generator capability - thesis da silva

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CAPABILITY CHARTSFOR

POWER SYSTEMS

A thesis presented for the degree ofDoctor of Philosophy in Electrical Engineering

in theUniversity of Canterbury,New Zealand,

by

James Ranil de Silva, B.E.(Hons 1)

March 1987


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ENGINEERINGLlBIl.ARY

taos

Mostly for my father,the late

Dr Fredrick Peter Rienzi de Silva


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1111111111111111111111111


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(i)Abstract

This thes is extends the concept of the t radi t ional synchronousgenerator capabil i ty chart to describe the steady state performance oftransmission l ines, HVOC l inks, and entire AC/OC power systems.

Each capabil i ty chart depicts an operating region on the complexpower plane that represents the real and react ive power that may besuppl ied to a load from a part icular busbar. The boundaries of theoperat ing region are defined by a web of contours that represent thecr i t ical operating constraints of the system.

The charts for small systems can be constructed by manipulating theoperat ing equations into a form suitable for drawing loci on the complexpower plane. This technique is used in th is thesis to construct chartsfor generators, transmission l ines , and HVOC l inks.

The operating equations of large systems are not easi ly manipulated,so a different approach is used to construct charts for large systems.This approach involves i terat ive powerflows and contour plott ing to avoidthe formulation of explici t closed form locus equations.

Two algorithms for drawing the capabil i ty charts of large AC powersystems are described. The f i r s t algorithm uses a contour tracingtechnique to plot the constraint loci . The knowledge gained from the useof this algorithm was then used to design the second, faster algorithmthat uses a region growing technique to help plot the constraint loci .

Capabili ty charts for large AC/OC systems are also discussed. Theoperating constraints of the AC/OC converters require specia l treatmentdue to discontinuit ies in the converter operating equations. An offshootof the work on AC/OC charts is the development of an improved sequentialAC/OC powerflow algorithm.

A pract ical example of the use of the capabil i ty charting algorithmsis given by drawing charts for a proposed second New Zealand HVOC l ink.

The capabil i ty charting algorithms wil l make suitable additions toexist ing power system interactive graphics programs. On-line displays ofcapabil i ty charts in system control centres could also provide usefulinformation to human dispatchers.


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List of ContentsAbstractList of ContentsAcknowledgements

Page( i)

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( ix)ublications Associated with this Thesis

CHAPTER 1 I N T R O D ~ C T I O N1.1 Graphics, Power Systems, and Capability Charts1 .2 Historical Background1 .3 Chapter Review

CHAPTER 2 CHARTS FOR SIMPLE POWER SYSTEMS 62.1 Introduction 62.2 The Capability Chart for a Synchronous Generator 7

2.2.1 Simplifying Assumptions 82.2.2 Generator Vector Diagrams 82.2.3 Construction of the Capability Chart 13

Turbine Power Limits 13Maximum Stator Current 13Rotor Current LimitsSteady State Stabil i ty Limit 14

2.2.4 Summary of Synchronous Generator Chart 182.3 The Capability Chart for a Transmission Line 18

2.3.1 Transmission Line Model 192.3.2 Operating Constraints 20

Load Busbar Voltage Limits 21Voltage Stabil i ty 23Maximum Transmission Line Current 26

2,3.3 Capability Chart on the Admittance Plane 29Voltage Limits 30Line Current Limit 30

2.3.4 Summary of Transmission Line Capability Charts 312.4 Conclusion 32


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CHAPTER 3 A CHART FOR AN HVDC LINK 333.1 Introduction 333.2 Circuit Model of the HVDC Link 333.3 Operating Constraints of the HVDC Link 38

Converter Transformer Current 38Converter Valve Current 38Harmonic Fil ter Current 38DC Voltage Rating 39Converter Control Angles 39Converter Commutation Angle 39Capability of the Benmore Generator 40Interconnecting Transformer Current 41

3.4 Loci of Operating Constraints 413.4.1 AC/DC Per Unit System 413.4.2 Basic AC/DC Converter Operating Equations 423.4.3 Loci of Haywards Converter 43

Commutation Overlap Locus 44Loci of DC Current Limits 44Locus of Maximum Converter Transformer Current 46Loci of Maximum DC Voltage 46Locus of Minimum Firing Angle 47Locus of Minimum Extinction Angle 473.4.4 Loci of Benmore Converter 48DC Link Power Transfer Mapping 48

3.4.5 Loci of South Island Power Generation 503.4.6 Complete Capability Chart 52

3.5 Conclusion 54


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CHAPTER CHARTS FOR LARGE AC SYSTEMS 56Introduction 56System Operating Constraints 57

~ . 2 . 1 Voltages and Currents on Transmission Lines 58~ . 2 . 2 Generator Capability 58

~ . 2 . 3 Steady State Stability 58~ . 2 . ~ Total Constraint Number 59Test System 59Capability Charting Algorithm 62

~ . ~ . 1 Definition of the Vicinity of the OperatingRegion 62

~ . ~ . 2 Structure of th e Capability Charting Algorithm 63System Data Input 63Seed Point 66Search for Vicinity Perimeter 66Contour Following Algorithm 67Tracing of Constraint Contours 70

~ . ~ . 3 Interpretation of Capability Chart 71~ . ~ . ~ Holes and Islands~ . ~ . 5 Shortened Labels for Diagram Dressing 76

~ . 5 Conclusion 76

CHAPTER 5 A FAST CAPABILITY CHARTING ALGORITHM 785.1 Introduction 785.2 Structure of the Fast Algorithm 79

5.2.1 Region Growing 805.2.2 Contour Plotting 82

5.3 Powerflow Convergence Boundary 865 . ~ Contour Maps 885.5 Conclusion 90


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CHAPTER 6 CHARTS FOR LARGE AC/DC SYSTEMS6.1 Introduction6.2 Operating Constraints of DC systems

6.2.1 Constraints of DC Rectifier6.2.2 Constraints of HVDC Link

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6.2.3 A l l o w ~ b l e Range of Values for DC system Variables 946.3 Improving the Sequential AC/DC Power flow 96

6.3.1 Conventional Sequential Powerflow 976.3.2 Improving the Sequential Powerflow 976.3.3 Comparison of Convergence Behaviour 99

6.4 AC/DC System Capability Charts 1046.5 Conclusion 108

CHAPTER 7 PLANNING FOR A SECOND N. Z. HVDC LINK 1097 . 1 Introduction 1097.2 South Island Terminal 110

7.2.1 Capability Chart for Bendigo Converter Busbar 1127.2.2 Capability Chart for Ohau A Generator Busbar 1147.2.3 Voltage Fluctuations at the Bendigo Converter

Busbar 11 67.3 North Island Terminal 118

7.3.1 Capability Chart for Runciman Converter Busbar 1207.3.2 Capability Chart for Huntly Generator Busbar 1227.4 Conclusion 1-24


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CHAPTER 8 PROPOSED DEVELOPMENTS 1258.1 Introduction 1258.2 More Detailed Models 1258.3 Generation Cost Contours 1268.4 Optimal Powerflow Algorithms 1278.5 Stochastic Power flow Algorithms 1278.6 Eigenvalue Representation 1288.7 Interactive Graphics Programs 1308.8 System Control Applications 1318.9 Conclusion 134

CHAPTER 9 MAIN CONCLUSIONS 135

References 138

Appendix: C rcu it Da ta 141A1 IEEE 14 Busbar Test System 141A2 HVDC Link Modification to IEEE 14 Busbar Test System 144A3 New Zealand South Island Primary System 145A4 New Zealand Upper North Island Primary System 149


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Acknowledgements.My thanks to Dr Christopher P. Arnold for his supervision during the

three years of research involved in this thes i s .

I am also indebted to my employer, the Electr ic i ty Division of theMinistry of Energy, who gave me leave to do the work.

Thanks to the s t a f f and postgraduate students a t Ilam, part icularlyBil l Kennedy and Andrew Earl for the effor t s they have put into thecomputing and graphics fac i l i t i e s . Also Professor Josu Arri l laga, DrPatrick Bodger, Enrique Achadaza, Gordon Cameron, Neville Watson, andChris Callaghan for thei r discussions and humour.

Special thanks to my family who completely and ut te r ly approve ofthis work without actual ly knoWing what i t means.


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Publications Associatedwith this Thesis

The following three publications are associated with the researchpresented in th is thesis .

Arnold, C.P. , and de Si lva , J .R . , "A Capab i l i ty Chart for PowerSystems", Second Internat ional Conference on Power System Monitoring andControl, Durham, UK, 1986.

de Silva, J .R. , and Arnold, C.P., "Capability Charts for Analyzing PowerSystems", to be presented a t the Conference of the Ins t i tute ofProfessional Engineers New Zealand (IPENZ), 1987.

de Si lva ,Char t fo r

J .R . , Arnold, C.P. , and Arr i l l aga , J . , "A Capab i l i tyan HVDC Link" , accepted fo r publ i ca t ion in the lEE

Proceedings-C Generation, Transmission, and Distribution, 1987.

The following paper has also been submitted for publicat ion but hasyet to be off ic ia l ly accepted (as of 11th March, 1987).

de Silva, J .R. , and Arnold, C.P., "A Simple Improvement to SequentialAC/DC Power flow Algorithms", submitted to the Electr ic Energy SystemsJournal.


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Chapter 1Introduction

1.1 GRAPHICS, POWER SYSTEMS, AND CAPABILITY CHARTS

An old adage that claims "A picture paints a thousand words" conveysthe essence of graphical presentation. A good diagram can be more easilyunderstood than many pages of writing and, in part icular , a good graph ispreferable to tables of technical data. The appeal of graphical displayshas grown as computer based graphics systems have progressively reducedthe labour required to produce an effective display. The blossomingpopulari ty of computer graphics is being reflected in the f ie ld of powersystems as more research effort is commi t ted to the development ofgraphics software specifically for power system analysis .

The best e x a m p l e ~ of the use of computer graphics for analyzing powersystems are the interacti ve graphics programs that have been recentlydeveloped to model power system operation. Two such programs are IPSA(Interactive Power Systems Analysis) described by Lynch and Efthymiadis( 197 9) and ADAPOS (Advanced Analyzer of Power Systems) descri bed byFujiwara and Kohno (1985). These programs allow the user to interactivelyconstruct a power system model by drawing the circui t diagram on agraphi cs termi nal. Powerflow, transient s tab i l i ty , and short circui tstudi es can then be performed wi th the resul ts appeari ng alongside thecircui t diagram.

Graphical displays of circuits are also commonly used in systemcontrol centres to monitor the operation of the network. These displayscan be used in terac t i vely to directly control circui t breakers andgenerators. More than a decade of use has proven that graphical displaysare beneficial for both modelling and monitoring power systems. Furtherimprovements in power system 'graphics can be expected from continuedresearch and development.

The particular development of power system graphics that is exploredin th is thesis involves the drawing of capability charts . The capabili tycharts represent another method of graphically displaying power systemperformance. The programs used to draw the capabili ty charts are


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therefore related to other power systems graphics software and could beprof i tab ly included in exist ing power system interactive graphicspackages.

The capability charts are drawn on the complex power plane and definethe real and reactive power that may be suppl ied from a bus bar duringsteady s ta te operation. The power available is depicted by a region onthe plane, the boundaries of the region represent the cri t ical operatingl imits of the system.

The bes t known example of a capabil i ty chart is the operating chartof a synchronous generator as shown in figure 1.1 . The power availablefrom the generator is res tr icted by the rotor current, s tator current ,turbine power, and synchronous stabi l i ty l imits . The work discussed inth is thesis can be considered to be a generalization of this concept ofa genera tor capabi l i ty chart . This work can be placed in his toricalcontext by considering the origins of capabil i ty charts .

1.2 HISTORICAL BACKGROUNDThe origins of the power system capabil i ty charts l ie with the power

circle diagrams that were introduced by Philip (1911) and la te r extendedby Evans and Sels (1921) and Dwight (1922). The circle diagrams provide agraphical solution to the problem of relating the voltages and the realand reactive powers associated with a transmission l ine. A compass andruler are the only tools required to construct the diagrams, providingthat the system studied is res tr icted to only a single transmission l ine .This approach has proved to be very successful and is s t i l l used to solvetransmission l ine problems.

Szwander (1944) applied the circle diagram approach to drawing acapabi l i ty chart for round rotor synchronous turbo-generators. Walker(1953) la ter modified this chart to accommodate the character is t ics ofsal ient pole generators. A simple equivalent ci rcui t was used to modelthe generators to simplify the derivat ion of equat ions and aidconstruction of the chart .

Kimbark (1971) used charts to describe the behaviour of the HVDC


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1 pu Reactive powerMinimum turbine power

Maximum turbine power

Maximum rotor current

Maximum stator current1 pu Real power

Synchronous stability limit

Figure 1.1 Capability Chart of a Synchronous Generator

The shaded operat ing region represents the real andreact ive power that may be supplied by the generator. Theboundaries of the region indicate the cri t i ca l operatingl imits of the generator.

3


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converters that were jus t beginning to proliferate at the time. Again asimple system representation was used to simplify the circui t analysis.

All of the charts mentioned so far are rest r ic ted to describingsmall , simple systems because the equations associated wi th each curve,must be explici t ly derived. I f the curves can be shown to be circles orother simple geometric shapes then the chart can be easily constructed.I f larger power systems are considered then the system equations rapidlybecome much more complicated and prevent the simple derivation ofexplici t curve .equations.

The use of i tera t ive solut ions to large power system equations avoidsthe problem of formulating expl ic i t derivat ions. A computer becomesnecessary to handle the calculat ion burden that is involved in thei tera t ive algorithms. A graphics output facil i ty is also required to drawcurves that are much more complicated than the simple circle diagrams.

Wirth e t a l . (1983) used the i tera t ive approach to draw contour mapsof system eigenvalues on the complex power plane. These contour mapsdescribe the steady state s tabi l i ty of the power system. Price (1984)also used the i terat ive approach to draw a contour map of the powerflowfunction for large systems on the complex power plane.

Pr ice ' s work is probably the closest relat ive to the capabil i tychart ing algorithms that are described in this thesis . These algorithmscombine the contours associated with the most cr i t ica l operating l imitsto form capabil i ty charts for large power systems.

1.3 CHAPTER REVIEW

The discussion on capabil i ty charts progresses from small, simplesystems to larger , more complicated networks. The capabil i ty charts oftwo small systems are discussed in chapter 2. First the standard methodof constructing the chart for a synchronous generator is reviewed. Amodi f i cation of the circl e diagram approach is then used to draw thecapabil i ty chart for a single transmission l ine .


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The capabi l i ty chart fo r an HVDC l ink is described in chapter 3. Agrowing in te res t in HVDC l inks makes th i s char t par t icu lar ly useful .The New Zealand Benmore-Haywards HVDC system is used as an example todemonstrate the technique for drawing the char t .

The genera tor , transmission l ine , and HVDC l ink systems a l l havesmal l , spec i f i c conf igura t ions . An algorithm fo r drawing capabi l i tychar t s for l a rger , general AC systems i s described in chapter 4. Thisalgorithm uses an i t e ra t ive powerflow solut ion to handle the la rge numberof system equations. The algorithm also incorporates a contour t rac ingtechnique to draw the char t boundaries.

The exper ience gained from drawing a la rge numoer of capabi l i tychar t s has suggested possible improvements to the original chartingalgor i thm described in chapter 4. These improvements are incorporatedinto a fas t capabi l i ty charting algorithm tha t is descr ibed in chapter 5.The f a s t algorithm uses a region growing procedure to plot the c r i t i c a loperat ing curves in preference to the contour t racing technique used bythe origina l algorithm.

The capabi l i ty char ts for la rge AC/DC systems are then examined inchap te r 6 . The char t ing algori thms are modified to incorporate theopera t ing const ra ints of r ec t i f i e r s and two terminal HVDC l inks . TheAC/DC systems are analyzed by a new sequent ia l AC/DC powerflow algorithmtha t converges to a solut ion fas te r than previous sequent ia l algorithms.

Chapter 7 describes the applicat ion of capabi l i ty charts to helps tudy a proposed second HVDC l ink between the North and South Is landsof New Zealand.

The future prospects of research in to capabi l i ty char ts are examinedin chapter 8. More detai led modelling of the power system is envisaged toallow a grea te r variety of operat ing l imi ts to be portrayed on thecha r t s . Genera t ion cost contour maps are suggested to help minimi zeopera t ing cos t s . Considerat ion i s given to the poss ib i l i ty of usingopt imal powerf lows and s tochas t i c powerflows wi thin the capabi l i tychart ing algorithms. The incorporat ion of capabi l i ty char ts into exist ingpower systems in te rac t ive graphics programs is also discussed. Final ly ,the potent ia l applicat ion of capabi l i ty char ts for system monitoring isexamined.


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Chapter 2Charts for Simple Power Systems

2.1 INTRODUCTION

The t rad i t ional capabil i ty chart for a synchronous generator wasintroduced by Szwander (1944) to display the relat ionship between theoperat ing l imits of a round rotor machine. The steady sta te operation ofa round rotor generator can be described by a few equations and modelledby a simple equivalent circuit . The generator capabil i ty chart is easilyconstructed by manipulating the operating equations into a suitable formfor drawing loci on the complex power plane.

The same approach can be used to construct the capabil i ty charts forother simple power system circui ts such as the transmission l ine chartand the HVDC l ink chart that are also described in this thesis .

The operat ion of larger power systems is described by a large numberof equations that cannot be easi ly manipulated into a suitable form fordrawing loci . A different technique is used to construct the charts forthese systems and this is treated in chapters 4 to 7 of this thesis .

The capabil i ty charts of small, simple systems s t i l l retain a valuablerole that is not negated by the development of the algori thms to drawcharts for larger, more complicated networks. The charts associated withlarge networks cannot offer the same insights that can be gained from theexplicit derivation of locus equations for small systems.

Many of the loci for small systems can be drawn by hand without theassistance of a computer. I f faci l i t ies are not available to implementthe programs for drawing char t s for large systems then i t may bepossible to model the system being studied by a simple equivalentcircui t that can be analyzed by hand.

This chapter describes the construction of ' capabil i ty charts for twosimpl e power systems. First the tradi t ional chart for a synchronousgenerator is reviewed then a char t i s developed to , describe theperformance of a transmission l ine .


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2.2 THE CAPABILITY CHART FOR A SYNCHRONOUS GENERATOR

The capability chart for a round rotor synchronous generator wasoriginally developed by Szwander (1944). The concept was la ter extendedby Walker (1953) to include the operation of salient pole synchronousgenerators . This section reviews the construction of a chart for asal ient pole generator and regards the round rotor generator as a specialcase of sal ient pole machines.

Figure 2.1 shows the ci rcui t model of a synchronous generatorsupplying power to an infinite bus bar The purpose of the capabilitychar t is to describe the range of complex power that may be deliveredfrom the generator to the busbar.

v

P I

Turbine Generator Infinite busbar

Figure 2.1 Circuit Model of Synchronous Generator

Power from the turbine drives the lossless generatorwhich then delivers the power to an inf in i te busbar. Thesynchronous generator capability chart portrays the real andreactive power ( Q) that may be supplied from the generatorto the infinite busbar.


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2.2.1 Simplifying Assumptions

To simplify the analysis the following assumptions have been made

1. The generator is connected to an infini te busbar. This is a strongsystem that is able to maintain a constant voltage and frequency.

2. Only steady state operation is considered. All changes take longerthan the machine's t ransient time constant .

3. Magnetic saturation i s neglected. All inductive reactances in themodel are therefore constant and independent of current.

4. Losses due to hysteresis , eddy currents, and winding resistance areneglected.

2.2.2,Generator Vector Diagrams

The vector diagram shown in f igure 2.2 i l lus t ra tes the relationshipbetween the voltage, current , and magnetic flux phasors of the generator.The phasors have been superimposed onto a diagram representing theposi t ion of the rotor and the position of the sta tor winding. The currentand f lux phasors are resolved into components along the direct andquadrature axes of the rotor. This resolut ion is necessary to deal withthe magnetic assymmetry of the sal ient pole rotor.

The rotor field current produces the flux Ff along the direct axisof the ro tor . This rotat ing flux varies sinusoidally a t the statorwinding and induces the sinusoidal open circui t terminal voltage E. Theinstantaneous value of E reaches a maximum when the vector Ff is cutt ingdirectly through the sta tor winding hence E l ies along the quadrature axis.

The vector Ff may ~ l t e r n a t i v e l y be regarded as a complex phasor tha trepresents the component of flux passing perpendicularly through theplane of the sta tor winding. The induced voltage E is equal to the timederivat ive of Ff by Faraday's law. Therefore the sinusoidal f lux Ff mustlead the sinusoidal voltage E by 90 degrees, and E must l ie along thequadrature axis . This is in agreement with the previous argument based onflux cutt ing through the sta tor windings.


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When the generator is loaded the sta tor current I lags the terminalvoltage V by the power factor angle rp. The current I can be resolved intoa direct axi s component Id and a quadrature axis component Iq .current Id produces a flux Fd that l ies along the direct axis.current Iq produces a flux Fq that l ies along the quadrature axis.and Fq are added to Ff to produce the effective rotating flux Fe .

\ \\

\

~

000stator \-lirding

Figure 2.2 Vector Diagram of Salient Pole Synchronous Generator

The voltage, current, and magnetic flux phasors have beensuperimposed on the rotor and the stator winding. The currentand flux phasors have been resolved into components along thedirect and quadrature axes of th e rotor.

TheTheFd


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The combined effect of Fd and Fq is called the 'armature reaction'which tends to reduce the magnitude of Fe and lower the induced voltageE . This can be conveniently simulated by one voltage drop due to Idflowing through a direct axis armature reactance jXad and another voltagedrop due to Iq flowing through a quadrature axis armature reactance jX

aq

In a sa l i en t pole machine the direct axi s has a lower magneticre luctance path than the quadrature axis due to the larger a ir gap alongthe quadrature axis. The value of jXad is therefore larger than the valueof jXaq In a round rotor machine the reluctances of the two paths arealmost identical and jXad is considered equal to jXaq .

The two components of the current I both flow through the stator leakagereactance jXI This reactance can be added to the armature reactances toform an effective direct axis reactance jXd and an effective quadratureaxis reactance jXq .

'XJ q

'X + 'XJ ad J IjX + 'X

aq J I

(2.2.2.1 )

(2.2.2.2)

The vectorial addition of the voltage drops jIdXd and jIqXq to theterminal voltage V produces the no load terminal voltage E. The angle 0between V and E is called the rotor angle. This angle tends to increaseas more power is delivered from the generator.

Inspection of the vector diagram leads to the following steady stateoperating equations that relate the magnitudes of the vectors.

Ivl sin(o) I I IXq q (2.2.2.3)

I VI . cos (0) lEI - IIdl,xd (2.2.2.4)

I I I . cos (


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I t i s convenient to directly re la te E, V, and I without the use ofth e orthogonal components Id and I q . To achieve th is , the voltage vectorsof f igure 2.2 can be modified as shown in the vector diagram of figure 2.3 .The voltage drop jIqXq has been extended to jIqXd so that the current Ican be direct ly incorporated into the diagram as the voltage drop jIXd

The no load vpltage E is translated by jIq(Xd-Xq ) to E' and a l ine isextended backward to intersect with the path of V. As the rotor angle 0changes the angle between E' and jIq(Xd-Xq ) i s maintained a t 90 degrees.Hence the intersection point of E' and jIq(Xd-Xq ) t ravels in a circ le as


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This geometrical manipulation serves to portray EI , V and jIX d aspar t of a simple t r iangle . This triangle can be mapped onto the complexpower plane by imposing the following transformation on each vector A tomap i t onto the corresponding vector AI on the complex power plane.

AI (2.2.2.7)

Where the symbol * represents complex conjugation.The t r iangle on the complex power plane is shown in f igure 2.4 . The

axes on the plane represent the real power P and the reactive power Qthat are delivered from the generator to the busbar.

P

Q

I VI . I I I . cos ( /> ) (2.2.2.8)IVIIIIsin(


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2.2.3 Construction of the Capability Chart

The vector diagram shown in figure 2.4 is in a suitable form foreasily constructing the capabil i ty chart of a sal ient pole generator. Agenerator with the following specificat ions is used to demonstrate theconstruction technique j

1. Turbine power range of 0 MW to 60 MW2. Generator apparent power rating of 75 MVA3. Excitation system capable of providing rotor f ie ld current

corresponding to a no load vol tage range of 0.1 to 2.0 pu4. Direct axis reactance Xd = 1.5 pu

Quadrature axi s reactance X =q 1.1pu

5. Infini te busbar voltage V = 1 .0 pu

A 100 MVA power base i s used for the per unit system.

The capabili ty chart is i l lustrated in figure 2.5 and depicts therea l and reacti ve power that can be supplied from the generator to theinf ini te busbar. Each cr i t ica l operating constraint is represented by as epar at e locus on the chart . The shaded area denotes the safe operatingregion of th e generator.

Turbine Power Limits

The generator has been assumed to be lossless so the ent i re turbinepower output is delivered to the infini te busbar. The maximum and minimumturbine power l imits are therefore represented as s traight vert ical l inesthat intersect the real power axis at 0.0 pu and 0.6 pu

Maximum stator Current

The apparent power rating of 75 MVA indicates that the stator currentI is limited to 0.75 pu to prevent overheating of the stator windings. I fI is fixed at this value the vector vr* in figure 2.4 traces out a circleof radius IVII centred on the origin. A small portion of this circlecontributes to the boundary of the operating region on the capabili tychart .


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Rotor Current Limits

The upper limi t of the rotor f ie ld current i s determined by theheat ing of the f i e l d winding and the maximum current that can bedelivered by the, excitat ion system. The exci tat ion system also imposes alower limi t on the rotor current . The no load termi nal vol tage E i sdi rec t ly proport ional to the f ie ld current i f magnetic saturat ion i sneglected. Therefore the upper and lower 1 imi ts of the rotor currentalso correspond to upper and lower l imits on the magnitude of E.

The loci tha t represent these l imits are constructed by maintaining Ea t i t s l imit ing value in f igure 2.11 an d varying the rotor angle o. Thelocus traced out by vr* is in the shape of a s l ight ly distorted ci rc lecalled a 'Limacon of Pascal ' .

On the capabil i ty chart the locus of maximum rotor current l imi t s thereac t i ve power tha t can be generated and the locus of minimum rotorcurrent l imi t s the react ive power tha t can be consumed a t small leadingpower factors .

I f Xq is increased to the same value as Xd to simulate a s imilarround rotor generator then the rotor current loci become c i rc les centred

The locus of maximum rotor current for a round rotorgenerator is almost indistinguishable from the locus of maximum rotorcurrent for a sa l ien t pole generator . The loc i of minimum rotor currentare more easily dist inguished because the effec t of saliency is morepronounced for small val ues of E.

Steady State Stabi l i ty Limit

The steady s t a t e s tab i l i ty of a generator is determined by i t sab i l i t y to respond to small disturbances without losing synchronism.During these disturbances generators with slow acting exci ters willmaintain a constant rotor current and a constant open circui t terminalvol tage E. Under these condi t ions the real power output P becomesstrongly dependent on the rotor angle o. The relat ionship between P and 0can be obtained by f i r s t subst i tut ing (2.2.2.5) into (2.2.2.8) to give

p IVI . I I cos (0) + I I. s i n(0) ]q d (2 .2 .3 .1 )


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pu Reactive powerMinimum turbine power

Maximum turbine power

Maximum rotor current.....

..... , Round rotor maximum rotor current

Round rotorminimum

rotor c u r r e n t ~-1/Xd ,..

- l /Xq

Figure 2.5

Maximum stator current1 pu Real power

Round rotor theoretical stability limit~ - - - - -Theoretical stability limitMinimum rotor current

Capability Chart of a Salient Pole GeneratorThe chart portrays the real and reactive power that may

be supplied from the generator to the infini te busbar. Theshaded area denotes the safe operating region which isbounded by loci that represent the c r i t i ca l operating l imitsof the generator. The chart of a similar round rotor generatordiffers in the positions of the rotor current and s tab i l i tyl oc i . For comparison these round rotor loci are drawn onthe chart with dashed l ines .


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then substitution of Iq and Id from (2.2.2.3) and (2.2.2.4) gives

P

+

+ . co s (


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In a round rotor generator the second term of (2.2.3.2) i s zero andthe corresponding power/rotor angle graph i s sinusoidal . In th is casesynchronism can be theoret ical ly maintained unt i l the rotor angle exceeds90 degrees. The locus of the theoret ical stClbili ty l imit for a roundro tor generator is shown on the chart as a horizontal s t ra ight l ineintersect ing the react ive power axis a t -1/X d

The theore t ica l s tab i l i ty limi t for a sal ient pole generator isobtained by different iat ing (2.2.3.2) with respect to 0 to find themaximum power output.

dP Iv 11 EI .cos (0 )do

+ IvI 2 .cOS(2.0).(Xd-Xq )XdX q

This can then be solved .for cos(o).

cos(o)

-I E1Xq + [-I - : - ~ - ~ - : ~ - X - qT 84(2.2.3.3)

(2.2.3.4)

The locus of the theoret ical s tab i l i ty l imit is drawn by graduallyvarying E in (2.2.3.4) and obtaining the corresponding s tab i l i ty angle o.The pairs of E and 0 values are then used in the vector diagram of figure2.4 to obtain the corresponding points on the complex power plane.

The theoret ical s tabi l i ty l imi t does not allow for a margin of safetyso a pract ical s tabi l i ty l imi t is defined. The pract ical l imit allows fora power increase of 10% of the turbine rating before the theoret icall imit is reached. Each point on the locus of pract ical s tab i l i ty i s foundby choosing a point on the theoret ical s tabi l i ty locus and reducing 0whils t keeping E constant unti l the real power has dropped by 10% of theturbine rat ing.

The theore t ica l and prac t i ca l s t ab i l i ty loci for sa l ien t polegenerators are asymptotic to the corresponding loci for round rotorgenerators. This behaviour ref lec ts the reduced effect of saliency as Eincreases.


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18

2.2.4 Summary of Synchronous Generator Chart

The capabil i ty chart for a sal ient pole generator is drawn on thecomplex power plane and portrays the real and reactive power that may bedelivered by the generator to an infini te busbar. The safe operatingregion of the chart is bounded by loci that represent the cr i t i ca loperating l imits of the generator.

The cr i t i ca l operating l imits considered are the maximum and minimumturbine power output, the maximum sta tor current, the maximum and minimumrotor current, and the synchronous s tabi l i ty l imi t .

The char t for a round rotor generator can be drawn by treating i t asa spec ia l case of a sal ient pole machine with the direct axis reactanceequal to the quadrature axis reactance.

2.3 THE CAPABILITY CHART FOR A TRANSMISSION LINE

Phil ip (1911) originally described a chart to graphical ly analyze thebehaviour of a transmission l ine . This work was la te r extended by Evansand Sels (1921) and Dwight (1922) and is now commonly referred to as the' c i rc le diagram' approach to analyzing transmission l ine behaviour.

The circle diagrams consist of a ser ies of circles drawn on thecomplex power plane. Each c i rc le represents the locus of real andreac t ive power that may be delivered by the l ine a t a particular voltagemagnitude. These diagrams can easily be constructed by using a compass sothey were particularly popular before computer analysis became availableand are s t i l l used to study the performance of transmission l ines .

The circ le diagrams can provide the voltage constraint loci that formpart of the capabil i ty chart for a transmission l ine . To complete thechar t the vol tage c i rc les must be supplemented by other loci thatrepresent the remaining operating l imits of the l ine .


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19

2.3.1 Transmission Line Model

The transmission l ine is modelled by an equivalent pi section asshown in f igure 2.7 The pi sect ion consists of a ser ies impedance Z anda shunt admittance Y at either end of the l ine. The l ine delivers powerfrom an in f in i te bus bar at voltage E to a load busbar a t voltage V. Theshunt admittance F at the load busbar is used to represent a reactivepower compensation capacitor or any other device that can be modelled asan admi t tance.

The complex power delivered to the load is given by

s (2.3.1.1 )The load busbar voltage can be related to the supply bus bar voltage byconsidering the voltage drop across the l ine.

E (2.3.1.2)

The current 12 that produces the voltage drop must be combined with thecurrents flowing in the shunt admittances to obtain the to ta l l inecurrent measured at either end of the l ine .

13 = 12 - V.YInfinite supply busbar

ELoad bus bar

V

F i g u ~ e 2.7 Circuit Model of Transmission Line

(2.3.1.3)(2.3.1.4)

F


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20

Equations (2.3.1.1) to (2.3.1 form the complete se t of operatingequa t i ons tha t model the behaviour of the transmission l ine . Theseequations can be manipulated to describe loc i that represent theoperating constraints of the l ine.

A 33 kV transmission l ine is used to demonstrate the construction ofthe chart. The circui t parameters correspond to a power base of 10 MVA.

E 1 puZ 0.118595 + jO .25911 puy 0.0 + jO.000216 puF 0.0 + jO.2 pu

2.3.2 Operating Constraints

Each operating constraint of the system must be represented by aseparate locus on the capabi l i ty char t . The constraints that areconsidered in this transmission l ine system are

1. Maximum (1.1 pu) and minimum (0.9 pu) voltage at the load busbar.2. Voltage stabil i ty a t the load busbar.

3. Maximum current of 1.9 pu entering and leaving the lirre.The magnitude of the l ine voltage and current is known to vary as a

hyperbolic function along the length of the l ine (Steinmetz, 1916). Onlong l ines the maximum voltage and the maximum current may occur atpoints par t way along the l ine and not necessarily at the busbars a t theends of the l ine .

To simplify this analysis the transmission l ine is assumed to besufficiently short so that the cri t ical voltages and currents do occur a tthe ends of the l ine. This assumption gradually loses i t s val idi ty as thelength of the l ine increases beyond a quarter wavelength (1500 km a t 50Hz) because the hyperbolic fUnction exhibits more maxima with increasingl ine length.


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21

Load Busbar Voltage Limits

The maximum and minimum acceptable voltage a t the load busbar can berepresented on the chart by using the circ le diagram approach. Equation(2.3.1.1) can be rewritten in terms of V by using (2.3.1.2) and (2.3.1.4).

s * *. (E/Z) - Ivl2..(1/Z + Y + F) (2.3.2.1)

I f E i s constant and the magnitude of V is constant then (2.3.2.1)describes a circular locus on the complex power plane as the phase angleof V changes. The centre of the circle is at -lvl2.(1/Z+Y+F)* ~ n d theradius is IVE/zl .

Figure 2.8 shows a series of load busbar voltage circ les drawn on thecomplex power plane; All of the circ les f i t into a parabolic region onthe plane. There are two possible busbar voltage solutions correspondingto each complex load power within the parabola. Only one possible voltagesolution exists along the edge of the parabola and no solutions existoutside the parabola.

Of the two possible voltage solutions that can be obtained within theparabola, only the voltage with the larger magnitude is associated with as table operating point . The smaller voltage is associated with anunstable operating point that will lead to a voltage collapse at the loadbusbar. The rate of collapse is dependent on the type of load and mayoccur over a period of several minutes (Venikov and Rozonov,1961 andWeedy, 1968) .


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22

//

/

//

//

10 pu Reactive power

II

II

Figure 2.8 Voltage Circle Diagram

10 pu Real power

stability boundary

Each circ le represents a locus of constant voltagemagnitude at the load busbar. All of the circ les f i t into aparabolic envelope which represents the voltage s tabi l i tyboundary.


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23

Voltage Stabi l i ty

The theoret ical boundary of the stable voltage region is defined bythe parabol ic envelope tha t contains the voltage c i r c l e s . It isconvenient to form a simple T h ~ v e n i n equivalent of the ci rcui t model toinvest igate the nature of the envelope. The Thevenin equivalent shown in

. figure 2.9 consists of a voltage source E' and a ser ies impedance Z' where

E' EI (1 + 1 . Z + F. Z) (2.3.2.2)Z' ZI(1 + Y.Z + F.Z) (2.3.2.3)

The load busbar vol tage V and the load power S both re ta i n the samesignificance in the Thevenin ci rcui t .

The variables of the Thevenin circui t are related by the equation

S

v~ - - - - - - - - - - - - r - ~ SL - -_ . . . J

E'= E1+YZ+FZ

Figure 2.9 Thevenin Equivalent of Transmission Line Model


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24

The parabolic envelope is characterized by the existence of only onesolution for V corresponding to a given S in (2.3.2.4) . This uniqueso lu t ion can be found by f i r s t expressing the complex variables of(2 .3 .2 .4) in terms of their real and imaginary components. E' is used asa real reference vector.E' e + jOS P + j qV u + jvZ' = r + jx

The solution for V in (2.3.2.4) in terms of u and v is then

u

v

ele2 - 4[pr + qx + (qr-px)2/e 2J}2

(qr-px)/e

(2.3.2.5)

(2.3.2.6)

The unique solution for V is identif ied by a zero valued argument in thesquare root term of (2.3.2.5) . Hence V l i es on the s tabi l i ty boundary i fu=e/2

The relat ionship between the stable and unstable solutions for V i sshown in the vector diagram of figure 2.10 The original supply voltageE is used as the reference vector and the theoretical s tabi l i ty boundaryi s shown as a s traight l ine bisecting the Thevenin voltage E '. As thetheoretical boundary is approached from the stable side the load voltagefluctuations will become more pronounced unt i l a total collapse occurs inthe unstable region.

A safety margin is provided by the practical s tabi l i ty boundary that1 ie s parallel to the theoretical boundary and intersects the Theveninvoltage a t 0.8E' . The factor of 0.8 1s a compromise between the need fora reasonable safety margin and the need to allow for low but stable loadvol tages . If a greater quality of voltage s tabi l i ty is required then thefactor of 0.8 should be increased.

If the shunt admittances are neglected then the Thevenin voltage E'i s ident ica l to the original supply voltage E and the stabil i ty boundarybecomes a vert ical l ine passing through E/2


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Unsta ble r ~ g l o n

T h ~ o r ~ t i c a l stability boundary//for V -4.-/ -+ S t a b l ~ r ~ g l o n for V//

/ /Practical stabi li ty boundary/

~ - - - - - - - - - - - - - r - - - - - - - - + - ~ E

//

I V

Figure 2.10 Relationship of Stabi l i ty Boundary to Voltage VectorsThe theoret ical voltage s tabi l i ty boundary bisects the

Thevenin equivalent voltage vector E' a t a r ight angle. Theload voltage vector V should normally l i e to the right ofthe prac t ica l voltage stabi l i ty boundary to allow for asafety margin.

25


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26

The theoretical and pract ical voltage stabi l i ty boundaries are drawnon the complex power plane by gradually shif t ing V along the appropriateboundary in f igure 2.10 and generating a sequence of S values from

"(2.3.2.4) . Both the theoretical and pract ical s tabi l i ty boundaries areshown on the capabi l i ty chart in figure 2.11 .

The maximum and minimum voltage limi ts are also represented on thecapabil i ty chart by arcs from the corresponding voltage circ les that weredrawn in figure 2.8 .

Maximum Transmission Line Current

The maximum l ine current has been assumed to flow at one of the endsof the l ine . The current entering the l ine from the supply end is 11 andthe current leaving the l ine at the load end is 1 3

The locus of the maximum current 11 can be obtained by usingequations (2.3.1.2) to (2.3.1.4) to rewrite (2.3.1.1) in terms of 1 1 Th is yi elds

S (2.3.2.7)

where the complex constants A,B,C, and 0 are given by

* * * * *A E .(2.Y.Z+Y.Y.Z.Z +F.Z + F.Y.Z.Z )* * * * * * * *B E.(1+Y.Z +Y .Z +Y.Y .Z.Z +F .Z +F .Y.Z.Z )

* * * *C Z+Y .Z.Z +F .Z.Z* * * * * * * * * *D E.E .(2.Y+Y.Y.Z+2.Y.Y .Z +Y.Y.Y .Z.Z +F+F.Y.Z+F.Y .Z +F.Y.Y .Z.Z )

Equation (2.3.2.7) describes an el l ipt ical locus as the magnitude of 11is held constant at i t s maximum value and the phase angle of 11 is variedthrough a fu l l 360 degrees.


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Minimum

2 pu Reactive power

Maximum line current

\IIIPractical voltage stability limit IITheoretical voltage stability limit I

Figure 2.11 Capability Chart of Transmission LineThe shaded area denotes the range of real and reactive

power that may be supplied to the load. This area is boundedby loci that represent the cr i t ica l operating l imits of thesystem.

27


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28

At the load end of the l ine the locus of 13 can be found in a similarfashion by rewrit ing (2 .3 .1 .1) in terms of 1 3 , This yields

s (2 .3 .2 .8)

where the complex constants A,B,C, and D are given by

* *A E .F Z11 + Y.Z/2

*. ( 1 + 1 . Z + F.Z)B11 + Y.Z/2

*.C Z. ( 1 + 1 . Z + F.Z)

11 + Y.zI2

IE 12 .F*D11 + Y.zI2

Equation (2 .3 .2 .8) also describes an e l l ip t i ca l locus as the magnitudeof 13 i s held constant and the phase angle is varied through 360 degrees.I f the shunt admittance F is zero then this locus becomes circular .

The two loc i of maximum current are indist inguishable on thecapabi l i ty chart of figure 2.11 The el l ipses become more dis t inct asthe value of the shunt admittances increases.


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29

2.3.3 Capabili ty Chart on the Admittance Plane.

I f the load i s a sof t admittance load rather than a s t i f f constantpower load then i t is more appropriate to draw the capabil i ty chart onthe complex admittance plane instead of the complex power plane. Thechart on the admittance plane is shown in f igure 2.12 .

The maximum and minimum load voltage l imits and the maximum l inecurrent 1 imit are shown on the admittance chart . A vol tage stabU i tyl imit is not shown because the voltage collapse phenomenon does not existfor pure admittance loads.

2 pu Susceptanceaximum voltage

voltage

Maximum line current

Figure 2.12 Capability Chart of Transmission l ine on theComplex Admittance Plane


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30

I f the complex power S is replaced by a complex load admittance Gthen G is given by

G (2.3.3.1 )

This equation is used in place of the complex power equation (2.3.1.1)to formulate the equations describing the constraint loci on the complexadmittance plane.

Voltage Limits

The loci corresponding to the maximum and minimum load voltage areobtained by combining (2.3.1.2), (2.3.1.4) and (2.3.3.1) to yield

G E1 ( V Z) - ( 11 Z + F + Y) (2.3.3.2)

I f the magnitude of V is held constant a t the maximum or minimum valueand the phase angle of V is varied through 360 degrees then (2.3.3.2)describes a circle on the complex admittance plane. The radius of thecircle is IE/(VZ)I and the centre is at -(1/Z+F+Y) .

Two arcs from the voltage circles are drawn on the admittance chartin figure 2 . 12 . The voltage circ les are concentric and their commoncentre represents the shunt admittance that would produce a resonancebetween the series impedance Z and the shunt admittances Y, F, and G atth e load busbar.

Line Current Limit

The locus of maximum current entering the l ine from the supply bus ba ris obtained by using (2.3.1.2) to (2.3.1.4) to rewri te (2.3.3.1) interms of I I '

G - Y - F (2.3.3.3)


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31

I f the magnitude of II is held at i t s maximum value and the phase angleof II i s rotated through 360 degrees then (2.3.3.3) describes an el l ipseon the admittance plane.

The locus of maximum current leaving the l ine at the load busbar isalso e l l ip t ica l and is found by rewrit ing (2.3.3.1) in terms of 13 togive

G 1 3 .(1 + Y.Z) ._ FE - 13'Z

(2.3.3.4)

The two e l l ip t ica l loci of maximum current flowing through the endsof the l ine are indistinguishable on the admittance capabil i ty chart . Aswith the power capabil i ty chart the loci gradually become separated i fthe l ine shunt admittances of Yare great ly increased in value.

2.3.4 Summary of Transmission Line Capability Charts

Two capabi l i ty char ts have been constructed to descr ibe theperformance of a transmission l ine . One chart is drawn on the complexpower plane and th e other is drawn on the complex admittance plane.

The operating l imits represented on the complex power chart are theminimum and m.aximum voltage a t the load busbar, the voltage s tabi l i tyboundary, and the maximum current flowing at ei ther end of th e l ine .

These l imits are also represented on the complex admittance chartapart from the voltage s tabi l i ty boundary which is not applicable.

Power system operators tend to regard loads from the complex powerviewpoint ra ther than the complex admittance viewpoint. This tendencyreduces the importance of the admittance capabil i ty chart . The predominanceof the complex power viewpoint has also influenced the rest of the workin this thesis to concentrate on the complex power capabil i ty charts.


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322. CONCLUSION

This chapter has described the capabil i ty charts for two simple powersystems. First the chart for a sal ient pole generator is constructed byusing the method described by Walker (1953). The round rotor generator isregarded as a special case of the sal ient pole machines.

The chapter then describes the capabili ty chart for a transmissionl ine . The t radit ional voltage circle diagrams are used to construct theloci that represent maximum and minimum voltage l imi ts . The equations ofother loci are then derived to represent the maximum l ine current andvoltage s tabi l i ty l imits .

A capabil i ty chart for the transmission l ine is also constructed onthe complex admittance plane which is more appropriate for admittanceloads. The admittance chart is not considered in the rest of this thesisbecause of the prevalent tendency to regard loads in complex power terms.

The equations of the loci for these simple systems are derived byalgebraic and geometrical manipulation of the basic operating equationsand vector diagrams. This technique is restr icted to small systemsbecause the algebraic manipulation rapidly becomes too unwieldy as thecomplexity of the system grows.


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33

Chapter 3A Chart for an HVDC Link

3.1 INTRODUCTION

The development of a capabil i ty chart to describe the steady stateperformance of an HVDC link is particularly attractive because of theincreasing use of HVDC l inks in power systems. Kimbark (1971) developedcharts to describe the operation of individual stat ic converters, but acapabil i ty chart fo r a complete HVDC link has not been considered before.The work described in this chapter has been accepted for publication bythe lEE (de Silva, Arnold, and Arri l laga, 1987).

The HVDC link capabil i ty chart is constructed by using a similartechni que to tha t used for simple systems, such as the synchronousgenerator and transmission l ine described in chapter 2. The basicoperating equations that describe the behaviour of the HVDC link aremanipulated into a suitable form for drawing loci on the complex powerplane.

A simplified model of the New Zealand Benmore-Haywards HVDC l ink hasbeen chosen to demonstrate the construction of the capabil i ty chart . Thetechnique used is also applicable to any other two terminal HVDC l ink.

3.2 CIRCUIT MODEL OF THE HVDC LINK

The New Zealand HVDC l ink can transfer 600 MW between the North andSouth Islands of New Zealand. The southern terminal of the link is atBenmore power station and the northern terminal is a t Haywards. 600 km ofoverhead transmission l ine and km of undersea cable connect the twoterminals.

A circui t diagram of the HVDC link is shown in figure 3.1 The locusequations described in chapter 2 for single generators and transmissionl ines were not simple to derive. This indicates that considerablema thematical diff icul t ies can be expected in the der iva tion of locus


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equations for the actual HVDC scheme which includes several transformers,generators, and AC/DC converters. Therefore the simplified circui t modelshown in figure 3.2 has been chosen instead for the development of thecapabil i ty chart .

At the Benmore converter terminal the six 90 MW hydro-electricgenerators are modelled by a single equivalent MW generator feeding asingle 16 kV busbar. The voltage level of the busbar is regulated by thegenerator AVR. The equivalent generator can also operate as a synchronouscompensator to deliver up to MVAR or consume a maximum of 378 MVAR.

The two three-windi ng t ransformers that interconnect the 16 kVbusbars and the South Island 220 kV system are modelled by a single, twowinding, MVA interconnecting transformer, having a 3.3 % reactance.This reactance is derived from a power base of 100 MVA that has beenchosen for the entire AC/DC system.

The two banks of harmonic f i l ters that are attached to the tert iarywindings of the three-winding transformers are modelled as a single bankof f i l t e r s attached to the 16 kV busbar. The f i l ters absorb t h ~ harmoniccurrents from the converter and also supply 100 MVAR of react ive power.The behaviour of these f i l t e rs is considered to be ideal so that thevoltage waveform on the 16 kV busbar is assumed to be sinusoidal . Thisassumption is adequate for a steady sta te , fundamental frequency analysisof th e HVDC l ink.

The converter terminal at Benmore consists of two 250 kV DC poles.Each pole consists of two six pulse bridges in ser ies. The mercury arcvalves in the bridges are rated to carry a continuous DC current of 1.2kA . A 30 degree phase difference between the converter transformersallows an overal l twelve pulse operation for each pole.

The four bridges operate under identical control so the bridges andconverter transformers can be modelled by a single equivalent twelvepulse bridge and transformer (Arrillaga e t a l . , 1983). The equivalentbridge is rated to carry 1.2 kA DC current. The equivalent losslessconverter transformer is rated at 750 MVA and has a reactance of 2%Since i t has been assumed that the converter transformer is fed from asinusoidal voltage source, the transformer reactance can also be regardedas the commutation reactance of the converter.


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An overhead transmiss ion 1 ine and an undersea cable transmit thepower from the Benmore converter to the Haywards converter. The largesmoothing reactors a t each end of the line are assumed to maintain aconstant DC current. The to tal resistance of the converters, smoothingreactors, transmission l ine, and cable is 25.56 o.

The Haywards converter terminal also consists of four six pulsebridges in series. The three winding converter transformers connect thebridges to a 110 kV busbar as well as to four synchronous compensators.This arrangement is modelled by a single equivalent bridge, and a twowinding converter transformer. The equivalent bridge and transformer haveth e same parameters as the equivalent Benmore converter.

In the model, the synchronous compensators are connected directly tothe 110 kV busbar instead of a tert iary transformer winding. Thispart icular simplif ication introduces the greatest error in th e modellingof the. Benmore-Haywards scheme, but also resul t s in a closercorrespondence to other HVDC schemes that use two winding convertert ransformers (Vancouver Is land, Pacific Inter t ie , Skagerrak, SquareButte, CU, Nelson River Bipole II , Inga-Shaba, and Gotland I I ) .

The voltage level of the 110 kV busbar is maintained by the AVR onthe equivalent synchronous compensator which can provide up to 260 MVARof reactive power. This is supplemented by 110 MVAR from the harmonicf i l ters which are assumed to maintain a sinusoidal voltage on the busbar.The 110 kV busbar feeds the North Island AC network.

Similarly to the capabil i ty chart of a synchronous generator, whichrepresents the complex power available from th e generator terminals, themost useful information to be derived from th e HVDC link capabil i ty chartis the complex power available to the Haywards 110 kV busbar from theequivalent converter transformer.


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220kV

SouthIsland

Line Cable 110 kV

1---11----11 ~ r

NorthIsland

l - - -+ - - - l l ~ I '

BENMORE HAY'w'AROS

Figure 3.1 New Zealand HVDC Scheme

w(J"I


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SouthIsland

540MW220 kV

16 kV

Vdc

1[ ) 260 MVAR

Vac

~ 7 >- > North4-00 MVA P + jQ Island

'1r--rrrY100 MVAR Fi lters

BENMORE

Figure 3. 2 Simplified Model Of New Zealand HVDC SchemeThe capability chart of th e HVDC l ink portrays th e realand reactive power (P+jQ) that can be supplied from th eHaywards converter transformer to th e Haywards 110 kV busbar.

~ I I110 MVAR

HAYWARDS

w......


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38

3.3 OPERATING CONSTRAINTS OF THE HVDC LINK

Each operat ing constraint of the HVDC l ink must be represented as alocus on the capabil i ty chart .

Converter Transformer Current

The heat dissipat ion capabil i ty of each 750 MVA converter transformerdetermines the maximum RMS current that may flow through the transformerwindings. This current consists of a fundamental frequency component aswell as several harmonic frequency components. All of these componentsmust be considered even though only the fundamental current contributesto the useful power available.

Converter Valve Current

The DC l ine current flows through the converter valves, smoothingreactors, transmission l ine, and cable. Of these four components, theconverter valves possess the smallest current rating of 1.2 kA, and theytherefore determine the maximum DC current that may flow in the l ink.

The converter valves also require a minimum holding current of 0.1kA. This is the DC current n e c e s s a ~ y to sustain valve conduction whilsteach valve is nominally on .

Harmonic Fil ter Current

The DC l ine current is also related to the harmonic currents flowingin the f i l t e r banks. If commutation effects are neglected then theharmonic current levels can be assumed to be direct ly proportional to theDC current . This is a safe assumption because the inclusion ofcommutating e f fec t s tends to reduce the calculated harmonic currentlevels (Arrillaga, 1983).

The f i l ters are designed to cope with the harmonic currentsassociated with the maximum expected DC current , as well as thefundamental frequency reactive current. Therefore operation within the


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39

maximum converter valve current locus will also ensure a safe level ofharmonic f i l ter current.

DC Voltage RatingThe DC voltage of the transmission l ine and cable has a reasonably

smooth waveform during steady state operation. The maximum continuous DCvoltage that may be borne is 525 kV. This is determined by the continuousvoltage ratings of the transmission l ine insulators, surge arrestors, andcable insulation. Each of these components will also tolerate a muchhigher transient overvoltage, but th is is not represented on the steadystate capabil i ty chart .

Converter Control Angles

A minimum l imit is set on both converter control angles. Duringrec t i f ica t ion , the firing angle a must be at least 3 degrees to ensurethat the valve turn-on voltage is suff icient to reliably establish valveconduction.

During inversion, the extinct ion angle 0 must be at least 18 degreesto allow the valve to recover from forward conduction in time to blockthe reverse voltage which wil l be impressed a t the next voltagecrossover.

Converter Commutation Angle

The reactance of the converter transformers causes a commutationoverlap to occur as one valve takes over conduction from another valve.When the overlap angle increases above 60 degrees, periodic phase tophase short circui ts occur through the valves.

Although the converter may continue to operate with commutationangles greater than 60 degrees, the simple equations used to model theconverter are only valid for angles up to 60 degrees. The requirementtha t the commutation angle be less than 60 degrees is therefore due tothe mathematical model used rather than a real converter constraint .However, as the commutation angle rarely exceeds 30 degrees during


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prac t ica 1 steady s ta te opera t ion, the simple mathematical model isperfectly adequate.

Capability of the Benmore Generator

The performance res t r ic t ions of the Benmore generator can berepresented on a capabil i ty chart as described in chapter 2. To simplifythe analysis , the generator chart is approximated by a rectangle on thecomplex power plane as shown in figure 3.3 . The minimum and maximum realpower 1 im i ts on the rectangle correspond to the minimum and maximumturbine power l imits. The maximum reactive power l imit corresponds to themaximum rotor current and the minimum reactive power l imit corresponds tothe synchronous stabi l i ty l imit .

I f the Benmore lake level is low then the genera tor is used as asynchronous compensator. In this ~ a s e the performance of the generator isres t r i c t ed to zero real power output and a react ive power variat ion of-378 MVAR to + 3 2 ~ MVAR.

Readive powerI'324 MVAR

,,::>:i,:!:::,,::::'::':::'::i,::::!!540 MW:>:>< Real power

i-378 MVAR

Figure 3.3 Rectangular Approximation to CapabilityChart of Benmore Generator


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Interconnecting Transformer Current

The power that can be suppl ied from the South Island network isres t r ic ted by the ra t ing of the ~ O O MVA Benmore interconnectingt ransformer. This rest r ic t ion is especial ly important when the Benmoregenera tors a re ac t ing as synchronous compensators and are notcontributing any real power to the HVDC l ink.

3 . ~ LOCI OF OPERATING CONSTRAINTS

The equations that describe the locus of each operating constraintare formulated from the basic steady sta te operating equations of theHVDC l ink. The operating equations can be conveniently writ ten in termsof the combined ACiDC pe r uni t system described by Arri l laga e t al.(1983).

3 . ~ . 1 ACiDC Per Unit System

In th is system the conventional per unit quanti t ies are used for theAC networks. The DC per unit quanti t ies are defined by choosing the samebase power and base vol tage for both the AC and DC sides of eachconverter bridge.

The power base Sb for the entire circui t model is 100 MVA. The DCvoltage base Vbdc is the same as the converter transformer secondaryvoltage rating of kV.

The DC current base Ibdc must be 13 times the AC current base inorder to maintain consistent quanti t ies of per uni t power throughout theentire network.

13. I bac 0.238 kA ( 3 . ~ . 1 . 1 )

Consequently the DC valve current rating of 1.2 kA is represented as 5 . 0 ~pu current.


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The DC resistance base Rbdc is th e same as the AC impedance baseZbac'

Zbac Vbdc2/Sb 1 kfl ( 3 . ~ . 1 . 2 )

Hence the to ta l 25.56 Q ser ies res is tance of the DC ci rcui t i srepresented as O . 0 1 ~ pu resistance.

3 . ~ . 2 Basic AC/DC Converter Operating Equations

According to Arrillaga et a l. (1983), the steady state operation ofthe Haywards converter can be described by the following equations.

I ac

[a.V 1(/2.X )].[cos(a) - c o s ( a + ~ ) ]ac c

w h e ~ e the converter variables are signified by

Vdc DC voltageIdc DC currenta Transformer tap rat ioXc Transformer reactance (=commutation reactance)lac AC currentP Real power output from converter transformerQ Reactive power output from converter transformerVac AC voltage

( 3 . ~ . 2 . 1 )

( 3 . ~ . 2 . 2 )

( 3 . ~ . 2 . 3 )

( 3 . ~ . 2 . 5 )


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lJ3

The constants K1 K2 and K3 are given by

K3 0.995 for twelve pulse converter operation with a commutation anglewell below 60 degrees.

3.lJ.3 Loci of Haywards Converter

The operating constraint loci of the Haywards converter are examinedf i r s t . All of the Haywards loci are shown on th e chart in figure 3.lJ The scale of the chart must accommodate th e large locus that representsthe maximum commutation angle hence the other smaller loci appearcramped. Figure 3.5 portrays the smaller loci on a larger scale forclar i ty.

-4000 MW 4000 MW.... ';: ' ," ,. ' :. "

", ."

-5000 MVAR 60 deg commutation overlap

Figure 3.lJ Loci of Haywards ConverterThe locus representing th e maximum commutation angledominates this chart. The other constraint loci drawn with

dashed l ines res t r ic t th e converter operation such that themaximum commutation angle is not approached.


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Commutation Overlap Locus

The locus representing the maximum commutation angle is drawn byusing a parametric approach. The commutation angle is held constant ati t s maximum value of 60 degrees and the fir ing angle a is chosen to bethe variable parameter. Under these conditions the mathematical model'svalid range of a is from 0 degrees to 120 degrees (at which point theext inct ion angle 6 is 0 degrees). a is gradually increased through i t sful l valid range and ( 3 . ~ . 2 . 5 ) is used to calculate a value of Idcfor each value of a.

( 3 . ~ . 2 . ~ ) is then used to find the corresponding values of Vdc and( 3 . ~ . 2 . 1 ) provides the real power coordinate on the chart for each valueof a. The reactive power coordinate is obtained by substi tuting ( 3 . ~ . 2 . 3 )into ( 3 . ~ . 2 . 2 ) to give

Q

The other loci shown in figure rest r ic t the converter operationsuch that the overlap angle is always much less than 60 degrees. Thisjus t i f ies the mathematical model used and also allows the commutationoverlap locus to be ignored in the rest of this analysis.

Loci of DC Current Limits

The loci representing the maximum DC valve current and minimum DCvalve holding current are obtained by substituting ( 3 . ~ . 2 . 3 ) into

( 3 . ~ . 2 . 2 ) to give

( 3 . ~ . 3 . 2 )

which is the equation of a circle centred on the origin with a radius ofa.VacKlK3Idc .


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1000 MW

Maximum rectifying DC voltageMinimum firing angle

Minimum holding current,- 1000 MWcurrent

Maximum inverting DC voltageMinimum extinction angle

Maximum transformer current-1000 MVAR

Figure 3.5 Loci of Haywards Converter on Large ScaleThe locus representing the maximum commutation angle has

been omitted.

.t=I.1l


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Locus of Maximum Converter Transformer Current

The maximum RMS converter transformer current I t is equal to the pe runi t MVA rating of the transformer. If the effects of commutation areignored then the converter transformer current can be related to the DCcurrent in pe r unit by (Arrillaga, 1983)


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Locus of Minimum Firing Angle

A parametric approach is used to draw the locus that represents theminimum f ir ing angle amin ' The variable parameter that has been chosenis the DC current Idc which is gradually increased from zero to s l ightlybeyond i t s maximum value. For each value of Idc the rea l power coordinateof the locus is obtained by substi tut ing (3.2.4.4) into (3.2.4.1) to give

p -[K1a.V . cos (a . ) . Id - Kz.X . Id zJac mIn c c c 0.4.3.5)

The reactive power coordinate is obtained from 0.4.3.1) . These twoequations describe a cycloid. As amin decreases the cycloid twists towardthe hor i zon tal .

Locus of Minimum Extinction Angle

The locus representing the minimum extinction angle 0min is obtainedby changing the sign of the power flow in (3.4.3.5) and replacing am inwith 0min to give a real power coordinate of

p K1a.V .cos(o . ) . I d - Kz.Xc.ldczac mIn c 0.4.3.6)

The reactive power coordinate is again provided by (3.4.3.1) hence thislocus is also a cycloid. Similarly to the minimum f ir ing angle locus, theminimum ext inct ion angle locus twis ts towards the hor izonta l as0min decreases.


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118

3.4.4 Loci of Benmore Converter

The const ra int loc i of the Benmore converter must be combined withthe Haywards converter loc i to complete the capabi l i ty char t of theHVDC l ink.

DC Link Power Transfer Mapping

The opera t ing c o n s t r a i n t s of the Benmore c i r cu i t can be easi lyr e f e r r ed to the Haywards end of the l ink by using a DC l ink powert rans fe r mapping. This is a point to point mapping tha t re la tes thecomplex power entering the l ink from the Benmore 16 kV busbar (Pi+jQi) tothe complex power leav ing the 1 ink a t the Haywards 110 kV busbar(P r+ jQ r ) ' The subscr ipt r re fers to the nominal rec t i f i e r a t Benmoreand the subscr ip t i re fers to the nominal inver ter a t Haywards.

The mapping i s obtained by consider ing the rea l power loss in the DCl ine res is tance (R)

P.1 P - IdC 2. R 0.4.4.1)

Combining (3.4.2.2), (3.4.2.3), and (3.4.4.1) provides the point to pointmapping

P.1 Pr 0.4.4.2)

0.4.4.3)

The const ra int locus equations of the Benmore converter can now bef i r s t formulated in the same manner as those of the Haywards conver ter .This produces a se t of Benmore converter loc i which re fe r to the poweren te r in g the l ink a t the Benmore 16 kV busbar. These loc i are iden t ica lto the Haywards converter loc i shown in f igures 3.4 an d 3.5 . The DC l inkpower t ransfer mapping i s then used to transform these loc i , poin t byp o i n t , to produce a new se t of Benmore converter const ra int loc i whichare referred to the Haywards 110 kV busbar .


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-1000 MW

Maximum inverting DC voltageMinimum extinction angle

Minimum holding current 1000 MW

Maximum valve currentrectifying DC voltage

Minimum firing angleMaximum transformer current

-1000 MVAR

Figure 3.6 Loci of Benmore ConverterThese Benrnore converter loci ar e referred to th e Haywards

11 0 kV busbar. Th is a l lows th e se loc i to be directlysuperimposed onto the Haywards loci shown in figure 3.5 .

cT';III;:ICJl1-1)0';3IIIcT......0;:I

01-1)()......

';()I-'CDCJl

;::r:0CD


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3.4.5 Loci of South Island Power Generation

The South Island AC system behind the Benmore converter also placesi t s own res t r id t ions on the power available to the Haywards 110 kVbusbar.

The complex power supplied from th e South Island 220 kV busbar to theBenmore 16 kV busbar is dependent on the voltage angle between thebusbars. This can be analyzed by using the approach described in chapter2 for drawing voltage loci for transmission l ines. In this case thetransmission line is replaced by a 3.3% reactance that represents the 400MVA interconnecting transformer.

I f the voltages a t the South Island 220 kV busbar and the Benmore 16kV busbar are both held at 1 pu then the power delivered to the Benmore16 kV busbar is represented by a circular locus. An arc of this circle isshown in figure 3.7 The power is also res tr icted by the maximum currentrating of the 400 MVA interconnect transformer. This is represented bythe area within another circular locus. The ful l range of power deliveryis therefore represented by th e portion of th e arc that l ies within themaximum current circle .

If the Benmore generator is run as a synchronous compensator then thegenerator reactive power variation of -378 MVAR to 324 MVAR will augmentthe power delivered from the South Island busbar. Taking into account the100 MVAR contribution of th e harmonic f i l ters , the range of reactivepower variation becomes -278 MVAR to 424 MVAR.

This reactive power variation is added to each point on the powerdeli very arc on figure 3.7 to produce the chart shown in figure 3.8 .The shaded region in this char t represents the power that may bedelivered to the Benmore converter from the Benmore 16 busbar.


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1000 MVAR

Maximum 400 MVA t ransformer current

-1000 MW 1000 MWPower delivered via 40 0 MVA transformer

-1000 MVAR

Figure 3.7 Power Delivered to Benmore from South IslandThe arc within the maximum current circ le represents the

power available to the Benmore 16 kV busbar from the SouthIsland 220 kV busbar via the 400 MVA transformer.

1000 MVAR

Maximum 40 0 MVA transformer currenttMaximum generator reactive power

-1000 MW 100 0 MW. . . . . . . . . . . . . . . . . .......... ................ ./))){ ://).Minimum generator reactive power

-1000 MVAR

Figure 3.8 Power to HVDC Link from Benmore 16 kV BusbarThis can be regarded as a capabil i ty chart of the South

Island when the Benmore generator is used as a synchronouscompensator.

51


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3.4.6 Complete Capability Chart

The complete capabil i ty chart of the HVDC link is constructed bycombining a l l of the constraint loci . The wedge shaped capabil i ty chartshown in figure 3.9 is formed by superimposing the Benmore converter locifrom figure 3.6 onto the Haywards converter loci on figure 3.5 Only thec r i t i c a l operat ing constraints that border the operating region areshown, the other constraints have been omitted for clar i ty .

This char t describes the rea l and react ive power that may bedel ivered to the North Island from the l ink, assuming that an infini telystrong 16 kV busbar exists at Benmore. The operating point can be movedaround the operating region by adjusting the converter control angles.Under normal operating conditions, with rea l power being delivered fromBenmore to Haywards, the operating point is located a t the corner of thewedge by the intersection of the maximum valve current locus and theminimum Haywards extinction angle locus.

The reactive power coordinate of this operating point (-316 MVAR)matches the reactive power consumption of the actual Haywards converter.The rea l power coordinate of the operating point (598 MW) differss l igh t ly from the 580 MW supplied from the actual converter. This 3%error is attr ibuted to the simplif ications made during the analysis.

The shaded operating region shows the theoretical abi l i ty of the HVDCl i nk to supply a wide range of complex power to the AC system. Thepresent operating modes of constant current or constant power control donot fully uti l ize the entire operating region. More sophisticated controls t ra teg ies , such as power modulation, are needed to make better use ofthe range of complex power available.


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-1000 MW Minimum holding currents 1000 MW

Maximum rectifying DCaywards)

Minimum extinction angle(Benmore)

-1000 MVAR

Figure 3. 9 Capabi Chart of HVDC Link

Maximum valve currentsMaximum rectifying DC voltage

(Benmore)Minimum extinction angle

(Haywards)

The shaded area represents th e real and reactive powerthat may be supplied from th e Haywards converter transformerto th e Haywards 110 kV busbar.


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I f the Benmore generators are run as synchronous compensators, andth e power generation capabi l i ty of the South Is land is considered, thenthe char t shown in figure 3.8 must be incorporated into the completecapabil i ty chart . This i s achieved by applying the DC l ink power transfermapping to the loci of figure 3.8, and superimposing the transformed locionto the chart in figure 3.9 The region of negative reactive power inf igure 3.8 cannot be transformed because the converters must alwaysconsume reactive power.

The resul tant capabi l i ty chart is shown in figure 3.10 . The shadedregion of t h i s char t shows that the maximum rea l power transfer i sr e s t r i c t ed by the maximum current r a t ing of the ~ O O MVA Benmoreinterconnecting transformer.

3.5 CONCLUS ION

This chapter has described the construction of a capabi l i ty chart toport ray the performance of an HVDC l ink . A simplif ied model of the NewZealand Benmore-Haywards HVDC scheme has been chosen to demonstrate theconstruction technique.

The approach used to construct th e char t involves th e expl ic i tder iva t ion of the equations tha t describe the constra in t loci . Thisapproach is s imilar to tha t used in chapter 2 for the construction ofcharts for a generator and a transmission l ine .

Unlike the generator and transmission l ine loci , the HVDC l ink loc icannot be easi ly drawn by hand and the assistance of computer plot t ingequipment is essent ia l .

The wedge shaped capabi l i ty chart shows the wide range of complexpower tha t is theoret ical ly avai lable from HVDC l inks.

While the approach of exp l i c i t equat ion derivat ions was againsuccessful , it has also emphasised the enormous amount of effor tthat i s associated with the derivation of equations for a moderatelycomplicated power system.


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-1000 MW

Maximum rectifying DC(Haywards)

Minimum extinction angle(Benmore)

Maximum 400 MVA transformer

Minimum holding currents 1000 MW

Maximum rectifying DC voltage(Benmore)

Minimum extinction angle(Haywards)

-1000 MVAR Maximum generator reactive power

Figure 3.10 Capability Chart of Combined HVDC Linkand South Island System

The operating constraints of the South Island systemprevent th e fu l l u t i l iza t ion of the weage shaped operatingregion of th e HVDC link.

U lU l


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56

Chapter 4Charts for Large AC Systems4.1 INTRODUCTION

The concept of a capabi l i ty chart can be extended to describe theperformance of any large general power system, in addition to the smallsystems previously described in chapters 2 and 3. In the context of alarge system, the capabi l i ty chart portrays the real and reactive poweravailable to a load from any par t icular busbar in th e system.

The two dimensional capabi l i ty chart associated with a par t icularbus bar can be regarded as being a s ingle s l ice of an overal l 2nd i ~ e n s i o n a l capabi l i ty chart for the n busbars that make up the generalpower system. This overa l l capabi l i ty char t describes a l l of thecombinations of complex power s imultaneously available from the nbusbars.

The c r i t i ca l operating constrain ts of the system are represented onthe overa l l char t by 2n-l dimensional hypersurfaces that bound theoperating region. Unfortunately we cannot easi ly visual ise regions ofsuch a high dimensional order, and must be content with observing twodimensional s l i ce s of the overa l l char t . Each sl ice describes theal lowable load var ia t ion a t one busbar whilst the loading a t otherbusbars is kept fixed.

The generator, transmission l ine , and HVDC l ink charts that weredescribed in chapters 2 and 3 were a l l constructed by using the techniqueof manipulating the system operating equations into a sui table form fordrawing loc i on the complex power plane. This technique is sat i sfactoryi f the power system is small, because only a few simple operatingequat ions need to be considered. The same technique cannot be used forlarge power systems because the large number of operating equationscannot be so easi ly manipulated.

The d i f f i c u l t y of forming e x p l i c i t ' l o c u s equat ions can becircumvented by using a contour plot t ing approach to gradually trace outeach locus on the complex power plane. A powerflow algorithm is used to


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57

i terat ively solve the operating equations at each point on the contour,without having to resort to an explici t closed form solution.

Contour tracing has been used before by Price (1984) to displayfamilies of contours for a power system. Price 's work concentrated onvoltage contour maps and showed that the contours were often looped andtwisted. Only one family was considered a t a time and the contourswere not combined to portray a capabili ty chart .

This chapter describes a capabili ty charting algorithm for largegeneral power systems that uses a similar contour tracing approach tocope with the expected twists and loops in the contours. This algorithmwas the subject of a paper presented a t the Second InternationalConference on Power System Monitoring and Control (Arnold and de Silva,1986) .

4.2 SYSTEM OPERATING CONSTRAINTS

The performance of a large power system is restr icted by numerousopera ting constraints. The categories of operating constraint that areconsidered by the capability charting algorithm are

1. Minimum/maximum voltage at a load busbar.2. Minimum/maximum reactive power generation from a generator busbar.3. Minimum/maximum real power generation from a slack generator busbar.4. Maximum current in a transmission l ine or transformer.5. Maximum voltage phasor angle difference across a transmission l ine or

transformer.6. Maximum angular deviation of any generator busbar voltage phasor from

the swing busbar voltage phasor.

This se t of operating constraints is adequate for most AC powersystems. In principle, any other constraint that can be numericallyevaluated may also be included in the algorithm i f necessary.


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4.2.1 Voltages and Currents on Transmission Lines

The transmission line capability chart described in chapter 2 assumedtha t the maximum values of voltage and current along the transmissionl ine occurred a t the busbars a t the ends of the l ine . To simplifycalculat ions, the same assumption is used for transmission l ines in largesystems;

Th is assumption becomes less valid as the length of transmissionl ines increasei beyond a quarter wavelength (1500 km a t 50 Hz). Thereforelong l ines should be modelled as a concatenated series of short l ineswith intermediate busbars. The values of voltage and current a t thein termedi ate bllsbars will then represent the corresponding values ofvoltage and current part way along th e l ine.

4.2.2 Generator Capability

The generator capabil i ty chart described in chapter 2 identif ies theoperating constraints that res t r ic t the performance of each generator inthe power system. In chapter 3, the true generator capabil i ty chart wasapproximated by a rectangular chart to simplify the analysis of the AC/DCsystem. The capabil i ty charting algorithm described in this chapter alsouses a rectangular approximation to the true generator chart . The bordersof each rectangle are represented by the maximum and minimum real andreactive power constraints on generators.

4.2.3 Steady State Stabil i ty

The voltage angle constraints imposed across branches and the voltageangle deviation of generator busbars are an attempt to impose a form ofsteady s ta te s tab i l i ty on the system. It is acknowledged that suchconstraints are extremely crude and do not guarantee stabi l i ty , howeverthe voltage angles have the advantage of being easily defined and evaluated.

The actual choice of values for the constraining angles depends onthe power system and must be l e f t to the judgement of the algorithm'suser. A large value of 30 degrees has been used for th e systems describedin this thesis.


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A bet t e r measure of steady sta te s tabi l i ty could be obtained from aneigenvalue analysis of th e system. This approach is briefly examined as aproposed development in chapter 8.

4.2.4 Total Constraint Number

Considering only the categories l is ted, the total number of possibleoperating constraints in a power system is

Constraint number+

+

+

4.3 TEST SYSTEM

4 x number of slack busbars3 x number of generator busbars2 x number of load busbars2 x number of branches (4.2.4.1)

The power system chosen for demonstrating the capabil i ty chartingtechniques is the well known IEEE 14 Busbar Test System. A single linediagram of this power system is shown in figure 4.1 The appendixprovide

generator capability - thesis da silva

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