(Advances in industrial control) giuseppe fusco adaptive voltage control in power systems -...

Advances in Industrial Control

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Giuseppe Fusco and Mario Russo

Adaptive VoltageControl inPower SystemsModeling, Design and Applications

With 79 Figures

123

Giuseppe Fusco, Dr. Eng.Mario Russo, Dr. Eng.

Università degli Studi di CassinoFacoltà di Ingegneriavia G. Di Biasio, 4303043 Cassino (FR)Italy

British Library Cataloguing in Publication DataFusco, Giuseppe

Adaptive voltage control in power systems : modeling,design and applications. - (Advances in industrial control)1. Voltage regulators 2. Adaptive control systems3. Electric power system stability 4. Electric powersystems - Mathematical modelsI. Title II. Russo, Mario621.3’1

ISBN-13: 9781846285646ISBN-10: 184628564X

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Advances in Industrial Control

Series Editors

Professor Michael J. Grimble, Professor of Industrial Systems and DirectorProfessor Michael A. Johnson, Professor (Emeritus) of Control Systemsand Deputy Director

Industrial Control CentreDepartment of Electronic and Electrical EngineeringUniversity of StrathclydeGraham Hills Building50 George StreetGlasgow G1 1QEUnited Kingdom

Series Advisory Board

Professor E.F. CamachoEscuela Superior de IngenierosUniversidad de SevillaCamino de los Descobrimientos s/n41092 SevillaSpain

Professor S. EngellLehrstuhl für AnlagensteuerungstechnikFachbereich ChemietechnikUniversität Dortmund44221 DortmundGermany

Professor G. GoodwinDepartment of Electrical and Computer EngineeringThe University of NewcastleCallaghanNSW 2308Australia

Professor T.J. HarrisDepartment of Chemical EngineeringQueen’s UniversityKingston, OntarioK7L 3N6Canada

Professor T.H. LeeDepartment of Electrical EngineeringNational University of Singapore4 Engineering Drive 3Singapore 117576

Professor Emeritus O.P. MalikDepartment of Electrical and Computer EngineeringUniversity of Calgary2500, University Drive, NWCalgaryAlbertaT2N 1N4Canada

Professor K.-F. ManElectronic Engineering DepartmentCity University of Hong KongTat Chee AvenueKowloonHong Kong

Professor G. OlssonDepartment of Industrial Electrical Engineering and AutomationLund Institute of TechnologyBox 118S-221 00 LundSweden

Professor A. RayPennsylvania State UniversityDepartment of Mechanical Engineering0329 Reber BuildingUniversity ParkPA 16802USA

Professor D.E. SeborgChemical Engineering3335 Engineering IIUniversity of California Santa BarbaraSanta BarbaraCA 93106USA

Doctor K.K. TanDepartment of Electrical EngineeringNational University of Singapore4 Engineering Drive 3Singapore 117576

Professor Ikuo YamamotoKyushu University Graduate SchoolMarine Technology Research and Development ProgramMARITEC, Headquarters, JAMSTEC2-15 Natsushima YokosukaKanagawa 237-0061Japan

To my mother Anna andin memory of my father Biagio

To Assunta, Flavia, Raffaele,Francesca and Luigi

Series Editors’ Foreword

The series Advances in Industrial Control aims to report and encourage tech-nology transfer in control engineering. The rapid development of control tech-nology has an impact on all areas of the control discipline. New theory, newcontrollers, actuators, sensors, new industrial processes, computer methods,new applications, new philosophies, new challenges. Much of this developmentwork resides in industrial reports, feasibility study papers and the reportsof advanced collaborative projects. The series offers an opportunity for re-searchers to present an extended exposition of such new work in all aspectsof industrial control for wider and rapid dissemination.

Since the Advances in Industrial Control monograph series began, it hasfeatured a consistent sequence of contributions on the topics of power systems,power generation and related topics. During the 1990s combined cycle powerplants found favour and the monograph due to Andrzej Ordys and others,Modelling and Simulation of Power Generation Plants (ISBN 3-540-19907-1,1994), was a valuable contribution in that area.

From the mid-1990s onwards, many in the power systems field have soughtthe best ways to control the power system given the new deregulated commer-cial regimes emerging in many national systems. The monograph HierarchicalPower Systems Control (ISBN 3-540-76031-8, 1996) by Marija Ilic and ShellLiu outlined a classical structure decomposition of the power system basedlargely on physical properties of the network. The methods outlined by Ilicand Liu formed a common paradigm in many national power system networks.The state of the art in power system control technology was then nicely cap-tured in a pair of Advances in Industrial Control monographs authored byEzio Mariani and Surabhi Murthy. These were Control of Modern IntegratedPower Systems (ISBN 3-540-76168-5, 1997) and Advanced Load Dispatch forPower Systems (ISBN 3-540-76167-5, 1997). The load dispatch monographwas concerned with top-level optimisation and the effective economic opera-tion of the power system. This top-level control theme was also part of thesubject of the monograph Price-based Commitment Decisions in the Electric-ity Market (ISBN 1-85233-069-4, 1998) by Eric Allen and Ilic that considered

x Series Editors’ Foreword

some of the impacts of commercial deregulation on national electricity mar-kets.

In the new millennium, the power system and power generation mono-graphs being published again reflect current preoccupations in the field. Al-ternative and renewable energy sources are topical and the series recentlypublished Control of Fuel Cell Power Systems (ISBN 1-85233-816-4, 2004) byJay Pukrushpan, Anna Stefanopoulou and Huei Peng and two monographson wind turbine systems will appear this year and next year.

A different industrial concern is that of how to obtain the best performancefrom existing installed plant. This is a generic question that applies acrossthe whole canopy of industrial activity and usually different industries adoptdifferent control solutions at different rates. In the power generation area,Doris Saez, Aldo Cipriano and Ordys. used model-based predictive controlmethods to improve power plant supervisory control and their experience andproposals were reported in the series as Optimisation of Industrial Processesat Supervisory Level (ISBN 1-85233-386-3, 2001).

For the enhanced control of voltage within the power system network,Giuseppe Fusco and Mario Russo report on their experience with adaptivecontrol techniques in this monograph. It is surprising to reflect that the tech-niques of adaptive control had their origins in the 1960s, were given a boost inthe mid-1970s with the self-tuning control method and have since taken a lowprofile in the toolbox of controller design methods; yet these methods are easyto understand and are supported by a useful body of theory. The difficultymight be that although these methods are prescriptive, engineering input andconfidence is needed to ensure success. Fusco and Russo demonstrate the ex-perience and confidence that are needed to apply adaptive methods to thecontrol of nodal voltage in typical power system networks. The reader willfind chapters on the physical context of the voltage control problem and themodelling of the system. A key step in the adaptive control method is pa-rameter identification and the volume has a chapter devoted to this task.This is followed by three chapters describing and verifying the designs forself-tuning regulators, model-reference adaptive regulators and finally tech-niques for adaptive nonlinear compensation. The value of this monograph tothe control engineer working in the power distribution industry does not needstating. However, the industrial control engineer and the academic working inthe control field will also find valuable applications experience in this welcomeaddition to the Advances in Industrial Control series.

M.J. Grimble and M.A. JohnsonGlasgow, Scotland, U.K.

Preface

This monograph aims at giving an extensive treatment of modeling issues,design methodologies and implementation aspects, arising when the adaptivecontrol theory is applied to nodal voltage control in power systems.

The monograph is structured into six chapters and an appendix. The firstthree chapters introduce the reader to the voltage control problem and thesubsequent three chapters illustrate different design methodologies. In par-ticular the first chapter describes the power system control problem and thevoltage control structures adopted in transmission, distribution and industrialelectrical systems. The second chapter presents power system models that canbe usefully employed in voltage control design. The third chapter studies theproperties of the Kalman filter used in the estimation of voltage and currentphasors. The subsequent two chapters deal with the discrete-time design of,respectively, self-tuning and model-reference adaptive voltage controllers. Thesixth chapter illustrates an adaptive design that is quite different from thosepresented in the two previous chapters, being based on an adaptive mechanismcompensating the power system model nonlinearities. Finally, the appendixdescribes the high-voltage transmission system and industrial network usedin the simulations reported in the monograph.

The authors wish to thank Professor Arturo Losi who has actively con-tributed to the start-up of the research activities. Finally, the authors aregrateful to all readers who will kindly contribute with their comments to fur-ther enhance of this research topic.

Cassino, Italy, Giuseppe FuscoSeptember 2006 Mario Russo

Contents

1 The Voltage Control Problem in Power Systems . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Power System Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Voltage Control in HV Transmission Systems . . . . . . . . . . . . . . . 41.4 Voltage Control in MV and LV Systems . . . . . . . . . . . . . . . . . . . . 6

1.4.1 Voltage Control at Fundamental Frequency . . . . . . . . . . . 71.4.2 Voltage Harmonic Distortion Containment . . . . . . . . . . . . 8

2 System Modeling for Nodal Voltage Regulator Design . . . . . 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Voltage Control Device Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Synchronous Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Static VAr Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 Active and Hybrid Shunt Filters . . . . . . . . . . . . . . . . . . . . . 19

2.3 Power System Equivalent Models . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Frequency Domain Models . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.2 Time Domain Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Voltage and Current Phasor Identification . . . . . . . . . . . . . . . . . 273.1 Techniques Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 Off-line Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.2 On-line Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Kalman Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.1 State-space Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.2 Estimation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.3 Convergence and Stability Properties . . . . . . . . . . . . . . . . 32

4 Self-tuning Voltage Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Indirect Self-tuning Voltage Regulator Design . . . . . . . . . . . . . . . 38

4.2.1 Recursive Least-squares Algorithm . . . . . . . . . . . . . . . . . . 39

xiv Contents

4.2.2 Pole-assignment Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.3 Pole-shifting Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.4 Generalized Minimum Variance Pole-assignment Design 484.2.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Direct Self-tuning Voltage Regulator Design . . . . . . . . . . . . . . . . 634.3.1 Pole-assignment Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.2 Generalized Minimum Variance Pole-assignment Design 724.3.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4 Properties of the Recursive Least-squares Algorithm . . . . . . . . . 78

5 Model-reference Adaptive Voltage Regulators . . . . . . . . . . . . . 875.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.2 Direct Model-reference Adaptive Voltage Regulator Design . . . 88

5.2.1 Model-reference Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2.2 Adaptive Law Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.2.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.3 Indirect Model-reference Adaptive Voltage Regulator Design . . 975.4 Properties of the Adaptive Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4.1 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.4.2 Robustness Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6 Adaptive Nonlinearities Compensation Technique . . . . . . . . . 1096.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2 Thevenin Circuit Parameters Estimation . . . . . . . . . . . . . . . . . . . 1106.3 Adaptive Voltage Regulator Design . . . . . . . . . . . . . . . . . . . . . . . . 114

6.3.1 Synchronous Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.3.2 Static VAr Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.3.3 Active and Hybrid Shunt Filters . . . . . . . . . . . . . . . . . . . . . 1236.3.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.4 Optimization Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

A Computer Models and Topology of Networks . . . . . . . . . . . . . . 141A.1 High-voltage Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

A.1.1 Computer Models of Components . . . . . . . . . . . . . . . . . . . 141A.1.2 Simulated Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147A.1.3 Network Equivalent Time Domain Model . . . . . . . . . . . . . 148

A.2 Industrial Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

1

The Voltage Control Problem in PowerSystems

1.1 Introduction

Modern electric power systems are large-scale systems with a complex struc-ture comprised of meshed and interconnected networks to guarantee adequateload supply. Power systems are continuously subject to unpredictable and sud-den operating point variations due to modifications of the topology, changesof generation and fluctuations of loads. Moreover environmental constraintsstrongly limit the possibilities of expansion and may reduce the capabilityof power systems, which are consequently operated near their technical andsafety limits. The aim of management and control is then to plan, coordinateand quickly perform suitable and effective actions on the system with respectto its limits.

Power system operation and control problems have been managed in thelast century by monopolistic structures in which a vertically-integrated utilityowned, managed and operated the whole electrical sector, from generation,through transmission, to distribution. In monopolistic structures there wasno clear separation between power plant management and power system op-eration. Nowadays, the electric power industry is moving towards deregulationall over the world: power generation is now being organized into a competitivemarket structure while distribution networks are operated locally by distribu-tion utilities which are independent from the regional/national transmissionsystem. Then, the adequacy of load supply is the result of the actions ofvarious operators at transmission and distribution levels.

The transmission system operation is assigned to an Independent SystemOperator (ISO) whose tasks may differ according to the chosen market struc-ture. The ISO’s main mission is to assure power system secure operation. Tothis aim, the ISO requires power plants to provide some services, the so-calledancillary services, to the transmission system. In particular, to support ade-quate system voltage profile and voltage stability, the ISO asks power plantsfor voltage/reactive power regulation services.

2 1 The Voltage Control Problem in Power Systems

At distribution level and in industrial electric systems, the main problemis assuring power quality while subject to a wide range of electromagneticphenomena. Among others, voltage regulation at fundamental frequency andharmonic distortion containment are particularly important issues, aiming atensuring a sinusoidal voltage waveform with adequate amplitude at all systemnodes. Their importance is due to the increasing number of power electronicequipments that are used to control the loads causing harmonic distortion,and to the high sensitivity of loads to voltage waveform characteristics. More-over, a new issue is arising in distribution systems: this concerns the impact onpower quality of dispersed (or embedded) generation, that is the connectionof small-sized energy conversion systems, mainly exploiting renewable energysources, to medium or low voltage networks. When comparing energy produc-tion costs, dispersed generation is not competitive compared with generationof large power plants connected to the transmission network. Nevertheless,environmental policies push for the exploitation of renewable energy sources.From the technical point of view, dispersed generation has a significant im-pact on distribution system operation and management. Always referring tothe problems of voltage regulation and harmonic distortion containment, themain issue is then to guarantee an adequate voltage amplitude and waveformall over the network while the dispersed generation varies randomly. This ob-jective should be achieved without altering the present structure of the voltageand reactive power control in distribution systems.

1.2 Power System Control

Electric power system control is a complex problem for several reasons: themulti-objective nature of the problem; the huge number of variables; the wideextent of the system; the nonlinearities that are present; and the unknownand unpredictable variations of the operating conditions.

The multi-objective nature is implicit in the strategic objective of theoverall control action: supplying loads with adequate levels of quality whileoptimizing the economic issues of system management. Indeed, the specifica-tion of what the terms “adequate” and “economic issues” practically meandepends on the power system operating status. Traditionally, three operatingstates are distinguished [41]: preventive, most commonly addressed as normal,emergency and restorative. Focusing on normal operation of the power system,economics and power quality are often contrasting objectives. In the previ-ous monopolistic structure the resulting contrast was coped with inside thesystem utility which operated the whole power system. Nowadays, in deregu-lated environments, the resulting contrast is managed by introducing variousoperators that care for different objectives [70]. The economic issues are oftenleft to market competition, which is claimed to increase economic efficiency ofthe electric power generation, and to environmental policies, which often en-courage dispersed generation using renewable energy, in spite of its high costs.

1.2 Power System Control 3

The power quality issues are left to the rules introduced by system operators,which must guarantee adequate supply service to customers.

The complexity caused by the huge number of variables and the wideextent of the system is traditionally managed with a decomposition-basedapproach. The overall control problem is split into a number of subproblems ofsmaller dimensions, which are easier to solve. The main issue is then to accountfor the overlapping among the various subproblems [111]. The problem istypically decomposed according to the following main criteria.

– Voltage level/geographical criteria: separating the high-voltage (HV) trans-mission system control problem at national/state level from medium-voltage (MV) and low-voltage (LV) distribution or industrial system con-trol problems referred to a limited area. This decomposition is reflectedin the organization structure, which is assigned the achievement of therequired power quality objective. The system operators are organized inISOs at national/regional transmission network level and distribution util-ities at limited-area distribution network level.

– Time-horizon criteria: separating different control subproblems accordingto the different system dynamics that are accounted for. For instance,when a fault occurs, the protection system problem faces electric transientsin a time horizon of a few periods of the fundamental frequency (short-term dynamics), the stability problem faces electromechanical transientsof electrically-nearby synchronous generators in a time horizon of about asecond (medium-term dynamics), the system voltage regulation problemfaces voltage profile recovery in several seconds (long-term dynamics).

– Decoupled variable criteria: separating control subproblems, which areweakly coupled due to the peculiar characteristics of power systems. Themost typical case is separation between the voltage/reactive power controlproblem and the active power flows in the transmission system. Such adecomposition is related to the peculiar feature of HV networks, in whichvoltage amplitudes are strongly dependent on reactive power flows andweakly coupled to active power flows.

The presence of nonlinearities is typically faced by separating the con-trol problem of response to small perturbations from the problem of responseto large perturbations. When accounting for small perturbations linear con-trol techniques can be adopted, although it must be kept in mind that powersystem nonlinearities are always present. Concerning large perturbations, non-linear control theory has been applied, for instance to design power systemstabilizers to prevent loss of stability in presence of faults, see among oth-ers [21,32,53,102,122,123,129].

Finally, the issue of system operating condition variations has to be facedadopting adequate control system design techniques that account for changesof the plant. An adaptive approach can be adopted [48–51] and it is the focusof this monograph.


The complexity of the power system control problem has been tackled bynon-conventional approaches. The fuzzy logic technique has been successfullyemployed both in power system stabilization with respect to large perturba-tions [105, 110, 118] and in the voltage control problem in transmission anddistribution networks [25, 26, 28, 35, 88, 106, 108]. The application of such atheory finds its motivation in two main items: the first is that fuzzy logic de-sign is not based on the mathematical model of the system; the second is thatnonlinear control laws can be coded in a natural way into fuzzy rules. Alsoexpert systems have been useful in dealing with complex large-scale intercon-nected power system as shown in [13, 14, 24, 90, 101], while the application oftechniques based on the artificial humoral immune response to the voltagecontrol can be found in [124,125].

The voltage control subproblem in normal operation of the power systemis briefly recalled in the following. The aim is to highlight the specific focusof this monograph, which is limited to the application of adaptive controltechniques to the following two subproblems:

– local nodal voltage control of small perturbations in HV transmission sys-tems,

– local nodal voltage control at fundamental and harmonic frequencies inMV and LV networks.

The former is defined in terms of the general hierarchical voltage controlstructure based on time-horizon/geographical separations, typically adoptedin HV transmission systems; the latter in the framework of the general problemof power quality assurance to the customers, which is an objective in planningand operation of MV and LV networks.

1.3 Voltage Control in HV Transmission Systems

As recalled in Sections 1.1 and 1.2, HV transmission systems are operatedby ISOs whose main mission is to assure power system secure operation. Tosupport adequate voltage profile throughout the system, the ISO directly op-erates some voltage control devices and requires from synchronous generatorsof power plants, voltage/reactive power regulation services.

Various structures for system voltage/reactive power regulation have beenadopted all over the world [30, 65, 128]. Control architectures vary accordingto the degree of centralization versus decentralization, and control implemen-tations vary according to the degree of automation versus manual operation.In most cases, control is organized in a three-level hierarchy [34,69,120]:

– the primary control level, which is based on local voltage regulation;– the secondary control level, which is based on regional/area voltage regu-

lation (RVR);

1.3 Voltage Control in HV Transmission Systems 5

– the tertiary control level, which is system centralized.

The objective of the primary control level is local control action: consid-ering a single node, the busbar voltage amplitude can be controlled to followa reference signal by acting on the reactive power injection at that node.In theory, voltage amplitude should be regulated at all nodes, but applyingdirect nodal voltage control at all system busbars is impracticable and un-economical. Direct nodal voltage control is performed at generation nodes bysynchronous generators and at some key nodes of the transmission systemby synchronous and static compensators, such as static VAr systems (SVS).Such devices rapidly vary their reactive power injection to control the voltageamplitude of the busbar at which they are connected following a referencesignal, which is determined by secondary voltage regulation.

The objective of the secondary control level is to control the voltage profilein an area or region according to a reference voltage profile. In practice, theRVR does not directly monitor the voltage amplitude at all nodes but achievesits objective by keeping at assigned values the voltage amplitude of someimportant nodes, named pilot nodes, which are in some way representative ofthe voltage profile of an area or a region [71, 98]. On the basis of the voltagevariations at the pilot nodes, the RVR generates reference signals for theprimary control level as well as for some other voltage control devices, suchas on-load tap changers (OLTC) of transformers, which are too slow for usein primary control action.

Finally, the objective of the tertiary control level is system-wide optimaloperation with respect to such objectives as maximizing voltage stability mar-gins, security indices or reactive power generation margins, or minimizingtransmission losses [34, 69, 87, 120, 121]. The optimal voltage profiles of thewhole system are determined at this level and sent to the RVRs as referencesfor the secondary control level. In addition, the connection/disconnection ofcapacitor banks is often programmed at this level on a daily basis.

With reference to the various aspect of control problem decomposition inpower systems described in Section 1.2, it is important to recall the followingconsiderations regarding the voltage control problem.

– The decoupling among the voltage control actions of the three levels isguaranteed by time-horizon decomposition. In fact, the primary controlresponse is as fast as possible (less then 1 s), the secondary control actionis in the time-horizon of 10−100 s and the optimization at tertiary controllevel is performed with a time period over of 5−60 minutes.

– The voltage control structure responds to small system perturbations; inthe case of large perturbations, such as electric faults, different actionsare studied and implemented. Small perturbations are introduced by vari-ations of loads and generations, changes of the network topology due toline connections and disconnections. In addition, primary voltage control,which responds very rapidly, may be involved in rejecting some slow tran-sient phenomena, such as voltage amplitude fluctuations caused by specific


loads (e.g. arc furnaces), or in improving the damping of electromechanicaloscillations and the small signal stability of synchronous generators [78,92].In both cases, some additional time-varying reference signals are addedto the RVR reference signal sent to primary voltage controllers of nearbycompensators. Consequently, the primary control action is not only voltageregulation at reference values but also tracking a time-varying referencesignal.

– The problem of voltage unbalances and/or distortions is usually negligi-ble in HV transmission systems. When specific loads or apparatus, suchas converters used in HV direct current transmission, may cause distor-tion, local filters are connected and controlled to avoid voltage waveformdistortion.

In conclusion, the voltage control methodologies analyzed in this mono-graph can be applied to the primary voltage control problem in HV trans-mission systems, aiming at regulating nodal voltage and tracking additionalreference signals to counteract slow transient phenomena and to damp elec-tromechanical oscillations.

1.4 Voltage Control in MV and LV Systems

As recalled in Sections 1.1 and 1.2, MV and LV systems are operated by dis-tribution utilities which are obliged to guarantee power quality to customers.A similar problem is the case of industrial electric systems, which are operatedby the industry itself to satisfy specific load requirements.

Power quality assurance aims at assuring an adequate supply for the op-eration of electric equipment connected to the network. Consequently, such aconcept is dependent on the sensitivity of the specific electric equipment thatis considered and power quality is not an absolute objective [16].

The term “power quality” usually refers to a wide variety of electromag-netic phenomena that characterize the voltage and current at a given timeand at a given location on the power system. It is then useful to classify theelectromagnetic phenomena into seven main categories [67]:

1. Transients2. Short duration variations3. Long duration variations4. Voltage unbalance5. Waveform distortion6. Voltage fluctuations7. Power frequency variations.

This monograph focuses on small perturbations and disturbances relatedto the steady-state voltage waveform. From this viewpoint, voltage control inMV and LV networks aims at counteracting mainly:

1.4 Voltage Control in MV and LV Systems 7

– type 3 disturbances, in particular, preventing undervoltages and overvolt-ages, by voltage regulation at fundamental frequency;

– type 5 disturbances, in particular, containing voltage harmonic distortion,by filtering actions.

The voltage control problems associated with these two issues and therelated control structures are briefly recalled in the following two subsections.Obviously, other disturbances, in particular those classified as types 4 and6, can be coped with by extending the results obtained in this monograph.For example, voltage fluctuations due to peculiar loads can be counteractedby sending appropriate additional reference signals to voltage control devices,such as static compensators; phase voltage waveforms can be balanced bycontrolling separately the three phases of a voltage control device.

1.4.1 Voltage Control at Fundamental Frequency

Overvoltages can be the result of load switching (e.g. switching off a largeload), or variations in the reactive compensation on the system (e.g. switch-ing on a capacitor bank). Undervoltages are the result of events that are thereverse of the events that cause overvoltages. A load switching on, or a ca-pacitor bank switching off, can cause an undervoltage until voltage regulationequipment on the system can bring the voltage back to within tolerances. Alsooverloaded circuits can result in undervoltages [67].

Improving nodal voltage profiles in MV networks is usually attained bychanging the HV/MV substation transformer ratio using the OLTC and byconnecting/disconnecting capacitors in the substation and along the feedersof the distribution system. In some cases, power electronic static compen-sators are introduced to cope with particular problems, such as peculiar loadswith large variations of absorbed reactive power. In LV networks the MV/LVsubstation transformer is not equipped with an OLTC and very rarely, directcontrol devices are present.

The voltage regulation problem in MV networks is traditionally split intotwo hierarchical levels: the off-line optimal setting problem and the on-linecontrol problem [64].

The off-line problem determines, typically on a daily schedule, the opti-mal settings for the on-line voltage control reference signals and the optimalsequences of connection/disconnection of the capacitors. The problem is usu-ally tackled adopting dynamic programming [80], and genetic [64] or simulatedannealing [81] algorithms.

The on-line problem aims at control of OLTC by closed-loop regulation tokeep the voltage amplitude close to the reference value. The controlled voltageis either the measured transformer secondary voltage or a voltage calculatedaccording to the Line Drop Compensation principle. In the latter case, frommeasurements of the transformer secondary voltage and current, a voltagealong the feeder is estimated; in this way the voltage drop along the feeder


is partially compensated. The on-line closed-loop voltage regulation variesthe transformer tap ratio, and consequently its steady-state response mustbe accounted for in distribution load flow studies [104]. Poor system voltageregulation capabilities or controls may result in under/overvoltages. Incorrecttap settings on transformers can also result in system under/overvoltages.The performance of the on-line voltage regulation may be strongly affectedby dispersed generation connected to the distribution system [18,27,93].

The OLTC control action is quite slow, due to current commutations alongthe windings. Consequently, in this monograph it is not investigated in detail.However, the results presented in the case of other voltage control devices caneasily be extended to the case of OLTC. The application of adaptive volt-age control to OLTCs is useful, especially in the case of distribution systemsincluding dispersed generation.

1.4.2 Voltage Harmonic Distortion Containment

Waveform distortion, that is type 5 disturbances in the classification reportedin Section 1.4, can always be included in the class of small perturbations of thesteady-state voltage waveform. In fact, waveform distortion is a steady-statedeviation from an ideal sine wave of power frequency principally characterizedby the spectral content of the deviation. Among various disturbances, voltageharmonic components are particularly dangerous.

Harmonic distortion is introduced by harmonic sources, that is nonlin-ear components, nonlinear loads and electronically-switched equipments [1].Such devices absorb current waveforms, which present sinusoidal componentsat harmonic frequencies, that is frequencies that are integer multiples of thefundamental frequency at which the supply system is designed to operate.The current harmonic components are injected by harmonic sources into theelectric system and interact with voltage harmonic components, resulting inharmonic distortion that propagates throughout the whole system. Harmonicdistortion has various dangerous effects [116]: increased system losses, compo-nent life-reduction, overcurrents or overvoltages resulting from resonant phe-nomena, interference with control, communication and protection equipment.

To contain harmonic distortion, shunt filters are typically used. Passive fil-ters are composed of shunt capacitors and inductances. Active shunt filters arepower electronic devices capable of generating current harmonic components.Unlike passive filters, active shunt filters are controllable devices. Hybrid shuntfilters are obtained by combining passive and active shunt filters, so as to keepthe feature of controllability while reducing the installation costs, which arehigh in the case of active shunt filters.

In distribution and industrial systems, it is often required to equip eachharmonic source with a dedicated filtering apparatus. If an active or hybridshunt filter is adopted, it must be controlled so as to inject current harmoniccomponents which have the same amplitude and opposite phase to those in-jected by the harmonic source.

1.4 Voltage Control in MV and LV Systems 9

Equipping each harmonic source with a filtering apparatus may be expen-sive, if many harmonic sources are present in the electric system. Conversely,it may be more convenient to install filters at strategically selected systemnodes. In such a way, advantages can derive from the mutual cancelation ofharmonics and the on-off switching state of the nonlinear loads [19, 20, 61].In this case, active or hybrid shunt filters must be controlled so as to locallycontain the voltage harmonic distortion at the node they are connected to.Appropriate selection of the node allows one to extend the benefits of thefiltering action to other nodes of the electric system.

2

System Modeling for Nodal Voltage RegulatorDesign

2.1 Introduction

A power system is composed of transformers, transmission lines, synchronousgenerators with their field excitation, static and dynamic loads and powerelectronic equipments. It is a complex nonlinear system with variable oper-ating conditions due to changes of set-points, increases or decreases of loads,connection or disconnection of lines, failures of components, as recalled inChapter 1.

Various power system models have been proposed according to the objec-tive of the control problem, which can be classified on the basis of

1. the amplitude of the operating point variations, considering different con-trol methodologies to minimize the effects of large perturbations, or ofsmall perturbations;

2. the controlled variables, separating the frequency/active power controlproblem from the voltage/reactive power control problem and, with refer-ence to the latter problem, separating the control problems at fundamentalfrequency and at harmonic frequencies.

Concerning the first item, it is important to underline that when largeperturbations occur, e.g. due to electric faults, the nonlinear behavior of thepower system must be accounted for as well as the strong coupling among allthe variables, that is frequency, voltages, active and reactive powers cannot betreated separately. Due to the complexity of the system the models that areadopted in the control design often assume significant approximations, such asreducing a system to a single equivalent machine connected to a bus with infi-nite short-circuit power. In these cases, nonlinear control schemes are typicallyadopted, such as ON/OFF control techniques, and significant improvementsin system performance can be attained only by improving the actuators’ in-trinsic capabilities. The main issues are control performance validation andsystem stability analysis, which are typically performed by accurate numeri-

12 2 System Modeling for Nodal Voltage Regulator Design

cal simulation. For these reasons, large perturbations are beyond the scope ofthis monograph.

On the other hand, during normal operation of the electrical power system,small perturbations are caused by changes of the loads, of the powers injectedby generators and of the topology of the network. Such operating conditionsare interesting to study because it is possible to adopt innovative controlschemes to improve system performance.

Referring to the second item, modeling for control design with referenceto normal operation assumes decoupling between frequency/active power andvoltage/reactive power controls [30]. Decoupled models are approximated butaccurate enough for control design and performance analysis. Decoupling ispossible due to the physical characteristics of some components, mainly trans-formers and transmission lines, and to the different closed-loop bandwidthof the control schemes: the frequency/active power control action is usuallymuch slower than the voltage/reactive power control response. Consequently,when analyzing the stability of synchronous generators with respect to fre-quency/active power, the effects of voltage/reactive power control must be ac-counted for; on the contrary, when analyzing the stability of voltage/reactivepower control the response of the frequency/active power control is not con-sidered. For these reasons, the models adopted for voltage control problemstypically neglect frequency variations and active power control actions. Con-sequently, the electrical variables, such as voltages, currents and powers, aremodeled by phasors at constant frequency [87].

A further decoupling is introduced in the voltage control problem by sep-arating the voltage amplitude regulation problem at fundamental frequencyfrom the voltage harmonic component cancelation problem at harmonic fre-quencies. Such decoupling is easily attained by Fourier decomposition of thevoltage and current waveforms, but requires the assumption of linear behaviorof the system components (a typical approximation is neglecting saturationof electrical machines and transformers) [5].

Summarizing, in the following, attention is focused on system modeling ofthe voltage amplitude control problem, neglecting frequency variations andactive power control actions and assuming decoupled models for fundamentaland harmonic frequencies. With reference to the hierarchical structures forvoltage control in power systems described in Chapter 1, it is worthwhilerecalling that only the local control problem is considered, that is the issueof regulating voltage amplitude at a single bus. In the following, first, modelsof the main actuators are briefly recalled; then, power system modeling istreated.

A final general comment about system modeling concerns three-phase un-balanced operation of power systems. In general, reference will be made in thefollowing to balanced operation, which is typical for high voltage transmis-sion and large-scale distribution systems. Consequently, equivalent one-phaseelectrical circuits will be assumed to model single components and the wholeelectrical system. On the other hand, in some cases, such as small-scale dis-

2.2 Voltage Control Device Models 13

tribution and industrial systems, unbalanced operation is frequent and mustbe analyzed. In such cases, voltage control actuators are sometimes assigneda further task, contributing to phase balancing. In this chapter, issues relatedto the extension of models derived from the equivalent one-phase electricalcircuits to three-phase unbalanced operation are also discussed. In the follow-ing chapters, control design techniques will be presented with reference to thebalanced operation of power systems.

2.2 Voltage Control Device Models

As recalled in Chapter 1, in power systems various types of devices are adoptedfor voltage control. The principal devices adopted for nodal voltage control intransmission systems are synchronous generators and compensators and staticVAr systems. In distribution systems, on-load tap changers (OLTC) of trans-former ratios are adopted, but this actuator is quite slow and, consequently,it is not analyzed in the following. Concerning voltage harmonic containment,the main controllable devices are active and hybrid filters. Other power elec-tronic devices, classified as Flexible AC Transmission Systems (FACTS), arepresent in power systems: most of them are used either for compensating cur-rents absorbed by specific loads, such as STATCOMs [72,119], or for increasingthe transmission capacity of the system or for stabilizing the generating ma-chines. Consequently, they are not considered in the following because theirtask is not strictly the control of nodal voltage phasors at fundamental andharmonic frequencies, which is the focus of the monograph. Indeed, STAT-COMs have been proposed also for nodal voltage regulation [97,103]; anyway,in this application a model similar to the one used for active filters can beadopted.

In addition to models of voltage control devices described in the following,in control design, models of voltage transducers should be accounted for. Theyare typically represented by a time constant related to their bandwidth. Fi-nally, the voltage rms value evaluating algorithm at fundamental frequency orthe harmonic phasor identification algorithm (described in Chapter 3) shouldalso be included in the voltage transducer.

2.2.1 Synchronous Machines

From the model standpoint, synchronous generators, which inject both activeand reactive power into the power system, are not significantly different fromsynchronous compensators, which do not inject active power, although thetwo machines are constructed in quite a different manner. For details aboutmodeling synchronous machines for power system analysis and control designrefer to [76,86,87].

Keeping in mind the considerations described in Section 2.1, the followingsimplifications are typically assumed [76] by neglecting


– the stator “transformer emfs”, that is the emfs due to magnetic flux timederivatives,

– the effects of speed variation on stator voltage,– the distortion of stator voltage and current waveforms.

The transformer emfs can be neglected because the electrical transients asso-ciated with them rapidly decay compared to the transients involved in voltagecontrol. The second assumption generally counterbalances the approximationsintroduced by the former one. The third assumption is very close to reality,especially for large-sized machines. With such assumptions, it is possible toadopt a model that uses time-varying phasors at fundamental frequency torepresent stator voltages and currents.

The synchronous machine phasor model is described by adopting the Parktransformation changing the three-phase phasors to the (d − q − 0) represen-tation. The (d− q) axes are fixed to the rotor flux magnetic axis. Concerningthe 0 axis, which is fixed in space, since the zero-sequence component of thestator current is usually null, in the above assumptions no voltage is present.The synchronous machine stator voltage v1(t) and current i1(t) phasors arerepresented, respectively, as

v1(t) = v1,d(t) + j v1,q(t), i1(t) = i1,d(t) + j i1,q(t) (2.1)

the stator voltage amplitude is

v1(t) =√

v21,d(t) + v2

1,q(t) (2.2)

In balanced operating conditions, the transformation of phasor coordinatesfrom (d − q) axes to ( − ) axes, that is the real and imaginary axes ofthe power system reference frame, is attained by the following (referring forexample to the current phasor)

i1,r(t) =

√23

(i1,d(t) sin

(δsm(t)

)+ i1,q(t) cos

(δsm(t)

))(2.3a)

i1,i(t) =

√23

(i1,q(t) sin

(δsm(t)

)− i1,d(t) cos(δsm(t)

))(2.3b)

where δsm(t) is the synchronous machine rotor angle in the reference frame.The inverse phasor transformation from (−) axes to (d− q) axes is givenby (referring, for example, to the voltage phasor)

v1,d(t) =

√32

(v1,r(t) sin

(δsm(t)

)− v1,i(t) cos(δsm(t)

))(2.4a)

v1,q(t) =

√32

(v1,i(t) sin

(δsm(t)

)+ v1,r(t) cos

(δsm(t)

))(2.4b)

In the above assumptions and neglecting the nonlinear effect of satura-tion, the following simplified transfer functions can be adopted to model thesynchronous machine in the Laplace operator s [86]:


V1,d(s) = xsm,q(s) I1,q(s) (2.5a)V1,q(s) = asm,f (s) Vsm,f (s) − xsm,d(s) I1,d(s) (2.5b)

where capital letters are used to represent the L-transformation of the time-varying components of stator voltage and current, xsm,d(s) are xsm,q(s) areoperational transfer functions modeling the machine reactance (sub-transient,transient and synchronous) along, respectively, d and q axis. The input vari-able is vsm,f (t), that is the rotor excitation voltage as seen from the statorwindings (Vsm,f (s) is the corresponding L-transformation). Then, in (2.5)asm,f (s) represents the stator voltage transfer function in no-load operatingconditions, that is with null stator current.

Synchronous machines contribute to voltage control by their excitationsystems, which allow variation of the excitation circuit feed voltage vsm,f (t).A basic scheme of a modern excitation system is shown in Figure 2.1: startingfrom the comparison between the reference set-point and the synchronous ma-chine voltage amplitude v1(t), a regulator generates a command input for theexciter, which consequently varies the supply voltage of the rotor excitationcircuit vsm,f (t).

In addition to (2.5), it is necessary to model also the dynamic response ofthe exciter to represent the actual variation of the exciting voltage vsm,f (t)in response to a variation of the command input u(t). Such a response isstrictly dependent on the exciter characteristics. Many different types of ex-citer configuration are used. A major classification is based on the type ofexciter power supply: it can be derived by a transformer from the power sys-tem and, consequently, it depends on the controlled voltage amplitude v1(t)(dependent supply); on the contrary, the exciter power can be supplied bya rotating machine – a dynamo in the past and, nowadays, an exciting syn-chronous generator, whose voltage does not depend on the power system oper-ating conditions (independent supply). In the former case, the exciter schemeis the one shown in Figure 2.1, adopting a pulse generator and a thyristorbridge rectifier to directly control vsm,f (t). In the latter case, two schemes aregenerally adopted: the one shown in Figure 2.1 or a more complex one, whichis usually referred to as a brushless exciter and is sketched in Figure 2.2. Inthe latter exciter type, the thyristor bridge rectifier does not directly generatevsm,f (t), it feeds the excitation circuit of the exciting synchronous generator,whose stator voltage feeds a diode bridge rectifier that generates vsm,f (t).

To model the dynamic response of the exciter, the thyristor bridge rectifiercan be represented by including a time delay Texc,d, which is obviously setequal to 1/6 of the time period of the fundamental frequency, and a timeconstant Texc,c, which is equal to 3−4 ms, yielding the following transferfunction:

EXC(s) Vsm,f (s)U(s)

=e s Texc,d

1 + s Texc,c(2.6)


Fig. 2.1. Basic scheme of a rectifier-based excitation system

If the power supply is derived from a transformer, the transfer functionmust be multiplied by the feeding voltage, which is subject to variations.Since only small perturbations are considered, the effect of such variations isnegligible. Finally, in the case of the brushless exciter scheme (Figure 2.2), thetransfer function (2.6) must be multiplied by two transfer functions modeling,respectively, the exciting synchronous generator and the diode bridge rectifier.The former is generally modeled by a simple time constant, equal to 0.5−1 s,whereas the latter is modeled by a simple time constant of 3−4 ms.

Fig. 2.2. Basic scheme of a brushless excitation system

2.2.2 Static VAr Systems

Static VAr compensators (SVCs) [29,68,76,91] are shunt connected static ma-chines capable of generating inductive or capacitive reactive power, measuredas volt ampere reactive (VAr). The term static is used to indicate that, un-like synchronous compensators, SVCs present no rotating components, beingbased on power electronic devices. The SVC consists of a static VAr generator


(SVG), which can derive lagging (inductive) and/or leading (capacitive) reac-tive currents, and of a suitable controller. Finally, a static VAr system (SVS)is a combination of SVCs and mechanically-switched banks of fixed capaci-tors and/or reactors, equipped with an adequate controller that coordinatesthe SVC control actions with switching of the banks. A SVS may include atransformer between the HV electrical network and the MV busbar where theSVC and the banks are connected.

The SVSs are usually referred to using a combination of the followingacronyms, thus indicating its components:

– FC = fixed capacitors,– SR = saturated reactors,– MSC = mechanically-switched capacitors,– MSR = mechanically-switched reactors,– TSC = thyristor-switched capacitors,– TSR = thyristor-switched reactors,– TCR = thyristor-controlled reactors.

Thus, a FC-TCR SVS is composed of fixed capacitors and a SVC equippedwith thyristor-controlled reactors, as shown in Figure 2.3. In the following,without lost of generality, reference will be made to the FC-TCR configuration,which is one of the most widely adopted.

The structure and the control laws of a SVS are chosen on the basis ofthe tasks assigned to the SVS. In normal operating conditions of a powersystem, the main and basic task of a SVS is to derive reactive current andpower, which vary as functions of the nodal voltage amplitude according to anassigned steady-state voltage-current characteristic. A typical characteristic isshown in Figure 2.4: in between points A and B the injected current presentsa linear variation according to a voltage slope (droop). In the case of the FC-TCR configuration, it is obtained by partially switching on the thyristors thuscontrolling the reactance value. Outside the range of linear operation, the SVScharacteristic saturates: below the capacitive limit (point A) the thyristors areswitched off and the SVS behaves like a fixed shunt capacitor, whereas abovethe inductive limit (point B) the thyristors are fully switched on and the SVSbehaves like a fixed shunt reactor.

In addition to nodal voltage regulation, the SVSs may contribute to achieveother objectives in power system operations, in particular:

1. balancing voltages and counteracting the effects of asymmetrical loads,2. compensating the reactive power absorbed by large-sized power electronic

converters,3. increasing the power transfer capability of the transmission lines and sys-

tem,4. damping electromechanical oscillations of synchronous generators,5. damping sub-synchronous resonances,6. increasing the transient stability of the power system during large pertur-

bations.


Fig. 2.3. Configuration of a FC-TCR SVS

Inductive currentCapacitive current I

V

0

B

A

Imax

Vmax

Vref

Fig. 2.4. Typical SVS steady-state characteristic

In case 1 the balancing action is attained by separately controlling theSVS phases. Case 6 refers to large perturbations of the power system oper-ating conditions in which the previously-described nodal voltage control isusually by-passed. In the other remaining cases, the additional performancerequirements are typically achieved by acting on the reference signal of theSVS nodal voltage controller. In conclusion, reference will be made to thenodal voltage control task, including both voltage regulation and the capabil-ity of tracking the variations of the voltage reference signal.

Using phasor modeling, the SVS can be represented by variable admittanceysvs

(α(t)

), which relates the current i1(t) injected by the SVS to the voltage

v1(t) of the busbar, where the SVS is connected to, according to the followingrelation:

i1(t) = −ysvs

(α(t)

)v1(t) (2.7)

The value of ysvs

(α(t)

)is composed of two terms: a fixed term yFC related

to the FC branches and a variable term yTCR related to the TCR branches,whose value depends on the thyristor firing angle α(t). Then, it is

ysvs

(α(t)

)= yFC + yTCR

(α(t)

)= yFC + f

(α(t)

)yR (2.8)


where yR is the total value of the admittance of the reactor branch when nopartialization is performed and the partialization term f(α) assumes valuesin the range [0,1] according to the following analytical expression:

f(α) = 2 − 2α

π+

sin(2α)π

(2.9)

where the firing angle α is measured starting from the zero-crossing of thephase-to-phase voltage and π/2 ≤ α ≤ π.

The partialization of the reactance causes the generation of a distortedcurrent waveform. In practice, the SVS injects a current phasor at fundamentalfrequency according to (2.7) together with other harmonic currents at oddmultiple frequencies of the fundamental one. The amplitude and the phase ofthe harmonic current phasors depend on the nodal voltage waveform and onthe value of the firing angle α(t). Assuming a perfectly-sinusoidal nodal voltageand indicating its amplitude with v1, the amplitude of the hth harmoniccurrent is given by

ih(t) =4yR

πv1(t)

(h sin

(α(t)

)cos(hα(t)

)− cos(α(t)

)sin(hα(t)

)h(h2 − 1)

)for h = 5, 7, 11, . . . (2.10)

Finally it must be noted that a complete SVS model should include thedynamic response of the power electronic apparatus, which represents theactual variation of the thyristor firing angle α(t) in response to variationof the SVS input command u(t). In the most general case such dynamicsinclude [52, 76] a time delay Tsvs,d, which is typically set equal to 1/6 of thetime period of the fundamental frequency, and a time constant Tsvs,c equal to3−4 ms, yielding the following transfer function:

TCR(s) L(α(t)

)U(s)

=e s Tsvs,d

1 + s Tsvs,c(2.11)

where capital letters always represent the L-transformation of time-varyingsignals.

2.2.3 Active and Hybrid Shunt Filters

Active shunt filters (ASFs) belong to the general category of active powerquality conditioners, which include series and shunt active devices [112]. ASFsare mainly adopted to compensate current distortion introduced by nonlinearloads. In this case, the ASF should inject harmonic currents which perfectlycompensate the harmonic currents absorbed by the nonlinear load. Two differ-ent approaches are then used for current waveform corrections, namely timedomain based correction and frequency domain based correction [60]. Con-cerning the ASF power converter, voltage source inverters (VSIs) as well as


current source inverters (CSIs) are adopted. In the case of VSI, which is oftenpreferred for economic reasons and whose three-phase topology is shown inFigure 2.5, a current control loop is added to attain the required current. Inboth converter types, various commutation techniques can be used, the mostcommon being pulse width modulation (PWM) and hysteresis-based switch-ing techniques.

Fig. 2.5. ASF topology with a VSI

For voltage harmonic control at a busbar of an electrical system, bothCSI and VSI provided with a current control loop can be modeled as idealharmonic current generators. In fact, the bandwidth frequency of such devicesis significantly higher than the frequencies considered in harmonic voltagecontrol design.

Hybrid shunt filters (HSFs) are characterized by the combination of ASFand passive filters. In their topology the ASF is connected to a passive shuntfilter rather than directly to the power system. Such a configuration is adoptedto reduce the cost of ASF. Referring to the general topology shown in Fig-ure 2.6 and to the reference phase in the assumption of balanced operatingconditions, the relationship between the controlled harmonic current iaf,h in-jected by the ASF and the harmonic current ihf,h injected by the hybrid filterinto the system busbar is given by

ihf,h =zf2,h iaf,h

zf1,h + zf2,h− vh

zf1,h + zf2,h(2.12)

2.3 Power System Equivalent Models

During normal operation the most significant nonlinearities are introduced bythe power system load-flow equations [76,109], which describe the steady-stateinteraction between the system variables. To account for the small variationsof the operating conditions, these equations are sometimes linearized aroundan assigned operating point to produce a linear dynamic model for control

2.3 Power System Equivalent Models 21

ASF

zf1,h

zf2,h

iaf,h

ihf,h

Fig. 2.6. HSF topology

design; see, for example, [22]. Such a system modeling approach is necessaryto design frequency/active power control. In fact, frequency is a system vari-able; that is, it represents the unique value of the frequency in the wholesystem. Consequently, all the local frequency/active power regulators of thesynchronous generators connected to the system cooperate to control the sys-tem frequency thus interacting through the load-flow equations. On the otherhand, when considering the voltage/reactive power control problem, the effectof local nodal voltage controller actions on local variables and their interac-tion through the power-flow balance equations is often weak enough to beneglected.

From previous considerations, in the case of local nodal voltage amplitudecontrol at fundamental and harmonic frequencies, power system steady-statemodels in the frequency domain [30,76] are usually adopted, such as those rep-resented by a simple short-circuit impedance or by the Thevenin equivalentcircuit as seen from the controlled bus. These models account for the non-linearities of the power system steady-state model but neglect its dynamics:they can be useful in the design of nonlinearities compensation in the controlscheme [49].

An alternative modeling approach assumes a state-space model in thediscrete-time domain starting from the power system response in terms of localnodal voltage amplitude in assigned operating conditions. Such an approachallows one to adopt classical linear or nonlinear control design techniques thataccount for system dynamics.

In the following the two system modeling approaches, namely the one inthe frequency domain and the other one in the discrete time domain, arepresented in detail. In both approaches, balanced operation of the systemis assumed and, then, extention of the models to unbalanced operation isdiscussed.


2.3.1 Frequency Domain Models

We consider frequency domain models of the power system steady-state be-havior as seen from the busbar at which the actuator device is connected. Thecontrolled nodal voltage and the current injected by the actuator are modeledas phasors at fundamental and, if the case stands, harmonic frequencies. Thenonlinear relationship between the voltage and the current phasors represent-ing the power system steady-state response at each frequency is modeled.

The Thevenin equivalent circuits as seen from the controlled busbar atfundamental and harmonic frequencies (Figure 2.7) usually provide enoughinformation about the power system response in terms of variations of thenodal voltage phasor vh = vh,r+j vh,i to changes of the current ih = ih,r+j ih,i

injected by the actuator. The phasor equation associated with the Theveninequivalent circuit shown in Figure 2.7 is

vh = v0,h + zeq,h ih (2.13)

The parameters to be identified are the no-load voltage phasor v0,h =v0,h,r + j v0,h,i and the network equivalent impedance zeq,h = req,h + j xeq,h. Ifit is needed to express the model with respect to the nodal voltage amplitude,the following equation is added to (2.13)

vh =√

v2h,r + v2

h,i

v0,h

req,h+jxeq,h

vh(t)

ih(t)

Actuatoru(t)

Thevenin equivalent circuit

Fig. 2.7. System frequency model using the Thevenin equivalent circuit

Many methods have been proposed to identify req,h and xeq,h at funda-mental and harmonic frequencies, using voltage and current data obtained byfield measurements or detailed numerical simulations. They basically utilizetwo approaches. The first approach [40,57] uses the existing load current vari-ations and the existing harmonic sources to identify the network equivalent


impedance at each frequency. The second approach [56, 84] uses the switch-ing transients caused by network equipment, e.g. capacitor banks, to iden-tify, first, the network equivalent transfer function and, then, the equivalentimpedance at each frequency. The first approach is usually simpler and al-ways applicable to the nodes at which power electronic devices are connected.In the literature, attention has been paid mainly to the network equivalentimpedance, assuming that the no-load voltage can be trivially obtained fromvoltage measurements when the actuator is not injecting any current. This isthe case in [40], which proposes practical analytical methods, whose applica-tion requires separate off-line measurements of the no-load voltages.

The Thevenin equivalent circuit model can easily be extended to accountfor unbalanced operation by either representing each one of the three phasesor using symmetrical component transformation or the Park transformationin the d − q − 0 axes.

2.3.2 Time Domain Models

A state-space model can be assumed to represent the power system responseto the action of the actuator in terms of nodal voltage amplitude in a givenoperating condition. In the literature, linearized dynamic models have beenused in [57, 84] to identify the network equivalent transfer function from thedata collected during normal operating conditions in response to small pertur-bations. The proposed techniques assume that the no-load voltage is knownand filtered out from the voltage data. Concerning the order of the dynamicmodel, in [57] the order is chosen so as to minimize the prediction error butmay therefore be very high, whereas in [84] a basic first-order model is as-sumed.

Referring to the voltage amplitude at fundamental frequency, a generalform of the discrete-time model describing the power system dynamics is (seeFigure 2.8)

A(z−1)(v1(tc,k) − v0,1(tc,k)

)= z−d B(z−1) u(tc,k) (2.14)

where tc,k = k Tc, k ∈ N0, Tc being the sampling period and z−1 the unitbackward shift operator. In model (2.14) the polynomials A(z−1) and B(z−1)take the form

A(z−1) = 1 + a1 z−1 + . . . + anAz−nA

B(z−1) = b0 + b1 z−1 + . . . + bnBz−nB

with b0 = 0; furthermore v1(tc,k) is the controlled nodal voltage, u(tc,k) is theinput to the actuator, d is a positive integer representing a delay and v0,1(tc,k)the no-load voltage representing the nodal voltage when u(tc,k) = 0, see [76].The delay d is assumed to be known because it is introduced by the voltage


control devices, see Section 2.2. The no-load voltage v0,1(tc,k) can be thoughtof as generated by the dynamical system

Ad(z−1) v0,1(tc,k) = (1 − z−1) v0,1(tc,k) = D δ(tc,k) (2.15)

where D δ(tc,k) is a pulse. It can be easily recognized that v0,1(tc,k) is a stepfunction.

z−d B(z−1)

A(z−1)

u(tc,k) v1(tc,k)

v0,1(tc,k)

+

Fig. 2.8. Block scheme corresponding to model (2.14)

Polynomial B(z−1) can be factorized into two terms B+(z−1) and B−(z−1)including, respectively, its stable and unstable roots, that is

B(z−1) = B−(z−1)B+(z−1) (2.16)

where

B+(z−1) = 1 + b+1 z−1 + . . . + b+

nB+z−nB+

B−(z−1) = b0 + b−1 z−1 + . . . + b−nB− z−nB−

and nB = nB+ + nB− . Usually it is realistic to assume nB− = 0, except forsome specific cases, such as voltage regulation at midpoint of a long transmis-sion line [97], which yield to a nonminimum phase model (nB− = 0).

Model (2.14) approximates the relationship between u(tc,k) and v1(tc,k) atthe controlled node. Finally, embedding (2.15) into (2.14) yields the followingmodel:

A(z−1) v1(tc,k) − A(z−1)Ad(z−1)

D δ(tc,k) = z−d B(z−1) u(tc,k) (2.17)

whose block scheme is shown in Figure 2.9.In some cases, further improvement of the accuracy of the time domain

model can be obtained by including also a white noise term, which takes intoaccount the noise due to measurement devices and to commutations in thepower electronic apparatus. Then, (2.14) is extended to the following model:


z−dB(z−1)u(tc,k)

A−1(z−1)v1(tc,k)

D δ(tc,k)

A(z−1)Ad(z−1)

v0,1(tc,k)

+


A(z−1)(v1(tc,k) − v0,1(tc,k)

)= z−d B(z−1) u(tc,k) + ν(tc,k) (2.18)

where ν(tc,k) is an uncorrelated zero mean random sequence with variance σ20 .

Embedding model (2.15) into (2.18) yields the following equation repre-senting the model shown in Figure 2.10:

A(z−1) v1(tc,k) = z−d B(z−1) u(tc,k) +A(z−1)Ad(z−1)

D δ(tc,k) + ν(tc,k) (2.19)

z−d B(z−1)u(tc,k)

A−1(z−1)v1(tc,k)

D δ(tc,k)

A(z−1)Ad(z−1)

+

ν(tc,k)


3

Voltage and Current Phasor Identification

The first task faced in voltage control design is to identify voltage and currentphasor components at various frequencies. The problem is essentially to evalu-ate the coefficients of the Fourier series of a periodic time signal, assuming itsperiodicity is known. It is a well-known problem and various approaches havebeen used to solve it. In this chapter, after briefly recalling the classificationof the methods, attention will be focused on the Kalman filtering technique.

3.1 Techniques Overview

We consider the generic periodic time signal c(t), which could be a nodalvoltage or an injected current. It is assumed to be composed of nh harmoniccomponents according to the Fourier series:

c(t) =nh∑

h=1

(ch,r(t) cos(hωt) − ch,i(t) sin(hωt)

)(3.1)

where the subscripts r and i indicate real and imaginary parts of phasor ch(t),and ω/(2π) is the fundamental frequency. The equivalent polar notation canbe used:

c(t) =nh∑

h=1

ch(t) cos(hωt + χh(t)

)standing, for h = 1, . . . , nh, the Cartesian to polar coordinates transformation:

ch(t) =√

c2h,r(t) + c2

h,i(t)

χh(t) = tan−1

(ch,i(t)ch,r(t)

)

28 3 Voltage and Current Phasor Identification

It should be noted that the choice of time reference, that is of the timeinstant t = 0, determines the coordinates reference in both Cartesian and polarnotation. If different signals have to be analyzed and their phasors comparedor used in the same model, the reference choice is arbitrary provided that itis synchronized among all the signals.

The problem is to identify the values of ch,r(t) and ch,i(t) or, equivalently,ch(t) and χh(t), starting from the sampled measurements of the signal c(t).The methods available for the solution of such problem can be classified into:

– off-line methods, which assume that the phasors do not change during agiven time interval of measurements, that is ch,r(t) = ch,r, ch,i(t) = ch,i,ch(t) = ch and χh(t) = χh;

– on-line methods, which keep on acquiring new sampled measurements soas to identify the phasors and track their variations in real time.

3.1.1 Off-line Methods

The discrete samples of the time evolution waveforms are analyzed a pos-teriori by off-line algorithms which typically calculate the fundamental andharmonic phasors using the Discrete Fourier Transform (DFT) or the FastFourier Transform (FFT) [5]. Unfortunately, to correctly apply DFT and FFTbased algorithms the following assumptions must be satisfied:

i. the signal components must be stationary (constant in magnitude andphase);

ii. interharmonics must not be present, that is the signal must be composedof harmonics that are all integer multiple of the fundamental frequency;

iii. the analyzed time interval must be an integer multiple of the fundamentalperiod of the signal;

iv. the sampling frequency of the analyzed data must be greater than twicethe highest harmonic frequency to be evaluated.

Assumption i is typical of off-line analysis, because an adequate data timeinterval can always be chosen such that the signal is stationary. However, ifthe analyzed time interval is too large and assumption i is not satisfied, thenthe application of DFT and FFT based algorithms has proven to be mislead-ing and yields incorrect results [55]. If assumptions ii and iii are not satisfied,application of DFT or FFT based algorithms usually yields the spectral leak-age phenomenon. To reduce spectral leakage, special window functions havebeen introduced [36,63], which are used to suitably weight the original wave-forms. Assumption iv is usually fulfilled due to the low harmonic frequenciesthe present estimation is concerned with. In conclusion, by choosing adequatesets of recorded sampled data signals with stationary components and by us-ing special window functions, off-line frequency analysis with DFT and FFTbased algorithms usually allows one to obtain accurate results.

3.2 Kalman Filtering 29

3.1.2 On-line Methods

In the case of on-line frequency analysis of nonstationary signals, e.g. in thecase of the current drawn by regulated power electronic devices or by time-varying nonlinear loads, DFT and FFT based algorithms cannot be applieddue to the limitations described in Section 3.1.1. Good results have beenobtained by adopting identification algorithms based on optimal estimationtechniques [94]. The aim is to accurately track the amplitude and phase ofthe fundamental and harmonic phasors. A widely-used optimal estimationtechnique is Kalman filtering [37,55,107]. Further studies [113] have proposeda different cost function, a weighted least absolute value function, for theoptimal design of the filter gain to obtain accurate performance also in the caseof gross errors in the measurements. Finally, other researchers are investigatingthe possibility of adopting neural networks for estimation and tracking ofharmonic components [38, 95]. In the following section an optimal discreteKalman filtering technique is adopted and applied to the sampled waveformsof the voltage and current.

3.2 Kalman Filtering

3.2.1 State-space Modeling

Referring to the Fourier series (3.1) of the time signal c(t), the discrete-timesystem to be estimated is described by the state equation

x(tf,k+1) = x(tf,k) + w(tf,k) (3.2)

and by the measurement equation

y(tf,k) = hTk x(tf,k) + . (tf,k) (3.3)

where tf,k = k TKf , TKf being the Kalman filter sampling period.In (3.2), x(tf,k) ∈ IR2nh is the state variable vector of real and imaginary

components of the fundamental and harmonic phasors, that is

x(tf,k) =[c1,r(tf,k) c1,i(tf,k) . . . cnh,r(tf,k) cnh,i(tf,k)

]Twhile w(tf,k) ∈ IR2nh is the random variable vector allowing for time vari-ation of the state variables and it is described by zero mean value, no timecorrelation and covariance matrix Qw,k defined as

Qw,k = E

w(tf,k)wT(tf,k)

where operator E represents the expected value.In (3.3), y(tf,k) is the kth sampled measurement of c(t), hk ∈ IR2nh is the

vector


hk =[cos(ω tf,k) − sin(ω tf,k) . . . cos(nhω tf,k) − sin(nhω tf,k)

]T (3.4)

. (tf,k) is measurement noise, assumed to be a white sequence with knownQ. ,k covariance and to be uncorrelated to the w(tf,k) sequence.

In Figure 3.1 a graphical representation of the state-space model is re-ported for the case nh = 1 and, consequently, 2-dimensional vector hk. Vectorh0 determines the fixed reference frame, whereas hk is a rotating referenceframe whose angular speed is equal to ω. The following correspondences caneasily be derived from the graphical representation shown in Figure 3.1:

OA ≡ cos(ω tf,k)

AB ≡ sin(ω tf,k)

OF ≡ c1,r(tf,k)

DF ≡ c1,i(tf,k)

OD ≡ c1(tf,k)

OG ≡ c(tf,k) = c1(tf,k) cos(ω tf,k + χ1(tf,k)

)3.2.2 Estimation Algorithm

The on-line state estimation x(tf,k) at the kth step is based on the followingrecursive equation [44]:

x(tf,k) = x(tf,k−1) + kk

(y(tf,k) − hT

k x(tf,k−1))

(3.5)

kk being a vector of blending factors optimally designed according to a cho-sen cost function to be minimized. The Kalman filtering approach assumesthe squared Euclidean norm of the diagonal of the estimate error covariancematrix Rk as the cost function; to this aim, at the kth step, define

Rk ≡ E(

x(tf,k) − x(tf,k))(

x(tf,k) − x(tf,k))T

(3.6)

Substituting (3.3) into (3.5) yields

x(tf,k) = x(tf,k−1) + kkhTk

(x(tf,k) − x(tf,k−1)

)+ kk. (tf,k) (3.7)

Using (3.7), the estimate error covariance matrix Rk (3.6) can be writtenas


Oh0

hk

c1(tf,k)

G

F

D

B

A

ω tf,k

. 1(tf,k)

Fig. 3.1. Graphical representation of the state-space model

Rk = E

((I2nh

− kk hTk

)(x(tf,k) − x(tf,k−1)

)− kk. (tf,k))

((I2nh

− kk hTk

)(x(tf,k) − x(tf,k−1)

)− kk. (tf,k))T

(3.8)

where I2nhis the 2nh-dimensional identity matrix. Equalizing to zero the

derivative of (3.8) with respect to kk, the recursive equation to update theKalman gains matrix is obtained as

kk =Rk hk

hTk Rk hk + Q. ,k

(3.9)

In (3.9), Rk is the prediction error covariance matrix at the kth step, thatis, the covariance matrix based on the predicted state x(tf,k−1) before themeasurement y(tf,k) is available; it is defined as

Rk ≡ E(

x(tf,k) − x(tf,k−1))(

x(tf,k) − x(tf,k−1))T

(3.10)

Since, from (3.2),

x(tf,k) = x(tf,k−1) + w(tf,k−1)


by substituting x(tf,k) into (3.10) the following recursive equation, used toupdate the prediction error covariance matrix, is derived:

Rk = Rk−1 + Qw,k−1 (3.11)

Finally, from (3.8) and (3.10) the estimation error covariance matrix isobtained as

Rk =(I2nh

− kk hTk

)Rk = Fk Rk (3.12)

where Fk ≡ I2nh− kk hT

k .

3.2.3 Convergence and Stability Properties

First of all, it is necessary to derive the conditions that guarantee stableasymptotic behavior of error covariance matrix Rk, that is an upper-boundedsequence of Rk.

Theorem 3.1. The sequence Rk is upper-bounded if and only if any set of 2nh

subsequent vectors hk+1, . . . hk+2nhrepresents an independent set in IR2nh .

Proof. Since from (3.2) it is apparent that all the system states are not asymp-totically stable, to assure that the sequence Rk is upper-bounded and hasstable asymptotic behavior it is necessary and sufficient that all the statesare observable [15]. To analyze state observability, consider the state-spacesystem response after 2nh steps, starting from a generic kth step. The follow-ing equivalent system with a sampling period equal to 2nh Tkf has the sameresponse as the original system (3.2),(3.3):

x(tf,k+2nh) = x(tf,k) + η(tf,k) (3.13a)

ζ(tf,k+2nh) = Hk+2nh

x(tf,k+2nh) + υ(tf,k+2nh

) (3.13b)

where

η(tf,k) =2nh−1∑

i=0

w(tf,k+i)

ζ(tf,k+2nh) =

[y(tf,k+1) . . . y(tf,k+2nh

)]T

Hk+2nh=[hk+1 . . .hk+2nh

]Tυ(tf,k+2nh

) =

[(. (tf,k+1) − hT

k+1

2nh−1∑i=1

w(tf,k+2nh−i))

. . . . (tf,k+2nh)

]T


The states in (3.13) are observable if and only if HTk+2nh

Hk+2nhis a

full rank matrix, that is the set of 2nh subsequent vectors hk+1, . . . hk+2nh

represents an independent set in IR2nh .

From Theorem. 3.1 it is trivial to derive the following simple yet con-servative and sufficient condition to guarantee that the sequence of Rk isupper-bounded and presents stable asymptotic behavior:

TKf <π

2n2h ω

In the remainder it is assumed that the observability conditions requiredby Theorem. 3.1 are satisfied.

The evolution of the estimation error covariance matrix Rk depends alsoon its initial value R0. Some properties of the initialized matrix are keptduring time evolution according to the following theorem.

Theorem 3.2. If the initialization matrix R0 is symmetric and positive defi-nite and, ∀ k ∈ N0, the covariance matrices Qw,k are symmetric and positivedefinite and Q. ,k > 0, then the error covariance matrices Rk and Rk aresymmetric and positive definite ∀ k ∈ N0.

Proof. The proof is induced starting from k = 1. From (3.11) for k = 1 itis trivial that R1 is symmetric and positive definite. Then, substituting (3.9)into (3.12) it is obtained, for k = 1,

R1 = R1 − R1h1hT1 R1

hT1 R1h1 + Q. ,1

(3.14)

which directly shows that R1 is symmetric. From (3.14) it can easily be shownthat

aTR1a > 0 ∀a ∈ IR2nh and a = 0

by applying the Schwartz inequality to the positive definite matrix R1(hT

1 R1h1

)(aTR1a

)≥(hT

1 R1a)2

It is interesting to analyze some further conditions to be imposed whichassure that, asymptotically, the error covariance matrix Rk is not only stablebut also reaches a constant value.

Theorem 3.3. If the initialization matrix R0 is symmetric and positive defi-nite, if ∀ k ∈ N0 Qw,k = Qw, with Qw a diagonal matrix whose diagonal el-ements satisfy the following condition: qw,j = qw,j−1 > 0 for j = 2, 4, . . . , 2nh,and if ∀ k ∈ N0 we have Q. ,k = Q. > 0, then the matrices Rk and Rk asymp-totically reach values Rk,∞ and Rk,∞ which are constant in the rotating frameof the observable direction hk.


Proof. The proof is based on the observability assumption that Rk is upper-bounded and on the following inequality, which can easily be derived from (3.9)and (3.12):

hTk Rkhk =

Q.

hTk Rkhk + Q.

hTk Rkhk < hT

k Rkhk (3.15)

which shows that the “projection” hTk Rkhk of the estimated error covari-

ance matrix Rk along the observable direction hk is strictly smaller than the“projection” hT

k Rkhk of the prediction error covariance matrix. In addition,from (3.11) it can be written

hTk Rkhk = hT

k Rk−1hk + hTk Qwhk (3.16)

Then, with A the rotation matrix such that hk+1 = Ahk, keeping inmind (3.4), due to the assumed property on the diagonal covariance matrixQw, it can be easily proved that

Qw = ATQwA

and consequently,

hTk Qwhk = hT

k−1Qwhk−1 =nh∑i=1

qw,2i . (3.17)

From (3.15), (3.16) and (3.17) one directly derives that hTk Rkhk and

hTk Rkhk reach constant asymptotic values hT

k Rk,∞hk and hTk Rk,∞hk, in-

dependently of the initial vector h0, that is independently of the assumedrotating reference frame of coordinates.

In the assumptions of Theorem. 3.3, it can be written

hTk Rk,∞hk = hT

k+1Rk+1,∞hk+1

and consequentlyRk+1,∞ = ARk,∞AT (3.18)

Keeping in mind (3.9), (3.11), (3.12) and (3.18), the constant asymptoticvalues can be obtained, respectively, Rk,∞ from solution of the steady-stateRiccati equation (SSRE) :

Rk,∞ = AT

(Rk,∞ − Rk,∞ hkhT

k Rk,∞hT

k Rk,∞ hk + Q.

)A + Qw (3.19)

and Rk,∞ from the following equation

Rk,∞ = Rk,∞ − Rk,∞hk

(hT

k Rk,∞ hk + Q.)−1

hTk Rk,∞


Assuming that the assumptions of Theorem 3.3 hold, we analyze theasymptotic behavior of the Kalman filter state estimation when, starting fromthe initial estimate x0, the discrete-time system (3.2),(3.3), with a constantexpected value E

x(tf,k)

= x0, is estimated.

The expected value of the estimation error at the kth step is derivedfrom (3.5) and (3.12), and is given by

Ex0 − x(tf,k)

= Fk E

x0 − x(tf,k−1)

(3.20)

Since

hTk Fkhk =

Q.

hTk Rk,∞ hk + Q.

< 1

the matrix Fk determines a reduction along the observable direction hk. Sincethe observability conditions described in Theorem 3.1 are verified so that allthe directions are observable, thanks to the rotation of hk, then the sequenceof expected values of estimation errors (3.20) cannot diverge. In addition,since the matrix Fk asymptotically constant in a frame of coordinates jointlyrotating with hk, the sequence (3.20) must asymptotically tend to a constantvalue in such a frame; that is

ek,∞ = Aek−1,∞ (3.21)

where ek,∞ and ek−1,∞ are the asymptotic values of, respectively, Ex0 −

x(tf,k)

and Ex0 − x(tf,k−1)

. Combining (3.20) and (3.21), the following

steady-state equation is obtained:

Fk,∞ ek−1,∞ = A ek−1,∞ (3.22)

where Fk,∞ is the asymptotic value of Fk. The solution of (3.22) yields theexpected value of the asymptotic bias of the estimates. Actually, the estimatesare unbiased because the solution of (3.22) is null, according to the followingcorollary of Theorem 3.3.

Corollary 3.4. Based on the same hypotheses as those of Theorem 3.3, theestimates are asymptotically unbiased.

Proof. If there were a nonzero solution of (3.22), then there would exist thenonzero vector ak−1,∞ defined as

ak−1,∞ ≡ R−1

k,∞ ek−1,∞ (3.23)

Using (3.23), (3.22) becomes

Fk,∞ Rk,∞ ak−1,∞ = ARk,∞ ak−1,∞ (3.24)

It is easy to prove that (3.24) is absurd. In fact the SSRE (3.19) can bewritten in the form


ARk,∞ AT − Qw = Fk,∞ Rk,∞

which shows that the eigenvalues of Fk,∞ Rk,∞ differ from those of Rk,∞ dueto Qw, and the eigenvectors of Fk,∞ Rk,∞ are the rotation of the eigenvectorsof Rk,∞ by A.

Finally, the correlation between subsequent estimation errors can easily bederived from (3.5) and (3.12) as follows:

Rk+m,k = E

(x0 − x(tf,k+m)

)(x0 − x(tf,k)

)T

=( m∏

i=1

Fk+i

)Rk

and, asymptotically,

Rk+m,k =( m∏

i=1

Fk+i,∞

)Rk,∞ (3.25)

Since each term Fk+i,∞ in (3.25) causes a reduction along the observabledirection hk+i, then, thanks to the observability conditions of Theorem 3.1,the sequence of matrices Gm, defined as

Gm

= m∏

i=1

Fk+i,∞

tends to zero as m → ∞. In conclusion, (3.25) enables to evaluate the mini-mum “time distance” mTkf , allowing one to neglect, with assigned accuracy,the correlation between two different estimates of the Kalman filter.

4

Self-tuning Voltage Regulators

The aim of this chapter is to illustrate the design of self-tuning voltage reg-ulators for the discrete-time linear model, presented in Chapter 2, which de-scribes the electrical power system dynamics from the regulation node. Themission of the self-tuning regulator is to guarantee regulation of the nodalvoltage amplitude in the presence of an unknown variable operating point ofthe power system. The design is developed both with reference to indirectand direct methods. In both methods the estimator adopts a recursive least-squares (RLS) algorithm with variable forgetting factor while the design ofthe voltage regulator is developed according to the pole-assignment techniqueand to the generalized minimum variance approach. The chapter ends with asection dedicated to illustration of the properties of the RLS algorithm.

4.1 Introduction

The purpose of self-tuning regulators is to control systems with unknown pa-rameters, unknown bounded disturbance and random noise. This is realizedby adding an automatic adjustment mechanism in the control loop. A possiblecriterion for obtaining an adjustment mechanism consists in identifying thesystem model parameters using measured input and output data and thento synthesize an appropriate regulator according to some design specifica-tions [10, 127]. Figure 4.1 shows a self-tuning control scheme. The inner loopconsists of the system and a linear feedback regulator. The outer loop is com-posed of a parameter estimator and a design calculation. The partitioningof the regulator, depicted in Figure 4.1, is also convenient from the imple-mentation point of view, because the parameter estimator and the regulatorparameters calculation are often conveniently time shared between severalloops.

Self-tuning regulators are divided into two classes: those based on indirectmethods, often named explicit self-tuning regulators, and those based on di-rect methods, often named implicit self-tuning regulators [9, 10]. In indirect

38 4 Self-tuning Voltage Regulators

methods, the regulator parameters are not updated directly, but rather indi-rectly via estimation of the system model. In direct methods, the regulatorparameters are directly estimated through a re-parameterization of the modelassumed to represent the system. In this case a significant simplification ofthe algorithm is obtained, because the design calculations are eliminated.

r

RegulatorPower

system

v

Estimator

EstimatedParameters

Design

Regulator

Parameters

u

Fig. 4.1. Overview of a self-tuning scheme

The class of self-tuning regulators can be thought of as composed of threemain parts: a parameter estimator, a linear controller and a third part, im-plementing the synthesis law, which relates the controller parameters to theestimated parameters. The true parameter values are replaced by their esti-mated values (certainty equivalence principle) when determining the controllaw using the design criteria. The regulator parameters are in general a nonlin-ear function of the estimated parameters. There are many possible self-tuningregulators, depending on the system to be controlled and the design and pa-rameter estimation techniques that are used [8–11,127]. Regarding parameterestimates, different methods can be employed such as least-squares, extendedleast-squares, and maximum likelihood [10,77,127], while the regulator can bedesigned according to well-known control techniques such as pole-assignment,linear quadratic theory and minimum variance control [2, 3, 11,127].

4.2 Indirect Self-tuning Voltage Regulator Design

The first task in the design of indirect self-tuning regulators consists in esti-mating the parameters of a prediction model of the output variable. Such atask is accomplished using a recursive least-squares algorithm with variableforgetting factor. The application of a such algorithm in the field of powersystem control enables the estimator to track variations in power system dy-namics caused by unexpected operating point changes [50,54,79,114].

4.2 Indirect Self-tuning Voltage Regulator Design 39

4.2.1 Recursive Least-squares Algorithm

Let us recall the power system discrete-time model (2.17)

A(z−1) v1(tc,k) = z−d B(z−1) u(tc,k) +A(z−1)Ad(z−1)

D δ(tc,k)

Unpredictable changes of the power system operating point lead to un-known variations of the

np = nA + nB + 2

parameters, that is the coefficients ai, bj and D. The estimator continuallyattempts to find the parameters of a prediction model of the output variablev1(tc,k), such that the modeling error

ε(tc,k) = v1(tc,k) − v1(tc,k) (4.1)

is small. To enhance the ability of the estimator to track parameter variationscaused by variability in the operating points of the actual power system itis necessary to discount old data. The use of a forgetting factor [9, 10, 127]ensures that data in the distant past are forgotten.

To derive the prediction model we write the output v1(tc,k) in recursiveform as

v1(tc,k) = −(a1 v1(tc,k−1) + . . . + anA

v1(tc,k−nA))

+ b0 u(tc,k−d) + . . .

. . . + bnBu(tc,k−d−nB

) + v0,1(tc,k)

which, expressed in compact form, becomes

v1(tc,k) = −nA∑i=1

ai v1(tc,k−i) +nB∑j=0

bj u(tc,k−d−j) + v0,1(tc,k) (4.2)

where

v0,1(tc,k) = v0,1(tc,k−1) + D δ(tc,k) + D

nA∑i=1

ai δ(tc,k−i)

represents the impulse response of filter A(z−1)/Ad(z−1). Such a responsereaches, after nA steps, a constant value v0,1 given by

v0,1 =(1 +

nA∑i=1

ai

)D = A(1) D (4.3)


in which A(1) is the value assumed by the polynomial A(z−1) for z−1 = 1. Forall k ≥ nA it is then possible to substitute v0,1 for v0,1(tc,k) into model (4.2)yielding

v1(tc,k) = −nA∑i=1

ai v1(tc,k−i) +nB∑j=0

bj u(tc,k−d−j) + v0,1 (4.4)

Model (4.4) is linear in the parameters ai, bj and v0,1. The adoption ofmodel (4.4) instead of the nonlinear (4.2) in the recursive least-squares algo-rithm leads to a modeling error for the first nA samples. However, it is worthrecalling that usually nA is not high (nA = 2−5) and, anyway, smaller thannp. In addition, since the recursive least-squares algorithm uses a forgettingfactor, the modeling error has no effect on the asymptotic estimates. At thispoint, model (4.4) put in compact form becomes

v1(tc,k) = φT(tc,k)θ

in which the regression vector φ and the unknown vector parameters θ aregiven by

φT =[− v1(tc,k−1) . . . − v1(tc,k−nA

) u(tc,k−d) . . . u(tc,k−d−nB) cv

](4.5)

θ =[a1 . . . anA

b0 . . . bnBv0,1/cv

]Twhere the constant fictitious input cv can be viewed as a scaling factor allowingan estimate of the aggregate parameter v0,1.

The prediction model of the output variable v1(tc,k) given by

v1(tc,k) = φT(tc,k) θ(tc,k−1)

is used to determine, according to (4.1), the a priori output prediction erroras

ε(tc,k) = v1(tc,k) − φT(tc,k) θ(tc,k−1) = φT(tc,k)(θ − θ(tc,k−1)

)(4.6)

The parameters vector θ is determined in such a way that the cost functionfor the exponentially weighted squared error is minimum

J = ε(tc,k)T Λk ε(tc,k) =k∑

i=1

λk−i ε2(tc,i) (4.7)

where

Λk = diag

λk−1 λk−2 . . . λ2 λ 1


with λ the forgetting factor, and

ε(tc,k) =[ε(tc,1) . . . ε(tc,k)

]TThe prediction error together with the gain vector m, calculated on the

basis of the following equation:

m(tc,k) =P(tc,k−1) φ(tc,k) β(tc,k)

1 + φT(tc,k)P(tc,k−1) φ(tc,k), (4.8)

in which P is the covariance matrix, are used to calculate θ according to

θ(tc,k) = θ(tc,k−1) + m(tc,k) ε(tc,k)

= θ(tc,k−1) + m(tc,k)(v1(tc,k) − φT(tc,k) θ(tc,k−1)

)(4.9)

An estimator dead-zone with hysteresis, represented by the functionβ(tc,k), is introduced in the RLS algorithm to switch off the estimator whenthe excitation level is low. The function β(tc,k) takes the form

β(tc,k) =

⎧⎪⎪⎨⎪⎪⎩1 if ε2(tc,k) > β0 ε2dz

β(tc,k−1) if ε2dz ≤ ε2(tc,k) ≤ β0 ε2dz

0 if ε2(tc,k) < ε2dz

with β(0) = 1, where ε2dz is an empirically selected threshold that activatesthe dead-zone and β0 is an arbitrary positive constant that sets the width ofthe hysteresis.

The simple block scheme depicted in Figure 4.2 illustrates how the estimateθ is recursively obtained at each tc,k.

m(tc,k)+

+θ(tc,k−1)

θ(tc,k)

z−1φT(tc,k)

v1(tc,k)

v1(tc,k)

−+

Fig. 4.2. Block scheme representing the implementation of (4.9)

The updating law of the covariance matrix is given by

P(tc,k) =1λ

(Inp

− m(tc,k) φT(tc,k))P(tc,k−1) (4.10)

where Inpis the np-dimensional identity matrix.


In (4.10) all elements of matrix P are scaled by the same amount. If onewishes to target the effect upon selected parameters it is possible to scaleonly the diagonal of P. Furthermore, it is useful to point out that in order toget smooth parameters estimation a minimum forgetting factor λmin has tobe specified. If one wishes to keep the estimator more sensitive to the powersystem changes at the steady-state operating point a maximum forgettingfactor λmax can be used, that is

0 < λmin ≤ λ ≤ λmax ≤ 1

However the use of a forgetting factor can cause the phenomenon knownas estimator wind-up. Specifically, if no new data enter into the estimator overa long time, as in the case of absence of set-point changes, the regressor φ issmall, and the matrix P grows exponentially if, as an extreme case, φ is zero.The occurrence of such a circumstance is then caused by poor excitation incombination with discounting of old data and it can be monitored by lookingat the trace of P. To prevent this behavior the use of a variable forgettingfactor represents a good solution [9,10,127]. A suitable approach to the designthe updating law of the variable forgetting factor has been proposed in [43]whose convergence properties have been discussed in [33]. According to thisapproach, the forgetting factor is updated as

λ(tc,k) = 1 −(1 − φT(tc,k) m(tc,k)

)ε2(tc,k) β(tc,k)

σ20 N0

(4.11)

In (4.11) σ20 is the expected measurement white noise covariance and N0

controls the speed of adaptation. A small value of N0 will give a large co-variance matrix and a sensitive estimator while a larger value will give a lesssensitive estimator thus leading to a slower adaptation.

Other approaches are available to select the variable forgetting factor aswell as other techniques such as the constant trace algorithm and directionalforgetting factor can be utilized to prevent estimator wind-up [77,83,126,127].

4.2.2 Pole-assignment Design

The design of the voltage regulator is developed by resorting to the pole-assignment technique [8–11,31,127]. With the application of such a techniquethe poles of the closed-loop transfer function are located in any desired loca-tions, defined by the roots of polynomial

T (z−1) = 1 + τ1 z−1 + . . . + τnTz−nT (4.12)

so that the desired characteristics, expressed in terms of closed-loop require-ments, are satisfied. The major points in favor of the pole-assignment tech-nique are that it provides a robust regulator against modeling errors and cancope with nonminimum phase systems [2, 12].


The first step in pole-assignment design is to decide the location of closed-loop poles. On this point, it is always useful to keep in mind that the controleffort required is related to how far the open-loop poles are moved by thefeedback. Large control effort is required if the feedback moves a pole neara zero. Such considerations are particulary important in the case in whichthe control input is constrained due to the presence of actuator saturation.Saturation of the control input may last for a long time interval and may leadto the identification of a false power system model. An alternative approachtakes into account the control effort in the regulator design relies on the useof the linear quadratic theory. In this approach the required control input canbe found by finding steady-state solutions of Riccati equations [3].

The second step aims to determine the coefficients of the polynomialsF (z−1), G(z−1) and H(z−1) which define the regulator law

F (z−1) u(tc,k) = −G(z−1) v1(tc,k) + H(z−1) r(tc,k) (4.13)

The structure of the closed-loop control system is sketched in Figure 4.3.

r(tc,k)

H(z−1)+

F−1(z−1) Model (2.17)v1(tc,k)

G(z−1)

−

u(tc,k)

Fig. 4.3. Closed-loop system

Since v0,1(tc,k) is modeled as in (2.15), according to the internal modelprinciple, the regulator will contain an integrator, that is Ad(z−1) is a factorof F (z−1)

F (z−1) = Ad(z−1) Fv(z−1) (4.14)

The regulator cancels the stable zeros of B(z−1) while the unstable onescannot be canceled; for this reason B(z−1) is factorized according to (2.16).

The polynomial Fv(z−1) assumes then the following factorization:

Fv(z−1) = B+(z−1) F (z−1) (4.15)

where

Fv(z−1) = 1 + fv,1 z−1 + . . . + fv,nFz−nFv

It is worth noting from (4.14) that polynomial F (z−1) is expressed by theconvolution of polynomials Ad(z−1) and Fv(z−1) in which only the coefficientsof the latter are adapted. Based on this consideration it is convenient to write


Fv(z−1) Ad(z−1) u(tc,k) = Fv(z−1) uv(tc,k)

in which

uv(tc,k) = Ad(z−1) u(tc,k) = u(tc,k) − u(tc,k−1) (4.16)

represents the control input not integrated.The coefficients of polynomials

F (z−1) = 1 + f1z−1 + f2 z−2 + . . . + fnF

z−nF

G(z−1) = g0 + g1 z−1 + g2 z−2 + . . . + gnGz−nG

are determined by solving, every steps with a fixed integer, the followingDiophantine equation:

A(z−1) Ad(z−1) F (z−1) + z−d B−(z−1)G(z−1) = T (z−1) (4.17)

obtained by substituting (4.14) and (4.15) in the following pole-assignmentidentity:

A(z−1) F (z−1) + z−d B(z−1)G(z−1) = T (z−1) B+(z−1)

In (4.17) A(z−1) and B−(z−1) are replaced with their estimates A(z−1)and B−(z−1).

The Diophantine equation (4.17) has a unique solution if A(z−1) andB(z−1) are co-prime and the degrees of polynomials F (z−1), G(z−1) andT (z−1) satisfy the constraints

nF = nB− + d − 1 (4.18a)

nG = nA + nAd− 1 = nA (4.18b)

nT ≤ nA + nB− + nAd+ d − 1 = nA + nB− + d (4.18c)

where nB− is the number of the unstables zeros of B(z−1). From (4.15) itresults

nFv= nB+ + nF (4.19)

Equalizing the coefficients of the power of z−i in (4.17) yields a set oflinear equations which can be cast in matrix form suitable to be solved forthe unknown coefficients by matrix inversion. From this point of view, it isimportant to remark that the Diophantine equation (4.17) is poorly condi-tioned if A(z−1) and B(z−1) have roots which are closer together [12]. Thiscircumstance often happens for models of high order.


Once the Diophantine equation has been solved, the polynomial Fv(z−1)is obtained from (4.15) where B+(z−1) substitutes for B+(z−1). PolynomialFv(z−1) is then substituted in (4.14) to give F (z−1).

Finally, to ensure unitary steady-state gain for the closed-loop transferfunction, the simplest choice for H(z−1) is

H(z−1) = h0 =T (z−1)

B−(z−1)

∣∣∣z−1=1

(4.20)

where B−(z−1) is replaced with B−(z−1).

4.2.3 Pole-shifting Design

With the design developed in Section 4.2.2, the roots of polynomial T (z−1)are selected at fixed locations. However, the amount of control required tocounteract the effects of disturbance depends on the distance between theopen-loop pole locations and closed-loop locations. If a large disturbance oc-curs, such as a power system fault, the open-loop pole locations are subjectto wide variations and then the control effort required after the disturbancecan be very large. If the control input is limited due to the presence of ac-tuator saturation, it is not possible to restore the pre-fault condition. Thesaturated control input leads to a wrong identification of the power systemmodel parameters. An effective solution to this problem is to adopt the pole-shifting algorithm [23, 73, 85, 96]. With the application of this algorithm, theclosed-loop pole positions are selected so that they are radially in line withthe position of the poles of the estimated plant and are closer to the centerof the unit circle in the z-plane by a fraction ρ, where ρ is the pole-shiftingfactor.

Conversely, in power system voltage regulation, the system undergoessmall disturbances. Such disturbances change the poles of the power systemmodel but such variations are not large because the power system is operat-ing under normal conditions. Staring from this consideration, the pole-shiftingtechnique can be usefully applied in the voltage regulation problem to varythe locations of the desired closed-loop poles, represented by the roots of poly-nomial T (ρ, z−1), in such a way that the limits imposed by the actuator aretaken into account in the design of the control input [51].

It is then admissible to suppose that the two dominant poles of T (ρ, z−1)belong to a domain in the z-plane characterized by a constant damping ratiolocus (cardioid) [99] to satisfy closed-loop specifications concerning the allowedlargest overshoot. To satisfy also requirements imposed on the settling time,the two dominant poles are forced to move along with the cardioid comprimesbetween the two circles of radius

Rmin = e−4 Tc/Ta,s , Rmax = e−4 Tc/Ta,m


with Ta,s < Ta,m, see Figure 4.4. In practice, the largest roots of T (ρ, z−1)are constrained to move along the paths AB and DC.

Fig. 4.4. Constant damping ratio locus into the z-plane

The circle of radius Rmin has been introduced to avoid the pole-shiftingalgorithm placing the two largest roots of T (z−1) in such a way that theresulting settling time is not feasible. In fact, it is useful to recall that, in thevoltage regulation task, the settling time assumes values equal to a few timesthe fundamental frequency cycle.

In the pole-shifting design, the Diophantine equation (4.17) takes the form

A(z−1) Ad(z−1) F (ρ, z−1) + z−d B−(z−1)G(ρ, z−1) = T (ρ, z−1) (4.21)

with ρ the pole-shifting factor, where

T (ρ, z−1) =(1 − z1(ρ)z−1)

)(1 − z2(ρ)z−1)

)nT∏i=3

(1 − zi(ρ)z−1)

)(4.22)

is a pre-specified polynomial which defines, at each sampling period Tc, theclosed-loop pole locations according to the pole-shifting approach. In (4.22)z1,2(ρ) represent the largest roots of T (ρ, z−1) given by

z1,2(ρ) = e−4ρ Tc/Ta

(cos(ρ ωnTc

√1 − ζ2

)± j sin(ρ ωnTc

√1 − ζ2

))


with Ta = 4/(ζ ωn), and ωn the natural frequency. The remaining (nT − 2)roots of T (ρ, z−1), either complex pairs or reals, are characterized by exponen-tial terms such as exp(−4 q ρ Tc/Ta) with q > 1. It is then easy to recognizethat increasing ρ decreases |zi|, i = 1 . . . nT .

As far as the calculation of ρ is concerned, it must be said that the pole-shifting factor depends on the operating conditions of the power system. Understeady-state conditions the value of ρ must be constant, while in the presenceof small disturbance or during set-point changes this factor must assumesappropriate values to match design specifications and control input limits. Thefactor ρ can be determined on the basis of a gradient approach to minimizethe normalized control effort given by

J(tc,k) =12

u2v(tc,k)

m2(tc,k)

where

m2(tc,k) = k1 +

(∂

∂ρuv(tc,k)

)2

(k1 > 0)

with uv(tc,k) given by

uv(tc,k) = −nF −1∑i=1

fv,i(ρ) uv(tc,k−i) + h0(ρ) vref(tc,k) −nG∑j=0

gj(ρ) v1(tc,k−j)

(4.23)see (4.13) and (4.15). Accordingly, the pole-shifting factor is varied as follows:

ρ(tc,k+1) = ρ(tc,k) − ∆ρ(tc,k)

with

∆ρ(tc,k) = γuv(tc,k)m2(tc,k)

∂

∂ρuv(tc,k) (4.24)

being γ a positive constant gain.The sensitivity of the control input with respect to ρ, namely ∂uv/∂ρ

in (4.24), can be calculated by expressing the derivatives of the coefficientsfv,i(ρ), gj(ρ) and h0(ρ) with respect to ρ in (4.23). To this aim it is necessaryto solve (4.21) in symbolic form to obtain the closed-form expression of fi(ρ),gj(ρ) and h0(ρ). In particular, if B+(z−1) = 1 then ∂fv,i/∂ρ = ∂fi/∂ρ, other-wise ∂fv,i/∂ρ are obtained by differentiating with respect to ρ the coefficientsof the convoluted polynomial B+(z−1) F (ρ, z−1).

The presence of limits acting on the control input, such as

umin ≤ u(tc,k) ≤ umax


can be taken into account by modifying the factor ∆ρ as follows:

∆ρ = γsat

( ∂

∂ρuv(tc,k)

)−1

|uv(tc,k)| (4.25)

If u(tc,k−1) = umax, to obtain u(tc,k) < umax, it is necessary, accordingto (4.16), to make uv(tc,k) < 0. That is uv(tc,k) must decrease. The pole-shifting factor ρ has to vary to pursue this objective. It is easy to recognizefrom (4.25) that, whatever the sign of ∂uv(tc,k)/∂ρ, decreasing uv(tc,k) isensured by choosing γsat > 0.

Conversely, if u(tc,k−1) = umin, the pole-shifting factor ρ varies to increaseuv(tc,k) so that u(tc,k) > umin. This is done with γsat < 0, whatever the signof ∂uv(tc,k)/∂ρ.

4.2.4 Generalized Minimum Variance Pole-assignment Design

As already pointed out, the presence of white noise in the power systemmodel (2.19) takes into account the noise due to commutation of electronicdevices and the measurement noise of the voltage signal.

A suitable approach to design the voltage regulator makes use of the gen-eralized minimum variance control [2, 31, 127]. According to the generalizedminimum variance approach, a generalized pseudo-output η(tc,k+d) is intro-duced, which includes appropriate filtering of the regulated output v1(tc,k+d),of the control input u(tc,k) and of the reference signal r(tc,k). The objectiveis to find a control law which minimizes the following cost function:

I = Eη2(tc,k+d)

where

η(tc,k+d) = P (z−1) v1(tc,k+d)+Q(z−1) Ad(z−1) u(tc,k)−R(z−1) r(tc,k) (4.26)

Figure 4.5 shows the generalized system output.The weighting polynomials P (z−1), Q(z−1) and R(z−1) are appropriately

selected to assign the closed-loop poles and to reduce the output and controlsignal variances. The presence of the difference filter Ad in η(tc,k+d) removesthe offset due to reference signal changes [31].

The rms voltage at time instant tc,k+d, namely v1(tc,k+d), in the gener-alized pseudo-output (4.26) must be predicted since it cannot be observedat the present time tc,k. To this aim we introduce the following Diophantineequation:

P (z−1) = p0 A(z−1) Ad(z−1) E(z−1) + z−d G(z−1) (4.27)

where the polynomials E(z−1), G(z−1) and P (z−1) are


z−d B(z−1)u(tc,k)

A−1(z−1)v1(tc,k)

D δ(tc,k)

A(z−1)Ad(z−1)

+

ν(tc,k)

P (z−1)+

r(tc,k)

z−d R(z−1)

−

η(tc,k)

z−d Ad(z−1)Q(z−1)

+

Fig. 4.5. Generalized system output

E(z−1) = 1 + e1 z−1 + . . . + enEz−nE

G(z−1) = g0 + g1 z−1 + . . . + gnGz−nG

P (z−1) = p0 + p1 z−1 + . . . + pnPz−nP

whose degrees are given by

nE = d − 1 (4.28a)

nP = nA (4.28b)

nG = maxnA + nAd− 1, nP − d = nA (4.28c)

In (4.27) the difference filter Ad removes the offset due to the no-loadvoltage disturbance [31]. Now, multiplying (2.19) by p0 Ad(z−1) E(z−1) andparticularizing it at time tc,k+d gives

p0 A(z−1) Ad(z−1) E(z−1) v1(tc,k+d) = p0 Ad(z−1) B(z−1) E(z−1) u(tc,k)

+p0 A(z−1) E(z−1) D δ(tc,k+d) + p0 Ad(z−1) E(z−1) ν(tc,k+d)(4.29)

Equation (4.27) multiplied by v1(tc,k+d) is used to rewrite model (4.29) as

P (z−1) v1(tc,k+d) = G(z−1) v1(tc,k) + p0 Ad(z−1) B(z−1) E(z−1) u(tc,k)

+ p0 A(z−1) E(z−1) D δ(tc,k+d) + p0 Ad(z−1) E(z−1) ν(tc,k+d)(4.30)


The last term in (4.30) is rewritten as

p0 Ad(z−1) E(z−1) ν(tc,k+d) = p0 E(z−1) ν(tc,k+d) − p0 ed−1 ν(tc,k)

where the sequence

E(z−1) ν(tc,k+d) =(e0 + (e1 − e0) z−1 + (e2 − e1) z−2 + . . .

. . . + (ed−1 − ed−2) z−(d−1))ν(tc,k+d)

relating to the future values of the noise is uncorrelated with the first twoterms appearing at the right-hand side of (4.30). Nevertheless, the termp0 ed−1 ν(tc,k) relating to the present value of the noise affects the predictionat time tc,k.

Regarding the disturbance term in (4.30),

p0 A(z−1) E(z−1) D δ(tc,k+d)

it is easy to recognize that it is equal to zero

∀ k ≥ nA + nE − d = nA + d − 1 − d = nA − 1 (4.31)

if, for the first nA −1 steps following a variation of the disturbance amplitudeD, polynomial E(z−1) has roots inside the unit circle, since p0 A(z−1) E(z−1)is a FIR filter (finite impulse response).

The sub-optimal d-steps ahead weighted prediction is then given by

P (z−1) vp(tc,k+d) = G(z−1) v1(tc,k) + p0 Ad(z−1) B(z−1) E(z−1) u(tc,k)

which substituted in (4.26) together with (4.14), gives the prediction of thegeneralized pseudo-output

ηp(tc,k+d) = G(z−1) v1(tc,k) + F (z−1) u(tc,k) − R(z−1) r(tc,k)

where

Fv(z−1) = Q(z−1) + p0 B(z−1) E(z−1) (4.32)

withQ(z−1) = q0 + q1 z−1 + . . . + qnQ

z−nQ

From (4.32) it useful to recognize that the following equality holds:

1 = q0 + p0 b0

The control law setting ηp(tc,k+d) to zero is then given by


up(tc,k) = −G(z−1)F (z−1)

v1(tc,k) +R(z−1)F (z−1)

r(tc,k) (4.33)

where, according to the regulator structure given by (4.13),

H(z−1) = R(z−1)

Substituting the control law (4.33) in model (2.19), using (4.14), (4.27) and(4.32), and solving with respect to v1(tc,k) yields the closed-loop equation

v1(tc,k) = z−d B(z−1) R(z−1)A(z−1) Ad(z−1) Q(z−1) + B(z−1) P (z−1)

r(tc,k)

+A(z−1) F (z−1)

A(z−1) Ad(z−1) Q(z−1) + B(z−1) P (z−1)v0,1(tc,k)

+F (z−1)

A(z−1) Ad(z−1) Q(z−1) + B(z−1) P (z−1)ν(tc,k)

The regulator is then able to regulate v1(tc,k) at the set-point in the pres-ence of the unknown no-load voltage v0,1(tc,k). To achieve zero steady-statetracking error,

R(z−1) = r0 = P (1) (4.34)

consequently one has h0 = r0. The closed-loop control structure is depictedin Figure 4.6.

r(tc,k)

h0

+

F−1v (z−1)

uv(tc,k)

G(z−1)

−A−1

d (z−1) z−dB(z−1) A−1(z−1)v1(tc,k)

D δ(tc,k)

A(z−1)Ad(z−1)

+

ν(tc,k)

Fig. 4.6. Block scheme representing the closed-loop control structure

The weighting polynomials P (z−1) and Q(z−1) are calculated recursivelyby solving, at each step with a fixed integer, the following Diophantineequation:


A(z−1) Ad(z−1) Q(z−1) + B(z−1) P (z−1) = T (z−1) (4.35)

subject to (4.28c) and

nQ = nB − 1 (4.36a)

nT ≤ nA + nB (4.36b)

where polynomials A(z−1) and B(z−1) are replaced with their estimatesA(z−1) and B(z−1).

Once the Diophantine equation (4.35) has been solved, it is possible both tocalculate r0 = P (1) and to proceed by determining the polynomials G(z−1)and E(z−1) as the solutions of the Diophantine equation (4.27) in whichA(z−1) is replaced with A(z−1). Finally, the polynomial F (z−1) is calcu-lated from (4.32) and (4.14), where B(z−1) substitutes for B(z−1). The tripleF (z−1), G(z−1) and r0 gives the control law (4.33).

An alternative design procedure employs the estimate of the amplitude Dof the unknown no-load voltage disturbance to realize a compensating actionin the control law. This approach obviously does not require the presence ofthe difference filter Ad(z−1) in the Diophantine equation (4.27); accordinglyone has

P (z−1) = p0 A(z−1) E(z−1) + z−d G(z−1) (4.37)

To illustrate this alternative approach it is necessary to determine theexpression P (z−1) v1(tc,k+d). To this aim, starting from (4.37) and using (2.19)it is simple to verify that one obtains

P (z−1) v1(tc,k+d) = G(z−1) v1(tc,k) + p0 B(z−1) E(z−1) u(tc,k)

+ p0A(z−1)Ad(z−1)

E(z−1) D δ(tc,k+d) + p0 E(z−1) ν(tc,k+d)(4.38)

It is now important to note that in this case the noise sequence

p0 E(z−1) ν(tc,k+d)

is uncorrelated with the first two terms appearing at the right-hand sideof (4.38).

Regarding the disturbance term in (4.38),

p0A(z−1)Ad(z−1)

E(z−1) D δ(tc,k+d)

it represents the impulse response of filter p0 A(z−1) E(z−1)/Ad(z−1) thatreaches the constant value


v0,1 = p0 A(1) E(1) D (4.39)

after nA − 1 steps.The sub-optimal d-steps ahead weighted prediction is then given by

P (z−1) vp(tc,k+d) = G(z−1) v1(tc,k) + p0 B(z−1) E(z−1) u(tc,k) + v0,1

which, substituted in (4.26), gives the prediction of the generalized pseudo-output

ηp(tc,k+d) = G(z−1) v1(tc,k) + F (z−1) u(tc,k) − R(z−1) r(tc,k) + v0,1

where F (z−1) takes the form

F (z−1) = Ad(z−1) Q(z−1) + p0 B(z−1) E(z−1) (4.40)

The control law, which in this case sets ηp(tc,k+d) to zero, is given by

up(tc,k) = −G(z−1)F (z−1)

v1(tc,k) +R(z−1)F (z−1)

r(tc,k) − v0,1

F (z−1)(4.41)

The control law (4.41) leads to the following closed-loop equation:

v1(tc,k) = z−d B(z−1) R(z−1)A(z−1) Ad(z−1) Q(z−1) + B(z−1) P (z−1)

r(tc,k)

− z−d B(z−1)A(z−1) Ad(z−1) Q(z−1) + B(z−1) P (z−1)

v0,1

+A(z−1) F (z−1)

A(z−1) Ad(z−1) Q(z−1) + B(z−1) P (z−1)v0,1(tc,k)

+F (z−1)

A(z−1) Ad(z−1) Q(z−1) + B(z−1) P (z−1)ν(tc,k) (4.42)

At steady-state the sum of the second and third terms on the right-handside of (4.42) is equal to

− v0,1

P (1)+

A(1) F (1)B(1)P (1)

D

Since

F (1) = p0 B(1)E(1)


(see (4.40)) and having in mind (4.39), one obtains that such a sum is equalto zero.

Then, in the ideal case, the control input is able to cancel the constantbias at the output. However, in real applications, the effectiveness of the com-pensation action of the disturbance term is strictly related to the estimateaccuracy of v0,1.

4.2.5 Numerical Simulations

The numerical simulations reported in this chapter and in the next one havebeen developed using as test system the high-voltage transmission networkdescribed in Section A.1.2. Since the variables of the simulated power systemare initialized far from their steady-state values, the first 3 s of simulationare not significant because they do not represent actual system operation.Consequently, in this chapter the time variation of the variables of interestare reported only after the time 3 s.

Hereafter, in each simulation study a polynomial T (z−1) has been assignedso that the closed-loop step response has a rise time between 10% and 90%equal to about 40 ms, and a settling time at ±2% equal to about 70 ms,corresponding to respectively two and three and half times the fundamentalfrequency cycle.

Concerning the implementation of the voltage regulator scheme, it is worthrecalling that the control input u(tc,k) is saturated between 0 and 1 due to thepresence of the SVS actuator. However, since saturation has been neglectedin the control design, the integrator of the regulator may suffer from wind-up.This causes low-frequency oscillations and leads to instability. The wind-up isdue to the controller states becoming inconsistent with the saturated controlsignal, and future correction is ignored until the actuator desaturates. Toachieve antiwind-up the desaturated scheme shown in Figure 4.7 has beenimplemented [17].

Fig. 4.7. Voltage regulator scheme avoiding wind-up

The first simulation presented aims to show the performance of the nodalvoltage regulator scheme designed according to the pole-shifting technique.

Since it is set nA = 4, nB+ = 2 and d = 4, see Section A.1.3, con-straints (4.18) are fulfilled with nF = 3, nG = 4 and nT = 4; moreover,from (4.19) it is nFv

= 5.


The recursive least-squares algorithm has been initialized as follows:

θ(0) =[0 0 0 0 0.01 0 0 0.7

]TP(0) = 200 I8

εdz = 2 10−4

σ20 N0 = 0.01

λmin = 0.9

In addition cv has been set to 1.The polynomial T (ρ, z−1) has four roots; the largest ones, representing a

complex pair, belong to the cardioid characterized by ζ = 1/√

2. In addition,Ta,s = 50 ms and Ta,m = 70 ms, which leads to Rmin = 0.9231 and Rmax =0.9445, with Tc = 1 ms. When ρ = ρmax = 1 the largest roots of T (ρ, z−1)lie on the circle of radius Rmin (points B and C in Figure 4.4), while whenρ = ρmin = 0.7143 such roots lie on the circle of radius Rmax (points A andD in Figure 4.4). Since the initial value of the pole-shifting factor has beenchosen equal to 0.9, the initial values of the four roots of the polynomialT (ρ, z−1) are

z1,2 = 0.9281 ± j 0.0669

z3 = 0.4428 (q = 8)

z4 = 0.3612 (q = 10)

corresponding to ωn = 113.1371 rad/s.The coefficients fv,i (i = 1, . . . 5), gj (j = 0, . . . 4), h0 and the pole-shifting

factor ρ are updated every 0.01 s. The initial values of fv,i and gj are set equalto zero, except for

g0 = h0 =T (0.9, 1)

0.01= 0.1476

Finally we set γ = 10 and γsat = 0.1.The adaptive loop is closed at time instant t = 5.5 s.The time variation of the regulated rms voltage at node 4, namely v#4,1(t),

is shown on the left-hand side of Figure 4.17 while the right-hand side evi-dences its time characteristics, reporting an enlarged view of the time varia-tion between time instants t = 12.998 s and t = 13.1 s in response to the stepvariation of r(t) applied at t = 13 s.

The self-tuning voltage regulator scheme is able to meet the closed-loopspecifications in all three simulated operating conditions. The first operatingpoint changes due to the variation of load Q1 required at time instant t = 12 s.The second operating point disappears because the line L2 is opened at timeinstant t = 14 s. This change in the topology of the network determines thebegin of the third operating point.


Fig. 4.8. Time variation of v#4,1(t) in the case of pole-shifting design (left); enlargedview of v#4,1(t) between the time instants t = 12.998 s and t = 13.1 s (right)

The time variation of the pole-shifting factor is reported in Figure 4.9.After closing the adaptive loop, ρ exhibits a large variation at time instant t =7 s in which the first change of the set-point is imposed. Due to the incorrectinitialization of the recursive least-squares algorithm, the pole-shifting factoris drastically varied with respect to its initial value. Afterwards ρ(t) assumesan almost constant value throughout the simulation.

The absence of large and sudden variations in the time variation of ρ(t),except for that at time instant t = 7 s, ensures that the control input u(tc,k)does not saturate. This is confirmed by looking at the time variation of thefiring angle α(t) reported in Figure 4.10.

Fig. 4.9. Time variation of ρ(t)

Regarding the output of the recursive least-squares algorithm, Figure 4.11shows the time variation of a1, a2, a3, a4, while Figure 4.12 shows the timevariation of b0, b1, b2 and D.


Fig. 4.10. Time variation of α(t) in the case of pole-shifting design

Fig. 4.11. Time variation of a1, a2, a3, a4 in the case of pole-shifting design

The estimated no-load voltage amplitude D has been obtained accord-ing to (4.3). From the engineering point of view, its time variation is moreinteresting than v0,m.

The time variation of coefficients fv,1, fv,2, fv,3, fv,4, fv,5 and g0, g1, g2,g3, g4, h0 are reported in Figure 4.13 and in Figure 4.14.


Fig. 4.12. Time variation of b0, b1, b2, D in the case of pole-shifting design

Finally for the sake of comparison, the same simulation previously de-scribed has been run without adapting ρ(t), so implementing the pole-assignment technique.

The results obtained in terms of time variations of the regulated nodalvoltage and of the firing angle are reported in Figure 4.15 and in Figure 4.16.

The time variation of the regulated nodal voltage is quite similar with andwithout adoption of the pole-shifting technique, as can be seen comparing Fig-ure 4.8 with Figure 4.15. A little difference can be noticed in the time variationof the regulated nodal voltage at the time of line opening. In fact, with thepole-shifting technique the variation of the regulated nodal voltage is slightlysmaller than that obtained in the case of the pole-assignment technique.

Moreover, with the adoption of the pole-shifting technique it is possibleto avoid saturation of the firing angle which appears at time instants corre-sponding to line opening and set-point step changes; see Figure 4.16.

The results of numerical simulation of the implementation of the indirectself-tuning generalized minimum variance nodal voltage regulator are now pre-sented. The recursive least-squares algorithm has been initialized as follows:

θ(0) =[0 0 0 0.1 0.01 0 0.1 0.77

]TP(0) = 200 I8

cv = 1


Fig. 4.13. Time variation of fv,1, fv,2, fv,3, fv,4, fv,5 in the case of pole-shiftingdesign

εdz = 10−4

λmin = 0.9

The imposed initial values of plant parameters ai and bj (i = 1, . . . 4) (j =0, 1, 2), are then used to solve the Diophantine equation (4.35) with nT = 6see (4.36b), whose solutions represent the initial values of the coefficients ofthe weighting polynomials P (z−1) and Q(z−1), where nP = 4 and nQ = 1,see (4.28b) and (4.36a). To complete the initialization step, the initial valuesof the coefficients of the regulator polynomials Fv(z−1) and G(z−1) have to bedetermined. The coefficients of polynomials E(z−1) and G(z−1), with nE = 3and nG = 4, see (4.28a) and (4.28c), have been calculated as the solution ofthe Diophantine equation (4.27) while Fv(z−1) is obtained from (4.32). Thisprocedure furnished the following polynomials:

Fv(z−1) = 1 − 0.7z−1 − 0.61 z−2 + 0.439 z−3 − 0.0008537 z−4 + 0.0217 z−5

G(z−1) = 0.6885 − 1.0468z−1 + 0.4369 z−2 − 0.0218 z−3 + 0.0217 z−4

h0 = 0.0785


Fig. 4.14. Time variation of of g0, g1, g2, g3, g4, h0 in the case of pole-shiftingdesign

The updating of the weighting polynomials P (z−1), Q(z−1) and r0 as wellas the regulator polynomials Fv(z−1), G(z−1) and h0 is accomplished every10 Tc = 0.01 s.

The adaptive loop is closed at time instant t = 6 s.The time variation of the regulated rms voltage at node 4, namely v#4,1(t),

is shown in the left-hand side of Figure 4.17 while the right-hand side evidencesits time characteristics, reporting an enlarged view of the time variation be-tween the time instants t = 13.998 s and t = 14.1 s in response to the stepvariation of r(t) applied at t = 14 s. The rise time obtained is equal to 30 mswhile the settling time is 71 ms. These characteristics have been evaluatedafter a 20% step increase of the load Q1. Observing the response to load in-creasing, it is possible to recognize that the voltage drop is limited to about0.3% of the reference value and the regulated nodal voltage quickly recoversto the nominal value.

The time variation of the firing angle α(t) is reported in Figure 4.18.Regarding the output of the RLS algorithm, Figure 4.19 shows the time

variation of a1, a2, a3, a4, while Figure 4.20 shows the time variation of b0,b1, b2 and D.


Fig. 4.15. Time variation of v#4,1(t) in the case of indirect pole-assignment design(left); enlarged view of v#4,1(t) between the time instants t = 12.998 s and t = 13.1 s(right)

Fig. 4.16. Time variation of α(t) in the case of indirect pole-assignment design

Fig. 4.17. Time variation of v#4,1(t) in the case of indirect generalized mini-mum variance design (left); enlarged view of v#4,1(t) between the time instantst = 13.998 s and t = 14.1 s (right)


Fig. 4.18. Time variation of α(t) in the case of indirect generalized minimum vari-ance design

The estimated no-load voltage amplitude D has been obtained accord-ing to (4.3). From the engineering point of view, its time variation is moreinteresting than v0,m.

Finally the time variation of the coefficients fv,1, fv,2, fv,3, fv,4, fv,5 andg0, g1, g2, g3, g4, h0 are reported in Figure 4.21 and in Figure 4.22.

Fig. 4.19. Time variation of a1, a2, a3, a4 in the case of indirect generalized mini-mum variance design

4.3 Direct Self-tuning Voltage Regulator Design 63

Fig. 4.20. Time variation of b0, b1, b2, D in the case of indirect generalized minimumvariance design

4.3 Direct Self-tuning Voltage Regulator Design

Section 4.2 has illustrated the design of a self-tuning voltage regulator using toan indirect method. Differently from this method, the design of a direct self-tuning regulator is realized in one step. Basically, the key idea is to exploit theclosed-loop specifications, assigned in terms of roots of the polynomial T (z−1),to re-parameterize the power system model so that it can be expressed interms of aggregate parameters which are function of the unknown coefficientsof the polynomials Fv(z−1) and G(z−1). At this point a constrained recursiveleast-squares algorithm (CRLS) with variable forgetting factor is adopted toestimate such aggregate parameters. Once such parameters have been esti-mated, the regulator polynomials are easily obtained by solving linear systemequations.

4.3.1 Pole-assignment Design

The direct self-tuning voltage regulator design will first be developed withreference to a power system model whose polynomial B(z−1) has only stablezeros, that is nB− = 0, and under the hypothesis neglecting the presence of


Fig. 4.21. Time variation of fv,1, fv,2, fv,3, fv,4, fv,5 in the case of indirect gener-alized minimum variance design

the white noise term ν(tc,k). The design in the presence of unstable zeros inB(z−1) will be treated afterwards as well as the case ν(tc,k) = 0. To begin weexpress polynomial B(z−1), according to (2.16), as

B(z−1) = b0 B+(z−1)

Application of the pole-assignment technique requires solution of the Dio-phantine equation (4.17), which in this case becomes

A(z−1) Ad(z−1) F (z−1) + z−d b0 G(z−1) = T (z−1) (4.43)

subject to the constraints (4.18) with n−B = 0, that is

nF = d − 1 (4.44a)

nG = nA + nAd− 1 = nA (4.44b)

nT ≤ nA + nAd+ d − 1 = nA + d (4.44c)

In particular, in the remainder, constraint (4.44c) will be satisfied assuming


Fig. 4.22. Time variation of g0, g1, g2, g3, g4, h0 in the case of indirect generalizedminimum variance design

nT = nA + d

that represents the general case. Trivially, if nT < nA + d, then some coef-ficients of the polynomial T (z−1) will be zero. Even in this case, due to thepresence of the disturbance term v0,1(tc,k) in the power system model, theregulator will contain an integral action, that is the polynomial F (z−1) takesthe form given by (4.14).

The first step in this design procedure aims to obtain the re-parameterizedpower system model (2.17). Multiplying the Diophantine equation (4.43) bythe voltage amplitude v1(tc,k) and using model (2.17) one obtains

A(z−1) F (z−1) D δ(tc,k) + z−d b0 B+(z−1) Ad(z−1)F (z−1)u(tc,k)

+ z−d b0 G(z−1) v1(tc,k) = T (z−1) v1(tc,k)(4.45)

At this point it can be recognized that the term A(z−1) F (z−1) appearingin (4.45) is a FIR filter; then we have


A(z−1) F (z−1) D δ(tc,k) = 0 ∀ k ≥ nA + d − 1

Based on this, it is then possible to consider the following model:

z−d b0

(Ad(z−1) F (z−1) B+(z−1) u(tc,k) + G(z−1) v1(tc,k)

)= T (z−1) v1(tc,k)

that, based on (4.15), can be rewritten as

z−d b0

(Ad(z−1) Fv(z−1) u(tc,k) + G(z−1) v1(tc,k)

)= T (z−1) v1(tc,k) (4.46)

At this point expressions for polynomials Fv(z−1), G(z−1) and Ad(z−1)have to be substituted in (4.46). Since nB− = 0, from (4.19) we have

nFv= nB+ + nF = nB+ + d − 1

Recalling constraint (4.44b), one obtains

b0

(u(tc,k−d) + (fv,1 − 1) u(tc,k−d−1) +

nB++d−1∑i=2

(fv,i − fv,i−1) u(tc,k−2i−1)

− fv,nB++d−1 u(tc,k+nB++2d) +nA∑i=0

gi v1(tc,k−d−1))

= v1(tc,k) +nA+d∑i=1

τi v1(tc,k−i)

which, solved with respect to v1(tc,k), yields

v1(tc,k) = −((τd − b0 g0) v1(tc,k−d) + . . . + (τnA+d − b0 gnA

) v1(tc,k−nA−d))

+ b0

(u(tc,k−d) + (fv,1 − 1) u(tc,k−d−1)

+nB++d−1∑

i=2

(fv,i − fv,i−1) u(tc,k−2i−1) − fv,nB++d−1 u(tc,k−nB+−2d))

−d−1∑i=1

τi v1(tc,k−i) (4.47)

Equation (4.47) put in compact form becomes


v1(tc,k) = φT(tc,k) θ −d−1∑i=1

τi v1(tc,k−i) (4.48)

in which the regression vector φ and the unknown parameters vector θ aregiven by

φT =[− v1(tc,k−d) . . . − v1(tc,k−nT

) u(tc,k−d) u(tc,k−d−1) . . .

. . . u(tc,k−nB+−2d)]

θ =[(τd − b0 g0) . . . (τnA+d − b0 gnA

) b0 b0(fv,1 − 1) b0(fv,2 − fv,1) . . .

. . . − b0 fv,nB++d−1

]T(4.49)

Analyzing the structure of vector (4.49) it is easy to recognize that thenumber of parameters to estimate is given by

nr = nA + nB+ + d + 2 (4.50)

When B(z−1) has unstable zeros, the Diophantine equation to solve isgiven by (4.17) under constraints (4.18).

Even in this case, constraint (4.18c) will be satisfied imposing nT = nA +nB− + d. Since in this case nB− = 0, (4.19) gives

nFv= nB+ + nF = nB + d − 1

By repeating the procedure illustrated in the case nB− = 0, after somemathematical manipulations it is possible to obtain the recursive expressionwhich gives v1(tc,k) in compact form as in (4.48). In particular, the regressionvector φ and the unknown parameters vector θ have the following structure

φT =[− v1(tc,k−d) − v1(tc,k−d−1) . . . − v1(tc,k−nT

)

u(tc,k−d) u(tc,k−d−1) . . . u(tc,k−2d−nB)]

θ =[(τd − b0 g0) (τd+1 − b0 g1 − b−1 g0) . . .

. . . (τnT− b−nB− gnA

) b0 (−b0 + b0 fv,1 + b−1 )

(−b0 fv,1 + b0 fv,2 − b−1 + b−1 fv,1 + b−2 ) . . .

. . . − b−nB− fv,d−1+nB+

]T(4.51)


In this case the number of parameters to estimate is expressed by

nq = nA + d + 2 + nB+ + 2nB−

It is important to remark that the re-parameterized model (4.48) has beenobtained by neglecting the term A F D δ(tc,k) in (4.45). This leads to modelingerror for the first nA + d − 1 samples. However, it is worth noticing thatnA + d − 1 < nr < nq. In addition, since θ is estimated by using a CRLSwith forgetting factor, the modeling error has no effect on the asymptoticestimates.

To estimate vector θ it is worth noting that the last d + nB+ + 1 elementsin (4.49), respectively, d + nB + 1 elements in (4.51), sum to zero. This con-sideration introduces into the estimation algorithm the following constraint:

nf∑i=ns

θi = 0 (4.52)

with ns = nA + 2nf = np

in the case nB− = 0, or ns = nA + 2 + nB−

nf = nq

otherwise.The parameters vector θ is then obtained by solving problem

minθ

J (4.53)

with J given by (4.7) subject to

wT1 θ = 0 (4.54)

where

wT1 =

[0 . . . . . . 0︸︷︷︸

nA+1

1 . . . . . . . . 1︸︷︷︸d+1+nB+

] (nB− = 0)

orwT

1 =[

0 . . . . . . 0︸︷︷︸nA+1+nB−

1 . . . . . . . . 1︸︷︷︸nB+d+1

] (nB− > 0)

Problem (4.53),(4.54) is a convex quadratic programming problem whichhas a global minimum that can be found by solving the well-known first-orderconditions on the Lagrangian function [100]


L = J + µ w1θ

µ being a multiplier.The solution to problem (4.53),(4.54) is

θ(tc,k) = θu(tc,k) − µ

2PT(tc,k) w1

in which

µ = 2θ

T

u (tc,k)w1

wT1 P(tc,k) w1

Vector θu, solution of the unconstrained problem (4.53), is obtained ac-cording to (4.8),(4.11).

The concluding step in the pole-assignment design consists in determin-ing the coefficients of polynomials Fv(z−1) and G(z−1) and the scalar h0

starting from the estimated vector θ. In the case nB− = 0 this task canbe easily accomplished. In fact, looking at the parameters vector (4.49) itcan be recognized that the last d + 1 + nB+ elements are functions of fv,i,(i = 1, . . . d−1+nB+) and b0. There are then d+1+nB+ equations in d+nB+

unknowns, i.e. fv,1 . . . fv,d−1+nB+ , b0. However since constraint (4.52) holds,the number of linearly independent equations is d + nB+ . In addition since b0

coincides with θnA+2, the number of equations becomes d − 1 + nB+ . Then,the set of d− 1 + nB+ linear equations in d− 1 + nB+ unknowns representedby the coefficients fv,i can be trivially solved by substitution.

Concerning the calculation of the coefficients gj , they can easily be ob-tained from the following equation:

gj =τj+d − θj+1

θnA+2

, j = 0, . . . nA

see the first nA + 1 elements of θ in (4.49).In the case nB− > 0, from the analysis of vector θ in (4.51) it is possible to

recognize that the nq estimated parameters yield a set of nq − 1 independentnonlinear equations in the nq − 1 unknowns, consisting of fv,i (i = 1 . . . nB +d − 1), gj (j = 0, . . . nA), b0 and b−q (q = 1, . . . nB−). Solving this nonlinearsystem of equations is a critical task in on-line implementations because theuniqueness of the solution is not guaranteed and the calculation time requiredto solve this nonlinear system is uncertain.

Finally, whether nB− = 0 or nB− > 0 is, h0 is computed as in (4.20).The illustrated design of a direct self-tuning voltage regulator has been

developed by neglecting the presence of the noise term in the power systemmodel. If this assumption is removed, the noise term appears in model (4.45),which becomes


A(z−1) F (z−1) D δ(tc,k) + z−d b0 B+(z−1) Ad(z−1)F (z−1) u(tc,k)

+ z−d b0 G(z−1) v1(tc,k) + Ad(z−1) F (z−1) ν(tc,k) = T (z−1) v1(tc,k)

After nA + d − 1 steps it is then possible to consider the following model:

z−d b0

(Ad(z−1) F (z−1) B+(z−1) u(tc,k) + G(z−1) v1(tc,k)

)+Ad(z−1) F (z−1) ν(tc,k) = T (z−1) v1(tc,k)

which, according to (4.15), can be rewritten as

z−d b0

(Ad(z−1) Fv(z−1) u(tc,k) + G(z−1) v1(tc,k)

)+ Ad(z−1) F (z−1) ν(tc,k)

= T (z−1) v1(tc,k)(4.55)

Writing (4.55) in extended form yields

b0

(u(tc,k−d) + (fv,1 − 1) u(tc,k−d−1) +

nB++d−1∑i=2

(fv,i − fv,i−1) u(tc,k−2i−1)

− fv,nB++d−1 u(tc,k+nB++2d) +nA∑i=0

gi v1(tc,k−d−1))

+ ν(tc,k)

+ (f1 − 1) ν(tc,k−1) +d−1∑i=2

(fi − fi−1

)ν(tc,k−i) − fd−1 ν(tc,k−d)

= v1(tc,k) +nA+d∑i=1

τi v1(tc,k−i)

which, solved with respect to v1(tc,k), gives

v1(tc,k) = −((τd − b0 g0) v1(tc,k−d) + . . . + (τnA+d − b0 gnA

) v1(tc,k−nA−d))

+ b0

(u(tc,k−d) + (fv,1 − 1) u(tc,k−d−1)

+nB++d−1∑

i=2

(fv,i − fv,i−1) u(tc,k−2i−1) − fv,nB++d−1 u(tc,k+nB+−2d))


+ ν(tc,k) + (f1 − 1) ν(tc,k−1) +d−1∑i=2

(fi − fi−1

)ν(tc,k−i)

− fd−1 ν(tc,k−d) −d−1∑i=1

τi v1(tc,k−i) (4.56)

Equation (4.56) put in compact form becomes

v1(tc,k) = φT(tc,k) θ −d−1∑i=1

τi v1(tc,k−1) + ν(tc,k) (4.57)

in which the regression vector φ and the unknown parameters vector θ aregiven by

φT =[− v1(tc,k−d) . . . − v1(tc,k−d) u(tc,k−d) . . . u(tc,k+nB+−2d)

ν(tc,k−1) . . . ν(tc,k−d+1) ν(tc,k−d)]

θ =[(τd − b0 g0) . . . (τnA+d − b0 gnA

) b0 . . . − b0 fv,nB++d−1

(f1 − 1) . . . (fd−1 − fd−2) − fd−1

]TIn order to give an estimate of the true parameters vector it is important

to note that model (4.57) is corrupted by coloured noise. The estimate ofvector θ requires the knowledge of the sequence

ν(tc,k−1) ν(tc,k−2) . . . ν(tc,k−d) (4.58)

that represents an unobservable model bias. To overcome this problem esti-mation procedures, such as recursive extended least-squares or recursive max-imum likelihood algorithms can be employed [10, 77, 127]. For example, inthe case of the recursive extended least-squares technique, the unknown se-quence (4.58) is substituted by the sequence

ε(tc,k−1) ε(tc,k−2) . . . ε(tc,k−d)

which leads to definition of the following new regressor vector

ϕT =[− v1(tc,k−d) . . . − v1(tc,k−d) u(tc,k−d) . . . u(tc,k+nB+−2d)

ε(tc,k−1) . . . ε(tc,k−d+1) ε(tc,k−d)]


where

ε(tc,k) = v1(tc,k) − ϕT(tc,k) θ(tc,k−1)

is the prediction error computed using the estimate θ(tc,k−1).In the recursive extended least-squares algorithm the estimates are deter-

mined exactly as in the recursive least-squares method, where the estimatedprediction error is included as another input signal. The asymptotic propertiesof the extended least-squares algorithm and the maximum likelihood methodare established in the literature on system identification [10,82,83,127].

4.3.2 Generalized Minimum Variance Pole-assignment Design

According to the direct approach, in the generalized minimum variance pole-assignment design the polynomials Fv(z−1) and G(z−1) are directly estimated.In particular polynomial Fv(z−1) will not be assumed monic; that is

Fv(z−1) = fv,0 + fv,1z−1 + . . . + fv,nFv

z−nFv

The motivation of this assumption will be given shortly.The first design step consists in forming the generalized pseudo-output,

see (4.26) and (4.34), at time instant tc,k

η(tc,k) = P (z−1) v1(tc,k) + Q(z−1) Ad(z−1) u(tc,k−d) − r0 r(tc,k−d)

This value is then used to estimate Fv(z−1) and G(z−1) from

η(tc,k) = Fv,k(z−1) uv(tc,k−d) + Gk(z−1) v1(tc,k−d) − h0,k r(tc,k−d) + e(tc,k)(4.59)

where

e(tc,k) = p0 A(z−1) E(z−1) D δ(tc,k) + p0 Ad(z−1) E(z−1) ν(tc,k)

in which subscript k denotes estimates at time instant tc,k.When estimating regulator polynomials from (4.59), one parameter needs

to be fixed [2, 11, 127]. If fv,0 is set to unit, the parameter estimation coulddiverge if the true value of fv,0 is too far from unity [2, 11]. Based on thismotivation it has been assumed that Fv(z−1) is not monic. From the analysisof the transfer function of the closed-loop scheme in Figure 4.6 it is easy torecognize that one can set h0,k = Gk(1) to ensure unitary steady-state gainfor the closed-loop transfer function.

Equation (4.59) is subsequently used to recursively estimate Fv,k(z−1) andGk(z−1). It is useful to recall that the disturbance term in e(tc,k) vanishes after


nA − 1 steps, see (4.31), and that the sole term p0 ed−1 ν(tc,k−d) appearingin e(tc,k) is correlated with the sequences uv(tc,k−d), v1(tc,k−d) and r(tc,k−d).Then, this term must be accounted for in the prediction error used by theRLS procedure, according to

ep(tc,k) = η(tc,k) − Fv,k−1(z−1) uv(tc,k−d) − Gk−1(z−1) v1(tc,k−d)

+h0,k−1 r(tc,k−d) + p0 ed−1 ν(tc,k−d)

in which the sequence ν(tc,k) can be recursively evaluated from

ν(tc,k) =1

p0 Ad(z−1) E(z−1)e(tc,k)

where e(tc,k) is obtained from (4.59).The estimated polynomials concur to define the regulator law

⎧⎪⎪⎪⎨⎪⎪⎪⎩uv(tc,k) =

1

fv,0,k

(−

nFv∑i=1

fv,i,k uv(tc,k−i) −nA∑i=0

gi,k v1(tc,k−i) + h0,k r(tc,k)

)

u(tc,k) = uv(tc,k) + u(tc,k−1)

To ensure that the closed-loop poles coincide with the zeros of the as-signed polynomial T (z−1) it is necessary to derive the equation which de-scribes the relationship between P (z−1), Q(z−1), Fv(z−1) and G(z−1). Wemultiply (4.27) by Q(z−1), (4.32) by P (z−1) and (4.35) by p0 E(z−1) to give

P (z−1) Q(z−1) = p0 A(z−1) Ad(z−1) E(z−1) Q(z−1)

+ z−d G(z−1)Q(z−1) (4.60a)

Fv(z−1) P (z−1) = p0 B(z−1) E(z−1) P (z−1) + P (z−1) Q(z−1) (4.60b)

p0 E(z−1) T (z−1) = p0 A(z−1) Ad(z−1) E(z−1) Q(z−1)

+ p0 B(z−1) P (z−1) E(z−1) (4.60c)

Substituting in (4.60b) the term P (z−1) Q(z−1) given by (4.60a) and theterm p0 B(z−1) E(z−1) P (z−1) obtained from (4.60c) yields

Fv(z−1) P (z−1) − z−d G(z−1) Q(z−1) = fv,0 p0 E(z−1) T (z−1) (4.61)

where the factor fv,0 has been introduced in the right-hand side becauseFv(z−1) has been assumed not monic.


By replacing in the Diophantine equation (4.61) the polynomials Fv(z−1)and G(z−1) with their estimates Fv(z−1) and G(z−1) it is possible to recur-sively compute at each k Tc time instant the coefficients qi/p0, (i = 0, . . . nQ)pi/p0, (i = 1, . . . nA) and ei, (i = 1 . . . d − 1). Finally the coefficient p0 is de-termined as

p0 =h0

1 +nA∑i=1

pi

p0

with h0 = G(1) and setting r0 = h0.


The results of the numerical simulations reported here aim to show the per-formance of the voltage regulator scheme designed using the direct self-tuningapproach. For this reason two simulation studies are reported. The first is de-veloped with reference to the pole-assignment design while the second studyis developed with reference to the generalized minimum variance design.

Regarding the required closed-loop specifications defined by the zeros ofpolynomial T (z−1), they are imposed equal to those illustrated in Section 4.2.5for the case of simulations of the indirect self-tuning approach. In particular,a closed-loop step-response is required exhibiting a rise time between 10% and90% equal to about 40 ms, and a settling time at ±2% no greater than 70 ms,respectively corresponding to two and three and half times the fundamentalfrequency cycle.

In the simulation study of the nodal voltage regulator scheme designedon the basis of the pole-assignment technique, the number of parameters toestimate, namely nr, is 12, see (4.50), for nA = 4, nB+ = 2 and d = 4. Theirinitial values have been assigned, starting from suitable initial polynomialsFv(z−1) and G(z−1) whose roots are inside the unit circle. The choice of suchpolynomials is an important task in the initialization step since wrong initialvalues lead to an algorithm that does not converge. In particular, the initialvalue of b0 has been determined as

b0 =T (1)G(1)

= 0.0021

since at the start-up of the CRLS algorithm h0 = G(1). To help the conver-gence of the algorithm it is advisable that the value obtained for b0 is not toofar from its true value. This consideration is the result of several simulationsconducted with different initializations. The degrees of such polynomials arenFv

= 5 and nG = 4. The constrained recursive least-squares algorithm hasbeen run with εdz = 10−5, σ2

0 N0 = 0.1 and λmin = 0.9.The adaptive loop is closed at time instant t = 5 s.


The time variation of the regulated rms voltage at node 4, namely v#4,1(t),is shown in the left-hand side of Figure 4.23 while the right-hand side evidencesits time characteristics, reporting an enlarged view of the time variation be-tween the time instants t = 8.998 s and t = 9.1 s in response to the stepvariation of r(t) applied at t = 9 s. From Figure 4.23 the effect on the regu-lated nodal voltage of the closing of the adaptive loop is evident. Within 0.1 s,v#4,1(t) assumes the value of the desired set-point.

Fig. 4.23. Time variation of v#4,1(t) in the case of direct pole-assignment design(left); enlarged view of v#4,1(t) between the time instants t = 8.998 s and t = 9.1 s(right)

In addition, at time instant t = 12 s the regulated voltage drops-out sincea 20% step increase of the load Q1 is imposed.

The time variation of α(t) obtained is shown in Figure 4.24.

Fig. 4.24. Time variation of α(t) in the case of direct pole-assignment design

The time variations of the coefficients fvi, gj and h0 obtained are respec-tively reported in Figure 4.25 and in Figure 4.26.


Fig. 4.25. Time variation of fv1, fv2, fv3, fv4, fv5 in the case of direct pole-assignment design

Concerning the implementation of the direct self-tuning generalized mini-mum variance approach, a critical aspect of the implementation is the ini-tialization of the estimation algorithm. In fact, this direct method can-not converge if the recursive least-squares algorithm is carelessly initial-ized [2,10,31,127]. In this simulation study, the initial values of the coefficientsof the polynomials E(z−1), P (z−1) and Q(z−1) were calculated solving (4.61)with nE = 3, nP = 4, nG = 4, nQ = 1, nFv

= 5 and nT = 6, see (4.28), (4.32)and (4.36).

Polynomials Fv(z−1) and G(z−1) are the same used to initialize the CRLSalgorithm in the case of the simulation of the pole-assignment design. In ad-dition, denoting with bM

0 a known upper bound on b0, the coefficient p0 hasbeen determined as

p0 =1

bM0 + q0/p0

obtained from (4.32) particularized for z−1 = 0 and fv0 = 1. The value of bM0

has been chosen equal to 0.1.The adaptive loop is closed at time instant t = 5 s. Moreover, equa-

tion (4.61) is solved every = 50 steps.


Fig. 4.26. Time variation of g0, g1, g2, g3, g4, h0 in the case of direct pole-assignmentdesign

The time variation of the regulated rms voltage at node 4, namely v#4,1(t),is shown in the left-hand side of Figure 4.27, while the right-hand side evi-dences its time characteristics, reporting an enlarged view of the time variationbetween the time instants t = 8.998 s and t = 9.1 s in response to the stepvariation of r(t) applied at t = 9 s.

Fig. 4.27. Time variation of v#4,1(t) in the case of direct generalized minimumvariance design (left); enlarged view of v#4,1(t) between the time instants t = 8.998 sand t = 9.1 s (right)


In addition, at time instant t = 12 s the regulated voltage drops out sincea 20% step increase of the load Q1 is imposed.

The time variation of α(t) obtained is shown in Figure 4.28.

Fig. 4.28. Time variation of α(t) in the case of direct generalized minimum variancedesign

Finally the time variations of the estimated coefficients fvi, gj and h0 arereported in Figure 4.29 and in Figure 4.30.

4.4 Properties of the Recursive Least-squares Algorithm

The design of self-tuning voltage regulators, using direct or indirect methods,employs the recursive least-squares algorithm (4.8)-(4.11) in the parameterestimation. For this reason it is advisable to study the properties of this algo-rithm.

Lemma 4.1. Algorithm (4.8)-(4.11) has the following properties:

(a) P(tc,k) = PT(tc,k), and P(tc,k) is bounded;

(b) limk→∞

ε2(tc,k)γ2(tc,k)

= 0, where γ2(tc,k) = 1 + φT(tc,k)P(tc,k−1) φ(tc,k);

(c) θ(tc,k) − θ(tc,k−1) ∈ L2;

(d) ∃Pλ,∞ and θ∞ such that

limk→∞

λp(k, 1)P(tc,k) = Pλ,∞

limk→∞

θ(tc,k) = θ∞

where λp(k, 1) =k∏

i=1

λ(tc,i) .

4.4 Properties of the Recursive Least-squares Algorithm 79

Fig. 4.29. Time variation of fv0, fv1, fv2, fv3, fv4, fv5 in the case of direct gener-alized minimum variance design

Proof. To demonstrate property (a) let us write the inverse of P(tc,k)

P−1(tc,k) = λ(tc,k)

(P(tc,k−1) − P(tc,k−1)φ(tc,k)φT(tc,k)PT(tc,k−1)

1 + φT(tc,k)P(tc,k−1)φ(tc,k)

)−1

(4.62)which, based on the the Matrix Inversion Lemma [58], can be rewritten as

P−1(tc,k) = λ(tc,k)P(tc,k−1) + λ(tc,k) φ(tc,k) φT(tc,k)

By reiterating this equality one has

P−1(tc,k) = λp(k, 1)P−1(0) +k∑

i=1

λp(k, i) φ(tc,i) φT(tc,i) (4.63)

where

λp(k, i) =k∏

j=i

λ(tc,j)


Fig. 4.30. Time variation of g0, g1, g2, g3, g4, h0 in the case of direct generalizedminimum variance design

Equation (4.63) means that P−1(tc,k) is nondecreasing, i.e.

P−1(tc,k) =(P−1(tc,k)

)T ≥ P−1(0) > 0

and P(tc,k) = PT(tc,k) > 0. So P(tc,k) is bounded.To prove property (b) let us introduce the residual e(tc,k) defined as

e(tc,k) = θ − θ(tc,k) (4.64)

Using (4.6) and (4.9), (4.64) becomes

e(tc,k) =

(I − P(tc,k−1)φ(tc,k)φT(tc,k)

γ2(tc,k)

)e(tc,k−1)

From (4.62) it follows that

e(tc,k) = λ(tc,k)P(tc,k)P−1(tc,k−1) e(tc,k−1) (4.65)

We introduce the positive definite Lyapunov function


V(e(tc,k−1), tc,k−1

)=

1λ(tc,k−1)

eT(tc,k−1)P−1(tc,k−1) e(tc,k−1)

in which

0 < λmin ≤ λ(tc,k) ≤ 1

The time increment ∆ V (tc,k) = V(e(tc,k), tc,k

) − V(e(tc,k−1), tc,k−1

)along (4.9) is

∆ V (tc,k) =eT(tc,k)λ(tc,k)

P−1(tc,k) e(tc,k) − eT(tc,k−1)λ(tc,k−1)

P−1(tc,k−1) e(tc,k−1)

=eT(tc,k)λ(tc,k)

P−1(tc,k) λ(tc,k)P(tc,k)P−1(tc,k−1) e(tc,k−1)

− eT(tc,k−1)λ(tc,k−1)

P−1(tc,k−1) e(tc,k−1)

=

(eT(tc,k) − eT(tc,k−1)

λ(tc,k−1)

)P−1(tc,k−1) e(tc,k−1)

Since

e(tc,k) = e(tc,k−1) − P(tc,k−1)φ(tc,k)ε(tc,k)γ2(tc,k)

(4.66)

one obtains

∆ V (tc,k) =

((1 − 1

λ(tc,k−1)

)e(tc,k−1) − εT(tc,k) φT(tc,k)P(tc,k−1)

γ2(tc,k)

)

P−1(tc,k−1) e(tc,k−1)

= −1 − λ(tc,k−1)λ(tc,k−1)

eT(tc,k−1)P−1(tc,k−1) e(tc,k−1)

−εT(tc,k) φT(tc,k) e(tc,k−1)γ2(tc,k)

and from (4.6)

∆ V (tc,k) = −1 − λ(tc,k−1)λ(tc,k−1)

eT(tc,k−1)P−1(tc,k−1) e(tc,k−1) − ε2(tc,k)γ2(tc,k)

≤ 0


Hence 0 ≤ V(e(tc,k), tc,k

) ≤ V(e(tc,k−1), tc,k−1

)and V

(e(tc,k), tc,k

)is

nonincreasing; then

limk→∞

V(e(tc,k), tc,k

)= V ∗ ≥ 0

Since

0 ≤ V(e(tc,k), tc,k

)= V

(e(tc,k−1), tc,k−1

)− 1 − λ(tc,k−1)

λ(tc,k−1)eT(tc,k−1)P−1(tc,k−1) e(tc,k−1) − ε2(tc,k)

γ2(tc,k)

≤ V(e(tc,k−1), tc,k−1

)− ε2(tc,k)γ2(tc,k)

one obtains

0 ≤ V ∗ ≤ V ∗ − limk→∞

ε2(tc,k)γ2(tc,k)

that is

limk→∞

ε2(tc,k)γ2(tc,k)

= 0 (4.67)

Proof of property (c). From (4.64) and (4.66) one has

θ(tc,k) − θ(tc,k) =P(tc,k−1)φ(tc,k)ε(tc,k)

γ2(tc,k)

hence

∥∥θ(tc,k) − θ(tc,k)∥∥2

2=∥∥P(tc,k−1)φ(tc,k)

∥∥2

2

ε2(tc,k)γ4(tc,k)

=

∥∥P(tc,k−1)φ(tc,k)∥∥2

2

1 + φT(tc,k)P(tc,k−1) φ(tc,k)ε2(tc,k)γ2(tc,k)

(4.68)

Now since the following inequality holds

0 ≤∥∥P(tc,k−1)φ(tc,k)

∥∥2

2

1 + φT(tc,k)P(tc,k−1) φ(tc,k)≤

∥∥PTs (tc,k−1)

∥∥2

2

∥∥Ps(tc,k−1)φ(tc,k)∥∥2

2

1 + φT(tc,k)PTs (tc,k−1)Ps(tc,k−1) φ(tc,k)

=∥∥PT

s (tc,k−1)∥∥2

2

∥∥Ps(tc,k−1) φ(tc,k)∥∥2

2

1 +∥∥Ps(tc,k−1) φ(tc,k)

∥∥2

2

≤ ∥∥Ps(tc,k−1)∥∥2

2≤ c


where P(tc,k−1) PTs (tc,k−1)Ps(tc,k−1) and c ∈ IR+, it is possible to affirm

that the term ∥∥P(tc,k−1)φ(tc,k)∥∥2

2

1 + φT(tc,k)P(tc,k−1) φ(tc,k)

is bounded. Then from (4.67) and(4.68) follows that

limk→∞

∥∥θ(tc,k) − θ(tc,k−1)∥∥

2= 0

and then

θ(tc,k) − θ(tc,k−1) ∈ L2

Finally to prove (d) we write

λp(k, 1)P(tc,k) = P(0) −k∑

i=1

λf (tc,i)P(tc,i−1) φ(tc,i)φT(tc,i)P(tc,i−1)

γ2(tc,i)

where

λf (tc,i) =λp(k, 1)λp(k, i)

=i−1∏j=1

λ(tc,j) ∀ i ≥ 2 with λf (tc,1) = 1

For any constant vector χ one obtains

χT λp(k, 1)P(tc,k)χ

= χT P(0) χ −k∑

i=1

λf (tc,i)χT P(tc,i−1) φ(tc,i)φT(tc,i)P(tc,i−1) χ

γ2(tc,i)≥ 0

that is

k∑i=1


γ2(tc,i)≤ χT P(0) χ

is nondecreasing and bounded from above. Then there exists and is finite

limk→∞

k∑i=1


γ2(tc,i)

andlim

k→∞χT λp(k, 1)P(tc,k)χ


That is, there exists a constant matrix Pλ,∞ such that

limk→∞

λp(k, 1)P(tc,k) = Pλ,∞

However, from (4.65) for backwards iterative substitutions one obtains

e(tc,k) = λp(k, 1)P(tc,k)P−1(0) e(0) (4.69)

and then

limk→∞

e(tc,k) = Pλ,∞ P−1(0) e(0)

which implies that

limk→∞

θ(tc,k) = θ + Pλ,∞ P−1(0) e(0) = θ∞

To address the issue of parameter convergence of the recursive least-squaresalgorithm it is necessary to recall the definition of persistent excitation forthe regressor φ(tc,k) which states that there exists ka > 0 and β > 0 suchthat [115]

ks+ka∑k=ks

φ(tc,k) φT(tc,k) ≥ β I ∀ks ≥ 0 (4.70)

Now, reconsider (4.63) written as

1λp(k, 1)

P−1(tc,k) = P−1(0) +k∑

i=1

1λf (tc,i)

φ(tc,i) φT(tc,i)

If the condition of persistancy of the excitation for φ(tc,k) is verified, then

limk→∞

min

eig

k∑

i=1

1λf (tc,i)

φ(tc,i) φT(tc,i)

= ∞

from which it follows that

limk→∞

min

eig

1

λp(k, 1)P−1(tc,k)

= ∞

Now, since

min

eig

1

λp(k, 1)P−1(tc,k)

=

1

max

eig

λp(k, 1)P(tc,k)


one has

limk→∞

max

eig

λp(k, 1)P(tc,k)

= 0

From (4.69) it follows that

limk→∞

e(tc,k) = 0

which implies that

limk→∞

θ(tc,k) = θ

The presence of the constant input cv in the regressor (4.5) does not influ-ence the fulfilment of condition (4.70). To show this, let us define the regressor

φTw =

[− v1(tc,k−1) . . . − v1(tc,k−nA) u(tc,k−d) . . . u(tc,k−d−nB

)]

which, for hypothesis, is persistently exciting. Obviously one has

φT =[φT

w cv]

Assuming without loss of generality ks = 0, condition (4.70) is rewrittenas

ka∑k=0

φ(tc,k) φT(tc,k) = Φ(tc,k)

=

⎡⎢⎢⎢⎢⎢⎣Φw(tc,k) cv

ka∑k=0

φw(tc,k)

cv

ka∑k=0

φTw(tc,k) c2

v (ka + 1)

⎤⎥⎥⎥⎥⎥⎦ ∈ IR(np×np)

where

Φw(tc,k) =ka∑

k=0

φw(tc,k) φTw(tc,k)

Let us now suppose that there exist αi real coefficients (i = 1, . . . np − 1),such that the last row in Φ(tc,k) can be expressed as a linear combination ofthe other np − 1 rows; that is


cv (ka + 1) =ka∑

k=0

βk

cv

ka∑k=0

φw,(tc,k) =ka∑

k=0

φw,(tc,k) βk ∀ = 1, . . . np − 1 (4.71)

with

βk =np−1∑j=1

αj φw,j(tc,k)

where φw,j(tc,k) is the jth component of φw(tc,k). From (4.71) we have

βk = cv ∀ k = 0, . . . ka (4.72)

Looking at the expression for βk it is easy to recognize from (4.72) thatthe vector φw(tc,k) is not persistently exciting. This conclusion, obviously,violates the hypothesis.

From the above demonstrated properties, it is possible to infer the bound-ness of the closed-loop signal using the stability analysis reported in [42,59].

Finally it is worth recalling that, if the power system model is corruptedby white noise (see (2.19)), the recursive least-squares algorithm gives an esti-mate that is asymptotically unbiased [10,77,83,127]. The closed-loop stabilityanalysis in presence of white noise can be found in [45,62,75].

5

Model-reference Adaptive Voltage Regulators

This chapter illustrates the design of nodal voltage regulators using model-reference adaptive control theory for the discrete-time linear model presentedin Chapter 3. Both indirect and direct methods are illustrated. As in thecase of self-tuning voltage regulator design developed in Chapter 4, indirectmodel-reference adaptive design is articulated in two steps. In the first step,the parameters of the power system model are estimated recursively using arecursive least-squares algorithm with forgetting factor. In the second step theestimated parameters are then used to solve a Diophantine equation in theunknown polynomials regulator.

Conversely, in direct model-reference adaptive design, assuming a knownoperating point of the power system, the regulator parameters are determinedby solving a model-reference design. Adaptive laws to update the regulatorparameters are then designed using a normalized gradient method based ona standard quadratic cost function.

This chapter ends with a robustness analysis carried out with reference toa no-load voltage disturbance.

5.1 Introduction

Voltage regulator design developed using model-reference adaptive systemstheory is based on the existence of a reference model which specifies the desiredperformance of the closed-loop voltage control scheme. The overall aim is toforce the regulated nodal voltage to correspond to the output of the referencemodel.

The block scheme illustrating how it is possible to realize a direct adap-tive voltage regulator scheme is reported in Figure 5.1. The regulator can bethought of as composed of two loops: the inner one is the classical feedbackloop in which the parameters of the regulator are adjusted by the outer loopin a such way that the tracking error is small.

88 5 Model-reference Adaptive Voltage Regulators

r

RegulatorPower

system

v

Adaptive

law

Reference

model vm

e −+

u

Regulatorparameters

Fig. 5.1. Overview of a direct model-reference adaptive voltage regulation scheme

Looking at Figure 5.1 and Figure 4.1 it is evident that the model-referenceadaptive system and self-tuning regulator are closely related.

Among well-established techniques for implementing adaptive laws for up-dating the voltage regulator parameters, we will refer in the direct methodto the gradient technique, which is based on minimization of a quadratic costfunction [8, 10,115]

5.2 Direct Model-reference Adaptive Voltage RegulatorDesign

In direct model-reference adaptive design, the parameters of the voltageregulator are updated recursively without the use of any estimation algo-rithm [10, 115]. In particular, the design will be developed by assuming toneglect the noise term ν(tc,k) in the power system model. When the operat-ing point of the power system is assigned, the values of the parameters ai, bj

and D appearing in model (2.17)

A(z−1) v1(tc,k) − A(z−1)Ad(z−1)

D δ(tc,k) = z−d B(z−1) u(tc,k)

are known and polynomials A(z−1) and B(z−1) are used to solve the non-adaptive problem that consists in determining the regulator parameters asthe solution of a particular Diophantine equation.

Conversely, in the presence of unknown operating points, the regulatorparameters are modified according to adaptive laws designed on the basis ofa gradient technique to minimize a normalized quadratic cost function of theaugmented error. In particular it will be assumed that the polynomial B(z−1)has only stable roots; then it can be factorized according to (2.16) as

5.2 Direct Model-reference Adaptive Voltage Regulator Design 89

B(z−1) = b0 B+(z−1)

where the sign of b0 is known and

|b0| ≤ bM0

with bM0 > 0.

5.2.1 Model-reference Design

The model-reference design has the objective of finding an output feedbackcontrol signal u(tc,k) for the power system model (2.17) with nA, nB , d, ai,bj and D known, such that v1(tc,k) tracks a given reference output vm(tc,k)so that the error

e(tc,k) = v1(tc,k) − vm(tc,k) (5.1)

is small.The reference signal vm(tc,k) is generated from a reference model system

Am(z−1) vm(tc,k) = z−d Bm(z−1) r(tc,k) (5.2)

where Am(z−1) and Bm(z−1) are assigned polynomials, bm,0 = 0, and r(k)is the command signal. A classical choice for the polynomials Am(z−1) andBm(z−1) leads to the following model-reference: vm(tc,k) = r(tc,k−d), that is,the output vm(tc,k) assumes the values of the reference r(tc,k) with d steps ofdelay [115].

The voltage control law assumes the form (see (4.13)),

F ∗(z−1)u(tc,k) = −G∗(z−1) v1(tc,k) + H∗(z−1) r(tc,k) (5.3)

where, according to (4.14) and( 4.15), the polynomials F ∗(z−1) and F ∗v (z−1)

are factorized asF ∗(z−1) = Ad(z−1)F ∗

v (z−1) (5.4)

withF ∗

v (z−1) = B+(z−1) F ∗(z−1) (5.5)

The polynomials

F ∗(z−1) = 1 + f∗1 z−1 + . . . + f∗

nFz−nF

G∗(z−1) = g∗0 + g∗1z−1 + . . . + g∗nGz−nG

are solutions of the Diophantine equation


A(z−1) Ad(z−1) F ∗(z−1) + z−d b0 G∗(z−1) = A0(z−1) Am(z−1) (5.6)

with A0(z−1) an assigned observer polynomial. Equation (5.6) has a uniquesolution if A(z−1) and B(z−1) are co-prime and the following compatibilityconditions are satisfied [10]:

nA0 ≥ 2 nA + nAd− nB+ − nAm

− 1 = 2nA − nB+ − nAm(5.7a)

nG < nA + nAd= nA + 1 (5.7b)

nF ≥ d − 1 (5.7c)

Finally F ∗(z−1) is obtained via (5.4) and (5.5) while H∗(z−1) is given by

H∗(z−1) = h∗0 A0(z−1) Bm(z−1) = A0(z−1) Bm(z−1)/b0

5.2.2 Adaptive Law Design

In the presence of operating points different from the one corresponding tothe model-reference design, the parameters ai, bj and D are unknown. Thusthe following adaptive version of law (5.3)

Fk(z−1) u(tc,k) = −Gk(z−1) v1(tc,k) + Hk(z−1) r(tc,k) (5.8)

will be implemented with

Fv,k(z−1) = 1 + fv,1(tc,k)z−1 + . . . + fv,nFv(tc,k)z−nFv

Gk(z−1) = g0(tc,k) + g1(tc,k)z−1 + . . . + gnG(tc,k)z−nG

Hk(z−1) = h0(tc,k) A0(z−1) Bm(z−1)

whereFk(z−1) = Fv,k(z−1) Ad(z−1) (5.9)

Moreover, let us define the following vector

θ =[fv,1 . . . fv,nFv

g0 . . . gnGh0

]T∈ IRnt

withnt = nFv

+ nG + 2

To design an adaptive law to update vector θ an expression for the er-ror (5.1) will be derived. Multiplying both sides of (5.6) by v1(tc,k) and addingand subtracting the quantity

A(z−1)Ad(z−1)Fv,k−d(z−1)B+(z−1)

v1(tc,k)


gives

A0(z−1) Am(z−1) v1(tc,k) = A(z−1)Ad(z−1)F ∗(z−1)v1(tc,k)

+b0 G∗(z−1)v1(tc,k−d) + A(z−1)Ad(z−1)(Fv,k−d(z−1)/B+(z−1)

)v1(tc,k)

−A(z−1)Ad(z−1)(Fv,k−d(z−1)/B+(z−1)

)v1(tc,k) (5.10)

At this point multiplying (2.17) by Fv,k−d(z−1)/B+(z−1), using (5.9),and (5.8) evaluated at tc,k−d yields

A(z−1)Ad(z−1)(Fv,k−d(z−1)/B+(z−1)

)v1(tc,k) = b0

(Hk−d(z−1) r(tc,k−d)

−Gk−d(z−1) v1(tc,k−d))

+ A(z−1)(Fv,k−d(z−1)/B+(z−1)

)Dδ(tc,k)

which substituted into (5.10) gives

A0(z−1)Am(z−1)v1(tc,k) = A(z−1)Ad(z−1)F ∗(z−1)v1(tc,k)

+ b0G∗(z−1)v1(tc,k−d) − A(z−1)Ad(z−1)

(Fv,k−d(z−1)/B+(z−1)

)v1(tc,k)

+ b0

(Hk−d(z−1) r(tc,k−d) − Gk−d(z−1) v1(tc,k−d)

)+A(z−1)

(Fv,k−d(z−1)/B+(z−1)

)Dδ(tc,k) (5.11)

Equation (5.11) can be rewritten as

A0(z−1) Am(z−1)v1(tc,k)

= A(z−1)Ad(z−1)(F ∗(z−1) − (Fv,k−d(z−1)/B+(z−1)

))v1(tc,k)

+ b0

(Hk−d(z−1) r(tc,k−d) + ∆Gk−d(z−1) v1(tc,k−d)

)+A(z−1)

(Fv,k−d(z−1)/B+(z−1)

)Dδ(tc,k) (5.12)

with

∆Gk−d(z−1) = G∗(z−1) − Gk−d(z−1)


Now, multiplying both sides of (2.17) by

F ∗(z−1) − Fv,k−d(z−1)B+(z−1)

using (5.5) and (5.9) yields

A(z−1) Ad(z−1)(F ∗(z−1) − Fv,k−d(z−1)/B+(z−1)

)v1(tc,k)

= b0∆Fv,k−d(z−1)uv(tc,k−d)

+ A(z−1)(F ∗(z−1) − Fv,k−d(z−1)/B+(z−1)

)Dδ(tc,k) (5.13)

where

∆Fv,k−d(z−1) = F ∗v (z−1) − Fv,k−d(z−1)

anduv(tc,k−d) = Ad(z−1)u(tc,k−d) (5.14)

see (4.16).Substituting (5.13) into (5.12), after few trivial manipulations, one has

A0(z−1) Am(z−1) v1(tc,k) = b0

(∆Fv,k−d(z−1) uv(tc,k−d)

+∆Gk−d(z−1) v1(tc,k−d) + Hk−d(z−1) r(tc,k−d))

+A(z−1) F ∗(z−1) D δ(tc,k) (5.15)

Finally multiplying (5.2) by A0(z−1) one has

A0(z−1) Am(z−1)vm(tc,k) = A0(z−1) Bm(z−1) r(tc,k−d) = b0H∗(z−1)r(tc,k−d)

which subtracted from (5.15) yields

A0(z−1) Am(z−1) e(tc,k) = b0

(∆Fv,k−d(z−1) uv(tc,k−d)

+∆Gk−d(z−1) v1(tc,k−d) − A0(z−1) Bm(z−1)(h∗0 − h0)r(tc,k−d)

)+A(z−1)F ∗(z−1)Dδ(tc,k)

At the kth step, the error e(tc,k) can be expressed in compact form as


e(tc,k) = b0

(θ∗ − θ(tc,k−d)

)T

ϕf (tc,k−d) + d(tc,k) (5.16)

where

θ∗− θ(tc,k−d) =[

f∗v,i − fv,i(tc,k−d)

g∗j − gj(tc,k−d)

h∗

0 − h0(tc,k−d)]T

ϕf (tc,k−d) =1

A0(z−1) Am(z−1)

[uv(tc,k−d−i)

v1(tc,k−d−j)

−Bm(z−1) A0(z−1) r(tc,k−d)]T

with i = 1, . . . nFvand j = 0, . . . nG.

In (5.16) the disturbance term

d(tc,k) =A(z−1)F ∗(z−1)

A0(z−1) Am(z−1)D δ(tc,k)

represents the contribution due to the no-load voltage. It coincides with theimpulse response of filter

A(z−1)F ∗(z−1)A0(z−1) Am(z−1)

In particular the signal d(tc,k) ∈ L2 and

limk→∞

d(tc,k) = 0

Defining the augmented error as

ε(tc,k) = e(tc,k) + ρ(tc,k) ξ(tc,k) (5.17)

where ρ(tc,k) is the estimate of b0 and

ξ(tc,k) =(θ(tc,k−d) − θ(tc,k)

)T

ϕf (tc,k−d) (5.18)

it is finally possible to rewrite (5.17) as

ε(tc,k) = d(tc,k) + b0 θ(tc,k)Tϕf (tc,k−d) + ρ(tc,k)ξ(tc,k) (5.19)

where

θ(tc,k) = θ∗ − θ(tc,k)

ρ(tc,k) = ρ(tc,k) − b0


By changing the parameters in the direction of the negative gradient ofthe normalized quadratic cost function given by

J(tc,k) =12

ε2(tc,k)m2(tc,k)

one obtains

θ(tc,k+1) = θ(tc,k) +signb0Γ ε(tc,k)ϕf (tc,k−d)

m2(tc,k)(5.20a)

ρ(tc,k+1) = ρ(tc,k) − γ ε(tc,k)ξ(tc,k)m2(tc,k)

(5.20b)

where

m2(tc,k) = k1 + ϕf (tc,k−d)Tϕf (tc,k−d) + ξ2(tc,k) (5.21)

with k1 > 0.In adaptive laws (5.20) Γ=diagγi ∈ IRnt , with γ and γi positive gains.The adaptive design has been developed neglecting the presence of the

noise term ν(tc,k) in power system model. If this assumption is removed, theterm

Ad(z−1) F ∗(z−1)A0(z−1) Am(z−1)

ν(tc,k) (5.22)

will appear in (5.16). However, having in mind that ν(tc,k) is mainly due tomeasurement noise and commutation in the electronic devices, it is quite re-alistic to assume that (5.22) represents a bounded disturbance not necessarilyin L2. To handle such a circumstance, adaptive laws (5.20) can be suitablymodified by adding, for example, a dead-zone [74].


The adaptive laws (5.20) have been implemented in a numerical simulationcase study aimed at analyzing the performance of the model-reference adap-tive voltage regulator control scheme.

The assumed model-reference is

vm(tc,k) = z−4 0.0081 − 1.82 z−1 + 0.828 z−2

r(tc,k)

whose output is shown in Figure 5.2Since nA = 4 and nB+ = 2, see Section A.1.3, the observer polynomial

A0(z−1) must have degree greater than 4 as prescribed by (5.7a); in particularit has been chosen such that


Fig. 5.2. Output of the model-reference

A0(z−1) = 1 − 1.27 z−1 + 0.626 z−2 − 0.149 z−3 − 0.0174 z−4 + 0.000792 z−5

Since nFv= 5 (see (5.5) and (5.7c) with d = 4), nG = 4 (see (5.7b)),

the number of parameters to update is nt = 11. The parameters are fv,i

(i = 1, . . . 5), gj (j = 0, . . . 4) and h0. The adaptive laws (5.20) have beenimplemented with

Γ = diagI5 2 I5 1γ = 1.5

k1 = 1

in addition we have set bM0 = 0.1. The adaptation mechanism starts at t = 0.

The initial value of θ has been set such that

θ(0) =[− 1.3068 0.3805 0.4239 − 0.9594 0.4663

41.8982 − 105.7492 95.9702 − 36.4702 4.6766 133.333]T

The time variation of v#4,1(t) obtained is shown in Figure 5.3 while Fig-ure 5.4 reports the time variation of v#4,1(t) and vm(t) between the timeinstants t = 15.998 s and t = 16.2 s to highlight the tracking properties of thedesigned adaptive voltage control scheme. Due to considerations of the initial-ization of power system variables reported in Sect. 4.2.5, during the first 1.7 sof simulation the control input u(tc,k) is greater than one and consequentlythe firing angle α(t) is saturated at its upper limit. Afterwards, u(tc,k) exitsfrom saturation, α(t) decreases and consequently the regulated nodal voltagev#4,1(t) is promptly forced to track the output of the reference model. Thetime variation of α(t) is reported in Figure 5.5

Comparing Figure 5.2 and Figure 5.3 it can be easily recognized that attime t = 10 s and t = 16 s the regulated nodal voltage v#4,1(t) undergoes a


Fig. 5.3. Time variation of v#4,1(t)

Fig. 5.4. Time variation of v#4,1(t) and vm(t) between the time instants t = 15.998 sand t = 16.2 s

variation while the output of the reference model is unchanged. This is due toa 20% step increase of the load Q1 at time t = 10 s and subsequently to theopening of the transmission line L2 at time t = 14 s. This opening represents achanges in the network topology which determines a wide variation of v#4,1(t).This can also be noticed by looking at the time variation of the tracking errorreported in the left-hand side of Figure 5.6. Moreover, the right-hand side ofthe same figure shows the tracking error between time instants t = 3 s andt = 11 s.

Regarding the adapted coefficients fv,i, gj and h0, since their most signifi-cant variations occur during the first seconds of simulation when the trackingerror is large, Figure 5.7 and Figure 5.8 report the time variations only be-tween the time instants t = 0 s and t = 5 s. It can be easily recognized fromFigure 5.7 and Figure 5.8 that after time t = 3 s these coefficients assumeconstant values. In the presence of load variations and line opening, negligiblevariations occur.

5.3 Indirect Model-reference Adaptive Voltage Regulator Design 97

Fig. 5.5. Time variation of α(t)

Fig. 5.6. Time variation of e(t) (left); enlarged view of e(t) between the time instantst = 3 s and t = 11 s (right)

In conclusion, the model-reference adaptive control can be usefully ap-plied when additional time-varying reference signals must be tracked; see Sec-tion 1.3.

5.3 Indirect Model-reference Adaptive VoltageRegulator Design

According to the indirect method illustrated in Section 4.2 with referenceto self-tuning voltage regulator design, the first step in the indirect model-reference adaptive design procedure requires estimation of the parameters ai,bj and v0,1 appearing in (4.4). The recursive least-squares algorithm withvariable forgetting factor (4.8)-(4.11) is adopted to pursue this task.

The estimates A(z−1) and B(z−1) are then used to solve every steps thefollowing Diophantine equation:


Fig. 5.7. Time variation of fv1, fv2, fv3, fv4, fv5 up to t = 5 s

A(z−1) Ad(z−1) F (z−1) + z−d B−(z−1) G(z−1) = A0(z−1) Am(z−1) (5.23)

under constraints (5.7), yielding the unknown polynomials F (z−1) and G(z−1).The polynomial F (z−1) is then obtained using (4.15), in which B+(z−1)

replaces B+(z−1), and from (4.14), while H(z−1) is given by

H(z−1) = A0(z−1) B+m(z−1)

where Bm(z−1) = B+m(z−1) B−(z−1).

It is easy to recognize that the illustrated indirect model-reference voltageregulator design slightly differs from the indirect self-tuning design. Both pro-cedures employ the recursive least-squares algorithm and solve a Diophantineequation. In detail, in the indirect model-reference design the characteristicpolynomial of the closed-loop system

A(z−1) F (z−1) + z−d B(z−1) G(z−1)

is designed to have three types of factors: the canceled power system ze-ros B+(z−1), the desired model poles Am(z−1) and the desired observer

5.4 Properties of the Adaptive Law 99

Fig. 5.8. Time variation of g0, g1, g2, g3, g4, h0 up to t = 5 s

poles A0(z−1). Letting T (z−1) = Am(z−1) A0(z−1), the Diophantine equa-tion (5.23) coincides with the Diophantine equation (4.17) used in the indirectself-tuning voltage regulator design. Although different estimation algorithmscan be used to estimate the power system model parameters [8, 10], the twoindirect design methods are quite similar. For this reason numerical simula-tions of the implementation of the indirect model-reference adaptive voltageregulator will not be presented.

5.4 Properties of the Adaptive Law

The analysis of the convergence of the adaptive law, and of the robustnesswith respect to the disturbance term v0,1(tc,k) is developed with reference tothe direct method. The analysis for the indirect method can be derived in asimilar way.

5.4.1 Convergence Analysis

The convergence analysis of the adaptive law (5.20) will be developed on thebasis of the following lemma.


Lemma 5.1. Law (5.20) has the following properties: θ(tc,k) ∈ L∞, ρ(tc,k) ∈L∞, ε(tc,k)/m(tc,k) ∈ L2

⋂L∞ and θ(tc,k+0) − θ(tc,k) ∈ L2 for any finite

integer 0.

Proof. Let us consider the following positive definite Lyapunov function [115]

V(θ(tc,k), ρ(tc,k)

)= b0 θ(tc,k)T Γ−1 θ(tc,k) + γ−1 ρ2(tc,k) (5.24)

Expressing (5.24) at tc,k+1 and using (5.20) yields

V(θ(tc,k+1), ρ(tc,k+1)

)= |b0|

(θ(tc,k) − signb0 ε(tc,k)

m2(tc,k)Γϕf (tc,k−d)

)T

Γ−1

(θ(tc,k) − signb0 ε(tc,k)


)

+1γ

(ρ(tc,k) − γε(tc,k)ξ(tc,k)

m2(tc,k)

)2

The time increment of function (5.24) along the trajectories (5.20) is thengiven by


)− V

(θ(tc,k), ρ(tc,k)

)

= − |b0|θ(tc,k)TΓ−1(signb0 ε(tc,k)


)

− |b0|(signb0 ε(tc,k)


)T

Γ−1θ(tc,k)

+ |b0| ε2(tc,k)m4(tc,k)

(Γϕf (tc,k−d)

)T

Γ−1(Γϕf (tc,k−d)

)

+ γε2(tc,k)m4(tc,k)

ξ2(tc,k) − 2ε(tc,k)

m2(tc,k)ξ(tc,k)ρ(tc,k) (5.25)

Choosing

0 < Γ = ΓT


the first two terms at the right-hand side of (5.25) are equal; their sum addedto the last term gives

−2ε(tc,k)

m2(tc,k)

(b0 θ(tc,k)Tϕf (tc,k−d) + ρ(tc,k)ξ(tc,k)

)which, based on (5.19), can be rewritten as

−2ε2(tc,k)m2(tc,k)

+ 2ε(tc,k) d(tc,k)

m2(tc,k)

At this point the time increment (5.25) is given by


)− V

(θ(tc,k), ρ(tc,k)

)

= − ε2(tc,k)m2(tc,k)

(2 − |b0|ϕf (tc,k−d)TΓϕf (tc,k−d)

m2(tc,k)+

γ ξ2(tc,k)m2(tc,k)

)

+2ε(tc,k) d(tc,k)

m2(tc,k).

Looking at (5.21) it is easy to recognize that the term in parentheses ispositive if

0 < Γ <2

bM0

Int0 < γ < 2

Consequently one has

V(

θ (tc,k+1), ρ(tc,k+1))− V

(θ(tc,k), ρ(tc,k)

)≤ −β1

ε2(tc,k)m2(tc,k)

+2ε(tc,k) d(tc,k)

m2(tc,k)(5.26)

for some β1 > 0. Now, since(β1

2ε(tc,k) − d(tc,k)

)2

≥ 0

it yields

2 ε(tc,k) d(tc,k) ≤ β1

2ε2(tc,k) +

2β1

d2(tc,k)

which substituted in (5.26) gives


V(

θ (tc,k+1), ρ(tc,k+1))− V

(θ(tc,k), ρ(tc,k)

)≤ −β1

2ε2(tc,k)m2(tc,k)

+2β1

d2(tc,k)m2(tc,k)

(5.27)

Since d(tc,k) ∈ L2, so does d(tc,k)/m(tc,k). Hence, from (5.27) one has


)+

β1

2

tc,k∑=0

ε2( )m2( )

≤ V(θ(0), ρ(0)

)+

2β1

tc,k∑=0

d2( )m2( )

which implies that V(θ(tc,k), ρ(tc,k)

)∈ L∞.

Consequently, θ(tc,k) ∈ L∞, ρ(tc,k) ∈ L∞ and ε(tc,k)/m(tc,k) ∈ L2.From (5.19) one has ε(tc,k)/m(tc,k) ∈ L∞, and from (5.20a) one has θ(tc,k+1)−θ(tc,k) ∈ L2.

Finally, using the inequality

∥∥θ(tc,k+0) − θ(tc,k)∥∥

2≤

0−1∑=0

∥∥θ(tc,k++1) − θ(tc,k+)∥∥

2

we obtain that θ(tc,k+0) − θ(tc,k) ∈ L2 for any finite integer 0.

5.4.2 Robustness Analysis

The robustness analysis with respect to the disturbance v0,1(tc,k) is based onthe following theorem.

Theorem 5.2. Regulator (5.8) and laws (5.20) guarantee that all signals inthe closed-loop system are bounded and limk→∞ e(tc,k) = 0

Proof. To proceed, multiplying the Diophantine equation (5.6) by v1(tc,k),using (5.4) and (5.14), solving with respect to uv(tc,k) gives

uv (tc,k) = −nFv∑i=1

f∗v,i uv(tc,k−i) −

nG∑j=0

g∗j v1(tc,k−j) +1b0

(v1(tc,k+d)

+nC∑q=1

cq v1(tc,k+d−q) − A(z−1) F ∗v (z−1)

B+(z−1)D δ(tc,k+d)

)(5.28)

in which cq are the coefficients of the polynomial C(z−1) = A0(z−1) Am(z−1)appearing at the right-hand side of (5.6) and where


nC = nA0 + nAm> 2 nA

(see constraint (5.7a)).At this point, defining the polynomial Ae(z−1) as

Ae(z−1) Ad(z−1) A(z−1) = 1 + ae,1z−1 + . . . + ae,nA+1 z−(nA+1)

from model (2.17) and using (5.14) it is possible to write

v1(tc,k+d) = −nA+1∑i=1

ae,i v1(tc,k+d−i) +nB∑j=0

bj uv(tc,k−j)

+A(z−1) D δ(tc,k+d) (5.29)

Substituting in (5.29) the expression for uv(tc,k) given by (5.28) one has

v1(tc,k+d) = −nA+1∑i=1

ae,i v1(tc,k+d−i) +nB∑j=1

bj uv(tc,k−j) − b0

nFv∑i=1

f∗v,i uv(tc,k−i)

−b0

nG∑j=0

g∗j v1(tc,k−j) + v1(tc,k+d) +nC∑q=1

cq v1(tc,k+d−q)

−A(z−1) F ∗v (z−1)

B+(z−1)D δ(tc,k+d) + A(z−1) Dδ(tc,k+d) (5.30)

To cast (5.30) in compact form let us introduce the vector

n(tc,k) =[uv(tc,k−1) . . . uv(tc,k−nFv

)︸︷︷︸nFv

v1(tc,k+d−1) . . . v1(tc,k)︸︷︷︸d

v1(tc,k−1) . . . v1(tc,k−µ︸︷︷︸µ

]T∈ IRnn

where nn = nFv+ d + µ, µ = maxnG, nC − d with nC > 2 nA ≥ nA + 1.

Equation (5.30) can then be expressed in matrix form as

n(tc,k+1) = N∗ n(tc,k) + b∗n

(v1(tc,k+d) − A(z−1) F ∗

v (z−1)B+(z−1)

Dδ(tc,k+d))

+d∗n A(z−1) Dδ(tc,k+d) (5.31)


where

N∗ =

⎡⎢⎢⎢⎢⎣n∗

1T n∗

2T

N∗1 0(nFv−1)×(d+µ)

n∗3T n∗

4T

0(d+µ−1)×(nFv ) N∗2

⎤⎥⎥⎥⎥⎦ ∈ IR(nn×nn)

b∗n =

[ 1b0

0 . . . . . . . . . 0︸︷︷︸nFv−1

1 0 . . . . . . . . . 0︸︷︷︸d+µ−1

]T

d∗n =

[0 . . . . . . . . . . . . 0︸︷︷︸

nFv

1 0 . . . . . . . . . 0︸︷︷︸d+µ−1

]Twith

N∗1 =

⎡⎢⎢⎢⎢⎣1 0 0 0 . . . 00 1 0 0 . . . 0...

......

......

...0 0 . . . 0 1 0

⎤⎥⎥⎥⎥⎦ ∈ IR(nFv−1)×(nFv )

N∗2 =

⎡⎢⎢⎢⎢⎣1 0 0 0 . . . 00 1 0 0 . . . 0...

......

......

...0 0 . . . 0 1 0

⎤⎥⎥⎥⎥⎦ ∈ IR(d+µ−1)×(d+µ)

n∗1 = −

[f∗

v,1 . . . fv,nFv

]T∈ IRnFv

n∗2 =

[c1

b0. . .

cd−1

b0

(cd

b0− g∗0

). . .(cd+µ

b0− g∗µ

)]T ∈ IRd+µ

n∗3 = b0

[(− f∗v,1 +

b1

b0

). . .(− f∗

v,nB+

bnB

b0

) − f∗v,nB+1

. . . − fv,nFv

]T∈ IRnFv

n∗4 =

[(c1 − ae,1

). . .(cd−1 − ae,d−1

) (cd − ae,d − b0 g∗0

). . .(cµ − ae,µ − b0 g∗µ

)]T ∈ IRd+µ


The eigenvalues of matrix N∗ are inside the unit circle; to show this prop-erty, consider the following equality:

A(z−1) Ad(z−1)uv(tc,k) = c∗n(zInn

− N∗)−1b∗

n b0 uv(tc,k)

= c∗nAdj

(zInn − N∗)

det(zInn

− N∗) b∗n (5.32)

where

c∗n =[0 . . . . . . . . .︸︷︷︸

nFv−1

1 0 . . . . . . . . . 0︸︷︷︸d+µ

]obtained using model (2.17) and (5.31) with D = 0. Due to the structure ofb∗n and c∗n one has

c∗n Adj(zInn

− N∗) b∗n =

1b0

(−1)nFv +1det(N 1,nFv

)+ (−1)2 nFv +1det

(N nFv +1,nFv

)=

1b0

zµ+d A(z−1) Ad(z−1) (5.33)

where N 1,nFvand N nFv−1,nFv

denote the (nn−1)×(nn−1) matrices obtainedby deleting respectively the first row and (nFv

)th column and the nFv− 1th

row and the (nFv)th column of

(zInn

−N∗). Substituting (5.33) in (5.32) onehas

det(zInn

− N∗) = znn−nB B+(z) (5.34)

which demonstrates that N∗ has nB eigenvalues coincident with the roots ofB+(z) and the remaining ones in the origin.

Now, replacing in (5.17) both the expression for ξ(tc,k) given by (5.18)and the error e(tc,k) defined in (5.1), solving with respect to v1(tc,k) andsubstituting in (5.31) yields

n(tc,k+1) = N∗ n(tc,k) + b∗n(vm(tc,k+d) + gn(tc,k)

−A(z−1) F ∗v (z−1)

B+(z−1)Dδ(tc,k+d)

)+ d∗

n A(z−1) Dδ(tc,k+d) (5.35)

where


gn(tc,k) = ε(tc,k+d) + ρ(tc,k+d)(θ(tc,k+d) − θ(tc,k)

)T

ϕf (tc,k)

Since ρ(tc,k) ∈ L∞, ε(tc,k)/m(tc,k) ∈ L∞, and recalling the expression form2(tc,k) given by (5.21) and for ξ(tc,k) given by (5.18) one has

|gn(tc,k)| ≤ |ε(tc,k+d)||m(tc,k+d)| |m(tc,k+d)| + k3

∥∥∥θ(tc,k+d) − θ(tc,k)∥∥∥

2

∥∥ϕf (tc,k)∥∥

2

≤√

k1|ε(tc,k+d)||m(tc,k+d)| +

|ε(tc,k+d)||m(tc,k+d)|

∥∥ϕf (tc,k)∥∥

2

+

(|ε(tc,k+d)||m(tc,k+d)| + k3

)∥∥∥θ(tc,k+d) − θ(tc,k)∥∥∥

2

∥∥ϕf (tc,k)∥∥

2

≤ k4 + xn(tc,k)∥∥ϕf (tc,k)

∥∥2

in which

xn(tc,k) =|ε(tc,k+d)||m(tc,k+d)| + k5

∥∥∥θ(tc,k+d) − θ(tc,k)∥∥∥

2(5.36)

for some positive constants k3, k4 and k5.Since

(θ(tc,k+d) − θ(tc,k)

) ∈ L2 and ε(tc,k)/m(tc,k) ∈ L2 also the signalxn(tc,k) belongs to L2.

Based on (5.34), it is possible to affirm that there exists a nonsingularmatrix T∗ such that ‖ T∗ N∗ T∗−1 ‖2 < 1. With matrix T∗ let us define thevector norm ‖ n∗ ‖= ‖ T∗ n∗ ‖2 that enables us to rewrite (5.35) as

∥∥n(tc,k+1)∥∥ ≤

∥∥∥T∗ N∗ n(tc,k)∥∥∥

2+∥∥∥T∗ b∗

n vm(tc,k+d)∥∥∥

2+∥∥∥T∗ b∗

n gn(tc,k)∥∥∥

2

+∥∥∥∥T∗ b∗n

A(z−1) F ∗v (z−1)

B+(z−1)D δ(tc,k+d)

∥∥∥∥2

+∥∥∥T∗ d∗

n A(z−1) D δ(tc,k+d)∥∥∥

2

At this point using (5.36) one obtains∥∥n(tc,k+1)∥∥ ≤ (k6 + k7 xn(tc,k)

) ∥∥n(tc,k)∥∥+ k8 (5.37)

Now, since xn(tc,k) ∈ L2, application of the Holder inequality gives

k0+kf∑k=k0

xn(tc,k) ≤√kf + 1

√√√√k0+kf∑k=k0

x2n(tc,k) ≤ k9

√kf + 1 (5.38)


for any kf ≥ 1 and some positive constant k9. Using (5.38) one has

k0+kf∏k=k0

(k6 + k7 xn(tc,k)

) ≤ (k6 +k7

kf + 1

k0+kf∑k=k0

xn(tc,k)

)kf+1

≤(

k6 +k9 k7√kf + 1

)kf+1

≤ kkf+16

(1 +

k9 k7

k6

√kf + 1

)kf+1

with k9 k7/k6 > 0. Since

limkf→∞

(1 +

k9 k7

k6

√kf + 1

)√kf+1

= ek9 k7

k6

monotonically, one has

kkf+16

(1 +

k9 k7

k6

√kf + 1

)kf+1

≤ kkf+16 e

k9 k7k6

√kf+1

from which it follows that

limq→∞

q∑kf=1

k0+kf∏k=k0

(k6 + k7 xn(tc,k)

)< ∞ (5.39)

Using (5.37) and (5.39) it is possible to affirm that n(tc,k) is bounded andthen ε(tc,k) ∈ L2,

(θ(tc,k+1) − θ(tc,k)

) ∈ L2 so that

limk→∞

ε(tc,k) = 0

limk→∞

(θ(tc,k+1) − θ(tc,k)

)= 0

limk→∞

ξ(tc,k) = 0

Finally from (5.17) we obtain

limk→∞

e(tc,k) = limk→∞

(v1(tc,k) − vm(tc,k)

)= 0

which shows that the adaptive voltage regulator ensures robustness with re-spect to the disturbance v0,1(tc,k).

6

Adaptive Nonlinearities CompensationTechnique

In the design techniques illustrated in Chapters 4 and 5, the power system hasbeen represented by a discrete-time linear model. This chapter illustrates theprocedure to design a nodal voltage control scheme employing a technique thatadaptively compensates for the nonlinearities of the power system model inthe frequency domain. The procedure is structured into three main tasks [49].In the first, two Kalman filters are adopted to identify and track the valuesof the voltage and current phasors at fundamental and harmonic frequenciesat the regulation node. These values are then used in the second task for theon-line estimation of the parameters of the equivalent model representing theelectrical power system at each phasor frequency. Finally, in the third task,the voltage regulator is designed using pole-assignment technique with anadaptive compensation mechanism. The last section of the chapter presentsan optimum design which minimizes the harmonic distortion level.

6.1 Introduction

The power system is modeled in the frequency domain by the steady-stateThevenin equivalent circuit; see Figure 2.7.

A block scheme giving an overview of the overall control system is shownin Figure 6.1. It assumes the most general scheme, including fundamental andharmonic frequencies and both real and imaginary phasor components. Othercases can be trivially derived from such general scheme. In particular, in thecase of voltage amplitude regulation at fundamental frequency, only one AVRblock is included and its output command uc,1 coincides with the commanduc sent to the actuator.

The first task consists in determining the voltage and current phasor atfundamental and harmonic frequencies

vh(tf,k) = vh,r(tf,k) + j vh,i(tf,k)for h = 1, 3, . . . , nh

ih(tf,k) = ih,r(tf,k) + j ih,i(tf,k)

110 6 Adaptive Nonlinearities Compensation Technique

CRLSh = 1

AVRh = 1

CRLSh = nh

AVRh = nh

v0,1,r v0,1,i

zeq,1

r1,r

r1,i

rnh,r

rnh,i

v0,nh,r v0,nh,i

zeq,nh

Kalmanfilter

i1,r i1,i

inh,r inh,i

Currenttransducer

Kalmanfilter

v1,r v1,i

vnh,r vnh,i

Voltagetransducer

uc,1

uc,nh

+ ZOHuc Actuator

u(t) Powersystem

v(t)

i(t)

Fig. 6.1. Overview of the generalized control system

starting from the sampled measurements of the waveforms of the nodal volt-age and of the current injected by the actuator. This task, represented inFigure 6.1 by the Kalman Filter blocks, adopts a filtering technique based ona discrete Kalman filter whose properties were studied in Chapter 3.

When power system operating conditions change, the values of parametersv0,h,r, v0,h,i and zeq,h = req,h + j xeq,h vary; see Figure 2.7. Consequently, theobjective of the second task is to guarantee fast parameters tracking; see CRLSblock (constrained recursive least-squares). Finally the third task concerns thedesign of the adaptive voltage controller; see AVR block. The structure of theAVR is strictly related to the voltage control device adopted. In the following,the CRLS and AVR blocks are analyzed.

6.2 Thevenin Circuit Parameters Estimation

To guarantee simple and fast tracking of the no-load voltage and short-circuit impedance a classical well-known recursive least-squares based tech-nique is adopted [44, 127], which is extended to take into account physicalconstraints [47]. The algorithm is performed at each te,k = k Te where Te

is the sampling period of the constrained recursive least-squares algorithm.It utilizes the estimates vh,r(tf,k), vh,i(tf,k) and ih,r(tf,k), ih,i(tf,k) given byKalman filters. In particular if Te is chosen according to

Te > m TKf m >> 1 (m integer) (6.1)

(see Section 3.2.3), negligible correlation relating subsequent phasor identifiedvalues utilized in the estimation procedure can be assumed.

Consider first a first-order model that includes dynamics only on currentphasors; it is described by the following equation:

6.2 Thevenin Circuit Parameters Estimation 111

yh(te,k) =[vh,r(te,k) vh,i(te,k)

]= ψh(te,k)Θh (6.2)

in which

ψh(te,k) =[ih,r(te,k) ih,i(te,k) cv,h ih,r(te,k−1) ih,i(te,k−1)

](6.3)

with Θh ∈ IR5×2. In (6.3) the quantity cv,h is a constant input; it can beviewed as a scaling factor.

Since in steady-state it is

ih,r(te,k) = ih,r(te,k−1) = ih,r(∞)

ih,i(te,k) = ih,i(te,k−1) = ih,i(∞)

the steady-state value yh(∞) of the output model (6.2) is given by

yh(∞) =[ih,r(∞) ih,i(∞) cv,h

] ⎡⎢⎢⎣θh,11 + θh,41 θh,12 + θh,42

θh,21 + θh,51 θh,22 + θh,52

θh,31 θh,32

⎤⎥⎥⎦ (6.4)

Comparing (2.13) with (6.4), it is possible to derive the following “physi-cal” conditions on the parameters⎡⎢⎢⎣

θh,11 + θh,41 θh,12 + θh,42

θh,21 + θh,51 θh,22 + θh,52

θh,31 θh,32

⎤⎥⎥⎦ =

⎡⎢⎢⎢⎣req,h xeq,h

−xeq,h req,h

v0,h,r

cv,h

v0,h,i

cv,h

⎤⎥⎥⎥⎦ (6.5)

To estimate matrix Θh, the following prediction model of the output vari-able yh(te,k) is considered:

yh(te,k) = ψh(te,k) Θh(te,k−1)

By taking into account constraints (6.5), matrix Θh can be obtained bysolving the following constrained least-squares problem:

minΘh

J(Θh

)(6.6)

subject to ⎧⎨⎩wT

1 Θh,1 − wT2 Θh,2 = 0

wT2 Θh,1 − wT

1 Θh,2 = 0(6.7)


where

J(Θh) =k∑

i=1

λk−ih

(εT

h,1(te,i) εh,1(te,i) + εTh,2(te,i) εh,2(te,i)

)Θh =

[Θh,1 Θh,2

]w1 =

[1 0 0 1 0

]Tw2 =

[0 1 0 0 1

]Twith

[εh,1(te,k) εh,2(te,k)

]= Yh(te,k) − Φh(te,k) Θh(te,k−1)

=[Yh,1(te,k) − Φh(te,k) Θh,1(te,k−1) Yh,2(te,k) − Φh(te,k) Θh,2(te,k−1)

]and

Yh(te,k) =[yh

T(te,1) . . . yhT(te,k)

]T =[Yh,1(te,k) Yh,2(te,k)

]Φh(te,k) =

[φh

T(te,1) . . . φhT(te,k)

]TConstraints (6.7) are forced due to the particular form of (6.5): they assure

that the elements of Θ are physically meaningful.Problem (6.6) subject to (6.7) is a convex quadratic programming problem

which has a global minimum that can be found by solving the well-knownfirst-order conditions on the Lagrange function. A quite similar problem hasalready been solved in the case of implicit self-tuning regulator design; seeSection 4.3.1. By repeating the procedure illustrated in that case, it is easyto show that the solution to problem (6.6),(6.7) is given by

Θh,1 = Θ∗h,1 −

µh,1

2Ph w1 +

µh,2

2Ph w2

Θh,2 = Θ∗h,2 +

µh,1

2Ph w2 − µh,2

2Ph w1 (6.8)

with

Θ∗h,1 = Ph ΦT

h Yh,1

Θ∗h,2 = Ph ΦT

h Yh,2 (6.9)

6.2 Thevenin Circuit Parameters Estimation 113

being the covariance matrix

Ph =(ΦT

h Φh

)−1

The expression of the Lagrangian multipliers µh,1 and µh,2 appearingin (6.8) is

µh,1 = 2wT

1 Θ∗h,1 − wT

2 Θ∗h,2

wT1 Phw1 + wT

2 Phw2

µh,2 = 2wT

2 Θ∗h,1 + wT

1 Θ∗h,2

wT1 Phw1 + wT

2 Phw2(6.10)

In practice, for the on-line identification of the Thevenin equivalent circuitparameters at each harmonic frequency, a classical unconstrained recursiveleast-squares algorithm can be adopted to estimate Θ

∗h,1 and Θ

∗h,2, see (6.9),

which substituted in (6.8) give Θh,1 and Θh,2 via (6.10).Finally from (6.5) it is obtained

req,h = wT1 Θh,1

xeq,h = wT1 Θh,2

v0,h,r = cv,h θh,31

v0,h,i = cv,h θh,32

Regarding the classical unconstrained recursive least-squares algorithmand its properties, reference can be made, respectively, to Section 4.2.1 andto Section 4.4. It should be noted that also in this case the presence of theconstant input cv,h in the regressor (6.3) does not influence the fulfilment ofthe condition of persistent excitation related to the regressor sequence.

In general, the problem to identify Θ∗h,1 and Θ

∗h,2 can be solved by other

well-known identification techniques, see e.g. [9, 43, 126, 127]. With referenceto this problem, in [46] three different identification approaches have beenadopted to estimate on-line Θ

∗h,1 and Θ

∗h,2, and a comparative performance

analysis between the considered techniques has been conducted.It is important to recall that the estimated values of the phasors used by

the parameter estimation, that is the elements of yh(te,k) and ψh(te,k) areaffected by error which can be considered as composed of two components:a stochastic error, whose sequence is uncorrelated if constraint (6.1) is satis-fied, and a transient bias. The effects of the former error are canceled by theclassical recursive least-squares algorithm; the latter error is due to the fact


that the Kalman filter estimations are only asymptotically unbiased. Then,since the system composed of the power system together with the Kalmanfilters includes dynamics, it might be interesting to investigate if higher ordermodels yield a better performance in the parameter estimation.

Higher-order models for the current input vector ψh(te,k) can easily bederived generalizing the proposed analytical procedure; linearly-constrainedleast-squares models can be obtained and their solutions have the same formas (6.8)-(6.10), provided that vectors and matrices with appropriate dimen-sions are assumed. However, in the authors’ experience, when compared tothe assumed first-order model, higher-order input models present a limitedimprovement in the performance of the identification algorithm, which doesnot seem to justify the higher computational burden. Another field of inves-tigation concerns the possibility of adopting models which include dynamicsalso in the output vector yh(te,k), that is, holding memory also of voltage pha-sors. Unfortunately, to force the steady-state response of the identified systemto be physically meaningful similarly to what has been previously done, non-linear constraints have to be forced in the least-squares models. Consequently,a nonlinear programming problem is obtained which has no analytical closed-form solution, and traditional least-squares algorithms cannot be adopted.The on-line application of such models would be particularly troublesome be-cause there is no certainty about finding the solution and about the requiredcomputational time. Finally, it should be stressed that adopting an uncon-strained least-squares approach with higher-order models obviously causesthe identified parameters to be not physically meaningful.

6.3 Adaptive Voltage Regulator Design

Using the estimated values of the Thevenin equivalent circuit parameters theAVR blocks in Figure 6.1 can be designed. The structure of the AVR is strictlydependent on the adopted voltage control device. The model representingthe specific actuator type that is being considered must be included into theelectrical circuit shown in Figure 2.7. In the following the models describedin Section 2.2 are used to perform the AVR design. The voltage regulatorsampling period is Tc whereas the adaptive task is performed with a samplingperiod equal to Tc, that is every steps with a fixed integer.

6.3.1 Synchronous Machines

Synchronous machines are used to regulate voltage amplitude at fundamentalfrequency by acting on the excitation voltage through the exciter; see Sec-tion 2.2.1. Then, in Figure 6.1 only one AVR block is included and its outputcommand uc,1 coincides with the command uc sent to the actuator.

6.3 Adaptive Voltage Regulator Design 115

The stator voltage amplitude to be regulated is expressed by (2.2). TheThevenin equivalent circuit Equation (2.13) can be written including the pa-rameter estimations given by the CRLS algorithm and transformed into the(d − q) axes frame using (2.3) and (2.4)

v1(t) = v0,1 + zeq,1 i1(t) (6.11)

where v1(t) and i1(t) are expressed by (2.1). It should be noted that in (6.11)not only the stator voltage and current but also the no-load voltage estimationv0,1 must be transformed from the ( − ) axes frame, which is used bythe CRLS algorithm estimations, into the (d − q) axes frame by the Parktransformation (2.4). On the contrary, the equivalent impedance estimationzeq,1 does not change from one frame to the other.

To write (6.11) in the (d−q) axes frame the machine rotor angle δsm mustbe estimated. A very simple way can be derived from (2.5) written in steady-state operation (s = 0) and applying (2.3) and (2.4). By trivial manipulationthe following estimation is derived:

δsm(te,k) =π

2− tan−1

(v1,i(te,k) + xsm,q(0) i1,r(te,k)v1,r(te,k) − xsm,q(0) i1,i(te,k)

)(6.12)

Using the steady-state expression of (2.5) introduces an approximation inthe estimation which anyway is acceptable for the AVR design development.

From (6.11) the (d − q) components of the stator current can be derived:

i1,d(t) = req,1

(v1,d(t) − v0,1,d

)+ xeq,1

(v1,q(t) − v0,1,q

)(6.13a)

i1,q(t) = −xeq,1

(v1,d(t) − v0,1,d

)+ req,1

(v1,q(t) − v0,1,q

)(6.13b)

which can be L-transformed and substituted into (2.5) to obtain the relation-ship between the stator voltage and the excitation voltage.

To this aim, it is assumed that the estimated parameters of the Theveninequivalent circuit (req,1, xeq,1, v0,1,d and v0,1,q) as well as of the machine rotorangle δsm are constant. In practice, it means that the equivalent circuit pa-rameters and the mechanical variables of the synchronous machine are subjectto slow variations with respect to their estimation, which is performed witha sampling period equal to Te by the CRLS algorithm and by (6.12), and tothe AVR design, which is performed with a sampling period equal to Tc.

Then, writing (6.13) in terms of variational equations and L-transformingyields

∆I1,d(s) = req,1 ∆V1,d(s) + xeq,1 ∆V1,q(s) (6.14a)∆I1,q(s) = −xeq,1 ∆V1,d(s) + req,1 ∆V1,q(s) (6.14b)


Writing (2.5) in terms of variations and substituting (6.14) allows one towrite

∆V1,d(s) = xsm,q(s)(− xeq,1 ∆V1,d(s) + req,1 ∆V1,q(s)

)(6.15a)

∆V1,q(s) = asm,f (s) ∆Vsm,f (s)

− xsm,d(s)(req,1 ∆V1,d(s) + xeq,1 ∆V1,q(s)

)(6.15b)

Equations (6.15) are solved with respect to ∆V1,d(s) and ∆V1,q(s) obtaining

∆V1,d(s) = z2eq,1 req,1

asm,f (s) xsm,q(s)zt,1(s)

∆Vsm,f (s) (6.16a)

∆V1,q(s) = z2eq,1

(z2eq,1 + xsm,q(s) xeq,1

)asm,f (s)

zt,1(s)∆Vsm,f (s) (6.16b)

where

zt,1(s) = r2eq,1 xsm,d(s) xsm,q(s)

+(z2eq,1 + xeq,1 xsm,q(s)

)(z2eq,1 + xeq,1 xsm,d(s)

)To design the AVR at a specific time instant tc,k, Equation (2.2) is lin-

earized around the operating point and then L-transformed obtaining

v1( tc,k) ∆V1(s) = v1,d( tc,k) ∆V1,d(s) + v1,q( tc,k) ∆V1,q(s) (6.17)

Substituting (6.16) into (6.17) yields the synchronous machine transferfunction around the operating point at time instant tc,k

∆V1(s)∆Vsm,f (s)

= z2eq,1

( (v1,d( tc,k)v1( tc,k)

req,1 +v1,q( tc,k)v1( tc,k)

xeq,1

)xsm,q(s)

+v1,q( tc,k)v1( tc,k)

z2eq,1

)asm,f (s)zt,1(s)

(6.18)

The transfer function to be used in the AVR design must include (6.18)together with the transfer functions of the exciter, that is the actuator, ofthe voltage transducer and of the Kalman filter (see Figure 6.1). Regardingthe exciter, refer to Section 2.2.1: the transfer function can be written in thegeneral form (2.6). The voltage transducer is usually characterized by such alarge bandwidth that its dynamics can be neglected. Concerning the Kalmanfilter refer to Chapter 3. Multiplying the various transfer functions yields


the overall transfer function Qsm,1(s) relating the stator voltage amplitudeestimation at fundamental frequency to the command input.

It is important to notice that (6.18) refers to variations and, consequently,the constant bias related to the no-load voltage is not accounted for. If nosteady-state error in the regulation is required, an integrating action can beintroduced into the AVR and the resulting AVR structure is depicted in Fig-ure 6.2 (see also Figure 6.1). The dashed box evidences the AVR blocks thatare subject to the adaptive design which is performed every steps accordingto the procedure described in the following. The inputs to the AVR are theamplitude r1(tc,k) of the voltage reference signal and the amplitude v1(tc,k) ofthe estimated voltage phasor at fundamental frequency v1(tc,k). The outputis the command u(t) which is sent to the exciter.

H1(z−1)r1(tc,k) +

F−1v,1 (z−1)

uv,1(tc,k)

G1(z−1)

−

v1(tc,k)

11 − z−1

v0,1,r( tc,k) v0,1,i( tc,k)req,1( tc,k) xeq,1( tc,k)

uc,1(tc,k)≡ uc(tc,k)

ZOHu(t)

Fig. 6.2. AVR structure in the case of synchronous machine

The transfer function Msm,1(z−1) that represents the sampled data trans-fer function between uv,1 and v1 is obtained applying the Z-transformationto Qsm,1(s) assuming a sampling period equal to Tc (see Figure 6.2)

Msm,1(z−1) =(

11 − z−1

)Z

ZOH(s)Qsm,1(s)

= ZQsm,1(s)

s

= z−d B+1 (z−1) B−

1 (z−1)A1(z−1)

(6.19)

In (6.19) the polynomial B+1 (z−1) has its roots inside the unit circle while

the roots of B−1 (z−1) are outside the unit circle. Moreover, the term z−d

appears since the model (2.6) that represents the exciter includes, in additionto poles and zeros, a pure time delay Texc,d which is not, in practice, an exact


multiple of the sampling period Tc. However, this time delay can be expressedas

Texc,d = d Tc − ρd Tc

with d a positive integer and 0 ≤ ρd ≤ 1.The AVR polynomials Fv,1(z−1), G1(z−1) and H1(z−1) in Figure 6.2 can

be designed applying the pole-assignment technique to the system schemeshown in Figure 6.3.

H1(z−1)r1(tc,k) + 1

Fv,1(z−1)

uv,1(tc,k)Msm,1(z−1)

v1(tc,k)

G1(z−1)

−

Fig. 6.3. Block scheme representation adopted in the pole-assignment design forsynchronous machine

Defining the desired characteristics, expressed in terms of controller perfor-mance, the required closed-loop poles are assigned at some specified locationsdefined by the zeros of the polynomial T (z−1), see (4.12).

Then, the coefficients of F1(z−1) and G1(z−1) are obtained every stepsby solving the Diophantine equation

A1(z−1) F1(z−1) + z−d B−1 (z−1) G1(z−1) = T (z−1) (6.20)

which has an unique solution if B1(z−1) and A1(z−1) are co-prime and con-straints

nF1= d − 1 + nB−

1(6.21a)

nG1 = nA1 − 1 (6.21b)

nT ≤ nA1 + nB−1

+ d − 1 (6.21c)

are fulfilled. From solution of the Diophantine equation (6.20) we obtain

Fv,1(z−1) = F1(z−1) B+1 (z−1)

and

H1(z−1) = h1,0 =T (z−1)

B−1 (z−1)

∣∣∣∣∣z−1=1

the latter ensuring a unity steady-state gain for the closed-loop transfer func-tion.


6.3.2 Static VAr Systems

Static VAr Systems are used to regulate voltage amplitude at fundamental fre-quency by varying their equivalent admittance, see Section 2.2.2. Also in thiscase, in Figure 6.1 only one AVR block is included and its output commanduc,1 coincides with the command uc sent to the actuator.

Referring to the FC-TCR configuration as in Section 2.2.2, the SVS modelto be included into the system equivalent circuit shown in Figure 2.7 is simplyrepresented by a shunt admittance, which varies according to the value of thefiring angle α. The Thevenin equivalent circuit equation, that is (2.13), canbe written including the parameter estimations given by the CRLS algorithmand combined with the SVS model equation, that is (2.7), yielding

v1(t) =v0,1

1 + zeq,1 ysvs

(α(t)

) (6.22)

where the function ysvs

(α(t)

)is described by (2.8) and (2.9).

It is apparent that (6.22) does not include any dynamics but representsa nonlinear algebraic relationship between the SVS firing angle α(t) and thenodal voltage amplitude. In the overall model the dynamics to be included(see Figure 6.1) are related only to the power electronic apparatus, to the volt-age transducer and to the Kalman filter. Regarding the former, the transferfunction (2.11) is used to model the response of the thyristor firing angle α(t)to the SVS input command u(t). Concerning the voltage transducer and theKalman filter the same considerations as those reported for the synchronousmachine AVR design in Section 6.3.1 can be made. Such dynamics are linearand time-invariant.

Consequently the AVR can be structured in two parts according to thescheme shown in Figure 6.4: a linear time-invariant control and adaptive non-linearities compensation [48]. The former is designed to account for the lineartime-invariant dynamics; the latter to counteract the algebraic nonlinearitiesrepresented by (6.22), (2.8) and (2.9), which depend on the power systemoperating point.

The linear control is formed by the time-invariant polynomials F ∗v,1(z

−1),G∗

1(z−1) and H∗

1 (z−1). The inputs to this part are the amplitude r1(tc,k) of thevoltage reference signal and the amplitude v1(tc,k) of the estimated voltagephasor at fundamental frequency v1(tc,k). The output of the linear control isthe amplitude uv,1(tc,k) of the phasor voltage error at fundamental frequency.

The output of the linear control is subsequently integrated, see Figure 6.4,introducing saturation whose bounds are adaptively adjusted. From the in-tegrated output usat,1(tc,k), which represents the required voltage amplitude,the signal uc,1(tc,k) is generated by the adaptive compensating mechanism:using the estimated parameters v0,1,r, v0,1,i, req,1 and xeq,1 which are updatedevery steps, the non linearities represented by (6.22) and (2.8) are com-pensated by the block “Adaptive compensation of the power system model”.Finally, the nonlinearity represented by (2.9) is compensated using the block


H∗1 (z−1)

r1(tc,k) +F ∗−1

v,1 (z−1)uv,1(tc,k)

G∗1(z−1)

−

v1(tc,k)

Adaptively

saturated

integrator

v0,1,r( tc,k) v0,1,i( tc,k)

req,1( tc,k) xeq,1( tc,k)

usat,1(tc,k)

Adaptive

compensation

of the power

system model

Inverse

of f(α)

α(tc,k)ZOH

u(t)

Fig. 6.4. AVR structure in the case of SVS

including the inverse of f(α). The output is the firing angle α(tc,k) which isconverted by the zero-order hold device (ZOH) [99] into a continuous-timecommand u(t) = α(t) to be sent to the SVS.

The following two sections are dedicated, respectively, to the design ofthe linear time-invariant part of the AVR according to the pole-assignmenttechnique, and to the development of the adaptive compensating mechanism.

Pole-assignment Design

The polynomials F ∗v,1(z

−1) and G∗1(z

−1) (see Figure 6.4) are assigned so asto shift the closed-loop poles in some specified locations defined by the zerosof the polynomial T (z−1), see (4.12), such that the desired characteristics,expressed in terms of controller performance, are satisfied.

A block scheme representation used in the design phase is depicted inFigure 6.5.

H∗1 (z−1)

r1(tc,k) + 1F ∗

v,1(z−1)

uv,1(tc,k)Msvs,1(z−1)

v1(tc,k)

G∗1(z

−1)

−

Fig. 6.5. Block scheme representation adopted in the pole-assignment design forthe SVS

The transfer function Msvs,1(z−1) represents the sampled data transferfunction between uv,1 and v1. The hypothesis assumed in the design is that thenonlinearities in the model represented by (6.22), (2.8) and (2.9) are exactlycompensated and that the integrator is not saturated. The transfer functionMsvs,1(z−1) can be represented in a general form as


Msvs,1(z−1) =(

11 − z−1

)Z

ZOH(s)Qsvs,1(s)

= ZQsvs,1(s)

s

= z−d B+1 (z−1) B−

1 (z−1)A1(z−1)

(6.23)

where Qsvs,1(s) represents the transfer function of the cascade composed bythe actuator, the voltage measurement device and the Kalman filter. Thetransfer function of the actuator is represented by (2.11). The voltage mea-surement device is usually characterized by a bandwidth which is high and,consequently, the device is modeled by a simple gain.

The term z−d in (6.23) appears because (2.11) includes a pure time delayTsvs,d which is not, in practice, an exact multiple of the sampling period Tc.However, this time delay can be expressed as

Tsvs,d = d Tc − ρd Tc

with d a positive integer and 0 ≤ ρd ≤ 1.The coefficients of polynomial F ∗

v,1(z−1) and G∗

1(z−1) are obtained by solv-

ing the Diophantine equation

A1(z−1) F ∗1 (z−1) + z−d B−

1 (z−1) G∗1(z

−1) = T (z−1)

which has an unique solution if B1(z−1) and A1(z−1) are co-prime, and con-straints

nF∗1

= d − 1 + nB−1

(6.24a)

nG∗1

= nA1 − 1 (6.24b)

nT ≤ nA1 + nB−1

+ d − 1 (6.24c)

are fulfilled.Finally

F ∗v,1(z

−1) = F ∗1 (z−1) B+

1 (z−1)

while

H∗1 (z−1) = h∗

1,0 =T (z−1)

B−1 (z−1)

∣∣∣∣∣z−1=1

which ensures a unity steady-state gain for the closed-loop transfer function.


Adaptive Compensation

In the AVR control scheme (Figure 6.4) the estimates v0,1,r( tc,k), v0,1,i( tc,k),req,1( tc,k) and xeq,1( tc,k) are used to adapt the saturation limits of the in-tegrator as well as to perform adaptive nonlinearities compensation of thesystem model.

The first adaptive action refers to the calculation of the saturation limits ofthe integrator. The limits umax

sat,1 and uminsat,1 imposed on the variable usat,1(tc,k)

are calculated every steps using (6.22) rewritten, respectively, as

umaxsat,1( tc,k) =

∣∣∣∣ v0,1,r( tc,k) + j v0,1,i( tc,k)1 +

(req,1( tc,k) + j xeq,1( tc,k)

)ysvs(π)

∣∣∣∣umin

sat,1( tc,k) =∣∣∣∣ v0,1,r( tc,k) + j v0,1,i( tc,k)1 +

(req,1( tc,k) + j xeq,1( tc,k)

)ysvs(π/2)

∣∣∣∣in which the estimated parameters are known and we impose, respectively,

ysvs(π) = yFC

for the upper limit and

ysvs(π/2) = yFC + yR

for the lower limit, see (2.8) and (2.9).The second adaptive action has the objective of compensating the power

system model nonlinearities. This task is accomplished by solving, for a givenvalue of the variable usat,1(tc,k), at time instant tc,k, the following nonlinearproblem

find f(α(tc,k)

) ∈ [0, 1] : (6.25)

usat,1(tc,k) =∣∣∣∣ v0,1,r(tc,k) + j v0,1,i(tc,k)1 +

(req,1(tc,k) + j xeq,1(tc,k)

)(yFC + f

(α(tc,k)

)yR

) ∣∣∣∣It should be noted that in (6.25) the values of the estimated parameters

v0,1,r(tc,k), v0,1,i(tc,k), req,1(tc,k) and xeq,1(tc,k) that are used at time instanttc,k are the most recent ones, since they are updated every steps and canbe bigger than 1. The existence of the solution of (6.25) is assured by theadaptive saturation action on the integrator that determines usat,1(tc,k). Thesolution can be trivially obtained in closed analytical form.

Eventually, the last compensation block in Figure 6.4 evaluates the valueof α(tc,k) from the solution of (6.25) using the inverse of f(α). Due to theform of (2.9), the inverse cannot be evaluated analytically and a numericalprocedure must be used, based on a look-up table whose points lie on thediagram shown in Figure 6.6.


Fig. 6.6. Plot of the inverse of f(α)

6.3.3 Active and Hybrid Shunt Filters

Active shunt filters and hybrid shunt filters act to suppress harmonic compo-nents of nodal voltage, see Section 2.2.3. In this case, an AVR block is includedfor each harmonic of interest, see Figure 6.1, and the output command uc sentto the actuator is the sum of the various uc,h.

As described in Section 2.2.3, in both cases of ASF and HSF the activesection can be modeled by an ideal harmonic current generator, injecting acontrolled phasor iaf,h(t).

The ASF model to be included into the system equivalent circuit shownin Figure 2.7 is simply represented by the controlled current iaf,h(t). TheThevenin equivalent circuit equation, that is (2.13), can be written includingthe parameter estimations given by the CRLS algorithm yielding

vh(t) = v0,h + zeq,h iaf,h(t) (6.26)

The HSF model to be included into the system equivalent circuit shownin Figure 2.7 depends on the HSF topology. Referring to the one shown inFigure 2.6, the relationship (2.12) must be included into the Thevenin equiv-alent circuit equation written including the parameter estimations given bythe CRLS algorithm, yielding

vh(t) =zf1,h + zf2,hzeq,h + zf1,h + zf2,h

v0,h +zeq,h zf2,hzeq,h + zf1,h + zf2,h

iaf,h(t) (6.27)

It is apparent that both (6.26) and (6.27) do not include any dynamicsbut represent a nonlinear algebraic relationship between the controlled cur-rent iaf,h(t) and the nodal voltage harmonic component. In the overall modelthe dynamics to be included (see Figure 6.1) are related only to the currentcontrolled power electronic inverter (VSI or CSI), to the voltage transducer


and to the Kalman filter. Regarding the inverter, its dynamics are usuallyvery fast with respect to the voltage harmonic frequencies and they can beneglected. Concerning the voltage transducer and the Kalman filter the sameconsiderations as reported for the AVR design in the case of synchronous ma-chines and SVS, Section 6.3.1 and Section 6.3.2, can be made. Such dynamicsare linear and time-invariant.

The AVR structure is shown in Figure 6.7 with reference to the hth-ordervoltage harmonics.

H∗h(z−1)

rh,r(tc,k) +F ∗−1

v,h (z−1)uv,h,r(tc,k)

G∗h(z−1)

−

vh,r(tc,k)

Adaptivelysaturatedintegrator

v0,h,r( tc,k) v0,h,i( tc,k)

req,h( tc,k) xeq,h( tc,k)

usat,h,r(tc,k) Adaptive compensationof the power system

model and ofASF or HSF

uc,h(tc,k)



H∗h(z−1)

rh,i(tc,k) ++F ∗−1

v,h (z−1)uv,h,i(tc,k)

G∗h(z−1)

−

vh,i(tc,k)

Adaptivelysaturatedintegrator

usat,h,i(tc,k)



Fig. 6.7. AVR structure in the case of ASF or HSF

The time-invariant parts of the AVR, composed of polynomials F ∗v,h(z−1),

G∗h(z−1) and H∗

h(z−1), are evidenced by the dashed boxes in Figure 6.7. Theinputs to first box are the real component rh,r(tc,k) of the voltage referencesignal rh(tc,k) and the real component vh,r(tc,k) of the estimated voltage pha-sor vh(tc,k). The output is the real component uv,h,r(tc,k) of the phasor volt-age error uv,h(tc,k). Concerning the second box, the inputs are the imaginarycomponent rh,i(tc,k) of the voltage reference signal rh(tc,k) and the imaginarycomponent vh,i(tc,k) of the estimated voltage phasor vh(tc,k). The output isthe imaginary component uv,h,i of the phasor voltage error uv,h(tc,k).

The outputs are subsequently integrated yielding a phasor voltage com-mand expressed in terms of real and imaginary components usat,h,r(tc,k) and


usat,h,i(tc,k). The adopted integrator has saturation whose bounds are adap-tively adjusted.

Then, from the phasor voltage command the signal uc,h is generated bythe adaptive compensating mechanism which, on the basis of the estimatedparameters v0,h,r, v0,h,i, req,h and xeq,h, compensates for the nonlinearities ofthe power system and actuator models. Finally the outputs uc,h are summedfor all the nh harmonic orders and the resulting output uc is converted into acontinuous-time command u by the ZOH.

The following two paragraphs are dedicated, respectively, to the design ofthe linear time-invariant section of the AVR according to the pole-assignmenttechnique, and to the development of the adaptive compensating mechanism.

Pole-assignment Design

We consider the real components in the most general case, see Figure 6.1 andFigure 6.7. The imaginary components can be treated similarly.

The polynomials F ∗v,h(z−1) and G∗

h(z−1) are assigned so as to shift theclosed-loop poles in some specified locations defined by the zeros of the poly-nomial Th(z−1), see (4.12).

A block scheme representation used in the design phase is depicted inFigure 6.8; Mh(z−1) represents the sampled data transfer function betweenuv,h,r and vh,r. It can be represented in a general form as

Mh(z−1) =(

11 − z−1

)Z

ZOH(s)Qh(s)

= ZQh(s)

s

= z−d B+h (z−1) B−

h (z−1)Ah(z−1)

(6.28)

where Qh(s) represents the transfer function of the cascade composed of theactuator, the voltage measurement device and the Kalman filter. The transferfunction Mh(z−1) assumes that the nonlinearities in the model that representsthe circuit shown in Figure 2.7 are exactly compensated and the integrator isnot saturated.

In (6.28) the polynomial B+h (z−1) has its roots inside the unit circle while

the roots of B−h (z−1) are outside the unit circle. Moreover, the term z−d

appears since the model that represents the actuator includes, in additionto poles and zeros, a pure time delay Td, which is not, in practice, an exactmultiple of the sampling period Tc. However, this time delay can be expressedas

Td = d Tc − ρd Tc 0 ≤ ρd ≤ 1

with d a positive integer.


H∗h(z−1)

rh,r(tc,k) + 1F ∗

v,h(z−1)uv,h,r(tc,k)

Mh(z−1)vh,r(tc,k)

G∗h(z−1)

−

Fig. 6.8. Block scheme representation adopted in the pole-assignment design forthe ASF and the HSF

The coefficients of polynomial F ∗v,h(z−1) and G∗

h(z−1) are obtained bysolving the Diophantine equation

Ah(z−1) F ∗h (z−1) + z−d B−

h (z−1) G∗h(z−1) = Th(z−1)

which has a unique solution if Bh(z−1) and Ah(z−1) are co-prime and con-straints

nF∗h

= d − 1 + nB−h

(6.29a)

nG∗h

= nAh− 1 (6.29b)

nTh≤ nAh

+ nB−h

+ d − 1 (6.29c)

are fulfilled.Finally

F ∗v,h(z−1) = F ∗

h (z−1) B+h (z−1)

while

H∗h(z−1) = h∗

h,0 =Th(z−1)B−

h (z−1)

∣∣∣∣∣z−1=1

which ensures a unity steady-state gain for the closed-loop transfer function.

Adaptive Compensation

In the AVR structure, see Figure 6.7, the estimates v0,h,r, v0,h,i, req,h andxeq,h are used to adapt both the saturation limits of the integrators and toperform adaptive nonlinearities compensation of the power system model.

The first adaptive action refers to calculation of the saturation limits ofthe integrators in Figure 6.7. For each integrator one limit is adaptively cal-culated as described in the following: the other limit is fixed to be equal inmodulus to the former one but with opposite sign. The limits ulim

sat,h,r andulim

sat,h,i imposed, respectively, on the variables usat,h,r(tc,k) and usat,h,i(tc,k)


are calculated every steps using (6.26) and (6.27). In particular, for ASFtopology (6.26) is rewritten as

ulimsat,h( tc,k) = v0,h( tc,k) + zeq,h( tc,k) ilimaf,h( tc,k) (6.30)

whereas for HSF topology (6.27) is rewritten as

ulimsat,h( tc,k) =

zf1,h + zf2,hzeq,h( tc,k) + zf1,h + zf2,h

v0,h( tc,k)

+zeq,h zf2,hzeq,h( tc,k) + zf1,h + zf2,h

ilimaf,h( tc,k) (6.31)

In (6.30) and (6.31) it is assumed

ilimaf,h( tc,k) = imaxaf,h

iaf,h( tc,k)iaf,h( tc,k)

where imaxaf,h is the maximum amplitude or rms value of the hth order harmonic

component of the current that the active section of the shunt filter can gener-ate. Concerning the phasor iaf,h( tc,k) and its amplitude iaf,h( tc,k), they areobtained by sampling every steps the corresponding required current phasor,which is evaluated in the second task described in the following.

Eventually, the limits ulimsat,h,r( tc,k) and ulim

sat,h,i( tc,k) are evaluated as,respectively, the real and the imaginary component of ulim

sat,h( tc,k) givenby (6.30) or (6.31).

The second adaptive task in Figure 6.7 has the objective of compensatingthe power system model nonlinearities. Starting from (6.26) and (6.27), therequired current phasor iaf,h(tc,k) can be evaluated, respectively, for the ASFtopology from

iaf,h(tc,k) =usat,h(tc,k) − v0,h(tc,k)zeq,h(tc,k)

(6.32)

and for the HSF topology from

iaf,h(tc,k) = − zf1,h + zf2,hzeq,h(tc,k) zf2,h

v0,h(tc,k)

+zeq,h(tc,k) + zf1,h + zf2,hzeq,h(tc,k) zf2,h

usat,h(tc,k) (6.33)

where usat,h(tc,k) is the phasor obtained from the real component usat,h,r(tc,k)and the imaginary component usat,h,i(tc,k). It should be noted that in (6.32)and (6.33) the values of the estimated parameters v0,h(tc,k), req,1(tc,k) andxeq,1(tc,k) used at time instant tc,k are the most recent, since they are updatedevery steps and can be bigger than 1.


From the real and imaginary components of the required current phasoriaf,h(tc,k) the output uc,h(tc,k) of the AVR structure is evaluated as:

uc,h(tc,k) = iaf,h,r(tc,k) cos(h ω tc,k) − iaf,h,i(tc,k) sin(h ω tc,k) .


In the following the results obtained applying the adaptive nonlinearities com-pensation technique will be illustrated with reference to three cases: the syn-chronous generator and the SVS connected to the HV network described inSection A.1 and the active shunt filter connected to the industrial networkdescribed in Section A.2.

Synchronous Generator

The synchronous generator characteristics are described in Section A.1.2.Concerning the implementation of the AVR scheme shown in Figure 6.2,

it is worth recalling that the control input uc(tc,k) is saturated to accountfor the limitation of the voltage excitation vsm,f . In particular, the exciteris assumed to guarantee an excitation voltage in the range 1.5 − 3.5 p.u. Toachieve antiwind-up a desaturated integrator scheme similar to the one shownin Figure 4.7 was implemented [17].

The constrained recursive least-squares algorithm described in Section 6.2was used only at the fundamental frequency (h = 1) and using per unit values.In (6.3) it is assumed that cv,1 = 10. The constrained recursive least-squaresalgorithm for the estimation of the Thevenin equivalent circuit parameters hasan adaptive forgetting factor according to (4.11) with σ2

0 N0 = 0.01, λmin =0.6 and λmax = 1.0. In addition, εdz = 0.0004.

Concerning the AVR design, see Section 6.3.1, the transfer function tobe used in the design, which is obtained from (6.18) and (2.6), is simplifiedto limit the order of Qsm,1(s), accounting only for its dominant poles. Thediscrete transfer function Msm,1(z−1) resulting from (6.19) with a samplingtime equal to 1 ms, gives nA1 = 5 and nB1 = 1. Consequently, from (6.21),nFv,1 = 4, nG1 = 4 and nT = 8. The polynomial T (z−1) has been assignedso that the closed-loop step response has a rise time between 10% and 90%equal to about 130 ms, and a settling time at ±2% equal to about 200 ms. Theadaptive design is performed every fundamental time period, that is = 20.

During the start-up phase of the numerical simulation the coefficients ofthe polynomials Fv,1(z−1), G1(z−1) and H1(z−1) are set equal to zero, exceptfor g1,0 = h1,0 = 0.2. At time instant t = 7 s the adaptive loop is closed.

The time variation of the regulated voltage v1(t) at the generator ter-minals and of the excitation voltage vsm,f (t) are reported, respectively, inFigure 6.9 and in Figure 6.10. It is evident that after closing the adaptiveloop the regulated voltage quickly reaches the reference value of to 1.04 p.u.


At time instants t = 10 s, t = 16 s and t = 19 s, at which step variations ofr1(t) are imposed, the excitation voltage saturates causing lower values of therise and settling times with respect to the desired ones. In addition, at timeinstant t = 13 s a 50% variation of load Q2 is imposed. From Figure 6.9 it isapparent that the adaptive mechanism limits the amplitude of the consequentvoltage variation in spite of the significant load change. This is due to theprompt estimation of the Thevenin equivalent circuit parameters as shown inFigure 6.11.

It is important to underline that the time variation of the excitation voltagevsm,f (t) presents oscillations in response to the mechanical damping action ofthe power system stabilizer. Yet, the effects of such oscillations on the timevariation of v1(t) are negligible.

Fig. 6.9. Time variation of v1(t)

Fig. 6.10. Time variation of vf (t)


Fig. 6.11. Time variation of the estimated parameters values v0,1, req,1, xeq,1 ofthe Thevenin equivalent circuit at fundamental frequency

Finally, in Figure 6.12 the time variation of the coefficients of polynomialFv,1(z−1) are reported, whereas Figure 6.13 shows the time variations of thecoefficients of polynomials G1(z−1) and H1(z−1). After a large variation atthe closure of the adaptive loop, the subsequent variations of the coefficientsof Fv,1(z−1) cannot be detected in the reported figures because of their smallvalues with respect to the first large variation. On the other hand, the vari-ations of the coefficients of polynomials G1(z−1) and H1(z−1) can be clearlyobserved.

Static VAr System

The closed-loop specifications imposed during this simulation case study areequal to the ones described in Section 4.2.5; that is a rise time between 10%and 90% equal to about 40 ms, and a settling time at ±2% equal to about70 ms.

The constrained recursive least-squares algorithm described in Section 6.2has been run setting εdz = 0.001, while the forgetting factor is updated ac-cording to (4.11) with σ2

0 N0 = 0.01, λmin = 0.6 and λmax = 1.0. We also setcv,1 = 10.


Fig. 6.12. Time variation of fv,1,1, fv,1,2, fv,1,3, fv,1,4

Fig. 6.13. Time variation of g1,0, g1,1, g1,2, g1,3, g1,4, h1,0


The coefficients of the polynomials regulator F ∗v,1(z

−1), G∗1(z

−1) andH∗

1 (z−1) have been designed under the hypothesis that the power systemmodel nonlinearities are fully compensated and the time-integrator shown inFigure 6.4 is unsaturated. The discrete transfer function Msvs,1(z−1), resultingfrom (6.23) with a sampling time equal to 1 ms, gives nA1 = 3 and nB1 = 2.Consequently, the fulfillment of constraints (6.24) leads to nFv,1 = 5, nG1 = 2and nT = 4.

Fig. 6.14. Time variation of v#4,1

The time variation of the regulated nodal voltage v#4,1(t) obtained inresponse to the step changes of the reference set-point is reported in the left-hand side of Figure 6.14. As already previously done, the right-hand side of thesame figure shows an enlarged view of the time variation between time instantst = 12.998 s and t = 13.1 s in response to the step variation of r(t) applied att = 13 s to allow calculation of the required closed-loop specifications.

At time instant t = 5 s the nodal voltage v#4,1(t) exhibits a large variationdue to the closing of the adaptive loop. This circumstance can also be noticedby looking at the time variation of the firing angle α(t) reported in Figure 6.15.By comparison of Figure 4.8 with Figure 6.15 it is seen that with application ofthe adaptive nonlinearities compensation technique, the variations undergoneby the nodal voltage v#4,1(t) caused by load variation and line opening aresmaller than those obtained with application of the self-tuning pole-shiftingtechnique. This is mainly due to the fact that a change of the estimatedparameters has a direct effect on the AVR output according to (6.25).

Moreover, it is worth noting that the adaptive compensation mechanismof the power system model nonlinearities does not lead to saturation of thefiring angle α(t).

Finally, Figure 6.16 reports the time variation of the estimated values of theparameters of the Thevenin equivalent circuit at the fundamental frequency.


Fig. 6.15. Time variation of α(t)

Fig. 6.16. Time variation of the estimated parameters values v0,1, req,1, xeq,1 ofthe Thevenin equivalent circuit at the fundamental frequency


Active Shunt Filter

The active shunt filter (ASF) considered in the simulation is an active shuntfilter connected to the 13.8 kV/60 Hz main busbar of the IEEE-industrialnetwork described in Section A.2. The ASF is composed of a voltage sourceinverter (VSI); the PWM commutation frequency is 10 kHz and the DC siderated voltage is 1000 V. The current that the ASF injects into the powersystem is controlled by means of a current feedback regulation loop.

The mission of the ASF is to control the main busbar voltage waveform byinjecting adequate current harmonics. In particular, without loss of generality,our attention has been focused on the 5-th order voltage harmonic.

In the self-tuning voltage regulator design, it has been assumed that TKf =8.33µs for the Kalman filters, Te = 250Tsf (m = 250) for the CRLS blocks, seeSection 6.1, and Ts = 0.6ms for the AVR blocks, see Figure 6.1. The voltageerror generator design has been performed so as to guarantee a closed-loopdamped step response with a settling time equal to 4 fundamental cycles. Thediscrete transfer function Mh(z−1) resulting from (6.28) presents nAh

= 3and nBh

= 2 with nB+h

= 1 and nB−h

= 1. Consequently, from (6.29) it is setnF∗

h= 2, nG∗

h= 2 and nTh

= 5.Three different operating conditions of the industrial system have been

simulated. In the first simulation (in the following, it is referred to as case 1),the step response of the adaptive voltage regulator is analyzed. In this case,the CRLS blocks do not detect any change in the values of the parameters ofthe electrical system equivalent circuit and, consequently, the adaptive mech-anism of the self-tuning voltage regulator is practically inactive. The stepresponse that is considered corresponds to the actual case that happens whenthe active shunt filter voltage regulation loop is instantaneously closed: thebusbar voltage harmonics are equal to the ambient harmonics before the reg-ulation insertion (no current is injected by the active shunt filter) and tend tozero after the voltage regulation action. In the left-hand side of Figure 6.17the time variation of the real and imaginary part of the 5-th order voltageharmonic is shown. It can be recognized that the adaptive voltage regula-tor design guarantees the required closed-loop performance. Indeed, from thepower quality point of view, it is interesting to analyze the plot reported inthe right-hand side of Figure 6.17 which shows the resulting response in termsof rms value of the same voltage harmonic.

To test the self-tuning voltage regulator with respect to changes of theindustrial distribution system, two different operating conditions have beensimulated. In detail, in the second simulation (it is referred to as case 2), thestarting operating condition is the basic IEEE-test standard case except forthe higher value of the harmonic currents injected by the nonlinear load. Attime instant t = 0.3 s, an additional linear load Q4 is suddenly connectedby means of a switch to the main medium voltage busbar (at which alsothe active shunt filter is derived): it yields a decrease both of the voltagedistortion level and of the Thevenin equivalent impedance. In the left-hand


Fig. 6.17. Time variation of the real and imaginary part (top) and of the rms value(bottom) of the 5th order voltage harmonic; case 1: step response of the adaptivevoltage regulator

side of Figure 6.18 the time variation of the real and imaginary part of the 5-thorder voltage harmonic is reported, while in the right-hand side of the samefigure the corresponding rms voltage harmonic time variation is represented.

Fig. 6.18. Time variation of the real and imaginary part (top) and of the rms value(bottom) of the 5-th order voltage harmonic; case 2: sudden connection of the linearload Q4

Regarding the identification of the Thevenin circuit parameters, Fig-ure 6.19 shows their time variation during the regulator response.

In the third simulation (it is referred to as case 3), the starting operatingcondition is the basic IEEE-test standard case. At time instant t = 0.3 s, theadditional nonlinear load Q2 is suddenly connected by means of a switch toa low voltage load busbar: it yields an increase of the voltage distortion leveland a decrease of the Thevenin equivalent impedance. In the left-hand sideof Figure 6.20 the time variation of the real and imaginary part of the 5-thorder voltage harmonic is reported, while in the right-hand side of the samefigure the corresponding rms voltage harmonic time variation is represented.


Fig. 6.19. Time variation of the estimated parameters values v0,5 and zeq,5 of the5-th order harmonic Thevenin equivalent circuit; case 2: sudden connection of thelinear load Q4

In Figure 6.21 the time variation of the identified parameters during the reg-ulator response is shown. From the analysis of Figure 6.18 and Figure 6.20 itis evident that the design requirement (a settling time equal to 4 fundamentalcycles) is satisfied in spite of the imposed load variations. From the identifi-cation point of view, Figure 6.19 and Figure 6.21 highlight the responsivenessof the constrained recursive least-squares algorithm which identifies the newvalues of the parameters of the Thevenin equivalent circuit in both the casesconsidered.

Fig. 6.20. Time variation of the real and imaginary part (top) and of the rmsvalue (bottom) of the 5-th order voltage harmonic; case 3: sudden connection of thenonlinear load Q3

Finally, an additional simulation has been run to severely test the robust-ness of the self-tuning voltage regulator scheme to the estimation errors. A20% estimation error on the values v0,5 and zeq,5 has been arbitrarily imposed.In particular, v0,5 and req,5 have been increased while xeq,5 decreased. Thetime variation of the real and imaginary part of the 5-th order harmonic volt-


Fig. 6.21. Time variation of the estimated parameters values v0,5 and zeq,5 of the5-th order harmonic Thevenin equivalent circuit; case 3: sudden connection of thenonlinear load Q3

age to the reference step variation in the presence of the considered estimationerrors is reported in Figure 6.22. For the sake of comparison, in the same Fig-ure 6.22 the time variation of the real and imaginary part of the 5-th ordervoltage harmonic in the absence of the imposed estimation errors, already re-ported in Figure 6.17, is also depicted. From the analysis of Figure 6.22 it canbe recognized that the imposed estimation errors slightly affect the adaptivenonlinearities compensation function. In particular, a small overshoot in thetime variation of the imaginary part appears, while the desired value of thesettling time is still satisfied. It can be finally observed that the large errorsimposed on |v0,5| and zeq,5 lead to a faster but still stable response of theself-tuning voltage regulator.

Fig. 6.22. Time variation of the real (left-hand side) and imaginary part (right-hand side) of the 5-th order harmonic voltage as reported in the left-hand side ofFigure 6.17 (x) and in presence of a 20% parameters identification error (xx)

Regarding the value of the imposed estimation error (20%), it represents asevere robustness test for the proposed self-tuning voltage regulator scheme.


In fact the adopted recursive least-squares technique is constrained by (6.7)representing physically meaningful constraints. Then, with respect to a clas-sical recursive least-squares algorithm, the implemented constrained versionof this algorithm has fewer degrees of freedom and leads to a better estimateof the unknown parameters.

6.4 Optimization Strategy

The adaptive nonlinearities compensation technique can be extended toachieve a multi-objective optimization control strategy for the voltage con-trol device. The estimated parameters of the Thevenin equivalent circuit canbe used to change the voltage reference values that are sent to the AVR blocks,see Figure 6.1, according to optimization criteria which account for additionalobjectives.

Consider, for example, the SVS: while regulating nodal voltage amplitudeat fundamental frequency, the SVS injects current harmonic components intothe power system. Then, expensive filters have been traditionally used tocompensate for harmonic pollution [39, 112]. To help reduce the rating ofsuch filters it is possible to adopt an optimization strategy which is based onminimization of an index accounting for the harmonic distortion level.

The optimization procedure, see Figure 6.23, receives the following in-puts: the voltage reference signal vd from the Regional/area voltage regulation(RVR), as described in Section 1.3, and the estimates v0,1,r, v0,1,i, req,1 andxeq,1 from the estimation task of the Thevenin equivalent circuit parameterdescribed in Section 6.2.

vd(1 ∓ c) Optimizationstrategy

v0,1,r v0,1,i

req,1 xeq,1

r1

Fig. 6.23. Optimization strategy block

If the RVR can successfully perform its regional voltage regulation taskallowing the SVS voltage amplitude within an assigned range [vd,m, vd,M ]where

vd,m = vd(1 − c) and vd,M = vd(1 + c)

then the optimization strategy utilizes such degrees of freedom to minimizethe harmonic distortion level produced by the SVS into the power system.The quantity c is equal to

6.4 Optimization Strategy 139

c =∆ v

2 vd

with ∆ v = vd,M − vd,m the width of the allowed voltage range.The first step in the optimization procedure realizes the transformation of

the assigned boundary values vd,m and vd,M in the corresponding limit valuesαm and αM , with αm < αM , to be forced to the firing angle α. The followingproblem, similar to (6.25), must be solved:

find f(αM ) : vd,M =∣∣∣∣ v0,1,r + j v0,1,i

1 +(req,1 + j xeq,1

)(yFC + f

(αM

)yR

) ∣∣∣∣The resulting value of f(αM ) must be limited to the range [0, 1] and then

substituted into (2.9) so that the required value of αM is obtained numerically.The same procedure can be adopted to evaluate αm from vd,m.

The second step of the optimization strategy concerns the problem ofminimizing, with respect to the variable α, an assigned function J

(ih(α)

)of

the amplitudes of the current harmonic phasors injected by the SVS. Amongdifferent functions available in the literature which give a measure of thedistortion level present in the power system (see for example [6]), the totalharmonic distortion index is widely accepted as one of the most significant

THD(α) =

√√√√ n∑i = 1

ih(α)2

in which ih(α) is given by (2.10) and v1(α) by (6.22).The one-dimensional problem to be solved in the second step is then

minα

THD(α)

with αm ≤ α ≤ αM (6.34)

The optimization function is nonlinear but continuous and differentiablewith respect to α. In addition only box constraints are present. Consequentlythe solution can be found numerically by applying well-established methodsfor one-dimensional numerical search.

Denote with αopt the value resulting from the solution of problem (6.34).The third and final step of the procedure strategy aims at determining thereference value r1(αopt). This can be easily obtained by

r1(αopt) =∣∣∣∣ v0,1,r + j v0,1,i

1 +(req,1 + j xeq,1

)(yFC + f

(αopt

)yR

) ∣∣∣∣It is worthwhile noting that all three steps of the optimization strategy

require the availability of the estimated parameter values v0,1,r, v0,1,i, req,1 andxeq,1. That is, topological changes in the power system, leading to variationsin the estimated parameter values, are taken into account to generate thereference value r1(αopt). Then the optimization strategy realizes an adaptiveaction with respect to the changes of the power system operating points.

A

Computer Models and Topology of Networks

In this appendix, we describe two different types of electrical system whichhave been employed in the numerical simulations reported in this book. Thefirst type is a high-voltage transmission network while the second is an indus-trial electric system.

A.1 High-voltage Network

For a simple software implementation on computers, a modular approach tothe modeling problem of a high-voltage transmission network is adopted. Itcomprises all the main components, namely synchronous generators, trans-formers, transmission lines, electronic actuators and loads.

A.1.1 Computer Models of Components

The synchronous generator is represented, in its d− q frame, by means of thefollowing 8th-order dynamic nonlinear model:

p δsm(t) = ω(t) − ωn (A.1a)

pω(t) =ωn

Ta

(Cm(t) − Ce(t)

)(A.1b)

p e′′q (t) = −k1 e′′q (t) + k1 e′q(t) − k2 id(t) + k3 vsm,f (t) (A.1c)

p e′q(t) = −k4 e′q(t) − k5 id(t) + k6 vsm,f (t) (A.1d)

p e′′d(t) = −k7 e′′d(t) + k7 e′d(t) + k8 iq(t) (A.1e)

p e′d(t) = −k9 e′d(t) + k10 iq(t) (A.1f)

p ψd(t) = ωn vd(t) + ωn r id(t) + ω(t) ψq(t) (A.1g)

pψq(t) = ωn vq(t) + ωn r iq(t) − ω(t) ψd(t) (A.1h)

142 A Computer Models and Topology of Networks

Ce(t) = ψd(t) iq(t) − ψq(t) id(t) (A.1i)

ψd(t) = e′′q (t) − x′′d id(t) (A.1j)

ψq(t) = −e′′d(t) − x′′q iq(t) (A.1k)

where p is the differential operator.Model (A.1) is written for a two-pole machine. It is characterized by two

state variables related to the mechanical dynamics, namely rotor angular posi-tion δsm(t) and speed ω(t) (which in the two-pole machine corresponds to theelectrical pulse), and by six variables related to the electromagnetic dynam-ics, namely fluxes ψd(t), ψq(t), subtransient emfs e′′d(t), e′′q (t) and transientemfs e′d(t), e′q(t). The inputs are the mechanical torque Cm(t) and the exci-tation voltage vsm,f (t). Details concerning the expressions for constants ki,(i = 1, . . . 10), can be found, for example, in [7, 86].

When model (A.1) is utilized to simulate the synchronous generator in asimulation scheme, a choice of inputs and outputs of this model is required. Inparticular, voltages and currents are related by the power system equations:voltages should be given by the remaining power system models and consid-ered as inputs to the machine model, whereas currents should be outputs.It is then difficult to cope with the presence of current components in theright-hand sides of all the electromagnetic equations in (A.1). However, sincethe present modeling aims at voltage control problem analysis, an approxi-mated model can be considered, neglecting the effects of pulse variations onthe flux components. Consequently, in (A.1g) and (A.1h) the nominal pulseωn substitutes for the electrical pulse ω(t). Such a simplification transformsmodel (A.1) into a linear differential model with nonlinear algebraic equations.Then, substituting algebraic expressions for flux components, that is (A.1j)and (A.1k), in the electromagnetic differential equations referring to the flux,that is (A.1g) and (A.1h), after trivial manipulations, yields the followingdifferential equations referred to the current components:

p id(t) =1x′′

d

(− k1 e′′q (t) + k1 e′q(t) − k2 id(t) + k3 vsm,f (t) − ωn rs id(t)

−ωnvd(t) + ωn e′′d(t) + ωn x′′q iq(t)

)(A.2a)

p iq(t) =1x′′

q

(k7 e′′d(t) − k7 e′d(t) − k8 iq(t) − ωn rs iq(t)

−ωnvq(t) + ωn e′′q (t) − ωn x′′d id(t)

)(A.2b)

The resulting 8th-order model given by (A.1a)–(A.1f),(A.2a) and (A.2b),presents a set of eight linear differential equations in the eight state variables,namely δsm(t), ω(t), e′′q (t), e′q(t), e′′d(t), e′d(t), id(t) and iq(t), and one nonlinearalgebraic equation, needed to evaluate the electromagnetic torque accordingto the following:

A.1 High-voltage Network 143

Ce(t) = e′′q (t) iq(t) + e′′d(t) id(t) −(x′′

d − x′′q

)iq(t) id(t)

The model inputs are the mechanical torque Cm(t), the excitation voltagevsm,f (t) and the voltage components vd(t) and vq(t). The torque is the out-put of the models of the governor and of the turbine, whereas the excitationvoltage is the output of the model of the excitation system, including alsopower system stabilizing signals. For this, well-known models established inthe literature [4, 7, 76,87] can be used.

Eventually, transformations from d−q to three-phase components and viceversa must be included, to exchange stator current and voltage componentswith the remaining power system models. Such transformations refer to therotor angular position δsm(t), which represents the phase displacement be-tween the machine q-axis and the real axis (positive when the former leadsthe latter) in the power system phasor axis frame, which rotates at ωn speed.Without loss of generalization, the zero component can be neglected becauseit is usually null in normal operating conditions of power generating units dueto the ∆−Y connection of the step-up transformer windings. Then, the trans-formations of d− q current components into three-phase r− s− t componentsare

ir(t) =

√23

(id(t) sin

(ωn t + δsm(t)

)+ iq(t) cos

(ωn t + δsm(t)

))

is(t) =

√23

(id(t) sin

(ωn t + δsm(t) − 2

3π)

+ iq(t) cos(ωn t + δsm(t) − 2

3π))

it(t) =

√23

(id(t) sin

(ωn t + δsm(t) +

23

π)

+ iq(t) cos(ωn t + δsm(t) +

23

π))

whereas the transformations of r − s − t voltage components into d − q com-ponents are

vd(t) =

√23

(vr(t) sin

(ωn t + δsm(t)

)+ vs(t) sin

(ωn t + δsm(t) − 2

3π)

+vt(t) sin(ωn t + δsm(t) +

23

π))

vq(t) =

√23

(vr(t) cos

(ωn t + δsm(t)

)+ vs(t) cos

(ωn t + δsm(t) − 2

3π)

+vt(t) cos(ωn t + δsm(t) +

23

π))


Three transmission lines are represented by a series of elementary differ-ential elements including only passive elements. Two types of transmissionline models are needed to allow the meshed network topology. The first typeadopts an elementary cell represented by the two-port shown in Fig. A.1. Thedynamic state-space model is given by

p

[i1(t)v2(t)

]=

[−r/ −1/

1/c 0

][i1(t)v2(t)

]+

[1/ 00 −1/c

][v1(t)i2(t)

]

y(t) =

[i1(t)v2(t)

]

cv1 v2

r i1 i2

Fig. A.1. Two-port circuit representing the line elementary cell: current-voltagetype

Such a line is called a current-voltage line because its inputs are v1(t)and i2(t). The second type adopts an elementary cell represented by the two-port circuit shown in Figure A.2; it is called a voltage-voltage line because itsinputs are v1(t) and v2(t). Its dynamic state-space model is expressed by

p

⎡⎢⎣ i1(t)vc(t)i2(t)

⎤⎥⎦ =

⎡⎢⎣−r/ −2/ 01/c 0 −1/c

0 2/ −r/

⎤⎥⎦⎡⎢⎣ i1(t)

vc(t)i2(t)

⎤⎥⎦+2

[1 00 −1

][v1(t)v2(t)

]

y(t) =

[1 0 00 0 1

]⎡⎢⎣ i1(t)vc(t)i2(t)

⎤⎥⎦Finally the third type, named current-current, is represented by the ele-

mentary cell depicted in Figure A.3Its dynamic state-space model is expressed by


c

v1 v2vc

r/2 /2i1 i2r/2 /2

Fig. A.2. Two-port circuit representing the line elementary cell: voltage-voltagetype

c/2 c/2

r

v1 v2

i1 i i2

Fig. A.3. Two-port circuit representing the line elementary cell: current-currenttype

p

⎡⎢⎣ v1(t)v2(t)i(t)

⎤⎥⎦ =

⎡⎢⎣ 0 0 −2/c

0 0 2/c

1/ −1/ −r/

⎤⎥⎦⎡⎢⎣ v1(t)

v2(t)i(t)

⎤⎥⎦+2c

⎡⎢⎣1 00 −10 0

⎤⎥⎦[ i12(t)i21(t)

]

y(t) =

[1 0 00 1 0

]⎡⎢⎣ v1(t)v2(t)i(t)

⎤⎥⎦In a high-voltage distribution network, transformers are usually simply

modeled by equivalent series resistance and inductance [7, 87]. In fact, morecomplex models are necessary to extract detailed information about trans-former operations but do not increase the accuracy of overall power systemsimulation.

In the case of step-up transformers in power plants, their equivalent resis-tance and reactance can be trivially included into the synchronous machinemodels by adding them respectively to the stator resistance and to the sub-transient reactances in (A.2a) and (A.2b). It is possible because the voltagedrops related to the three phases r − s − t on the transformer resistance rtr

and inductance tr


rtr ir(t) + tr p ir(t)

rtr is(t) + tr p is(t)

rtr it(t) + tr p it(t)

can be transformed into d − q voltage components according to

rtr id(t) + tr p id(t) −ωn tr iq(t)

rtr iq(t) + tr p iq(t) +ωn tr id(t)

Step-down transformers in substations are usually included into the equiv-alent model of the loads, which are described in the following.

The electronic actuator considered in the computer simulations is a fixed-capacitor thyristors controlled reactor (FC-TCR); see Figure 2.3. The follow-ing equation represents the capacitive branches:

p

[ic(t)vc(t)

]=

[−r/ −1/

1/c 0

][ic(t)vc(t)

]+

[1/

0

]v(t)

where v(t) is the busbar phase-to-phase voltage, that is the input, and thestate variables are the ideal capacitance current ic(t) and voltage vc(t); rand respectively account for the real capacitor active losses and the smallinductances of the electrical circuit connections.

The following simple equation represents the inductive branches:

p i(t) = −r i(t) + v(t)

where the state variable is the inductance current i(t) and r accounts for theinductor active losses. Such an equation is integrated only when thyristors areswitched-on.

The thyristor firing pulse is driven according to the input angle α whichrepresents the delay of the firing instant of each thyristor with respect tothe zero crossing of the corresponding voltage [68]. In this way, the FC-TCRdevice varies the value of its equivalent admittance at fundamental frequencywith the input firing angle α according to law (2.8)

Finally, nodal loads are represented by means of equivalent shunt admit-tances (static loads). This is a common assumption in high voltage transmis-sion networks [7, 87]. The active component ıG(t) and reactive ıB(t) of thecurrent driven by load can be expressed as

ıG(t) =v(t)r

=PL,n

v2n

v(t)

p ıB(t) =v(t)

= −QL,n

v2n

ωn v(t)

where v(t) is the nodal voltage, PL,n the rated value of the active power drivenby load, QL,n the rated value of the reactive power driven by load and vn thenominal voltage amplitude.


A.1.2 Simulated Network

The high-voltage distribution network shown in Figure A.4 has been simulatedas a test system.

It is a three-phase 220 kV - 50 Hz system which is connected at busbar 1 toa larger 220/380 kV network, which is represented by a Thevenin equivalentcircuit assuming a short-circuit power equal to 11, 000 MVA and an open-circuit voltage equal to 1.01 p.u.

At busbar 2 a 180 MVA synchronous generator, coupled with a 150 MWturbine, is connected to the network by a 200 MVA 20/220 kV step-up trans-former. The synchronous machine parameters are reported in Table A.1.

At busbars 3 and 4 two 50 MW loads named, respectively, Q1 and Q2are derived both with a power factor equal to 0.9. In addition, at busbar 4 aFC-TCR SVS is connected, which can vary its reactive power injection in therange ±50 MVAr.

Three-phase simulation is performed accounting for ∆/Y transformerwinding connections, but the power system has been assumed to be balancedin all its components.

The 220 kV overhead transmission lines are modeled by a series of ele-mentary four-terminal circuits described in the previous section; each circuitis equivalent to 10 km of the line. The lumped parameter of the circuits areevaluated assuming the following values per unit length: r = 84 [mΩ/km], = 1.3 [mH/km] and c = 8.6 [nF/km].

Fig. A.4. High-voltage test network


Table A.1. Synchronous generator data

Rotor Sn Vn fn rs number

type [MVA] [kV] [Hz] [p.u.] of poles

massive 180 20 50 0.001 2

Xd X′d X′′

d Xq X′q X′′

q

[p.u.] [p.u.] [p.u.] [p.u.] [p.u.] [p.u.]

1.90 0.302 0.204 1.70 0.50 0.30

T′d T′′

d T′q T′′

q TAA Ta

[s] [s] [s] [s] [s] [s]

1.27 0.027 0.235 0.012 0.02 7.5

Concerning the generator unit control, a brushless excitation system hasbeen modeled according to [4] with a voltage reference value equal to 1.02 p.u.and a no-load steady-state gain equal to 400 p.u. The turbine/governor hasbeen modeled assuming a 5% frequency droop and a power reference valueequal to 50 MW. The power system stabilizer has been modeled accordingto [87].

Finally, in the numerical simulations reported in this book, reference ismade to the phase voltage rms values expressed in per unit on a 220 kV base.

A.1.3 Network Equivalent Time Domain Model

The high-voltage test network has been simulated at the described operatingpoint to determine the orders nA and nB of polynomials A(z−1) and B(z−1)appearing in model (2.17). Let us consider the block scheme reported in Fig-ure A.5. It represents the simulated system in the open-loop configuration.

The output v#4,1(t) is the rms value at the fundamental frequency of thevoltage at the regulation node 4 obtained by a Kalman filter, see Chapter 3.A white noise with standard deviation equal to 0.14 kV is added to the inputof the Kalman filter. The block before the FC-TCR represents the simulatedfiring pulse generator, essentially including the voltage zero-crossing detectorand the logic circuit for the gate signal generation.

In Figure A.5 the nonlinear function f−1(1 − u) represents the inverseof (2.9). It should be recalled that π/2 ≤ α ≤ π, 0 ≤ f(α) ≤ 1 and 0 ≤u(tc,k) ≤ 1. To ensure that u(tc,k) = 0 corresponds to α(tc,k) = π/2 and,vice versa, u(tc,k) = 1 corresponds to α(tc,k) = π, the inverse of (2.9) isapplied to 1 − u(tc,k). In practice, for a given value u of the input u(tc,k)in the considered interval, the corresponding value of the firing angle α isdetermined by numerically solving the implicit equation

f(α) = 1 − u

or by resorting to a look-up table whose points lie on the diagram reportedin Figure A.6.


u(tc,k)f−1(1 − u)

α(tc,k)Firingpulse

generatorFC-TCR

Powersystem

v#4(t)

Tc

Kalmanfilter

v#4,1(tc,k)

Fig. A.5. Simplified block-scheme of the system in open-loop configuration

Fig. A.6. Plot of α = f−1(1 − u)

The input u(tc,k) ∈ [0, 1] is a square waveform with a period of 0.4 s,50% duty cycle and amplitude of 0.4. The values of u(tc,k) and v#4,1(tc,k)have been stored with a sampling period equal to Tc.

The time delay introduced by the SVS is approximately equal to Td =3.4 ms [68]. Since Tc = 0.001 s, d = 4.

An off-line least-squares algorithm has been run several times, each onecorresponding to a different value of nA and nB . The least-squares algorithmhas been initialized with θ(0) = 0 and P(0) = 104 Inp

. cv was set to 10.For each pair nA, nB , the estimated parameter values ai, bj and D are

used to calculate the residual prediction error R0 as

R0 =1N

k=N∑k=1

(v1(tc,k) − v#4,1(tc,k)

)2where N is the number of samples. The results obtained are depicted in Fig-ure A.7, which reports the values of R0 versus nA+nB , respectively for nB = 1,nB = 2 and nB = 3.

By looking at the reported figure, it was decided to choose nA = 4 andnB = 2. This choice is the result of a compromise between accuracy andorder of model. Moreover, higher-order models would be more accurate athigh frequencies which do not usually affect the closed-loop stability underthe designed voltage regulator [12].


Fig. A.7. Residual prediction error R0 versus nA + nB

Fig. A.8. Industrial test network

A.2 Industrial Network

The industrial network considered consists of 13 buses and is representativeof a medium-sized industrial plant. The system is extracted from a commonsystem that is being used in many of the calculations and examples in theIEEE Color Book Series [66, 117]. The network is fed from a utility supplyat 69 kV, 60 Hz and the local plant distribution system operates at 13.8 kV.The system is shown in Figure A.8 and described by the data reported inTables A.2–A.4. In Figure A.8 the block ASF represents the active shunt filter.

A.2 Industrial Network 151

Due to the balanced nature of the network considered, only positive sequencedata are provided, moreover the capacitances of all cables are neglected.

The industrial network has been simulated in the Matlab/Simulink [89]programming environment also using Power System Blockset package.

Table A.2. Cable impedance data (base values: 13.8 kV, 10000 kVA)

Cable %R %X

L1 0.00122 0.00243

L2 0.00139 0.00296

L3 0.00075 0.00063

L4 0.00157 0.00131

L5 0.00109 0.00091

Table A.3. Transformer data

Transformer Voltage Tap kVA %R %X

T1 13.8 : 0.48 13.45 1500 0.9593 5.6694

T2 69 : 13.8 69 15000 0.4698 7.9862

T3 13.8 : 0.48 13.45 1250 0.7398 4.4388

T4 13.8 : 4.16 13.11 1725 0.7442 5.9537

T5 13.8 : 0.48 13.45 1500 0.8743 5.6831

T6 13.8 : 0.48 13.8 1500 0.8363 5.4360

T7 13.8 : 2.4 13.11 3750 0.4568 5.4810

T8 13.8 : 0.48 13.8 500 0.4000 4.0000

Table A.4. Load data

Load Type PL,n [kW] QL,n [kvar]

Q1 Linear 600 530

Q2 Nonlinear 1150 290

Q3 Linear 1310 1130

Q4 Linear 2240 2000

Q5 Nonlinear 810 800

Q6 Linear 370 300

Q7 Linear 2800 2500

References

1. Acha E, Madrigal M (2001) Power Systems Harmonics. John Wiley & Sons,West Sussex, UK.

2. Allidina AY, Hughes FM (1980) Generalised self-tuning controller with poleassignment. IEE Proc. Pt D 127: 13–18

3. Anderson BDO, Moore JB (1989) Optimal Control Linear Quadratic Methods.Prentice-Hall, Inc., Englewood Cliffs, New Jersey

4. Anderson PM, Fouad AA (1994) Power System Control and Stability. IEEEPress, Inc, New York, USA

5. Arrillaga J, Bradley DA, Bodger PS (1985) Power System Harmonics. JohnWiley & Sons, Chichester New York Brisbane Toronto Singapore

6. Arrillaga J, Watson NR, Chen S (2000) Power System Quality Assessment,John Wiley & Sons, Chichester, West Sussex, UK

7. Arrillaga J, Arnold CP, Harker BJ (1991) Computer Modelling of ElectricalPower Systems, John Wiley & Sons, USA

8. Astrom KJ (1983) Theory and application of adaptive control - a survey .Automatica 19: 471–486

9. Astrom KJ, Borisson U, Ljung L, Wittenmark B (1977) Theory and applicationof self-tuning regulators. Automatica 13: 457–476

10. Astrom KJ, Wittenmark B (1989) Adaptive Control. Addison-Wesley Publish-ing Company, New York, USA

11. Astrom KJ, Wittenmark B (1973) On self-tuning regulators. Automatica 9:185–199

12. Astrom KJ (1980) Robustness of a design method based on assignment of polesand zeros. IEEE Trans. Automatic Control 25: 588-591

13. Azzam M, Abdul-Sadek Nour M (1996) An expert system for voltage controlof a large-scale power system. Energy Convers. Mgmt: 37:81–86

14. Bansilal DT, Parthasathy K (1998) An expert system for voltage control ina power system network. Proc. Int. Conf. on Energy Management and PowerDelivery, 364–369

15. Balakrishnan AV (1987) Kalman Filtering Theory. Optimization Software Inc.Publications Division, New York, USA

16. Beaulieu G. Bollen MHJ, Malgarotti S, Ball R, et al (2002) Power quality indicesand objectives ongoing activities in CIGRE WG 36-07. IEEE PES SummerMeeting: 789–794

154 References

17. Bolzern P, Scattolini R, Schiavoni N (2004) Fondamenti di Controlli Auto-matici. McGraw-Hill, Milan, Italy.

18. Bonhomme A, Cortinas D, Boulanger F, Fraisse J-L (2001) A new voltage con-trol system to facilitate the connection of dispersed generation to distributionsystems. IEEE Int. Conf. CIRED, IEE Conference Publication 482

19. Chang WK, Grady WM, Samotji MJ (1994) Meeting IEEE-519 harmonic volt-age and voltage distortion constraints with an active power line conditioner.IEEE Trans. Power Delivery 9: 1531–1537

20. Chang WK, Grady WM, Verde P (1994) Determining the optimal current in-jection and placement of an active power line Conditioner for several harmonic-related network correction strategies. Proc. IEEE ICHPS VI, Bologna, Italy:351–355

21. Chapman JW, Ilic MD (1994) A new model formulation for power systemvoltage control during large disturbances. Proc. IEEE Int. Conf. on Decisionand Control: 4049–4054

22. Chaudhuri B, Majumder R, Pal BC (2004) Application of multiple-model adap-tive control strategy for robust damping of interarea oscillations in power sys-tems. IEEE Trans. Control Systems Technology 12: 727–736

23. Chen GP, Malik OP (1995) Tracking constrained adaptive power system sta-biliser. IEE Proc. Gener. Transm. Distrib. 142: 149–156

24. Cheng SJ, Malik OP, Hope GS (1998) An expert system for voltage and reactivepower control of a power system. IEEE Trans. Power Systems 4: 1449–1455

25. Ching-Tzong S, Chien-Tung L (1996) A new fuzzy control approach to voltageprofile enhancement for power systems. IEEE Trans. Power Systems 11: 1654–1659

26. Ching-Tzong S, Chien-Tung L (2001) Fuzzy-based voltage/reactive powerscheduling for voltage security improvement and loss reduction. IEEE Trans.Power Delivery 16: 319–323

27. Choi J-H, Kim J-C (2001) Advanced voltage regulation method of power distri-bution systems interconnected with dispersed storage and generation systems(revised). IEEE Trans. Power Delivery 16: 329–334

28. Mu-Chun S, Chih-Wen L, and Chen-Sung C (1999) Rule extraction for voltagesecurity margin estimation. IEEE Trans. Industrial Electronics 46: 1114–1122

29. CIGRE WG 38-01 TF 2 (1986) Static VAR Compensators. Edited by I.A.Erinmez, Cigre, Paris, France.

30. CIGRE TF 39/02 (1992) Voltage and reactive power control. Cigre Meetingpaper 39–203, Paris, France

31. Clarke DW, Gawthrop PJ (1979) Self-tuning control. Proc. IEE 126: 633–64032. Cong L, Wang Y, Hill DJ (2005) Transient stability and voltage regulation en-

hancement via coordinated control of generator excitation and SVC. ElectricalPower & Energy Systems 27: 121–130

33. Cordero Osorio A, Mayne DQ (1981) Deterministic convergence of a self-tuningregulator with variable forgetting factor. IEE Proc. Pt D 128: 19–23

34. Corsi S, Pozzi M, Sabelli C, Serrani A (2004) The coordinated automatic voltagecontrol of the Italian Transmission Grid – Part I: Reasons of the choice andoverview of the consolidated hierarchical system. IEEE Trans. Power Systems19: 1723–1732

35. Curk J, Gubina F (1999) Fuzzy sets - a natural way to describe power systemvoltage profile. Proc. IEEE Int. Fuzzy Systems Conf. Seoul, Korea, 974–979

References 155

36. Daponte P, Falcomata G, Testa A (1993) A multiple deep attenuation frequencywindow for harmonic analysis in power systems. IEEE Trans. Power Delivery9: 863–871

37. Dash PK, Sharaf AM (1988) A Kalman filtering approach for estimation ofpower system harmonics. Proc. Third Int. Conf. Harmonics in Power Systems:34–40

38. Dash PK, Swain DP, Liew AC, Rahman S (1996) An adaptive linear combinerfor on-line tracking of power system harmonics. IEEE Trans. Power Systems11: 1730–1735

39. De Battista H, Mantz RJ (2000) Harmonic series compensator in power sys-tems: their control via sliding mode. IEEE Trans. Control Systems Technology8: 939–947

40. De Oliveira A, De Oliveira JC, Resende JW, Miskulin MS (1991) Practicalapproaches for AC system harmonic impedance measurements. IEEE Trans.Power Delivery 6: 1721–1726

41. Dy Liacco TE (1967) The adaptive reliability control system. IEEE Trans.Power Apparatus and Systems 86: 517–531

42. Egardt B (1980) Stability analysis of discrete-time adaptive control schemes.IEEE Trans. Automatic Control 25: 710–716

43. Fortescue TR, Kershenbaum LS, Ydstie BE (1981) Implementation of self-tuning regulators with variable forgetting factors. Automatica 17: 831–835

44. Franklin GF, Powell JD, Workman ML (1990) Digital Control of DynamicSystems. Addison-Wesley, New York, USA

45. Fuchs JJ (1982) Indirect stochastic adaptive control: the general delay-whitenoise case. IEEE Trans. Automatic Control 27: 219–223

46. Fusco G, Russo M (1998) Parameter tracking techniques applied to harmonicequivalent circuit identification in power systems. Proc. IEEE Int. Conf. ControlApplications, Trieste, Italy, 1388–1393

47. Fusco G, Losi A, Russo M (2000) Constrained least squares methods for pa-rameter tracking of power system steady-state equivalent circuits. IEEE Trans.Power Delivery 15: 1073–1080

48. Fusco G, Losi A, Russo M (2001) Adaptive voltage regulator design for staticVAr systems. Control Engineering Practice 9: 759–767

49. Fusco G, Russo M (2003) Self-tuning regulator design for nodal voltage wave-form control in electrical power systems. IEEE Trans. Control Systems Tech-nology 11: 258–266

50. Fusco G, Russo M (2005) Nodal voltage regulation employing an indirect self-tuning approach. Proc. IEEE Int. Conf. on Control Applications, Toronto,Canada, 797–802

51. Fusco G, Russo M (2006) Self-tuning nodal voltage regulation in power sys-tems based on optimal pole-shifting technique. Proc. IEEE Int. Conf. PowerEngineering Society General Meeting, PES Conference, Montreal, Canada

52. Gaglioti E, Santagostino G, Leone F, Rosati R, Valfre L (1984) Evaluation oftransient performance of a SVC for voltage support in a transmission systemwith low short-circuit power. Proc. Int. Conf. on Large High Voltage Electricsystem: 1–10

53. Gao L, Chen L, Fan Y, Ma H (1992) A nonlinear control design for powersystems. Automatica 28: 975–979

156 References

54. Ghandakly AA, Farhoud AM, (1992) A parametrically optimized self-tuningregulator for power system stabilizers. IEEE Trans. Power Systems 7: 1245–1250

55. Girgis AA, Chang WB, Makram EB (1991) A digital recursive measurementscheme for on-line tracking of power system harmonics. IEEE Trans. PowerDelivery 6: 1153–1160

56. Girgis AA, McManis RB (1989) Frequency domain techniques for modelingdistribution or transmission networks using capacitor switching induced tran-sients. IEEE Trans. Power Delivery 4: 1882–1890

57. Girgis AA, Quaintance III WH, Qiu J, Makram EB (1993) A time-domainthree-phase power system impedance modeling approach for harmonic filteranalysis. IEEE Trans. Power Delivery 8: 504–510

58. Golub GH, Van Loan CF (1996) Matrix Computations, Third Edition. TheJohns Hopkins University Press. USA

59. Goodwin GC, Ramadge PJ, Caines PE (1980) Discrete-time multivariableadaptive control. IEEE Trans. Automatic Control 25: 449- -456

60. Grady WM, Samotji MJ, Noyola AH (1990) Survey of active power line cond-tioning methodologies. IEEE Trans. Power Delivery 5: 1536–1542

61. Grady WM, Samotji MJ, Noyola AH (1992) The application of network objec-tive functions for actively minimizing the impact of voltage harmonics in powersystems. IEEE Trans. Power Delivery 7: 1379–1386

62. Guo L, (1995) Convergence and logarithm laws of self- tuning regulators. Au-tomatica 31: 435–450

63. Harris FJ (1978) On the use of windows for harmonic analysis with the discreteFourier transform. Proc. IEEE 66: 51–83

64. Hu Z, Wang X, Chen H, Taylor GA (2003) Volt/var control in distribution sys-tems using a time-inverval based approach. IEE Proc. Gener. Transm. Distrib.150: 548–554

65. IEE Colloquium (1993) International practices in reactive power control. IEEDigest No. 79, London, UK.

66. IEEE Standard 399-1990 (1990) IEEE Recommended Practice for Industrialand Commercial Power System Analysis. IEEE, New York, USA

67. IEEE Standard 1159-1995 (1995) IEEE Recommended Practice for MonitoringElectric Power Quality. IEEE, New York, USA

68. IEEE Working Group Special Stability Controls (1994) Static var compensatormodels for power flow and dynamic performance simulation. IEEE Trans. PowerSystems 9: 229–240

69. Ilic MD, Liu X, Leung G, Athans M, Vialas C, Pruvot P (1995) Improvedsecondary and new tertiary voltage control. IEEE Trans. Power Systems 10:1851–1862

70. Ilic M, Skantze P (2000) Electric power systems operation by decision andcontrol. The case revisited. IEEE Control Systems Magazine 20: 25–39

71. Ilic-Spong M, Christensen J, Eichorn KL (1988) Secondary voltage control usingpilot point information. IEEE Trans. Power Systems 3: 660–665

72. Jain AK, Behal A, Zhang XT, Dawson DM, Mohan N (2004) Non linear con-trollers for fast voltage regulation using STATCOMs. IEEE Trans. ControlSystems Technology 12: 827–842

73. Kothari ML, Bhattacharya K, Nanda J (1995) Adaptive power system stabiliserbased on pole shifting technique. IEE Proc. Gener. Transm. Distrib. 143: 96–98.

References 157

74. Kreisselmeier G, Anderson BDO (1986) Robust model-reference adaptive con-trol. IEEE Trans. Automatic Control 31: 127–133

75. Kumar PR (1990) Convergence of adaptive control scheme using least-squaresparameter estimates. IEEE Trans. Automatic Control 35: 416–424

76. Kundur P (1994) Power System Stability and Control. McGraw-Hill,Inc. NewYork, USA

77. Landau ID (1990) System Identification and Control Design. Prentice Hall In-formation and System Sciences Series, New Jersey

78. Larsen EV, Clark K, Hill AT, Piwko RJ, Beshir MJ, Bhuiyan M, Hormozi FJ,Braun K (1996) Control design for SVS’s on the Mead-Adelanto and Mead-Phoenix Transmission Project. IEEE Trans. Power Delivery 11: 1498–1506.

79. Li XY, Malik OP (1993) Estimation of equivalent models for emergency statecontrol of interconnected power-systems based on multistage recursive least-squares identification. IEE Proc. Pt C 140: 319–325

80. Liang R-H, Cheng C-K (2001) Dispatch of main transformer ultc and capacitorsin a distribution system. IEEE Trans. Power Delivery 16: 625–630

81. Liang R-H, Wang Y-S (2003) Fuzzy-based reactive power and voltage controlin a distribution system. IEEE Trans. Power Delivery 18: 610–618

82. Ljung L (1977) On positive real transfer functions and the convergence of somerecursive schemes. IEEE Trans. Automatic Control 22: 539-551

83. Ljung L (1999) System Identification: Theory for User. Prentice Hall PTR,Upper Saddle River, New Jersey, USA

84. Lucero EM, Rios SM, Dan AM, Czira Z (1996) Modeling of harmonic networkimpedance using field data and identification methods. Proc. 7th Int. Conf. onHarmonics and Quality of Power, Las Vegas, USA, 481–486

85. Malik OP, chen GP, Hope GS, Qin YH, Xu GY (1991) Adaptive self-optimisingpole shifting control algorithm. IEE Proc. Pt D 139: 429–438

86. Marconato R (2002) Electric Power System: Background and Basic Concepts.Second Edition, vol. 1, CEI Italian Electrotechnical Committee, Milan, Italy

87. Marconato R (2004) Electric Power System: Steady-State Behaviour, Controls,Short Circuits and Protection Systems. Second Edition, vol. 2, CEI ItalianElectrotechnical Committee, Milan, Italy

88. Marques BA, Taranto GN, Falcao DM (2005) A knowledge-based system forsupervision and control of regional voltage profile and security. IEEE Trans.Power Systems 20: 400–407

89. MATLAB Reference Guide, The MathWorks Inc., Natick, MA, 1999.90. Matsuda S, Ogi H, Nishimura K, Okataku Y, Tamura S (1988) Power system

voltage control by distributed expert systems. Proc. IEEE Int. Workshop onArtificial Intelligence for industrial application: 303–308

91. Mohan N, Undeland TM, Robbins WP (1995) Power Electronics - Converters,Applications and Design. John Wiley & Sons, New York, USA

92. O’Brien M, Ledwich G (1987) Static reactive-power compensator controls forimproved system stability. IEE Proc. Pt C 134: 38–42

93. O’Gorman R, Redfern MA (2004) Voltage control problems on modern distri-bution systems. IEEE PES General Meeting: 662–667

94. Orfanidis SJ (1989) Optimum Signal Processing. Second Edition, MacMillanPublishing Co, New York, USA

95. Osowski S (1992) Neural network for estimation of harmonic components in apower system. IEE Proc. Pt C 139: 129–135

158 References

96. Pahalawatha NC, Hope GS, Malik OP, Wong K (1990) Real time implementa-tion of a MIMO adaptive power system stabiliser. IEE Proc. Pt C 137: 186–194

97. Padiyar KR, Kulkarni AM (1997) Design of reactive current and voltage con-troller of static condenser. Electric Power & Energy System 19: 397–410

98. Paul JP, Leost JY, Tesseron JM (1987) Survey of the secondary voltage controlin France: Present realization and investigation. IEEE Trans. Power Systems2: 505–511

99. Phillips CL, Troy Nagle H (1995) Digital Control System Analysis and Design,Third Edition. Prentice Hall, New York, USA

100. Pierre DA (1986) Optimization Theory with Applications, Dover Pubblica-tions, Inc., New York, USA

101. Pimpa C, Premrudeepreeechacharn S (2002) Voltage control in power systemusing expert system based on scada system. Proc. IEEE Power EngineeringSociety Winter Meeting: 1282–1286

102. Popovic DS (2002) Real-time coordination of secondary voltage control andpower system stabilizer. Electrical Power & Energy Systems 24: 405–413

103. Rao P, Crow ML, Yang Z. (2000) STATCOM control for power system voltagecontrol applications. IEEE Trans. Power Delivery 15: 1311–1317

104. Roytelman I, Ganesan V (2000) Modeling of local controllers in distributionnetwork applications. IEEE Trans. Power Delivery 15: 1232–1237

105. Ruhua Y, Eghbali HJ, Nehrir MH (2003) An online adaptive neuro-fuzzy powersystem stabilizer for multimachine systems. IEEE Trans. Power System 18: 128–135

106. Ruey-Hsun L, Yung-Shuen W (2003) Fuzzy based reactive power and voltagecontrol in a distribution system. IEEE Trans. Power Delivery 18: 610–618

107. Saiz VMM, Guadalupe JB (1997) Application of Kalman filtering for continu-ous real-time tracking of power system harmonics. IEE Proc. Gener. Transm.Distrib. 144: 13–20

108. Saric AT, Calovic MS, Strezoski VC (2003) Fuzzy multi-objective algorithmfor multiple solution of distribution systems voltage control. Electrical Power& Energy Systems 25: 145–153

109. Sauer P, Pai M (1998) Power System Dynamics and Stability. Prentice Hall,Englewood Cliffs, NJ, USA

110. Senjyu T, Toyohiro A, Katsumi U (2000) Adaptive power system stabilizerbased on fuzzy logic technique with frequency domain analysis. Proc. IEEEInt. Conf. on Power System Technology, 1299–1304

111. Siljak DD (1991) Decentralized Control of Complex Systems. Academic Press,Inc., San Diego, USA

112. Singh B, Al-Haddad K, Chandra A (1999) A review of active filters for powerquality improvment. IEEE Trans. Industrial Electronics 46: 960–971

113. Soliman SA, Al-Kandari AM, El-Nagar K, El-Hawary ME (1995) New dynamicfilter based on least absolute value algorithm for on-line tracking of powersystem harmonics. IEE Proc. Gener. Transm. Distrib. 142: 37–44

114. Soos A, Malik OP (2002) An H2 optimal adaptive power system stabilizer.IEEE Trans. Energy Conversion 17: 143–149

115. Tao G (2003) Adaptive Control Design and Analysis. John Wiley & Sons, NewYork, USA

116. Task Force on Harmonics Modeling and Simulation (1996) Modeling and sim-ulation of the propagation of harmonics in electric power networks – Part I:

References 159

Concepts, models, and simlation techniques. IEEE Trans. Power Delivery 11:452–465

117. Task Force on Harmonics Modeling and Simulation Transmission & distribu-tion Committee (1999) Test systems for harmonics modeling and simulation.IEEE Trans. Power Delivery 14: 579–587

118. Tavares da Costa C, Barreiros JAL, de Oliveira RCL, Barra W (2001) Gen-eralized predictive power system stabilizer with fuzzy supervision from a localmodel network. Proc. IEEE Int. Power Tech. Conf., Porto, Portugal

119. Valderrama GE, Mattavelli P, Stankovic AM (2001) Reactive power and un-balance compensation using STATCOM with dissipativity-based control. IEEETrans. Control Systems Technology 9: 718–727

120. Vu H, Pruvot P, Launay C, Harmand Y (1996) An improved voltage controlon large-scale power system. IEEE Trans. Power Systems 11: 1295–1303

121. Wang HF, Li H, Chen H (2003) Coordinated secondary voltage control toeliminate voltage violations in power system contingencies. IEEE Trans. PowerSystems 18: 588–595

122. Wang Y, Hill DJ, Middleton RH Gao L (1994) Transient stabilization of powersystems with an adaptive control law. Automatica 30: 1409–1413

123. Wang Y, Hill DJ (1996) Robust nonlinear coordinated control of power sys-tems. Automatica 32: 611–618

124. Wang HF, Li H, Chen H (2002) Power system voltage control based on artificialhumoral immune response. Proc. IEEE Int. Conf. on Power System Technology:2091–2095

125. Wang HF, Li H, Chen H (2002) Power system voltage control by multipleSTATCOMs based on learning humoral immune response. IEE Proc. Gener.Transm. Distrib. 149: 416–426

126. Wellstead PE, Sanoff P (1981) Extended self-tuning algorithm. InternationalJournal of Control 34: 433–455

127. Wellstead PE, Zarrop MB (1991) Self-tuning Systems Control and Signal Pro-cessing. John Wiley & Sons, New York, USA

128. Yu CN, Yoon YT, Ilic MD, Catelli A (1999) On-line voltage regulation: thecase of New England. IEEE Trans. Power Systems 14: 1477–1484

129. Zhu C, Zhou R, Wang Y, Ma H (1997) A new nonlinear voltage controller forpower systems. Electrical Power & Energy Systems 19: 19–27

Index

Active shunt filters 8, 19, 123, 150

Diophantine equations 44, 48, 51, 89,97, 118, 121, 126

Dispersed generation 2, 8

FACTS 13Fourier analysis

DFT 28, 29FFT 28, 29series 27

Generalized minimum variance 48, 72Gradient technique 47, 88, 94

Harmonic distortioncontainment 8THD index 139

Hybrid shunt filters 8, 20, 123

Interharmonics 28ISO 1, 3

Kalman filteralgorithm 30, 110observability 32properties 32state-space modeling 29

Lagrangian function 68, 112Load models 146Lyapunov function 80, 100

Model-reference adaptive voltageregulators

convergence 99direct 87, 88indirect 87, 97robustness 102

Multi-objective optimization 138

No-load voltage 22, 23, 52, 93, 110Noise

coloured 71white 24, 42, 48, 69, 86

OLTC 5, 7, 13

Park transformation 14, 23, 115Persistency of excitation 84, 113Phasor representation 12, 22, 109, 110Pole-assignment 42, 48, 63, 72, 118,

120, 125Pole-shifting 45Power quality 6Power system models

frequency domain 22, 109nonlinearities compensation 122,

126time domain 23, 148

Recursive least-squaresalgorithm 39constrained 68, 110dead-zone 41extended 71forgetting factor 42properties 78

Riccati equation 34

162 Index

Self-tuning voltage regulatorsdirect 37, 63indirect 37, 38

STATCOM 13Static VAr Systems 5, 16, 119, 146Synchronous machines 13, 114, 141

Thevenin equivalent circuit 22, 109,110

Transformers 145Transmission lines 144

Wind-up 54

Other titles published in this Series (continued):

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Model-based Process SupervisionBelkacem Ould Bouamama andArun K. SamantarayPublication due February 2007

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Continuous-time Model Identificationfrom Sampled DataHugues Garnier and Liuping Wang (Eds.)Publication due May 2007Process ControlJie Bao, and Peter L. LeePublication due June 2007

Distributed Embedded Control SystemsMatjaz Colnaric, Domen Verber andWolfgang A. HalangPublication due October 2007

Optimal Control of Wind Energy SystemsIulian Munteanu, Antoneta Iuliana Bratcu,Nicolas-Antonio Cutululis andEmil CeangaPublication due November 2007

Model Predictive Control Design andImplementation Using MATLAB®Liuping WangPublication due November 2007

(Advances in industrial control) giuseppe fusco adaptive voltage control in power systems -...

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