resonant overvoltages caused by transformer energization

Civilingenjörsprogrammet i energisystem

Uppsal a universitets l ogotyp

UPTEC ES 21032

Examensarbete 30 hp

september 2021

Resonant overvoltages caused by transformer energization and saturation Two EMT case studies conducted using models of the grid in Stockholm and an off-shore wind farm

Gustav Sundberg Civilingenj örspr ogrammet i energisystem

Teknisk-naturvetenskapliga fakulteten

Uppsala universitet, Utgivningsort Uppsala/Visby

Handledare: Robert Rogersten och Stefan Råström Ämnesgranskare: Cecilia Boström

Examinator: Petra Jönsson

Uppsal a universitets l ogotyp

Resonant overvoltages caused by transformer saturation

Gustav Sundberg

Abstract

This thesis investigates the impact of resonant overvoltages and their origin. Series and parallel

resonances are present in any electrical grid. The frequency of which this resonance occurs is

called resonance frequency. For parallel resonance, which is mainly being studied in this thesis,

a high impedance peak can be found at the resonance frequency. This impedance peak in

conjunction with a harmonic rich current cause a kind of temporary overvoltages called resonant

overvoltages. The harmonic content of the current is high following a fault clearing in the grid,

due to transformer saturation. The resonance frequency is heavily dependent on the amount of

reactance present in the grid, which entail that a change in reactance causes a change in the

resonance frequency. The electromagnetic transient tool PSCAD has been used to investigate

resonant overvoltages following transformer energization caused by faults and switching in

Stockholm. Secondly, a model was created of a grid connecting off-shore wind power to the

mainland via long AC submarine transmission cables. These cables, having a high capacitance,

lower the resonance frequency. Faults in this model were simulated to investigate the

phenomenon of resonant overvoltages in such a grid. This was especially interesting due to

Swedens planned expansion of wind power in the Baltic sea.

While resonant overvoltages were found in Stockholm they were not deemed significant due to

their low magnitude and longevity. However, severe resonant overvoltages were found in the

off-shore wind farm model. The worst resonant overvoltages had a maximum amplitude of the

2nd order harmonic voltage of 130 kV which, while eventually damped, were significant for up to

50 periods. Lastly, the phenomenon of an increased resonance frequency during the saturation

of a transformer was studied. The most severe resonant overvoltages occured in a model where

the frequency scans showed a resonance frequency of 98 Hz. Indicating, caution needs to be

had during EMT-studies of resonant overvoltages while choosing what resonance frequency to

study. Teknisk-naturvetenskapliga fakulteten, Uppsala universitet . Utgivningsort U ppsal a/Visby . H andledare: Robert R ogersten och Stefan R åström , Ämnesgranskare: Cecili a Bos tröm, Examinator: Petra Jönsson

Popularvetenskaplig sammanfattning

Det har arbetet undersokte resonanta overspanningar i Stockholms elnat samt i en modellmed havsbaserad vindkraft. For att gora detta gjordes en EMT-studie (elektromagnetiskatransienter) i programmet PSCAD. pa vartenda av de tre casen som arbetet delades in i.Case ett undersokte olika fenomen for att faststalla att modellen over Stocholm samt PSCADgav de forvantade resultaten. Fenomenen som testades var resonans i elnatet, serie/parallellresonans samt magnetiseringsstrom. Magnetiseringsstrom uppkommer da en transformatorblir magnetiserad snabbt, som vid en inkoppling eller vid bortkoppling av ett fel. Underfelet i elnatet gar spanningen runt transformatorn ner mot noll, for att sedan aterga tillsitt ursprungliga varde vid bortkoppling av felet. Denna magnetiseringsstrom ar ofta rikpa overtoner. I ett elnat med parallellresonans kommer det att skapas en impedanstopp paen viss frekvens, resonansfrekvensen. Om denna resonansfrekvens skulle sammanfalla mednagon av overtonerna i strommen, fran exempelvis en magnetiseringsstrom, kan det skapasresonanta overspanningar. Resonansfrekvensen ar direkt relaterad till mangden induktansoch kapacitans i natet, vilket betyder att en andring pa elnatet ocksa skapar en andring paresonansfrekvensen.

I case tva undersoktes resonanta overspanningar kring en utvald kraftstation i Stockholmda fel intraffar i elnatet. Olika sorters fel samt olika platser dar felen intraffar testades. Harstuderades overspanningarna som skapas och deras overtoner. Den hogsta overspanningensom hittades var 576 kV. Denna overspanning hade en tredje overton pa 115 kV som inom0.15 sekunder sjunkit till 34 kV. Slutsatsen fran case tva var att problemet med resonantaoverspanningar i Stockholm inte ar stort.

I case tre studerades en model over en mindre stad an Stockholm dar fyra hogspanningsk-ablar kopplats in. Dessa kablar sammanlankade fastlandet med vindkraftsparken till havs.Case tre ar relevant i dagslaget da Sverige har planer pa att bygga ut vindkraften i Ostersjon.Langden pa de fyra kablarna justerades mellan 50 km och 70 km for att pa sa satt andra deraskapacitans som i sin tur andrade resonansfrekvensen. Detta for att fa en impedanstopp somsammanfoll med den andra overtonen pa 100 Hz. De resonanta overspanningarna som upp-kom i denna modell hade en mycket langre livslangd an de resonanta overspanningarna francase tva. Det intressanta resultatet fran case tre var den andra overtonens overspanningar,i och med impedanstoppen pa 100 Hz. Den hogsta andra overtonen av spanningen somfanns var 130 kV och tog 0.354 sekunder for att sjunka till 37% av dess ursprungliga varde.Slutsatsen fran detta blev att i en stad som modellen i case tre utformades efter kan detuppkoma problem med resonanta overspanningar. Slutligen testades fenomenet att resonans-frekvensen okar da transformatorer blir mattade. Resultatet fran detta blev att den varstaresonanta overspanningen inte intraffade da resonansfrekvensen var 100 Hz, utan istallet daresonansfrekvensen var 98 Hz.

III

Exekutiv sammanfattning

Rapporten beskriver studier som utfordes pa en modell over Stockholm samt en modell overen stad med en ansluten havsbaserad vindkraftspark. Det som undersoktes var amplitudoch livslangd pa resonanta overspanningar vid fel i natet. Slutsatserna ar att resonantaoverspanningar inte ar ett problem i modellen over Stockholm, daremot kan det uppkommaproblem med resonanta overspanningar i nat dar havsbaserad vindkraft kopplas in. Dettabor darfor tas i beaktning nar havsbaserade vindkraftsparker projekteras.

IV

Acknowledgement

This project has been a constant challenge to my academic knowledge and with this smallsegment to start my master thesis i would like to thank everyone who has helped me, withoutwhom I would not have been able to reach the end results I did. Firstly, I would like tothank my two advisors on Svenska kraftnat. Robert Rogersten and Stefan Rastrom, whofirst and foremost gave me the exciting opportunity to pursue this master thesis project atSvenska kraftnat. Robert and Stefan have helped me throughout the entire process withadvise on reading material, guidance when it comes to theory, problems concerning PSCADand technical issues but also with inspiration and their vast experience. In addition to thisthey, with the rest of the people at Svenska kraftnat, have given me the warmest of welcomesdespite the ongoing pandemic.

Secondly, I would like to give a special thanks to Oscar Lennerhag from IndependentInsulation Group, who has been of great help. Oscar has contributed with his deep insightinto resonant overvoltages, transformer magnetization and EMT-studies. He has also beenexcellent at describing the theory on a level that, a soon to be, newly graduated student canunderstand.

Lastly, I would like to thank my academic advisor at Uppsala Universitet, CeciliaBostrom. Cecilia has guided me throughout the process of writing this master thesis paper.

Thank you.

V

Contents

1 Introduction 11.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theory 32.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Resonance in the grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Series resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.3 Parallel resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.4 Changed resonance frequency due to saturated transformers . . . . . 8

2.3 Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.1 Circuit breakers and fault clearing time . . . . . . . . . . . . . . . . . 9

2.4 Inrush current and resonant overvoltages . . . . . . . . . . . . . . . . . . . . 102.4.1 Inrush current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4.2 Sympathetic inrush . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.3 Mitigation of the Inrush current . . . . . . . . . . . . . . . . . . . . . 132.4.4 Impact of resonant overvoltages . . . . . . . . . . . . . . . . . . . . . 142.4.5 Mitigation of the resonant overvoltages . . . . . . . . . . . . . . . . . 14

3 Method and model development 163.1 Case one - Testing and validating the model of Stockholm . . . . . . . . . . 16

3.1.1 Resonance and impedance . . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 Inrush Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.3 Sympathetic inrush current . . . . . . . . . . . . . . . . . . . . . . . 183.1.4 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Case two - Faults in the Stockholm grid . . . . . . . . . . . . . . . . . . . . 193.2.1 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.2 Case without fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.3 Single-line-to-ground fault . . . . . . . . . . . . . . . . . . . . . . . . 203.2.4 Three phase to ground fault . . . . . . . . . . . . . . . . . . . . . . . 203.2.5 Location of fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.6 Fault on a cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.7 Fault during a case with downstream load . . . . . . . . . . . . . . . 21

3.3 Case three - Off-shore wind farm model . . . . . . . . . . . . . . . . . . . . . 213.3.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.2 Matching of the cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.3 Standard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.4 Strength of the grid altered . . . . . . . . . . . . . . . . . . . . . . . 233.3.5 N-2 scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

VI

3.3.6 Sensitivity analysis and changed resonance frequency with saturatedtransformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.7 Evaluating the changed resonance frequency due to saturated trans-formers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Results 254.1 Case one - Testing and validating the model of Stockholm . . . . . . . . . . 25

4.1.1 Resonance and impedance . . . . . . . . . . . . . . . . . . . . . . . . 254.1.2 Inrush Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1.3 Sympathetic inrush current . . . . . . . . . . . . . . . . . . . . . . . 294.1.4 Validation of transfomrer . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Case two - Faults in the grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.1 Fault close to Power station B . . . . . . . . . . . . . . . . . . . . . . 36

4.2.1.1 Three Phase to ground fault . . . . . . . . . . . . . . . . . . 364.2.1.2 Phase to ground fault . . . . . . . . . . . . . . . . . . . . . 37

4.2.2 Location of the fault . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2.3 Fault on a Cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.4 Impact of a load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2.5 Summarizing the overvoltages and their longevity for the different faults 45

4.3 Case three - Off-shore wind farm model . . . . . . . . . . . . . . . . . . . . . 464.3.1 Initial scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3.2 Strength of the electric grid altered . . . . . . . . . . . . . . . . . . . 484.3.3 N-2 incident . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.4 96 Hz vs. 104 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3.5 Evaluating the changed resonance frequency due to transformer satu-

ration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Discussion 555.1 Case one - Testing and validating the model of Stockholm . . . . . . . . . . 555.2 Case two - Faults in the grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.3 Case three - Off-shore wind farm model . . . . . . . . . . . . . . . . . . . . . 56

6 Conclusion 59

VII

1 Introduction

The Swedish Transmission System Operator (TSO), Svenska kraftnat, has decided to improvethe grid in and around Stockholm. This is a result of an increasing demand for electric powerin the area. The plan involves building new connections as well as replacing the old ones.Due to political reasons and technical difficulties, the new grid will contain a larger portionof cables than before. This change will contribute with more capacitance to the grid that inextension leads to lowered resonance frequencies. With a lower resonance frequency problemsmay arise with resonant overvoltages, especially during non-business-as-usual scenarios, suchas energization of transformers, and faults [1].

Furthermore, significant developments of off-shore wind power in the Baltic sea are ex-pected in order to meet the increasing demand of renewable energy in Sweden. The expansionof off-shore wind power requires long AC submarine transmission cables, which are presumedto impact the resonance frequency more than if the same cables were introduced into thegrid in Stockholm. This because of the difference in strength between the two grids.

The resonance phenomenon is inevitable in a power system but the severity of its impactcan be managed. By studying what the resonance frequency becomes and ensuring that itdoes not overlap with the fundamental frequency or any harmonic frequency in the system,some resonant overvoltages can be avoided.

The studies presented in this thesis are divided into three cases. The first case investi-gates how the phenomena of resonance and inrush current following transformer energizationbehave in a model of Stockholm using the electromagnetic transient tool, PSCAD. Secondly,case two evaluates the impact of faults occurring in the same model used in case one witha focus on resonance and inrush current. The last case of the thesis is case three, wheresimulations were made in a model specifically designed to investigate resonance and inrushcurrent in a grid connected to off-shore wind power.

1.1 Scope

When designing a model of a real grid, and the grid is too big to fit into the model, i.e.the creation of the model would be too time-consuming and simulations made in the modelwould be too computational heavy, boundaries to the model has to be set. This was donein this thesis, by choosing a suitable location for the boundary, and representing the rest ofthe grid with Thevenin equivalents. This will not be as accurate as having the entire gridin the model, however that would not be feasible in this project in regards to time. Thiswas done for both the model of Stockholm and off-shore wind farm model. In the off-shorewind farm model, only the presence of one wind farm was studied. This thesis focuses onthe amplified impedance that comes with parallel resonance, series resonance was thereforesomewhat neglected. In case one and two the fact that the resonance peak decrease duringsaturation of the transformer was ignored.

When studying the overvoltages, a suitable bus to take measurements from was chosen.The cases studied will undoubtedly impact other parts of the grid. But in this thesis the

1

location where measurements were taken has been chosen in order to focus on equipment inclose proximity to the fault/connection.

The phenomenon of sympathetic inrush current was only studied to the extent of itspresence in the model. This was done to evaluate the credibility of the models. Any furtherstudies on how large the impact of the sympathetic inrush current in case two and three washowever excluded from the study.

No techniques for mitigating the temporary overvoltages have been implemented in thestudies.

Single line to ground fault and three phase to ground fault have exclusively been chosento study in this thesis.

1.2 Limitations

The data needed for the models are in some cases hard to find. This due to the fact thatthey do not exist, are hard to measure or that the model uses a simplified version of reality.These parameters were realistically approximated in the case of their absence.

The complexity of the grid causes a problem when creating an absolute accurate model.The model was therefore approximated and validated to be sufficiently accurate.

Since the study was made in PSCAD, any limitations found in the program will prop-agate into the end results. The general approach when evaluating the impact of resonantovervoltages is uncertain. Therefore the verdict on how severe the impact of the resonantovervoltages studied in this thesis is left somewhat open.

2

2 Theory

2.1 Overview

In transmission and distribution networks, resonance may occur at specific frequencies de-pending on the system components and network topology. In case of a large amount ofunderground cables, the parallel resonance frequency may become close to the 2nd or 3rd

order harmonics.A sudden shift in voltage in a transformer causes it to be saturated. This happens when

a transformer gets connected to the grid or when the voltage level gets restored followingthe disconnection of a fault. The latter is what is primarily being studied in this thesis. Atransformer being saturated causes 2nd and 3rd order harmonics to enter the grid. If theresonance frequency in the grid is close to the 2nd and/or 3rd order harmonics, i.e. 100 Hzand 150 Hz respectively, temporary overvoltages (TOVs) could emerge. These overvoltagesare henceforth called resonant overvoltages.

This is taken into consideration when designing transmission and distribution networks.However by introducing new cables into the grid, as planned in Stockholm and with off-shorewind power, the resonance frequencies get lowered due to larger capacitance. This couldcause the resonance frequencies to coincide with the 2nd and 3rd order harmonic current, andif so, resonant overvoltages could occur [2].

Temporary overvoltages are a problem in any electrical grid and arise due to differentactivities and states of operation: such as a disconnect of a load, dysfunctional power elec-tronics, ground faults, and transformer saturation in the presence of low-order resonances.Because of their regular occurrence, insulation coordination methods are often done to iden-tify what temporary overvoltages can occur, where, and to what magnitude. This to ef-ficiently select an insulation strength strong enough to minimize the risk of failure to anacceptable level. However, resonant overvoltages can not be studied as a traditional TOVsince they are hard to compare to the conventional technical specifications of the insulationin normal apparatuses. Because of this, the impact of resonant overvoltages is generallystudied by simulations in power system models [2] [3].

2.2 Resonance

To understand resonant overvoltages one first must understand resonance. Resonance is aphenomenon that occurs in a system where energy easily can be transferred and/or convertedbetween different energy reservoirs and/or forms. In an electrical system the energy is storedin magnetic and electric fields. Therefore, if an electrical system contains an electric anda magnetic field it is capable of resonance. The capacitive and inductive reactance of thesystem is getting charged and discharged in tandem and the frequency of this exchange iscalled resonance frequency. The reactance for a capacitive and an inductive component in asystem is calculated using equation 1 and 2.

XC =1

2πf · C(1)

3

XL = 2πf · L (2)

Resonance occurs when the capacitive and the inductive parts of the reactance are ofequal magnitude and thereby cancel each other out leaving only a resistive component leftin the impedance. Using this knowledge in conjunction with equation 1 and equation 2, andsolving for the frequency gives an expression for the resonance frequency:

XL = XC ⇐⇒ 2πf · L =1

2πf · C(3)

⇐⇒ f 2 =1

22π2 · L · C−→ f =

1

2π ·√L · C

(4)

In a system that is capable of resonance, the impedance is altered at certain frequenciesdue to parallel and series resonance. This is explained further below.

2.2.1 Resonance in the grid

In an alternating current grid there is an impedance in every point that changes with fre-quency. Both components of the impedance changes with the frequency, the resistancebecause of skin effect and the reactance because of frequency dependency of its components.Both the resistance and the reactance are interesting to study regarding resonance, the re-sistance because of its dampening effect and the reactance because of its ability to decidethe resonance frequencies.

Equation 4 shows that a larger product of the inductance and the capacitance will cre-ate a lower resonance frequency than before, and vice versa. Meaning that the resonancefrequencies in a grid introduced to more capacitance or inductance decrease. Examples ofsources of inductance in the grid are overhead lines and transformers, while capacitance oftenoriginates from cables. The capacitance is larger for cables compared to overhead lines, andthe opposite for inductance. [2] [4].

VsR L Z

C

Figure 1: Schematic picture of a theoretical grid connected to a transformer.

Figure 1 shows a theoretical grid to illustrate the impedance Z seen from the transformer.The placement of the resonance frequency in the grid is decided by the product of theinductance and the capacitance as shown earlier in equation 4. In the following chapters

4

it will be shown that at the resonance frequency for a circuit with parallel resonance animpedance peak is formed. The magnitude of this peak is determined by the impedancein the grid. Since the inductance is connected to the strength of the grid, a weaker gridmeans a higher inductance and vice versa, the impedance peak is connected to the strengthof the grid. A common way of determine grid strength is by measuring a fault current atthe location in question.

Figure 2: Two traces illustrating the different impedance (positive sequence) peaks caused by reso-nance depending on the strength of the grid. The capacitance in the model is adjusted to keep theresonance frequency constant as the strength of the grid changes. The figure contains simulatedvalues from the model used in case three.

In figure 2 it can be seen that, in accordance with equation 4, the same model can havethe same resonance frequency with different combinations of inductance and capacitancepresent in the grid. However, what should be noted is that when the grid is weaker, andthe inductance is larger, the amplitude of the impedance peak is greater. [5] [6]. Resonancein transmission and distribution networks is generally divided into two sub-classes: seriesresonance and parallel resonance.

2.2.2 Series resonance

The series resonance is the name of resonance that occur between components that areconnected in series. Circuits with series resonance have the ability to amplify the current ata desired frequency, which is useful in components like band pass filters, but in power systemsit could cause problems. Figure 3 illustrates a simple circuit with a resistance, inductanceand a capacitance connected in series.

5

R LC

Vs

Figure 3: Schematic picture of a RLC-circuit connected in series.

In figure 3, there is only one path for the current to take, which means that the impedancescan be described as the following equation:

Ztot = R−XC +XL = R +1

jωC+ jωL (5)

The circuits total impedance Ztot changes with the frequency. Equation 5 shows that thesmallest absolute value that the reactance can assume is zero, when the inductive and thecapacitive part of the reactance cancel out, leaving a solely resistive impedance. This canbe seen in figure 4. Where a resonance frequency can be found at approximately 145 Hz.Moving away from the resonance frequency causes the reactance to grow, i.e. at the resonancefrequency for a series resonance a local minimum of the impedance can be found.

6

Figure 4: A graph showing the intersection between the inductive and capacative part of the re-actance. The frequency for which the two graphs intersect is the resonance frequency. Arbitraryvalues were used to create the graph.

If the resistance, which is the only part of the impedance left at the resonance frequency,in the circuit is small, a large current will run through the capacitance and inductance andwith time charging them to high voltages. The high voltages caused in the inductance andcapacitance here are called resonant overvoltages [4].

2.2.3 Parallel resonance

The other subclass of resonance in electrical circuits is parallel resonance. This subclassis similar to the series resonance in many ways but the parallel resonance occurs betweencomponents that are connected in parallel. An example of a circuit like this can be seen infigure 5.

Vs

iC

CR

iL

L

Figure 5: Schematic picture of a RLC-circuit connected in parallel.

When the circuit is connected to a power source, energy will be stored in the inductor

7

and the capacitor. The energy stored in these components is then transferred back and forth,creating a circulating current. At the resonance frequency the instantaneous current of iLand iC will be of equal amplitude with opposing sign, meaning the energy and current drawn,by the circulating current, from the source is zero. This can be illustrated in a circuit, seefigure 6, where both the inductor and the capacitor act as an open circuit.

Vs R

Figure 6: Schematic picture of a RLC-circuit connected in parallel where the inductor and capacitoract as an open circuit.

At a certain frequency, the resonance frequency, this oscillation will coincide with the fre-quency of the voltage source, Vs which then causes current resonance. The total admittanceof the circuit presented in figure 5 is:

Ytot = G+ jωC +1

jωL, where G =

1

R(6)

From equation 6 it can be seen that during low frequencies the inductive susceptance isdominating. With an increasing frequency the inductive susceptance drops off while thesusceptance in the capacitor gets stronger, until, at the resonance frequency, they are ofequal size and cancel out. Leaving an admittance that only contains a conductance. Atfrequencies above the resonant frequency the inductive susceptance goes towards zero, whilethe capacitive susceptance increases [2].

The admittance in equation 6 is a dual to the impedance in equation 3. With the knownrelationship between impedance and admittance it can be concluded that where there is localminimum for the impedance in a circuit with series resonance, there is a local maximum ofimpedance at the resonance frequency for a circuit with parallel resonance [2].

2.2.4 Changed resonance frequency due to saturated transformers

According to a study published in 1991 by the international council on large electrical systems(Cigre), it was found that the maximum overvoltages were not found precisely at a multiple ofthe fundamental frequency, but slightly bellow the resonance frequency [7]. An explanationwas given to this in the report; a transformer being saturated can be seen as an ideal harmoniccurrent source with an equivalent inductance of the magnetizing branch connected in parallel.This inductance, in accordance with equation 4 can alter the resonance frequency [7]. Furtherstudies on how the resonance frequency changes with the saturation of a transformer havebeen done [8]. The study showed that the equivalent inductance of the magnetizing branch

8

decreases with voltages above 1.0 pu, and subsequently increases the resonance frequency.Meaning, the peak in the resonance profile found from impedance frequency scans, is notentirely accurate. The impedance profile is in reality found to be more like a plateau spanningfrom the resonance frequency down to a few Hz below, than the peak presented earlier infigure 2 [7] [8].

2.3 Faults

Short-circuits occur in power systems when the insulation cannot contain the power that itis supposed to. This may be because of faulty insulation or, overvoltages that create a largedifference in electrical potential, resulting in a higher chance for the electricity to jump/arc.The voltage in the grid around a fault goes towards zero while the current becomes verylarge. [9].

In a power system with three phases there are several combinations of faults that canoccur. They are [9]:

• Single line to ground fault

• Line to line fault

• Double line to ground fault

• Three phase fault

• Three phase to ground fault

The most common of these is the single line to ground fault and the largest voltage dropoccurs during a three phase fault [9].

2.3.1 Circuit breakers and fault clearing time

To minimize the impact of faults, circuit breakers are installed in power systems. A circuitbreaker is designed to open if a fault is detected, effectively removing the fault from thegrid. The time it takes for the fault to get detected and the circuit breakers to receive theinformation to remove the fault is called fault clearing time. In the Swedish transmissiongrid the maximum fault clearing time in an over head line is 130 ms [1] [10].

During the fault clearing time, the voltage in the grid adjacent to the fault goes towardszero. When the fault then gets disconnected the voltage level in the nearby grid gets restoredand any transformer connected to the impacted part of the grid gets energized. When afault occurs, the possibility of a well timed transformer energization is impossible, since theenergization occurs simultaneously with the disconnection of the fault, which needs to bedone as soon as possible [11].

9

2.4 Inrush current and resonant overvoltages

One part of resonant overvoltages is the amplified impedance characteristic discussed above.However, for resonant overvoltages to occur, the system also needs to be excited by somekind of energy source. The impedance in the system is frequency dependent, and thereforethe harmonic content of the exciting source is important. A typical example of such asource is a transformer. This, partly due to the non-linearity of the magnetic core in atransformer which is present during normal operations. But especially during the transientstate, following the energization of a transformer. During this time the current is rich inharmonics which is called inrush current [11] [12]. A parallel resonance circuit is exited by acurrent source and a series resonance circuit is exited by a voltage source. The energizationof a transformer acts as a current source, which means that the parallel resonance will getexited following a disconnection of fault or the connection of a transformer [2].

2.4.1 Inrush current

Faraday’s Law describes the realationship between the voltage of the coil, E, the number ofturns in the coil N and the change of magnetic flux, φ, equation 7.

N · dφdt

= E · sin(ωt) (7)

According to equation 7, if the voltage, E, is a sinusoidal waveform then the magnetic flux isas well. From equation 7 it can also be derived that the magnetic flux waveform lags behindthe voltage by 90◦. This is visualised in figure 7 [13].

Figure 7: A typical magnetic flux and voltage waveform for an ideal transformer.

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Equation 7 is then rearranged and integrated:

φ = − E

Nω· cos(ωt) + C (8)

Where C is the integration constant. In a worst case scenario, in which the highest possiblevoltage is reached, the transformer is being energized when there is no remanence of theflux present in the magnetic core, i.e. setting the left side of equation 8 to zero, which thendetermines the integration constants value.

φ = K · (1− cos(ωt)), K =E

Nω(9)

If the transformer is energized at a time that creates a relationship that satisfies equation9. Then the magnetic flux does not oscillate about the zero value as in zero state, it oscillateswith the amplitude K (and with the same frequency as the voltage source) between the values0 and 2K. This is illustrated in figure 8.

Figure 8: A typical magnetic flux and voltage waveform for an ideal transformer when the trans-former is connected to the grid when the voltage is zero.

The integration constant’s value causes a need for a higher magnetic flux to create thesame voltage as before. And by studying equation 9, illustrated in figure 8, it can be seen thatthe maximum magnetic flux needed is doubled in comparison to the transformer energizationpresented in figure 7. From a typical saturation curve for a transformer seen in figure 9a inconjunction with the knowledge that magnetic field density is proportional to the magneticflux density, and that the magnetic field strength is proportional to the current in the coil,the phenomenon of inrush current can be understood.

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(a) Saturation curve of a transformer. (b) Magnetic flux linkage and magnetizing current.

Figure 9: Two graphs illustrating the creation of an inrush current due to the saturation of atransformer. In the right graph the absolute value was taken for the magnetic flux linkage forclarity.

It can be seen in figure 9a that when the flux is doubled from φ0 to φ1 (K to 2K),the current i1 drawn from the transformer becomes large because the transformer becomessaturated. This large current due to the heightened magnetic flux from a poorly timedconnection is called inrush current. Figure 9b illustrates data from a simulation made incase one. The transformer in the model that created figure 9b had a knee-point voltageof 1.14 pu. During normal operations when the magnetic flux linkage stays below 1 puno significant magnetizing current is created, but immediately as the magnetic flux linkageexceeds 1.14 pu, a large magnetizing current can be seen.

In reality the waveform is not sustained forever but will gradually decrease under theeffect of resistance of the winding, Ri. Instead of the simplified equation 7, the completeequation of the voltages can be seen in equation 10:

N · dφdt

+Ri = E · sin(ωt) (10)

A thorough solution of the differential equation is complicated because flux and currenthave different curve shapes. However, approximate solutions are sufficient for most cases butare left out in this thesis.

2.4.2 Sympathetic inrush

The phenomenon sympathetic inrush current is relevant today even though it has been well-known since its first mention in 1941 [14]. Sympathetic inrush current is the interactionbetween two or more transformers or shunt reactors connected in parallel/series. The focusof this thesis is inrush current from transformers, and therefore sympathetic inrush will beexplained using transformers. Also, sympathetic inrush current in shunt reactors is rarebecause of the high knee-point voltage in shunt reactors in relation to transformers [6].

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Parallel sympathetic inrush can occur in a system with two transformers connected inparallel with their primary windings on the same bus. Where Transformer 2 is connectedto the grid and Transformer 1 gets connected to the grid at a point in time, see figure 10.If Transformer 1 gets connected at a point on the voltage waveform that causes an inrushcurrent as described in the previous section, it will not only cause a high saturation level initself, but also gradually draw Transformer 2 to saturation. The inrush current created inTransformer 1 in conjunction with the series resistance in the circuit causes an asymmetricalvoltage at the bus. The magnetic flux in Transformer 2, which before the connection ofTransformer 1 is in steady-state, gradually gets drawn towards saturation of opposite po-larity to Transformer 2 by the asymmetrical voltage at the bus. Once both transformershave reached saturation, a magnetizing current can be seen in both transformers. Since thetransformers are connected to the same bus in parallel, the voltage seen from both trans-formers is the same. If this is true and the current peaks are of opposite polarity, there needsto be a time difference to when the peaks occur. The current peaks in Transformer 1 andTransformer 2 are therefore reached with half a period apart [6] [14] [15].

Sympathetic interaction between two transformers connected in series occurs when thesecondary winding of Transformer 2, which is already connected to the grid, is connectedin series with the primary side of Transformer 1. When Transformer 1 gets connected tothe grid an inrush current is created in the primary side of Transformer 1, which is thenseen by the secondary winding of Transformer 2. When the inrush current flow throughthe secondary winding of Transformer 2 an inrush current is created in its primary side.The addition of this inrush current to the steady-state-current in Transformer 2 causes aflux change per cycle, which gradually causes saturation. A simulation has shown that theimpact from sympathetic inrush in parallel and series connected transformers is similar [6].Since the sympathetic inrush current in the transformer connected in series stems from theinrush current reaching the secondary winding of Transformer 1, a logical conclusion wouldbe that any dampening effect in the series, such as resistance would minimize the impact ofthe phenomenon [6].

Another form of sympathetic inrush is called Pseudo-inrush and occurs after a fault hasbeen cleared in a system where transformers are already connected to the grid. This causesthe voltage in the grid to restore to a normal wave form. However during the fault voltagesag occurs, meaning the RMS-voltage in the transformers go down. In addition to this thevoltage waveform may also lag behind the voltage source. When the fault then is cleared thesystem tries to recover, causing pseudo-inrush currents in order to restore the phase angleand magnitude of the voltage in the transformers [6].

2.4.3 Mitigation of the Inrush current

Inrush currents are undesirable and can cause problems in many parts of the electricalsystem. Problems such as: large voltage dips, transients and temporary overvoltages. Theinrush current that is often filled with low order harmonics, can also cause somewhat unusualproblems like: irregular torques in generators or large motors, mechanical vibrations, slippageof shaft couplings and shifting of windings [6]. Some methods to reduce the inrush current

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are: controlled switching, de-fluxing the transformer core before re-energization, reducingthe voltage in the system before energizing the transformer or adjusting the on-load tapbefore energizing the transformer [6].

2.4.4 Impact of resonant overvoltages

The impact of temporary overvoltages and resonant overvoltages is hard to evaluate. Thisbecause of the different parameters to be studied such as, the amplitude, longevity andharmonic content of the overvoltages. At the time of writing this report there is workbeing done within CIGRE on how to evaluate the impact of resonant overvoltages. Whatshould be noted regarding the impact of resonant overvoltages is that the amplitude ofthe overvoltages is often not the issue since it is generally small compared to lightning orswitching overvoltages. However, the longevity of the overvoltages can be longer than thoseof other overvoltages, which could cause problems. Also, there are different components ofthe electrical system that can be impacted by the TOVs. The components are also impacteddifferently depending on the composition of the temporary overvoltages.

One kind of equipment that can be damaged by temporary overvoltages is componentscontaining a magnetic core: power transformers, phase angle regulators, shunt reactors andpotential transformers. The ability for these components to withstand TOVs is determinedby the magnitude of the voltage and the duration of the overvoltages. The prolonged over-voltages and over-excitation of the equipment could lead to high temperatures that coulddamage the equipment. High voltages could also cause partial or full flashovers [16].

The capabilities of equipment without a magnetic core such as: capacitor banks, circuitbreakers, disconnect switches, power cables, bushings and external insulation are mainlydetermined by the strength of the insulation. Capacitor banks could be affected by hightemperatures following large currents originating in voltages with high amounts of harmoniccontent [16].

Lastly different kinds of arresters are also exposed to temporary overvoltages. The in-creased voltage causes an amplified current which in turn yields a larger power for the arresterto dissipate. If the resonant overvoltages are too severe the dissipating capabilities of themetal oxide surge arrester could be too low, which causes high temperatures and the risk ofdamaged equipment [6] [7].

2.4.5 Mitigation of the resonant overvoltages

Since the resonant overvoltages are caused by the inrush current. Techniques to mitigate theinrush current will also mitigate resonant overvoltages. There are however some additionalmethods to mitigate resonant overvoltages [6]. In general there exists techniques to mitigateresonance overvoltages when energizing a transformer, but there are few to no techniques ofmitigating resonant overvoltages when the cause is a fault clearing.

With EMT-studies, such as this one, problematic areas can be found where the resonancefrequency seen from a bus is close to a low order harmonic present in the inrush current.If such a location is found, and the technique is feasible to implement at the location, the

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resonance frequency can be de-tuned in preparations for an energized transformer. This byaltering the reactance by connecting or disconnecting components like shunt capacitor banksor shunt reactors [6].

The resonant overvoltages can be reduced by adding more load to the system beforeenergizing a transformer. This works in two ways, by dampening the overvoltages as well asreducing the impedance peak caused by the resonance [6].

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3 Method and model development

Throughout this work, the studies were conducted using the electromagnetic transient simu-lation tool PSCAD. The studies were divided into three cases, where case one and two weresimulated in a model of a Stockholm and case three was simulated using a newly developedmodel. The new model was developed based on future off-shore wind farms, expected to beconnected using long AC cables in southern Sweden.

In case one the theory was tested and the output of the model was validated. Case onematched the output from the model with real data to find parameters for future simulations.

Case two was conducted to test the impact of uncontrolled re-energizing due to faultclearing. Different scenarios were tested altering the kind of fault, location of the fault andamount of load present in the grid.

Lastly, the goal of case three was to test the impact of resonant overvoltages in a weakergrid. Since the length of the off-shore cables had a large impact on the resonance frequency,different scenarios could be tested.

3.1 Case one - Testing and validating the model of Stockholm

This case was made as a starting point for the project where the goal was to get a betterunderstanding of the model and the theory. As well as test the central theory of the projectto enforce the confidence in the model. In addition to this, a comparative study was madeto find parameters related to the saturation of the transformers used in the model.

3.1.1 Resonance and impedance

The resonance frequency’s dependency on the reactance present in the grid was tested. Thiswas done by finding a suitable location in the grid, closely connected to a high amount ofcables. A 400 kV power staion was chosen, henceforth known as Power station A. Powerstation A is directly connected to two large cables, Cable 1 and 2, as is shown in figure 10.

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Figure 10: Illustrative picture of Power station A, Cable 1, Cable 2. The picture also showsTransformer 1 and Transformer 2. Transformer 1 is connected to Power station A with a breaker.(This to illustrate which transformer is being connected in case one).

Firstly, a frequency scan was done on a scenario where Cable 1 and 2 had their correctlengths. Secondly, the same simulation was made, with the sole difference of an alteredcapacitance. This was done by changing the length of Cable 2 in the model to 10 % of itsoriginal length. This approach was chosen to keep all other things equal. The output fromboth frequency scans were then compared to find the change in resonance frequency thatthe altered length of Cable 2 had caused.

3.1.2 Inrush Current

The phenomenon of inrush current was central in the project and was subsequently simulatedin the model to ensure the compatibility of the model with the phenomenon. This was doneby simulating a connection of a fully disconnected transformer to a power station, Powerstation B, see figure 11. The focus of this simulation was not to investigate how differenttimings of the connecting transformer affected the overvoltages, since this had been done inearlier studies [17].

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Figure 11: Illustrative picture of Power station B, Cable 3, Overhead line 1 and Transformer A.Transformer A is connected to Power station A with a breaker. (This to illustrate which transformeris being connected in case one).

The chosen power station with a voltage level of 400 kV, Power station B, was suitablefor the simulation because of a 2nd parallel connected transformer to Transformer A. Inaddition to this, Power station B is located in close proximity to another power station withtwo transformers connected.

According to theory the worst inrush current would occur if a connection was made whenthe voltage in a phase passes zero. A business-as-usual scenario was simulated to find thetime where a phase passes zero.

To compare the impact of the inrush current two simulations were made where Trans-former A was connected at a point in time where one of the voltage phases passes zero. Thefirst simulation was done where the knee-point of the transformer was kept at a reasonablelevel. The knee-point voltage was doubled in the 2nd simulation to remove the possibility ofany saturation in the magnetic core of the transformer, i.e, the system will not experienceany current harmonics originating from the transformer being connected. The current andvoltage were then measured in Power station B as well as in an adjacent overhead line,Overhead line 1.

3.1.3 Sympathetic inrush current

Sympathetic inrush current was tested by a simulation on the Power station A, see figure10. A second equal transformer to the one being connected is present at Power station A,Transformer 2. This to be able to test the interaction between two parallel coupled trans-formers during the connection of one of them to the grid. The transformer being connected,henceforth known as Transformer 1, was connected at a time where one of the voltage phasespassed zero. This to create a large inrush current from Transformer 1. Transformer 2 wasconnected to the grid during the entire simulation. Neither of the two transformers wereconnected with their secondary side to the low-voltage grid. The impact of sympatheticinrush was tested by running two simulations of the model with all things equal but one,

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Transformer 2’s ability to saturate. The reason for this was to visualize the sympatheticinteraction between the transformers. The current in Transformer 2 was measured for bothsimulations.

To understand the origin of the sympathetic inrush current, additional measurementswere done. For all measurements taken where one phase was studied, the phase with thelargest inrush current in Transformer 1 was chosen. Firstly the flux linkages in both trans-formers were plotted against each other with the purpose of illustrating the 2nd transformerbeing dragged towards its saturation after the magnetization of Transformer 1. Then themagnetizing current for both transformers was plotted.

Lastly the impact of the sympathetic inrush current was evaluated by studying the currentand voltage and their respective harmonic content in Cable 1.

3.1.4 Model Validation

To find suitable parameters for the transformers in the model a comparative study was madebetween a simulation done in a model of a small isolated part of the grid and data from areal connection of a transformer. The data that was used came from a real connection of atransformer at a power station very similar to Power station A. The test that the data comesfrom was done without a phase synchronizing device, resulting in a badly timed connectionon the voltage wave form. This also led to that the connection of the different phases was notseparated by the usual 120 degrees, but instead all occurred in a span of 2 milliseconds. Forsimplicity, this was modeled as a connection of all phases simultaneously. The parametersstudied were the air core reactance and the knee-point voltage. These two parameters areseldom known when it comes to EMT-studies and were therefore chosen to be derived froma comparative simulation.

Firstly a simulation was conducted to determine a suitable time for the connection tooccur. This was done by studying the voltage waveform at the Power station A. With theknowledge of how inrush current work and by studying the phase current of the data fromthe real connection, an initial time for the connection in the simulation could be derived.The time for the connection in the simulation was adjusted by comparing the simulated andthe actual data of the phase current on the primary side of the transformer.

Once the time for the transformer connection had been found the parameters for thetransformer in the model could be approximated. This was done by matching the size of thesimulated and the actual inrush current in the transformer, by iteratively altering the valuesof the air core reactance and the knee-point voltage. Lastly, parameters for the knee-pointvoltage and air core reactance were found.

3.2 Case two - Faults in the Stockholm grid

The goal of case two was to evaluate and compare different faults and different locations fora fault in and around Power station B, see figure 11. To do this several simulations wereconducted where the same measurements were acquired each time. Firstly the frequency de-pendent impedance for the specific setup was measured. Then the voltage and its respective

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harmonic content on Power station B was measured. Lastly, the current from one of thetransformers connected to Power station B was measured as well as its harmonic content.Any harmonic in the voltage that was affected by the impedance peak was studied further.The measurements that were done on the harmonic content of the voltage were: the max-imum amplitude and also the average amplitude at the chosen time, 0.3 seconds into thesimulation. This to evaluate the decay of the TOVs in each simulation.

3.2.1 Model setup

The parameters for air-core reactance, 0.14 pu, and knee-point voltage, 1.13 pu, which werefound in the comparative simulation were used in case two and three.

Two circuit breakers were modeled at each end of the overhead line or cable where thefault occurs to enable disconnection the fault. To model the fault a suitable time was foundto maximise inrush current, henceforth known as t0. The fault clearing time was chosen tobe 100 ms. The circuit breakers were modeled as ideal, meaning that the arc that is createdat t0 + 100ms when the circuit breakers are opened is not modeled. The fault clearing willtherefore occur at a later time, at the first zero pass for the current. Resulting in a faultclearing at t1 ≥ t0 + 100ms.

3.2.2 Case without fault

A simple simulation of the system was conducted where no fault occurred. This was done tocreate measurements from normal operations to compare with the measurements from thesimulations where faults occur.

3.2.3 Single-line-to-ground fault

The fault was modeled on an 20.3 kilometers long overhead line, Overhead line 1, which wasone of the connections from Power station B to the rest of the grid. The location of the faultwas chosen to be right outside of the circuit breaker between Power station B and the line.

3.2.4 Three phase to ground fault

This fault was created in the same way and at the same location as the phase to groundfault. The only exception being the number of phases that the fault included.

3.2.5 Location of fault

Since the three phase to ground fault gave the the largest impact it was chosen as the faultto use when testing different locations for the fault. A convenient place in the middle ofthe overhead line was chosen as the new place for the fault to occur. This ended up being9.3 kilometers away from Power station B, where the measurements are made. The secondlocation that was chosen for a fault to occur at was at the end of the overhead line, i.e. 20.3kilometers away from Power station B.

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3.2.6 Fault on a cable

Since a fault in the grid entails a disconnection of the segment of the grid where the faultis, and removing a cable will impact the resonance frequency, a fault in a nearby cable wastested, Cable 3. The fault was modeled as a three phase to ground fault just outside of thecircuit breaker between Power station B and Cable 3.

3.2.7 Fault during a case with downstream load

To study a more realistic case, some loads were added to the model. The loads were modeledto correspond to 15 % of the rated power in all the transformers in the system. This sizeof the load was chosen to mimic a light load scenario as well as take into consideration thatnot all loads can be assumed to be resistive [1].

The loads were then distributed onto the power stations in the system with a voltagelevel of 220 kV. Between the distributed loads and the power stations there were also addedtransformers from 220 kV to 22 kV. This was done to mimic a real grid where the loadsoften are positioned on lower voltage levels.

The model was then simulated with a three phase to ground fault located close to Powerstation B at Overhead line 1.

3.3 Case three - Off-shore wind farm model

The goal of case three was to investigate any problems which could arise when connectingoff-shore wind farms to the mainland. The model was designed to mimic a smaller town inthe southern part of Sweden. The simulations in case three were also conducted to studyworst-case-scenarios, the severity of their impact and the likelihood of the incidents to occur.This was done by studying resonant overvoltages in a weaker grid than the one studied incase one and two. In a weaker grid the resonance frequency changes more with a change inreactance, making it easier to test different resonance frequencies.

3.3.1 The model

Figure 12 illustrates a schematic picture of the model used in case three.

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Figure 12: Schematic picture illustrating the general components and their placement in the model.

The model contains 3 buses: A, B and C which are connected with 3 overhead transmis-sion lines, OHLs. The OHLs are named after the buses they create a connection between.The voltage on all three busses was 400 kV. Four 400/220 kV transformers, indicated by thenumber 1 in figure 12, were placed on Bus C to connect 4 cables to the grid. The cables,number 2 in the figure, being placed on the secondary side of the transformers had a voltageof 220 kV. The cables were connected in such a way it could occur in reality when off-shorewind farm is connected to the mainland grid. The connection of such a park to the mainlandwould need long cables, in this model the length of the four cables were kept between 50 and70 kilometers per cable.

Number 3 in the figure shows the position of a fault. This fault was kept for the sake ofsimplicity as a three phase to ground fault in this model for all cases. A time for the faultto create the largest inrush current was found in the same way as in case one and two. Oneach end of the line with the fault a circuit breaker was modelled, each with a opening timeof 100 milliseconds with the fault clearing occurring at the first zero crossing of the current,the same way as described in case two. Lastly the numbers 5 and 6 in figure 12 show theconnection to the rest of the grid. The rest of the grid is modeled as Thevenin equivalentssupplying the isolated part of the grid that is being studied. The voltage sources components

22

in the model, used as Thevenin equivalents, have a built-in impedance connected in series.This impedance can be altered to change the strength of the grid, and in extension theresonance frequency.

To better evaluate the longevity of the harmonics found in all simulations of case threethe harmonics decay are represented by a time constant. This time constant is the time ittakes for the harmonic to reach 37% of its initial value.

3.3.2 Matching of the cases

For the results from this study to be as dependable as possible the voltage on the transformerbefore the fault had to be matched in every case. This, because altering the model in somecases changed the voltage level. This is important since the voltage level in the transformerbefore the fault determines the gap between a saturated and a non-saturated transformer. Inthis case this was done on the primary side of on of the four transformers connected on BusC. The voltage level to match was an RMS line to line voltage of 420 kV, which correspondsto the upper operational limit for the voltage.

3.3.3 Standard model

Firstly the frequency dependent impedance was studied for the model. This was done withfour maximum length cables, 70 kilometers, connected to Bus C. This to create a scenariowith the maximum amount of capacitance present from the cables. From the impedanceprofile found in this scenario, in conjunction with equation 4 the impedance profile could bedesigned for further scenarios.

It was found that the desired impedance profile with a peak at the 2nd harmonic was notpossible to occur with just the added capacitance from the 4 cables connecting the off-shorewind power. To reach the desired scenario the strength of the adjacent grid had to be altered.This was done in two ways, by decreasing the power contributed from the southern and thenorthern Thevenin equivalents or by removing one of the connections. Effectively creatinga N-2 scenario in conjunction with the fault. To evaluate the strength of the grid for thedifferent cases the fault current was measured.

3.3.4 Strength of the grid altered

The strength of the adjacent gird was decreased in order to lower the frequency that theimpedance peak occur at to 100 Hz. This was done by increasing the positive sequenceimpedance in the northern and southern Thevenin equivalents.

3.3.5 N-2 scenario

The other way of decreasing the strength of grid was to disconnect one of the connectionsto the rest of the grid, the southern connection was chosen to be disconnected. This discon-nection in combination with the disconnection occurring after a fault creates a N-2 scenario.Lastly the lengths of the four cables were altered to achieve a resonance frequency of 100 Hz.

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3.3.6 Sensitivity analysis and changed resonance frequency with saturatedtransformers

A scenario was tested where the model from the N-2 scenario was used to investigate theimportance of the precision of the cable lengths. Doing this by simulating two scenarios withcable lengths resulting in resonance frequencies slightly above or below the 2nd harmonic.The scenario was also used to study the phenomenon of an altered resonance frequency withthe saturation of transformers.

3.3.7 Evaluating the changed resonance frequency due to saturated transform-ers

To evaluate the change in resonance frequency that occur in a saturated transformer severaladditional simulations were made. In these simulations the lengths of the cables were alteredto find resonance frequencies below 100 Hz. The 2nd order harmonic was then simulated forresonance frequencies at and below 100 Hz. This to investigate the range of the impedance”plateau”.

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4 Results


From case one the model was deemed credible for studies of EMT as done in case two andthree.

4.1.1 Resonance and impedance

The impedance profile was studied for Power station A with varying amount of capacitance.The results are presented in figure 13, where the blue trace illustrates the impedance from theoriginal model where 100 % of the capacitance is present. The red trace is the impedanceseen from Power station A but from a model where the magnitude of the capacitance inCable 2 is reduced to 10% of its original value.

Figure 13: A figure illustrating the positive sequence impedance seen from Power station A for theinitial scenario and a scenario where the length of a nearby cable has been decreased.

It can be seen from figure 13 that when the capacitance in the system decreases, inaccordance with equation 4, the resonance frequency increases.

4.1.2 Inrush Current

The connection of the transformer, at Power station B to the grid, was chosen to occur after0.05 seconds into the simulation. This was done primarily to have the voltage of the phaseillustrated by the blue trace to be exactly zero, see figure 14.

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Figure 14: Voltage on the Power station B during connection of a transformer at 0.05 secondsindicated by the dotted line.

The connection displayed in figure 14 shows the point on wave where the transformer isenergized. The transformer being energized is forced into its saturation zone and causes aninrush current, as can be seen in figure 15a. Figure 15b illustrates the same connection butwith a transformer with a higher knee-point voltage. This means that the amplified magneticflux will not cause the transformer to be saturated and thus it will not cause an extra largecurrent, including harmonics, to be drawn from the transformer. Note the different scaleson the y-axis in the two graphs of figure 15.

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(a) 1.14 pu knee-point voltage. (b) 2.28 pu knee-point voltage.

Figure 15: Two graphs illustrating the difference between a transformer forced into its saturationzone, figure 15a, and a transformer that is not, figure 15b. Note the different scales on the y-axisin the two graphs.

According to the theory, a voltage close to zero during the connection of the transformerwould cause the highest inrush current. The phase illustrated by the blue curve in figure 14should therefore cause the largest inrush current. This can be seen to be true in figure 15a.


Figure 16: The current in line 1 connecting the power stations B with an adjacent power stationduring the connection of a transformer at Power station B.

The connection of a saturated transformer affects the nearby Overhead line 1, figure 16a,much more than the connection of a transformer that is not being saturated 16b.

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Figure 17: The harmonic content of the currents in figure 16.

Figure 17 shows the harmonic content of the phase with the largest inrush current, theblue trace in figure 16. Figure 17a shows an amplified fundamental, a large 2nd harmonicand some small 3rd and 4th order harmonics.


Figure 18: The harmonic content of the voltage at the Power station B during the connection of atransformer.

In figure 18 the harmonic content of the voltage at Power station B can be seen. Thepower station is directly connected to Overhead line 1 and will therefore see and get affectedby the amplified currents displayed in figure 17. Lastly, it should be noted that with thepossibility of saturation in the transformer, a connection if badly timed could cause a highinrush current rich of harmonics.

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4.1.3 Sympathetic inrush current

The current in one phase of Transformer 2, the transformer that was already connected tothe grid, was tested with and without the ability to saturate.

(a) Without the ability saturate. (b) With the ability to saturate.

Figure 19: One phase current in Transformer 2 during a connection of Transformer 1.

Figure 19 illustrates the difference between the phase current in Transformer 2 when itsaturates and when it does not. Figure 19b shows a large sympathetic inrush current as aresult of Transformer 2 being dragged into saturation by the connection of Transformer 1.

(a) Full-size image. (b) Zoomed in during start of saturation.

Figure 20: Two graphs showing the flux linkage in Transformer 2 that gives rise to the magnetizingcurrent at saturation.

In figure 20 it can be seen that the magnetic linkage in Transformer 2 is a sinusoidalwaveform centered at zero with the amplitude of one pu, before the connection of Transformer

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1 at 0.5 seconds into the simulation. After the connection, the magnetic flux in Transformer2 gradually gets drawn towards saturation. In figure 20b the moment that Transformer 2reaches saturation can be seen. The knee-point voltage in the transformer was set to 1.14pu as indicated by the black dotted line. As the flux linkage reaches the dotted line amagnetizing current can be seen, starting out small and growing rapidly as Transformer 2gets more saturated. The time it takes for Transformer 2 to get saturated causes a timedifference for when the inrush current in the two transformers occurs. This can be seen infigure 21.

(a) Full-size image. (b) Zoomed in during start of saturation.

Figure 21: Two graphs showing the magnetizing current in the two transformers.

In figure 21 it can be seen that the two transformers creates magnetizing currents inopposite polarity and reaches their respective maximum with a half period apart.

(a) Without sympathetic inrush. (b) With sympathetic inrush.

Figure 22: Two graphs showing the three phase current in the nearby cable, Cable 1, during theconnection of Transformer 1.

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Cable 1, that is connected to the Power station A, got two different profiles for the currentduring a connection depending on the presence/absence of sympathetic inrush. The currentwaveforms look similar until the magnetization current from transform two appears.


Figure 23: Graphs showing the harmonic content of the two graphs in figure 22. Harmonics ofhigher order than four were excluded.

Figure 23 more clearly illustrates the longevity of the current in Cable 1 in the case ofsympathetic inrush. What can be seen is that the fundamental and the 3rd harmonic areamplified even further in figure 23b. The saturation of Transformer 2 also dampens the 2nd

harmonic.


Figure 24: Two graphs showing the harmonic content of the voltage in Cable 1. The graphs arezoomed in on the harmonics of the voltage but both graphs have similar fundamental waveforms ofaround 240 kV.

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Figure 24b shows that the larger 3rd harmonic in the current that occur in the case withsympathetic inrush causes a higher 3rd harmonic in the voltage. However the 2nd and 4th

harmonic are dampened.

4.1.4 Validation of transfomrer

Figure 25: The current in the three phases of the primary side of the transformer being energizedat Power station A. The data in the graph was taken from a real connection of a transformer.

In figure 25 it was observed that the amplitude of the inrush current in two of the phaseswas similar and larger than the third. This knowledge proved two things, the time for theconnection was when the voltage corresponding to the two largest phase currents had to besimilar. And that the voltage of the third phase current would be further away from zerothan the other two.

Two times that matched both criteria were found in a chosen time period and subse-quently tested. These times were 0.4456 and 0.4556 seconds. The times when the red andyellow phase of figure 26a cross one another. The phase current resulting from a connectionat 0.4456 seconds was found to be the correct and can be seen in figure 26b.

32

(a) Simulated voltage at Power station A whilethe transformer is disconnected.

(b) Simulated current in the transformers primary sideduring connection.

Figure 26: Two figures illustrating the procedure of finding the point on the voltage waveform whenthe transformer is being connected.

Since the simulation was done with all phases being connected simultaneously two ad-ditional runs were made. One where the connection occurred 2 milliseconds before the realconnection and one 2 milliseconds after. The phases were still all connected simultaneously.

(a) Connection 2 ms earlier. (b) Connection 2 ms later.

Figure 27: The current of the transformer being connected 2 ms before and after the originalconnection seen in figure 26b.

Since the phase current in the simulated case, as can be seen in figure 26b, was smallerthan the real data in figure 25, the air core reactance and the knee-point voltage were reduced.With the air core reactance of 0.14 pu and the knee-point voltage of 1.13 pu a simulatedphase current similar to figure 25 was found in figure 28. The magnitude of the first peak ofeach phase was 3.22 kA and 2.39 kA. The magnitude of the respective peaks from the real

33

data was 3.13 kA and 2.57 kA.

Figure 28: The current in the three phases of the primary side of the transformer connected toPower station A. Parameters of transformer; Knee-Point Voltage: 1.13 pu, Air core reactance:0.14 pu.

4.2 Case two - Faults in the grid

The setup was first monitored during normal operations.

34

Figure 29: Positive sequence impedance as a function of frequency with a visual peak close to 150 Hz.

An impedance peak seen from Power station B was found at around 160 Hz, presentedin figure 29. The voltage in and around Power station B during normal operations can beseen in figure 30.

Figure 30: Three phase voltage at Power station B during normal operations.

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4.2.1 Fault close to Power station B

4.2.1.1 Three Phase to ground fault

Firstly a three phase fault was simulated close to Power station B.

Figure 31: The voltage on the Power station B during the connection of a three phase fault at 0.06seconds. Fault gets disconnected 100 milliseconds later at 0.16 seconds.

In figure 31 the maximum voltage observed was 576 kV. 122 kV and 115 kV was themaximum amplitude found in the 2nd and 3rd harmonic voltage in figure 32.

36

Figure 32: The harmonic content of the voltage on the Power station B after the fault had beendisconnected.

4.2.1.2 Phase to ground fault

A phase to ground fault in Overhead line 1 was simulated.

Figure 33: The voltage on the Power station B during a phase to ground fault at 0.06 seconds. Theline and the fault gets disconnected 100 milliseconds later at 0.16 seconds.

37

The maximum amplitude of the voltage in figure 33 was 473 kV.

(a) Current in the phase with the fault. (b) Current in the phase without the fault.

Figure 34: The harmonic content of the current from a transformer connected on the Power stationB after the fault had been disconnected.

The inrush current in the phase with the fault is approximately two times larger thanthe inrush current in the phase without a fault, as can be seen in figure 34.

(a) Voltage in the phase with the fault. (b) Voltage in the phase without the fault.


The 2nd, 3rd and 4th order harmonic is larger in the phase with the fault than in thephase without a fault. The 2nd and 3rd harmonic both had a maximum amplitude of 106kV, see figure 35.

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4.2.2 Location of the fault

A three phase fault occurring in Overhead line 1, 9.3 kilometers away from the Power stationB was simulated. The entire length of Overhead line 1 is 20.3 kilometers, i.e. the fault occursapproximately in the middle of the line. The maximum voltage that was measured was 525kV, as can be seen in figure 36.

Figure 36: The voltage on the Power station B during a three phase fault at 0.06 seconds. Thefault gets disconnected 100 milliseconds later at 0.16 seconds.

The largest 2nd and 3rd harmonics found in the voltage were 75 kV and 58 kV, seefigure 37.

39


Next, a three phase fault occurring at the end Overhead line 1 close to the adjacentpower station was simulated, where the largest voltage measured was 492 kV, as shown infigure 38. There was 20.3 kilometer of overhead line between the fault and the transformersstudied and the point of measurements.

40

Figure 38: The voltage on the Power station B during a three phase fault at 0.06 seconds. Thefault gets disconnected 100 milliseconds later at 0.16 seconds.

The largest 2nd and 3rd harmonics found in the voltage were 79 kV and 74 kV, seefigure 39.


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4.2.3 Fault on a Cable

A fault in Cable 3, connected to the Power station B causes a shift in the frequency dependentimpedance seen from Power station B, see figure 40.

Figure 40: Positive sequence impedance seen from Power Station B after the disconnection ofCable 3.

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Figure 41: The voltage on the Power station B during a three phase fault on Cable 3 at 0.06 seconds.The fault gets disconnected 100 milliseconds later at 0.16 seconds.

When the fault occured in a cable the maximum voltage that was measured was 425 kV,see figure 41. The largest harmonic voltage in the 2nd, 3rd and 4th was 100 kV, 70 kV and80 kV, see figure 42.

Figure 42: The harmonic content of the voltage on the Power station B after the fault in the cablehad been disconnected.

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4.2.4 Impact of a load

A three phase fault was simulated in the same way as in chapter 4.2.1.1 with the additionof a downstream load, where the maximum voltage measured was 532 kV, see figure 43.

Figure 43: The voltage on the Power station B during a three phase fault on a nearby overhead lineat 0.06 seconds in a model with a downstream load, The fault gets disconnected 100 millisecondslater at 0.16 seconds.

In figure 44 it can be seen that the largest 2nd and 3rd harmonics in the voltage were 100kV and 67 kV.

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Figure 44: The harmonic content of the voltage on the Power station B after the fault in Overheadline 1 had been disconnected.

4.2.5 Summarizing the overvoltages and their longevity for the different faults

The relevant data was then summarized in a table to evaluate the resonant overvoltages,table 1.

Table 1: Table summarizing the results from the faults simulated in case two. [1] Single-line-to-ground fault. [2] Three phase to ground fault. [3] Average voltage measured at the end of theobserved time window, 0.3 seconds into the simulation.

Maximum Voltage[kV]

3rd harmonicmaximum voltage

[kV]

3rd harmonicaverage voltage[3]

[kV]

SLG[1] Fault close to,Power station B

473 106 23

3L-to-G[2] Fault close to,Power station B

576 115 34

3L-to-G Fault on OHL 1,middle of line

525 58 23

3L-to-G Fault on OHL 1,end of line

492 75 8

3L-to-G Fault on Cable 3 425 70 103L-to-G Fault witha downstream load

532 67 6

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In table 1 it can be seen that the largest voltages measured occur in the cases wherethere is a thee phase to the ground fault. The largest being 576 kV and occurs when thefault occurs close to the point of measurement. The same fault also had the largest 3rd

order harmonic voltage left after 0.3 seconds. The maximum voltage observed goes downwhen the fault occurs farther away from the point of measurement. A fault in the middleof the line causes a maximum voltage of 525 kV, and at the end of the line the maximumvoltage reaches 492 kV. It can be seen that a downstream load in the system causes boththe maximum voltage and 3rd order harmonic to decrease, and also a drastic decrease in thelongevity of the harmonic content. The results from the fault on the cable is best studiedfrom figure 42, where a large 2nd, 3rd and 4th harmonic can be seen.


4.3.1 Initial scenario

The fault current in the initial scenario was 21 kA, and a impedance peak of 350 Ohm canbe found at 127 Hz.

Figure 45: Positive sequence impedance for the initial scenario seen from Bus C.

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Figure 46: Phase voltage on the primary side of the transformer.

In figure 46 a maximum overvoltage of 521 kV can be seen. A 2nd order harmonic voltagecan initially be found in figure 47 of 105 kV, but disappears soon after the fault clearing.The time constant (as defined in section 3.3.1) for the harmonic in this case was 0.01 s.

Figure 47: Graph illustrating the harmonic content of the voltage on the primary side of one of thetransformers connecting a cable with Bus C.

47

4.3.2 Strength of the electric grid altered

The fault current in the scenario with an altered characteristic of the adjacent grid was 8 kA.The strength of the adjacent grid was altered to achieve a impedance peak at 100 Hz, whichhas an amplitude of 522 ohm.

Figure 48: Graph illustrating the positive sequence impedance seen from Bus C in the model withan altered strength of the electric grid.

The maximum amplitude of the voltage seen in figure 49 was 525 Kv.

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The harmonic content found in the voltage in figure 50 is almost exclusively of the 2nd

order. This harmonic reaches a peak amplitude of 114 kV with a time constant of 0.274 s.


49

4.3.3 N-2 incident

In the scenario with two disconnected OHLs the maximum fault current was 10 kA. Thefinal length of the cables to achieve the impedance profile in figure 51 was 65.5 km. Theimpedance peak at 100 Hz had an amplitude of 676 ohm.

Figure 51: Graph illustrating the positive sequence impedance seen from Bus C in the model withtwo disconnected OHLs.

In figure 52 a maximum voltage of 641 kV can be seen.

50


The 2nd order harmonic of the voltage was found to have an amplitude of 130 kV with atime constant of 0.354 s.


51

4.3.4 96 Hz vs. 104 Hz

The fault current (system strength) for the two simulations below were both 10 kA. Theimpedance peak that occur at 104 Hz has an amplitude of 713 ohm while the peak at 96 Hzhas an amplitude of 636 ohm. The cable lengths in the models that have an impedance peakat 96 Hz and 104 Hz were 70.0 km and 59.6 km respectively.

(a) 96 Hz. (b) 104 Hz.

Figure 54: Positive sequence impedance seen from Bus C in the model for two resonance frequencies.

The maximum amplitude of the voltage found with a impedance peak at 96 Hz was 614kV and for the case with a impedance peak 104 Hz the maximum voltage was 635 kV, seefigure 55.

(a) 96 Hz. (b) 104 Hz.

Figure 55: Phase voltage on the primary side of the transformer for the two models with differentresonance frequencies.

In figure 56a the maximum voltage of the 2nd harmonic was 130 kV with a time constant

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of 0.273 s. The maximum voltage of the 2nd harmonic in the case with an impedance peakat 104 Hz was 120 kV with a time constant of 0.258 s, see figure 56b.

(a) 96 Hz. (b) 104 Hz.

Figure 56: Two graphs illustrating the harmonic content of the voltage on the primary side of oneof the transformers connecting a cable with Bus C.

A last comparison was made of the 2nd harmonic voltage found in the N-2 model withimpedance peaks at three different frequencies: 96 Hz, 100 Hz and 104 Hz.

Figure 57: The 2nd order harmonic voltage on the primary side of the transformer for three modelswith different resonance frequencies, 96Hz, 100 Hz and 104 Hz.

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4.3.5 Evaluating the changed resonance frequency due to transformer satura-tion

In figure 58 it can be seen that initially the largest 2nd order harmonic of the voltage is foundwhen the system has a resonance frequency of 98 Hz. It can also be seen that the 2nd orderharmonic of the voltage decrease as the resonance frequency decrease below 98 Hz.

Figure 58: The 2nd order harmonic voltage on the primary side of the transformer from simulationswith resonance frequencies at and below 100 Hz in increments of 2 Hz.

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5 Discussion


All the phenomena tested in case one work as intended in the model. In the case withinrush current, as can be seen in figure 18a, small low order harmonics were present in thevoltage. The absence of a high 2nd order harmonic, related to the current in figure 17a,can be explained by an impedance profile similar to figure 13, which has a impedance peakrelating to the 3rd harmonic but a small impedance for the 2nd harmonic.

What should be noted about the results for the sympathetic inrush is that the longevityof the harmonics in the inrush current increases when a system is capable of sympatheticinrush, see figure 23. Especially the 3rd harmonic which correlates to the large 3rd harmonicin the resonant overvoltages, see figure 24, even though this results comes from the connectionof a transformer, it will logically act similarly when fault clearing occurs, as in case two andthree.

Good values for the parameters for the knee-point voltage and the air-core reactanceare difficult to obtain. This because in reality a transformer acts differently at varyingamount of saturation, which means that choosing a constant that satisfies the entire processof transformer saturation perfectly is nearly impossible. This can be seen when comparingthe inrush current with the simulated current. One thing that should be noted is that growthof inrush current in the real data is steep, see figure 25, indicating that the air-core reactancein a real transformer increases as you enter more deeply into saturation. This in comparisonto the simulated inrush current that can be seen in figure 28 to decrease in growth, as if itis linearly following the magnetic flux waveform seen in figure 15b. Because of this, it wasdecided to find the parameters for transformer saturation from a comparative study, and theresult of that study should increase the confidence of said parameters.

5.2 Case two - Faults in the grid

The first two simulations done in case two were used to decide what kind of fault should beused in the subsequent studies. This choice was made on the basis of which of the single lineto ground fault or three phase to ground fault caused the largest overvoltage. This resultedin a three phase fault being simulated for the subsequent simulations. While a three phaseto ground fault was also used for the simulation of the fault on a cable, this is improbable tohappen. However, since the fault occur close to the bus, the fault can be assumed to occuron the connection between cable and bus, and therefore be somewhat more probable.

The harmonic content of the voltages observed in case two are not decaying stationaryas the time goes, but instead have some form of periodicity. Which, makes studies of thelongevity of the harmonic content of the voltage complicated and unprecise.

It should be noted that the large 4th harmonic voltage when the fault occurs in a cablecomes from the altered impedance peak, see figure 40. This due to the fault clearing removinga large source of capacitance, thereby shifting the resonance upwards in frequency. Thatthe large 4th order harmonic created decays fast could be explained by the impedance peak

55

being shifted towards a higher resonance frequency during high saturation of the transformerimmediately after the fault, to then return to the impedance peak seen in figure 40. Whichwould mean a high impedance at the 4th harmonic during the initial time after the faultclearing and then the impedance goes down with the saturation.

The decreasing overvoltage with three phase to ground faults occurring farther away fromPower station B on the overhead line can be explained by the voltage level during the fault.In figure 36 and figure 38 the voltage before the disconnection of the fault can be seen to below but not zero. In figure 31 however, the three phase to ground fault occur next to Powerstation B, which causes the voltage level during the fault to be zero. Which according tothe theory, should cause the largest inrush current, which in turn causes the largest voltage.During these simulations the time difference from the location on where the fault occurs wasneglected. This added time would in theory cause a miss-match between the zero-passing ofthe voltage, which also would cause a lower inrush current. But, the time difference was, atthe farthest location, 0.07 ms, which was neglected.

Furthermore, it should be noted that a system containing a downstream load causesa smaller 3rd harmonic voltage, table 1. The downstream load also causes the 3rd orderharmonic to decay at a faster rate relative to the decay in the case without a downstreamload, table 1. In addition to this it should be noted that designing a model with a downstreamload to be used for EMT-studies is complicated. This due to the fact that the magnitude ofthe load changes and that not all loads can be assumed to be purely resistive. Because ofthis, the loads being modeled to be 15 % of the rated power in all transformers should bethought of more as an approximation than anything else. Lastly the maximum amplitudeof the resonant overvoltages found in case two are not severe. The long lived harmonicsfound in some of the simulations, especially in the three phase to ground fault close to Powerstation B, could be a concern regarding the TOV-capabilities of arresters.


It could be argued that the use of time constant to measure the longevity of the harmonicsis a superior way than the method used in case two. It was decided not to be used in casetwo however, because of the variety of harmonics found in the different simulations. Sincethis would make the method tedious and unprecise. For the sake of simplicity the cableswere all kept at the same length for the different scenarios. Which is unrealistic but doesnot affect the result.

Because of the sudden change in voltage following a fault clearing, the signal processingwhen creating graphs could cause higher voltage spikes than what is actually simulated. Thiscould cause uncertainties closely after the fault clearing occurs in the figures presented inthis thesis. These uncertainties were however deemed negligible.

The maximum voltage does not change drastically between the initial case, figure 46,and the case with an altered strength of the grid, figure 49, from 521 kV to 525 kV. Thelongevity of the harmonics are however increased from a time constant of 0.01 s to 0.274 s.This is in line with the theory that overvoltages due to resonance have a longer lifespan thanother overvoltages. The model with an altered strength had a fault current of 8 kA which is

56

low, and unrealistic. It could however occur during a the sudden loss of a big power sourcelocated close by.

The N-2 model is a more realistic scenario to occur in reality. During the maintenanceon one overhead line the grid is vulnerable to be exposed to the resonant overvoltages thatare presented in figure 52 and 53, in the case of a fault. Since both the maintenance andthe fault could occur due to unforeseen incidents, enforcement measures should be made tominimize the impact of such an outcome. The fault current of 10 kA that was found in theN-2 model is more realistic than the fault current in the model with an altered strength ofthe grid. The resonant overvoltages are also more severe.

One of the goals of studying the N-2 model with resonance frequencies at 96 Hz and104 Hz was to find the impact of the resonant overvoltages if the resonance frequency didnot perfectly align with the 2nd harmonic. This because the likelihood of the cables havingthe precise length to cause the grid to have a resonance frequency at 100 Hz is low. Thecapacitance from all cables would have to add up to the amount that they had if all cableswere of the length 65.5 km. From the model with a resonance frequency at 104 Hz to 96 Hzthe cable lengths can differ by 10.4 km. Which is far more likely to occur. And since theresonant overvoltages presented in figure 56 were of comparable severity to the resonantvoltages found in a grid with a resonance frequency of 100 Hz, as can be seen in figure 53,the likelihood of an incident is more probable.

The second goal of studying the model for frequencies above and below 100 Hz was toinvestigate the phenomenon of an altered resonance frequency during the saturation of atransformer. It can be seen in figure 56 that the simulation with a resonance frequencyof 96 Hz causes a larger 2nd order harmonic voltages with a higher longevity than for thesimulation with a resonance frequency at 104 Hz. This despite the fact that the impedancepeak when the resonance frequency was 104 Hz, see figure 54, is larger than that for themodel with a resonance frequency of 96 Hz. This indicates that during the saturation of thetransformer the impedance peak gets shifted towards a higher resonance frequency, pushingthe impedance peak at 96 Hz towards the 2nd harmonic, while the impedance peak at 104 Hzis pushed away from the 2nd harmonic. This is clearly visualized in figure 57, where the 2nd

order harmonic of the voltage can be seen for three resonance frequencies, 96 Hz, 100 Hzand 104 Hz. The 2nd harmonic of the voltage for the models with resonance frequencies at96 Hz and 100 Hz is initially equal, and both of them larger than the the model where theresonance frequency incur at 104 Hz.

This was investigated further to see what resonance frequency actually caused the largestresonant overvoltage. In figure 58 the 2nd order harmonic of the voltage can be seen forsimulations made where the resonance frequency was 100 Hz and below, in increments of2 Hz. The 2nd order harmonic of the voltage was initially the largest for the model witha resonance frequency of 98 Hz, which then gets overtaken by the resonance frequency at100 Hz. This could be due to that during the high saturation of the transformer, theresonance frequency of 98 Hz goes towards 100 Hz, while the resonance frequency at 100 Hzincrease, i.e. moving away from the 2nd harmonic. As the saturation of the transformerdecreases with time, the shunt inductance caused by the saturation also decrease, which

57

causes the resonance frequencies to retract towards their original positions. This wouldexplain the overtaking of the 2nd harmonic voltage from the resonance frequency at 100 Hzover 98 Hz, which occur at approximately 1.15 s.

The arrester is one of the components in the grid that is most affected by resonantovervoltages. This because of the high amount of energy the arrester needs to dissipatedue to the longevity of resonant overvoltages. Since the 2nd order harmonic voltage for themodel with a resonance frequency at 98 Hz is largest during the high voltage portion of theresonant overvoltage. Since this portion would be the period where the largest energy needsto be dissipated, it can be assumed that a resonance frequency of 98 Hz would be the mostharmful for a arrester in the studied case.

What should be noted between case two and three is the time observed of the harmoniccontent of the voltages. Where the time window studied in case two ranged from 0.15 s to0.30 s, while the time window i case three was 0.8s to 2.5 s. The longevity of the harmonicsin case three is much longer than in case two.

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6 Conclusion

From case one it can be conclude that the model of Stockholm is credible to use in a studyabout resonant overvoltages.

In case two the grid in Stockholm was tested for different faults at different locations. Thisresulted in resonant overvoltages that are not severe enough to be cause for concern. Theresult from this limited study is that the probability for destructive resonant overvoltagesfollowing transformer energization and saturation is low in the Stockholm grid.

In case three the four high voltage off-shore cables connecting wind power to the grid inconjunction with a N-2 scenario caused the resonance frequency to go down towards the 2nd

harmonic. This resulted in severe resonant overvoltages, which could be a concern consid-ering their long longevity. It was also concluded that the resonant overvoltages could occurwith cable lengths ranging from, at least, 59.6 km to 70.0 km. Lastly, it was shown thatspecial caution needs to be taken during EMT-studies regarding the increased resonancefrequency during the saturation of transformers. Meaning, the largest impact from the reso-nant overvoltage might not occur at a resonance frequency of a multiple of the fundamentalfrequency, but somewhere below.

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