.4)

While (6.4) is the exact solution of the optimization problem (6.1), itsevaluation requires centralized knowledge of the matrices S, Q, Wb and thevectors c and b. If any of these parameters change or evolve over time, theoptimizer should be recomputed. In many multi-agent system applications,the cost matrix Q is diagonal or block-diagonal and J(x) = ∑i

qi2 x2

i + cixiis, therefore, a sum of local costs. Finally, the constraints encoded in S areoften sparse, mirroring the topology of an interaction or communicationnetwork between agents. These factors motivate the solution of (6.1) in anonline distributed fashion, where agents in the network communicate andcooperate to calculate the global optimizer.

A simple continuous-time algorithm to seek the optimizer is the saddle-point or primal-dual method [99]–[101], [124], [125]

Tx x = − ∂

∂xLϑ(x, ν), Tνν = +

∂

∂νLϑ(x, ν),

which here reduces to the affine dynamical system

Tx x = −Qx− S>ν− c, (6.5a)

Tνν = Sx−Wbb, (6.5b)

where Tx, Tν � 0 are positive definite diagonal matrices of time constants.By construction, the equilibrium points of (6.5) are in one-to-one correspon-dence with the solutions of the KKT conditions (6.3) and the system isinternally exponentially stable [101].

Lemma 8 (Global Convergence to Optimizer) The unique equilibrium point(x?, ν?) given in (6.4) of the saddle-point dynamics (6.5) is globally exponentiallystable, with exponential convergence rate ∝ 1/τmax where

τmax = max( maxi∈{1,...,n}

Tx,ii, maxi∈{1,...,r}

Tν,ii).

6.4 H2 performance of saddle-point methods 91

With stability settled, we will next focus exclusively on quantifying tran-sient input-output performance of (6.5) in the presence of exogenous dis-turbances.

6.4 H2 performance of saddle-point methods

We now subject the saddle-point dynamics (6.5) to disturbances in boththe primal and dual equations. Specifically, we assume the vectors b and care subject to disturbances ηb ∈ Rnb , ηc ∈ Rnx and make the substitutionsb 7→ b+ tbηb and c 7→ c+ tcηc in the saddle-point dynamics (6.5). The scalarparameters tb, tc ≥ 0 parameterize the strength of the disturbances andwill help us keep track of which terms in the resulting norm expressionsarise from which disturbances. As an example, when we study distributedresource allocation problems in Section 6.7, b will have the interpretationof a vector of demands for some resource, and ηb will, therefore, model afluctuation or disturbance to this nominal demand.

After translating the nominal equilibrium point (6.4) of the system to theorigin1, we obtain the LTI system[

Tx x

Tνν

]=

[−Q −S>

S 0

]︸︷︷︸

= A

[x

ν

]−[

tc Inx 0

0 tbWb

] [ηc

ηb

], (6.6a)

z =[C1 0

]︸︷︷︸= C

[x

ν

], (6.6b)

where C1 ∈ Rnx×nx is an output matrix.

As the system (6.6) is written in error coordinates, convergence to thesaddle-point optimizer (x?, ν?) from (6.4) is equivalent to convergence of(x(t), ν(t)) to the origin. How should we measure this convergence? Anatural way to is to use the cost matrix Q from the optimization problem(6.1) as a weighting matrix, and to study the performance output ‖z(t)‖2

2 =

x(t)>Qx(t), which is obtained by choosing C1 = Q12 . For example, in the

context of the resource allocation problems in Section 6.7, these weightsdescribe the relative importance of the various resources.

1 We assume the change of state variables ∆x = x − x?, ∆ν = ν − ν? and with an abuse ofnotation we drop the ∆s and simply refer to the error coordinates as x and ν.


Theorem 9 (Saddle-Point Performance) Consider the input-output saddle-pointdynamics (6.6) with Q = Q> � 0 diagonal, and let C1 = Q

12 so that ‖z(t)‖2

2 =x(t)>Qx(t). Then the squared H2 norm of the saddle-point system (6.6) is

‖G‖2H2

=t2c2

Tr(T −1x ) +

t2b2

Tr(W>b T−1

ν Wb). (6.7)

Proof of Theorem 9: We will directly construct the unique positive definiteobservability Gramian; since the system is internally stable (Lemma 8), thisalso indirectly establishes observability (see Exercise 4.8.1 of [126]).

Formally, since C = [Q12 0] and Q is positive definite, ker(C) =

{(x, ν)|x = 0n}. Suppose that (0n, ν) ∈ ker(C) is an eigenvector of Awith eigenvalue λ, i.e., [

−Q −S>

S 0

] [0

ν

]= λ

[0

ν

].

Since Re(λ) < 0 by stability, these equations are consistent if and only ifν = 0r. Thus no right-eigenvector of A belongs to ker(C) and observabilityfollows by the eigenvector test.

We can rewrite the system dynamics (6.6) as[x

ν

]=

[−Tx

−1Q −Tx−1S>

Tν−1S 0

] [x

ν

]−[Tx−1tc 0

0 Tν−1Wbtb

]︸︷︷︸

= B

[ηc

ηb

],

z =[

Q12 0

] [x

ν

].

For this system, assuming a block-diagonal observability Gramian X =blkdiag(X1, X2), the Lyapunov equation (2.4) yields the two equations

X1T −1x Q + QT −1

x X1 − C21 = 0, (6.9a)

X2T −1ν S− ST −1

x X1 = 0, (6.9b)

with the third independent equation trivially being 0 = 0. By inspection,the solution of (6.9a) is diagonal and given by

X1 =12Tx Q−1C2

1 =12Tx,


since Q is diagonal and C1 = Q12 . Clearly X1 is positive definite and

symmetric. Since S has full row-rank, and X2 can be uniquely recoveredfrom (6.9b) as

X2 = ST −1x X1S†Tν =

12

SS†Tν =12Tν.

It follows that X2 is positive definite, and therefore X = 12 blkdiag(Tx, Tν) is

the unique positive definite solution of (2.4). Since X is block-diagonal, wefind from (2.3) and (6.6)

‖G‖2H2

= t2c Tr(T −1

x X1T −1x ) + t2

b Tr(W>b T−1

ν X2T −1ν Wb),

from which the result follows.

We make two key observations about the result (6.7). First, (6.7) is in-dependent of both the cost matrix Q and the constraint matrix S; neithermatrix has any influence on the value of the system norm. Second, theexpression in (6.7) scales inversely with the time constants Tx and Tν, whichindicates an inherent trade-off between convergence speed and input-outputperformance. As a special case of Theorem 9, suppose that Tx, Tν are multi-ples of the identity matrix, i.e., Tx = τx Inx , Tν = τν Inr , and that Wb = Inr ,meaning there is one disturbance for each constraint. Then (6.7) reduces to

‖G‖2H2

=t2c

2τxnx +

t2b

2τνnr, (6.10)

meaning the squared H2 norm scales linearly in the number of disturbancesto the primal dynamics and the number of disturbances to the dual dynam-ics. While this scaling is quite reasonable, the lack of tuneable controllergains other than the time constants means that convergence speed andinput-output performance are always conflicting objectives. Finally, wenote that (6.10) can be immediately reinterpreted as a design equation forthe time constants. That is, given a specified level γp > 0 of desired H2performance, one has ‖G‖H2 ≤ γp if

min{τx, τν} ≥1

γ2p

(t2c nx

2+

t2bnr

2

).


6.4.1 Regularized Saddle-Point Methods

A common variation of the Lagrangian optimization framework includes aquadratic penalty [127]–[129] on the dual variable ν in the Lagrangian (6.2).The so-called regularized Lagrangian assumes the form

Lϑreg(x, ν) =

12

x>Qx + c>x + ν>(Sx−Wbb)− ε

2‖ν‖2

2, (6.11)

where ε > 0 is small. The regularization term adds concavity to the La-grangian and has been shown to increase the convergence rate of optimiza-tion algorithms. However, this regularization alters the equilibrium of theclosed-loop system, which moves from the value in (6.4) to the new value[

x?reg

ν?reg

]=

[−Q−1(S>ν?reg + c)

−(SQ−1S> + εInr )−1(Wbb + SQ−1c)

]. (6.12)

The penalty coefficient ε is chosen to strike a balance between the conver-gence rate improvement and the deviation of (x?reg, ν?reg) from (x?, ν?). TheLagrangian (6.12) also admits a continuous-time saddle-point algorithmwith regularized saddle-point dynamics

Tx x = −Qx− S>ν− c

Tνν = Sx−Wbb− εν.(6.13)

Quite strikingly, we shall observe that a small regularization which has aminor effect on the equilibrium, achieves a tremendous improvement inperformance.

As we did with the standard saddle-point dynamics, we can shift the equi-librium point (x?reg, ν?reg) of (6.13) to the origin and introduce disturbancesto the parameters b and c, leading to the input-output model[

Tx x

Tνν

]=

[−Q −S>

S −εInr

] [x

ν

]−[

tc Inx 0

0 tbWb

] [ηc

ηb

],

z =[

Q12 0

] [x

ν

],

(6.14)

where we consider the time constant matrices Tx, Tν, and disturbances ηb,ηc as in (6.6).


Theorem 10 (Regularized Saddle-Point Performance) Consider the input-output regularized saddle-point dynamics (6.14) denoted by Greg with Q = Q> �0 a diagonal matrix. Then the squared H2 norm ‖Greg‖2

H2of the system (6.14) is

upper bounded by (6.7) with strict inequality.

Proof of Theorem 10: We rewrite (6.14) in the standard state-space formGreg := (Areg, B, C, 0), where[

x

ν

]=

[−Tx

−1Q −Tx−1S>

Tν−1S −Tν

−1εInr

]︸︷︷︸

= Areg

[x

ν

]

−[Tx−1tc 0

0 Tν−1Wbtb

]︸︷︷︸

= B

[ηc

ηb

], z =

[Q

12 0

]︸︷︷︸

= C

[x

ν

].

One may verify that Areg is Hurwitz and that Greg is observable. Considerthe observability Gramian from (6.9), i.e., X = 1

2 blkdiag(Tx, Tν). An easycomputation shows that

X Areg + A>regX + C>C =

[0 0

0 −εInr

]� 0. (6.16)

We conclude that X is a generalized observability Gramian for the regularizedsystem Greg. If Xε is the true observability Gramian for Greg, then Xε 6= Xand Xε � X (see Chapter 4.7 of [126]) and we conclude that

‖Greg‖2H2

= Tr(B>XεB) ≤ Tr(B>XB) = ‖G‖2H2

. (6.17)

It remains only to show that the above inequality holds strictly. Proceedingby contradiction, assume that Tr(B>XεB) = Tr(B>XB), which implies thatTr(B>(X− Xε)B) = 0. Since X− Xε � 0, we may write X− Xε = F>F forsome matrix F, and

0 = Tr(B>(X− Xε)B) = Tr(B>F>FB).

Since B has full row rank, this implies that F must be zero and thus, X = Xε

which is a contradiction.


Corollary 11 (Regularized Saddle-Point Performance) Consider the case withone constraint (nr = 1) and one constraint disturbance (nb = 1) with uniformproblem parameters Q = qInx , Tx = τx Inx , Tν = τν Inr for scalars q, τx, τν > 0and Wb = 1. Then, we have

‖G‖2H2− ‖Greg‖2

H2= αεt2

c + γεt2b, (6.18)

where s = ‖S‖2 and

αε =εs2

2(εq + s2)(ετx + qτν),

γε =ε(τxqε + q2τν + τxs2)

2τν(εq + s2)(ετx + qτv).

Proof of Corollary 11: Let ∆ := X − Xε, where X = 12 blkdiag(Tx, Tν) is the

observability Gramian of (6.6) with C1 = Q12 , and Xε is the observability

Gramian of Greg. As noted in the proof of Theorem 10, X � Xε, and thusthe matrix ∆ is positive semi-definite. Clearly, ∆ satisfies

A>reg(X− ∆) + (X− ∆)Areg + C>C = 0.

By (6.16), this reduces to

A>reg∆ + Areg∆ +

[0 0

0 εInr

]= 0. (6.19)

Then it is easy to see that

‖Greg‖2H2

= ‖G‖2H2− ‖Gε‖2

H2, (6.20)

where ‖G‖2H2

is as in (6.7), and Gε(s) := Cε(sI − Areg)−1B with Cε =

[0 εInr ]. Therefore, the improvement in the H2 norm performance is equalto the squared H2 norm of the axillary system given by Gε. Next, wecalculate the H2 norm of Gε, which requires computing the observabilityGramian from (6.19). Consider the matrix

∆ =

[αS>S βS>

βS γ

], (6.21)


where α, β, γ are constant and positive. A straightforward calculation showsthat by choosing

α

β

γ

=

s2

τxε

τν+ q

τx− 1

τν

qτx

− 1τν

0

0 τνs2

τxε 1

−1

0

0

τv2

, (6.22)

the matrix ∆ is a solution of the Lyapunov equation (6.19). Given the factthat Areg is Hurwitz, this solution is unique and the matrix ∆ = ∆ is theobservability Gramian of the system given by Gε. The proof is completedby calculating the inverse in (6.22), using (2.3), and noting Tr(S>S) = s2.

Corollary 11 quantifies the performance improvement resulting fromthe regularization in the special case of a single constraint and uniformparameters. For sufficiently large ε, this improvement is approximated

by t2b

2τν, which coincides with the second term in the right-hand side of

(6.10). This means that, as expected, the constraints do not contribute tothe H2 norm in case the penalty term in the regularized Lagrangian (6.11)tends to infinity. On the other hand, for ε → 0+, the H2 norm of theregularized dynamics (6.14) clearly converges to the H2 norm of (6.6). Ingeneral, the improvement in the input-output performance obtained due toregularization is not a monotonic function of the regularization parameterε. To illustrate this, Figure 6.1 and Figure 6.2 plot the system norm forthe regularized dynamics as a function of ε. It is noteworthy that for boththe plots, even a modest ε improves the performance. Depending on thespecific values of the parameters, the maximum performance gain may beachieved as ε→ ∞ (Figure 6.1) or at a finite value of ε (Figure 6.2).

6.4.2 Augmented Saddle-Point Methods

Another option for improving the H2 performance of saddle-point methodsis to return to the Lagrangian function (6.2) and instead consider theaugmented Lagrangian [130]–[132]

Lϑaug(x, ν) , Lϑ(x, ν) +

ρ

2‖Sx−Wbb‖2

2, (6.23)

where we have incorporated the squared constraint Sx−Wbb = 0nr intothe Lagrangian with a gain ρ ≥ 0. One way to interpret this is that the termρ2‖Sx−Wbb‖2

2 adds additional convexity to the Lagrangian in x.


0 5 10 15 202

3

4

5

6

7

ε

‖Gre

g‖2 H

2

Figure 6.1: System norm for regularized dynamics as a function of ε, for pa-rameters τx = 1, τν = 1, tc = 1, tb = 3, Q = 3I5, and S =[0.82 0.90 0.13 0.91 0.63].

0 5 10 15 202

2.1

2.2

2.3

2.4

2.5

ε

‖Gre

g‖ H

2

Figure 6.2: System norm for regularized dynamics as a function of ε, for pa-rameters τx = 1, τν = 1, tc = 1, tb = 1, Q = 0.05I5, andS = [0.82 0.90 0.13 0.91 0.63].

It follows that (x, ν) is a saddle point of Lϑaug(x, ν) if and only if it is a

saddle point of Lϑ(x, ν), and hence the optimizer is unchanged. Applyingthe saddle-point method to the augmented Lagrangian Lϑ

aug(x, ν), weobtain the augmented saddle-point dynamics

Tx x = −(Q + ρS>S)x− S>ν− c + ρS>Wbb,

Tνν = Sx−Wbb.(6.24)


One may verify that as before, the unique stable equilibrium point of (6.24)is given by (6.4). We again consider disturbances ηb and ηc, and make thesubstitution b 7→ b+ tbηb and c 7→ c+ tcηc. After translating the equilibriumpoint to the origin, we obtain the MIMO system[Tx x

Tνν

]=

[−(Q + ρS>S) −S>

S 0

] [x

ν

]−[

tc Inx −ρtbS>Wb

0 tbWb

] [ηc

ηb

],

z =[

Q12 0

] [x

ν

].

(6.25)

The additional term −ρS>S in the dynamics (6.25) complicates the solutionof the Lyapunov equation, and we require additional assumptions to obtainan explicit formula. We consider the parametrically uniform case whereQ = qInx , Tx = τx Inx , and Tν = τν Inr for scalars q, τx, τν > 0.

Theorem 12 (Augmented Saddle-Point Performance) Consider the input- out-put augmented saddle-point dynamics (6.25), denoted by Gaug under the aboveassumptions, with performance output ‖z(t)‖2

2 = x(t)>Qx(t) = q‖x(t)‖22. Then

the squared H2 norm of the augmented saddle-point system (6.25) for identicallyweighted disturbances Wb = Inr is

‖Gaug‖2H2

=t2c

2τx(nx − nr) +

(t2b

2τν+

t2c

2τx

) nr

∑i=1

qq + ρσ2

i

+t2b

2τx

nr

∑i=1

qρ2σ2i

q + ρσ2i

,

(6.26)

where σi is the ith non-zero singular value of S.

Proof of Theorem 12: Under the given assumptions, the system (6.25) simpli-fies to x

ν

=

− 1τx(qInx + ρS>S) − 1

τxS>

1τν

S 0

x

ν

−

tcτx

Inx − tbρτx

S>Wb

0 tbτν

Wb

ηc

ηb

, z =[q

12 Inx 0

] x

ν

.

Let S = UΣV> be the singular value decomposition of S, where U ∈ Rnr×nr

and V ∈ Rnx×nx are both orthogonal matrices, Σ = [Σ 0nr×(nx−nr)] and


Σ ∈ Rnr×nr is the diagonal matrix of non-zero singular values. Considernow the invertible change of variables x = V>x, ν = U>ν. In these newcoordinates, the dynamics become ˙x

˙ν

=

− 1τx(qInx + ρΣ>Σ) − 1

τxΣ>

1τν

Σ 0

︸︷︷︸

= A

x

ν

−

tcτx

V> − tbρτx

V>S>Wb

0 tbτν

U>Wb

ηc

ηb

, z =[q

12 V 0

]︸︷︷︸

= C

x

ν

.

We now show the observability of the pair (C, A); note that this is equiv-alent to observability of (C>C, A). First note that ker(C>C) is spannedby [0>nx ν>]>. Now, suppose that [0>nx ν>]> is an eigenvector of A witheigenvalue λ:[

− 1τx(qInx + ρΣ>Σ) − 1

τxΣ>

1τν

Σ 0

] [0nx

ν

]= λ

[0nx

ν

].

Since by stability Re(λ) < 0, the above relation only holds for ν = 0nr ,which shows observability by the eigenvector test. Assuming a block-diagonal observability Gramian X = blkdiag(X1, X2), the Lyapunov equa-tion (2.4) yields the two independent equations

X11

τx(qInx + ρΣ>Σ) +

1τx

(qInx + ρΣ>Σ)X1 = qInx , (6.29a)

X21τν

Σ− Σ1

τxX1 = 0, (6.29b)

where we have used the fact that V>V = Inx . Noting that

Σ>Σ = blkdiag(Σ, 0(nx−nr)×(nx−nr)),

we find by inspection that the solution of (6.29a) is diagonal and given by

X1ii =12

qq + ρσ2

iτx, i ∈ {1, . . . , nr}

X1ii =12

τx, i ∈ {nr + 1, . . . , nx}.


Observe that X1 is positive definite and symmetric. The matrix equation(6.29b) admits a solution X2 if and only if ker(Σ) ⊆ ker(ΣX1), which holdsin this case since X1 is diagonal. The lower block X2 can, therefore, beuniquely recovered from (6.29b) as

X2 = τ−1x (ΣX1Σ>)(ΣΣ>)−1τν.

A straightforward calculation shows that this is equivalent to

X2ii =12

qq + ρσ2

iτν, i ∈ {1, . . . , nr}.

It follows that X2 is diagonal and positive definite, and therefore X =blkdiag(X1, X2) is the unique positive definite solution. A calculation using(2.3) now shows that

‖Gaug‖2H2

=t2c

τ2x

Tr(VX1V>) +t2b

τ2ν

Tr(W>b UX2U>Wb)

+t2bρ2

τ2x

Tr(W>b SVX1V>S>Wb).

For the special case of one disturbance per constraint, i.e., Wb = Inr , theresult (6.26) follows by the cyclic property of the trace operation.

Under the assumed restrictions on parameters, Theorem 12 generalizesTheorem 9, since when ρ = 0 the expression (6.26) reduces to (6.10). Con-sider now the dependence of the expression (6.26) on the augmentationgain ρ. First, in the case when tb = 0 (meaning the vector b is not subject todisturbances), then as ρ→ ∞ the expression (6.26) reduces to only the firstterm: in this case, augmentation unambiguously improves input-outputperformance. In particular, note that a more favourable scaling than in(6.7) is achieved when nr is comparable to nx; see the resource allocationproblem in Section 6.7.2 On the other hand, if tb 6= 0, then as ρ becomeslarge, the second term in the expression vanishes, while the third termgrows without bound. Therefore, a large augmentation gain will lead topoor input-output performance. This behavior is explained by examining(6.24): the vector b enters the primal dynamics multiplied by ρ, and hence

2 As an observation, we note that even when S> is a sparse matrix, S>S typically will notbe, and hence the augmented dynamics (6.24) may not be immediately implementable as adistributed algorithm. A notable exception is when S is the transposed incidence matrix of asparse graph, which gives S>S as the corresponding sparse graph Laplacian; this will occur inSection 6.7.


any noise in b is amplified as ρ grows. To remedy this deficiency in theaugmented approach, the next section exploits a dual formulation of theoptimization problem (6.1).

6.5 dual and distributed dual methods

This section develops an approach to overcome the performance issuesof augmented Lagrangian methods observed in Theorem 12 when distur-bances enter the constraints. To focus in on these problematic disturbances,in this section we ignore possible disturbances to the vector c and set tc = 0.Section 6.5.1 contains a quick examination of dual ascent, before proceedingto a distributed dual formulation in Section 6.5.2.

6.5.1 Centralized Dual Ascent

To begin, we return to the Lagrangian (6.2) of the optimization problem(6.1), and compute

x?(ν) = arg minx∈Rn

Lϑ(x, ν) = −Q−1(c + S>ν).

It follows quickly that the dual function Φ(ν) is given by

Φ(ν) = minx∈Rn

Lϑ(x, ν)

= −12

ν>SQ−1S>ν− ν>(SQ−1c + Wbb)− 12

c>Q−1c.(6.30)

With the primal variable eliminated, a possible approach is to simply maxi-mize the dual function Φ(ν) via gradient ascent. Introducing disturbanceinputs b 7→ b + tbηb and performance outputs similar to before, and shiftingthe unique equilibrium point to the origin, one quickly obtains for Tν � 0,the input-output dual ascent dynamics

Tνν = −SQ−1S>ν− tbWbηb,

z = −Q−12 S>ν.

(6.31)

The performance of (6.31) is characterized by the following result.

Proposition 13 (Dual Ascent Performance) TheH2 norm of the input-outputdual ascent dynamics (6.31) is given by

‖G‖2H2

=t2b2

Tr(W>b T−1

ν Wb).

6.5 dual and distributed dual methods 103

Proof of Proposition 13: The Lyapunov equation (2.4) for this problem takesthe form

−XT −1ν SQ−1S> − SQ−1S>T −1

ν X + SQ−1S> = 0,

from which we find the unique solution X = 12Tν � 0. With B = [tbT −1

ν Wb],the result follows by applying (2.3).

Comparing the result of Proposition 13 to the unaugmented saddle-point result of Theorem 9, we observe that the terms proportional to t2

bare identical. Therefore, when considering algorithm performance withdisturbances entering the constraints, the primal-dual and pure dual-ascentalgorithms achieve identical performance.

6.5.2 Distributed Dual Augmented Lagrangian

Building off the dual function (6.30), we now derive a modified augmentedLagrangian algorithm, which can overcome the performance issues posedby disturbances affecting the vector b. The particulars of the derivationbelow are tailored towards distributed solutions, which will be discussedfurther in Section 6.7 in the context of distributed resource allocation. Withthis application in mind, we will focus in on the case where nb = nx, sothat the ith component of the disturbance b can be uniquely associated tothe ith primal variable xi; this assumption can be relaxed in the derivationbelow as long as one uniquely assigns components of b and the associatedcolumns of Wb to a particular agent. We partition each of the followingmatrices according to their columns as

S = [s1 s2 · · · snx ], Wb = [w1 w2 · · · wnx ].

With this partitioning, one may quickly see that for Q = diag(q1, . . . , qnx ),the dual function (6.30) may be written as

Φ(ν) =

nx

∑i=1

[− 1

2qiν>sis>i ν− ν>

(ciqi

si + wibi

)−

c2i

2qi

]︸︷︷︸

:= Φi(ν)

.

The dual function appears to separate into a sum, except for the com-mon multiplier ν. To complete the separation, for each i ∈ {1, . . . , nx}we introduce a local copy νi ∈ Rnr of the vector of Lagrange multipliersν ∈ Rnr , and require that νi = νj for all i, j ∈ {1, . . . , nx}. To enforce these


so-called agreement constraints, let E ∈ Rnx×|E| be the oriented node-edgeincidence matrix (see Chapter 8 of [133]) of a weakly connected acyclic3

graph G = ({1, . . . , nx}, E), where E is the set of oriented edges. The dualproblem maxν∈Rnr Φ(ν) is then equivalent to the constrained problem4

minν∈R(nxnr)

Jdual(ν) := −nx

∑i=1

Φi(νi) (6.32a)

subject to (E> ⊗ Inr )ν = 0|E |nr , (6.32b)

where ν = col(ν1, ν2, . . . , νnx ) ∈ R(nxnr). Since the graph is acyclic, E> hasfull row rank, and therefore satisfies our assumption concerning the con-straint matrix (Section 6.3). The key observation now is that the parametersb do not enter into the equality constraints of the optimization problem(6.32); this permits the full application of augmented Lagrangian techniquesfor improving the H2 performance of the saddle-point algorithm. Buildingthe augmented Lagrangian (6.23) for the problem (6.32) we have

Lϑaugdual = −

nx

∑i=1

Φi(νi) + µ>(E> ⊗ Inr )ν +

ρ

2ν>(L⊗ Inr )ν,

where µ = col(µ1, µ2, . . . , µ|E |) ∈ R|E |nr is a stacked vector of Lagrangemultipliers µ` ∈ Rnr for ` ∈ {1, . . . , |E |}, and L = L> = EE> ∈ Rnx×nx isthe Laplacian matrix of the undirected graph Gu, obtained by ignoring theorientation of the edges in G; we let N (i) denote the neighbours of vertex iin the graph Gu. Applying the saddle-point method under the assumption

3 The acyclic assumption implies that rank(E>) = |E |, in line with our assumption from Section6.3 that the constraint matrix has full row rank. This assumption can be relaxed at the expenseof more complex stability/performance proofs.

4 Another equivalent formulation can be obtained by using the Laplacian matrix EE> =L = L> ∈ Rnx×nx of the graph, and using instead the constraint (L⊗ Inr )ν = 0(nx nr). Thisformulation is sometimes preferable for multi-agent implementations, the analysis of whichrequires only small modifications from the present analysis. We focus instead of formulationsinvolving the incidence matrix.

6.5 dual and distributed dual methods 105

of identical time constants, the dynamics may be written block-component-wise as

τννi = −sis>i

qiνi −

(ciqi

si + wibi

)− ∑

j:(i,j)∈Eµij + ∑

j:(j,i)∈Eµji

− ρ ∑j∈N (i)

(νi − νj), i ∈ {1, . . . , nx}

τµµij = νi − νj, (i, j) ∈ E .

(6.33)

The algorithm (6.33) is distributed, in that the ith update equation requiresonly the local parameters si, wi, bi, ci, qi along with communicated statevariables νj, µij which come from adjacent nodes and edges in the graphG. We refer to this dynamical system as the augmented dual distributedsaddle-point (ADD-SP) dynamics. Following (6.31), we equip this systemwith disturbance inputs and performance outputs

bi 7→bi + tbηi, zi = −q−12

i s>i νi, i ∈ {1, . . . , nx}. (6.34)

While a closed-form expression for the H2 norm of the this system isdifficult to compute, we can state the following comparative result.

Corollary 14 (Augmented Dual Distributed Saddle-Point Performance)Consider the ADD-SP dynamics (6.33), denoted by GADD, with disturbance inputsη(t) and performance output z(t) as in (6.34), under the same assumptions asTheorem 12. Then the squared H2 norm of the system (6.33)–(6.34) satisfies theupper bound

‖GADD‖2H2≤

t2b

2τνTr(W>b Wb), (6.35)

where Wb = blkdiag(w1, w2, . . . , wnx ) ∈ R(nxnr)×nx . Moreover, (6.35) is satis-fied with equality if and only if ρ = 0.

Proof of Corollary 14: To begin, we note that

nx

∑i=1

Φi(νi) = −1

2ν>SQ−1S>ν− ν>SQ−1c− ν>Wbb


where S = blkdiag(s1, s2, . . . , snx ) ∈ R(nxnr)×nx . In vector notation aftershifting the equilibrium to the origin, the system (6.33)–(6.34) is

τνν = −SQ−1S>ν− tbWbηb − (E⊗ Inr )µ− ρ(L⊗ Inr )ν,

τµµ = (E> ⊗ Inr )ν,

z = −Q−12S>ν,

where η = col(η1, . . . , ηnx ) ∈ Rnx . In state-space this translates to[ν

µ

]=

− 1τνSQ−1S> − 1

τνρ(L⊗ Inr ) − 1

τν(E⊗ Inr )

1τµ(E> ⊗ Inr ) 0

︸︷︷︸

= AADD

[ν

µ

]

−[

tbτνWb

0

]︸︷︷︸= BADD

ηb, z =[−Q−

12S> 0

]︸︷︷︸

= CADD

[ν

µ

].

(6.36)

As in our previous results, one may verify that AADD is Hurwitz and that(AADD, CADD) is observable. Consider the observability Gramian candidateXADD = 1

2 blkdiag(τν I(nxnr), τµ I|E |∗nr ). A straightforward algebra showsthat

XADD AADD + A>ADD XADD + QADD =

[−ρ(L⊗ Inr ) 0

0 0

]� 0, (6.37)

where QADD = C>ADDCADD. We conclude that XADD is a generalized ob-servability Gramian for the system GADD. Furthermore, if Xt is the true ob-servability Gramian for GADD, then as in Theorem 10, we have Xt � XADD,and we conclude that

‖GADD‖2H2≤ Tr(B>ADDXADDBADD) =

t2b

2τνTr(W>b Wb). (6.38)

When ρ = 0, XADD is the exact observability Gramian and hence

‖GADD‖2H2

=t2b

2τνTr(W>b Wb). (6.39)

To complete the proof, it remains to show that (6.39) in fact implies thatρ = 0. Suppose that (6.39) holds, and define ∆ := XADD − Xt � 0. Then,

6.6 application to optimal frequency regulation 107

following a similar argument as in the proof of Theorem 10, one may showthat ∆BADD = 0. From (6.37) and the fact that Xt is the actual observabilityGramian of the system, we may subtract equations to obtain

∆ AADD + A>ADD ∆ =

[−ρ(L⊗ Inr ) 0

0 0

].

Now, by multiplying each side of the equality above from the left by B>ADDand from the right by BADD, we find that

0 = ρW>b (L⊗ Inr )Wb

= ρW>b (E⊗ Inr )(E> ⊗ Inr )Wb = ρWeB>We

B,

where WeB = (E>⊗ Inr )Wb. Since the graph G is connected and Wb is square

and of full rank, it always holds that WeB 6= 0, and therefore we conclude

that ρ = 0.

The point of interest from Corollary 14 is that the bound on the H2performance is independent of ρ. In particular then, as ρ becomes large thenorm does not grow without bound, which resolves the issue observed inthe result of Theorem 12. When applied to the resource allocation problemin Section 6.7, we will, in fact, be able to make the stronger statement that thenorm is a strictly decreasing function of ρ, and augmentation can thereforebe used successfully to improve saddle-point algorithm performance.

6.6 application to optimal frequency regulation

We now focus on the performance of primal-dual controllers for power sys-tem dynamics under and in particular on the optimal frequency regulationoptimization problem.

6.6.1 Power Network Model and Frequency Regulation

We model a power network as a weighted graph (V , E) where V ={1, . . . , n} is the set of nodes (buses), and E ⊂ V × V is the set of edges(branches) with associated edge weights bij > 0 for {i, j} ∈ E . To eachbus i ∈ V we associate state variables (θi, ωi) corresponding to the voltagephase angle and the frequency deviation from nominal. Under the linear


DC Power Flow approximation, the system evolves according to the swingdynamics

θi = ωi,

miωi = −diωi + P?i − pe,i + pi,

(6.40)

where mi > 0 represents inertia or inverter filter time constants, di > 0 mod-els damping and/or droop control, P?

i is the constant nominal active powerinjection (nominal generation minus nominal load), pe,i = ∑n

j=1 bij(θi − θj)is the active power injected at bus i, and pi is the control input, correspond-ing to additional power generation from reserves. Throughout we ignorereactive power and voltage dynamics; these assumptions are standard insecondary frequency control studies.

When p = 0n, the dynamics (6.40) converge from every initial condition toa common steady-state frequency ω → ωss1n which can be easily calculatedto be ωss = (∑n

i=1 P?i )/(∑

ni=1 di). When ωss 6= 0, this represents a static

deviation from nominal, which we will eliminate by appropriately selectingthe reserve power inputs p. Moreover, since the variables P?

i will tendto vary, feedback control should be used to provide real-time frequencyregulation. To determine the the steady-state values for pi, an optimalfrequency regulation problem (OFRP) can be formulated as

minp∈Rn

n

∑i=1

12

ki p2i (6.41a)

subject to 0 = 1>n (P? + p), (6.41b)

where we seek to minimize the total cost5 (6.41a) of reserve generationpi ∈ R, for some coefficients ki > 0. The minimization is subject to network-wide balancing of power injections (6.41b). One may deduce from (6.40)that the constraint (6.41b) also ensures that ω = 0n in steady-state, i.e., thefrequency returns to its nominal value.

5 A linear term could also of course be added to the cost, but we omit it here for simplicity. Weassume that any inequality constraints on p are non-binding, and subsequently drop themfrom the problem (6.41), which is then similar to the classic economic dispatch (see Page 412

of [134]).


6.6.2 Performance of Primal-Dual Frequency Controllers

Beginning from the OFRP (6.41), we roughly follow [24], [103] to derive thecontroller dynamics. The Lagrangian of the OFRP (6.41) is given by

Lϑ(p, µ) =12

p>Kp + µ1>n (P? + p),

where K = diag(ki) and µ ∈ R is a multiplier. One finds that p = −µK−11nis the unique minimizer, on computing arg minp∈Rn Lϑ(p, µ). From this one

quickly calculates the dual function Φ(µ) = infp∈Rn Lϑ(p, µ), and the dualOFRP is

maxµ∈R

Φ(µ) =n

∑i=1

µP?i −

12ki

µ2, (6.42)

where we seek to maximize Φ(µ) over the common variable µ ∈ R. Todistribute (6.42), we introduce local variables µi ∈ R for each bus, andconsider the equivalent constrained optimization problem

maxµ∈Rn

n

∑i=1

µiP?i −

12ki

µ2i (6.43a)

subject to 0 = µi − µj, {i, j} ∈ Ec, (6.43b)

where Ec is the edge set of a connected, undirected, and acyclic6 communica-tion graph (V , Ec) between the buses. The additional constraints µi − µj = 0force the local variables µi to agree at optimality. Letting Ec ∈ Rn×|Ec|

denote the incidence matrix of the communication graph, the dual OFRP(6.43) is written in vector notation as

minµ∈Rn

12

µ>K−1µ− (P?)>µ (6.44a)

subject to 0|Ec| = E>c µ, (6.44b)

where now µ = (µ1, . . . , µn). The problem (6.44) is a linearly constrained,strictly convex quadratic program of the form (6.1) from Section 6.1, withQ = K−1, c = −P?, S = E>c and b = 0|Ec|. The corresponding Lagrangian

6 The acyclic assumption is made for consistency with our assumption that rank(S) = r fromSection 6.3.


is Lϑ(µ, ν) = 12 µ>K−1µ− (P?)>µ + ν>E>c µ, where ν ∈ R|Ec| is a vector of

multipliers and the primal-dual algorithm with control output p is

Tµµ = −K−1µ + P? − Ecν,

Tνν = E>c µ,

p = −K−1µ,

(6.45)

where as before Tµ and Tν are positive diagonal matrices of controller gains.Since the graph (V , Ec) is acyclic, the incidence matrix Ec has full columnrank. Therefore, by Lemma 8, the controller (6.45) converges exponentiallyto the global optimizer (µ?, ν?) of the problem (6.44). The output p(t) of(6.45) is the input to the swing dynamics (6.40); the interconnection is acascade. Since p(t) an exponentially converging input to the exponentiallystable linear system (6.40), the cascade is exponentially stable; we omit thedetails. It follows from the cascade structure that the map from P? to pgiven by (6.45) is the same as the map from P? to p after the systems areinterconnected.

To evaluate the input-output performance of the primal-dual controller(6.45), we consider the case where P? is subject to an additive distur-bance, modeling fluctuating generation/load, noise, or other uncertainty.As in Section 6.3, we shift the undisturbed equilibrium point of (6.45) tothe origin, and following Section 6.4 we define the performance outputz(t) = 1√

2K

12 p(t), such that ‖z(t)‖2

2 = 12 p(t)>Kp(t). Since p(t) = −K−1µ(t),

the performance output becomes z(t) = − 1√2

K−12 µ(t) or alternatively,

‖z(t)‖22 = 1

2 µ(t)>K−1µ(t). We now apply Theorem 9 to obtain the follow-ing result.

Theorem 15 (Primal-Dual OFRP Performance) For the primal-dual OFRPdynamics (6.45), consider the corresponding shifted, input-output dynamics[

τµµ

τνν

]=

[−K−1 −Ec

E>c 0

] [µ

ν

]+

[B1

0

]η,

z = − 1√2

K−12 µ,

(6.46)

with disturbances η and performance outputs y. Then the squared H2 norm of(6.46) is

‖G‖2H2

= Tr(B>1 T −1µ B1)/4. (6.47)


Moreover, assuming that Tµ = τµ In and B1 = bIn for some τ, b > 0, we havethat

‖G‖2H2

=b2

4τµn. (6.48)

The result indicates that the input-output performance of the primal-dualfrequency controller (6.45) is completely independent of the cost coefficientski and the incidence matrix Ec used to implement the distributed control; itdepends only on the controller time constants Tµ, the disturbance strengthB1, and the number of buses subject to disturbances.

Finally, we can consider an augmented Lagrangian leading to the aug-mented primal-dual OFRP dynamics

Tµµ = −K−1µ + P? − Ecν− ρEcE>c µ,

Tνν = E>c µ,

p = −K−1µ.

(6.49)

The matrix ρEcE>c is, in fact, a Laplacian matrix for the graph (V , Ec). Theadditional term arising from the augmentation is, therefore, a distributedproportional consensus-type term on the µ variables, complementing theintegral consensus-type term −Ecν. We now apply Theorem 12 to obtainthe following result.

Theorem 16 (Augmented OFRP Performance) For the primal-dual OFRP dy-namics (6.49) with the uniform parameters K = kIn, Tµ = τµ In and Tν = τν I|Ec|for constants k, τµ, τν > 0, consider the corresponding shifted, input-outputdynamics τµµ

τνν

=

−( 1k In + ρEcE>c ) −Ec

E>c 0

µ

ν

+

bIn

0

η,

z = − 1√2k

µ,

(6.50)

with disturbance inputs η and performance outputs y. Then the squared H2 normof (6.50) is

‖G‖2H2

=b2

4τµ+

b2

4τµ

n−1

∑i=1

11 + ρkσ2

i, (6.51)


where σi is the ith non-zero singular value of Ec. Moreover, in the high augmenta-tion gain limit ρ→ ∞, we have that

limρ→∞‖G‖2

H2=

b2

4τµ.

Proof of Theorem 16: The proof is immediate by applying Theorem 12 andnoting that r = n− 1 for E>c µ = 0r, since the graph (V , Ec) is acyclic.

Remarkably, we find that by designing the frequency controller basedon the augmented Lagrangian and making the gain ρ sufficiently high, theperformance of the primal-dual OFR controller (6.49) becomes independentof network size, converging to a constant b2/4τµ. Moreover, due to thespecial structure of graph incidence matrices, the algorithm remains dis-tributed. Theorem 16 demonstrates that the performance characteristicsof unaugmented primal-dual frequency controllers (Theorem 15) do notrepresent a fundamental performance limit for the approach.

6.7 application to resource allocation problems

We now apply the results from the previous sections to a particular classof problems. As a special case of the problem (6.1), consider the resourceallocation problem

minx∈Rn

n

∑i=1

12

qix2i + cixi (6.52a)

subject ton

∑i=1

xi =n

∑i=1

di, (6.52b)

where qi > 0, ci ∈ R, and di ∈ R. Comparing (6.52) to (6.1), we haveQ = diag(q1, . . . , qn), S = 1>n , Wb = 1>n , and d := col(d1, . . . , dn) = b. Theinterpretation of (6.52) is that a resource must be obtained from one of nsuppliers in amount xi, subject to a total demand satisfaction constraint.The objective function of (6.52) can be interpreted as the sum of the utilities−cixi minus the sum of the costs qix2

i /2. In a multi-agent context, eachvariable xi is assigned to an agent, the parameters qi, ci, di are availablelocally to each agent, and the agents must collectively solve the problem(6.52) through local exchange of information. As a concrete example, in thecontext of power system frequency control, the objective function models thecost of producing an auxiliary power input xi– as discussed in Section 6.6.1.

6.7 application to resource allocation problems 113

We will consider (6.52) along with several equivalent reformulations, andapply our results from the previous sections to assess the input-outputperformance of the resulting saddle-point algorithms. External disturbancesηd will be integrated into the algorithms as di 7→ di + ηi, where ηi modelsthe disturbances in demand di. For simplicity, all time constant matricesare assumed to be multiples of the identity matrix. To most clearly indi-cate which algorithms require communication of which variables, in thissection algorithms are not written in deviation coordinates with respect tothe optimizer. In all cases, the performance output z is chosen such that‖z(t)‖2

2 = (x(t) − x?)>Q(x(t) − x?), where x? is the global primal opti-mizer. With this choice of performance output, the transient performanceof the algorithm is measured using the same relative weightings as thesteady-state performance.

(a) To begin, the augmented Lagrangian of (6.52) in vector notation is

Lϑ(x, ν) =12

x>Qx + c>x + ν1>n (x− d) +ρ

2‖1>n (x− d)‖2

2, (6.53)

where ν ∈ R. Applying the saddle-point method to the Lagrangian Lϑ(x, ν)and attaching the same disturbances η ∈ Rn and performance outputsz ∈ Rn as before, we obtain the centralized saddle-point dynamics

RAcent(ρ) :

τx x = −Qx− c− ν1n − ρ1n1

>n (x− d− η),

τνν = 1>n (x− d− η),

z = Q12 (x− x?).

(6.54)

When ρ = 0, the algorithm (6.54) is of a gather-and-broadcast type [135],where all states xi and disturbances di are collected and processed by acentral agent with state ν. When ρ > 0, the additional term 1n1

>n in the

algorithm requires all-to-all communication of the local imbalances xi − di.

(b) We now consider a reformulation that results in a distributed optimiza-tion algorithm. Let G = ({1, . . . , n}, E) denote a weakly connected acyclicgraph over the agent set {1, . . . , n}, and let E ∈ Rn×|E| denote the orientednode-edge incidence matrix of G. The constraint 1>n x = 1>n d in the resourceallocation problem (6.52) is equivalent to the existence7 of a vector ξ ∈ R|E |

7 This follows since ker(E>) = span(1n), and hence Im(E) is the subspace orthogonal to thevector 1n.


such that Eξ = x− d. The resource allocation problem (6.52) can thereforebe equivalently rewritten as

minx∈Rn ,ξ∈R|E |

n

∑i=1

12

qix2i + cixi (6.55a)

subject to Eξ = x− d, (6.55b)

with associated augmented Lagrangian

Lϑ ′(x, ξ, ν) =12

x>Qx + c>x + ν>(Eξ − x + d) +ρ

2‖Eξ − x + d‖2

2,

where ν ∈ Rn. This reformulation can be interpreted as a version of (6.1)with an expanded primal variable (x, ξ) and an expanded dual variableν = col(ν1, . . . , νn). By applying the saddle-point method to the problem(6.55), we obtain the distributed saddle-point dynamics

RAdist(ρ) :

τx x = −Qx− c + ρ(Eξ − x + d + η) + ν,

τξ ξ = −E>ν− ρE>(Eξ − x + d + η),

τνν = Eξ − x + d + η,

z = Q12 (x− x?).

(6.56)

When ρ = 0, the algorithm (6.56) is distributed with the topology ofthe graph G, with states (xi, νi) associated with each node and a state ξiassociated with each edge. When ρ > 0, the algorithm contains the so-callededge Laplacian matrix E>E [136], which under our acyclic assumption ispositive definite.

(c) Our third formulation is the dual ascent algorithm (6.30). The substi-tution of the appropriate matrices into (6.30) leads to the centralized dualascent dynamics

RAdualcent :

{τνν = −(1>n Q−11n)ν− 1>n (Q−1c + d + η),

z = Q12 (x− x?) = −Q−

12 (c + ν1n)−Q

12 x?,

(6.57)

where ν ∈ R. The Algorithm (6.57) is again centralized, with a singlecentral agent with state ν performing all computations and broadcastingxi = −q−1

i (ci + ν) back to each agent. We note that this algorithm isindependent of the augmentation and therefore, also independent of theparameter ρ.


(d) For our fourth and final formulation, we apply the ADD-SP methoddeveloped in Section 6.5.2. For the problem (6.52), one quickly deducesthat S =Wb = In, and the algorithm (6.33) reduces to

RAdualdist (ρ) :

τνν = −Q−1ν− (d + η)−Q−1c− Eµ− ρLν,

τµµ = E>ν,

z = Q12 (x− x?) = −Q−

12 (ν + c)−Q

12 x?,

(6.58)

When ρ = 0, this algorithm is distributed with the graph G associated withthe incidence matrix E, with states νi associated with nodes and states µijassociated with edges. When ρ > 0, the algorithm additionally contains theundirected Laplacian matrix L = EE> of G, and thus remains distributed.

For each of the four formulations above, we compute the H2 norm fromthe disturbance input η to the performance output z. For RAcent(ρ), RAdual

cent ,and RAdual

dist (ρ) this follows immediately from Theorem 12, Proposition 13,and Corollary 14, respectively. The algorithm RAdist(ρ) requires a modifica-tion of the proof of Theorem 12, since the objective function is no longerstrongly convex in the primal variables (x, ξ); we omit the details.

System ρ = 0 ρ→ ∞

RAcent(ρ) n/(2τν) +∞

RAdist(ρ) n/(2τν) +∞

RAdualcent n/(2τν) independent of ρ

RAdualdist (ρ) n/(2τν) 1/(2τν)

Table 6.1: Comparison of squared H2 norm expressions.

The first column of Table 6.1 shows the H2 system norms for the fourformulations when ρ = 0, i.e., the unaugmented versions of the varioussaddle-point algorithms. Despite substantial differences between the algo-rithms in terms of information structure and number of states, all four havethe same input-output performance in the H2 norm. This implies that adistributed implementation will perform no worse than centralized.

While these four formulations all possess identical system norms underthe basic primal-dual algorithm, augmentation differentiates these meth-ods from one another and substantial differences between the algorithms


begin to appear as ρ is increased. The limiting results are tabulated in thesecond column of Table 6.1. The input-output performance of the first twoformulations becomes arbitrarily bad as the augmentation gain ρ increases,while the performance of the ADD-SP algorithm improves substantially,becoming independent of the system size in the limit ρ→ ∞.

We illustrate the results in Table 6.1 via time-domain simulations inFigure 6.3 and Figure 6.4 for the system in (6.55) with n = 2 and anunderlying line graph with E = [1 − 1]>. In the following, we summarizeour observations:

(i) With unit variance white noise as inputs, the unaugmented implemen-tation in Figure 6.3 for the four different algorithms results in identicalsteady-state output variance, numerically computed as the squared H2norm in (2.2).

(ii) In Figure 6.4, a sufficiently large augmentation factor ρ is introducedto penalize the constraint violations. It is observed that with the augmen-tation, the steady-state variance of the outputs for the centralized anddistributed implementations in RAcent(ρ), RAdist(ρ) worsens, while that ofthe distributed dual implementation from RAdual

dist (ρ) improves.

(iii) Figure 6.5 illustrates how the choice of communication graph topologyinfluences the performance of the algorithm RAdual

dist (ρ). We consider n = 4agents, and implement the algorithm with line, ring, and complete com-munication graphs. While Corollary 14 and the results of Table 6.1 holdonly for acyclic graphs, Figure 6.5 shows that in all cases the algorithm’sperformance improves as ρ increases. For a given value of ρ, graphs withhigher connectivity show a greater improvement. This behavior is explainedby noting that the algorithm (6.58) has the same form as the augmentedsaddle-point dynamics (6.24), and an analysis similar to that performed forTheorem 12 can, in fact, be performed for (6.58). For uniform cost functionparameters and an acyclic graph, this leads to the expression

‖GRAdualdist (ρ)

‖2H2

=1

2τν

1 +n

∑i=2

qq + ρλi

, (6.59)

where 0 = λ1 < λ2 ≤ · · · ≤ λn are the eigenvalues of the Laplacian matrix.As graph connectivity increases, so does λ2, and performance thereforeimproves. From a design perspective, note that for a fixed time constant τν


020

040

060

080

010

0012

0014

0016

0018

0020

00

−202

RAcent

050

100

150

−202

020

040

060

080

010

0012

0014

0016

0018

0020

00

−202

RAdist

050

100

150

−202

020

040

060

080

010

0012

0014

0016

0018

0020

00

−202

RAdualcent

050

100

150

−202

020

040

060

080

010

0012

0014

0016

0018

0020

00

−202

t[s

]

RAdualdist

050

100

150

−202

sam

ples

Fig

ur

e6

.3:S

tead

y-st

ate

vari

ance

for

the

unau

gmen

ted

case

,for

para

met

ers

n=

2,Q

=di

ag(4

,25)

,τx=

1,τ

ξ=

1,τ

ν=

1,τ

µ=

1,E=

[1−

1]>

;the

rem

aini

ngpa

ram

eter

sdo

not

influ

ence

the

resu

lts.


0200

400600

8001000

12001400

16001800

2000−

5 0 5RAcent

060

120180

−5 0 5

0200

400600

8001000

12001400

16001800

2000−

5 0 5

RAdist

060

120180

−5 0 5

0200

400600

8001000

12001400

16001800

2000−

5 0 5

t[s]

RAdualdist

x1

x2

060

120180

−5 0 5

samples

Fig

ur

e6.

4:Steady-state

variancefor

theau

gmented

case,forp

arameters

n=

2,ρ=

100,Q=

diag(4,25),

τx=

1,τ

ξ=

1,τ

ν=

1,τ

µ=

1,E=

[1−1] >

;therem

ainingparam

etersdo

notinfluence

theresults.


0 0.1 0.2 0.3 0.4 0.50.5

0.8

1.1

1.4

1.7

2

2.3

ρ

‖GR

Adu

aldi

st‖2 H

2

lineringcomplete

Figure 6.5: RAdualdist (ρ) for Q = diag(4, 25, 16, 49).

and a desired level of performance γ ∈ (1/√

2τν,√

n/√

2τν), examinationof (6.59) shows that a sufficient condition for ‖GRAdual

dist (ρ)‖H2 ≤ γp is that

ρ ≥ qλ2

n− 2τνγ2p

2τνγ2p − 1

.

In particular, this shows that the augmentation gain should be chosen ininverse proportion to the algebraic connectivity λ2 of the Laplacian matrix.A strongly connected graph will require a lower augmentation gain thana weakly connected graph to achieve a desired level of H2 performance.Achieving an H2 norm lower than 1/

√2τν requires an increase in τν.

(iv) Finally, Figure 6.6 plots the system norm as a function of ρ for the threeaugmented algorithms, for a test case with n = 4 agents. The norm is nota monotonic function of the augmentation factor ρ for the implementa-tions in RAcent(ρ) and RAdist(ρ), but is monotonic for RAdual

dist (ρ) appliedto resource allocation problems, in agreement with the result for the para-metrically uniform case in equation (6.59). We recall here that the DualAscent algorithm does not involve the augmentation term and is henceindependent of the parameter ρ. Therefore, the plot for the performance ofsuch an algorithm as a function of ρ is redundant.


0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

ρ

‖GR

A‖2 H

2

RAcentRAdist

RAdualdist

Figure 6.6: RAcent, RAdist, RAdualdist for Q = diag(4, 4, 4, 9) and line graph.

6.8 conclusions

We studied the input-output performance of continuous-time saddle-pointmethods for solving linearly constrained convex quadratic programs, pro-viding an explicit formula for the H2 norm under a relevant input-outputconfiguration. We then studied the effects of Lagrangian regularization andaugmentation on this norm and derived a distributed dual version of theaugmented algorithm which overcomes some of the limitations of naiveaugmentation. We finally applied the results to compare several imple-mentations of the saddle-point methods to distributed optimal secondaryfrequency regulation problems and to generic resource allocation problems.

7C O N C L U S I O N S A N D O U T L O O K

In this thesis, we primarily investigated robustness and frequency stabilityof low-inertia power systems. We advocated system norms as a powerfulalternative approach for analyzing system resilience. Further, we adoptedthis method for a comprehensive input-output performance analysis ofprimal-dual saddle-point algorithms, with special attention to optimalfrequency regulation.

7.1 stability analysis and frequency control in future low-inertia power systems

In Chapters 3–4, we presented performance metrics for low-inertia powersystem stability analysis and studied the optimal placement and tuning ofcontrol devices providing virtual inertia and fast frequency response. Weintroduced two non-trivial algorithms that tractably formalized this controlproblem– these algorithms were based on the control system notion of anH2 system gain characterizing the amplification of a disturbance.

In particular, we developed a computational approach based on an ex-plicit gradient formulation and compared the performance of the proposedapproach for a three-region power system test case based on a stylisticswing equation model in Chapter 3. Our results highlighted that it is thelocation of disturbance and the placement of inertia in the system, ratherthan the total inertia in the power system by itself that dictates its resilience.The focus in this chapter was more on general problem formulations, e.g.,the `1-regularization and the min-max optimization.

To broaden our understanding and to concentrate on scaling our analysisto real world systems, in Chapter 4, we extended our analysis to non-linear,large-scale power system models with detailed power converter modelsin the grid-following and grid-forming VI implementations. A compu-tationally efficient tuning algorithm (with an order of magnitude lowercomplexity) in comparison to some of the other existing algorithms in theliterature was proposed. This approach relied on gradient computation viaobservability and controllability Gramians. We showcased the capabilities ofsuch VI devices on a high-fidelity model of the South-East Australian power

121

122 conclusions and outlook

system. Our results again indicated that the system robustness stronglydepended on the specific implementation and location of virtual inertiaand damping. This was in contrast to conventional paradigms of ancillaryservice markets which valued energy or the total damping and inertiabut not location or specifics of the implementation. This motivated us toinvestigate the problem of markets for virtual inertia.

In Chapter 5, we explored a decentralized market mechanism for vir-tual inertia on the lines of ancillary service markets for power systems, byconsidering a non-standard combination of VCG mechanism and the gener-alized concept of system norms. A social welfare maximization problemwas formulated and the proposed mechanism ensured that truthful biddingwas the dominant strategy for the participating agents. In contrast to con-ventional regulatory approaches, the designed mechanism simultaneouslycompensated the agents providing virtual inertia and met performancerequirements.

Based on the analysis in the above three chapters, the grid-formingVI devices appear to be plausible building blocks for a workable, renew-able energy-dominated power grid. However, as these devices offer fasterresponse times, negligible inertial response, and involve less storage, acentralized dispatch as in conventional power system operation no longerseems feasible or viable. This prompted us to evaluate the prospects of real-time distributed optimization with robustness guarantees as an alternative.An exhaustive analysis of the class of distributed algorithms that coulddeliver the requisite performance was therefore crucial.

Finally, in order to understand such algorithms, in Chapter 6, a compre-hensive study of the input-output performance of primal-dual saddle-pointalgorithms was carried out under the framework of H2 norms. Severalextensions, including regularization and augmentation were analyzed forperformance improvement. Two specific applications in the context of opti-mal frequency regulation and resource allocation were discussed in-depth–with a motive to address the issues relating to the integration of the renew-able devices and evolving a new paradigm of grid operation.

Table 7.1 and Table 7.2 summarize some of the key results on the basis ofperformance bounds and optimizer characteristics for various coherencymetrics discussed in the preceding chapters.

7.2 future outlook 123

Coherency Metric Result

Primary control effort Globally optimal

Generic energy cost Locally optimal

Robust min-max Locally optimal

Regularized `1 cost Locally optimal

Centralized planninga Globally optimal

Table 7.1: Summary of results on the basis of the optimizer characteristics.

a For the primary control effort

Metric/Implementation Performance

Primary control effort Exact closed-from

Generic energy cost Upper and lower bounds

Robust min-max Upper bound

Regularized `1 cost Upper and lower bounds

Centralized Planninga Exact closed-from

Regularized saddle-point Upper bound

Augmented saddle-point Exact closed-form

Primal-Dual OFRP Exact closed-form

Table 7.2: Summary of results on the basis of the performance bounds.

a For the primary control effort

7.2 future outlook

In this section, we comment on some of the open problems and futuredirections of research pertinent to the ideas developed in this thesis.

The past few years have seen a significant interest in the broad areaof low-inertia power systems and their stability. With power converter-interfaced renewable energy poised to assume a preeminent role in thenear future, it is only expected that research in this direction will assumegreater importance. As discussed in the previous chapters, we envision

124 conclusions and outlook

future power systems operating entirely based on converter-interfaced gen-eration. In addition, services that are provided by synchronous machinestoday (e.g., voltage regulation) need to be provided by grid-forming powerconverters. Therefore, an interesting direction for future research wouldinvolve extending the proposed framework by incorporating suitable per-formance metrics for ancillary services apart from inertia, damping, andfast frequency response.

With the increasing scale of deployment for virtual inertia and fast fre-quency response, matters relating to optimal procurement and markets forthese services are also expected to gain further prominence. Though weillustrated our approach and formulation for a market-based mechanismthrough stylistic swing equation-based models, the construct can be ex-tended to more complex system dynamics. Also, it is not clear as to hownon-convex cost functions can be incorporated in the current framework.Other problems not considered here, pertain to robust auction mechanismswhich counteract shill-bidding and collusion among various bidders.

With regards to performance analysis of saddle-point algorithms, opendirections for future research include input-output performance metrics forproblems involving inequality constraints, for distributed implementationswhere communication delays occur between agents, and for other classesof distributed optimization algorithms. An analogous study in an H∞ per-formance framework and a comparison thereof would also be insightful.Another interesting question is how to further improve the H2 performanceof saddle-point methods by designing auxiliary feedback controllers; aug-mentation is but one approach. Finally, extending these results beyondthe case of quadratic cost functions with linear constraints may requirenon-linear, robust control approaches, as in [118]–[122].

AA P P E N D I X

a.1 expression for the H2 norm

For a linear system G = (A, B, C), we interpret the H2 norm ‖G‖2H2

as

• The squared H2 norm of G measures the energy amplification, i.e.,the sum of L2 norms of the outputs yj(t), for unit impulses at all inputsηi(t)=δ(t). From state-space analysis, we have

x(t) = eA(t−t0)x0 +

∫ t

t0

eA(t−τ)Bu(τ)dτ. (A.1)

For x0 = 0, ui(t) = eiδ(t), let the output vector be yi(t). This output can beevaluated as

yi(t) = Cx(t) = C∫ t

t0

eA(t−τ)Bδ(τ)dτ = CeAtBei. (A.2)

The system norm ‖G‖2H2

can thus be evaluated as

‖G‖2H2

=n

∑i=1

m

∑j=1

∫ ∞

0yi

j(t)2dt

=n

∑i=1

∫ ∞

0yi(t)

>yi(t)dt

=n

∑i=1

∫ ∞

0(CeAtBei)

> CeAtBei dt

=n

∑i=1

∫ ∞

0ei>B>eA>tC>CeAtBei dt

=n

∑i=1

ei>(

B>∫ ∞

0eA>tC>CeAt dt︸︷︷︸

= P

B)

ei

= Tr(B>PB),

125

126 appendix

where P is the observability Gramian and the solution of the Lyapunovequation

PA + A>P + C>C = 0.

• The squared H2 norm of G quantifies the steady-state total varianceof the output for a system subjected to unit variance stochastic whitenoise inputs η(t). As the input is unit variance white noise, we haveE{

η(t)η(t)>}= In and E

{η(t1)η(t2)

>} = δ(t1 − t2)In, where E denotesthe expectation operator. From the definition of the system norm and (A.1)

‖G‖2H2

= Tr(

limt→∞

E{

y(t) y(t)>})

= Tr

(limt→∞

E

{(C∫ t

0eA(t−t1)Bη(t1)dt1) (C

∫ t

0eA(t−t2)Bη(t2)dt2)

>})

= Tr

(limt→∞

E

{∫ t

0

∫ t

0CeA(t−t1)Bη(t1)η(t2)

>B>eA>(t−t2)C> dt1 dt2)

})

= Tr

(limt→∞

{∫ t

0

∫ t

0CeA(t−t1)B E{η(t1)η(t2)

>}B>eA>(t−t2)C> dt1 dt2

})

= Tr

(limt→∞

{∫ t

0

∫ t

0CeA(t−t1)Bδ(t1 − t2)B>eA>(t−t2)C> dt1 dt2

})

= Tr

(limt→∞

{∫ t

0CeA(t−t1)BB>eA>(t−t1)C> dt1

})

= Tr

(limt→∞

{∫ t

0CeAτ BB>eA>τC> dτ

})

= Tr({∫ ∞

0CeAτ BB>eA>τC> dτ

})= Tr(B>PB),

where P is the observability Gramian and the solution of the Lyapunovequation

PA + A>P + C>C = 0.

A.2 proof of observability gramian lemma 127

a.2 proof of observability gramian lemma

Proof of Lemma 1: Following the derivation of the H2 norm for state-space systems [31], we have ‖G‖2

H2= Tr(B>PB), where P is the observ-

ability Gramian P =∫ ∞

0 eA>tC>CeAt dt. Note from (3.5) that the modez0 = [1>n 0>n ]

> associated with the marginally stable eigenvalue of A is notdetectable, i.e., it holds that CeAtz0 = Cz0 = 02n for all t ≥ 0. Because theremaining eigenvalues of A are stable, the integral is finite.

Next, we show that P is a solution for both (3.8) and (3.9). By taking thederivative of eA>tC>CeAt with respect to t, and then integrating from t = 0to t→ +∞, we obtain

A>P + PA =[eA>tC>CeAt

]∞

0.

Using the fact that Cz0 = Az0 = 02n, we conclude that[eA>tC>CeAt

]∞

0=

−C>C and therefore (3.8) holds for P. The fact that P satisfies (3.9) can beverified by inspection, as

Pz0 =

∫ ∞

0eA>tC>CeAtz0 dt =

∫ ∞

0eA>tC>Cz0 dt = 02n.

It remains to be shown that P is the unique solution of (3.8) and (3.9). Asrank

(A>)= 2n− 1, the rank–nullity theorem implies that the kernel of A>

is given by a vector ζ ∈ R2n. It can be verified that A>ζ = 02n holds forζ = [(D1n)> (M1n)>]>. The solutions of (3.8) can hence be parametrizedby

P(τ) = P + τζζ>,

for τ ∈ R. Finally, (3.9) holds if (P + τζζ>)z0 = 02n. In combination withPz0 = 02n this implies τ = 0. With this choice of τ, P equals the positivesemi-definite matrix P.

128 appendix

a.3 gradient computation for H2 norms via perturbation

An algorithm to compute the gradient of the H2 norm via perturbation.

Algorithm 1: Gradient computation via perturbationInput current value m of the decision variablesOutput numerical evaluation of the gradient ∇ f (m)

A(0) ←[

0 In

−M−1L −M−1D

];

B(0) ←[

0

M−1 Π1/2

];

P(0) ← Lyap(

A(0), C>C)

;

for i = 1, . . . , n doΦ← eie

>i ;

A(1) ←[

0 0

ΦM−2L ΦM−2D

];

B(1) ←[

0

−ΦM−2 Π1/2

];

P(1) ← Lyap(

A(0), P(0)A(1) + A(1)>P(0))

;

∇i f (m)← Tr(

2B(1)>P(0)B(0) + B(0)>P(1)B(0))

.

A.4 gradient computation for output feedback 129

a.4 gradient computation for output feedback

J(K) = Tr(B>g PKBg) (A.3)

To evaluate the gradient of J(K) with respect to the matrix K, we use theimplicit function theorem on the function Φ(PK, K), where

Φ(PK, K) = PK(A + BKC) + (A + BKC)>PK + Cp>Cp = 0.

On taking partial derivatives, we have

0 = ΦK(PK, K)dK + ΦPK (PK, K)dPK

0 = PKB dK C + C>dK>B>PK + dPK(A + BKC) + (A + BKC)>dPK

0 = PKB dK C + C>dK>B>PK + PK′dK(A + BKC) + (A + BKC)>PK

′dK,

where PK′ denotes the derivative of PK with respect to K. This yields

−(PKB dK C + C>dK>B>PK) = PK′dK(A + BKC) + (A + BKC)>PK

′dK.(A.4)

The gradient of the H2 norm can, therefore, be computed as follows,

dJ(K) = Tr(∇J(K)>dK) = Tr(PK′ dK BgBg

>).

Let LK be the closed-loop controllability Gramian of the same system, i.e.,

(A + BKC)LK + LK(A + BKC)> + BgBg> = 0. (A.5)

From the relations (A.4) and (A.5), we have

dJ(K) = −Tr(P′K dK ((A + BKC)LK + LK(A + BKC)>))

= −Tr(P′K dK (A + BKC)LK)− Tr(P′K dK LK(A + BKC)>)

= −Tr(P′K dK (A + BKC)LK)− Tr((A + BKC)>P′K dK LK)

= −Tr((P′K dK (A + BKC) + (A + BKC)>P′K dK) LK)

= Tr((PKB dK C + C>dK>B>PK)LK)

= Tr(2(B>PK)LKC>dK>),

i.e., ∇J(K) = 2(B>PK)LKC>.

130 appendix

a.5 gradient-based optimization of H2 norms

By using the implicit linearization technique from [137] and shown inAppendix A.4, the gradient of the norm ‖G‖2

H2with respect to K is given

by

∇K‖G‖2H2

= 2(B>PK)LKC>, (A.6)

where LK is the positive semi-definite controllability Gramian obtainedfrom the linear (in LK) Lyapunov equation

LAcl> + Acl L + BgBg

> = 0, (A.7)

parameterized in K for the given system matrices A, B, C, and Bg. Thus,computing the norm ‖G‖2

H2and its gradient ∇K‖G‖2

H2for a given K re-

quires solving the Lyapunov equations (4.18) and (A.7). Moreover, thenumber of decision variables of the optimization problem (4.20) can bereduced by projecting the gradient ∇K‖G‖2

H2on the sparsity constraint S .

Using the vector of non-zero parameters φ = [m1, d1, . . . , mnc , dnc ] for thegrid-following or alternatively φ = [α1, β1, . . . , αnc , βnc ] for the grid-formingimplementation, the projected gradient is given by e.g.,

projφ(∇K‖G‖2

H2

)= (

∂

∂m1‖G‖2

H2,

∂

∂d1‖G‖2

H2, . . . ,

∂

∂dnc

‖G‖2H2

).

Similar projections can be performed for the constraint set C.Because the H2 norm is infinite for unstable systems, both the system

norm ‖G‖2H2

as well as its gradient (A.6) are only well defined for a stableclosed-loop system (4.17). Thus, to optimize the control gain K, an initialguess for K is required that stabilizes (4.17) and satisfies the constraints Sand C. Assuming that the system without VI devices is stable, it followsthat mk = 0, dk = 0 suffices as an initial guess for the grid-followingimplementation. Moreover, the H2 norm cost is smooth and approachesinfinity as the control gains K approach the boundary of the set of stabilizinggains. In other words, any sequence of control gains K with non-increasingcost is guaranteed to be stabilizing.

Assuming that the projections onto C can be efficiently computed, theprojected gradient method [130] and gradient computation outlined abovecan be used to find a locally optimal solution of the optimization problem(4.20) even for systems of very large dimension. For instance this is the casewhen C encodes upper and lower bounds on mk and dk. If the projectiononto C cannot be computed efficiently, the above gradient computation canstill be used to speed up the computation times of higher-order methods.

A.6 proof of global convergence to optimizer lemma 131

a.6 proof of global convergence to optimizer lemma

Proof of Lemma 8: The proof of Lemma follows by using the Lyapunovcandidate V(x, ν) = 0.5(x − x?)>Tx(x − x?) + 0.5(ν − ν?)>Tν(ν − ν?) +ε(ν− ν?)>STx(x− x?) for ε > 0, i.e., consider the Lyapunov candidate

P =12

[Tx εTxS>

εSTx Tν

],

which is positive definite since for ε sufficiently small. With A as the systemmatrix in (6.5), we compute that

AT P + PA = −[

∆ εQS>/2

εSQ/2 εSS>

],

where ∆ = Q − ε2 (TxS>T −1

ν S + S>T −1ν STx). Since ε > 0 and S has full

row-rank, εSS> is positive definite. Moreover, ∆ is positive definite if εis sufficiently small. Standard Schur complement results then yield thatA>P + PA ≺ 0 if and only if

SS> − ε

4SQ∆−1QS> > 0,

which holds for ε sufficiently small as limε→0 ∆ = Q.As the convergence rate is dictated by the eigenvalues, it follows from a

simple variant of Theorem 3.6 of [124] that the rate is

∝ minii

(blkdiag(T −1x , T −1

ν ))

or∝ 1/ max( max

i∈{1,...,n}Tx,ii, max

i∈{1,...,r}Tν,ii).

B I B L I O G R A P H Y

[1] J. Slootweg and W. Kling, “Impacts of distributed generation onpower system transient stability”, in Proc. IEEE Power EngineeringSociety Summer Meeting, 2002, pp. 862–867.

[2] G. Lalor, J. Ritchie, S. Rourke, D. Flynn, and M. O’Malley, “Dynamicfrequency control with increasing wind generation”, in Proc. IEEEPower Engineering Society General Meeting, 2004, pp. 1715–1720.

[3] A. S. Ahmadyar, S. Riaz, G. Verbic, A. Chapman, and D. J. Hill, “Aframework for assessing renewable integration limits with respect tofrequency performance”, IEEE Transactions on Power Systems, vol. 33,no. 4, pp. 4444–4453, 2018.

[4] RG-CE System Protection & Dynamics Sub Group, “Frequency sta-bility evaluation criteria for the synchronous zone of continentaleurope”, ENTSO-E, Tech. Rep., 2016.

[5] AEMO, “Black System South Australia 28 September 2016- FinalReport”, Tech. Rep., 2017.

[6] Svenska Kraftnät, Statnett, Fingrid and Energinet.dk, “Challengesand opportunities for the nordic power system”, Tech. Rep., 2016.

[7] W. Winter, K. Elkington, G. Bareux, and J. Kostevc, “Pushing thelimits: Europe’s new grid: Innovative tools to combat transmissionbottlenecks and reduced inertia”, IEEE Power and Energy Magazine,vol. 13, no. 1, pp. 60–74, 2015.

[8] M. Milligan, B. Frew, B. Kirby, M. Schuerger, K. Clark, D. Lew, P.Denholm, B. Zavadil, M. O’Malley, and B. Tsuchida, “Alternativesno more: Wind and solar power are mainstays of a clean, reliable,affordable grid”, IEEE Power and Energy Magazine, vol. 13, no. 6,pp. 78–87, 2015.

[9] P. Tielens and D. V. Hertem, “The relevance of inertia in powersystems”, Renewable and Sustainable Energy Reviews, vol. 55, pp. 999–1009, 2016.

[10] F. Milano, F. Dörfler, G. Hug, D. Hill, and G. Verbic, “Foundationsand challenges of low-inertia systems”, in Proc. Power Systems Com-putation Conference (PSCC), 2018, pp. 1–25.

133

134 bibliography

[11] A. Ulbig, T. S. Borsche, and G. Andersson, “Impact of low rotationalinertia on power system stability and operation”, in Proc. IFAC WorldCongress, 2014, pp. 7290–7297.

[12] N. Soni, S. Doolla, and M. C. Chandorkar, “Improvement of transientresponse in microgrids using virtual inertia”, IEEE Transactions onPower Delivery, vol. 28, no. 3, pp. 1830–1838, 2013.

[13] H. Bevrani, T. Ise, and Y. Miura, “Virtual synchronous generators: Asurvey and new perspectives”, International Journal of Electrical Powerand Energy Systems, vol. 54, pp. 244–254, 2014.

[14] S. D’Arco and J. A. Suul, “Virtual synchronous machines- classi-fication of implementations and analysis of equivalence to droopcontrollers for microgrids”, in Proc. IEEE PowerTech, 2013.

[15] B. Kroposki, B. Johnson, Y. Zhang, V. Gevorgian, P. Denholm, B.-M.Hodge, and B. Hannegan, “Achieving a 100% renewable grid: Oper-ating electric power systems with extremely high levels of variablerenewable energy”, IEEE Power and Energy Magazine, vol. 15, no. 2,pp. 61–73, 2017.

[16] J. Morren, S. W. De Haan, W. L. Kling, and J. Ferreira, “Wind turbinesemulating inertia and supporting primary frequency control”, IEEETransactions on Power Systems, vol. 21, no. 1, pp. 433–434, 2006.

[17] M. Koller, T. S. Borsche, A. Ulbig, and G. Andersson, “Review of gridapplications with the zurich 1 MW battery energy storage system”,Electric Power Systems Research, vol. 120, pp. 128–135, 2015.

[18] P. Ashton, C. Saunders, G. Taylor, A. Carter, and M. Bradley, “Inertiaestimation of the GB power system using synchrophasor measure-ments”, IEEE Transactions on Power Systems, vol. 30, no. 2, pp. 701–709, 2015.

[19] E. Ela, V. Gevorgian, A. Tuohy, B. Kirby, M. Milligan, and M. J.O’Malley, “Market designs for the primary frequency responseancillary service Part I: Motivation and design”, IEEE Transactionson Power Systems, vol. 29, no. 1, pp. 421–431, 2014.

[20] S. Henry, “System security market frameworks review, final report”,AEMC, Tech. Rep., 2017.

[21] S. Guggilam, C. Zhao, E. Dall’Anese, Y. C. Chen, and S. Dhople,“Optimizing der participation in inertial and primary-frequencyresponse”, IEEE Transactions on Power Systems, vol. 33, no. 5, pp. 5194–5205, 2018.

bibliography 135

[22] N. Li, L. Chen, C. Zhao, and S. H. Low, “Connecting automaticgeneration control and economic dispatch from an optimizationview”, in Proc. American Control Conference, Portland, OR, USA, 2014,pp. 735–740.

[23] E. Mallada, C. Zhao, and S. Low, “Optimal load-side control forfrequency regulation in smart grids”, IEEE Transactions on AutomaticControl, vol. 62, no. 12, pp. 6294–6309, 2017.

[24] E. Mallada and S. Low, “Distributed frequency-preserving optimalload control”, IFAC Proceedings Volumes, vol. 47, no. 3, pp. 5411–5418,2014.

[25] T. Stegink, C. D. Persis, and A. van der Schaft, “A unifying energy-based approach to stability of power grids with market dynamics”,IEEE Transactions on Automatic Control, vol. 62, no. 6, pp. 2612–2622,2017.

[26] K. Zhou and J. C. Doyle, Essentials of Robust Control. New Jersey:Prentice Hall, 1998.

[27] P. Kundur, Power System Stability and Control. McGraw-Hill, 1994.

[28] D. P. Kothari and I. Nagrath, Modern power system analysis. TataMcGraw-Hill Education, 2011.

[29] RG-CE System Protection and Dynamics Sub Group, “Frequencystability evaluation criteria for the synchronous zone of continentaleurope”, ENTSO-E, Tech. Rep., 2016.

[30] T. S. Borsche, T. Liu, and D. J. Hill, “Effects of rotational inertia onpower system damping and frequency transients”, in Proc. IEEEConference on Decision and Control, 2015, pp. 5940–5946.

[31] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control.Prentice Hall, 1996.

[32] E. Lovisari and S. Zampieri, “Performance metrics in the averageconsensus problem: A tutorial”, Annual Reviews in Control, vol. 36,no. 1, pp. 26–41, 2012.

[33] B. Bamieh, M. R. Jovanovic, P. Mitra, and S. Patterson, “Coherencein large-scale networks: Dimension-dependent limitations of localfeedback”, IEEE Transactions on Automatic Control, vol. 57, no. 9,pp. 2235–2249, 2012.

136 bibliography

[34] M. Fardad, F. Lin, and M. R. Jovanovic, “Design of optimal sparseinterconnection graphs for synchronization of oscillator networks”,IEEE Transactions on Automatic Control, vol. 59, no. 9, pp. 2457–2462,2014.

[35] M. Fardad, X. Zhang, F. Lin, and M. R. Jovanovic, “On the prop-erties of optimal weak links in consensus networks”, in Proc. IEEEConference on Decision and Control, 2014, pp. 2124–2129.

[36] T. Summers, I. Shames, J. Lygeros, and F. Dörfler, “Topology designfor optimal network coherence”, in Proc. European Control Conference,2015, pp. 575–580.

[37] M. Siami and N. Motee, “Systemic measures for performance androbustness of large-scale interconnected dynamical networks”, inProc. IEEE Conference on Decision and Control, 2014.

[38] E. Sjödin, B. Bamieh, and D. F. Gayme, “The price of synchrony:Evaluating the resistive losses in synchronizing power networks”,IEEE Transactions on Control of Network Systems, vol. 2, no. 3, pp. 254–266, 2015.

[39] F. Dörfler, M. R. Jovanovic, M. Chertkov, and F. Bullo, “Sparsity-promoting optimal wide-area control of power networks”, IEEETransactions on Power Systems, vol. 29, no. 5, pp. 2281–2291, 2014.

[40] X. Wu, F. Dörfler, and M. R. Jovanovic, “Input-output analysis anddecentralized optimal control of inter-area oscillations in powersystems”, IEEE Transactions on Power Systems, vol. 31, no. 3, pp. 2434–2444, 2016.

[41] F. Dörfler, J. W. Simpson-Porco, and F. Bullo, “Breaking the hierarchy:Distributed control and economic optimality in microgrids”, IEEETransactions on Control of Network Systems, vol. 3, no. 3, pp. 241–253,2016.

[42] P. W. Sauer and M. A. Pai, Power System Dynamics and Stability.Prentice Hall, 1998.

[43] F. Dörfler and F. Bullo, “Kron reduction of graphs with applicationsto electrical networks”, IEEE Transactions on Circuits and Systems I:Regular Papers, vol. 60, no. 1, pp. 150–163, 2013.

[44] Q.-C. Zhong and T. Hornik, Control of Power Inverters in RenewableEnergy and Smart Grid Integration. Wiley-IEEE Press, 2013.

bibliography 137

[45] J. Schiffer, D. Zonetti, R. Ortega, A. Stankovic, T. Sezi, and J. Raisch,“A survey on modeling of microgrids - from fundamental physics tophasors and voltage sources”, Automatica, vol. 74, pp. 135–150, 2016.

[46] J. Schiffer, D. Goldin, J. Raisch, and T. Sezi, “Synchronization ofdroop-controlled autonomous microgrids with distributed rotationaland electronic generation”, in Proc. IEEE Conference on Decision andControl, 2013.

[47] I. A. Hiskens and E. M. Fleming, “Control of inverter-connectedsources in autonomous microgrids”, in Proc. American Control Con-ference, 2008, pp. 586–590.

[48] M. R. Jovanovic and M. Fardad, “H2 norm of linear time-periodic sys-tems: A perturbation analysis”, Automatica, vol. 44, no. 8, pp. 2090–2098, 2008.

[49] E. J. Candès, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity byreweighted `1 minimization”, Journal of Fourier Analysis and Applica-tions, vol. 14, no. 5, pp. 877–905, 2008.

[50] M. R. Jovanovic and N. K. Dhingra, “Controller architectures: Trade-offs between performance and structure”, European Journal of Control,vol. 30, pp. 76–91, 2016.

[51] F. Lin, M. Fardad, and M. R. Jovanovic, “Design of optimal sparsefeedback gains via the alternating direction method of multipliers”,IEEE Transactions on Automatic Control, vol. 58, no. 9, pp. 2426–2431,2013.

[52] Q.-C. Zhong and G. Weiss, “Synchronverters: Inverters that mimicsynchronous generators”, IEEE Transactions on Industrial Electronics,vol. 58, no. 4, pp. 1259–1267, 2011.

[53] C. Arghir, T. Jouini, and F. Dörfler, “Grid-forming control for powerconverters based on matching of synchronous machines”, Automatica,vol. 95, pp. 273–282, 2018.

[54] A. Tayyebi, F. Dörfler, F. Kupzog, Z. Miletic, and W. Hribernik, “Grid-forming converters – inevitability, control strategies and challengesin future grid applications”, in Proc. CIRED Workshop, 2018.

[55] J. Morren, S. de Haan, W. Kling, and J. Ferreira, “Wind turbinesemulating inertia and supporting primary frequency control”, IEEETransactions on Power Systems, vol. 21, no. 1, pp. 433–434, 2006.

138 bibliography

[56] J. A. Taylor, S. V. Dhople, and D. S. Callaway, “Power systems with-out fuel”, Renewable and Sustainable Energy Reviews, vol. 57, pp. 1322–1336, 2016.

[57] G. Denis, T. Prevost, M. S. Debry, F. Xavier, X. Guillaud, and A.Menze, “The migrate project: The challenges of operating a trans-mission grid with only inverter-based generation. a grid-formingcontrol improvement with transient current-limiting control”, IETRenewable Power Generation, vol. 12, no. 5, pp. 523–529, 2018.

[58] D. Groß, S. Bolognani, B. K. Poolla, and F. Dörfler, “Increasingthe resilience of low-inertia power systems by virtual inertia anddamping”, in Proc. IREP Bulk Power System Dynamics and ControlSymposium, 2017.

[59] M. Pirani, E. Hashemi, B. Fidan, and J. W. Simpson-Porco, “H∞performance of mechanical and power networks”, in Proc. IFACWorld Congress, 2017, pp. 5196–5201.

[60] B. K. Poolla, S. Bolognani, and F. Dörfler, “Optimal placement ofvirtual inertia in power grids”, IEEE Transactions on Automatic Control,vol. 62, no. 12, pp. 6209–6220, 2017.

[61] A. Mešanovic, U. Münz, and C. Heyde, “Comparison of H∞, H2,and pole optimization for power system oscillation damping withremote renewable generation”, in Proc. IFAC Workshop on Control ofTransmission and Distribution Smart Grids, 2016, pp. 103–108.

[62] F. Paganini and E. Mallada, “Global performance metrics for syn-chronization of heterogeneously rated power systems: The roleof machine models and inertia”, arXiv:1710.07195 [math.OC], 2017,Available at https://arxiv.org/abs/1710.07195.

[63] T. S. Borsche and F. Dörfler, “On placement of synthetic inertia withexplicit time-domain constraints”, arXiv:1705.03244 [math.OC], 2017,Available at https://arxiv.org/abs/1705.03244.

[64] S.-K. Chung, “A phase tracking system for three phase utility inter-face inverters”, IEEE Transactions on Power Electronics, vol. 15, no. 3,pp. 431–438, 2000.

[65] R. Ofir, U. Markovic, P. Aristidou, and G. Hug, “Droop vs. virtualinertia: Comparison from the perspective of converter operationmode”, in Proc. IEEE International Energy Conference (ENERGYCON),2018.

bibliography 139

[66] M. Ashabani and Y. A. I. Mohamed, “Novel comprehensive controlframework for incorporating VSCs to smart power grids using bidi-rectional synchronous-VSCs”, IEEE Transactions on Power Systems,vol. 29, no. 2, pp. 943–957, 2014.

[67] I. Cvetkovic, D. Boroyevich, R. Burgos, C. Li, and P. Mattavelli,“Modeling and control of grid-connected voltage-source convertersemulating isotropic and anisotropic synchronous machines”, in Proc.IEEE Workshop on Control and Modeling for Power Electronics, 2015.

[68] D. J. Hill and I. M. Y. Mareels, “Stability theory for differen-tial/algebraic systems with application to power systems”, IEEETransactions on Circuits and Systems, vol. 37, no. 11, pp. 1416–1423,1990.

[69] S. Golestan, M. Monfared, F. D. Freijedo, and J. M. Guerrero, “Perfor-mance improvement of a prefiltered synchronous-reference-framePLL by using a PID-type loop filter”, IEEE Transactions on IndustrialElectronics, vol. 61, no. 7, pp. 3469–3479, 2014.

[70] N. Pogaku, M. Prodanovic, and T. C. Green, “Modeling, analysis andtesting of autonomous operation of an inverter-based microgrid”,IEEE Transactions on Power Electronics, vol. 22, no. 2, pp. 613–625,2007.

[71] J. Schiffer, D. Goldin, J. Raisch, and T. Sezi, “Synchronization ofdroop-controlled microgrids with distributed rotational and elec-tronic generation”, in Proc. IEEE Conference on Decision and Control,2013, pp. 2334–2339.

[72] S. Curi, D. Groß, and F. Dörfler, “Control of low-inertia power grids:A model reduction approach”, in Proc. IEEE Conference on Decisionand Control, 2017, pp. 5708–5713.

[73] Nordic Analysis Group, Future System Inertia project, “Future sys-tem inertia”, ENTSO-E, Tech. Rep., 2015.

[74] M. Beck and M. Scherer, “Overview of ancillary services”, SwissgridLtd., Switzerland, Tech. Rep., 2010.

[75] B. K. Poolla, D. Groß, T. S. Borsche, S. Bolognani, and F. Dörfler,“Virtual inertia placement in electric power grids”, in Energy Marketsand Responsive Grids, Springer, 2018, pp. 281–305.

[76] M. Gibbard and D. Vowles, “Simplified 14-generator model of thesouth east australian power system, revision 4”, The University ofAdelaide, South Australia, Tech. Rep., 2014.

140 bibliography

[77] A. Moeini, I. Kamwa, P. Brunelle, and G. Sybille, “Open data IEEEtest systems implemented in SimPowerSystems for education andresearch in power grid dynamics and control”, in Proc. InternationalUniversities Power Engineering Conference (UPEC), 2015.

[78] S. Püschel-Løvengreen and P. Mancarella, “Frequency response con-strained economic dispatch with consideration of generation contin-gency size”, in Proc. Power Systems Computation Conference (PSCC),2018, pp. 1–7.

[79] B. K. Poolla and D. Groß, Australian10gen-lowinertia: Implementa-tion of grid-forming and grid-following virtual inertia, Git repository,https://github.com/bpoolla/Australian10gen-lowinertia, 2018.

[80] A. W. Berger and F. C. Schweppe, “Real time pricing to assist in loadfrequency control”, IEEE Transactions on Power Systems, vol. 4, no. 3,pp. 920–926, 1989.

[81] J. Zhong and K. Bhattacharya, “Frequency linked pricing as aninstrument for frequency regulation in deregulated electricity mar-kets”, in Proc. Power Engineering Society General Meeting, IEEE, vol. 2,2003, pp. 566–571.

[82] J. A. Taylor, A. Nayyar, D. S. Callaway, and K. Poolla, “Consolidateddynamic pricing of power system regulation”, IEEE Transactions onPower Systems, vol. 28, no. 4, pp. 4692–4700, 2013.

[83] T. Tanaka, A. Z. W. Cheng, and C. Langbort, “A dynamic pivotmechanism with application to real time pricing in power systems”,in Proc. American Control Conference, 2012, pp. 3705–3711.

[84] W. Tang and R. Jain, “Dynamic economic dispatch game: The valueof storage”, IEEE Transactions on Smart Grid, vol. 7, no. 5, pp. 2350–2358, 2016.

[85] ——, “Market mechanisms for buying random wind”, IEEE Transac-tions on Sustainable Energy, vol. 6, no. 4, pp. 1615–1622, 2015.

[86] W. Lin and E. Bitar, “A structural characterization of market powerin power markets”, arXiv:1709.09302 [math.OC], 2017, Available athttps://arxiv.org/abs/1709.09302.

[87] P. Milgrom, “Putting auction theory to work: The simultaneous as-cending auction”, Journal of Political Economy, vol. 108, no. 2, pp. 245–272, 2000.

[88] W. Vickrey, “Counterspeculation, auctions, and competitive sealedtenders”, Journal of Finance, vol. 16, no. 1, pp. 8–37, 1961.

bibliography 141

[89] T. Groves, “Incentives in teams”, Econometrica: Journal of the Econo-metric Society, pp. 617–631, 1973.

[90] P. G. Sessa, N. Walton, and M. Kamgarpour, “Exploring the vickrey-clarke-groves mechanism for electricity markets”, IFAC-PapersOnLine,vol. 50, no. 1, pp. 189–194, 2017.

[91] Y. Xu and S. H. Low, “An efficient and incentive compatible mecha-nism for wholesale electricity markets”, IEEE Transactions on SmartGrid, vol. 8, no. 1, pp. 128–138, 2017.

[92] W. Tang and R. Jain, “Aggregating correlated wind power with fullsurplus extraction”, IEEE Transactions on Smart Grid, 2017.

[93] M. Pirani, J. W. Simpson-Porco, and B. Fidan, “System-theoreticperformance metrics for low-inertia stability of power networks”, inProc. IEEE Conference on Decision and Control, 2017, pp. 5106–5111.

[94] U. Münz, A. Mešanovic, M. Metzger, and P. Wolfrum, “Robust opti-mal dispatch, secondary, and primary reserve allocation for powersystems with uncertain load and generation”, IEEE Transactions onControl Systems Technology, pp. 475–485, 2017.

[95] H. Thiesen, C. Jauch, and A. Gloe, “Design of a system substitutingtoday’s inherent inertia in the european continental synchronousarea”, Energies, vol. 9, no. 8, 2016.

[96] T. Basar and G. J. Olsder, Dynamic noncooperative game theory, 1998.

[97] T. Kose, “Solutions of saddle value problems by differential equa-tions”, Econometrica, vol. 24, no. 1, pp. 59–70, 1956.

[98] K. Arrow, L. Hurwicz, and H. Uzawa, Studies in linear and non-linearprogramming. Stanford University Press, 2006.

[99] D. Feijer and F. Paganini, “Stability of primal–dual gradient dynam-ics and applications to network optimization”, Automatica, vol. 46,no. 12, pp. 1974–1981, 2010.

[100] J. Wang and N. Elia, “A control perspective for centralized anddistributed convex optimization”, in Proc. IEEE Conference on Decisionand Control and European Control Conference, 2011, pp. 3800–3805.

[101] A. Cherukuri, E. Mallada, and J. Cortés, “Asymptotic convergenceof constrained primal–dual dynamics”, Systems and Control Letters,vol. 87, pp. 10–15, 2016.

[102] J. Machowski, J. Bialek, and J. Bumby, Power system dynamics: stabilityand control. John Wiley & Sons, 2011.

142 bibliography

[103] C. Zhao, U. Topcu, N. Li, and S. Low, “Design and stability of load-side primary frequency control in power systems”, IEEE Transactionson Automatic Control, vol. 59, no. 5, pp. 1177–1189, 2014.

[104] A. Cherukuri and J. Cortés, “Initialization-free distributed coordi-nation for economic dispatch under varying loads and generatorcommitment”, Automatica, vol. 74, pp. 183–193, 2016.

[105] E. Dall’Anese, S. V. Dhople, and G. B. Giannakis, “Regulation ofdynamical systems to optimal solutions of semidefinite programs:Algorithms and applications to ac optimal power flow”, in Proc.American Control Conference, Chicago, IL, USA, 2015, pp. 2087–2092.

[106] K. C. Kosaraju, S. Mohan, and R. Pasumarthy, “On the primal-dualdynamics of support vector machines”, arXiv:1805.00699 [math.OC],2018, Available at https://arxiv.org/abs/1805.00699.

[107] D. Ding and M. R. Jovanovic, “A primal-dual Laplacian gradientflow dynamics for distributed resource allocation problems”, in Proc.American Control Conference, Milwaukee, WI, USA, 2018, pp. 5316–5320.

[108] B. Gharesifard and J. Cortes, “Distributed continuous-time convexoptimization on weight-balanced digraphs”, IEEE Transactions onAutomatic Control, vol. 59, no. 3, pp. 781–786, 2014.

[109] N. K. Dhingra, S. Z. Khong, and M. R. Jovanovic, “A second orderprimal-dual method for nonsmooth convex composite optimization”,in Proc. IEEE Conference on Decision and Control, 2017, pp. 2868–2873.

[110] S. H. Low and D. E. Lapsey, “Optimization flow control I: Basicalgorithm and convergence”, IEEE/ACM Transactions on Networking,vol. 7, no. 6, pp. 861–874, 1999.

[111] J. T. Wen and M. Arcak, “A unifying passivity framework for net-work flow control”, IEEE Transactions on Automatic Control, vol. 49,no. 2, pp. 162–174, 2004.

[112] J. Wang and N. Elia, “Control approach to distributed optimization”,in Proc. Allerton Conference on Communications, Control and Computing,Monticello, IL, USA, 2010, pp. 557–561.

[113] G. Droge, H. Kawashima, and M. B. Egerstedt, “Continuous-timeproportional-integral distributed optimisation for networked sys-tems”, Journal of Control and Decision, vol. 1, no. 3, pp. 191–213, 2014.

bibliography 143

[114] G. Droge and M. Egerstedt, “Proportional integral distributed opti-mization for dynamic network topologies”, in Proc. American ControlConference, Portland, OR, USA, 2014, pp. 3621–3626.

[115] T. Hatanaka, N. Chopra, T. Ishizaki, and N. Li, Passivity-based dis-tributed optimization with communication delays using pi consensus algo-rithm, 2018.

[116] H. D. Nguyen, T. L. Vu, K. Turitsyn, and J. Slotine, “Contraction androbustness of continuous time primal-dual dynamics”, IEEE ControlSystems Letters, vol. 2, no. 4, pp. 755–760, 2018.

[117] H. Mohammadi, M. Razaviyayn, and M. R. Jovanovic, “Varianceamplification of accelerated first-order algorithms for strongly con-vex quadratic optimization problems”, in Proc. IEEE Conference onDecision and Control, 2018, pp. 5753–5758.

[118] A. Cherukuri, E. Mallada, S. Low, and J. Cortés, “The role of strongconvexity-concavity in the convergence and robustness of the saddle-point dynamics”, in Proc. Allerton Conference on Communications,Control and Computing, Monticello, IL, USA, 2016, pp. 504–510.

[119] ——, “The role of convexity on saddle-point dynamics: Lyapunovfunction and robustness”, IEEE Transactions on Automatic Control,vol. 63, no. 8, pp. 2449–2464, 2018.

[120] J. W. Simpson-Porco, “Input/output analysis of primal-dual gradientalgorithms”, in Proc. Allerton Conference on Communications, Controland Computing, 2016, pp. 219–224.

[121] L. Lessard, B. Recht, and A. Packard, “Analysis and design of opti-mization algorithms via integral quadratic constraints”, SIAM Journalon Optimization, vol. 26, no. 1, pp. 57–95, 2016.

[122] B. Hu and L. Lessard, “Control interpretations for first-order opti-mization methods”, in Proc. American Control Conference, Seattle, WA,USA, 2017, pp. 3114–3119.

[123] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge Uni-versity Press, 2004.

[124] M. Benzi, G. H. Golub, and J. Liesen, “Numerical solution of saddlepoint problems”, Acta Numerica, vol. 14, pp. 1–137, 2005.

[125] A. Nedic and A. Ozdaglar, “Subgradient methods for saddle-pointproblems”, Journal of Optimization Theory and Applications, vol. 142,no. 1, pp. 205–228, 2009.

144 bibliography

[126] G. E. Dullerud and F. Paganini, A Course in Robust Control Theory,ser. Texts in Applied Mathematics 36. Springer, 2000.

[127] M. B. Khuzani and N. Li, “Distributed regularized primal-dualmethod: Convergence analysis and trade-offs”, arXiv:1609.08262[math.OC], 2016, Available at https://arxiv.org/abs/1609.08262.

[128] A. Simonetto and G. Leus, “Double smoothing for time-varyingdistributed multiuser optimization”, in Proc. IEEE Global Conferenceon Signal and Information Processing, Atlanta, GA, USA, 2014, pp. 852–856.

[129] E. Dall’Anese and A. Simonetto, “Optimal power flow pursuit”,IEEE Transactions on Smart Grid, vol. 9, no. 2, pp. 942–952, 2018.

[130] D. Bertsekas, Nonlinear Programming, 2nd ed. Athena Scientific, 1995.

[131] F. Lin, M. Fardad, and M. R. Jovanovic, “Augmented Lagrangianapproach to design of structured optimal state feedback gains”, IEEETransactions on Automatic Control, vol. 56, no. 12, pp. 2923–2929, 2011.

[132] N. K. Dhingra, S. Z. Khong, and M. R. Jovanovic, “The proximal aug-mented Lagrangian method for nonsmooth composite optimization”,IEEE Transactions on Automatic Control, 2018, In Press.

[133] F. Bullo, Lectures on Network Systems. Version 0.85, 2016, With contri-butions by J. Cortés, F. Dörfler, and S. Martinez.

[134] A. R. Bergen, Power Systems Analysis. Pearson Education India, 2009.

[135] F. Dörfler and S. Grammatico, “Gather-and-broadcast frequencycontrol in power systems”, Automatica, vol. 79, pp. 296–305, 2017.

[136] D. Zelazo and M. Mesbahi, “Edge agreement: Graph-theoretic per-formance bounds and passivity analysis”, IEEE Transactions on Auto-matic Control, vol. 56, no. 3, pp. 544–555, 2011.

[137] T. Rautert and E. W. Sachs, “Computational design of optimal outputfeedback controllers”, SIAM Journal on Optimization, vol. 7, no. 3,pp. 837–852, 1997.

P U B L I C AT I O N S

journal articles :

[1] B. K. Poolla, S. Bolognani, and F. Dörfler, “Optimal placement ofvirtual inertia in power grids”, IEEE Transactions on Automatic Control,vol. 62, no. 12, pp. 6209–6220, 2017.

[2] B. K. Poolla, D. Groß, and F. Dörfler, “Placement and implementationof grid-forming and grid-following virtual inertia”, IEEE Transactionson Power Systems, 2019, In Press.

[3] J. W. Simpson-Porco, B. K. Poolla, N. Monshizadeh, and F. Dörfler,“Input-output performance of linear-quadratic saddle-point algo-rithms with application to distributed resource allocation problems”,IEEE Transactions on Automatic Control, To Appear.

book chapters :

[1] B. K. Poolla, D. Groß, T. S. Borsche, S. Bolognani, and F. Dörfler,“Virtual inertia placement in electric power grids”, in Energy Marketsand Responsive Grids, S. Meyn, T. Samad, I. Hiskens, and J. Stoustroup,Eds., vol. 162, Springer, 2018, pp. 281–305.

conference proceedings :

[1] B. K. Poolla, S. Bolognani, and F. Dörfler, “Placing rotational inertiain power grids”, in Proc. American Control Conference, 2016, pp. 2314–2320.

[2] J. W. Simpson-Porco, B. K. Poolla, N. Monshizadeh, and F. Dörfler,“Quadratic performance of primal-dual methods with applicationto secondary frequency control of power systems”, in Proc. IEEEControl and Decision Conference, 2016, pp. 1840–1845.

[3] D. Groß, S. Bolognani, B. K. Poolla, and F. Dörfler, “Increasingthe resilience of low-inertia power systems by virtual inertia anddamping”, in Proc. IREP Bulk Power System Dynamics and ControlSymposium, 2017.

bibliography

preprints/under review :

[1] B. K. Poolla, S. Bolognani, L. Na, and F. Dörfler, “A market mecha-nism for virtual inertia”, Available at https://arxiv.org/abs/1711.04874.

[2] B. K. Poolla, J. W. Simpson-Porco, N. Monshizadeh, and F. Dör-fler, “Quadratic performance analysis of secondary frequency con-trollers”, Under Review.

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