Impact of EHV/HV Underground Power Cables on Resonant … · Cables on Resonant Grid Behavior ......

Impact of EHV/HV underground power cables on resonantgrid behaviorWu, L.

DOI:10.6100/IR781433

Published: 01/01/2014

Document VersionPublisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differencesbetween the submitted version and the official published version of record. People interested in the research are advised to contact theauthor for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

Citation for published version (APA):Wu, L. (2014). Impact of EHV/HV underground power cables on resonant grid behavior Eindhoven: TechnischeUniversiteit Eindhoven DOI: 10.6100/IR781433

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ?

Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Download date: 30. May. 2018

https://doi.org/10.6100/IR781433

https://research.tue.nl/en/publications/impact-of-ehvhv-underground-power-cables-on-resonant-grid-behavior(91bb5301-9132-4ba4-a83d-9df843451106).html

Impact of EHV/HV Underground PowerCables on Resonant Grid Behavior

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van de

rector magnificus, prof.dr.ir. C.J. van Duijn, voor eencommissie aangewezen door het College voor

Promoties in het openbaar te verdedigenop dinsdag 21 oktober 2014 om 16.00 uur

door

Lei Wu

geboren te Hubei, China

Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van depromotiecommissie is als volgt:

voorzitter: prof.dr.ir. A.C.P.M. Backx1e promotor: prof.dr.ir. E.F. Steenniscopromotor: dr. P.A.A.F. Woutersleden: prof.dr.ir. M.H.J. Bollen (Luleå University of Technology)

prof.dr.Dipl.-Ing. V. Terzija (The University of Manchester)prof.ir. W.L. Klingdr.ir. M. Popov (TUD)

adviseur: dr. G.R. Kuik (TenneT TSO B.V.)

To my parents and my wife Jin

This research was financially supported by TenneT TSO B.V. within the frameworkof the Randstad380 cable research project, Arnhem, the Netherlands.

Printed by Ipskamp drukkers, Enschede.Cover design by Lei Wu.

A catalogue record is available from the Eindhoven University of Technology Library.

ISBN: 978-90-386-3698-6

Copyright c© 2014 Lei Wu, Eindhoven, the NetherlandsAll rights reserved.

Contents

Summary v

1 Introduction 11.1 Background and Challenges . . . . . . . . . . . . . . . . . . . . . . . 11.2 Research Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Research Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Dissertation Contribution . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Components Modeling 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Underground Cable System Modeling . . . . . . . . . . . . . . . . . 10

2.2.1 Single Cable Model . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Multiple Cable Model . . . . . . . . . . . . . . . . . . . . . . 142.2.3 ABCD-Matrix of Cable System . . . . . . . . . . . . . . . . . 152.2.4 Chain Matrix Formulation . . . . . . . . . . . . . . . . . . . . 20

2.3 Overhead Line System Modeling . . . . . . . . . . . . . . . . . . . . 212.3.1 Shunt-Admittance Matrix Y (S/m) . . . . . . . . . . . . . . . 222.3.2 Series-Impedance Matrix Z (Ω/m) . . . . . . . . . . . . . . . 23

2.4 Transformer Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4.1 Transformer Inductances . . . . . . . . . . . . . . . . . . . . . 252.4.2 Transformer Resistances . . . . . . . . . . . . . . . . . . . . . 352.4.3 Transformer Capacitances . . . . . . . . . . . . . . . . . . . . 372.4.4 Transformer Saturable Core . . . . . . . . . . . . . . . . . . . 392.4.5 ABCD-Matrix for Transformer . . . . . . . . . . . . . . . . . 40

2.5 Shunt Reactor Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 432.5.1 Single-Phase Model . . . . . . . . . . . . . . . . . . . . . . . 432.5.2 Three-Phase Model . . . . . . . . . . . . . . . . . . . . . . . . 45

i

ii CONTENTS

2.6 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.6.1 Transmission Line Model . . . . . . . . . . . . . . . . . . . . 462.6.2 Transformer and Shunt Reactor . . . . . . . . . . . . . . . . . 48

3 Frequency Domain Transient Analysis 533.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.1 Application on Simplified Connection . . . . . . . . . . . . . 553.2.2 Effect of Simplifications . . . . . . . . . . . . . . . . . . . . . 57

3.3 Switching Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.3.1 Line Energization Transients . . . . . . . . . . . . . . . . . . 593.3.2 Fault Transients . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.4 Transformer Inrush Current . . . . . . . . . . . . . . . . . . . . . . . 653.4.1 Derivation of Transformer Impedance Matrix . . . . . . . . . 653.4.2 Inrush Current Modeling . . . . . . . . . . . . . . . . . . . . 683.4.3 Remanence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Study on Small-Scale Network 754.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2 Resonances in Cascaded Circuit . . . . . . . . . . . . . . . . . . . . . 76

4.2.1 Categories of transfer functions . . . . . . . . . . . . . . . . . 764.2.2 π-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3 Parameters Affecting Resonant Behavior . . . . . . . . . . . . . . . . 804.3.1 Mixed OHL-Cable Versus Full Cable Connection . . . . . . . 804.3.2 Sensitivity Study . . . . . . . . . . . . . . . . . . . . . . . . . 814.3.3 Impact Dominant Parameters . . . . . . . . . . . . . . . . . . 88

4.4 Transients with External Components . . . . . . . . . . . . . . . . . 894.4.1 Line Energization . . . . . . . . . . . . . . . . . . . . . . . . . 904.4.2 Line with Transformer and Shunt Reactor . . . . . . . . . . . 93

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5 Study on Large-Scale Network 975.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2 Directly Imposed Transients . . . . . . . . . . . . . . . . . . . . . . . 99

5.2.1 Line Energization . . . . . . . . . . . . . . . . . . . . . . . . . 995.2.2 Fault and Fault Clearing . . . . . . . . . . . . . . . . . . . . . 103

5.3 Transients Indirect to Line . . . . . . . . . . . . . . . . . . . . . . . . 1045.3.1 Transformer Energization . . . . . . . . . . . . . . . . . . . . 1045.3.2 Capacitor Bank . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4 Requirements on Level of Model Detail . . . . . . . . . . . . . . . . . 107

6 Conclusion and Recommendation 1156.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

CONTENTS iii

6.2 Recommendation for Future Work . . . . . . . . . . . . . . . . . . . 117

A Configuration of OHL-Cable-OHL 119A.1 Configuration of Cable and OHL . . . . . . . . . . . . . . . . . . . . 119A.2 Geometry of Single Cable and Line . . . . . . . . . . . . . . . . . . . 121A.3 Validation of Model for Cable Screen . . . . . . . . . . . . . . . . . . 125

B Matrix Manipulation 129B.1 ABCD-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

B.1.1 Impedances and Admittances . . . . . . . . . . . . . . . . . . 129B.1.2 Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . 131

B.2 Manipulation of Matrix Elements . . . . . . . . . . . . . . . . . . . . 133B.2.1 Row-Rearrangement of Matrix Equation . . . . . . . . . . . . 133B.2.2 Exchange Elements in Matrix Equation . . . . . . . . . . . . 133B.2.3 Add Boundary Conditions to Matrix Equations . . . . . . . . 135B.2.4 Application Examples . . . . . . . . . . . . . . . . . . . . . . 136B.2.5 Screen Layer Elimination Method . . . . . . . . . . . . . . . 140B.2.6 Solving a Matrix System . . . . . . . . . . . . . . . . . . . . . 141

B.3 Parallel Connection of Multiple Lines . . . . . . . . . . . . . . . . . . 142B.3.1 Multiple Lines for Single Phase . . . . . . . . . . . . . . . . . 143B.3.2 Multiple Lines for Three Phases . . . . . . . . . . . . . . . . 144

Bibliography 147

Acknowledgment 155

Curriculum Vitae 157

List of Publications 159

Summary

Impact of EHV/HV Underground Power Cables on ResonantGrid Behavior

Recently, the Dutch 380 kV transmission network has been extended with new lineswhere underground power cables are integrated in the connection to reinforce itstransmission capacity. Research is initiated to understand the influence of cablesin the fields of: steady state, resonant transient (up to 10 kHz), fast transient (10kHz to 10 MHz), and reliability. The focus of this dissertation is on the resonantbehavior and its consequences, e.g. upon switching, faults, and uses the Randstad380south-ring as study reference.

The Randstad380 south-ring is the first connection in the Netherlands witha considerable amount of underground power cable at the 380 kV level. Thesecables, 12 in parallel, are laid over a distance of 10.8 km between two overhead line(OHL) sections connecting substations Wateringen and Bleiswijk. This is a totalcable length of 130 km. In addition to the aspect of integrating underground cableand OHL, experience needs to be gained on dealing with the increased complexityintroduced by the specific arrangement of the south-ring connection:

• Large number of mutually coupled underground cables, two three-phasecircuits with two cables per phase and two compensating wires.

• Cable connection is subdivided in regularly cross-bonded minor sections whichare subdivided in several segments with different lengths and trench types.

• Design of overhead line tower with two compensating wires and a parallel 150kV OHL at Wateringen side.

• 500 MVA three-phase three-winding three-limb transformer with 100 MVAshunt reactor connected to its tertiary winding.

v

vi Summary

To investigate the complete connection in full detail and efficiently analyzethe sensitivity on all design parameters a frequency domain approach isdeveloped, tailored for transient analysis up to the desired frequency range.PSCAD/EMTDC software is used as comparison after applying simplifications toenable implementation: no mutual coupling between cable circuits, no compensatingwires, all cable sections are identical, no inter-phase coupling in transformer magneticcore.

For the frequency domain analysis, OHLs and cables are represented by thewell-developed frequency-dependent models. Segments are coupled by means ofcascading ABCD-matrices for each segment. Methods to represent the double-layer metal screen of the real cable by an equivalent single-layer screen and forhandling parallel connected cables are proposed. Transformers and shunt reactorsare modeled by the combination of their inductances, parasitic capacitances, andresistances. Based on the unified magnetic equivalent circuit (UMEC) model of thetransformer, formulas for the used three-phase three-winding three-limb transformerare developed only using public-accessible datasheet information. Parasiticcapacitances of the transformer are categorized as winding-to-earth capacitance,inter-winding capacitances, and winding cross-over capacitance. Likewise, due to thelayer-wise configuration, the parasitic capacitances of the shunt reactor contain layer-to-earth capacitances, inter-layer capacitances, and layer cross-over capacitances.The models of transmission lines are validated with PSCAD/EMTDC simulation;the models of transformer and shunt reactors are compared with measurements usingsweep frequency response analysis (SFRA).

The analyzed transients are from switching surge response, switching operationtransients (opening and closing), fault and fault clearing transients, and inrushcurrent of transformer. By means of (inverse) discrete Fourier transform(DFT/IDFT) time domain responses are extracted. Switching and faults aremodeled by introducing virtual voltage and current sources in the frequency model.Results are compared with PSCAD/EMTDC simulation. A common method totreat the nonlinearity of transformer magnetic core by one fixed inductance and aparallel compensating current source is adopted. This invariant inductance allowsthe frequency-domain approach, suitable for linear systems, to be applied here. Theinrush current of a transformer is also compared with measurement including theeffect of remanence.

Resonances in the Randstad380 south-ring are studied by considering theconnection as a stand-alone system. The impact of all design parameters isinvestigated. These parameters are categorized according:

• Cable type: conductor and insulation dimensions;

• Layout: soil properties, trench type, phase sequence;

• Configuration: cable joints, number of parallel cables;

Summary vii

• Topology: cable length, sequence of OHL and cable parts.

In addition to varying parameters, the consequence of a completely cabled circuitbetween Wateringen and Bleiswijk is considered. The effect of components likebusbar short-circuit inductances, transformer (with and without shunt reactor),resonances between the line capacitance and external inductance connected to thecircuit are investigated with emphasis on the impact of integrating undergroundcable. Generally, introducing power cable affects capacitance and inductance of theconnection. For a mixed cable-OHL circuit the total inductance is still dominatedby the OHL section, whereas the total capacitance is increased by the cable. Shiftof resonances is therefore mainly determined by parameters changing the cablecapacitance (length, conductor radius, and insulation thickness). For a completecable connection, trench types and phase order becomes more important parameterbecause of the mutual coupling between the cables. With cables the resonancesdamp slower and higher short-circuit currents are observed.

The impact of underground power cable between Wateringen and Bleiswijkis then investigated as being part of the full Netherlands 380 kV transmissiongrid. PSCAD/EMTDC is used to analyze the complete Dutch 380 kV transmissionsystem including OHLs, cables, transformers, shunt reactors, capacitor banks, etc.(including abovementioned simplifications). Four common transients occurringnearby the cable location are studied:

• line energization and reclosing,

• fault and fault clearing,

• transformer energization, and

• capacitor bank energization.

The responses are evaluated by comparing with standards. For overvoltages,switching impulse withstand voltage (SIWV) is used; for voltage distortion upontransformer energization, the “3 %”-limit from the Dutch grid code is used. Amongthe studied cases, all responses were within the standards; adding cable had strongerimpact for transients occurred direct to the line (line energization and fault/faultclearing) than for transients occurred indirect to the line (transformer energizationand capacitor bank energization).

Analysis of a complete grid requires effort to acquire all detailed data on circuitcomponents. However, many details have hardly effect on the transients especiallywhen they concern components far from the occurrence of an event. A methodfor determining the necessary level of modeling detail to build a sufficient accuratelarge-scale network is presented. Reduction of model complexity for transmissionlines can reduce amount of information needed for pylon specification or trenchconfiguration. The relationship between minimal needed level of transmissionline model and distance to interested area is exemplified for a switching surge at

viii Summary

Wateringen substation. Only in the direct environment of interested area mostaccurate modeling is required. For further areas simpler models suffice. The extentof “direct environment” depends on the objectives: how accurate should the modelingbe and what aspect is most relevant. It is shown that depending on the objective,e.g. prediction of the overvoltage magnitude or the value of the main resonanceleads to different requirements for modeling detail.

Chapter 1

Introduction

1.1 Background and Challenges

An EHV/HV transmission system (or network) is beneficial in transferring electricalenergy over long distance [1, 2]. It consists of many components: overhead lines(OHLs), underground power cables, transformers, switching devices, compensationreactors, capacitor banks, etc.. A safe and stable system is paramount and cannever be constructed if the behavior of the system is not well understood [3, 4].To guarantee sufficient electric power for the densely populated Randstad regionin the Netherlands, it was decided in 2009 (pkb1 — Key Planning Decision) toextend the 380 kV grid with two connections, referred to as the Randstad380 project.The ‘south-ring’ extends from Wateringen (municipality of Westland) to Bleiswijk(municipality of Lansingerland) [5]. The ‘north-ring’ will connect Bleiswijk withBeverwijk. Besides safeguarding electrical energy supply, the connections providesufficient transport capacity for future off-shore wind farms near generation sitesand for the BritNed interconnector.

The south-ring has become operational as from 2013. It contains a mixedOHL and underground power cable connection between substation Wateringen andsubstation Bleiswijk. From substation Wateringen there is a 4.4 km OHL to theOHL-cable transition point. The line continues as underground cable for 10.8 km.From the second transition point there is a 6.8 km OHL to substation Bleiswijk.The double circuit connection is schematically illustrated in Fig. 1.1. The maincharacteristics are:

• OHL1: The overhead line section from Wateringen comprises two circuits(‘a’ and ‘b’). The phase conductors are vertically arranged in the so calledWin-track pylons (Fig. 1.2-a). The phase arrangement is optimized for lowmagnetic field just above ground level. In addition to the 380 kV circuits, there

1In Dutch it is: de planologische kernbeslissing

1

2 Introduction

Figure 1.1: Transmission line configuration between substations Wateringen andBleiswijk

is a double parallel 150 kV circuit mounted on the same pylons (not shown inFig. 1.1).

• Cable: To match the transport capacity of the overhead line, each phase ofboth circuits is connected to two parallel 380 kV cables. This results in 12parallel single-phase cables. Cross-bonding is applied to reduce induced earthscreen currents (Fig. 1.2-b-c). The total single-cable length amounts to about130 km for the south-ring. Two bare copper wires are laid underground alongthe whole cable connection to limit step voltages of human beings above theground.

• OHL2: The overhead line to Bleiswijk consists two 380 kV circuits (Fig. 1.2-d).In addition, two compensating wires (CW, bundles each with 2 conductors)are present. The induced current in these wires from the currents in the phaseconductors compensates the magnetic field from the lines just above groundlevel.

Since there is no experience in the Netherlands on integrating power cables inthe grid on this voltage level, the cable circuit length was limited to about 10km. At the same time it was decided to initiate an extensive research programto investigate possible consequences of underground cables in the 380 kV network.The generated knowledge is to be used for future projects where cables are consideredas an alternative for overhead lines. Four main research areas were defined at firststage:

• Steady-state analysis (fundamental frequency — 50 Hz, e.g. load flow [1,6]).

1.1. Background and Challenges 3

(a) (b)

(c) (d)

Figure 1.2: Photos of Win-track pylon for OHL from substation Wateringen tocable (a), cable (b), cross-bonding joint (c), Win-track pylon for OHL from cable tosubstation Bleiswijk (d)

• Resonant transients analysis (low frequency transients — up to 10 kHz, e.g.line energization [1, 7–9]).

• Fast transients analysis (high frequency transients — 10 kHz to 10 MHz, e.g.lightning strike [1, 7, 8, 10]).

• Reliability analysis [1, 11].

The present dissertation reports on resonant transients analysis. A value of 10 kHzis set as (somewhat arbitrary) limit of the frequency range to be considered for allcomponent models to be applied. The focus is on the Randstad380 south-ring as areference system. Variation in design is studied by comparing the response to thisreference. The complexity of this specific connection involves:

4 Introduction

• The 12 power cables integrated in the connection of this double circuit couplemutually. The cross-bonding of the screens affect system resonances which canoccur in the system.

• The cable minor segments between cross-bonding joints consist of segmentswhich are not identical in length (minimal length is about 100 m) and in thetype of trench in which there are laid. Implementing this level of detail shouldnot lead to excessive computation time (e.g. days instead of minutes).

• Along both the overhead line and the cable there are compensating wires, aswell as a 150 kV double-circuit connection partly in parallel. They all causeadditional coupling.

• The 500 MVA three-phase three-winding three-limb transformers have 100MVA shunt reactors connected to their tertiary windings. A UMEC (UnifiedMagnetic Equivalent Circuit) model only using public-accessible data for thedesired frequency range is needed.

• The Randstad380 south-ring is a part of the Dutch 380 kV network. Resonanttransients can transmit deep in the complete network. Guidelines are neededfor the level of detail the components need to be modeled.

Detailed design parameters of the connection are presented in Appendix A.1.

1.2 Research Goals

Substitution of HVAC overhead lines by underground HVAC power cables increasesprimarily the line capacitance and reduces the line inductance typically with an orderof magnitude. These variations affect inherent resonances in the circuit. Naturalfrequencies of a network can match one of the frequencies in an external sourcecaused by e.g. a switching event. In literature, several case studies for EHV/HVAC underground/submarine connections of several tens of kilometers are available,analyzing the impact of resonances on network behavior (e.g. [12,13]). The researchproject aims to deliver insights how resonances are impacted by the introduction oflong EHV AC underground cables in the electric power system and derive guidelinesfor finding the limits of utilizing cables. The Dutch 380 kV network, more specificallythe Randstad380 south-ring, will be taken as study reference.

Power system transient analysis has been developed over a long period of time.It origins go back to 1854 for analyzing wave propagation on telecommunicationcable [14]. At early times, symbolic analysis approaches like Heaviside OperationalCalculus and Laplace Transformation [15–18] were popular. After 1960s transientanalysis is mostly performed by Electromagnetic Transient Program (EMTP [19]).The basic idea of EMTP-type simulation is using a difference equation to representa differential equation, with defined time-steps. Accuracy and execution speed

1.3. Research Approach 5

depend on their choice. Analysis with PSCAD/EMTDC simulation is applied forthis dissertation. EMTP-type simulation software designed for general applications,may lack flexibility to consider certain level of details for specific systems. For theRandstad380 connection these limitations are the mutually coupling between largenumber of parallel power cables, and between phases of the transformer using theUMEC model. The abovementioned complexity calls for a hands-on analysis besidesusing the well-established approach adopted in PSCAD/EMTDC.

To tackle the main research challenges stated before, the following ‘sub-goals’ (orresearch questions) are derived:

• Establish models of key components in the system: cables, overhead lines,power transformers, and shunt reactors.

• Develop a method that can effectively analyze complex networks and the effectof its parameters for analyzing resonances.

• Evaluate the Randstad380 south-ring connection upon resonant transientovervoltages, considering it as a stand-alone (small-scale) connection.

• Analyze the Randstad380 south-ring as part of the whole Dutch 380 kV (large-scale) network.

Apart from EMTP-type time domain simulation, many approaches in frequencydomain have been introduced [20–29]. All of them only consider applications onsimple connections consisting of either overhead lines or underground cables.

1.3 Research Approach

To realize the goals, a research is conducted from building up the models of networkkey components to reconstructing transients in time-domain, followed by analysisof resonant behavior in small-scale and large-scale networks. Comparisons withmeasurements and PSCAD/EMTDC simulations are applied to verify to models,as well as the approach used to reconstruct time-domain transients. Within thesmall-scale network, sensitivity study on the influence of detailed design parametersin the connection is performed to find the parameters which have larger influence onresonant behavior of the connection. Within the large-scale network, the impactsof cables on transients occurring both directly and indirectly to the transmissionconnection containing cables are investigated.

1.4 Dissertation Outline

The dissertation content consists of two major parts. The first part, coveringChapters 2 and 3, develops the theoretical model and the frequency domain approachapplicable for analysis of resonances. The second part applies this approach together

6 Introduction

with simulations in PSCAD/EMTDC to analyze the cable impact on resonant gridbehavior in Chapters 4 and 5.

Chapter 2 focuses on models of network components and their connection. Fortransmission lines (overhead lines and underground cables), the ABCD-matrices areestablished and compared with PSCAD/EMTDC on system impedance at differentfrequencies and with measurement on positive/zero sequence impedance at 50 Hz.For transformer and shunt reactor, together with their parasitic capacitances, theY-matrices are constructed and compared with measurement at different frequencies.

Chapter 3 describes transient analysis approach in frequency-domain. It focuseson switching surge response, switching operation transients (opening and closing),fault transients, and inrush current of transformer. Comparison is made with resultsobtained by PSCAD/EMTDC simulations. In addition, the approach is applied toreconstruct inrush current of a MV/LV transformer to verify its capability. For allswitching scenarios in this dissertation, the circuit breaker is modeled by a variableresistor.

Chapter 4 shows a sensitivity study of cable parameters for its impact onresonant behavior using the transmission line of OHL-Cable-OHL in the south-ring part of Randstad380 network. This study includes parameters related to thecable design, layout, configuration, and whole connection topology. The resonancesbetween transmission lines and external components like source inductance,transformer (with and without shunt reactor) are investigated.

Chapter 5 studies the cable influence on resonant behavior of large scalenetwork. The concerned transients include line energization and reclosing, fault andfault clearing, transformer energization, and capacitor bank energization. Methodsfor determining the necessary level of modeling detail to build an accurate largescale network are presented.

Chapter 6 summarizes the main conclusions and gives recommendations forfuture work.

1.5 Dissertation Contribution

To be able to address the research questions stated in Section 1.2, this dissertationaddresses innovations for modelling transmission connections. These contributionsin particular apply for integrating underground cable circuits:

• Commercial power system software has limitations in modelling a largenumber of parallel conductors which all are mutually coupled, like 12 parallelcables and two compensating wires (26 conductors in total) in the referencenetwork addressed in this dissertation. A frequency-domain approach has beendeveloped to be used next to PSCAD/EMTDC. It handled 48 mutually coupledconductors in Chapter 4.

1.5. Dissertation Contribution 7

• The implementation of a mixed cable line system consisting of short segments(down to 100 m) with cross-bonding can be dealt with in an efficient way withthe developed method. For the frequency range up to 10 kHz, the developedmethod in frequency domain has shown to be computational efficient.

• Models for cables with equivalent cable structure is extended to represent thereal cable and employed successfully to a real case study. Based on well-developed single-cable model in terms of impedance and admittance per unitlength, a model has been developed to represent a double-metal-layer screenby an equivalent single-layer screen.

• For transformer modelling the UMEC model has been tailored for “three-phase three-winding three-limb” transformers using publicly-accessible data.The parasitic capacitive couplings are included in the model as well. Similarmodeling is applied to the shunt reactor connected to the tertiary winding ofthe transformer.

• To analyze electromagnetic transients like those due to line energizing, fault,and transformer energizing in realistic networks, a frequency-domain approachhas been developed. In addition, this method is used to analyze (nonlinear)inrush current and the effect of remanence.

• The importance of design details of an existing 380 kV connection withlarge amount of AC cables has been investigated. Variations of design areconsidered to predict and understand the effect of cable lengths and differentconfigurations of cable and OHL. The methodology can be applied in the studyof future cable and overhead line connections.

• The impact of cable in an existing transmission network on energizing andfault transients is analyzed with PSCAD/EMTDC, complemented by thedeveloped frequency-domain approach. A method is proposed for determiningthe necessary modelling detail for the study of transients in large-scale network.This method has been applied to an existing transmission network.

Chapter 2

Components Modeling

2.1 Introduction

Transmission connections in electric networks comprise connections above ground,the overhead lines, and connections underground, the underground power cables.They can be modeled as a set of conductors along which electromagnetic energy istransported. Within certain approximations the Telegrapher’s equations apply [30].Section 2.2 uses this approach for underground cables having a coaxial structure.Models from literature describing the impedance and admittance matrices andcoupling of parallel cables, see e.g. [31–34], are summarized. For the series connectionof line segments the ABCD formalism is employed. Special attention is paid tocascading cable segments with cross-bonded earth screens. In Section 2.3 similarapproach is made for modeling the overhead line segments. Both Section 2.2 an 2.3provide a comprehensive step-wise overview of the frequency-dependent transmissionline modeling applicable for the Randstad380 south-ring. Compensation shuntreactors in the Randstad380 south-ring are placed on the tertiary windings ofthe 380/150/50 kV transformers in the substations. Parasitic elements contributeto resonances within the transformer and to resonances between the completetransmission line and the transformer with reactors. Models for the three-phasethree-winding three-limb transformer and for the shunt reactor are developed inSection 2.4 and 2.5, respectively. Especially, the inductive elements in transformersare mainly described in terms of magnetic circuits [35,36]. The complete transformerand reactor models are constructed from the constituting blocks by means of theABCD formalism. Comparison is made with PSCAD simulation on a transmissionline closely resembling the actual connection in the south-ring. As the underlyingmodels are the same, results from frequency domain approach and PSCAD shouldbe equal. Concerning transformer and reactor, measurements were undertaken atSMIT-Transformers B.V. on the (combined) transformer and reactor [37]. Section2.6 reports on the model validation results. All models are intended for frequencies

9

10 Components Modeling

up to 10 kHz.

2.2 Underground Cable System Modeling

To analyze a complete HV connection, it is dissected into segments which canbe mathematically considered as translational symmetric units (longitudinallyhomogeneous). For the Randstad380 cable connection each segment contains 2circuits with three-phases, but the model can be extended to an arbitrarily number ofN three-phase circuits, mutually electromagnetically-coupled. Cross-bonding, use ofdifferent trench geometries, etc., limit the distance over which translation symmetrycan be assumed. Earth screens of cascaded segments are terminated at the ends.The following definitions are adopted:

• Cable segment: one continuous cable with homogeneous longitudinaldistribution, i.e. any fraction of a segment has the same distributed shuntadmittance-matrix Y and series impedance-matrix Z satisfying [28,31,34,42]

− d

dzI = YU, − d

dzU = ZI (2.1)

• Cable minor section: one or more cascaded cable segments, whose coreconductors and screens are directly connected.

• Cable major section: three successive cable minor sections, two neighboringminor sections are connected via a cross-bonding joint. Major sections areconnected via a straight-through joint.

• Cable termination: terminal connections of a whole cable system.

2.2.1 Single Cable Model

The basic frequency-dependent model of a single cable consists of homogeneousconducting and insulating parts, see Fig. 2.1a.

1. core-conductor: with radius rC

2. insulation: with thickness rI − rC

3. screen: with thickness rS − rI

4. outer-sheath: with thickness rO − rS .

HVAC cables usually have a more complicated structure, e.g. they include semi-conducting layers. The methods presented in Appendix A.2 and in references[33, 38–40] simplify the real structure to fit the model of Fig. 2.1a. These methodsare constructed such that the equivalent model has equal per unit length impedances

2.2. Underground Cable System Modeling 11

(a) (b)

Figure 2.1: Cable Configuration: (a) cross-section; (b) impedances of the cablemodel. (Note that the position y is negative for underground positions.)

and admittances. In particular, the semi-conducting layers are replaced by the XLPEinsulation material. To account for the increased insulation thickness, the equivalentpermittivity is increased accordingly. By means of numerical electromagnetic fieldsoftware it was verified that the per unit length capacitance remained the samewithin 0.5 %. The following discussion summarizes the key aspects in establishing Zand Y matrices in (2.1) for this cable model. Details can be found in [31–34], hereonly the key aspects are shown.

Shunt-Admittance Matrix Y (S/m) The Y matrix of a single cable with twoconducting parts (core-conductor C and screen S) can be constructed as

− d

dz

[ICIS

]= Y

[UC

US

](2.2)

Y =

[Y CC Y CS

Y CS Y SS

]=

[y1

−y1−y

1y1+ y

2

]where UC , US , IC , and IS are the voltages and currents of core-conductor andscreen, respectively. The ground is the voltage reference.

y1= g1 + jω

2πεIln(r2/r1)

, y2= g2 + jω

2πεOln(r4/r3)

εI and εO are the permittivity of insulation layer and outer-sheath layer, respectively.g1 and g2 represent the losses in the insulating parts, they are not considered furtherin this study [26,31,39], because of the high quality insulation material being used theconductive current in the insulation layer is negligible comparing to the capacitivecurrent already as from 1 Hz, and the occurrence of resonances is not affected.


Series-Impedance Matrix Z (Ω/m) The series impedance matrix of the singlecable can be represented by seven components (see Fig. 2.1b), [31, 32,34].

• z1, the internal impedance of the conductor

z1 =ρCmC

2πrCcoth(0.733mCrC) +

0.3179ρCπr2C

(2.3)

mC =

√jωμC

ρC

where ρC , μC , and mC are the resistivity, permeability, and inverse penetrationdepth of core-conductor, respectively. Note that the constants in this formulaare adopted from reference [31]. For frequencies up to 10 kHz they give moreaccurate approximation than those in the formula given by [34].

• z2, the impedance due to the time-varying magnetic field in the XLPEinsulation

z2 = jωμI

2πln

rIrC

(2.4)

where μI is the permeability of insulation layer;

• z3, the inner screen internal impedance (= voltage drop on the inner surfaceof the screen per unit current which returns via the inner conductor)

z3 =ρSmS

2πrIcoth [mS(rS − rI)]− ρS

2πrI(rI + rS)(2.5)

mS =

√jωμS

ρS

where ρS , μS , and mS are the resistivity, permeability, and inverse penetrationdepth of screen, respectively;

• z4, the sheath mutual impedance (= voltage drop along the outer (inner)surface of the screen per unit current returns via the inner (outer) conductor,here the outer conductor is earth)

z4 =ρSmS

π(rI + rS)· 1

sinh [mS(rS − rI)]; (2.6)

• z5, the outer screen internal impedance (= voltage drop along the outer surfaceof the screen when current returns via earth)

z5 =ρSmS

2πrScoth [mS (rS − rI)] +

ρS2πrS(rS + rI)

; (2.7)


• z6, the impedance due to the time-varying magnetic field in outer-sheath

z6 = jωμO

2πln

rOrS

(2.8)

where μO is the permeability of outer-sheath;

• z7, the self impedance of the earth-return path

z7 =ρE ·m2

E

2π

[K0(mErO) +

2

4 +m2E · r2O

e2mE ·y]

(2.9)

mE =

√jωμE

ρE

where ρE , μE , and mE are the resistivity, permeability, and inverse penetrationdepth of earth, respectively. K0 is the modified Bessel function of the secondkind [41].

For z1, z3, z4, and z5 approximated expressions are used for the modified Besselfunctions [41] of the first and second kind to reduce evaluation time [31, 42]. Theseven impedance components can be assembled in two equations for the circuit loopsshown in Fig. 2.2.

Figure 2.2: Z-matrix equivalent circuit diagram

• The first loop gives

− d

dzUC = IC(z1 + z2 + z3 − z4) + (IC + IS)(z5 + z6 + z7 − z4)

• The second loop gives

− d

dzUS = ISz4 + (IS + IC)(z5 + z6 + z7 − z4)


The resulting Z-matrix is

− d

dz

[UC

US

]= Z

[ICIS

]

Z =

[ZCC ZCS

ZCS ZSS

](2.10)

=

[z1 + z2 + z3 + z5 + z6 + z7 − 2z4 z5 + z6 + z7 − z4

z5 + z6 + z7 − z4 z5 + z6 + z7

]

2.2.2 Multiple Cable Model

Figure 2.3: Geometric parameters of a multi-cable system

The Z and Y matrices for an n-cable system will be described according to thescheme in Fig. 2.3.

Shunt-Admittance Matrix Y (S/m) The Y-matrix has the form

Y =

⎡⎢⎢⎢⎣

Y11 O · · · OO Y22 · · · O...

.... . .

...O O · · · Ynn

⎤⎥⎥⎥⎦ (2.11)

Components in (2.11) are 2-by-2 matrices with elements relating the core-conductorand screen. Each main-diagonal component represents the self shunt-admittance ofa cable (see (2.2)). Since the earth can be considered as electrostatic shield betweencables [34], every off-diagonal component in the Y-matrix is

O =

[0 00 0

].


Series-Impedance Matrix Z (Ω/m) The series-impedance matrix Z of a multi-cable system can be written as

Z =

⎡⎢⎢⎢⎣

Z11 Z12 · · · Z1n

Z21 Z22 · · · Z2n

......

. . ....

Zn1 Zn2 · · · Znn

⎤⎥⎥⎥⎦ (2.12)

Each main-diagonal component is the self series-impedance of a cable. The off-diagonal components represent the mutual coupling between cables. For example:the current in cable k (either in core-conductor or screen) returning through theearth will generate a voltage drop with respect to cable m [34]. This voltage isapplied to both core-conductor and screen of cable m. The relationship between thecurrent in cable k and voltage in cable m is described by Zkm (k �= m),

Zkm =

[zkm zkmzkm zkm

]

where

zkm =ρE ·m2

E

2π

[K0(mE · dkm) +

2

4 +m2E · (xk − xm)

2 emE ·(yk+ym)

]

dkm =

√(xk − xm)

2+ (yk − ym)

2

2.2.3 ABCD-Matrix of Cable System

Fig. 2.4 shows a cable system with N three-phase groups. Establishing ABCD-matrix and specific matrix manipulation methods needed to combine conductorsand screens of parallel and series connections are summarized in Appendix B.

ABCD-Matrix for Minor Sections The ABCD-matrix of one minor section canbe constructed by multiplying the ABCD-matrices of all segments in the sequenceas they occur within a particular minor section. The ABCD-matrix of each segmentneeds to be obtained individually according to (B.6). Consider a major section withthree consecutive minor sections J , K, and L (Fig. 2.5); their ABCD-matrices canbe denoted as[

Uηp

Iηp

]=

[A BC D

]η

[Uηq

Iηq

], (2.13)

[A BC D

]η

=

[A BC D

]1η

· · ·[

A BC D

]nη

.


Figure 2.4: Configuration of a cable system with N three-phase groups

where η = J, K, or L, nη is the number of segments contained in the correspondingminor section. The voltage “U” and current “I” e.g. for minor section J at terminalp, UJp and IJp, are given by:

UJp =[U1Jp · · · UNJp

]T, IJp =

[I1Jp · · · INJp

]T

UgJp =[UgJApC UgJApS UgJBpC UgJBpS UgJCpC UgJCpS

]IgJp =

[IgJApC IgJApS IgJBpC IgJBpS IgJCpC IgJCpS

]where g = 1 . . . , N represents a three-phase group. The remaining four subscriptsindicate respectively: minor section (J/K/L), phase (A/B/C), terminal of a minorsection (p/q), conductor core or screen (C/S).

Cross-Bonding between Minor Sections The relationship between voltagesand currents at terminal p of minor section J and terminal q of the consecutiveminor section K, including the cross-bonding system, is established. As an example,Fig. 2.6 shows the configuration of cross-bonding joint within three-phase group grelated to Fig. 2.4 [39].


Figure 2.5: An arbitrary minor section η composed of n segments with twoconductive parts (C for core-conductor, S for screen)

Figure 2.6: Configuration of cross-bonding joint

A convenient way to obtain this ABCD-matrix is to reorganize the voltages andcurrents in (2.13) such that the core-conductor and screen related quantities aregrouped together. This is accomplished by row-arrangement matrix R1, which canbe obtained by the method in Appendix B.2.1. For example, the column vector ofterminal p in minor section J becomes

[UJp

IJp

]C,S

= R1

[UJp

IJp

](2.14)


where[UJp IJp

]TC,S

=[UJpC UJpS IJpC IJpS

]Twith the sub-vectors

UJpC =[U1JpC · · · UNJpC

], UJpS =

[U1JpS · · · UNJpS

]IJpC =

[I1JpC · · · INJpC

], IJpS =

[I1JpS · · · INJpS

]

UgJpC =[UgJApC UgJBpC UgJCpC

]UgJpS =

[UgJApS UgJBpS UgJCpS

]IgJpC =

[IgJApC IgJBpC IgJCpC

]IgJpS =

[IgJApS IgJBpS IgJCpS

]The ABCD-matrix of each minor section within a major section η (η = J, K, L)should be transformed according to

Mη = R1

[A BC D

]η

R−11 (2.15)

Having grouped the screen related voltages and currents, the cross-bonding can bemade — a cyclic permutation of the phase sequence of the screen only. Matrix R2

permutates the phase sequence from A-B-C to C-A-B for the screen conductors.The re-ordered quantities are denoted as U′ and I′, e.g. for terminal p of section K:

U′KpS = R2UKpS , I′KpS = R2IKpS ,

where

U′KpS =

[U′

1KpS · · · U′NKpS

]T, I′KpS =

[I′1KpS · · · I′NKpS

]T

U′gKpS =

[UgKCpS UgKApS UgKBpS

]I′gKpC =

[IgKCpS IgKApS IgKBpS

].

The impedances of the cross-bonding connection itself can be included by matrixMCB,⎡

⎢⎢⎣UJqC

UJqS

IJqCIJqS

⎤⎥⎥⎦ = MCB

⎡⎢⎢⎣

UKpC

U′KpS

IKpC

I′KpS

⎤⎥⎥⎦ (2.16)


where

MCB =

⎡⎢⎢⎣

I O O OO I O ZCB

O O I OO O O I

⎤⎥⎥⎦ ,

I is identity matrix and O is null matrix. ZCB is a diagonal matrix of ZCB; for theexample in Fig. 2.6,

ZCB = Zcb + Zcb

The relation for two minor sections which are cross-bonded becomes⎡⎢⎢⎣

UJpC

UJpS

IJpCIJpS

⎤⎥⎥⎦ =

[A BC D

]JK

⎡⎢⎢⎣

UKqC

U′KqS

IKqC

I′KqS

⎤⎥⎥⎦

with [A BC D

]JK

= MJ ·MCB · (R2MKR−12

)

ABCD-Matrix for Major Section To construct the ABCD-matrix of acomplete major section, the third minor section, L, is added to the terminal q ofminor section K. Similar to the operation for minor section K, minor section L alsorequires rearrangement by R1 and R2. Since the phase sequence of the screen layerof minor section L is changed from A-B-C to B-C-A (reverse operation on minorsection K, meaning applying the inverse of R2, indicated by “ ′′ "). The equationfor one major section becomes⎡

⎢⎢⎣UJpC

UJpS

IJpCIJpS

⎤⎥⎥⎦ =

[A BC D

]JKL

⎡⎢⎢⎣

ULqC

U′′LqS

ILqC

I′′LqS

⎤⎥⎥⎦ (2.17)

with [A BC D

]JKL

=

[A BC D

]JK

·MCB · (R−12 MLR2

)According to Fig. 2.7 [39], the screen related rows in the ABCD-matrix equation

can be eliminated with the method in Appendix B.2.5. The resulting ABCD-matrixonly refers to the voltage and currents for 3N core-conductors.


(a)

(b)

Figure 2.7: Screen earthing connection: straight-through joint (a) and cabletermination (b)

The ABCD-matrix for the whole cable system can be established by multiplyingthe ABCD-matrices of all major sections in the same sequence as they are connected.

2.2.4 Chain Matrix Formulation

Constructing the ABCD-matrix of the whole cable connection involves chainmatrix formulation (multiplying matrices), which could potentially cause numericalproblems [43]. Even though a direct implementation of the considered configurationdid not experience numerical instability, a measure to avoid them is implemented.An example is given by the two networks connected as shown in Fig. 2.8, eachdescribed by:

[Ukp

Ikp

]=

[Ak Bk

Ck Dk

] [Ukq

Ikq

]

where k = 1 or 2.

2.3. Overhead Line System Modeling 21

Figure 2.8: Two series connected networks

Suppose the aim is to obtain U2q when U1p is the known source and I2q = 0.Instead of multiplying the two ABCD-matrices, they can be stacked as⎡

⎢⎢⎣U1p

I1pU2p

I2p

⎤⎥⎥⎦ =

⎡⎢⎢⎣

A1 B1 0 0C1 D1 0 00 0 A2 B2

0 0 C2 D2

⎤⎥⎥⎦⎡⎢⎢⎣

U1q

I1qU2q

I2q

⎤⎥⎥⎦

With the method e.g. given in [44], it can be transformed to⎡⎢⎢⎣

I1pI2pI1qI2q

⎤⎥⎥⎦ =

⎡⎢⎢⎣

M11 M12 M13 M14

M21 M22 M23 M24

M31 M32 M33 M34

M41 M42 M43 M44

⎤⎥⎥⎦⎡⎢⎢⎣

U1p

U2p

U1q

U2q

⎤⎥⎥⎦

With known condition U1q = U2p, the following equation holds⎡⎣ I1p

I1q − I2pI2q

⎤⎦ = K

⎡⎣ U1p

U1q

U2q

⎤⎦

K =

⎡⎣ M11 M12 +M13 M14

M21 −M31 (M22 +M23)− (M32 +M33) M24 −M34

M41 M42 +M43 M44

⎤⎦

Since U1p is the source and

I1q − I2p = 0, I2q = 0

U2q can be solved. This method is similar to node-equation analysis as in[19,31], which avoids the numerical problem caused by chain-matrix multiplication.Drawback is the increased size of the matrix to be processed.

2.3 Overhead Line System Modeling

Like underground cables, overhead lines can be modeled by distributed shuntadmittance Y and series impedance Z matrices [31, 33]. Here the key aspects arepresented.


2.3.1 Shunt-Admittance Matrix Y (S/m)

Each element in a Y matrix of aerial conductors is a complex number, whose realpart represents the leakage losses from corona and leakage currents along the surfaceof overhead line insulators. Its imaginary part represents the conductor-to-groundor conductor-to-conductor capacitive couplings.

Among n conductors supported by an overhead line tower, the correspondingpotential matrix P relates the voltages and charges on all conductorsU = −P ·dQ/dz (note that along the overhead line in the direction of z, the amountof charges decreases)

⎡⎢⎢⎢⎣

U1

U2...

Un

⎤⎥⎥⎥⎦ = −

⎡⎢⎢⎢⎣

P 11 P 12 · · · P 1n

P 21 P 22 · · · P 2n...

.... . .

...Pn1 Pn2 · · · Pnn

⎤⎥⎥⎥⎦ · d

dz

⎡⎢⎢⎢⎣

Q1

Q2...

Qn

⎤⎥⎥⎥⎦ (2.18)

where, according to Fig. 2.9,

Figure 2.9: Configuration of two aerial conductors

P kk =1

2πε0· ln 2yk

rk

P km =1

2πε0· ln Dkm

rkm(k �= m) (2.19)

k,m = 1, 2, . . . , n

2.3. Overhead Line System Modeling 23

Eliminating Earth Wires Earth wires are normally applied to protect overheadline phase conductors against lightning strokes. For frequencies up to 10 kHz [31],the voltage along earth wires is usually assumed to be zero. In (2.18) elementsrelated to phase wires (PW) and earth wires (EW) can be grouped:[

UPW

UEW

]= −

[P11 P12

P21 P22

]· d

dz

[QPW

QEW

](2.20)

With UEW ≡ 0, (2.20) reduces to

UPW = −Peq · d

dzQPW (2.21)

Peq =(P11 −P12P

−122 P21

)The admittance matrix of the overhead line becomes

Y = G+ jωP−1eq (2.22)

In addition to the capacitive coupling conductive losses also contribute to theadmittance. The conduction through air to the environment is incorporated bymatrix G, a diagonal matrix with the same dimension as Peq

G =

⎡⎢⎢⎢⎢⎣

g11 0 · · · 0

0 g22...

.... . . 0

0 0 0 gnn

⎤⎥⎥⎥⎥⎦

Here g1 = g2 = · · · = gn = 10−11 (S/m) [31,45].

2.3.2 Series-Impedance Matrix Z (Ω/m)

The n aerial conductors are assumed to be independent in the sense of self-impedance, but are mutually coupled via the earth return path [31,46,47].

Z = Zself + Zmutual

Self Impedance The impedance of conductor k in Fig. 2.9 consists of twocomponents: conductor Zkk,C and earth-return path Zkk,E :

Zkk = Zkk,C + Zkk,E

Zkk,C =ρkmk

2πrkcoth(0.733mkrk) +

0.3179ρkπr2k

(2.23)

Zkk,E = jωμ0

2πln

(2 · (yk + dE)

rk

)


where ρk and μk are respectively the resistivity and permeability of the conductor,mk =

√(jωμk) /ρk is the inverse penetration depth in the conductor, dE =√

ρE/ (jωμE) is the penetration depth in earth with ρE and μE being respectivelythe resistivity and permeability of the earth.

Consequently the self-impedance matrix of n aerial conductors is:

Zself =

⎡⎢⎢⎢⎢⎣

Z11 0 · · · 0

0 Z22

. . ....

.... . .

. . . 00 · · · 0 Znn

⎤⎥⎥⎥⎥⎦ (2.24)

Mutual coupling Every pair of two conductors (e.g. k and m in Fig. 2.9)mutually couples also via earth-return path:

Zmutual =

⎡⎢⎢⎢⎣

0 Z12 · · · Z1n

Z21 0 · · · Z2n...

.... . .

...Zn1 Zn2 · · · 0

⎤⎥⎥⎥⎦ (2.25)

where,

Zkm = jωμ0

2πln

⎛⎝√(yk + ym + 2 · dE)2 + (xk − xm)

2√(xk − xm)2 + (yk − ym)2

⎞⎠ (2.26)

Eliminating Earth Wires Similar to (2.20), the earth wire related rows can beeliminated from the following equation

− d

dz

[UPW

UEW ≡ 0

]=

[Z11 Z12

Z21 Z22

]·[IPW

IEW

](2.27)

so that

− d

dzUPW = Zeq · IPW (2.28)

Zeq =(Z11 − Z12Z

−122 Z21

)

2.4 Transformer Modeling

The modeling of a transformer for transients requires knowledge of all transformerinductances (linear elements, Section 2.4.1), transformer resistances (linear elements,Section 2.4.2), transformer capacitances (linear elements, Section 2.4.3), and

2.4. Transformer Modeling 25

transformer saturable core (nonlinear element, Section 2.4.4), [19, 31, 48–51]. Thesevalues can be obtained by either precise measurements or by extracting them from thetransformer datasheet. Detailed transformer parameters are normally only availablefor the manufacturer. Therefore, this section deals with methods to retrieve allinformation needed from publicly accessible datasheets. In Section 2.4.5 theseparameters are used to construct the ABCD-formulation for the transformer.

2.4.1 Transformer Inductances

There are generally two methods to model inductances in a transformer: ClassicalModel [1, 31, 52, 53] and Unified Magnetic Equivalent Circuit (UMEC) Model[31,35,37,54,55].

2.4.1.1 Classical Model

The classical model is analyzed for a two-winding transformer. The datasheetinformation needed involves:

• Sr, rated apparent power

• fr, rated frequency

• U1,r, rated voltage of winding 1


• uk,r, rated impedance voltage (from short-circuit test)

• im,r, rated magnetizing current (from open-circuit test)

Single-Phase Two-Winding Transformer The generalized configuration of asingle-phase two-winding transformer is shown in Fig. 2.10. Its inductive coupling(in frequency-domain) is given by:[

U1

U2

]= jω ·

[L11 M12

M21 L22

]·[I1I2

](2.29)

where L11 and L22 are the self-inductances of winding 1 an 2, respectively; M12 andM21 are the mutual-inductances of winding 1 and 2, M12 = M21.

Figure 2.10: Single-phase two-winding transformer configuration


Figure 2.11: Single-phase two-winding transformer T-model

The equivalent T-Model is shown in Fig. 2.11, where a is the turns-ratio of thetwo windings. Usually a is not directly available from the datasheet, so it is assumedto be the ratio of the rated voltages of the two windings:

a =U1,r

U2,r. (2.30)

For the short-circuit test, the magnetizing branch aM12 is ignored:

L1 + L2 =uk,r · U2

1,r

Sr

2πfr(2.31)

For the open-circuit test,

U1,r

im,r · Sr

U1,r

= 2πfr · L11, L11 = L1 + aM12 (2.32)

U2,r

im,r · Sr

U2,r

= 2πfr · L22, L22 =L2 + aM12

a2

Combining I2,r = aI1,r, (2.30), and (2.32) results in

L1 = L2 (2.33)

Consequently, the inductive coupling matrix in (2.29) can be constructed:

• L1 and L2 can be calculated by combining (2.33) and (2.31);

• L11 and L22 can be obtained by (2.32);

• M12 can be found by inserting ‘L11 and L1’ or ‘L22 and L2’ into (2.32).


Three-Phase Two-Winding Transformer The model of a three-phase two-winding transformer inductance can be represented by three single-phase two-winding transformer inductance models (2.29):

⎡⎢⎢⎢⎢⎢⎢⎣

U1,A

U2,A

U1,B

U2,B

U1,C

U2,C

⎤⎥⎥⎥⎥⎥⎥⎦= jω ·

⎡⎢⎢⎢⎢⎢⎢⎣

L11,A M12,A 0 0 0 0M21,A L22,A 0 0 0 0

0 0 L11,B M12,B 0 00 0 M21,B L22,B 0 00 0 0 0 L11,C M12,C

0 0 0 0 M21,C L22,C

⎤⎥⎥⎥⎥⎥⎥⎦·

⎡⎢⎢⎢⎢⎢⎢⎣

I1,AI2,AI1,BI2,BI1,CI2,C

⎤⎥⎥⎥⎥⎥⎥⎦

2.4.1.2 UMEC Model

Unlike the classical model, the UMEC model describes the inductive couplingsaccording to the physical configuration of the transformer magnetic core. Althoughdifferent formulas are needed for different core configuration, they are based on thesame concept of modeling.

Single-Phase Two-Winding Two-Limb Transformer Data from datasheetneeded for the UMEC model are







• rA, ratio of areas between yoke and limb

• rL, ratio of lengths between yoke and limb


(a) (b)

Figure 2.12: Single-phase two-winding transformer; a) core configuration; b) UMECModel

Analog to an electric circuit, the magnetic circuit can be constructed. The current,resistance, and electromotive force (EMF) are replaced by magnetic flux (φ),reluctance (R), and the magnetomotive force (MMF). A simple example of single-phase two-winding two-limb transformer is shown in Fig. 2.12, where:

• φ1 and φ2 are the fluxes in the limbs of winding 1 and 2, respectively; φ3 isthe flux in the yoke; φ4 and φ5 are the leakage fluxes in the air from winding1 and 2, respectively.

• The reluctances R1 to R5 correspond to the flux φ1 to φ5.

• N1i1 and N2i2 are the MMF of winding 1 and 2, respectively.

The reluctance can be expressed as [36]

R =length

μ · area,

With this definition, the reluctance in the limb of winding 1 and yoke are correlatedvia

R3 =2 rlengthrarea

· R1 (2.34)

where the yoke and limb are assumed to have the same permeability μ and

rlength =lengthyokelengthlimb

rarea =areayokearealimb


The factor “2” in (2.34) indicates that R3 represents the total reluctance of the topyoke and bottom yoke in Fig. 2.12. It is reasonable to assume that{

R4 = R5 >> R1 = R2, R3

N1 = U1,r, N2 = U2,r

(2.35)

meaning

• reluctances of limbs are equal, reluctances via air are equal and much largerthan that from the magnetic core (limb and yoke),

• with the so-called “normalized core concept” [31,35], the winding turns numberis assumed equal to its rated voltage.

This magnetic circuit can be solved by applying the MMF on either limb. Take limb1 (with winding 1) for example. Its equivalent circuits corresponding to open-circuitand short-circuit test are shown in Fig. 2.13 [54].

(a)

(b)

Figure 2.13: Single-phase two-winding transformer equivalent circuit for two tests;a) open-circuit test; b) short-circuit test


Open-circuit test The open-circuit test provides:

L1,OC =U1,r

2πfr · im,r · Sr

U1,r

(2.36)

N2i2 = 0 when winding 2 is open-ended (Fig. 2.12-b), since i2 = 0. Hence N2i2 isremoved (short-circuited) in Fig. 2.13-a. The resulting equivalent reluctance Req,OC

of the rest of the circuit can be obtained with (2.35) as

Req,OC = 2R1 +R3

= 2

(1 +

rlengthrarea

)· R1 (2.37)

Thus, by combining

N1i1,OC = Req,OC · φ1,OC = 2

(1 +

rlengthrarea

)· R1 · φ1,OC

with (2.36) and the definitions

• flux-linkage in open-circuit test λ1,OC = N1 · φ1,OC

• inductance in open-circuit test L1,OC = λ1,OC/i1,OC,

R1 can be calculated from

R1 =N2

1

L1,OC · 2(1 +

rlengthrarea

)=

2πfr · im,r · Sr

2(1 +

rlength

rarea

) (2.38)

Short-circuit test The short-circuit test provides:

L1,SC =uk,r · U1,r

2πfr · Sr

U1,r

(2.39)

φ2 = 0 when winding 2 is short-circuited (Fig. 2.12-b), since u2 = 0. It is impossibleto establish any flux in the limb (magnetic core) of winding 2, and the flux has togo via air (leakage reluctance). The resulting equivalent reluctance Req,OC of therest of the circuit can be obtained with (2.35) as

Req,SC =R4

2(2.40)


Figure 2.14: Single-phase two-winding transformer UMEC node equationconfiguration

Thus, combining

N1i1,SC = Req,SC · φ1,SC =R4

2· φ1,OC

with (2.39) and the definitions

• flux-linkage in short-circuit test λ1,SC = N1 · φ1,SC,

• inductance in short-circuit test L1,SC = λ1,SC/i1,SC,

R4 can be found by

R4 =2 ·N2

1

L1,SC

=2 · 2πfr · Sr

uk,r(2.41)

Since the values of R1, R3, and R4 are known from (2.38), (2.34), and (2.41),the magnetic circuit in Fig. 2.12-b can be solved analogue to the node-equation inan electric circuit, see Fig. 2.14. With the definition of

• self-inductance of winding k: Lkk = λk/ik = (Nkφk/ik),

• mutual-inductance of windings k and m: Mmk = λmk/ik = (Nmφmk/ik),where φmk is the flux through winding m caused by the flux (φk) generated inwinding k,


all windings should be treated individually. The full inductance matrix in (2.29) forthe transformer of Fig. 2.12 is obtained by successively applying current to a singlewinding.

1. assume only winding 1 carries current (i2 = 0), then the node equation is

⎡⎣(

1R1

+ 1R3

+ 1R4

)−(

1R1

+ 1R4

)−(

1R1

+ 1R4

)2(

1R1

+ 1R4

)⎤⎦[ Θ1

Θ2

]=

1

R1·[

N1i1−N1i1

](2.42)

where Θ is the MMF at each node referring to the reference node 0© in Fig.2.14. Then, Θ can be obtained by

[Θ1

Θ2

]=

[K11

K21

]· i1 (2.43)

where,

[K11

K21

]=

N1

R1·⎡⎣(

1R1

+ 1R3

+ 1R4

)−(

1R1

+ 1R4

)−(

1R1

+ 1R4

)2(

1R1

+ 1R4

)⎤⎦−1

·[

1−1

](2.44)

With the nodal MMF it is possible to obtain the flux in each branch causedby the flux generated by winding 1, and then the self-inductance and mutualinductance related to winding 1.

L11 =N1

i1· φ1 =

N1

i1· (Θ2 −Θ1 +N1i1)

R1

=N1 (K21 −K11 +N1)

R1(2.45)

M21 =N2

i1· φ2 =

N2

i1· −Θ2

R1

= −N2K21

R1(2.46)

2. assume only winding 2 has current (i1 = 0) and repeat the procedure aboveto obtain L22 and M12.


(a)

(b)

Figure 2.15: Three-phase three-winding three-limb transformer: a) coreconfiguration; b) UMEC-model


Three-Phase Three-Winding Three-Limb Transformer A generalizedthree-phase three-winding three-limb transformer core together with its UMEC-model is shown in Fig. 2.15. It is assumed that all limb-related branches havethe same reluctance R1 and all winding-leakage branches have the same reluctanceR2. R3 is the reluctance of the yoke. R4 is the zero-sequence reluctance. It onlyrefers to windings on the same limb, and is assumed to have the same value asR2 [37, 55]. The relevant data from the datasheet are:



• UHV,r, rated voltage of winding HV

• UMV,r, rated voltage of winding MV

• ULV,r, rated voltage of winding LV



• rA, ratio of areas between yoke and limb

• rL, ratio of lengths between yoke and limb

The inductance matrix of the transformer in Fig. 2.15 can be constructed with theUMEC form, see Fig. 2.16. Columns HA to LC indicate the inductive couplingsreferring to the specific winding. For example, the column HA comprises the self-inductance of winding HA and the mutual couplings between winding HA and theother eight windings:

Figure 2.16: Inductance matrix configuration of a three-phase three-winding three-limb transformer


Column HA =[L1 L2 L3

]T

L1 =[LHAHA LMAHA LLAHA

]L2 =

[LHBHA LMBHA LLBHA

]L3 =

[LHCHA LMCHA LLCHA

]Between single-phase two-winding transformer and three-phase three-winding three-limb transformer, one major difference is that the former one has two symmetricallimbs while the latter one has two side limbs (left for phase A and right for phase C)and one middle limb (for phase B). However, information from datasheet applies toall three phases. To build the UMEC model only with available data, the followingprocess can be applied.

1. obtain the values of R1 to R4 based on the equivalent magnetic circuit foropen-circuit test and short-circuit test, see Fig. 2.17, where only the windingof HV level in phase A is connected to a external voltage source.

2. calculate the self-inductances and mutual-inductances referring to eachwinding on the side limbs in Fig. 2.16: columns HA, MA, LA, HC, MC,and LC

3. repeat process 1 but this time only the winding of HV level in phase B isconnected to a external voltage source.

4. calculate the self-inductances and mutual-inductances referring to eachwinding on the middle limb in Fig. 2.16: columns HB, MB, and LB.

2.4.2 Transformer Resistances

Analysis of the transformer losses is based on the following data from datasheet:



• Uk,r, rated voltage of winding k, k = 1, . . . , n, n is the number of windings

• PSC, rated short-circuit power loss

• POC, rated open-circuit power loss


(a)

(b)

Figure 2.17: Three-phase three-winding three-limb transformer UMEC configurationfor two tests: a) Open-circuit test; b) Short-circuit test


There are two major causes of losses in a transformer: Cu-loss resistance (representslosses in the conductor) and Fe-loss resistance (represents losses in the magneticcore). Since the datasheet provides only the total loss of n windings, they areassumed to be evenly distributed to each winding (see Fig. 2.18). Therefore, theresistance of every winding can be evaluated by

Figure 2.18: Transformer resistance equivalent circuit

RCu,k =PSC(

Sr

U1,r

)2· n

·(Uk,r

U1,r

)2

=PSC

n·(Uk,r

Sr

)2

(2.47)

RFe,k =U2k,r · nPOC

(2.48)

2.4.3 Transformer Capacitances

Three kinds of capacitive couplings, depending on transformer construction, can bedistinguished:

• winding cross-over capacitance

• inter-winding capacitance

• winding-earth capacitance (normally the tank and core of a transformer areearthed)

This section describes the construction of the capacitance matrix for a specific500 MVA three-phase three-winding three-limb transformer [37]. Its windingconfiguration is shown in Fig. 2.19. The HV windings are composed of two pairs ofparallel connected windings HV1 and HV2 (Fig. 2.19).


Figure 2.19: Winding configuration of each phase

The parasitic capacitances for single phase are depicted in Fig. 2.20-a. C1 isthe capacitance between LV winding to earth; C2 and C3 are the inter-windingcapacitances of MV-LV and HV-MV respectively; C4, C5, and C6 are the cross-overcapacitances of winding LV, MV, and HV, respectively.

(a)

(b)

Figure 2.20: Three-phase three-winding three-limb transformer capacitanceconfiguration in single-phase view

Particularly, the HV winding in Fig. 2.20-a is equivalent to the four windings(two HV1 and two HV2), meaning C6 can be approximately obtained according to


Fig. 2.20-b, as:

C6 ≈ 2×[(1/2C6,3 + C6,1) (1/2C6,3 + C6,2)

C6,1 + C6,2 + C6,3

]

The terminals p and q of MV winding are equally close to the terminal p of HVwinding, while the q terminal of HV winding is further away. Thus, the totalcapacitive coupling between HV and MV windings is represented by two identicalcapacitances (1/2C3) from both terminals p and q of MV winding only to terminalsp of HV winding, Fig. 2.20-a.

2.4.4 Transformer Saturable Core

The saturable core of a transformer causes a nonlinear relationship between itswinding flux-linkage (λ) and current (i), which can be represented by either a smoothcurve or a piece-wise linear curve depending on datasheet information [31, 56].Although the transformer magnetizing inductance is nonlinear, the nonlinearity iscommonly modeled by a fixed inductance value with a compensation current sourceacross it as shown in Fig 2.21a).

(a) (b) (c)

Figure 2.21: (a) Model of nonlinear transformer core; (b) smooth curve; (c) piece-wise linear curve

Smooth Curve

If the datasheet gives no information for nonlinearity, the smooth curve can be used(Fig. 2.21b). The curve given by (2.49) contains parameters which can be related


to datasheet information of transformer.

i =

√(λ− λ1)

2+ 4 ·D · La + λ− λ1

2 · La− D

λ1(2.49)

D =−b−√

b2 − 4 · a · c2 · a

a =La

λ21

b =La · imr − λmr

λ1

c = imr · (LA · imr − λmr + λ1)

λ1 = K · λmr, 1.15 < K < 1.25

λmr =

√2Ur√32πfr

La = 2 · ukr · Ur

2πfrSr√3Ur

Piece-Wise Linear Curve

If the datasheet gives the information for nonlinearity: usually being a list of npoints (ik, λk), a piece-wise linear curve can be used (Fig. 2.21c). Neighboringpoints are linked with a straight line.

2.4.5 ABCD-Matrix for Transformer

The ABCD-matrix formulation can facilitate modeling of the transformer. Thespecific transformer configuration of Fig. 2.15-a, 2.18 and 2.20 can be groupedinto five blocks, and their ABCD-matrices can be individually obtained using themethods described in Appendix B.1.1.

Figure 2.22: Scheme to combine inductances, resistances, and capacitances of atransformer. The symbol “//” stands for parallel connection


Fig. 2.22 shows the arrangement of these blocks which comprises the followingABCD-matrices (indicated by ‘K’)

• inductance matrix L, (matrix KL);

• Cu-loss and Fe-loss resistances matrix RCu and RFe, (KRCu and KRFe);

• winding cross-over capacitances matrix CWCO, (KCWCO);

• inter-winding and winding-earth capacitances matrix at winding terminals pand q, CIW−WE,p and CIW_WE,q, (KCIW−WE,p

and KCIW−WE,q).

Using ABCD-matrix manipulations for parallel connection described in AppendixB.3 [44], the ABCD-matrix of the transformer can be constructed

K = KCIW−WE,p · [KRFe//KCWCO// (KRCu ·KL)] ·KCIW−WE,q

Matrix K (dimension 18-by-18) correlates the voltages and currents at two terminals(p and q) of the nine windings:[

Up

Ip

]= K

[Uq

Iq

](2.50)

Up =[UHAp UHBp UHCp UMAp UMBp UMCp ULAp ULBp ULCp

]TIp =

[IHAp IHBp IHCp IMAp IMBp IMCp ILAp ILBp ILCp

]TUq =

[UHAq UHBq UHCq UMAq UMBq UMCq ULAq ULBq ULCq

]TIq =

[IHAq IHBq IHCq IMAq IMBq IMCq ILAq ILBq ILCq

]TIn this particular transformer, each LV winding is connected with two series reactors,see Fig. 2.23. The series reactors are represented by lumped resistances andinductances: RS,1 and LS,1, RS,2 and LS,2. CS,1, CS,2, and CS,3 are parasiticcapacitances. The terminals p and q of each LV winding are extended to p′ andq′. With matrix manipulation techniques in Appendix B.2, the series reactors canbe implemented, modifying the LV related entries of (2.50) into[

Up′

Ip′

]= K′

[Uq′

Iq′

](2.51)

Up′ =[UHAp UHBp UHCp UMAp UMBp UMCp ULAp′ ULBp′ ULCp′

]TIp′ =

[IHAp IHBp IHCp IMAp IMBp IMCp ILAp′ ILBp′ ILCp′

]TUq′ =

[UHAq UHBq UHCq UMAq UMBq UMCq ULAq′ ULBq′ ULCq′

]TIq′ =

[IHAq IHBq IHCq IMAq IMBq IMCq ILAq′ ILBq′ ILCq′

]T


(a) (b)

Figure 2.23: Series reactors with LV windings: (a) scheme; (b) single-phase view

Figure 2.24: Transformer vector groups

Vector Groups Vector groups of a transformer describe the connection styleof two winding terminals at the same voltage level. The vector groups of thetransformer in Fig. 2.20 is YNyn0D5, as shown in Fig. 2.24. It gives additionalboundary conditions:

• for Y-connection (HV and MV)

UHAq = UHBq = UHCq = 0

UMAq = UMBq = UMCq = 0

2.5. Shunt Reactor Modeling 43

• for D-connection (LV)

ULAp′′ = ULAp′ = ULCq′ ILAp′′ = ILAp′ − ILCq′

ULBp′′ = ULBp′ = ULAq′ ILBp′′ = ILBp′ − ILAq′

ULCp′′ = ULCp′ = ULBq′ ILCp′′ = ILCp′ − ILBq′

By inserting them into (2.51) with the methods in Appendix B.2 impedance-matrixZ can be derived

Up′′ = ZIp′′ (2.52)

Up′′ =[UHAp UHBp UHCp UMAp UMBp UMCp ULAp′′ ULBp′′ ULCp′′

]TIp′′ =

[IHAp IHBp IHCp IMAp IMBp IMCp ILAp′′ ILBp′′ ILCp′′

]TThe resulting matrix (9-by-9) only corresponds to the 9 terminals of the transformerthat are used to connect with other components like transmission lines, shunt reactor,etc..

2.5 Shunt Reactor Modeling

A shunt reactor can be represented by its inductances, resistances, and capacitancesdepending on its particular configuration. This section describes how to derivetheir values from the datasheet of a specific 100 MVA three-phase air-cored shuntreactor [37]. Each winding has 11 layers of 35 turns as shown in Fig. 2.25.

2.5.1 Single-Phase Model

The winding of each phase is constructed by connecting all layers in series (totally385 turns per phase). Fig. 2.25b represents the single-phase model with inductances,resistances, and capacitances modeled as lumped components:

• the resistance and inductance of layer k (k = 1, . . . , 11) can be calculated fromthe per-phase values which are public accessible

Rk =Rphase

11, Lk =

Lphase

11

• the capacitance between layer 1 and earthed screen C1_E and all the inter-layer capacitances Ck_k−1 (k = 2, . . . , 11) are divided by 2 and placed at twoterminals of each layer.

• the cross-over capacitance of each layer Ck (k = 1, . . . , 11)


(a)

(b)

Figure 2.25: (a) Layers configuration of each winding; (b) Equivalent circuit diagram

These components can be grouped into blocks and arranged as indicated in Fig.2.26:

• layer inductances matrix: L;

• layer resistances matrix: R;

• layer cross-over capacitances matrix: CLCO;

• inter-layer and layer-earth capacitances matrix at winding terminals p and q:CIL−LE,p and CIL−LE,q.

The ABCD-matrix of one phase, which can be established in a similar way as forthe transformer (Section 2.4.5), correlates the voltages and currents at terminal p of

2.6. Model Validation 45

layer 11 and terminal q of layer 1:[Up

Ip

]=

[A BC D

] [Uq

Iq

](2.53)

Figure 2.26: ABCD-matrix of one phase

2.5.2 Three-Phase Model

The voltages and currents of three-phase windings can be assembled by stackingthree single-phase equations (2.53) as:⎡

⎢⎢⎢⎢⎢⎢⎣

UAp

IAp

UBp

ICp

UCp

ICp

⎤⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎣

A BC D

A BC D

A BC D

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

UAq

IAq

UBq

ICq

UCq

ICq

⎤⎥⎥⎥⎥⎥⎥⎦

The boundary conditions given by the Y-connection style of the three phase windingscan be inserted into the equation above producing the impedance matrix⎡

⎣ UAp

UBp

UCp

⎤⎦ =

⎡⎣ ZAA ZAB ZAC

ZBA ZBB ZBC

ZCA ZCB ZCC

⎤⎦⎡⎣ IAp

IBp

ICp

⎤⎦ (2.54)

The procedure is the same for the Y-connection of transformer HV or MV windings(see Section 2.4.5).

2.6 Model Validation

The models introduced in the sections above are intended for resonant transientanalysis (up to 10 kHz). To validate accurate modeling at any frequency within thatrange, a comparison is made with sweep frequency response analysis (SFRA). For the


considered configuration OHL1-Cable-OHL2 (in Fig. 1.1), a comparison between themodel-based calculation and PSCAD/EMTDC simulation is made (Section 2.6.1).Besides, reference [58] reports on the comparison between model predictions andmeasured capacitance, DC resistance, positive-sequence impedance (at 50 Hz only),and zero-sequence impedance (at 50 Hz only) of the connection [59, 60]. Matchingresults were obtained. Transformer and shunt reactor models were validated bySFRA measurements [61] on these components up to the MHz range (Section 2.6.2).

2.6.1 Transmission Line Model

The frequency spectrum can be conveniently obtained from the ABCD-matrixformalism in frequency domain. An inductive load (arbitrary chosen value fordemonstration) is connected to enforce a resonance with line capacitance at relativelylow frequency, see Fig. 2.27. The frequency scan function at terminal p of thetransmission line is obtained from the ABCD-matrix to calculate the impedancematrix Zp which connects Up and Ip.

Transmission Line Load

p q A

B C

Ip

Up

Figure 2.27: Transmission system configuration for frequency scan analysis

The ABCD-matrix of the complete transmission line, denoted as[Up

Ip

]=

[A BC D

] [Uq

Iq

]contains the “U" and “I" quantities at p and q terminals, e.g.

Up =[UAp UBp UCp

]T, Ip =

[IAp IBp ICp

]TApplying the load connection (Uq = ZLoadIq), where

ZLoad =

⎡⎣ ZLoad,A 0 0

0 ZLoad,B 00 0 ZLoad,C

⎤⎦ (2.55)

expressions for the components of Zp, satisfying Up = ZpIp, are obtained:

Zp = (A · ZLoad +B) · (C · ZLoad +D)−1

Zp =

⎡⎣ ZAA ZAB ZAC

ZBA ZBB ZBC

ZCA ZCB ZCC

⎤⎦ . (2.56)


The comparison is illustrated on the connection between Wateringen andBleiswijk (Fig. 1.1 and Appendix A.1) with slight simplifications. It is assumedthat 1) mutual coupling of both cable circuits can be neglected; 2) no compensatingwires; 3) variations in length and trench types of cable minor sections are ignored(all 900 m long; 4) trench type O1 with cable depth as -1.4 m in Table A.12).The first two simplification are needed for the present version of PSCAD/EMTDC;the others reduce the simulation time from tens of hours to tens of minutes. Toestablish a scenario for comparison, terminals of the 150 kV circuits in “OHL1” areopen-ended, and each phase of the transmission line is loaded with LLoad = 0.26 H.The resulting diagonal element ZAA is plotted in Fig. 2.28, where frequency is variedin 10-base logarithmic scale with 2000 points. Note that the high magnitude of ZAA

in the figure is caused by the purely inductive load, the only resistive part is in theOHLs and cables. The deviations from the results obtained by PSCAD/EMDTCsimulation are indicated by “Error" defined as

Error =Zp,ReCon − Zp,PSCAD

Zp,PSCAD

0

5x 10

4

|ZAA|(Ω

)

0

500

1000

|ZAA|(Ω

)

101

102

103

104

−100

0

100

Frequency (Hz)

�Z

AA(◦)

Figure 2.28: SFRA of the OHL-Cable-OHL obtained by model-based calculation:(top) full view of magnitudes; (middle) zoomed view of magnitudes; (bottom) phaseangle.

2-1.4 m is adopted, since the original values of O1 trench violate the numerical stability inPSCAD/EMTDC


Table 2.1: Transformer rated electrical properties

Power (MVA) 500/500/167Voltage (kV) 380/150/50

No-load current (%) 0.032No-load loss (kW) 130

rL 0.5rA 1.0

Table 2.2: Transformer leakage reactances

Related Windings Values (p.u.)437 kV - 150 kV 0.213150 kV - 50 kV 0.177437 kV - 50 kV 0.414

From Fig. 2.29, where the “Error” is plotted on logarithmic scale, it can beclearly seen that the deviations are negligible.

101

102

103

104

−10

−5

0

Frequency (Hz)

log10|Error|

Figure 2.29: Comparison of the SFRA results from the proposed method andPSCAD/EMTDC

2.6.2 Transformer and Shunt Reactor

The relevant data of the transformer and shunt reactor under measurement are givenin Tables 2.1 to 2.7, and Fig. 2.30.


Table 2.3: Transformer parasitic capacitances (nF)

C1 8.6 C5 0.054C2 5.0 C6,1 0.37C3 5.0 C6,2 4.3C4 0.017 C6,3 2.4

Table 2.4: Transformer series reactor

RS,1 = RS,2 (mΩ) 4.5LS,1 = LS,2 (mH) 1.9

CS,1 20.0CS,2 25.8CS,3 3.1

Table 2.5: Transformer saturable core behavior1. Since the voltages and flux-linkage λ are all in p.u., their values are identical. The curve is plotted in Fig.2.30.

u = λ (p.u.) 0.90 1.00 1.10 1.15i (p.u.) 2.58× 10−4 3.20× 10−4 8.19× 10−4 1.72× 10−3

u = λ (p.u.) 1.20 1.30 1.80 2.10i (p.u.) 3.01× 10−3 9.22× 10−2 1.40 2.55

1 In PSCAD/EMTDC, the slope of the nonlinear curve has to becontinuously decreasing as current increases. In the real datasheet,three points (u,i) violating this rule are ignored: (1.4, 9.21 × 10−2),(1.5, 0.73), and (2.4, 3.45).

Table 2.6: Shunt reactor electrical rated properties

Power (MVAr) 100Voltage (kV) 50

Resistance (mΩ, at 15 ◦C) 3.4


0 1 2 30

1

2

3

i (p.u.)

λ(p.u.)

(a)

0 1 2 3 4

x 10−3

0

0.5

1

1.5

i (p.u.)

λ(p.u.)

(b)

Figure 2.30: (a) Saturation Curve of the transformer core; (b) zoomed view.

Table 2.7: Shunt reactor parasitic capacitances (nF)

C1_E 8.6 C1 0.047C2_1 31.1 C2 0.049C3_2 32.6 C3 0.051C4_3 34.1 C4 0.054C5_4 35.6 C5 0.056C6_5 37.1 C6 0.058C7_6 38.6 C7 0.061C8_7 40.1 C8 0.063C9_8 41.6 C9 0.065C10_9 43.0 C10 0.068C11_10 44.5 C11 0.070

The measurement scheme is depicted in Fig. 2.31, where the tap-changer inthe transformer is at maximum position (437 kV). Note that the transformer andthe shunt reactor used for the SFRA shown in Fig. 2.31-c are not the units fromwhich Fig. 2.31-a and Fig. 2.31-b are obtained, but they are built with the samespecification and share the same datasheet. The investigated response is the absolute


value of the transfer function H(f) between the input and output voltage values asa function of frequency:

|H(f)| =∣∣∣∣Uout(f)

U in(f)

∣∣∣∣The results based on measurements and calculation (with all parameters assumed tobe frequency invariant) are shown in Fig. 2.31, where the transfer function is plottedin dB (20 log10 |H|): Figure 2.31-d shows that, for the transformer, the simulatedcurve based on the model with parasitic capacitances matches the shape of themeasured curve, especially at frequencies below 10 kHz. Similar conclusion can bedrawn for the model of shunt reactor according to Figure 2.31-e, but the deviationbecomes larger for frequencies above 10-100 kHz. Extension of the model used inthis paper is required for higher frequencies; e.g. the capacitive coupling within theair-core limbed windings can be changed from layer-to-layer level to turn-to-turnlevel. Figure 2.31-f confirms both models of transformer and shunt reactor, sincethe simulation has similar shape even though the measurement was performed onanother transformer and shunt reactor (but with the same specifications). In allcases, the resonances in the simulated results are less damped than the measuredones. In reality at higher frequences, the resistances in transformer and shunt reactorshould be larger due to skin effect, but the parameters of transformer and shuntreactor used in the models are only based on 50 Hz. When the parasitic capacitancesare ignored, models of both transformer and shunt reactor are purely inductive andtheir simulated curves continue going downwards as frequency increases. They startto lose accuracy as from around 3 kHz. Note that the measured values above about500 kHz must be ignored since the accuracy of the measuring device is affected bythe precise connection details to the transformer.


(a)

(b)

(c)

101

102

103

104

105

106

−100

−80

−60

−40

−20

0

Frequency (Hz)20log10|H

|(dB)

ReConMeasuredReCon NoC

(d)

101

102

103

104

105

106

−100

−80

−60

−40

−20

0

Frequency (Hz)

20log10|H

|(dB)

ReConMeasuredReCon NoC

(e)

101

102

103

104

105

106

−100

−80

−60

−40

−20

0

20

Frequency (Hz)

20log10|H

|(dB)

ReConMeasured

(f)

Figure 2.31: Comparison of measurement and simulation: a) on transformer only;b) on shunt reactor only; c) on transformer with shunt reactor connected to its LVwinding. Uin (100 V) is the source voltage, Uout the measured voltage. R0 (50 Ω)is the terminal resistor of the measuring device; d) - e) are the results correspondingto a) - c) respectively. Lines of ‘ReCon’ and ‘ReCon_NoC’ are the reconstructedresponse with and without parasitic capacitances, respectively.

Chapter 3

Frequency Domain TransientAnalysis

3.1 Introduction

Power system transients are associated with events both in voltage and current ofa short duration. Although ‘short’ is not precisely defined, it is usually referred toanything shorter than one power cycle [62]. Switching transients are caused by asudden change of voltage or current by e.g. the operation of a circuit breaker (CB) ora fault in the system. Switching on transformers is associated with inrush currentshaving a strong non-linear behavior.

This chapter discusses modeling approaches in frequency domain and resultsare compared with PSCAD/EMTDC simulation. In Section 3.2 the transformationbetween time and frequency domain as applied throughout this chapter is brieflysummarized. Section 3.3 discusses the implementation of sudden changes in currentor voltage to analyze line energization and fault situations. Section 3.4 deals withmodeling methods for transformer energization.

3.2 Concept

The general concept of transient analysis in frequency domain is depicted in Fig.3.1.

53

54 Frequency Domain Transient Analysis

Figure 3.1: General concept of frequency domain transient analysis

The time domain response sout(t) caused by disturbance sin(t) can be obtainedfrom the frequency domain transfer function. Both sout(t) and sin(t) are representedby N discrete components, with time step Δt in a time-frame from 0 to Tw. Applyingthe Discrete Fourier Transformation (DFT, [63,64]) to sin(t) gives

SDFT (m) =

N∑n=1

sin(n)e−j2π(m−1)

(n−1)N , m = 1, . . . , N

where each value of index n corresponds to time t = Δt · (n − 1). The sequenceSDFT(m) contains frequencies from 0 Hz (at index m = 1) to Nyquist frequencyfnyq = 1/(2Δt) (at index �m = N/2 + 1�). Since sin(t) is real-valued, componentsin SDFT with frequencies higher than fnyq can be considered as negative frequenciesin reverse order. Therefore, the frequency spectrum of sin(t) can be obtained by

Sin (k) =

{1N · SDFT (k) , k = 12N · SDFT (k) , 2 ≤ k ≤ N

2 + 1

The frequency step is Δf = 1/Tw, and each index k corresponds to frequencyf = Δf · (k − 1). The frequency spectrum of sout(t) can be obtained by applyingfor each frequency in Sin:

Sout(k) = H(k) · Sin(k) (3.1)

where the H(k) is the transfer function, which can be obtained from the ABCD-matrix as described in Chapter 2. Likewise, constructing real-valued response sout(t)can be realised via Inverse Discrete Fourier Transformation (IDFT) by

sout (n) =1

N

N∑m=1

SIDFT (m) ej2π(m−1)(n−1)

N (3.2)

3.2. Concept 55

where (∗ means complex conjugate)

SIDFT (m) =

⎧⎪⎨⎪⎩N · Sout (k) , m = 1, k = 1N2 · Sout (k) , 2 ≤ m ≤ N

2 + 1, m = kN2 · S∗

out (k) ,N2 + 1 < m ≤ N, k = N −m+ 2

3.2.1 Application on Simplified Connection

The concept of frequency domain transient analysis is illustrated on a slightlysimplified circuit with respect to the Wateringen and Bleiswijk connection. Thesimplifications are the same as taken in Section 2.6.1. Fig. 3.2 illustrates a voltage-related disturbance uS(t) at terminal p of this transmission line with its terminal qopen-ended [65].

(a)

0 0.005 0.01 0.015 0.02

0

0.2

0.4

0.6

0.8

1

Time (s)

uAp(t)(p.u.)

0 0.5 1

x 10−3

0

0.5

1

(b)

Figure 3.2: (a) Equivalent circuit diagram for analysis of response upon switchingsurge; (b) 250/2500μs surge signal

The ABCD-matrix of this transmission line can be written as⎡⎢⎢⎢⎢⎢⎢⎣

UAp

UBp

UCp

IAp

IBp

ICp

⎤⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎣

K11 · · · K16...

. . ....

K61 · · · K66

⎤⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

UAq

UBq

UCq

IAq

IBq

ICq

⎤⎥⎥⎥⎥⎥⎥⎦

(3.3)

By applying the boundary conditions

IBp = ICp = IAq = IBq = ICq = 0

in (3.3), the transfer function between output and input voltages can be established

UAq(fk) = H(fk) · UAp(fk), k = 1, . . . , n.


The time domain response uAq(t) can be calculated from its frequency domaincomponents UAq(fk).

For a standard switching surge (Fig. 3.2b), the resulting response is shown inFig. 3.3. The input signal in frequency domain (Fig. 3.3a) peaks at 50 Hz becauseof the chosen time frame3. The transfer function in Fig. 3.3b has its first peak at800 Hz. In the response two main peaks appear (50 Hz and 800 Hz, Fig. 3.3c).The time domain response in Fig. 3.3d plotted over the time frame of 20 ms clearlyshows the 800 Hz resonance. It also shows that the frequency domain analysis hasgood agreement with PSCAD/EMTDC simulation.

0

0.1

0.2

|UAp|(p.u.)

101

102

103

104

−200

−100

0

Frequency (Hz)

�U

Ap(◦)

(a)

0

5

10

15|H

|

101

102

103

104

−200

0

200

Frequency (Hz)

�H

(◦)

(b)

0

0.1

0.2

|UAq|(p.u.)

101

102

103

104

−200

0

200

Frequency (Hz)

�U

Aq(◦)

(c)

0 0.005 0.01 0.015 0.02−0.5

0

0.5

1

1.5

2

Time (s)

uAq(t)(p.u.)

ReConPSCAD

(d)

Figure 3.3: Switching surge response: a) input in frequency domain; b) transferfunction H; c) output in frequency domain; d) reconstructed output in time domain(‘ReCon’), compared with PSCAD/EMTDC simulation

For the analysis, time steps of Δt = 50 μs were taken resulting in a maximum3The time frame is from 0 to 0.02 s; the frequency step in frequency domain is 50 Hz

3.2. Concept 57

frequency in the spectrum as (2Δt)−1 = 10 kHz. The needed 201 discrete frequencies(resolution 50 Hz) can be efficiently handled without accumulating numerical errors.In PSCAD/EMTDC the simulation time-step should be a fraction of the time neededfor a traveling wave to propagate along the minor section [31]. Steps of 0.5 μsare advised for about 1 km length, resulting in 40000 timings to reach 20 ms.The frequency-domain approach, implemented in MATLAB code, runs thereforeappreciably faster (over 50 times4) than PSCAD/EMTDC for this example.

3.2.2 Effect of Simplifications

In order to make use of PSCAD/EMTDC to model the Randstad380 connections,model simplifications had to be introduced as described in Section 2.6.1. Thesesimplifications mainly concerned mutual coupling between cable circuits, omittingcompensating wires and ignoring difference between cable segments which makeup minor sections between cross-bonding. The frequency domain approach canbe applied to the connection including all details: 12 mutually coupled cables, 28different segments, 2 compensating wires. Fig. 3.4 shows the comparison betweenthe response upon the switching surge of Fig. 3.3 for the simplified and completemodel. Both simulations are conducted with the frequency domain approach. Forthe complete connection, the proposed approach only requires computational timeclose to one minute, about a factor of 10 more than the simplified connection. Itshows that ignoring these details results in minor deviations for the specific case ofthe Randstad380 south-ring. This means that the simplified model can be employed,e.g. when the cable is simulated as part of the complete Dutch 380 kV grid usingPSCAD/EMTDC (Chapter 5).

0 0.005 0.01 0.015 0.02−0.5

0

0.5

1

1.5

2

Time (s)

uAq(t)(p.u.)

SimplifiedComplete

(a)

0.008 0.009 0.01 0.011 0.012−0.2

0

0.2

0.4

0.6

Time (s)

uAq(t)(p.u.)

SimplifiedComplete

(b)

Figure 3.4: Influence of model simplifications needed for implementation inPSCAD/EMTDC: overview (a) and zoomed-view (b).

4The model-based calculation and PSCAD/EMTDC simulation are both performed on a samecomputer. Its processor is Intel(R) Core(TM)2 Quad CPU Q9500 @ 2.83GHz with 4 GB RAM.Operating system is Windows 7, 32-bit. The model-based calculation needed less than 5 seconds.


3.3 Switching Operation

Switching operation, e.g. to open or close a circuit breaker (CB), is a mainsource of transient overvoltages in power systems. In time-domain simulation likePSCAD/EMTDC, a switch is represented by a resistor RCB, whose value changesbetween very high (RCB_Open = 109 Ω) to represent open state and very low(RCB_Close = 10−3 Ω) for closed state [19, 31]. This representation can also beused in frequency domain analysis. An equivalent voltage or current source, seeFig 3.5, is applied to emulate switching by injecting a “compensating” voltage andcurrent at the position of the switch.

Figure 3.5: Equivalent circuit diagrams of switching operation: closing (left) andopening (right)

Note that RCB_Close and RCB_Open are added just in order to follow the samerepresentation as used in time-domain simulation. RCB_Close can be set to zero,and the branch for RCB_Open can be removed. Across an opened CB a voltageuCB_Open (t) arises. The closing action can be emulated by adding a virtual voltagesource uCB_Close (t) whose waveform is opposite to uCB_Open (t) after the switchingmoment t0 [8, 66].

uCB_Close (t) =

{0 t < t0

−uCB_Open (t) t ≥ t0

The total voltage across the switch after closing becomes zero. Similarly, the opening

3.3. Switching Operation 59

action of CB can be emulated by an equivalent current source iCB_Open (t).

iCB_Open (t) =

{0 t < t0

−iCB_Close (t) t ≥ t0

Note that the CB opening moment represents the extinction of the current, not themoment of physical separation of the CB contacts.

Section 3.3.1 elaborates on line energization transients; Section 3.3.2 analyzestransients from single-phase short-circuit fault.

3.3.1 Line Energization Transients

A basic circuit diagram for line energization in a three-phase (A, B, C) system (seeFig. 3.6) consists of voltage source (uS), source inductance (LS), circuit breaker(CB), and transmission line.

Figure 3.6: Equivalent circuit diagram of line energization in a three-phase system

Generally, the CB contacts of three phases can be closed simultaneously orindividually. This subsection will focus on single-phase closure (phase A), sincethe case of three-phase concurrent closure is just an extension of single-phase case.Individual three-phase closures can be considered as three individual instances ofsingle-phase closures.

The ABCD-matrix of the system before closure can be written as⎡⎢⎢⎢⎢⎢⎢⎣

ULS,p,A

ULS,p,B

ULS,p,C

ILS,p,A

ILS,p,B

ILS,p,C

⎤⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎣

K11 · · · K16...

. . ....

K61 · · · K66

⎤⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

ULine,q,A

ULine,q,B

ULine,q,C

ILine,q,AILine,q,BILine,q,C

⎤⎥⎥⎥⎥⎥⎥⎦

where K = KLS · KCB_Open · KLine, being the product of ABCD-matrices of thesource inductance, circuit breaker, and the transmission line. The subscripts inthe voltages and currents indicate “Device”, “Terminal”, and “Phase”. The ABCD-matrices KLS

and KCB_Open can be constructed by the methods described in


Appendix B.1.1; specifically, KCB_Open is based on the resistance matrix

RCB_Open =

⎡⎣ RCB_Open 0 0

0 RCB_Open 00 0 RCB_Open

⎤⎦

Assume the closure takes place under steady state condition (at power frequencyfp = 50 Hz). With the known conditions:⎡

⎣ ULS,p,A

ULS,p,B

ULS,p,C

⎤⎦50Hz

=

⎡⎣ US,A

US,B

US,C

⎤⎦50Hz

,

⎡⎣ ILine,q,A

ILine,q,BILine,q,C

⎤⎦50Hz

=

⎡⎣ 0

00

⎤⎦

the steady-state three-phase voltages at terminal q of the transmission line, whichare

[ULine,q,A ULine,q,B ULine,q,C

]50Hz

, can be obtained (see Appendix B.2.6).The time-domain waveform of voltage in phase A caused by voltage source uS is

uLine_S,q,A(t) =∣∣ULine,q,A

∣∣ · sin (2πfp · t+ ∠ULine,q,A

).

Therefore, the voltages at two terminals of CB can be obtained by⎡⎢⎢⎢⎢⎢⎢⎣

UCB,p,A

UCB,p,B

UCB,p,C

ICB,p,A

ICB,p,B

ICB,p,C

⎤⎥⎥⎥⎥⎥⎥⎦50Hz

= KCB ·KLine ·

⎡⎢⎢⎢⎢⎢⎢⎣

ULine,q,A

ULine,q,B

ULine,q,C

ILine,q,A (= 0)ILine,q,B (= 0)ILine,q,C (= 0)

⎤⎥⎥⎥⎥⎥⎥⎦50Hz

and ⎡⎢⎢⎢⎢⎢⎢⎣

UCB,q,A

UCB,q,B

UCB,q,C

ICB,q,A

ICB,q,B

ICB,q,C

⎤⎥⎥⎥⎥⎥⎥⎦50Hz

= KLine ·

⎡⎢⎢⎢⎢⎢⎢⎣

ULine,q,A

ULine,q,B

ULine,q,C

ILine,q,A (= 0)ILine,q,B (= 0)ILine,q,C (= 0)

⎤⎥⎥⎥⎥⎥⎥⎦50Hz

These equations provide the time-domain voltage over the circuit breaker in phaseA: (

UCB,A

)50Hz

=(UCB,p,A

)50Hz

− (UCB,q,A

)50Hz

(uCB,A (t))50Hz =∣∣UCB,A

∣∣ · sin (2πfp · t+ ∠UCB,A

)The equivalent voltage source for closure is

uCB_Close,A (t) =

{0 0 ≤ t < t0

− (uCB,A (t))50Hz t0 ≤ t ≤ Tw


where 0 to Tw is the time span for analysis (time frame) and t0 is the closing moment.To calculate the output voltage contributions uLine_CB,A (t) caused by

uCB_Close,A (t), the three-phase source uS is removed. This is similar to the responseupon switching surge described in Section 3.2 and can be analyzed in the same way.The ABCD-matrix of the CB is

RCB_Close =

⎡⎣ RCB_Close 0 0

0 RCB_Open 00 0 RCB_Open

⎤⎦

The total time-domain waveform of the voltage at terminal q of the transmissionline in phase A caused by the closure operation is the sum of the responses causedby uS (t) and uCB_Close,A (t):

uLine,q,A (t) = uLine_S,q,A (t) + uLine_Close,q,A (t)

The response for the transmission line from substation Wateringen to substationBleiswijk, where the simplifications mentioned in Section 3.2 are involved, is shown inFig. 3.7. Note that the phase-to-ground source voltage is 1 p.u., and the followingtiming settings are chosen t0 = 0.205 s, Tw = 0.26 s, Δt = 50μs. The circuitbreaker closes when its voltage has reached the maximum value which determinesthe waveform of the equivalent voltage source for closure, Fig. 3.7a. The mainresonances in UCB_Close,A (around 50 Hz, Fig. 3.7b) and in H (around 430 Hz, Fig.3.7c) determine the main oscillations in the output voltage component ULine_Close,A

based only on uCB_Close,A, Fig. 3.7d. The total output voltage uLine,q,A withinthe full time frame (0.26 s) is shown in Fig. 3.7e, the nonzero part before 0.1 scan be considered as a residual response to previous periods in the process of DFT(Section 3.2). When transforming to frequency domain, the waveform is essentiallyconsidered as a periodic signal with period Tw = 0.26 s. To guarantee correct resultsany system oscillation after excitation should have damped within this time frame.Fig. 3.7e shows the calculated response over full time frame, repeated twice. Theoscillation of about 430 Hz from t = 0 which repeats after t = 0.26 s needs to vanishwithin the chosen time frame. The part describing the response is represented by thepart between 0.20 s and 0.26 s (end of time frame). The 50 Hz component originatesfrom the equivalent source after closure. It is not part of a natural oscillation ofthe system and therefore is absent after Tw. The need of sufficient damping can beconsidered as a drawback of the proposed frequency domain approach. In powersystem analysis sufficient damping is usually present. Nevertheless, it perfectlyrepresents the target response: the transients upon switching (see the comparisonwith PSCAD/EMTDC simulation in Fig. 3.7f).


0 0.05 0.1 0.15 0.2 0.25

−1

−0.5

0

0.5

1

1.5

Time (s)

Voltage(p.u.)

uCB Close,A (t) uCB,A (t)

(a)

0

0.1

0.2

|U|(p.u.)

101

102

103

104

−200

0

200

Frequency (Hz)�U

(◦)

(b)

0

50

|H|

101

102

103

104

−200

0

200

Frequency (Hz)

�H

(◦)

(c)

0

0.1

0.2

|U|(p.u.)

101

102

103

104

−200

0

200

Frequency (Hz)

�U

(◦)

(d)

0 0.1 0.2 0.3 0.4 0.5−2

0

2

uLine,q,A

(p.u.)

0.2 0.22 0.24 0.26 0.28−2

0

2

Time (s)

uLine,q,A

(p.u.)

(e)

0.2 0.22 0.24 0.26−2

−1

0

1

2

Time (s)

Voltage(p.u.)

ReConPSCAD

(f)

Figure 3.7: Single phase closing transients: a) equivalent voltage source uCB_Close,A

for closure of CB; b) uCB_Close,A in frequency domain; c) Transfer function H infrequency domain; d) component of voltage at terminal q of transmission line inphase A caused only by uCB_Close,A, uLine_Close,A; e) (top) two times repeated fulltime frame; (bottom) zoomed part: just before switching till just past the boundarybetween both time frames; f) Comparison with PSCAD/EMTDC simulation


The time to frequency transformation is implemented as DFT. This circumventsoscillations from the Gibbs phenomenon [63, 67]. Fig 3.8-top shows the responsenear the switching moment. If the vertical scale is expanded with a factor about 200(Fig 3.8-bottom), around the switching moment, a small oscillation appears withnegligible amplitude compared to the actual response. With applying the DFT,concerns on aliasing can arise. For the oscillations upon switching in power systemsthe observed oscillations are within the considered range up to 10 kHz, but generallycare must be taken. With the adopted settings, problems referred to aliasing arenot experienced in the analysed scenarios.

0.202 0.204 0.206 0.208−2

0

2

uLine,q,A

(p.u.)

0.202 0.203 0.204 0.205 0.206−0.01

0

0.01

Time (s)

uLine,q,A

(p.u.)

Figure 3.8: Gibbs phenomenon in the response

Further improvement can be achieved by implementing Fast Fourier Transform(FFT) [68]. In the present application the number of frequencies is relatively lowand the applied DFT is only a minor fraction of the total computation time. Thiscan be different when opting for higher frequencies.

3.3.2 Fault Transients

A circuit fault can basically be considered as the closure of a switch at the faultlocation. For instance, under steady state operation, a single phase-to-ground faultwith an arbitrary fault impedance occurs at location G in phase A (see Fig. 3.9)at time tFault [66]. The circuit breaker contacts start to separate at tSeparate. Afterthe currents in the circuit breaker reach zero the circuit breaker opens. The aim isto calculate the three-phase time-domain voltages and currents at the fault location(uG(t), iG(t)) and at the circuit breaker (uEF (t), iEF (t)).


Table 3.1: Parameters for fault transient analysis

US 1 p.u. (50 Hz) RLoad 55 Ω (50 % loaded) tFault 0.105 sLS 10 mH RFault 5 Ω [69] tSeparate 0.11 s

(a)

(b)

Figure 3.9: Equivalent circuit diagram of single-phase-to-ground fault

Besides the three-phase voltage source uS(t) a number of virtual sources areneeded. The fault is emulated by uFault,A(t) which opposes the normal voltage atlocation G after tFault. The opening of CB can be implemented by three-phasecurrent sources iOpen(t) acting as from zero current instances. All responses shouldbe calculated individually and added to obtain the total responses uG(t), iG(t),uEF (t), and iEF (t). Determination of a later acting virtual source has to includethe influences of all previous sources. Note that after the CB detects the fault, itwill open and the current will be interrupted at its natural zero-crossing moment.Since the currents reach zero point at different timings, the three-phase componentsin iOpen(t) have to be applied sequentially. For example, which phase opens firstdepends on the time-domain currents via CB caused by uS(t) and uFault,A(t).

Consider the transmission line from substation Wateringen to substationBleiswijk, with the simplifications mentioned in Section 2.6.1. The study scenariodefined in Table 3.1 results in the responses shown in Fig. 3.10. Transients startfrom tFault = 0.105 s. The current in phase A changes abruptly, and it distorts thevoltages and currents in phase B and C. After the currents of the circuit breakerare interrupted, the current in each phase extinguishes accordingly, immediatelyfollowed by establishing the voltages over the circuit breaker. Comparison withPSCAD/EMTDC simulation confirms the results, see Fig. 3.11 for all voltages andcurrents depicted in Fig. 3.10.

3.4. Transformer Inrush Current 65

−1

0

1

uG(t)(p.u.)

A B C

0.09 0.1 0.11 0.12 0.13−2

0

2

4

Time (s)

i G(t)(p.u.)

(a)

−1

0

1

uEF(t)(p.u.)

A B C

0.09 0.1 0.11 0.12 0.13−2

0

2

4

Time (s)

i EF(t)(p.u.)

(b)

Figure 3.10: Fault transients from single-phase fault. A, B, and C represent thethree phases.

3.4 Transformer Inrush Current

The proposed frequency domain approach is based on a sequence of matrixmanipulations representing the connection of different transformer ports. The idea isfirst to obtain the transformer impedance matrix according to the connection schemeof the transformer windings, and then calculate the inrush current via frequencydomain analysis. This section demonstrates the approach on a three-phase 1.8 MVAMV/LV transformer, see Fig. 3.12, for which measurements were made at DNV-GL.

3.4.1 Derivation of Transformer Impedance Matrix

The original matrix of a three-phase two-winding transformer is (see Section 2.4):

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

I1,p...

I6,pI1,q...

I6,q

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎣

Y 1,1 · · · Y 1,12...

. . ....

Y 12,1 · · · Y 12,12

⎤⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

U1,p...

U6,p

U1,q...

U6,q

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3.4)

where subscripts “1 to 6” refer to MV phases A B C followed by LV phases A B C.The following conditions apply


−1

0

1

uG,A

(t)(p.u.)

ReCon PSCAD

0.09 0.1 0.11 0.12 0.13

0

2

4

Time (s)

i G,A

(t)(p.u.)

(a)

−1

0

1

uEF,A

(t)(p.u.)

ReCon PSCAD

0.09 0.1 0.11 0.12 0.13

0

2

4

Time (s)i E

F,A

(t)(p.u.)

(b)

−1

0

1

uG,B

(t)(p.u.)

ReCon PSCAD

0.09 0.1 0.11 0.12 0.13−1

0

1

Time (s)

i G,B

(t)(p.u.)

(c)

−1

0

1

uEF,B

(t)(p.u.)

ReCon PSCAD

0.09 0.1 0.11 0.12 0.13−1

0

1

Time (s)

i EF,B

(t)(p.u.)

(d)

−1

0

1

uG,C

(t)(p.u.)

ReCon PSCAD

0.09 0.1 0.11 0.12 0.13−1

0

1

Time (s)

i G,C

(t)(p.u.)

(e)

−1

0

1

uEF,C

(t)(p.u.)

ReCon PSCAD

0.09 0.1 0.11 0.12 0.13−1

0

1

Time (s)

i EF,C

(t)(p.u.)

(f)

Figure 3.11: Comparison of fault transient results in Fig. 3.10 withPSCAD/EMTDC simulation. For phases A (top), B (middle), and C (bottom); left:voltage and current to ground; right: voltage and current related to CB.


(a)

(b)

(c) (d)

Figure 3.12: (a) Network configuration for inrush current experiment; (b) photoof MV transformer composed of three single-phase transformers; (c) delta connectionscheme of MV windings; (d) star connection of LV windings


• delta connection on MV level (Fig. 3.12c);

U1,ex,p=̂U1,p = U3,q I1,ex,p=̂I1,p − I3,q

U2,ex,p=̂U2,p = U1,q I2,ex,p=̂I2,p − I1,q

U3,ex,p=̂U3,p = U2,q I3,ex,p=̂I3,p − I2,p

• star connection on LV level (Fig. 3.12d);

UN =̂ U4,q = U5,q = U6,q

IN =̂ I4,q + I5,q + I6,q

• no-load state and neutral terminal earthed,

I4,p = I5,p = I6,p = 0, UN = 0;

With these conditions and by exchanging elements between the vectors, (3.4) canbe written as (see Appendix B.2.2),⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

I1,ex,pI2,ex,pI3,ex,pU4,p

U5,p

U6,p

IN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎣

K1,1 · · · K1,7...

. . ....

K7,1 · · · K7,7

⎤⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

U1,ex,p

U2,ex,p

U3,ex,p

I4,p = 0I5,p = 0I6,p = 0UN = 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3.5)

From (3.5), the relationship between the line voltage and line current at thetransformer MV terminals is established⎡

⎣ U1,ex,p

U2,ex,p

U3,ex,p

⎤⎦ = ZTransformer

⎡⎣ I1,ex,p

I2,ex,pI3,ex,p

⎤⎦ (3.6)

ZTransformer =

⎡⎢⎣

K1,1 · · · K1,3...

. . ....

K3,1 · · · K3,3

⎤⎥⎦−1

3.4.2 Inrush Current ModelingFor the closing action during steady-state operation of the (linear) system inFig. 3.13, the total time-domain transient voltages of the transformer terminal(uA,ex,p (t), uB,ex,p (t), and uC,ex,p (t)) can be obtained using the method describedin Section 3.3.


Table 3.2: Transformer data: parameters are given as three-phase values “3ph”

Sr,3ph 1.80 MVA PCu,3ph 6.50 kWU1r,3ph 12.50 kV PFe,3ph 2.97 kWU2r,3ph 0.50 kV US 10/

√3 kV

fr 50 Hz RS 50.50 mΩukr,3ph 8.72 % LS 1.60 mHimr,3ph 0.23 % tclosing 0.01 s

Figure 3.13: Circuit diagram of Fig. 3.12a: transformer is represented by itsequivalent impedance matrix

The flux linkage of transformer three-phase MV windings can be obtained byintegrating the winding voltages as:

λAB (t) =

∫ t

0

[uA,ex,p (t)− uB,ex,p (t)] dt

λBC (t) =

∫ t

0

[uB,ex,p (t)− uC,ex,p (t)] dt (3.7)

λCA (t) =

∫ t

0

[uC,ex,p (t)− uA,ex,p (t)] dt

With the flux-current curve from Fig. 2.21 (Section 2.4.4), the compensationcurrent source value can be established. In principle, the total inrush currentsof a transformer is the sum of three components: 1) the currents (linear) causedby the busbar voltage sources with RCB_Open; 2) the currents (linear) caused bythe equivalent virtual voltage sources for the closure of the circuit breaker withRCB_Close; 3) the currents (nonlinear) of compensation current sources. The currentsfrom compensation are much larger than the other contributions, which will beignored. The drawback of this approach is that it causes low damping of theinrush current for the compensation current source. The construction from timedomain transient voltages at the transformer terminal to inrush current involves onlyintegrals over time (3.7) and the curve of Fig. 2.21. The resulting inrush currentis accurate only for the first few cycles after energization, but these periods are ofconcern [70–72]. Results shown in Fig. 3.14 are based on the input data in Table3.2. The “per unit (p.u.)” quantities refer to the rated values of the transformer


u(1 p.u.) =

√2√3· 12.5× 103 (V) ≈ 10.2 (kV)

λ(1 p.u.) =

√2√3· 12.5× 103

2π · 50 (Vs) ≈ 32.5 (Vs)

i(1 p.u.) =

√2 · 1.8× 106√3 · 12.5× 103

(A) ≈ 117.6 (A)

The voltages at the MV terminals start to build up after the closure of the CB (Fig.3.14a). The flux linkage in each winding is established (Fig. 3.14b) by integratingthe winding voltage. The inrush currents drawn from the grid are shown in Fig.3.14c. The frequency spectra of voltages and currents are shown in Fig. 3.15. Thepeak at 50 Hz in the currents arises from the voltage excitation. The resonances athigher frequencies in the currents correlate with the distortion as the flux linkageexceeds 1 p.u., see Fig. 3.15 and Fig. 3.14b. A time-domain comparison is madewith simulation in PSCAD/EMTDC. Fig. 3.16 shows that within the first fewperiods, the frequency domain approach produces similar results. However, bothsimulation techniques deviate significantly from the measured inrush currents. Amore detailed comparison between the simulation techniques is shown in Fig. 3.17.Gradually, deviation increases due to the underestimated damping. The measurefor “damping” used in the bottom plots of Fig. 3.17 is obtained by normalizing thepeak value in every power cycle on the peak value in the first cycle. The voltagesfrom both methods remain closely equal (Fig. 3.17-a), but inrush current deviatesabout 7 % after 25 power cycles.

0 0.02 0.04 0.06−1

−0.5

0

0.5

1

Time (s)

u(t)(p.u.)

A B C

(a)

0 0.02 0.04 0.06

−2

0

2

Time (s)

λ(t)(p.u.)

AB BC CA

(b)

0 0.02 0.04 0.06−2

−1

0

1

2

Time (s)

i(t)

(p.u.)

A B C

(c)

Figure 3.14: Transients upon energization: voltages at MV terminals (a); flux linkageof MV windings (b); currents at MV terminals (c). A/B/C represent the three phases.


0

0.5

1

|UA|(p.u.)

101

102

103

104

−200

0

200

Frequency (Hz)

�U

A(◦)

(a)

0

0.5

1

|IA|(p.u.)

101

102

103

104

−200

0

200

Frequency (Hz)

�IA(◦)

(b)

0

0.5

1

|UB|(p.u.)

101

102

103

104

−200

0

200

Frequency (Hz)

�U

B(◦)

(c)

0

0.2

0.4|I

B|(p.u.)

101

102

103

104

−200

0

200

Frequency (Hz)

�IB(◦)

(d)

0

0.5

1

|UC|(p.u.)

101

102

103

104

−200

0

200

Frequency (Hz)

�U

C(◦)

(e)

0

0.2

0.4

|IC|(p.u.)

101

102

103

104

−200

0

200

Frequency (Hz)

�IC(◦)

(f)

Figure 3.15: Three-phase voltages and currents at the MV terminals in frequencydomain.


−1.5

−1

−0.5

0

i A(t)(p.u.)

ReCon PSCAD Measurement

0

0.5

1

i B(t)(p.u.)

0 0.02 0.04 0.06−0.5

0

0.5

1

Time (s)

i C(t)(p.u.)

(a)

−1

0

1

uA(t)(p.u.)

ReCon PSCAD Measurement

−1

0

1

uB(t)(p.u.)

0 0.02 0.04 0.06−1

0

1

Time (s)

uC(t)(p.u.)

(b)

Figure 3.16: Comparison between the reconstructed results via frequencydomain (‘ReCon’), PSCAD/EMTDC simulation (‘PSCAD’) and measurement(‘Measurement’): (a) three-phase voltages at MV terminals; (b) three-phase currentsat MV terminals

−1

0

1

uC(t)(p.u.)

ReCon PSCAD

0 0.1 0.2 0.3 0.4 0.595

100

105

Time (s)

damping(%

)

(a)

0

0.5

1

1.5

i C(t)(p.u.)

ReCon PSCAD

0 0.1 0.2 0.3 0.4 0.590

95

100

Time (s)

damping(%

)

(b)

Figure 3.17: Damping of voltage (a) and current (b) in phase C

3.4.3 Remanence

The calculated inrush currents largely deviate from measurement (Fig. 3.16a). Thiscan be attributed to remanence, which is not considered in the simulations. Other


reasons for the deviation, e.g. “the input data can be different from real measurementsetup”, are unlikely, since the resulting voltages match perfectly in Fig. 3.16b.

In field measurement, the transformer core usually has remanence which canlargely influence the waveform of inrush currents. Therefore, it is important forinrush current calculation to consider remanence. However, the remanence of thetransformer is unknown when conducting measurements. By assuming initial vlauesin (3.8) the effect of remanance is investigated. The values are chosen to have amatch between simulated and the measured inrush currents.

λAB (0) = 40.0% · λmr

λBC (0) = −8.0% · λmr (3.8)λCA (0) = −48.9% · λmr

λ (t) =

∫ t

0

u (t) dt+ λ (0) (3.9)

With these values, the resulting currents resemble measurements well (Fig. 3.18).The derived remanence values seem reasonable since normally they should be within±60%, according to [72].

−1

0

1

i A(t)(p.u.)

ReCon Measurement

−1

0

1

i B(t)(p.u.)

0 0.02 0.04 0.06−0.5

0

0.5

Time (s)

i C(t)(p.u.)

Figure 3.18: Comparison between frequency domain approach with assumed three-phase remanence (‘ReCon’) and measurement (‘Measurement’)

Chapter 4

Study on Small-Scale Network

4.1 Introduction

The Dutch 380 kV network extends over the whole country and has severalconnections abroad. Substations in this network can be considered as topologicalnodes in a branched grid. To systematically investigate the impact on resonances byintroducing underground power cable in one of these branches, the Randstad380south-ring — the connection Wateringen-Bleiswijk (small-scale network) — isconsidered as a “stand-alone” connection in this chapter. Resonances arise basicallyfrom “L-C” interactions. Obviously, a power cable contribute to the capacitance,and the compensation reactors to the inductance, but many other factors affect theactual resonances occurring in a single branch. To mention a few:

• The cable is embedded between overhead line sections.

• Double circuit with mutually coupled cables.

• Cables are regularly cross-bonded.

• Effect of load at the receiving ends.

Understanding observed resonances is not always straightforward. The frequencydomain approach can reveal resonances, irrespective whether the resonance is excitedby an arbitrary chosen input signal. In Section 4.2 a main categorization of transferfunctions is made. The effect of cascading different section is studied in which theπ-model with lumped parameters [30] is adopted to represent transmission lines.This simplified description helps to identify the origin of resonances both for anopen ended and a terminated cascaded connection. Section 4.3 explores designparameters that affect resonances in the Randstd380 south-ring. Focus is on thesensitivity upon these parameters. Section 4.4 studies transient behavior whenexternal components are added. Line energization upon connection to a busbar

75

76 Study on Small-Scale Network

and connection of transformer with compensation reactors are looked into. Section4.5 summarizes the main influencing factors which should be taken care of whendesigning a connection involving underground power cable segments.

4.2 Resonances in Cascaded Circuit

This section aims to identify resonances occurring in two cascaded segments.To facilitate the interpretation, only single-phase circuits are considered and atransmission line is described by the π-model.

4.2.1 Categories of transfer functions

The four components in the ABCD matrix for the general transmission line of Fig.4.1 correspond to four transfer functions in H describing input-output relations [65]:

[Up

Ip

]=

[A BC D

] [Uq

Iq

](4.1)

U q =HUUUp, HUU=A−1(relatesUq andUp, when Iq = 0

)Iq = HIUUp, HIU =B−1

(relates Iq andUp, whenUq = 0

)U q = HUIIp, HUI =C−1

(relatesUq and Ip, when Iq = 0

)Iq = HIIIp, HII =D−1

(relates Iq and Ip, whenUq = 0

)(4.2)

HUU and HIU refer to transients at terminal q caused by voltage-related disturbance(e.g. closing action of a circuit breaker) at terminal p. For example, HUU can be usedto calculate the switching surge response under no-load condition (no impedance isconnected to terminal q) [58] and HIU is associated with fault current. Likewise,HUI and HII refer to transients at terminal q caused by current-related disturbances(e.g. opening action of a circuit breaker) at terminal p.

Figure 4.1: Single phase transmission line

Frequency-dependent models of cables and overhead lines result in rathercomplex relations which cannot be expressed analytically. A simplified formulationin terms of the π-model allows to identify resonances in an analytical expression forthe transfer functions in lower frequency range.

4.2. Resonances in Cascaded Circuit 77

4.2.2 π-Model

If a transmission line is short compared to the wavelength associated with themaximum frequency to be studied, the π-model can be adopted [30]. For the π-model (Fig. 4.2), the ABCD-matrix becomes[

Up

Ip

]=

[1 + Zα · Y α Zα

(2 + Y α · Zα)Y α 1 + Y α · Zα

] [Uq

Iq

](4.3)

Zα = Z ·D, Y α =1

2Y ·D

where Z and Y are the series impedance and shunt admittance (per unit length) ofthe transmission line, respectively. D is its length.

Figure 4.2: Single phase transmission line in π-model

The four transfer functions in (4.2) of the transmission line using the π-modelare

HUU =1

1 + Zα · Y α

HIU =1

Zα

HUI =1

(2 + Y α · Zα)Y α

HII =1

1 + Y α · Zα

(4.4)

Resonances can occur when the absolute value the denominator has a minimum.Regarding a homogeneous transmission line, its resonant frequency (f0,UU) in HUU isdetermined by the inductance in series impedance Zα (j2πf ·Lα) and the capacitancein shunt admittance Y α (j2πf · Cα).

f0,UU =1

2π· 1√

Lα · Cα

(4.5)

HIU is determined by series impedance only, and has no resonance. Resonance inHUI is determined by

f0,UI =1√2π

· 1√Lα · Cα

(4.6)

Both HIU and HUI have a dominant 1/f behavior at low frequency. For HIU it isrelated to the dominant inductance in Zα and for HUI to the dominant capacitancein Y α. The resonant behaviors of HII and HUU are equal.


Two Cascaded Lines

When the transmission line is composed of two cascaded segments (as shown in Fig.4.3), its total transfer function using the π-model is

HUU =[(1 + Zα,1 · Y α,1

) (1 + Zα,2 · Y α,2

)+Zα,1 · Y α,2

(2 + Zα,2 · Y α,2

)]−1

HIU =[Zα,2

(1 + Zα,1 · Y α,1

)+ Zα,1

(1 + Zα,2 · Y α,2

)]−1

HUI =[Y α,1

(2 + Zα,1 · Y α,1

) (1 + Zα,2 · Y α,2

)+Y α,2

(1 + Zα,1 · Y α,1

) (2 + Zα,2 · Y α,2

)]−1

HII =[(1 + Zα,1 · Y α,1

) (1 + Zα,2 · Y α,2

)+Zα,2 · Y α,1

(2 + Zα,1 · Y α,1

)]−1

(4.7)

Figure 4.3: Two cascaded transmission lines

With cascaded lines, the four transfer functions are different from the case of ahomogeneous transmission line. Each transfer function can have several resonances,determined by the interaction of the inductances and capacitances of differentsections. The term

Zα,1 · Y α,2

(2 + Zα,2 · Y α,2

)in HUU indicates that the sequence of the cascaded lines (cascade pattern) matters.HII and HUU no longer have exactly the same resonant behavior.

The Randstad380 south-ring connection consists of a number of cascadedsections. Most obvious is the OHL-Cable-OHL topology. Further, the cable ismade up of totally 28 segments — 4 major sections each with 3 cross-bonded minorsections which are not identical (see Appendix A.1). This results in a spectrum ofresonances of which only some major ones can be easily identified in terms of L’sand C’s.

Line with External Impedance

When the transmission line is energized through a source impedance ZS or when itis terminated by a load ZL new resonances arise.

4.2. Resonances in Cascaded Circuit 79

Source Impedance If an impedance ZS, e.g. source impedance which usuallyhas a main inductive component (Fig. 4.4), is connected to a transmission line, thetransfer functions of the whole system can be formulated as

HUU = [1 + ZαY α + ZSY α (2 + ZαY α)]−1

HIU = [Zα + ZS (1 + ZαY α)]−1

HUI = [Y α (2 + ZαY α)]−1

HII =(1 + ZαY α)−1

(4.8)

Figure 4.4: Transmission line cascaded to an impedance

In HUU resonance affected by the inductance of ZS and line capacitance isobserved, in addition to resonances determined by the inductance and capacitance ofthe line. Similar resonance is also present in HIU. The other two transfer functions,HUI and HII are not influenced by ZS.

Load Impedance If an impedance ZL, e.g. a load (Fig. 4.5), is connected to atransmission line, the whole system can be represented by a equivalent impedanceat terminal p, Zp

Zp =(1 + Zα · Y α) · ZL + Zα

(2 + Zα · Y α) · (Y α · ZL + 1)− 1(4.9)

Figure 4.5: Transmission line terminated by an impedance

It shows a possible natural frequency of the system determined by the capacitanceof the line and the inductance of the load.


4.3 Parameters Affecting Resonant Behavior

Resonances are influenced by component type, component layout, circuitconfiguration, and network topology. The focus is on parameters that are related tointegrating underground power cable into the connection. For such a system thesecategories include:

• Cable type: conductor and insulation dimensions.

• Layout: soil properties, trench type, phase sequence.

• Configuration: cable joints, number of parallel cables.

• Topology: cable length, sequence of OHL and cable parts.

In Section 4.3.1 the four transfer functions defined in (4.2) are analyzed for theRandstad380 south-ring and compared to a hypothetical connection for which theconnection is completely underground power cable. A sensitivity study on designparameters is performed in Section 4.3.2. The parameters that have significant effecton the resonant behavior are identified. Section 4.3.3 elaborates on the importanceof these parameters for future design of similar connections.

4.3.1 Mixed OHL-Cable Versus Full Cable Connection

The impact of the power cable on resonant behavior can best be observed by varyingits length. Based on the real Randstad380 south-ring design, two configurations aredefined.

• Combined OHL-cable system: The actual OHL1-Cable-OHL2 configuration isanalyzed.

• Full cable system (see Fig. 4.6): The OHL sections are replaced by cablesections resulting in a total of 24 minor sections.

All cable minor sections are assumed to have a single segment, each minor sectionis 0.9 km long and with trench type ‘O1’ (Table A.1). To focus on the influenceof cables complicating factors like compensating wires along cable and OHL2 (seeFig. A.1 and A.2) are not considered here. Their influence are in detail analyzedin [58]. Earth resistivity is assumed to be 100 Ωm. The transient behavior of thesystems is analyzed in terms of the four transfer functions in (4.2). The analyzedtransfer functions shown in Fig. 4.7 refer to phase A, with terminals of phases Band C open-ended. From Fig. 4.7 it can be observed that transient behavior ofboth systems differ; e.g. the first resonance in HUU in the ‘combined OHL-cablesystem’ occurs at about 800 Hz, while in ‘full cable system’ the resonance is atabout 1000 Hz. This resonance is related to the capacitances and inductances of thetransmission line itself as well as the cascade pattern. The shift is mainly related

4.3. Parameters Affecting Resonant Behavior 81

Figure 4.6: Full cable system for sensitivity study

to the inductance of the OHLs which is dominant in the mixed configuration andabsent in a full cable connection. Apparently, the order of cascading OHLs and cableaffect resonance frequencies (see Section 4.3.2). The resonant peaks for the full cableconnection are higher and narrower than for the mixed connection. This points tolower losses in cable compared to overhead lines. Within each configuration, thefour transfer functions can have different resonant frequencies. This means thatfor one configuration, transient behavior depends on the kind of excitation event.For example, a resonant frequency in transient overvoltage upon line energization(related to HUU) is not necessarily present in transient current upon a fault (relatedto HIU). As suggested already by (4.4) for a homogeneous transmission line themain resonances in HUU and HII are similar. For the mixed configuration the firstpeak in HII is slightly shifted with respect to the one in HUU.

4.3.2 Sensitivity Study

The impact of parameter variation on transfer functions is investigated for themixed OHL-cable configuration above. In this sensitivity study, we mainly focusedon overvoltages and currents upon line energization (HUU) and upon short faults(HIU). The other two transfer functions can be analyzed in similar fashion. Thefirst resonance in HUU and the low frequency part (around 50 Hz) in HIU willbe focused on, because they are primarily of concern in common transients, seeChapter 3. The parameters are subdivided in categories related to cable type, layout,configuration, and to circuit topology. A reference parameter value and referenceconfiguration (identified as ‘Ref’ in the further text) is based on the Wateringen -Bleiswijk connection. Variations (e.g. half or double parameter value) are taken toobserve their impact.


0

10

20

|HUU|

101

102

103

104

−200

0

200

Frequency (Hz)

�H

UU(◦)

0

50

100

|HUU|

101

102

103

104

−200

0

200

Frequency (Hz)�H

UU(◦)

0

1

2

|HIU|

101

102

103

104

−200

0

200

Frequency (Hz)

�H

IU(◦)

0

2

4|H

IU|

101

102

103

104

−200

0

200

Frequency (Hz)

�H

IU(◦)

0

1000

2000

|HUI|

101

102

103

104

−200

0

200

Frequency (Hz)

�H

UI(◦)

0

500

1000

|HUI|

101

102

103

104

−200

0

200

Frequency (Hz)

�H

UI(◦)

0

10

20

|HII|

101

102

103

104

−200

0

200

Frequency (Hz)

�H

II(◦)

0

50

100

|HII|

101

102

103

104

−200

0

200

Frequency (Hz)

�H

II(◦)

Figure 4.7: Transfer functions of the combined OHL-cable system (left) and fullcable system (right). HIU is in (S), HUI is in (Ω).


Parameters related to cable type Fig. 4.8 shows that by increasing the core-conductor radius of each cable, the first resonant frequency in the transfer functionHUU is lowered since the cable capacitance is increased (about 0.1 μF/km for‘50%·Ref’, 0.2 μF/km for ‘Ref’, 0.4 μF/km for ‘200%·Ref’). Although this variationalso decreases the inductance of the cable, in the ‘combined OHL-cable system’, theinductance of OHL parts usually dominates the total inductance. In contrast, thefirst resonant frequency in HUU lowers if the thickness of the insulation decreases(see Fig. 4.9), which increases the cable capacitances. Since this variation has lessinfluence on the inductances, HIU is almost unchanged at low frequencies. Fig.4.10 and Fig. 4.11 show that the variations in the thickness of screen and thicknessof outer-sheath have minor influences on both HUU and HIU. In the concernedconnections, these parameters at most slightly affect screen impedance and mutualcoupling between cables, see cable modeling in Section 2.2.

Parameters related to cable layout Soil resistivity differs depending on the soiltype: loam clay 100 Ωm, swamp marl 30 Ωm, humid sand 200 Ωm [73]. Fig. 4.12shows that variation in earth resistivity has minor influence. Different trench typesproduce different electromagnetic coupling between parallel cables and have effect onthe self-earth-return path of cables. Fig. 4.13 shows that in ‘combined OHL-cablesystem’, the effect is limited by the large inductances of the OHL1 and OHL2. Forthe mixed connection the effect can be neglected, but it can have influence for afull cable connection. Different phase sequences of the cables slightly influence theelectromagnetic coupling between cables but are irrelevant to cable capacitances. In‘combined OHL-cable system’, HUU and HIU are almost unchanged (Fig. 4.14).Their influence can however be larger in a full cable system, see Section 4.3.3.

Parameters related to cable configuration Cable joints add small amountof inductance in the cable screen connection, but they hardly influence HUU andHIU in the cable core-conductor, see Fig. 4.15. Doubling the number of cables willdouble the capacitance but halve the inductance. For a full cable connection thefirst resonance is expected to be unchanged. However, for a mixed connection thetotal inductance is mainly determined by the OHL-section and the resonance willshift because of the doubled cable capacitance as observed in Fig. 4.16. The lowfrequency part of HIU has minor deviation because OHL1 and OHL2 are unchanged.The influences will be significant in ‘full-cable system’ because the cable determinesthe total inductance.

Parameters related to connection topology The relative distance covered bycable compared to OHL in a connection and the order in which these componentsare placed affect the frequency spectrum. Fig. 4.17 shows the influence of thecable length which also changes line capacitance and inductance. In HUU, the firstresonant frequency in HUU decreases with longer cable. Given the total length


101

102

103

104

0

10

20

Frequency (Hz)

|HUU|

Ref50 %200 %

(a)

101

102

103

104

0

0.5

1

1.5

Frequency (Hz)

|HIU|

Ref50 %200 %

(b)

Figure 4.8: Transfer functions with different radius of core-conductor: 30.7 mm(Ref), 50%·Ref, and 200%·Ref. HIU is in (S).

101

102

103

104

0

10

20

Frequency (Hz)

|HUU|

Ref50 %200 %

(a)

101

102

103

104

0

0.5

1

1.5

Frequency (Hz)

|HIU|

Ref50 %200 %

(b)

Figure 4.9: Transfer functions with different thickness of insulation: 32.5 mm (Ref),50%·Ref, and 200%·Ref. HIU is in (S).

of ‘combined OHL-cable system’ is constant, longer cable means shorter OHL2resulting in higher magnitude of HIU at low frequencies. Different connectionsequences of the ‘combined OHL-cable’ should also be treated as different systemschemes. Fig. 4.18 shows that shifting the cable to the end part of the transmissionline (from OHL1-Cable-OHL2 to OHL1-OHL2-Cable) produces lower frequency ofthe first resonance in HUU. The first resonance in HUU of the cable part itselfbecomes the first resonance of the whole connection in the scheme Cable-OHL1-OHL2. This frequency is about 2 kHz; approximately twice as that in the full cablesystem where the cable length is nearly doubled, see the first resonance of HUU

in Fig. 4.7-right column. Altering the connection sequence will not influence thetotal series impedance, therefore HIU is hardly influenced at low frequencies. Fig.4.19 depicts that varying mixing level of cable and OHLs in ‘combined OHL-cablesystem’ influences the first resonance in HUU but not the magnitudes of HIU at lowfrequencies. Mixing cable and OHL2 by alternating cable and OHL sections causeslower first resonant frequency in HUU.


101

102

103

104

0

10

20

Frequency (Hz)

|HUU|

Ref50 %200 %

(a)

101

102

103

104

0

0.5

1

1.5

Frequency (Hz)

|HIU|

Ref50 %200 %

(b)

Figure 4.10: Transfer functions with different thickness of screen: 0.6 mm (Ref),50%·Ref, and 200%·Ref. HIU is in (S).

101

102

103

104

0

10

20

Frequency (Hz)

|HUU|

Ref50 %200 %

(a)

101

102

103

104

0

0.5

1

1.5

Frequency (Hz)

|HIU|

Ref50 %200 %

(b)

Figure 4.11: Transfer functions with different thickness of outer-sheath: 7.8 mm(Ref), 50%·Ref, and 200%·Ref. HIU is in (S).

101

102

103

104

0

10

20

Frequency (Hz)

|HUU|

Ref30200

(a)

101

102

103

104

0

0.5

1

1.5

Frequency (Hz)

|HIU|

Ref30200

(b)

Figure 4.12: Transfer functions with different earth resistivity: 100 Ω for loam clay(Ref), 30 Ω for swamp marl, and 200 Ω for humid sand. HIU is in (S).


101

102

103

104

0

10

20

Frequency (Hz)

|HUU|

RefH1H2

(a)

101

102

103

104

0

0.5

1

1.5

Frequency (Hz)

|HIU|

RefH1H2

(b)

Figure 4.13: Transfer functions with different trench types: O1 (Ref), H1, and H2(see Fig. A.1 in Section A.1). HIU is in (S).

101

102

103

104

0

10

20

Frequency (Hz)

|HUU|

RefABCCBA

(a)

101

102

103

104

0

0.5

1

1.5

Frequency (Hz)

|HIU|

RefABCCBA

(b)

Figure 4.14: Transfer functions with different phase sequence: ABC−CBA−CBA−ABC (Ref), ABC −ABC −ABC −ABC (ABC), and CBA−CBA−CBA−CBA(CBA). HIU is in (S).

101

102

103

104

0

10

20

Frequency (Hz)

|HUU|

Ref50 %200 %

(a)

101

102

103

104

0

0.5

1

1.5

Frequency (Hz)

|HIU|

Ref50 %200 %

(b)

Figure 4.15: Transfer functions with different values of the inductances in joints: 1μH for cross-bonding joint (Fig. 2.6) and 10 μH for straight-through joint (Fig. 2.7)(Ref), 50%·Ref, and 200%·Ref. HIU is in (S).


101

102

103

104

0

10

20

Frequency (Hz)

|HUU|

Ref624

(a)

101

102

103

104

0

0.5

1

1.5

Frequency (Hz)

|HIU|

Ref624

(b)

Figure 4.16: Transfer functions with different cable number: 12 (Ref), 6 (1 cableper phase per circuit), 24 (4 cables per phase per circuit). HIU is in (S).

101

102

103

104

0

10

20

Frequency (Hz)

|HUU|

Ref50 %150 %

(a)

101

102

103

104

0

1

2

Frequency (Hz)

|HIU|

Ref50 %150 %

(b)

Figure 4.17: Transfer functions with different cable length: OHL1(4.4 km)-Cable(10.8 km, 12 minor sections)-OHL2(6.8 km) (Ref), OHL1(4.4 km)-Cable(5.4km, 6 minor sections)-OHL2(12.2 km) (50%·Ref), and OHL1(4.4 km)-Cable(16.2 km,18 minor sections)-OHL2(1.4 km) (150%·Ref). HIU is in (S).

101

102

103

104

0

50

100

150

Frequency (Hz)

|HUU|

RefSeq1Seq2

(a)

101

102

103

104

0

0.5

1

1.5

Frequency (Hz)

|HIU|

RefSeq1Seq2

(b)

Figure 4.18: Transfer functions with different sequence of OHL1, Cable, and OHL2:OHL1-Cable-OHL2(Ref), OHL1-OHL2-Cable, Cable-OHL1-OHL2. HIU is in (S).


101

102

103

104

0

10

20

Frequency (Hz)

|HUU|

RefSeq1Seq2

(a)

101

102

103

104

0

0.5

1

1.5

Frequency (Hz)

|HIU|

RefSeq1Seq2

(b)

Figure 4.19: Transfer functions with different mixing of Cable and OHL2, see Fig.4.20: OHL1-Cable-OHL2(Ref), Seq1, and Seq2. HIU is in (S).

Figure 4.20: Different mixing of cable and OHL2, with OHL1 unchanged. TheOHL2 is evenly divided into 12 sections and mixed with 12 cable minor sections intwo different sequences: Seq1 and Seq2.

4.3.3 Impact Dominant Parameters

Combined OHL-cable connection

According to the analysis performed before, based on the real connection topology— OHL1-Cable-OHL2, the following design parameters influence HUU and HIU

significantly: radius of cable core-conductor (r), thickness of cable insulation (t),cable length (D), and number of cables (N). Fig. 4.21 compares their impact.Note that the cable length is varied by changing the number of major sections, andchanging the length OHL2 accordingly. The length of OHL1 as well as the totallength of the connection are kept constant.

4.4. Transients with External Components 89

50 75 100 125 150 175 200600

800

1000

1200

Variation (%)

Frequency

(Hz) r t D N

(a)

50 75 100 125 150 175 2000.2

0.3

0.4

0.5

Variation (%)

|HIU|

r t D N

(b)

Figure 4.21: Comparison on the impact of different cable design parameters: (a)frequency of the first resonance in HUU; (b) the magnitude of HIU at 50 Hz

Generally, resonance frequency is of concern if the exciting source has a matchingfrequency. Resonance with power frequency harmonics should be avoided. Alsoevents in connected networks which excite resonances of the 380 kV connection canjeopardize safe operation. To avoid a specific frequency, any parameter analyzedabove can be adjusted. Particularly, to shift the first resonant frequency so thatthe influence of any external source (e.g. switching surge) is limited, the effect ofcable length should be considered. Cable length also affects the 50 Hz short-circuitcurrent. Note that for any cable length the reactive power compensation should beconsidered accordingly.

Full-cable connection

The mutual coupling of cables via earth-return-path can be influenced by usingdifferent trench types, which represent different depths of each cable and differentdistances between cables, as well as different phase sequences. Changes in the mutualcoupling result in different series impedances of the cable. Their influence becomesobvious in a full-cable system. Fig. 4.22 and 4.23 depict the resulting HUU and HIU

of the full cable system in Fig. 4.6. For mixed systems where the inductance of theOHL is dominant, the mutual coupling has less effect.

4.4 Transients with External Components

Components connected to the line essentially contribute to resonant behavior.Section 4.4.1 looks into line energization where the source inductance is part ofthe transient response. In Section 4.4.2 the load by transformer and shunt reactoris considered.


101

102

103

104

0

50

100

Frequency (Hz)

|HUU|

RefH1H2

(a)

101

102

103

104

0

2

4

6

Frequency (Hz)

|HIU|

RefH1H2

(b)

Figure 4.22: Impact of trench types in full cable connection: O1 (Ref), H1, and H2(see Fig. A.1 in Section A.1). HIU is in (S)

.

101

102

103

104

0

50

100

Frequency (Hz)

|HUU|

RefABCCBA

(a)

101

102

103

104

0

2

4

6

Frequency (Hz)

|HIU|

RefABCCBA

(b)

Figure 4.23: Impact of phase sequence in full cable connection. HIU is in (S).

4.4.1 Line Energization

When connecting a transmission line to a busbar (see Fig. 4.24), the busbar canbe considered as a voltage source with an inductance whose value can be extractedfrom the busbar voltage and short-circuit current at 50 Hz. The circuit breaker isclosed when the voltage source reaches its positive peak (1 p.u.), e.g. at t1 = 0.205 sin a time frame of Tw = 0.26 s. For the reference configuration of OHL-Cable-OHL(‘Ref’) in Section 4.3.2 along with source inductance LS = 10 mH, the overvoltage(uAG (t)) upon line energization in phase A is shown in Fig 4.25-a. The transientreaches up to 2 p.u. and contains a resonance at around 435 Hz (Fig 4.25-b), which isdetermined by the source inductance and line capacitance according to Section 4.2.2.Only the first resonance in the transfer function is clearly excited in the responses,because of the low magnitude of the input spectrum U eq (t) at higher frequencies.To investigate the influence of cable parameters specifically on the overvoltage uponenergization, a sensitivity study is useful. Parameters are selected that have clear


Figure 4.24: Equivalent circuit diagram of line energization in a three-phase system

0 0.05 0.1 0.15 0.2 0.25

−2

0

2

uAG(t)

0.2 0.22 0.24 0.26

−2

0

2

Time (s)

uAG(t)

(a)

101

102

103

104

0

50

|H|

0 200 400 600 8000

50

Frequency (Hz)

|H|

(b)

Figure 4.25: Line energization in phase A with view in full considered range (top)and zoomed view (bottom): (a) resulting time-domain overvoltages at location ‘G’;(b) transfer function between the voltage at location ‘G’ and the equivalent voltagesource to the closure of circuit breaker

effect on the first resonance in transfer function HUU. From the results in Section4.3, the radius of the cable core-conductor, the thickness of the cable insulation layer,the number of cables, and cable length are selected, see Fig. 4.26. Their values arevaried according to section 4.3.2. Besides varying the cable length, three extremescenarios are looked into: 1) “Ref” — OHL1(4.4 km)-Cable(10.8 km)-OHL2(6.8 km);2) “No cable” — OHL1(4.4 km)-OHL2(17.6 km); 3) “Full cable” — Cable (21.6 km,24 minor sections). Since the line energization also contains source inductance,its impact is also analyzed. To focus on the responses after energization and itsdominant frequency component, only the time-domain waveforms around closingmoment (within 20 ms) are depicted, together with the first resonance in the transferfunction. With larger radius of cable core-conductor, thinner cable insulation layer,higher number of cables, longer cable length, cable being positioned at the end ofthe line (OHL1-OHL2-Cable), and larger source inductance (weaker system), thefrequency of first resonance in HUU is lowered. Especially, when the configurationof the transmission line is changed as in Fig. 4.26-e (from combined OHL-cable topurely cable) the magnitude of first resonance is significantly increased.


0 0.005 0.01 0.015 0.02

−2

0

2

Time (s)

uAG(t)

Ref 50 % 200 %

0 200 400 600 8000

50

Frequency (Hz)

|H|

(a)

0 0.005 0.01 0.015 0.02

−2

0

2

Time (s)

uAG(t)

Ref 50 % 200 %

0 200 400 600 8000

50

Frequency (Hz)

|H|

(b)

0 0.005 0.01 0.015 0.02

−2

0

2

Time (s)

uAG(t)

Ref 6 24

0 200 400 600 8000

50

Frequency (Hz)

|H|

(c)

0 0.005 0.01 0.015 0.02

−2

0

2

Time (s)

uAG(t)

Ref 50 % 150 %

0 200 400 600 8000

50

Frequency (Hz)

|H|

(d)

0 0.005 0.01 0.015 0.02

−2

0

2

Time (s)

uAG(t)

Ref No Full

0 500 1000 1500 2000 25000

100

200

300

Frequency (Hz)

|H|

0 1000 20000

50

(e)

0 0.005 0.01 0.015 0.02

−2

0

2

Time (s)

uAG(t)

Ref 20 mH 30 mH

200 300 400 5000

50

100

150

Frequency (Hz)

|H|

(f)

Figure 4.26: Impact of parameters on line energization in phase A: (a) core-conductor radius, (b) insulation thickness, (c) number of cables, (d) cable length,(e) with and without cable, (f) source inductance


4.4.2 Line with Transformer and Shunt Reactor

Fig. 4.27 shows an unloaded three-phase three-winding three-limb transformer(described in Section 2.4.1.2) connected to a transmission line referring to thereference configuration of ‘combined OHL-Cable system’ (‘Ref’) in Section 4.4.1.The transformer specifications are given in Section 2.6.2.

Figure 4.27: Equivalent circuit diagram of transmission line and transformer

This configuration can be represented by impedances at terminal p of the line(Section 4.2.2). Fig. 4.28 illustrates the impedance in phase A (ZAp = UAp/IAp)when phases B and C at terminal p are open-ended.

0

5000

10000

∣ ∣Z

p

∣ ∣(Ω

)

101

102

103

104

−100

0

100

Frequency (Hz)

�Z

p(◦)

(a)

Figure 4.28: Frequency spectrum of ZAp.

The resonance at about 95 Hz is caused by the line capacitance and the leakageinductance of the transformer. The resonance caused by line capacitance andthe magnetizing inductance of the transformer is not depicted in the figure, sinceit occurs at very low frequency (less than 1 Hz). Moreover, the magnetizinginductance of the transformer can be changed by saturation of the magnetic core,but the frequency spectrum only considers linear components. Fig 4.29 shows thatthe first resonant frequency in ZAp is lowered with larger radius of cable core-conductor, thinner cable insulation layer, more cables, longer cable length, and largertransformer leakage inductance.

The frequency spectrum of ZAp is shown in Fig. 4.30 for the transformer withconnected shunt reactor (Section 2.6.2) to its tertiary windings. The resonanceat around 20 Hz is caused by the line capacitance and the inductance of the


0

5000

10000

∣ ∣Z

p

∣ ∣(Ω

)

Ref 50 % 200 %

0 50 100 150 200−100

0

100

Frequency (Hz)

�Z

p(◦)

(a)

0

5000

10000

∣ ∣Z

p

∣ ∣(Ω

)

Ref 50 % 200 %

0 50 100 150 200−100

0

100

Frequency (Hz)�Z

p(◦)

(b)

0

5000

10000

∣ ∣Z

p

∣ ∣(Ω

)

Ref 6 24

0 50 100 150 200−100

0

100

Frequency (Hz)

�Z

p(◦)

(c)

0

5000

10000

∣ ∣Z

p

∣ ∣(Ω

)

Ref 50 % 150 %

0 50 100 150 200−100

0

100

Frequency (Hz)

�Z

p(◦)

(d)

0 200 400 600 8000

5

10x 10

4

∣ ∣Z

p

∣ ∣(Ω

)

Ref No Full

0 200 400 600 800−100

0

100

Frequency (Hz)

�Z

p(◦)

0 100 2000

1000

2000

Ref No Full

(e)

0

5000

10000

∣ ∣Z

p

∣ ∣(Ω

)

Ref 50 % 200 %

0 50 100 150 200−100

0

100

Frequency (Hz)

�Z

p(◦)

(f)

Figure 4.29: Sensitivity study on ZAp: (a) core-conductor radius, (b) insulationthickness, (c) number of cables, (d) cable length, (e) with and without cable; (f)different transformer leakage inductance

4.5. Discussion 95

shunt reactor. The shunt reactor can be considered as parallel impedance to themagnetizing inductance of transformer. Because the inductance of the shunt reactor(about 3 H if transferred to the 380 kV level) is much less than the magnetizinginductance of the transformer (about 3 kH), the magnetizing inductance of thetransformer can be ignored. The resonance between the line capacitance and thetransformer leakage inductance remains at about 95 Hz (Fig. 4.30-middle).

0

1

2x 10

5

∣ ∣Z

Ap

∣ ∣(Ω

)

0

4000

8000

∣ ∣Z

Ap

∣ ∣(Ω

)

101

102

103

104

−100

0

100

Frequency (Hz)

�Z

Ap(◦)

Figure 4.30: Impact of shunt reactor connected to the transformer on ZAp: full viewof magnitudes (top), vertically zoomed view (middle); phase angle (bottom)

4.5 Discussion

The study in this chapter shows that in a given configuration, different transferfunctions (e.g. HUU and HUI) have different resonant behavior. Therefore, foranalysis of a specific transient the corresponding transfer function should be focusedon. For systematic analysis, all transfer functions should be investigated.

To apply power cables in transmission systems, the following scenarios need tobe considered:

• different cable parameters for a fixed system scheme: radius of cable core-conductor, thickness of cable insulation layer, number of cables, cable length.

• different system schemes: combined OHL-cable system or purely OHL/cablesystem, different connection sequence of OHL1, Cable, and OHL2, withexternal components (e.g. source inductance, transformer/shunt reactor).

If the series impedance of a transmission line is dominated by the series impedanceof the cable (e.g. the line is purely composed of cable), cable trench types and phasesequences are also recommended to be investigated.

Chapter 5

Study on Large-Scale Network

5.1 Introduction

The impact of underground power cable between Wateringen and Bleiswijk isinvestigated as part of the full Netherlands 380 kV transmission grid. As referencepoint the Wateringen substation is taken. The distances of all other substations ofthe 380 kV grid to substation Wateringen are listed in Table 5.1. The overview ofthe network is shown in Fig. 5.1.

Table 5.1: Distance to Wateringen Substation (in km)

Westerlee (WL) 7 Lelystad (LLS) 148Bleiswijk (BWK) 22 Borssele (BSL) 168

Maasvlakte (MVL) 27 Ens (ENS) 168Krimpen (KIJ) 41 Maasbracht (MBT) 187

Simonshaven (SMH) 50 Zwolle (ZL) 200Crayestein (CST) 55 Boxmeer (BMR) 247

Geertruidenberg (GTB) 74 Hengelo (HGL) 261Diemen (DIM) 98 Dodewaard (DOD) 287

Oostzaan (OZN) 113 Meeden (MEE) 308Beverwijk (BVW) 128 Doetinchem (DTC) 319Eindhoven (EHV) 138 Eemshaven (EEM) 346

The whole 380 kV grid is implemented in PSCAD/EMTDC software. It includesoverhead lines, underground power cables, power transformers, shunt reactors, seriesreactors, and capacitor banks. The surge arrestors are not modeled because theymay suppress overvoltages and thereby possibly disguise effects of changed circuitconfigurations. The loads are implemented as lumped impedances based on asnapshot situation of the real grid. The network (transmission routes) is based

97

98 Study on Large-Scale Network

Maasvlakte

Simonshaven

WesterleeWateringen

Borssele

Meeden

Eemshaven

Zwolle

Doetinchem

Maasbracht

Geertruidenberg

Dodewaard

LelystadOostzaan

Ens

Eindhoven

Diemen

Beverwijk

Boxmeer

Crayenstein

Hengelo

BleiswijkKrimpen a.d. IJssel

Kreekrak

Zandvliet

Figure 5.1: Substations in Dutch 380 kV transmission system with Randstad south-ring (cables are located between Wateringen and Bleiswijk)

5.2. Directly Imposed Transients 99

on the network state in 2013. More information can be found in the capacity andquality plan (KCD5) published by the Dutch TSO TenneT. In the modeled system,switches can be opened or closed, forming different topologies. This chapter focuseson the topology where all switches are closed, meaning all components are in serviceand all double-circuit busbars in substations are parallel connected.

The Z and Y matrices of cables and overhead lines, as well as the configurations oftransformer and shunt reactor described in Chapter 2 are used in PSCAD/EMTDCfor time-domain simulation. The mutual couplings between two cable circuits,compensating wires, mutual coupling between phases in transformers are all ignoredfor implementation. Capacitor banks are modeled by their main capacitance withlumped filter (see Section 5.3.2). Each generator and connection abroad is modeledas an ideal source with its subtransient reactance in series [74]. Each load, placeddirectly on the secondary side of a transformer, is modeled as impedance based onthe KCD.

To analyze the impact of cable on resonant behavior, this chapter studies fourtypes of transients, which are referred to slow-front overvoltages in IEC 60071-4. Lineenergization and fault/fault-clearing induce directly transients on the line. Theyare studied in Section 5.2. Indirect transients are caused by external components.Section 5.3 considers transformer energization and energization of a capacitor bank.In that respect, for each situation, comparison is made for three different cablelengths in the circuit between Wateringen and Bleiswijk: 10.8 km (reference —‘Ref’), 0 km (‘No’), and 16.2 km (‘150%’), see Fig. 5.2. For the study on transients,there is need for a systematic approach to decide on the level of details required inmodeling the complete network. Section 5.4 presents a methodology to determinethe necessary detail level for adequate prediction of transient phenomena.

5.2 Directly Imposed Transients

5.2.1 Line Energization

The transients upon line energization can be influenced by the phase-angle of thebusbar voltage at the switching moment. In a three-phase system, the three-phasecontacts of a circuit breaker usually close asynchronously, even if they receive theclosing command simultaneously. To find the worst switching moment giving highestovervoltage (worst case), the concept of statistic switch in EMTP is adopted, whichconsiders two uncertainties in timing:

• mean switching timing (see Fig. 5.3): 20 timings over 20 ms (uniformdistribution);

5KCD: Kwaliteits- en Capaciteits Document


Figure 5.2: Configurations with different cable lengths for studying the influence byunderground cables

• closing timing difference between phases: 20 timings around each meanswitching timing (normal distribution, assumed standard deviation σ = 0.1ms).

0 0.005 0.01 0.015 0.02

−1

0

1

Time (s)

uBusbar(p.u.)

Figure 5.3: Uniformly distributed command timings over one power cycle

Such number of simulations requires excessive computation time. Notice that thetime frame of each simulation is from 0 to 0.14 s. The duration 0 to 0.08 s is neededfor the simulation evolve to steady-state. The duration to investigate the transientsis from 0.08 to 0.14 s. Due to the length of cable minor sections, the simulationtime steps are taken as 0.5 μs. Therefore, each simulation consumes about 3 hours.Alternatively, the concept of frequency domain transient analysis can be adopted.The different switching moments can be realized by introducing ‘equivalent voltagesources’ at specific timings to a single system configuration. Transfer functionsbetween the voltages at all busbars and the equivalent voltage source can be obtainedjust from a single simulation. With the frequency domain approach, outputs from


all scenarios can be calculated in matter of seconds. The worst switching momentis the one producing the highest overvoltage. Since the frequency domain approachhere considers neither the responses to the real voltage sources (generators) nor thenonlinear components in the system, it is necessary to apply the obtained worstswitching timing for a PSCAD/EMTDC simulation including these factors.

The statistic switch simulation is separately applied at Wateringen and Bleiswijk,the nearest substations to the cable section. In this double-circuit transmission line,it is assumed that one circuit is energized while the other circuit is either in serviceor out of service (see Fig. 5.4). The per-unit voltage, in which the overvoltage isexpressed, is defined based on a value of 420 kV: 1 p.u. =

√2/

√3 ·420 kV (according

to IEC 60071-1), and the nominal voltage (380 kV) of the system is 0.9 p.u..

Figure 5.4: Study cases of line energization

From all switching scenarios, the worst switching timing is when the busbarvoltage at the switching substation reaches either its positive peak or negative peak inone phase. To compare the switching transients for different system configurations,the following situation is chosen: the three-phase contacts of the circuit breaker areclosed when the busbar voltage in phase A reaches its positive peak. For all studiedcases, the location with highest overvoltage occurs in the substations nearby and atthe open-end of the line being energized. The highest overvoltage is about 1.65 p.u.at the open-end obtained in two situations (referring to circuit configuration andenergization case)

• the line has no cable, switching at Wateringen side when the other circuit isout of service (Fig. 5.4, bottom-left);

• the line has no cable, switching at Bleiswijk side when the other circuit is inservice (Fig. 5.4, top-right).

For the latter energization case (shown in Fig. 5.4, top-right), Fig. 5.5 shows theovervoltages for the three configurations with different cable lengths (see Fig. 5.2).


0.07 0.08 0.09 0.1 0.11 0.12

−1

0

1

2

Time (s)

u(p.u.)

Ref No 150 %

0.084 0.086 0.088 0.09 0.092 0.094 0.096

−1

0

1

2

Time (s)

u(p.u.)

101

102

103

104

0

0.02

0.04

Frequency (Hz)

|U|(p.u.)

Figure 5.5: Overvoltage at open-end upon line energization at Bleiswijk: (top)overview; (middle) zoomed view of transients after energization; (bottom) frequencyspectrum

The short-circuit current at the busbar in Bleiswijk is 54 kA (rms). The cable reducesthe overvoltages but causes oscillation at lower frequency — from over thousands ofHertz to hundreds of Hertz, see Fig. 5.5-bottom. For line energization, the highestovervoltage is about 1.65 p.u., far below the Switching Impulse Withstand Voltage(SIWV 1050 kV ≈ 3.06 p.u. or 913 kV ≈ 2.66 p.u.)6.

Reclosure

Reclosing a circuit breaker after disconnection of an unloaded line can have acumulative overvoltage. It can be modeled with an equivalent voltage source whose

61050 kV is standard value; while 913 kV is the value with safe margin indicated by IEC 60071-1for inner insulation of system devices


magnitude reaches twice the busbar voltage, resulting in even higher overvoltages[8, 75]. Suppose one unloaded circuit between Wateringen and Bleiswijk, initiallyconnected to Bleiswijk, is disconnected when the line voltage reaches its peak. Forinstance at t = 0.085 s, since the unloaded line is mainly capacitive, the currentis zero when the voltage reaches its peak. As the busbar voltage continues alonga 50 Hz sinusoidal waveform, the line voltage remains. After 10 ms, the busbarvoltage will reach its negative peak, meaning the voltage difference between busbarat Bleiswijk and line is about twice the busbar voltage. If at this moment, the circuitbreaker recloses, the resulting overvoltages at the open-end are as shown in Fig. 5.6.Note that in the whole process, the other circuit is in service. The case withoutcable (‘No’) produces the highest overvoltage with about 2.6 p.u.. Moreover, afterdisconnection and before reclosure (between 0.085 s and 0.095 s) the line voltagerises. This is related to the capacitive coupling from the other circuit which isin service. With cables, the soil acts like a electrostatic shield reducing the totalcoupling between the circuits. The line voltage remains unchanged when there arecables.

0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12

−2

0

2

u(p.u.)

Ref No 150 %

0.094 0.096 0.098 0.1 0.102 0.104

−2

0

2

Time (s)

u(p.u.)

Figure 5.6: Overvoltage at open-end upon reclosure: (top) overview; (bottom)zoomed-view of transients after reclosing

5.2.2 Fault and Fault Clearing

Upon a network fault, high short-circuit currents trigger circuit breakers to open andthe line with the fault will be disconnected. The sudden drop of the short-circuitcurrent can cause overvoltages in nearby busbars. This section studies the transientsbased on a single-phase-to-ground fault in phase A. The fault impedance is taken


zero. The fault occurs at one circuit of the line between Krimpen and Bleiswijkat Bleiswijk side (see Fig. 5.7) when the busbar voltage of Bleiswijk in phase Areaches its positive peak (0.085 s). The other circuit is in service to allow for a largeshort-circuit current. The fault clearing action occurs 20 ms later.

Figure 5.7: Configuration for study of fault and fault clearing transients

Fig. 5.8 depicts the currents via the fault point and the voltage at the busbarof Bleiswijk in phase A. The highest short-circuit current is about 70 kA (peak) atthe fault location. With longer cable, the short-circuit current and the overvoltageupon fault clearing increases slightly. No overvoltage exceeds the SIWV.

−50

0

50

i(kA)

Ref No 150 %

0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14

−1

0

1

Time (s)

u(p.u.)

Figure 5.8: Fault and fault clearing transients

5.3 Transients Indirect to Line

5.3.1 Transformer Energization

To investigate the cable influence under large inrush current upon transformerenergization, the circuit breaker is closed when the busbar voltage (e.g. in phase A)

5.3. Transients Indirect to Line 105

is zero. Fig. 5.9 shows the inrush current and busbar voltage caused by energizinga transformer in substation Bleiswijk, with the other two, similar, transformers inservice. No residual flux in transformers is considered.

−1

0

1

2

3

i(p.u)

A B C

0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12−1

0

1

Time (s)

u(p.u.)

Figure 5.9: Transients in three phases (A, B, C) upon transformer energization whenthe line between Wateringen and Bleiswijk has the ‘Ref’ configuration

The inrush current reaches more than 2 p.u. The value of 1 p.u. current (peakvalue) is 1.1 kA, defined according to the rated current of the transformer in HVwinding. Busbars experience no overvoltages but the waveforms are distorted froma purely sinusoidal shape. The effect of different cable lengths in the Wateringen-Bleiswijk connection is illustrated in Fig. 5.10. With longer cable, the inrushcurrents are slightly increased, but no overvoltage is caused. The voltage distortioncan be checked by converting the real-time value of voltage waveforms into RMSvalue versus time. Two definitions are commonly used

ΔU =Urms[k]− Urms[0]

Un, or ΔU =

Urms[k]− Urms[0]

Urms[0](5.1)

Where Urms[0] is the value at the energizing moment and Un is the system nominalvoltage (here 380 kV). For the voltage at any moment Urms[k], the distortion shouldbe |ΔU | < 3%. Fig. 5.11 shows the results based on the left formula in (5.1), whichrefers to the Dutch grid code of 2013. The deviation occurring after the enerigzationmoment from the value before energization is less than 2 %, which is within theregulatory limit. This distortion marginally increases with longer cable.


−1

0

1

2

3

i(p.u.)

Ref No 150 %

0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12−1

0

1

Time (s)

u(p.u.)

Figure 5.10: Transients in phase A upon transformer energization

0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12−2.5

−2

−1.5

−1

−0.5

0

0.5

Time (s)

ΔUrm

s(%

)

Ref No 150 %

Figure 5.11: Distortion of phase-to-phase voltage AB upon transformer energization

5.3.2 Capacitor Bank

The capacitor bank located in Westerlee substation at 150 kV level (see Fig. 5.12) isnearest to the cable location. The transformer (380/150/50 kV) in between containsinductances which can interact with the capacitor bank and power cable. The ratingsof the capacitor bank are 151 MVA, 165 kV, and 747 A (peak). The parameters forthe capacitor bank model are presented in Table 5.2. Since the neutral point of thecapacitor bank is floating, the energization occurs when the phase-to-phase voltage(e.g. phase A to phase B) reaches its positive peak. The 150 kV busbar has a

5.4. Requirements on Level of Model Detail 107

Figure 5.12: Configuration of capacitor bank energization

Table 5.2: Parameters of capacitor bank

RS LS RP C0.4 Ω 1.2 mH 34.0 Ω 17.7 μF

short-circuit current of 22 kA (RMS). The transients upon energization are depictedin Fig. 5.13, where 1 p.u. voltage at 150 kV is defined as

√2/√3 · 170 kV. The

inrush current can exceed 5 p.u.. The overvoltages at the 150 kV level are nearly1.5 p.u.; at the 380 kV busbar they are just above 1 p.u.. No overvoltage violatesthe SIWV7. The effect of cable length is shown in Fig. 5.14. Longer cable lengthslightly increases the inrush current and lowers the damping of the oscillations.

5.4 Requirements on Level of Model Detail

Implementing the whole Dutch 380 kV transmission system in PSCAD/EMTDCis time consuming, especially for collecting information like: overhead linepylon specifications, routes of transmission lines, substations configurations, etc..Investigation on particular transients (e.g. transients upon line energization at acertain substation, for instance Wateringen) does not need all details for the completegrid. References [76, 77] propose a general concept dividing the whole network intothree areas: the study area (with detailed line models), the external area (with asimpler line model), and the rest of the network (not important for the transientstudy), see Fig. 5.15. A common way to estimate what detail level is necessary isbased on an iterative process, in which the complexity is gradually increased untilsimulation results remain unaltered. To have a better view on necessary details,this section presents an approach starting from a “complete” model (as the referencemodel) and decreasing the complexity gradually. The concept of this method canbe considered as a basis for establishing more systematic rules to model a systemomitting unnecessary details to study transients.

The reference model of the network contains all routes and substations at 380kV level in the Netherlands, see Section 5.1. The reference model is simplified and

7The SIWV for 150 kV level is 600 kV (standard, ≈ 4.3 p.u.) and 513 kV (with safe margin, ≈3.7 p.u.)


0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12

−5

0

5

i(p.u.)

A B C

0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12−2

0

2

u(p.u.)

0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12−2

0

2

Time (s)

u(p.u.)

Figure 5.13: Transients upon capacitor bank energization: top — three-phase inrushcurrents at 150 kV; middle — three-phase busbar voltages at 150 kV; bottom — three-phase busbar voltages at 380 kV

reduced in size in successive steps. The effect of the reductions and simplificationsis then compared with the reference model to analyze the reduction of accuracy.Differentiation between models aims to compare the models quantitatively. Thefocus for this study is on the frequencies up to 10 kHz, caused by events like switchingtransients, often referred to as slow front surges [78].

Model Comparison

A convenient way to determine the accuracy of a model is to check transferfunctions. Whether inaccuracy in a particular resonance actually influences theaccuracy of time-domains waveforms significantly depends on the type of excitation.It means that an accurate model for one objective (e.g. accurate switching transient


0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12

−5

0

5

i(p.u.)

Ref No 150 %

0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12−2

0

2

u(p.u.)

0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12−2

0

2

Time (s)

u(p.u.)

Figure 5.14: Cable impact on transients in phase A: top — inrush current at 150kV; middle — busbar voltage at 150 kV; bottom — busbar voltage at 380 kV

waveforms in time domain) may be inaccurate for another objective (e.g. an accuratefrequency spectrum). Therefore, the purpose of study is linked to find quantitativecriteria for comparison between reference model and simplified models (model understudy, mus). Two important objectives are

• accurate peak value of overvoltages upon line energization — e.g. the peakvalue in in time-domain transient waveform can tolerate ±1 % deviation

Δumax =|umax,ref − umax,mus|

umax,ref≤ 1%

• accurate value of the first resonant frequency — e.g. the frequency of the main


External Area(simpler line models)

Study area(detailed line models)

Rest of the network

Figure 5.15: A network can be subdivided in three areas: The study area, theexternal area, and the rest of the network.

resonance in a transfer function can be displaced within ±10 Hz

Δf = |fref − fmus| < 10Hz

The chosen values of accuracy are somewhat arbitrary. The peak voltage deviation of1 % refers to comparison of models, not to deviations between model and reality. Thelatter deviation is usually much larger, since applied models have their limitationsfor describing all electromagnetic interactions. The value of 10 Hz is taken, havinginteraction with system harmonics in mind [79]. Five models for transmission linescan be used, (listed in descending order in terms of accuracy):

• Frequency-dependent model, (FD)This model is most accurate to model a transmission line for transient behavior.It needs the geometric and material properties of the line, pylon or cable trench,and earth.

• Bergeron model, (B)The Bergeron model uses distributed line inductances and capacitances, butwith lumped resistance representing the total loss. It uses one specificfrequency (e.g. the fundamental frequency). For studying transients, onlyresults at the specified frequency are reliable.

• π-Model, (Pi)The π-model is a lumped representation of the line in terms of capacitancesand inductances.

• Infinite busbar connection, (IBB)An even simpler representation of a connection is considering it as an ideal


busbar. No resistance, inductance, and capacitance are considered. This modelensures that the related substations are modeled.

• Omitted, (Omitted)The simplest representation of a transmission line is by removing it completelyfrom the network, including the connected substations.

The reference model is reduced systematically starting with the lines connectedto substation furthest to substation Wateringen or Bleiswijk, see Table 5.3.

Results

One of the two parallel circuits between Wateringen and Bleiswijk is studied withthe other circuit in service. A standard switching surge (250/2500 μs) is applied atWateringen side of the circuit. The “1 %” criterion is checked for all busbars in thesouth-ring and for the two nearest substations (Westerlee and Bleiswijk) only. Theresults are summarized in Table 5.4. Note that the resulting simplified network onlyworks well upon the switching surge response. For a switching operation, e.g. circuitbreaker closes when busbar voltage at substation Wateringen reaches its positivepeak, the resulting Δumax at the busbar in substation Bleiswijk is close to 3 %. Tokeep the deviation of Δf of the first resonance in the transfer functions of voltagesat all busbars in the ring below 10 Hz, the configuration of Fig. 5.16 is a minimalrequirement. This figure shows which connections should be considered and whattype of model is at least required. It confirms that different simplifications shouldbe considered for different purposes of study. The area nearby should be modeledwith more complex models, where the rest of the system can be omitted.

To highlight the advantage of the proposed method which starts from aextensively detailed network model (reference model), an opposite example ispresented with the aim of “accurate voltage magnitudes of busbars in two closestsubstations upon a switching surge applied at substation Wateringen (WTR)”. Table5.4 already gives the results if a reference model is available. Suppose there is noreference model, and the process is starting from modeling of local area and graduallyadding details until results almost remain the same. Based on the network modelshown in Fig. 5.17, changing the model of transmission line between substationsMaasvlakte (MVL) and Westerlee (WL) from Bergeron model (in Fig. 5.17-a) tofrequency-dependent model (in Fig. 5.17-b) will alter the voltage magnitude about0.06 % — referred to the results based on the model in Fig. 5.17-b. However, bothof them deviates from the results obtained by reference model more than 10 %.

To conclude, the proposed approach to determine necessary parameters foradequately modeling a large-scale network is starting from an extensively detailednetwork model, then gradually removing the parameters which have small influenceon objectives.


Table 5.3: Order of reduction. The distance refers to the shortest path

Distance toSubstation BWK- line to eliminate

WTR (km)Eemshaven (EEM) 323.6 EEM-MEEDoetinchem (DTC) 297.2 DOD-DTC DTC-HGLMeeden (MEE) 286.0 ZL-MEEDodewaard (DOD) 264.6 BMR-DODHengelo (HGL) 238.5 ZL-HGLBoxmeer (BMR) 224.6 MBT-BMRZwolle (ZL) 178.4 ENS-ZLMaasbracht (MBT) 164.6 MBT-EHVEns (ENS) 146.3 LLS-ENSBorssele (BSL) 145.9 KRK-BSLLelystad (LLS) 126.4 DIM-LLSEindhoven (EHV) 116.1 GT-EHVKreekrak (KRK) 108.1 KRK-ZVL GTB-KRKBeverwijk (BVW) 106.3 BVW-OZNOostzaan (OZN) 91.2 OZN-DIMDiemen (DIM) 76.0 KIJ-DIMGeertruidenberg (GTB) 52.4 KIJ-GTBSimonshaven (SMH) 50.1 SMH-CST MVL-SMHCrayestijn (CST) 33.4 CST-KIJMaasvlakte (MVL) 26.5 MVL-WLKrimpen a.d. IJssel (KIJ) 18.6 KIJ-BWKWesterlee (WL) 6.8 WTR-WL

Table 5.4: Simplified network configuration for accurate peak values in overvoltagesupon switching transients

Shortest Full Randstad Two closestdistance Lines South ring substationsto BWK- 1% 1%

WTR (km) Line model type323.6 EEM-MEE297.2 DOD-DTC DTC-HGL286.0 ZL-MEE264.6 BMR-DOD238.5 ZL-HGL omitted224.6 MBT-BMR178.3 ENS-ZL164.6 MBT-EHV IBB146.3 LLS-ENS IBB145.9 KRK-BSL Pi126.4 DIM-LLS B116.1 GTB-EHV B108.1 KRK-ZVL GTB-KRK B B106.3 BVW-OZN B B91.2 OZN-DIM B B76.0 KIJ-DIM B B52.4 KIJ-GTB FD B50.1 SMH-CST MVL-SMH FD B33.4 CST-KIJ FD B26.5 MVL-WL FD FD18.6 KIJ-BWK FD FD6.8 WTR-WL FD FD


MVL

WL

WTR BWK

KIJ

CST

SMH

FD FD

FD FD

FD FD

FD

DIM

LLS ENS

CST

D

GTB BSL EHV

MBT

OZN

BVW

FD

IBB

IBB

IBB

IBB

IBB IBB IBB

IBB

Figure 5.16: Simplified network configuration for accurate frequency of the firstresonance in transfer functions. FD : frequency-dependent model; IBB : infinite Busbarmodel

MVL

WL

WTR BWK

KIJ

CST

SMH

FD FD

B FD

B B

B

(a)

MVL

WL

WTR BWK

KIJ

CST

SMH

FD FD

FD FD

B B

B

(b)

Figure 5.17: Example of adding modeling details which gains minor accuracy butstill can largely deviates from the reference model. FD : frequency-dependent model;B : Bergeron model.

To demonstrate its application of this approach, a scheme of Dutch 380 kVnetwork is applied. Generally, the components near the focused area should bemodeled with more details. Based on the study for responses to a switching surgeapplied at busbar in Wateringen, the three adopted objectives require differentmodeling details.

Chapter 6

Conclusion and Recommendation

6.1 Conclusion

This dissertation investigates the impact of integrating 380 kV underground powercable in the network on the resonant grid behavior. For this purpose, models ofcables, overhead lines, transformers and shunt reactors are established. An approachof frequency-domain transient analysis is developed by which detailed configurationparameters in the connections can be considered. The impact of integrating powercables into 380 kV grid is analyzed based on the Randstad380 south-ring.

Components modeling

The models of cables and overhead lines based on well-known frequency-dependentmodels are extended to deal with connections with large number of conductors inparallel (up to 48 have been applied for simulations). Further, it can deal witha large number of segments, handle cross-bonding and be combined with overheadlines. Application to the Randstad380 south-ring could solve the applied 12 mutuallycoupled cables with 2 compensating wires, 28 different segments. It showed that,though minor differences were observed when applying simplifications needed forimplementation in PSCAD/EMTDC, both methods gave comparable results forthe particular case of the Randstad380 south-ring. Because the frequency rangeconsidered in this study was limited to 10 kHz, the frequency domain approach wascomputational efficient. The model for cables refers to four homogeneous parts:core-conductor, insulation, screen, and outer sheath. In particular, it was shownthat the cable with double-metal layer (copper/lead) screen could be replaced bya cable with single layer (copper) screen with adapted thickness of the screen, thethickness, permittivity, and permeability of the outher-sheath.

The models for transmission lines are validated by PSCAD/EMTDC simulation.Models on transformers and shunt reactors are verified by SFRA (Sweep Frequency

115

116 Conclusion and Recommendation

Response Analysis) measurements. Up to the target range of 10 kHz the frequencydomain responses could be predicted.

Frequency-domain transient analysis

The presented approach of frequency-domain transient analysis is developed focusingon the details in the Randstad380 south-ring. Switching operations and faultsituations are implemented in frequency domain by introducing equivalent voltageor current sources. The response in time-domain is retrieved from the frequencyresponse upon these excitations. The investigated transient scenarios include:switching surge responses, line energization, short-circuit fault, and transformerenergization. This method not only allows to include all the design details, it isalso time-efficient for analyzing so-called slow transients (up to 10 kHz).

The nonlinear inrush current for transformer energization was modelled byintroducing a compensation current source based on the flux-current relationship.Comparison with PSCAD/EMTDC validates the proposed approach. To match theresults from measurements, the influence of remanence has been considered.

Cable impact on small-scale network

Generally, cables increase the line capacitances and reduces the line inductances.Detailed impact of cables depends on cable types, cable layout, circuit configuration,and connection topology. In the analyzed OHL-Cable-OHL schemes, the firstresonance in the transfer function between the voltages at two terminals of thewhole line can shift significantly. Sensitive parameters are the radius of cablecore-conductor, the thickness of cable insulation, the number of cables, and mostimportantly cable length. Different schemes, e.g. OHL/cable sequences or whetherthe connection is a combined OHL-cable or a full-cable connection, should beconsidered separately. Full cable connection is more sensitive to electromagneticcoupling between cables, as well as to the self-earth-return path of each cable.Impedances from connected components to the circuit contribute to additionalresonances.

Cable impact on large-scale network

Regarding the OHL-Cable-OHL between substation Wateringen and Bleiswijk, thecable impact is analyzed with three different cable lengths: 0 km (no cable), 10.8km (real situation), 16.2 km (cable length increased by 50 %). This whole networkis implemented in PSCAD/EMTDC software with necessary simplifications. Theadopted transients are line energization with reclosing, fault and fault clearing,transformer energization, and capacitor bank energization. The worst location (witheither highest voltage or current) is always near the source of disturbance. Noovervoltage exceeding SIWV (Switching Impulse Withstand Voltage) is observed,

6.2. Recommendation for Future Work 117

but the main oscillation frequency is reduced by the cable. In the extreme case,with repeated reclosing it is theoretically possible to have overvoltage higher thanSIWV. Regarding the overvoltages, the presence of power cables decreases its valueat the open-end upon line energization, but increases its value at busbar upon faultclearing. For transformer energization, the distortion in busbar voltage becomesworse with longer cable, although the distortion remains within the 3 % limit. Careshould be taken, e.g. in case of a weaker system. Introducing long cable lengths cancause distortions in excess of 3 %. Particularly for the study using “statistic switch”,the complementation with the proposed frequency-domain approach reduces thesimulation time from months to hours.

With the proposed method to determine the necessary modeling details toaccurately analyze resonant transients, a configuration of the Dutch 380 kVtransmission system is used to demonstrate responses caused by a switching surgeapplied at substation Wateringen. In the simplified model, components that havesmall impact are removed gradually from further area to nearby area, significantlyreducing the time needed for data collection. Generally, components near theinterested area should be modeled in more detail. The requirements for modelingdetails depends on the objective: prediction accuracy of the value of over-voltagesor prediction accuracy of a resonance frequency leads to different requirements.

6.2 Recommendation for Future Work

As part of transmission system, transmission lines can have resonance with externalcomponents. Therefore, it is important to analyze both of the resonances ofthe line itself and the resonances between lines with external components. Asexternal components also connected networks, e.g. at 150 kV, should be considered.Switching or fault in these circuits should not excite resonances in the 380 kV gridand vice versa. Analysis of connected grids and their interaction with the 380 kVnetwork is advised. This information can then be used to adjust some naturalfrequencies of the system to avoid unwanted frequency ranges. The reconstructedtime-domain transients can be used as a reference for applying protective deviceslike surge arrestor, whose effects are then recommended to study.

The proposed approach of frequency-domain transient analysis has beenvalidated in terms of its accuracy and efficiency. Since EMTP-type simulation isalready well-developed for general applications, this approach can be improved as acomplementary technique to EMTP-type simulation.

The proposed approach for determining the necessary modeling details in large-scale network is applied for Dutch 380 kV grid. It can be applied for grids at othervoltage levels, as well as in other countries to see whether general guidelines can bemade.

Appendix A

Configuration of OHL-Cable-OHL

A.1 Configuration of Cable and OHL

The transmission lines between substation Wateringen and substation Bleiswijkconsist of combined overhead lines and cables: OHL1 (4.4 km) - Cable (10.8 km)- OHL2 (6.8 km), as illustrated in Fig. 1.1. Each overhead-line pylon in OHL1comprises a double circuit at 380 kV level and a double circuit at 150 kV level.In the cable part, in total 12 mutually coupled parallel cables (two cables perphase on 380 kV level) are divided into 28 segments with lengths varying from0.1 km up-to 0.7 km. Different trench types are used: 5 kinds of open trench and 2kinds of horizontal directional drilling. Cable segments are grouped into 12 minorsections; three successive minor sections form one major section. Within a majorsection, the minor sections are connected via cross-bonding joints, while two majorsections are connected via a straight-through joint. Two bare 300 mm2 copper wires(compensating wires) are laid underground along the whole cable connection (0.3 mabove the cables belonging to either circuit) to limit step voltages of human beingsabove the ground to a safe level. OHL2 comprises a double circuit at 380 kV leveland 2 compensating wires (bundles each with 2 sub-conductors) to limit the ambientelectromagnetic field at 50 Hz. Note, that in trench types of “horizontal directionaldrilling”, cables and compensating wires are laid in pipes (to be filled with coolingwater), they can be laid deep, e.g. below the bedding of a small river. In this study,the pipes and cooling water and rivers are ignored; in each segment cables are onlyhorizontally laid in homogeneous soil; compensating wires in all trenches along cablesystem are assumed to have the same position as those in trench type O1 (see Fig.A.1 and Table A.1); the depths of cables in trench types H1 and H2 are assumed tobe similar as in open trenches.

119

120 Configuration of OHL-Cable-OHL

Figure A.1: Three main trench types applied in the Randstad380 three-phase(A − B − C) cable system (CW is compensating wire): open trench with five types(Table A.1), (a); horizontal directional drilling (H1, b) and (H2, c).

Figure A.2: Configuration of OHL1 and OHL2 pylons, with 380 kV and 150 kVcircuits, earth wires, and compensating wires (CW)

Configuration of OHL-Cable-OHL 121

Table A.1: Parameters for cables in open trenches (O)

yCable (m) ΔxCable,1 (m) ΔxCable,2 (m)O1 -0.90 0.60 8.45O2 -1.15 0.60 8.45O3 -0.90 0.75 6.95O4 -1.15 0.75 6.95O5 -0.90 0.75 6.95

Table A.2: Parameters for 12 series-connected cable minor sections (MS), OHL1,and OHL2

Trench Length (km) Trench Length (km)

MS01 O5 0.7 MS07 O4 0.6O3 0.2 H1 0.2

MS02 H1 0.4 MS08 O2 0.4O4 0.4 H1 0.5

MS03O2 0.2 MS09 O1 1.0H1 0.5

MS10O2 0.3

O1 0.3 H1 0.1

MS04 H1 0.5 O4 0.5O1 0.4 MS11 H1 0.5

MS05H1 0.3 O1 0.4O1 0.3

MS12

O1 0.3O3 0.3 H1 0.2

MS06 H1 0.5 O3 0.1O4 0.4 H2 0.3

OHL1 - 4.4 OHL2 - 6.8

A.2 Geometry of Single Cable and Line

Power cables contain additional layers aimed to control electrical and mechanicalstresses. These layers cannot directly be implemented in PSCAD/EMTDC andequivalent values for permittivity, permeability, and radii are needed. The overheadlines consist of bundle conductors to reduce electric field stress and to reduce lossesfrom ‘increased resistivity’ by the skin effect. Their models are described below.

Single Cable

The applied 380 kV cable has 7 parts or layers as shown Fig. A.3a. The cablemodel adopted in Section 2.2 only comprises four parts (see Fig. A.3b). Solid inner


conductor and screen separated by an insulation layer, and an outer-sheath aroundit. The actual layers should therefore be replaced by equivalent parts. Specifically,attention should be paid to

• the core-conductor is composed of stranded copper wires;

• there are two semi-conducting layers;

• the copper-wires in earth screen;

• the presence of two metal layers: copper and lead

(a)

(b)

Figure A.3: Single cable configuration: practical (a) and equivalent (b)

Methods for establishing the four equivalent parts in Fig. A.3b are given in[31, 33, 38–40, 80]. Fig. A.4 presents the approach to reduce the actual cableconfiguration into a four-layer model, the key aspects are presented based on theschematics in Fig. A.4.


Figure A.4: Equivalent screen layer

1. The conductor-core of stranded copper wires with resistivity ρCu, has a radiusof rC . Its nominal area (A1) is less than πr2C . In order not to affect the radiiof other layers its radius should be unchanged. An equivalent resistivity ρ1 isobtained instead:

ρC = ρCuπr2CA1

2. The two semi-conducting layers are combined with XLPE insulation layerto form one equivalent insulation layer. The copper wires in the screen aretranslated into a solid copper tube whose thickness is chosen to have the sameDC resistance. The outer radius of this equivalent layer is set to be equal tothe inner radius of lead sheath rx. The inner radius rI and the equivalentpermittivity εI are:

rI =

√r2x − A2

π

εI = εxlpeln (rI/rC)

ln (rv/ru)

where A2 is the nominal area of copper wires in screen. The copper wires ofthe screen are not straight along the cable but laid helical with N of turnsper meter. An equivalent permeability μI of the equivalent insulation layer iscalculated:

μI = μxlpe +μxlpe

ln (rI/rC)· 2π2N2

(r2I − r2C

)


3. One equivalent layer represents both copper and lead parts in the screen. Thislayer has the same resistivity as copper (ρS = ρCu) with adapted thickness toretain the DC resistance of the whole screen.

rS =

√ρSρPb

(r2y − r2x

)+ r2x

where ρPb is the resistivity of lead. To keep the total cable radius constant,the equivalent cable model the thickness of the outer-sheath is changed fromrO − ry to rO − rS . For equal admittance in (2.2) and impedance in (2.8),the permittivity and permeability of the equivalent outer-sheath (εO and μO)should be:

εO = εpe · ln (rO/rS)ln (rO/ry)

μO = μpe · ln (rO/ry)ln (rO/rS)

Table A.3 shows the parameter values for the equivalent cable model used forthe 380 kV cable applied in the Randstad380 project. The use of these equivalentlayers as compared to a double layer (inside copper outside lead, Fig. A.4 middle)is analyzed in Appendix A.3 and for the frequency range of interest for resonantresponse up to 10 kHz it is accurate within 0.1%.

Table A.3: Parameters for cable equivalent model. Permittivity and permeabilityare relative values (N ≈ 1.4 turns/m)

rC (mm) 30.7 ρC (Ωm) 1.98× 10−8 εC,r - μC,r 1rI (mm) 63.1 ρI (Ωm) - εI,r 2.79 μI,r 1.17rS (mm) 63.7 ρS (Ωm) 1.68× 10−8 εS,r - μS,r 1rO (mm) 71.5 ρO (Ωm) - εO,r 3.05 μO,r 0.76

Overhead Line Data

The data for the overhead lines are given in Table A.4. The relative permeability ofevery conductor is 1.


Table A.4: Parameters of overhead line conductors. EW and CW are abbreviationsfor Earth Wires and Compensating Wires, respectively. The values for 380 kV levelare equivalent to a bundle of 4 sub-conductors. The values for CW are equivalent toa bundle of 2 sub-conductors

380 kV 150 kV EW CWresistivity (Ωm) 2.05× 10−6 4.02× 10−8 4.78× 10−8 1.10× 10−6

radius (mm) 231 16.2 10.9 73.8

Commonly, one equivalent conductor is used to represent a bundle of sub-conductors, see Fig. A.5. The value of the equivalent radius can be found by[19,73,81]

req =n√nrRn−1

Figure A.5: A bundle of n subconductors

A.3 Validation of Model for Cable Screen

The impedance matrix of a single cable with one solid screen layer contains 7impedance components z1 to z7, see Section 2.2.1. Only impedances z3, z4, andz5 depend on the material of the screen. For a cable with double-layer screen (inFig. A.4), these three impedance components should be replaced by z3d, z4d, andz5d [38, 80]:

z3d =mS1ρS1

2πrIM(mS1ρS1FQ+mS2ρS2EP ) (A.1)

z4d =ρS1ρS2

2πrIrxryM(A.2)

z5d =mS2ρS2

2πryM(mS1ρS1GR+mS2ρS2HS) (A.3)


Table A.5: Radii of the double-layer screen of the cable

rI (mm) rx (mm) ry (mm)63.1 63.5 35.5

where subscripts “S1” and “S2” represents the inner and outer layer of the double-layer screen (layers rx − rI and ry − rx in Fig. A.4), respectively.

M = mS1ρS1FG+mS2ρS2EH

E = I0(x3)K1(x4) + I1(x4)K0(x3)

F = I1(x4)K1(x3)− I1(x3)K1(x4)

G = I0(x2)K1(x1) + I1(x1)K0(x2)

H = I1(x2)K1(x1)− I1(x1)K0(x2)

P = I0(x1)K1(x2) + I1(x2)K0(x1)

Q = I0(x2)K0(x1)− I0(x1)K0(x2)

R = I0(x4)K1(x3) + I1(x3)K0(x4)

S = I0(x4)K0(x3)− I0(x3)K0(x4)

mS1 =

√jωμS1

ρS1, mS2 =

√jωμS2

ρS2

x1 = mS1rI , x2 = mS1rx, x3 = mS2rx, x4 = mS2ry

Validation of the cable model with equivalent single-layer screen for a cablewith double-layer screen (see Appendix A.2) is needed, so that the applied cablein Randstad380 project can be implemented into softwares whose cable model hasonly one screen layer, for instance PSCAD/EMTDC. Consider one segment of asingle cable with double-layer screen shown in Fig. A.6 and Table A.5, laid at 1.4m under the ground with earth resistivity assumed as 100 Ωm. The used resistivityof copper and lead are 1.68× 10−8 Ωm and 2.20× 10−7 Ωm, respectively.

Figure A.6: One segment of single cable with 1 km long. Each terminal of the cablescreen is earthed via an resistor of 1 m Ω.


With the approach in Appendix B, this cable can be described by its ABCD-matrix at each frequency.[

UCp

ICp

]=

[Ad Bd

Cd Dd

] [UCq

ICq

], for cablewith double− layer screen (A.4)

[UCp

ICp

]=

[A BC D

] [UCq

ICq

], for cablewith single− layer screen (A.5)

Fig. A.7 illustrates the validation for frequencies up to 500 kHz. The error for eachelement in ABCD-matrices of the two models

ErrX = |Xd −X

Xd

|, with X = A,B,C, D.


100

102

104

106

0

0.5

1

1.5

Frequency (Hz)

|A|

100

102

104

106

10−15

10−10

10−5

100

Frequency (Hz)

Err

A

100

102

104

106

0

10

20

30

Frequency (Hz)

|B|

100

102

104

106

10−10

10−5

100

Frequency (Hz)

Err

B

100

102

104

106

0

0.02

0.04

0.06

Frequency (Hz)

|C|

100

102

104

106

10−15

10−10

10−5

100

Frequency (Hz)

Err

C

100

102

104

106

0

0.5

1

1.5

Frequency (Hz)

|D|

100

102

104

106

10−15

10−10

10−5

100

Frequency (Hz)

Err

D

Figure A.7: Validation of cable model with equivalent single-layer screen for a cablewith double-layer screen: (left) components in ABCD-matrix of cable model withequivalent single-layer screen, (right) errors comparing to cable model with double-layer screen.

Appendix B

Matrix Manipulation

This appendix provides an overview of manipulation methods applied for Chapter2 and 3. The construction of the ABCD-matrix is described and ways to exchangeand combine matrix elements, e.g. to deal with parallel and series connected cables.

B.1 ABCD-Matrix

The ABCD-matrix equation formulated as[Up

Ip

]=

[A BC D

] [Uq

Iq

](B.1)

provides the relationship of voltages and currents on the terminals p and q of anysystem (see Fig. B.1).

Figure B.1: A network with 2 terminals and n ports

This section demonstrates a systematic approach to obtain the ABCD-matrixof impedances and admittances (lumped components), and transmission lines(distributed components).

B.1.1 Impedances and Admittances

A network (dashed box in Fig. B.2) containing either n series impedances or n shuntadmittances can be described as

129

130 Matrix Manipulation

(a) (b)

Figure B.2: Illustration of impedances (a) and admittances (b)

Uk,p = Zk · Ik,q + Uk,q

Ik,p = Ik,q

}⇒ for series impedances, Fig. B.2a

Uk,p = Uk,q

Ik,p = Y k · Uk,q + Ik,q

}⇒ for shunt admittances, Fig. B.2b

where k = 1 . . . n. Thus, the corresponding ABCD-matrix equation of the networkis [

Up

Ip

]=

[A BC D

] [Uq

Iq

](B.2)

Uη =[U1,η · · · Un,η

]T, Iη =

[I1,η · · · In,η

]T, η = p or q

where

• for series impedances (Fig. B.2a)

[A BC D

]=

[I ZO I

], Z =

⎡⎢⎣

Z1

. . .

Zn

⎤⎥⎦ (Z is diagonal)

• for shunt admittances (Fig. B.2b)

[A BC D

]=

[I OY I

], Y =

⎡⎢⎣

Y 1

. . .

Y n

⎤⎥⎦ (Y is diagonal)

I and O are the n-by-n identity matrix and null matrix, respectively.

Matrix Manipulation 131

B.1.2 Transmission Lines

Transmission lines (e.g. Fig. B.3) are commonly modeled by their series impedancematrix Z (Ω/m) and shunt admittance matrix Y (S/m), satisfying the following twomatrix-equations [28]

− d

dzU = ZI, − d

dzI = YU (B.3)

where

Z =

⎡⎢⎣

Z11 · · · Z1n...

. . ....

Zn1 · · · Znn

⎤⎥⎦ , Y =

⎡⎢⎣

Y 11 · · · Y 1n...

. . ....

Y n1 · · · Y nn

⎤⎥⎦

U =[U1 · · · Un

]T, I =

[I1 · · · In

]T

Figure B.3: Illustration of general transmission line

Solving these differential equations can be facilitated by rearranging thecomponents into the form:

d

dzV = KV (B.4)

K =

[O −Z−Y O

]V =

[UI

],

where O is a null matrix with the same dimension as Z and Y.With KT = TΛ, where Λ is a diagonal matrix of all eigenvalues of K,

Λ =

⎡⎢⎣

λ1

. . .

λ2n

⎤⎥⎦

the columns of matrix T are the corresponding eigenvectors [82]. If T is invertible,K can be diagonalized as

Λ = T−1KT. (B.5)


Assume V = TS, where S = [s1, · · · , s2n]T, then (B.4) can be rewritten as

Td

dzS = KTS ⇒ d

dzS = T−1KTS = ΛS.

Since Λ is diagonal, the equation above can be decoupled into

ds1dz

= λ1s1, · · · , ds2ndz

= λ2ns2n

Their solutions can be written in matrix form

S(z) = eΛzS(0)

where S(0) is the boundary value of S for z = 0. Thus, the solution to (B.4) is

V(z) = TS(z)

= TeΛzS(0)

= TeΛzT−1V(0)

Consequently, the voltage and current at any location on the line, U(z) and I(z)(0 ≤ z ≤ D), can be related to the voltage and current at z = 0:[

U(z)I(z)

]= TeΛzT−1

[U(0)I(0)

]

Referring to the two terminals of the line (p: z = 0, q: z = D), the correspondingABCD-matrix is[

Up

Ip

]=

[A BC D

]·[Uq

Iq

](B.6)[

A BC D

]=(TeΛDT−1

)−1

π-Model

If the transmission lines in Fig. B.3 are short, they can be approximately representedby the π-model [30],[

Up

Ip

]=

[I+ Zα ·Yα Zα

(2 · I+Yα · Zα)Yα I+Yα · Zα

] [Uq

Iq

](B.7)

where

Zα = Z ·D, Yα =1

2Y ·D


B.2 Manipulation of Matrix Elements

This section describes methods to manipulate an arbitrary matrix equation (g =kx): ⎡

⎢⎢⎢⎢⎢⎢⎢⎣

...ga...gb...

⎤⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

......

......

...· · · ka,a · · · ka,b · · ·...

......

......

· · · kb,a · · · kb,b · · ·...

......

......

⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎣

...xa

...xb

...

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(B.8)

where vectors g and x have m and n elements respectively; k is an m-by-n matrix.

B.2.1 Row-Rearrangement of Matrix Equation

Exchanging rows a and b of matrix k in (B.8) can be realized by left-multiplyinga row-rearranging matrix R (also called permutation matrix), which is an m-by-midentity matrix with re-ordered rows:

⎡⎢⎢⎢⎢⎢⎢⎢⎣

......

......

...· · · kb,a · · · kb,b · · ·...

......

......

· · · ka,a · · · ka,b · · ·...

......

......

⎤⎥⎥⎥⎥⎥⎥⎥⎦= Rk, R =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1. . .

0 1b. . .

1a 0. . .

1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

The transformation allows exchanging components a and b in both g and x of(B.8). Here g and x are assumed to have the same dimension and k is a squarematrix.⎡

⎢⎢⎢⎢⎢⎢⎢⎣

...gb...ga...

⎤⎥⎥⎥⎥⎥⎥⎥⎦= RkR−1

⎡⎢⎢⎢⎢⎢⎢⎢⎣

...xb

...xa

...

⎤⎥⎥⎥⎥⎥⎥⎥⎦

B.2.2 Exchange Elements in Matrix Equation

Exchange Two Components from Left and Right Vectors To exchange gaand xb in (B.8), we can extract the corresponding equation (containing ga and xb)


from the matrix equation

ga = · · · + ka,axa + · · · + ka,bxb · · ·and solve xb

xb =1

ka,b[· · · − ka,axa − · · · + ga − · · · ] .

Inserting xb into the remaining equations in (B.8) results in

G = KX

where

Gi =

{gi if i �= a

xb if i = a

Ki,j =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

ki,j − ka,j

ka,bki,b if i �= a, j �= b

1ka,b

ki,b if i �= a, j = b

−ka,j

ka,bif i = a, j �= b

1ka,b

if i = a, j = b

Xj =

{xj if j �= b

ga if j = b

with i = 1, . . . ,m, and j = 1, . . . , n.

Exchange Complete Left and Right Vectors Exchanging the whole left andright vectors g and x, by solving equation g = kx, in (B.8) has two situations:

• if k is a n-by-n square matrix and is invertible (determinant is not zero), thenthe aimed equation matrix can be achieved by taking the inverse of k

x = k−1g

• if k is not a square matrix but m-by-n (m �= n), it needs kTk, since k cannotbe inverted. First, multiply kT at both side of the matrix equation (left-multiplication)

kTg = kTkx

Then, x and g can be exchanged as

x =(kT k

)−1kT g


Exchange ABCD-Matrix and Y-Matrix Notation Suppose (B.8) has theform of ABCD-matrix (B.1). It can be transformed to Y-matrix form or vice versa:[

Up

Ip

]=

[A BC D

] [Uq

Iq

]⇔

[IpIq

]=

[Y11 Y12

Y21 Y22

] [Up

Uq

]

where

A = −Y−121 ·Y22 Y11 = DB−1

B = Y−121 Y12 = C−DB−1A

C = Y12 −Y11 ·Y−121 ·Y22 Y21 = B−1

D = Y11 ·Y−121 Y22 = −B−1A

B.2.3 Add Boundary Conditions to Matrix Equations

This subsection describes the method to add boundary conditions to (B.8).

Condition for Equality: Assigning xa = xb =̂ xc to (B.8) can be achieved by:

1. adding column b to column a followed by elimination of column b in matrix k;

2. replacing xa by xc and eliminating xb;

3. keeping vector g unchanged.

The resulting matrix equation is

G = KX

where

Gi = gi

Ki,j =

⎧⎪⎨⎪⎩ki,j if j < b and j �= a

ki,j + ki,b if j = a

ki,j+1 if j ≥ b and j �= a

(B.9)

Xj =

⎧⎪⎨⎪⎩xj if j < b and j �= a

xc if j = a

xj+1 if b ≤ j ≤ (n− 1) and j �= a

with i = 1, . . . ,m and j = 1, . . . , n. Therefore, vectors G and X have m and n− 1elements, respectively; K has dimension of m-by-(n− 1).


Condition for Summation: Assigning ga + gb =̂ gc to (B.8) is achieved by:

1. adding row b to row a followed by eliminating row b in matrix k;

2. replacing ga by gc and eliminating gb;

3. keeping vector x unchanged.

The resulting matrix equation is

G = KX

where

Gi =

⎧⎪⎨⎪⎩gi if i < b and i �= a

gc if i = a

gi+1 if b ≤ i ≤ (m− 1) and i �= a

Ki,j =

⎧⎪⎨⎪⎩ki,j if i < b and i �= a

ki,j + kb,j if i = a

ki,j+1 if i ≥ b and i �= a

(B.10)

Xj = xj

with i = 1, . . . ,m and j = 1, . . . , n. Therefore, vectors G and X have m− 1 and nelements, respectively; K has dimension of (m− 1)-by-n.

B.2.4 Application Examples

To demonstrate the manipulations above in the field of power system analysis, thissubsection uses them for the network in Fig. B.4, whose corresponding ABCD-matrix equation is

[ · · · Ua,p · · · U b,p · · · Ia,p · · · Ib,p · · · ]T=

[A BC D

]· [ · · · Ua,p · · · U b,p · · · Ia,p · · · Ib,p · · · ]T (B.11)


Figure B.4: Network with two terminals, each terminal has several ports

Example 1The boundary conditions given in Fig. B.5 are

Ua,q = U b,q =̂ U c,q

Ia,q + Ib,q =̂ Ic,q

Figure B.5: Two ports at same terminal in a network are connected

The procedure to combine them with (B.11) is:

1. change (B.11) to Y-matrix form using the method described in Section B.2.2.

[ · · · Ia,p · · · Ib,p · · · Ia,q · · · Ib,q · · · ]T= Y · [ · · · Ua,p · · · U b,p · · · Ua,q · · · U b,q · · · ]T (B.12)

2. apply the methods for “Equality” and “Summation” of Section B.2.3:[ · · · Ia,p · · · Ib,p · · · Ic,q · · · ]T= Ynew · [ · · · Ua,p · · · U b,p · · · U c,q · · · ]T (B.13)


Note that Ynew has one row and one column less than Y in (B.12).

Example 2The boundary conditions given by Fig. B.6 are

Ua,p = U b,q =̂ U c,q

−Ia,p + Ib,q =̂ Ic,q

Figure B.6: Two ports at different terminals in a network are connected

The procedure to combine them with (B.11) is similar to Example 1, but thesign of Ia,p is minus.

Example 3Fig. B.7 can been seen as the port c in Fig. B.5, which is connected to a subsequentnetwork (network 2) with ABCD-matrix

[Ud,p

Id,p

]=

[A2 B2

C2 D2

] [Ud,q

Id,q

].

The boundary conditions contain:

• inter-connected ports, which can be facilitated by Y-matrix formulation

• cascaded networks, which can be facilitated by ABCD-matrix formulation

Therefore the procedure is:


Figure B.7: Two connected ports are connected to another network

1. starting from the Y-matrix form of (B.13). Exchange Ic,q with the quantityafter U c,q (i.e. Ua+1,q, since Ua,q is replaced by U c,q) with the methoddescribed in Section B.2.2:

[ · · · Ia,p · · · Ib,p · · · U c+1,q Ic+1,q · · · ]T= K · [ · · · Ua,p · · · U b,p · · · U c,q Ic,q · · · ]T

2. extend the ABCD-matrix of network 2 with the identity matrix to reach thesame dimension as matrix K, so that the quantities U c,q and Ic,q can bereplaced by Ud,q and Id,q.8

[Ud,p

Id,p

]=

[A2 B2

C2 D2

] [Ud,q

Id,q

]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

...Ua−1,q

U c,q

Ic,qUa+2,q

...

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

...Ua−1,q

Ud,p

Id,pUa+2,q

...

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

. . .. . .

......

... . ..

. . . 1 0 0 0 · · ·· · · 0 A2 B2 0 · · ·· · · 0 C2 D2 0 · · ·· · · 0 0 0 1

. . .

. .. ...

......

. . .. . .

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

...Ua−1,q

Ud,q

Id,qUa+2,q

...

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

8Intention is to replace port c at terminal q of network 1 by port d at terminal q of network 2.


3. consequently, all the unconnected ports can be correlated by:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

...Ia−1,q

Ua+1,q

Ia+1,q

Ia+2,q...

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦= K

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

. . .. . .

......

... . ..

. . . 1 0 0 0 · · ·· · · 0 A2 B2 0 · · ·· · · 0 C2 D2 0 · · ·· · · 0 0 0 1

. . .

. .. ...

......

. . .. . .

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

...Ua−1,q

Ud,q

Id,qUa+2,q

...

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

Note that Ua+1,q is the only voltage quantity on left side vector, and Id,q isthe only current quantity on the right side vector. Exchanging these two quantitiesgenerates the Y matrix form of the combined networks 1 and 2, the correspondingABCD-matrix form can also be obtained by the method described in Section B.2.2.

B.2.5 Screen Layer Elimination Method

In a cable system consisting of n cables with cable screens earthed via an impedanceat both terminals, the rows related to the screen layer in the ABCD-matrix can beeliminated.

Figure B.8: Illustration of cable screen elimination

Assume the ABCD-matrix equation for the cable configuration in Fig. B.8 hasthe form

⎡⎢⎢⎣

UpC

UpS

IpCIpS

⎤⎥⎥⎦ =

⎡⎢⎢⎣

A11 A12 B11 B12

A21 A22 B21 B22

C11 C12 D11 D12

C21 C22 D21 D22

⎤⎥⎥⎦⎡⎢⎢⎣

UqC

UqS

IqCIqS

⎤⎥⎥⎦ (B.14)


The corresponding terminal conditions are UpS = −ZpSIpS and UqS = ZqSIqS ,where ZpS and ZpS are both diagonal matrices

ZpS =

⎡⎢⎣

ZpS

. . .

ZpS

⎤⎥⎦ , ZqS =

⎡⎢⎣

ZqS

. . .

ZqS

⎤⎥⎦

The reduced matrix equation involving only conductor related quantities becomes[UpC

IpC

]=

[AnoS BnoS

CnoS DnoS

] [UqC

IqC

](B.15)

where the subscript “noS” means “no screen” and

AnoS = A11 − (A12ZqS +B12) ·E · (ZpSC21 +A21)

BnoS = B11 − (A12ZqS +B12) ·E · (ZpSD21 +B21)

CnoS = C11 − (C12ZqS +D12) ·E · (ZpSC21 +A21)

DnoS = D11 − (C12ZqS +D12) ·E · (ZpSD21 +B21)

E = [(A22ZqS +B22) + ZpS (C22ZqS +D22)]−1

B.2.6 Solving a Matrix System

Solving a system means obtaining the unknown voltages or currents of all ports ofa system. This subsection describes a systematic approach to solve a system witharbitrary ports (N), which can be represented by its Y-matrix equation:

⎡⎢⎣

I1...IN

⎤⎥⎦ =

⎡⎢⎣

Y 1,1 · · · Y 1,N...

. . ....

Y N,1 · · · Y N,N

⎤⎥⎦⎡⎢⎣

U1...

UN

⎤⎥⎦ (B.16)

There are only 2 conditions for any port (m):

1. Um is known, Im is unknown: Um = αm, Ik = βm

a) if port m is connected to a voltage source US, then Um = US

b) if port m is earthed, then Um = 0

2. Im is known, Um is unknown: Im = αm, Um = βk

a) if port m is connected to a current source IS, then Im = IS

b) if port m is open-ended, then Im = 0


Note that even if a port is earthed via an impedance, it can be represented by anextended earthed port, see the following example. Therefore, applying the techniquesin Section B.2.2 to (B.16), the known and unknown quantities (α1 to αN and β1 toβN ) can be assembled separately so that all the unknowns can be obtained:⎡

⎢⎣β1

...βN

⎤⎥⎦ =

⎡⎢⎣

K1,1 · · · K1,N...

. . ....

KN,1 · · · KN,N

⎤⎥⎦⎡⎢⎣

α1

...αN

⎤⎥⎦ (B.17)

Example

Suppose a system is composed of a three-phase transmission line with two sourcesand a load, see Fig. B.9.

Figure B.9: A three-phase system example

Consider the system defined by the dashed box with totally 5 ports, whoseresulting matrix equation (β = Kα) is:⎡

⎢⎢⎢⎢⎣UA,ex,p

IC,ex,q

IC,ex,p

IA,ex,q

UB,ex,q

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎣

K1,1 · · · K1,5...

. . ....

K5,1 · · · K5,5

⎤⎥⎦⎡⎢⎢⎢⎢⎣

IA,ex,p = ISUC,ex,q = US

UC,ex,p = 0UA,ex,q = 0IB,ex,q = 0

⎤⎥⎥⎥⎥⎦

Note that methods presented in Section B.2.3 and B.2.4 can be applied to tacklethe connection of phases A and B at terminal p as well as the connection with animpedance in phase A at terminal q.

B.3 Parallel Connection of Multiple Lines

The parallel connection of multiple lines is a special condition for portinterconnections of a network, but is very common in practise. The methods inSection B.2 can of course be applied to the parallel connection cases, but theprocedure can be cumbersome when there are large number of parallel connectedlines. Therefore, this section describes an alternative method particularly to handleparallel connection.


B.3.1 Multiple Lines for Single Phase

pU

pI

qU

qICable 1

Cable 2

Cable N

Figure B.10: Parallel Connection of N Cables

Fig. B.10 shows N parallel connected lines. Its ABCD-matrix equation⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

U1p...

UNp

I1p...

INp

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

KA11 · · · KA1N KB11 · · · KB1N...

. . ....

.... . .

...KAN1 · · · KANN KBN1 · · · KBNN

KC11 · · · KC1N KD11 · · · KD1N...

. . ....

.... . .

...KCN1 · · · KCNN KDN1 · · · KDNN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

U1q...

UNq

I1q...

INq

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B.18)

can be transformed into Y-matrix form:⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

I1p...

INp

I1q...

INq

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

MA11 · · · MA1N MB11 · · · MB1N...

. . ....

.... . .

...MAN1 · · · MANN MBN1 · · · MBNN

MC11 · · · MC1N MD11 · · · MD1N...

. . ....

.... . .

...MCN1 · · · MCNN MDN1 · · · MDNN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

U1p...

UNp

U1q...

UNq

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B.19)

Inserting the boundary conditions of parallel connection

U1p = · · · = UNp=̂Up

U1q = · · · = UNq=̂U q

I1p + · · ·+ INp=̂Ip

I1q + · · ·+ INq=̂Iq

describes the parallel connection in the form of Y-matrix equation.[IpIq

]=

[MA MBMC MD

] [Up

U q

]


where MA, MB, MC, and MD are respectively the sum of all the elements in thesub-matrices

⎡⎢⎣

MA11 · · · MA1N...

. . ....

MAN1 · · · MANN

⎤⎥⎦ ,

⎡⎢⎣

MB11 · · · MB1N...

. . ....

MBN1 · · · , MBNN

⎤⎥⎦

⎡⎢⎣

MC11 · · · MC1N...

. . ....

MCN1 · · · MCNN

⎤⎥⎦ ,

⎡⎢⎣

MD11 · · · MD1N...

. . ....

MDN1 · · · MDNN

⎤⎥⎦

With the method in Section B.2.2, the Y-matrix equation can be changed to thecorresponding ABCD-matrix equation:

[Up

Ip

]=

⎡⎣ −M−1

C MD M−1C

MB −MAM−1C MD MAM

−1C

⎤⎦[ Uq

Iq

](B.20)

B.3.2 Multiple Lines for Three Phases

This equation can directly be generalized when each line in Fig. B.10 represents a3-phase system (phases A, B, and C). Since only the lines in the same phase willbe parallel connected, all the “U” and “I” quantities in (B.18) must be replaced by3-by-1 column vectors, for example:

U1p =

⎡⎣ UA1p

UB1p

UC1p

⎤⎦ , I1p =

⎡⎣ IA1p

IB1p

IC1p

⎤⎦

Every element in the K-matrix, for example KA11 in (B.18), becomes a 3-by-3matrix.

Fig.B.11 illustrates that a 3-phase system can have different parallel connectionstyles. For each style, in order to use the Eq.B.20, the rows rearrangement in ABCD-matrix must be applied accordingly. For style α the phase order of I-III is inverseto IV-VI, while for style β the phase order of I-III is equal to IV-VI. As an example,only style α is considered. The analysis is similar for style β.


I II III

IV V VI

A B C

A B C

Style α

Style β

Terminal P Terminal Q

I II III

IV V VI

Figure B.11: Two Examples of Parallel Connection Style

The complete model can be constructed according to the following steps.

1. construct the ABCD-matrix, indicated with K, equation:⎡⎢⎢⎣

U1p

U2p

I1pI2p

⎤⎥⎥⎦ = K

⎡⎢⎢⎣

U1q

U2q

I1qI2q

⎤⎥⎥⎦ (B.21)

The “U” and “I” components are 3-by-1 column vectors containing elementsof the three phases; the subscripts “1” and “2” indicate the two three-phasecircuits.

U1p =[U Ip U IIp U IIIp

]TU2p =

[U IVp UVp UVIp

]TI1p =

[IIp IIIp IIIIp

]TI2p =

[IIVp IVp IVIp

]T2. Lines I and VI are parallel connected, as well as lines II and V, III and IV.

Thus the rows in U2p, U2q, I2p, and I2q must be rearranged accordingly. Therow-rearrangement matrix R//, obtained using the method in Section B.2.1,results in⎡

⎢⎢⎣U1p

U′2p

I1pI′2p

⎤⎥⎥⎦ = R// ·K ·R−1

//

⎡⎢⎢⎣

U1q

U′2q

I1qI′2q

⎤⎥⎥⎦ (B.22)


where

U′2p =

⎡⎣ UVI1p

UV1p

U IV1p

⎤⎦ , U′

2q =

⎡⎣ UVI1q

UV1q

U IV1q

⎤⎦ , I′2p =

⎡⎣ IVI1p

IV1p

IIV1p

⎤⎦ , I′2q =

⎡⎣ IVI1q

IV1q

IIV1q

⎤⎦

R// =

⎡⎢⎢⎣

I O O OO E O OO O I OO O O E

⎤⎥⎥⎦ , I =

⎡⎣ 1 0 0

0 1 00 0 1

⎤⎦ , E =

⎡⎣ 0 0 1

0 1 01 0 0

⎤⎦

3. The ABCD-matrix for the parallel connection is achieved by following themethod in Section B.3.1.

Bibliography

[1] L.L. Grigsby. The Electric Power Engineering Handbook. CRC Press LLC,2001.

[2] L.M. Faulkenberry and W. Coffer. Electrical Power Distribution andTransmission. Prentice-Hall, 1996.

[3] N. Watson and J. Arrillaga. Power Systems Electromagnetic TransientsSimulation. The Institute of Electrical Engineers, London, 2003.

[4] P.M. Anderson. Analysis of Faulted Power Systems. IEEE Press, 1995.

[5] TenneT. Randstad380 south-ring project. the Netherlandshttp://www.randstad380kv-zuidring.nl/.

[6] L. Powell. Power System Load Flow Analysis. McGraw-Hill, 2005.

[7] L. van der Sluis. Transients in Power Systems. John Wiley & Sons. LTD, 2001.

[8] A. Greenwood. Electrical transients in power systems. 2ed ed., John Wiley &Sons, Inc., 1991.

[9] M.G. Ippolito, F. Massaro, G. MORANA, and R. Musca. No-load energizationof very long ehv mixed overhead-cable lines. In Universities Power EngineeringConference (UPEC), 2010 45th International, pages 1–6, Aug 2010.

[10] D. Chanda, N.K. Kishore, and A.K. Sinha. A wavelet multiresolution-basedanalysis for location of the point of strike of a lightning overvoltage on atransmission line. IEEE Transactions on Power Delivery, 19(4):1727–1733, Oct2004.

[11] M. Cepin. Assessment of Power System Reliability: Methods and Applications.Springer-Verlag London Limited, 2011.

147

148 Bibliography

[12] L. Colla, M. Rebolini, and F. Iliceto. 400 kv ac new submarine cable linksbetween sicily and the italian mainland. outline of project and special electricalstudies. pages C4–116, 2008.

[13] W.L. Weeks and Y.M. Diao. Wave propagation characteristics in undergroundpower cable. IEEE Transactions on Power Apparatus and Systems, PAS-103(10):2816–2826, Oct. 1984.

[14] A. Ametani. The history and recent trends of transient analysis in transmissionlines. In IPST, Vancouver, 2013.

[15] D.H. Moore and E.T. Whittaker. Heaviside operational calculus: an elementaryfoundation. American Elsevier Publishing Company New York, 1971.

[16] J.R. Carson. Electric circuit theory and the operational calculus. ChelseaPublishing Company, 1953.

[17] S. Goldman. Laplace Transform Theory and Electrical Transients. DoverPublications, New York, 1966.

[18] G.W. Carter. The Simple Calculation of Electrical Transients. CambridgeUniversity Press, New York, 1944.

[19] H.W. Dommel. Electromagnetic Transients Program: Reference Manual:(EMTP Theory Book). Bonneville Power Administration, 1986.

[20] C.G. Kaloudas, C.G. Papadopoulos, and G.K. Papagiannis. Spectrum analysisof transient responses of overhead transmission lines. In 45th Int. Conf.Universities Power Engineering Conference (UPEC), pages 1–5, Cardiff, Wales,Aug. 31 2010-Sept. 3 2010.

[21] L.M. Wedepohl and S.E.T. Mohamed. Multiconductor transmission lines:Theory of natural modes and fourier integral applied to transient analyhsis.In Proc. IEE, volume 116, pages 1553–1563, Sept. 1969.

[22] L.M. Wedepohl and S.E.T. Mohamed. Transient analysis of multiconductortransmission lines with special reference to nonlinear problems. In Proc. IEE,volume 117, pages 979–988, Sept. 1970.

[23] N. Nagaoka and A. Ametani. A development of a generalized frequency-domaintransient program - ftp. IEEE Transactions on Power Delivery, 3(4):1996–2004,October, 1988.

[24] P. Moreno, R. de la Rosa, and J.L. Naredo. Frequency domain computationof transmission line closing transients. IEEE Transactions on Power Delivery,6(1):275–281, Janurary, 1991.

Bibliography 149

[25] F.A. Uribe, J.L. Naredo, P. Moreno, and L. Guardado. Electromagnetictransients in underground transmission systems trough the numerical laplacetransform. International Journal of Electrical Power & Energy Systems, 24:215–221, March, 2002.

[26] B. Gustavsen. Validation of frequency-dependent transmission line models.IEEE Transactions on Power Delivery, 20(2):925–933, April, 2005.

[27] P. Moreno and A. Ramirez. Implementation of the numerical laplace transform:a review. IEEE Transactions on Power Delivery, 23(4):2599–2609, October,2008.

[28] R.P. Clayton. Analysis of multiconductor transmission lines. John Wiley &Sons, Inc., ISBN:0-471-02080-X, 1994.

[29] P. Gómez, P. Moreno, and J.L. Naredo. Frequency domain transient analysis ofnonuniform lines with incident field excitation. IEEE Transactions on PowerDelivery, 20:2273–2280, July, 2005.

[30] H. Saadat. Power system analysis. WCB/McGraw-Hill Singapore, 1999.

[31] Manitoba HVDC Research Centre Inc. Emtdc user’s guide. Winnipeg, MB,Canada, 2010.

[32] O. Saad, G.Gaba, and M. Giroux. A closed-form approximation for groundreturn impedance of underground cables. IEEE Transactions on PowerDelivery, 11(3):1536–1545, 1996.

[33] S.A. Schelkunoff. The electromagnetic theory of coaxial transmission line andcylindrical shields. Bell Syst. Tech. J., 13:532–579, 1934.

[34] L.M. Wedepohl and D.J. Wilcox. Transient analysis of underground power-transmission systems. system-model and wave-propagation characteristics.Electrical Engineers, Proceedings of the Institution of, 120(2):253–260, 1973.

[35] W. Enright, O.B. Nayak, G.D. Irwin, and J. Arrillaga. An electromagnetictransients model of multi-limb transformers using normalized core concept. InIPST, Seattle, pages 93–98, 1997.

[36] S.N. Vukosavic. Electrical Machines. Springer, 2013.

[37] L. Wu, H. Fonk, P.A.A.F. Wouters, and E.F. Steennis. Influence by parasiticcapacitances on frequency response of a 380-150-50 kv transformer with shuntreator. In ISH, Seoul, pages 56–60, August 2013.

[38] B.J.H. de Bruyn, L. Wu, P.A.A.F. Wouters, and E.F. Steennis. Equivalentsingle-layer power cable sheath for transient modeling of double-layer sheaths.In PowerTech (POWERTECH), 2013 IEEE Grenoble, pages 1–6, June 2013.

150 Bibliography

[39] U.S. Gudmundsdottir, B. Gustavsen, C.L. Bak, and W. Wiechowski. Field testand simulation of a 400-kv cross-bonded cable system. IEEE Transactions onPower Delivery, 26(3):1403–1410, July 2011.

[40] B. Gustavsen. Panel session on data for modeling system transients insulatedcables. In Power Engineering Society Winter Meeting, 2001. IEEE, volume 2,pages 718–723 vol.2, 2001.

[41] A.B. George, H.J. Weber, and F.E. Harris. Mathematical Methods forPhysicists. A Comprehensive Guide. Academic Press, ELsevier, 7th Edition,2013.

[42] A. Ametani. A general formulation of impedance and admittance of cables.IEEE Transactions on Power Apparatus and Systems, pages 902–910, 1980.

[43] L.M. Wedepohl and C.S. Indulkar. Wave propagation in nonhomogeneoussystems. properties of the chain matrix. In Proceedings of the Institution ofElectrical Engineers, volume 121, pages 997–1000, September 1974.

[44] L. Wu, P.A.A.F. Wouters, and E.F. Steennis. Model of a double circuit withparallel cables for each phase in a hv cable connection. In IEEE InternationalConference on Power System Technology (POWERCON), Auckland, pages 1–5,November 2012.

[45] A.B. Fernandes, W.L.A. Neves, E.G. Costa, and M.N. Cavalcanti. The effect ofthe shunt conductance on transmission line models. In IPST, Rio de Janeiro,2001.

[46] A. Deri, G. Tevan, A. Semlyen, and A. Castanheira. The complex ground returnplane - a simplified model for homogeneous and multi-layer earth return. IEEETransactions on Power Apparatus and Systems, (8):3686–3693, August 1981.

[47] W.D. Stevenson and J.J. Grainger. Power System Analysis. McGraw-HillInternational Editions, 1994.

[48] A.D. Theocharis, J. Milias-Argitis, and T. Zacharias. Three-phase transformermodel including magnetic hysteresis and eddy currents effects. IEEETransactions on Power Delivery, 24(3):1284–1294, July, 2009.

[49] J.A. Martinez and B.A. Mork. Transformer modeling for low frequencytransients-a review. IEEE Transactions on Power Delivery, 20(2):1625–1632,April, 2005.

[50] F. de Leon and A. Semlyen. Transformer model for electromagnetic transients.IEEE Transactions on Power Delivery, 9(1):231–239, January, 1994.

Bibliography 151

[51] H.W. Dommel. Transformer models in the simulation of electromagnetictransients. In 5th Power Systems Computation Conference, 1975.

[52] H.W. Dommel. Digital computer solution of electromagnetic transients in singleand multiphase networks. IEEE Transactions on Power Apparatus and Systems,(4):388–399, April,1969.

[53] V. Brandwajn, H.W. Dommel, and I.I. Dommel. Matrix representation ofthree phase n-winding transformers for steady state transient studies. IEEETransactions on Power Apparatus and Systems, (6):1369–1378, June,1982.

[54] W. Enright. Transformer models for electromagnetic transient studies withparticular reference to hvdc transmission. 1996.

[55] L. M. R. Oliveira and A. J. M. Cardoso. Three-phase, three-limb, steady-state transformer model: the case of a ynzn connection. In Proceedings ofthe IASTED International Conference “Power and Energy Systems”, Marbella,pages 467–472, September 2000.

[56] M. Salimi, A.M. Gole, and R.P. Jayasinghe. Improvement of transformersaturation modeling for electromagnetic transient programs. In IPST,Vancouver, July 2013.

[57] E. Acha, A. Semlyen, and N. Rajakovic. A harmonic domain computationalpackage for nonlinear problems and its application to electric arcs. IEEETransactions on Power Delivery, 5(3):1390–1397, July, 1990.

[58] L. Wu, M. Achterkamp, J.P.W. de Jong, P.A.A.F. Wouters, W.L. Kling,M. Popov, and E.F. Steennis. Frequency domain analysis of the influence ofcompensating wires and earthing resistances in mixed overhead / undergroundhv transmission. CIGRE-Paris, pages C4–309, 2014.

[59] U. Klapper, M. Kruger, and S. Kaiser. Reliability of transmission by means ofline impedance and k-factor measurement. 18th International Conference andExhibition on Electricity Distribution, CIRED, pages 1–4, June 2005.

[60] J.L. Blackburn. Symmetrical Components for Power Systems Engineering.Marcel Dekker, 1993.

[61] CIGRE Working Group A2.26. Mechanical condition assessment of transformerusing frequency response analysis (fra). CIGRE-Paris, April, 2008.

[62] M.H.J. Bollen, E. Styvaktakis, and I.Y. Gu. Categorization and analysis ofpower system transients. IEEE Transactions on Power Delivery, 20(3):2298–2306, July, 2005.

152 Bibliography

[63] A.V. Oppenheim, A.S. Willsky, and I.T. Young. Signals and Systems. Prentice-Hall, 1983.

[64] C.L. Philips, J.M. Parr, and E.A. Riskin. Signals, Systems, and Transforms.Pearson, Prentice-Hall, 2008.

[65] L. Wu, P.A.A.F. Wouters, and E.F. Steennis. Frequency domain transientanalysis of resonant behavior for different hv overhead line and undergroundcable configurations. In IPST, Vancouver, July 2013.

[66] L. Wu, P.A.A.F. Wouters, and E.F. Steennis. Application of frequency domainanalysis to fault transients in complex hv transmission lines. In ATEE,Bucharest, pages 1–6, May 2013.

[67] L.F.W. de Souza and E.H. Watanabe. Eliminating gibbs phenomenon fromswitching functions for power electronics circuit analysis. IEEE Transactionson Power Delivery, 24(2):970–971, April, 2009.

[68] A. Ametani. The application of the fast fourier transform to electrical transientphenomena. Int. J. Elect. Engng. Educ., 10:277–287, 1973.

[69] J.C. Das. Power System Analysis: Short-Circuit Load Flow and Harmonics.CRC Press, 2002.

[70] N. Chiesa and H.K. Høidalen. Transformer model for inrush currentcalculations: Simulations, measurements and sensitivity analysis. IEEETransactions on Power Delivery, 25(4):2599–2608, October, 2010.

[71] J. Jesus, E. Acha, and M. Madrigal. The study of inrush current phenomenonusing operational matrices. IEEE Transactions on Power Delivery, 16(2):231–237, April, 2010.

[72] N. Rajakovic and A. Semlyen. Investigation of the inrush phenomenon: a quasi-stationary approach in the harmonic domain. IEEE Transactions on PowerDelivery, 4(4):2114–2120, 1989.

[73] D. Oeding and B.R. Oswald. Elektrische kraftwerke und netze. In 6 Auflage,Springer-Verlag Berlin Heidelberg New York, pages 282,287, 2004.

[74] D.W. Durbak and A.M. Gole. Modeling guidelines for switching transients.IEEE Modeling and Analysis of System Transients Working Group, 2000.

[75] L. Wu, P.A.A.F. Wouters, and E.F. Steennis. Aspects related to replacing hvlines by hv cables on resonant grid behavior. In IPST, Delft, 2011.

[76] F. Faria da Silva, C.L. Bak, and P.B. Holst. Study of harmonics in cable-basedtransmission networks. CIGRE-Paris, pages C4–108, 2012.

Bibliography 153

[77] G. Alvarez-Cordero, A. Bachiller Soler, A. Gómez-Expósito, J.A. RosendoMacías, and C. Gómez-Simón. A methodology for harmonic impedance in largepower systems. application to the filters of a vsc. CIGRE-Paris, pages C4–112,2012.

[78] Working Group 02. Guidelines for representation of network elements whencalculating transients. CIGRE, 1985.

[79] J. Arrillaga and N.R. Watson. Power System Harmonics. John Wiley & Sons.LTD, 2003.

[80] N. Amekawa, N. Nagaoka, Y. Baba, and A. Ametani. Derivation ofa semiconducting layer impedance and its effect on wave propagationcharacteristics on a cable. In Generation, Transmission and Distribution, IEEProceedings, volume 150, pages 434–440, July 2003.

[81] Univ.-Prof.-Dr.-Ing. H.J. Haubrich. Power system I, Lecture book. IAEWInstitute, RWTH-Aachen University, 2007.

[82] G. Strang. Introduction to Linear Algebra. Wellesley-Cambridge Press, 4thEdition, 2009.

Acknowledgment

My dissertation could never be accomplished without the support from peoplearound me. To specifically thank each one of them, I could write another book.So here only a part of them are mentioned.

I am extremely grateful to my esteemed supervisors Fred Steennis and PeterWouters for all their guidence, help, and encouragement during my whole four-yearstudy.

My highest honor firstly goes to Fred. Not only did he accept me as a Ph.D.candidate in the very beginning, but also recommended me to a job in DNV GL inthe very end.

Peter is a person that I could not thank enough. The deepest impression of himis his intelligence and diligence. I am enlightened by his detailed technical and literalcomments in my papers and presentation slides.

My initial application to the Ph.D. position was communicated to Wil Kling.Here I want to give my highest appreciation to him for carrying on my application.

My greatest thanks to my all the other members in the Committee for my defense:prof.dr.ir. A.C.P.M. Backx, prof.dr.ir. M.H.J. Bollen, prof.dr.Dipl.-Ing. V. Terzija,dr.ir. M. Popov, and dr. G.R. Kuik. Without their useful comments, this dissertationcan never reach this level.

With this chance, I want to again express my thankfulness to TenneT TSO B.V.,especially to those who directly supported me: Jan de Jong, Frans van Erp, and NelaNenadovic.

Measurements in power systems are big events and require high professionalskills. I want to thank Smit Transformers (especially Henk Fonk) and DNV GL(especially Marc Achterkamp) for their kind support in measurements.

My great thanks also go to colleagues from DNV GL for their support: Petervan der Wielen, Paul Wagenaars, and Bernd van Maanen.

My gratitude also goes to E.J.M. van Heesch, P.F. Ribeiro, J.F.G. Cobben, fortheir kind support and information during my research.

155

156 Bibliography

I also want to thank all the other colleagues from EES-group for creating such awonderful environment for working and studying. I especially want to thank Anna,Arno, Ballard, Bart (de Bruyn), Bart (Kruizinga), Chai, Gerben (from TU Delft),Gu, Helder, Ioannis, Jayati, Jerom, Jos, Pavlo, Phuong, Tom, Vindhya, Vladimir,Vuong, Yan, Yu.

Last but not the least, I want to thank my parents and my wife. Thank you forall you have done for me.

Curriculum Vitae

Lei Wu was born on 11-October-1984 in Hubei, China. He obtained his Bachelorof Science degree of electrical engineering from Chongqing University in Chongqing,China, in 2007. He graduated from RWTH-Aachen University in Aachen, Germany,as a Master of Science of electrical power engineering in 2010. In the sameyear, he joined the Electrical Energy Systems group at Eindhoven University ofTechnology, as a Ph.D. candidate under the supervision of prof.dr.ir. E.F. Steennisand dr. P.A.A.F. Wouters. His research topic is “Impact of EHV/HV UndergroundPower Cables on Resonant Grid Behavior”. In 2014, he starts to work at DNV GLin Arnhem, the Netherlands.

157

List of Publications

Journal Papers

L. Wu, P.A.A.F. Wouters, and E.F. Steennis. Frequency domain transient analysisin double-circuit mixed hv overhead line-cable connection including cross-bonding.International Transactions on Electrical Energy Systems, 2014. (submitted)

L. Wu, Y. Xiang, P.A.A.F. Wouters, and E.F. Steennis. Transformer inrushcurrent: measurement and frequency-domain analysis including remanence. IEEETransactions on Power Systems, 2014. (submitted)

A.R.A. Haverkamp, L. Wu, P.A.A.F. Wouters, and E.F. Steennis. Tradeoffbetween level of detail and accuracy for modeling a large-scale hv transmissionnetwork. (In preparation)

Y. Li, L. Wu, P.A.A.F. Wouters, P. Wagennars, P.C.J.M. van der Wielen, andE.F. Steennis. Effect of ground return path on high frequency signal propagationalong single-core and three-core power cable. IEEE Transactions on PowerDelivery, 2014. (submitted)

Cigre Paper

L. Wu, M. Achterkamp, J.P.W. de Jong, P.A.A.F. Wouters, W.L. Kling, M. Popov,and E.F. Steennis. Frequency domain analysis of the influence of compensatingwires and earthing resistances in mixed overhead / underground hv transmission.CIGRE-Paris, C4-309, 2014.

Conference Papers

L. Wu, H. Fonk, P.A.A.F. Wouters, and E.F. Steennis. Influence by parasiticcapacitances on frequency response of a 380-150-50 kv transformer with shunt

159

160 Bibliography

reator. In ISH, Seoul, pages 56-60, August 2013.

L. Wu, P.A.A.F. Wouters, and E.F. Steennis. Frequency domain transient analysisof resonant behavior for different hv overhead line and underground cableconfigurations. In IPST, Vancouver, July 2013.

B.J.H. de Bruyn, L. Wu, P.A.A.F. Wouters, and E.F. Steennis. Equivalentsingle-layer power cable sheath for transient modeling of double-layer sheaths. InPowerTech (POWERTECH), 2013 IEEE Grenoble, pages 1-6, June 2013.

L. Wu, P.A.A.F. Wouters, and E.F. Steennis. Application of frequency domainanalysis to fault transients in complex hv transmission lines. In ATEE, Bucharest,pages 1-6, May 2013.

L. Wu, P.A.A.F. Wouters, and E.F. Steennis. Model of a double circuit withparallel cables for each phase in a hv cable connection. In IEEE InternationalConference on Power System Technology (POWERCON), Auckland, pages 1-5,November 2012.

L. Wu, P.A.A.F. Wouters, E.J.M. van Heesch, and E.F. Steennis. On-site voltagemeasurement with capacitive sensors on high voltage systems. In PowerTech(POWERTECH), 2011 IEEE Trondheim, pages 1-6, June 2011.

L. Wu, P.A.A.F. Wouters, and E.F. Steennis. Aspects related to replacing hv linesby hv cables on resonant grid behavior. In IPST, Delft, 2011.

Impact of EHV/HV Underground Power Cables on Resonant … · Cables on Resonant Grid Behavior ......

Documents

Transcript of Impact of EHV/HV Underground Power Cables on Resonant … · Cables on Resonant Grid Behavior ......