On-line monitoring of a transmission line using synchrophasor measurements

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On-line monitoring of a transmission line using

synchrophasor measurements

FA de Jager

orcid.org 0000-0001-8918-7163

Dissertation submitted in fulfilment of the requirements for the

degree

Master of Engineering in Electrical and Electronic

Engineering

at the North-West University

Supervisor:

Prof APJ Rens

Graduation May 2018

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Declaration

I, F.A. de Jager hereby declare that the dissertation “On-line monitoring of a transmission line using synchrophasor measurements” and the material therein is my own original work except where specific reference is made by name or in the form of a numbered reference. The work herein has not been submitted to any other university or institution for examination.

F.A. de Jager

Student number: 22189556

Signed on the 5th_{day of}

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Abstract

The research assessed the possibility of using a wide area measurement system that uses synchrophasor data to calculate operational limits of an overhead line. These applications have become achievable to utility companies without the need of expensive phasor measurement units. Phasor measurements can be obtained by applying the IEEE C37-118 standard in general power system measuring equipment.

This dissertation presents the use of synchrophasors in the on-line modelling of a transmission line. Using time coherent measurements, the line parameters are continuously derived, which is then also used to monitor line temperature, sag, power transfer and voltage stability.

This enables the system operator to base decisions on real field data and less on estimated values as produced by state estimators. The methodology were verified by means of simulation and then validated by field data obtained across a 330 kV, 521 km line.

Promising results were obtained and analysis of the results confirmed that comprehensive transmission line performance monitoring is attainable. The electrical characteristic of the line can be made visible by tracking the variation in line parameters, line temperature and also line sag can be estimated. From a stability point of view, further application was shown to be the tracking of power transfer to visualise utilisation of installed line capacity and voltage stability across the line by means of tracking the locus of the operating points on P-V curves.

Key words— Synchrophasor applications, Transmission line modelling, Transmission line parameter

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Opsomming

Die navorsing assesseer die moontlikheid om 'n wye area metingstelsel te gebruik wat sinchrophasor data gebruik om operasionele perke van 'n oorhoofse lyn te bereken. Hierdie toepassings het haalbaar geword sonder die behoefte aan duur fasormetingeenhede. Fasormetings kan verkry word deur die IEEE C37-118-standaard in algemene kragstelsel meetapparatuur toe te pas.

Hierdie proefskrif bied die gebruik van sinchrophasors in die aan-lyn modellering van 'n transmissielyn aan. Deur sinkrone metings te gebruik, word die lynparameters voortdurend afgelei, wat dan ook gebruik word om lyntemperatuur, insakking, drywings-oordrag en spanningstabiliteit te monitor.

Dit stel die stelseloperateur in staat om besluite te neem gebasseer op werklike velddata en minder op beraamde waardes. Die metodologie is geverifieër deur middel van simulasie en dan gevalideer deur velddata verkry oor 'n 521 km, 330 kV lyn.

Belowende resultate is verkry, en analise van die resultate het bevestig dat omvattende monitering van ‘n transmissielyn bereikbaar is. Die elektriese eienskap van die lyn kan sigbaar gemaak word deur die variasie in lynparameters te monitor, lyn temperatuur en insakking kan ook geraam word. Uit 'n stabiliteitsoogpunt is verdere toepassing getoon om drywings-oordrag te monitor om die gebruik van geïnstalleerde lynkapasiteit en spanningstabiliteit oor die lyn te visualiseer deur die lokus van die bedryfspunte op P-V krommes te analiseer.

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Structure of dissertation

Introduction

The topic of synchrophasor application and other current trends in modern power systems will be discussed to motivate the need for research on innovative monitoring of transmission lines.

Performance monitoring of transmission lines

The opportunity for using synchrophasors to monitor transmission lines are introduced by studying the theoretical principles of transmission line small signal stability, voltage stability and equivalent line parameters.

Calculating transmission line parameters from field data

Different methods from the literature to calculate transmission line parameters by means of field data are analysed.

Verification of transmission line monitoring by network coherent data

The performance of the transmission line parameter calculation and monitoring methods discussed in previous chapters is verified with the use of simulation data.

Validation of transmission line monitoring by network coherent data

The application of transmission line monitoring is evaluated by the use of field data.

Conclusion

The research results verified and then validated how synchrophasor data can find innovative application complimentary to traditional small-signal stability applications by means of tracking transmission line performance comprehensively.

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Table of Figures

Figure 1: System stability categories [9]... 21

Figure 2: Simplified power system ... 22

Figure 3: Power transfer between generator and motor ... 22

Figure 4: Power angle across transmission line ... 23

Figure 5: Power transfer curve [14] ... 24

Figure 6: P-V curve principle [21] ... 28

Figure 7: Conductor bundling configurations ... 34

Figure 8: Two-port network model ... 36

Figure 9: Distributed nature of a transmission line ... 37

Figure 10: Equivalent  transmission line model [15] ... 38

Figure 11: Line sag representation ... 41

Figure 12: Concept of a synchrophasor [32] ... 44

Figure 13: Wide area monitoring system infrastructure [32] ... 45

Figure 14: Transmission line equivalent π impedance model... 48

Figure 15: Simulink simulation... 65

Figure 16: Active- and reactive power loading of line ... 66

Figure 17: Derived resistance error by window length ... 67

Figure 18: Derived resistance error by window length ... 68

Figure 19: Average and standard deviation of resistance error by window length... 68

Figure 20: Derived inductance error by window length ... 69

Figure 21: Error in inductance error by window length (histogram) ... 69

Figure 22: Average and standard deviation of inductance error by window length ... 70

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Figure 24: Derived capacitance error by window length (histogram) ... 71

Figure 25: Average and standard deviation of capacitance error by window length ... 71

Figure 26: Error analysis due to the measurement uncertainty in the magnitude of IR ... 73

Figure 27: Error analysis due to the measurement uncertainty in the magnitude of VS ... 74

Figure 28: Error analysis due to the measurement uncertainty in the magnitude of VR ... 76

Figure 29: Error analysis due to the measurement uncertainty in the phase angle of IR ... 77

Figure 30: Error analysis due to the measurement uncertainty in the phase angle of VS ... 78

Figure 31: Error analysis due to the measurement uncertainty in the phase angle of VR ... 79

Figure 32: Simulated active-power transferred over transmission line ... 80

Figure 33: Simulated sending- and receiving-end active power ... 81

Figure 34: Simulated active power losses over transmission line ... 82

Figure 35: Simulated power angle over transmission line ... 82

Figure 36: Simulated transmission power transfer stability ... 83

Figure 37: Simulated voltage stability P-V curve ... 84

Figure 38: Power factor vs. Transferred active power ... 85

Figure 39: Simulated increased loading - Active- and reactive power loading ... 86

Figure 40: Simulated increased loading – Sending- and receiving end active power ... 86

Figure 41: Simulated increased loading – Power angle across transmission line ... 87

Figure 42: Simulated increased loading – Power transferred over transmission line ... 87

Figure 43: Simulated increased loading - Transmission power transfer stability ... 88

Figure 44: Simulated increased loading - Voltage stability monitoring ... 89

Figure 45: Dynamic step change simulation - Active- and reactive power loading ... 90

Figure 46: Dynamic step change simulation - Sending- and receiving end active power ... 90

Figure 47: Dynamic step change simulation - Sending- and receiving end active power (Zoomed) 91 Figure 48: Dynamic step change simulation – Power angle over transmission line... 91

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Figure 49: Dynamic step change simulation – Power angle over transmission line (Zoomed)... 92

Figure 50: Dynamic step change simulation – Power transferred over transmission line ... 92

Figure 51: Dynamic step change simulation – Power transferred over transmission line (Zoomed) 93 Figure 52: Dynamic step change simulation – Transmission power transfer stability ... 94

Figure 53: Dynamic step change simulation – Voltage stability monitoring ... 95

Figure 54: Field test measurement schematic ... 97

Figure 55: Metering setup [39] ... 98

Figure 56: PMU phase error margin ... 100

Figure 57: Line of best fit of CT of Class 0.2 (amplitude) ... 102

Figure 58: Line of best fit of CT of Class 0.2 (phase angle) ... 102

Figure 59: ADC current measurement error contribution ... 103

Figure 60: ADC voltage measurement error contribution ... 103

Figure 61: Resistance error results from field measurements ... 116

Figure 62: Inductance error results from field measurements ... 118

Figure 63: Capacitance error results from field measurements ... 119

Figure 64: Case study 1 - Sending- and receiving end active power for the 24 h period ... 120

Figure 65: Case study 1 - Receiving end reactive power for the 24 h period ... 121

Figure 66: Case study 1 - Positive sequence resistance for the 24 h period ... 122

Figure 67: Case study 1 - Positive sequence inductance for the 24 h period ... 123

Figure 68: Case study 1 - Positive sequence capacitance for the 24 h period ... 124

Figure 69: Case study 1 - Transmission line temperature for the 24 h period ... 125

Figure 70: Case study 1 - Transmission line sag for the 24 h period ... 126

Figure 71: Case study 1 - Active power transferred for the 24 h period ... 127

Figure 72: Case study 1 - Reactive power transferred for the 24 h period ... 128

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Figure 74: Case study 1 - Power angle across transmission line for the 24 h period ... 129

Figure 75: Case study 1 - Transmission power transfer for the 24 h period ... 130

Figure 76: Case study 1 - Voltage stability monitoring for the 24 h period ... 131

Figure 77: Case study 2 - Sending- and receiving end active power for the 7-day period ... 132

Figure 78: Case study 2 - Receiving end reactive power for the 7-day period ... 132

Figure 79: Case study 2 - Positive sequence resistance for the 7-day period ... 133

Figure 80: Case study 2 - Positive sequence inductance for the 7-day period ... 134

Figure 81: Case study 2 - Positive sequence capacitance for the 7-day period ... 135

Figure 82: Case study 2 - Transmission line temperature for the 7-day period ... 136

Figure 83: Case study 2 - Transmission line sag for the 7-day period ... 138

Figure 84: Case study 2 - Active power transfer during the 7-day period ... 139

Figure 85: Case study 2 - Reactive power transfer for the 7-day period ... 140

Figure 86: Case study 2 - Active power loss in transmission line for the 7-day period ... 140

Figure 87: Case study 2 - Power angle across transmission line for the 7-day period ... 141

Figure 88: Case study 2 - Transmission power transfer stability for the 7-day period ... 142

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List of Tables

Table 1: Acceptable error margins for line parameter calculation [34] ... 56

Table 2: Method 1 performance [5] ... 56

Table 3: Method 1 performance analysis [5] ... 57

Table 10: Simulation distributed line parameter input values ... 66

Table 11: Sensitivity coefficient analysis margins ... 72

Table 12: Sensitivity coefficient analysis adjusting magnitude of IR ... 74

Table 13: Sensitivity coefficient analysis adjusting magnitude of VS ... 75

Table 14: Sensitivity coefficient analysis adjusting magnitude of VR ... 76

Table 15: Sensitivity coefficient analysis adjusting phase angle of IR ... 77

Table 16: Sensitivity coefficient analysis adjusting phase angle of VS... 78

Table 17: Sensitivity coefficient analysis adjusting phase angle of VR ... 79

Table 18: Impedo DUO Specifications [47] ... 99

Table 19: SANS61869-2 CT accuracy classes ... 101

Table 20: Current measurements systematic errors ... 105

Table 21: Voltage measurements systematic errors ... 105

Table 22: Uncertainty budget for current magnitude ... 106

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Table 24: Uncertainty budget for voltage magnitude ... 108

Table 25: Uncertainty budget for voltage phase angle ... 108

Table 26: Uncertainty budget for resistance estimation... 109

Table 27: Uncertainty budget for inductance estimation ... 110

Table 28: Uncertainty budget for capacitance estimation ... 111

Table 29: Transmission line configuration and parameter data ... 112

Table 30: Transmission line technical data ... 112

Table 31: Line resistance at different temperatures ... 113

Table 32: Simulated line parameters ... 115

Table 33: Field measurement window sizes ... 116

Table 34: Resistance calculation accuracy by window length ... 117

Table 35: Inductance calculation accuracy by window length ... 118

Table 36: Capacitance calculation accuracy by window length ... 119

Table 37: Case study 1 - Resistance error ... 122

Table 38: Case study 1 - Inductance error ... 123

Table 39: Case study 1 - Capacitance error ... 124

Table 40: Case study 1 – Line temperature results for the 24 h period ... 125

Table 41: Case study 1 - Recorded ambient temperatures for the 24 h period [52] ... 126

Table 42: Case study 1 – Line sag results for the 24 h period ... 126

Table 43: Case study 2 - Resistance error ... 134

Table 44: Case study 2 - Inductance error ... 135

Table 45: Case study 2 - Capacitance error ... 135

Table 46: Case study 2 - Sending-end recorded temperatures ... 136

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List of Acronyms

PMU Phasor measurement unit

DC Direct current

AC Alternating current

GMR Geometric mean radius

GMD Geometric mean distance

KVL Kirchoff’s voltage law

KCL Kirchoff’s current law

LTC Load tap changing transformer

TSO Transmission system operator

WAMS Wide Area Monitoring System

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List of Symbols

𝑅𝑑𝑐,𝑇 DC Resistance of conductor at specific temperature

R AC Resistance at 50 Hz

𝜌𝑇 Conductor resistivity

𝑙 Length

𝐴 Conductor cross-section area

𝑇ref Reference temperature of conductor

L Inductance

𝜆 Number of flux linkages

I Current

𝑟 Radius of solid conductor

𝑟′ _{GMR of stranded conductor}

𝐷 Distance between conductors

𝜇0 Permeability of air

𝑞 Electric charge

V Voltage

C Capacitance

𝜀0 Dielectric constant of air

X Reactance 𝛿 Power angle P Active power Q Reactive power Y Admittance Z Impedance

y Distributed parameters admittance

z Distributed parameters impedance

ZC Characteristic impedance

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𝜌 Level span length

𝐷_𝑟𝑒𝑓 Reference line sag

𝐷𝑒𝑠𝑡 Estimated line sag

𝑤 Per unit line weight

H Horisontal tension component

Hmax Breaking load

𝛼𝐴𝑆 Thermo elongation coefficient of conductor

𝛼 Thermo-resistivity coefficient of conductor

𝐿_𝑟𝑒𝑓 Reference line length between towers

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1 Introduction

Synchrophasors1 can be used in innovative smart grid applications. This topic and other current trends in modern power systems will be discussed to motivate the need for research on innovative monitoring of transmission lines.

1.1 Background

The US National Energy Technology Lab (NETL) gave the definition of a Smart Grid as, “Smart

Grid is the integration of technologies that allow us to rethink electric grid design and operations”

[1]. Synchrophasors can be one opportunity for “rethinking electric grid design and operations” as it can do more than tracking small signal stability.

Phasor Measurement Units (PMUs) are traditionally used to record synchronised voltage phasors (synchrophasors). If both voltage and current phasors are recorded coherently across a line (i.e. transmission or distribution), then the performance of the line can be monitored in detail, examples are:

- The continuous verification of the line parameters - Dynamic rating of line

- Tracking voltage stability

- Monitoring small signal stability, and - Power transfer stability.

Innovative application of network coherent data (such as synchrophasors) can significantly enhance the knowledge with regards to power system steady-state and dynamic conditions, reducing the dependence on estimating the state of the power system by derived parameters [2].

1.2 Scope of the research

The availability of network coherent data is perceived to have significantly improved. This is due to the fact that new power system measuring equipment can time-stamp measured parameters by a referenced time source, such as GPS time synchronisation. The IEEE C37-118 standard on the recording of synchrophasors can be implemented with relative ease in general power system

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measuring equipment. Not only can the fundamental frequency voltage and current phasors (synchrophasors) be recorded in coherent fashion, the time-stamping of all other parameters can allow innovation in power system operation and control.

The research presented in this dissertation evaluates how network coherent data can improve the observability of transmission line performance. The IEEE C37-118 protocol for recording synchrophasors was implemented in parallel with the recording of Class A IEC61000-4-30 (edition 3) power quality parameters. Modern Intelligent Electronic Devices (IEDs) can be multifunctional at a fraction of the cost of a traditional PMU system.

The scope of this research is limited to:

- The continuous on-line tracking of transmission line parameters, and

- The continuous on-line tracking of thermal, power transfer- and voltage stability.

Only the fundamental frequency components were considered and used to attain visibility on a transmission line to support network operators that operate and control the performance of the transmission system, now based on decisions reflecting the measured state of the network, not only an estimation thereof.

1.3 Problem Statement

PMU data can be used to derive line parameters if voltage and current synchrophasor data are available across an impedance. After calculating the line parameters, additional information such as line temperature, sag, power transfer and voltage stability monitoring can also be derived.

Line parameters are traditionally calculated from fundamental scientific principles reflecting the geometry, the type of conductor, tower construction, soil type and others. The inverse is possible in principle as the line parameters can be derived when the input and output phasors are known (synchronously measured) [3].

It is important that the calculation method is robust and that the measurement uncertainty introduced by factors such as noise or bias (calibration) errors caused by voltage and current transformers or transducers, are identified and minimised.

Various synchrophasor-based impedance calculation methods have been proposed in the literature [4], [5], [6]. Closed-form calculation methods are the simplest by only utilizing one or two measured sets of phasors in the calculation process [4]. These methods perform well when subjected to perfect

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conditions without any uncertainty in measurements. When this is not the case, the calculated parameters deteriorate in accuracy [5], illustrating the need for other, more robust, numerical calculation methods.

In practice, all phasor measurements will be subjected to some form of noise, bias errors or synchronisation inaccuracies in the instrumentation channel [7]. Systematic errors that are caused by metering and instrumentation transformers can be corrected by applying a calibration constant. This constant can be derived from the analysis of current and voltage measurements over a long period reflecting all possible states of the transmission line [8].

Impedance calculations using simulation packages are mostly regarded as accurate and can be used to test if the calculation method was implemented correctly from a mathematical point of view. Validation of the line parameters, if the calculation method was found suitable, can then be done based on actual field data.

With the availability of accurate line parameter values, along with coherent voltage and current phasors, transmission line stability monitoring techniques can also be applied to visibly track the thermal-, power transfer- and voltage-stability limit as additional improvements to the observability of transmission line performance.

1.4 Research aim and objectives

Objectives for the research is derived from the research aim and then structured for realisation.

1.4.1 Research aim

The aim of the research is to validate the opportunity for on-line tracking of transmission line parameters, thermal-, power-transfer- and voltage stability by means of synchronised voltage and current phasors recorded under real-world field conditions across a transmission line.

1.4.2 Research objectives

A critical analysis of the literature is performed to select an optimal non-invasive method for calculating line parameters.

The chosen method is then verified by means simulation data (laboratory tests) and sensitivity analyses, in order to evaluate the robustness of the calculation method when subjected to noise and

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bias errors. The simulation data is used to verify the application of transmission line monitoring techniques under steady-state and dynamic conditions.

Validation of the line parameter calculation method is done by application of PMU field data, obtained across an actual transmission line.

Additional information such as transmission line temperature, sag, power transfer- and voltage stability are then derived from the available data.

1.5 Summary

The opportunity for using synchrophasors to track the performance of a transmission line is to be validated based on field data obtained from a real transmission line.

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2 Performance monitoring of transmission

lines

The opportunity for using synchrophasors to monitor transmission lines are introduced by studying the theoretical principles of transmission line small signal stability, voltage stability and equivalent line parameters.

2.1 Introduction

Theoretical principles of power system stability with emphasis on power transfer and voltage stability are analysed in the context of synchrophasors. The application of network equivalent modelling in the measurement of transmission line parameters by means of synchrophasors, are presented.

2.2 Power system stability

Power system stability is defined as: “The ability of an electric power system, for a given initial

operating condition, to regain a state of operating equilibrium after being subjected to a physical disturbance, with most system variables bounded so that practically the entire system remains intact

[9]”.

The importance of power system stability has been recognised since the 1920s. This was due to various major blackouts resulting from power system instability [10], [11]. The concept of power system stability is presented below.

2.2.1 Introduction to power system stability

The continuous growth of power systems due to interconnections between different systems, as well as the integration of new technologies and control methods, has led to the current infrastructure being operated under conditions that caused transmission operators and planners to be more concerned about frequency, voltage and small signal stability [9] than in the past.

A power system can be seen as a continually changing nonlinear system [10]. Parameters such as loading and generation are continually subjected to disturbances and demand changes and respond according to their dynamic parameters. When the power system is subjected to a disturbance, stability depends on the initial operating condition and the type of disturbance it is being subjected to [12].

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Load changes and switching events that occur frequently in a power system are classified as small disturbances. The system must be able to withstand small disturbances without any signs of instability. Larger disturbances, such as short-circuit events, loss of significant load or generation can have a more severe effect on system stability. The power system must be planned to withstand these disturbances by contingency and stability analyses [13].

The response to a disturbance could involve a number of system components. For instance; a fault on a critical element will be followed by the operation of protection relays, isolating the affected section. This will cause a variation in power flow, bus voltages and machine rotor speeds. Voltage variations will trigger generator (depending on the type of excitation control) and network voltage regulators, which in turn will lead to variations in rotor speeds (affecting prime mover governors), bus voltages and frequencies at network loads.

Voltage and frequency variations may further cause individual protective devices to trip out more system equipment, depending on its transient characteristics and protection settings [9].

Power system stability cannot be understood, or effectively dealt with, by considering it as a single problem. This is why different categories exist as discussed next [12].

2.2.2 Categories of stability

Analysis of power system stability and the identification of factors contributing to instability and methods of improving it can be simplified by the categories shown in Figure 1 [13].

The following factors are considered:

 The physical nature of the resulting mode of instability, as specified by the main system variable in which instability can be observed.

 The magnitude of the disturbance, influencing the method of calculation and prediction of stability.

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Figure 1: System stability categories [9]

Voltage stability and its subcategories are presented in section 2.4. Rotor angle stability and monitoring is presented next.

2.3 Rotor Angle Stability

Rotor angle stability is the ability of interconnected synchronous machines in a power system to remain synchronised under normal conditions or after being subjected to a disturbance [9]. These scenarios could lead to increased angular swings between generators and unstable conditions due to loss of synchronism between the various machines. This includes synchronous machines in the same power plant or power system, as they also have an effect on each other.

Power transfer stability is determined by the rotor angle difference of the synchronous machines in different areas due to the transfer of power as shown in Figure 2, with power flowing from the generator (Machine 1) to the motor (Machine 2). In order to simplify the equations, only reactance and no capacitance and resistance are considered in this model of the transmission system.

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Figure 2: Simplified power system

The power transferred from the generator to the motor is a function of the angular separation between the rotors of the different machines. This angular separation has three components:

a) The generator’s internal angle 𝛿𝐺: Angle by which the rotor leads the stator field.

b) The motor’s internal angle 𝛿_𝑀: Angle by which the motor’s rotor lags the rotating stator field. c) Angular difference between the terminal voltages of the generator and motor across the line

𝛿𝐿: Angular difference between the generator and motor stator field.

The power angle model in Figure 3 includes the internal reactance of the machines.

Figure 3: Power transfer between generator and motor

Where:

 EG = Voltage at generator

 EM = Voltage at load

 XG = Generator’s internal reactance

 XL = Line reactance

 XM = Motor’s internal reactance

 𝛿 = Power angle difference between EG and EM (𝛿𝐺 + 𝛿𝑀 + 𝛿𝐿)

Power transferred from generator to the motor is defined as: 𝑃_{𝑡𝑟𝑎𝑛𝑠}= 𝐸𝐺𝐸𝑀

𝑋𝑇 sin 𝛿 W

(2.1)

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 Ptrans: Active power transferred over the line

And:

𝑋𝑇 = 𝑋𝐺 + 𝑋𝐿+ 𝑋𝑀 (Ω) (2.2)

These equations can be applied on a transmission line by utilising the voltages ET1 and ET2 as the

sending (VS) and receiving end (VR) voltage magnitudes, with only XL being applicable on the

transmission line in equation (2.1), as shown in Figure 4 below [14].

Figure 4: Power angle across transmission line

Coherent measurement of the positive sequence phasors VS and VR can be used to track the transfer

of power over a transmission line. The frequency of measurement is dependent on the system characteristics. When monitoring a volatile system, it would be needed to decrease the measurement interval to ensure sufficient visibility of unstable events [14].

The difference between the phase angles of VS and VR is defined as the power angle across the line:

𝛿 = 𝜃_𝑉_𝑆 − 𝜙_𝑉_𝑅 (2.3)

The theoretical maximum power transfer will occur at 𝛿 = 90º:

𝑃

_𝑚𝑎𝑥

=

𝑉𝑆𝑉𝑅

𝑋

W

(2.4)

The per unit power being transferred:

𝑃

_{𝑡𝑟𝑎𝑛𝑠 (𝑝.𝑢).}

=

𝑃𝑡𝑟𝑎𝑛𝑠

𝑃𝑚𝑎𝑥

p.u.

(2.5)

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This relation between per unit power transferred and the power angle across the transmission line can be tracked on a power transfer curve, as shown in Figure 5. This technique enables monitoring line utilisation and critical operating points, as discussed below.

Figure 5: Power transfer curve [14]

When the power angle is zero, no power will be transferred across the system. As the power angle increases, the power transferred will increase until it reaches a theoretical maximum at 90˚. A further increase in the angle leads to a decrease in power transferred and the system to become unstable due to a loss in synchronism between the machines at the sending- and receiving-end [15].

There is a maximum steady-state power that can be transferred from the generator to the motors, directly proportional to the machine terminal voltages and inversely proportional to the reactance of the system. In practise, the power angle is normally kept below 35° to ensure the system stays within the power transfer limit [13].

The transmission system operator (TSO) can develop an on-line application to supervise the power transfer over a transmission line or transmission system based on coherent data.Information on line

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loading can prevent a system collapse by means of an early warning system that warns the TSO when the system is nearing the practical transfer limit.

Another limiting factor of a power system, is voltage stability. This is the next category of stability that will be discussed.

2.4 Voltage stability

Voltage stability refers to the ability of a power system to maintain steady state voltages at all busses during normal operation, or after being subjected to a fault or sudden change in state [9]. This depends on the ability of the system to maintain/restore equilibrium between the load demand and supply of the system. Instability could lead to a loss of load in certain areas or tripping of transmission lines and other equipment by protection schemes, leading to cascading outages also known as voltage collapse.

Voltage instability can be caused or contained by generators, transmission lines and loads, based on the reactive power requirements of the power system [16]. Some examples of these various voltage instability contributors are discussed below.

2.4.1 Generators

The maximum reactive power output of a synchronous generator is limited by the current and thermal limits of the exciter and stator windings. When the excitation limiter reaches its limit, the terminal voltage can no longer be maintained. This causes high voltages on the exciter terminals, which will limit the capability of the generator, leading to medium term voltage instability [17]. The consequence of high reactive power demand is presented next with the emphasis now on transmission lines.

2.4.2 Transmission lines

A major contributor to voltage instability is the voltage drop caused by the current flowing through the inductive reactance of a transmission line. The voltage stability of a system becomes threatened when the reactive power transfer rises above the reactive power capability of the generators [9]. The reactive power transferred over a transmission line will be determined by the power factor, which is the reactive portion of the current demand by the load. The voltage drop caused by the active and reactive current flowing through a transmission line is calculated by:

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∆𝑉 = 𝑅_𝐿𝐼_𝐿cos 𝜃 + 𝑋_𝐿𝐼_𝐿sin 𝜃 (2.6)

This means in order to maintain acceptable voltage levels at the load, either the load power factor has to be kept close to unity, or the voltage regulators have to be able to keep the voltage above the required limit during all operating conditions, or series compensation has to be added to lower the line reactance and consequently lowering the voltage drop.

2.4.3 Loads

Sudden changes in load demand can also significantly affect stability.

After a change in the demand, the rotating loads will firstly adjust according to its instantaneous (transient) characteristics where after it will adjust the current drawn from the system until the power supplied by the system meets the load demand [17].

A sudden loss or increase in load demand could lead to significant voltage rises or drops. These short-term voltage stability scenarios have to be considered during network planning and operation. In order to perform sufficient contingency analysis during network planning, it is needed to categorise voltage stability events.

2.4.4 Voltage stability event categories

Voltage stability is categorised according to the duration of an event:

a) Short term: Electromechanical transients, caused by devices like induction motors (starting direct-on-line), generators or voltage regulators, as well as power electronics may cause a voltage collapse in the range of seconds.

b) Medium term: Discrete switching devices such as tap changers or excitation limiters will act in an interval of tens of seconds.

c) Long term: Load recovery can take up to several minutes [13].

In order to identify and mitigate these voltage events, monitoring methods has to be implemented.

2.5 Voltage stability monitoring

Voltage magnitude monitoring is the most intuitive index of analysing system stability because of the availability of voltage data all over a power system. Monitoring voltage magnitudes at different

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busses is widely accepted to monitor the need for preventative actions such as under voltage load shedding schemes to prevent voltage collapse [13].

Low voltages lead to an increase in load current demand to be sustained by generation and other electrical infrastructure (transformers, overhead lines and cables). A minimum set point of between 85% and 90% of nominal voltage is typically used [18] by protection relays and system operators to reduce loading in order to keep the system stable. Through load modelling, voltage levels can be categorised by means of a voltage index giving operators an indication of approaching instability conditions [19].

This voltage-based approach has limitations. A voltage index cannot determine how close the system is to the stability limit. The bus with the lowest voltage is not necessarily the closest to a voltage collapse event, i.e. a bus with high reactive power compensation will appear to have a normal voltage magnitude, even though its power transfer could be close to the transmission limit and the system close to its stability limit [20]. This statement is further elaborated in section 2.5.1.

It is needed to implement methods utilising both voltage and current measurements to accurately analyse the voltage stability of a power system. These methods can be categorised as steady-state or dynamic, depending on the sampling rate of measurement devices installed.

A higher resolution in the update of information for the analysis of transient conditions in voltage stability is needed, as fast changes in network parameters are expected during transient conditions [20]. These methods are defined as dynamic, where-as slower, less data intensive methods utilising i.e. 2 minute data windows is defined as steady-state monitoring methods.

Control and coordination of dynamic voltage stability events is normally based on a simulation of possible contingency scenarios when a network event occurs. This is due to the fact that it is mostly not possible to acquire and process sufficient measurements to control these events in real-time. Based on simulation studies, the proper coordination of protective equipment can be done to contain the impact of a system disturbance.

Steady-state analysis with slower changing network circumstances can be implemented based on time-stamped field data that track voltage stability in near real-time. A Wide Area Measurement

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System (WAMS) aims to produce synchronised data for stability analysis. Synchrophasors2 can be used in on-line or off-line applications for post-mortem analysis of significant system events [20]. On-line steady-state voltage analysis methods and the implementation thereof are discussed in the following section.

2.5.1 Steady-state voltage stability monitoring methods

Voltage stability can be analysed by the relationship between power and voltage. The voltage drop over a transmission line is determined by loading (power transferred) of the line and the power factor at the load, as discussed in section 2.4.2.

Steady-state voltage stability can be monitored with the use of P-V curves as shown in Figure 6 [13].

Figure 6: P-V curve principle [21]

A P-V curve can represent two possible points of operation for VR at a given power delivered (PR),

the higher value of VR being a stable point of operation and the other an unstable point of operation

(shown in Figure 6). As the load or transmission line parameters changes, so does the shape of the

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V curve affecting the position of the maximum deliverable power (Pmax). Operating points must be

kept above the nose (Pmax) of the P-V curve to ensure stability of the transmission line.

Monitoring voltage stability by means of a P-V curve can help network operators to control and mitigate instability concerns by changes to for example, generation and loading in compensation of reactive power [22].

When considering a simple 2-bus transmission line, as shown in Figure 6, the active power transferred across the line was given by equation (2.1). The reactive power transferred is defined through equation (2.7) below [21]. 𝑄𝑡𝑟𝑎𝑛𝑠 = − 𝑉_𝑅2 𝑋 + 𝑉_𝑆𝑉_𝑅 𝑋 cos 𝛿 (2.7) Where:

 Qtrans = Reactive power transferred over the line

Continuous measurement of voltage stability between two areas in a power system can be based on the relation between active- and reactive power [23]. By normalizing the active and reactive power transfer (equation (2.1) and (2.7)) over the line with:

𝑣 = 𝑉𝑅/𝑉𝑆 (2.8)

𝑝 = 𝑃_{𝑡𝑟𝑎𝑛𝑠}. 𝑋/𝑉_𝑆2 (2.9)

𝑞 = 𝑄𝑡𝑟𝑎𝑛𝑠. 𝑋/𝑉𝑆2 (2.10)

The following equations are obtained [21]:

𝑝 = (𝑉𝑆𝑉𝑅 𝑋 sin 𝛿) ( 𝑋 𝑉𝑆2 ) (2.11) 𝑝 = 𝑣 sin 𝛿 (2.12) And: 𝑞 = (−𝑉𝑅 2 𝑋 + 𝑉𝑆𝑉𝑅 𝑋 cos 𝛿) ( 𝑋 𝑉_𝑆2) (2.13)

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𝑞 = −𝑣2_{+ 𝑣 cos 𝛿} _(2.14)

By squaring equations (2.12) and (2.14) above:

(𝑣 sin 𝛿)2_{+ (𝑣 cos 𝛿)}2 _{= 𝑝}2_{+ (𝑞 + 𝑣}2₎2 _(2.15) 𝑣2_{(𝑠𝑖𝑛}2_{𝛿 + 𝑐𝑜𝑠}2_{𝛿) = 𝑝}2_{+ (𝑞 + 𝑣}2₎2 _(2.16) 𝑣4_{+ 𝑣}2_{(2𝑞 − 1) + (𝑝}2_{+ 𝑞}2_{) = 0} _(2.17) Solving equation (2.17): 𝑣 = √1 2− 𝑞 ± √ 1 4− 𝑝2− 𝑞 (2.18)

The P-V curve is then calculated for each measurement of coherent voltage and current phasor at the sending- and receiving end. Because of changing load conditions, the shape of the curve continuously changes.

In order to display the curve in terms of v and p, setting the ratio of q/p = k, the P-V curve from (2.18)

is defined with the functions (top and bottom part of the curve):

𝑣(𝑝) = √1 2− 𝑝𝑘 + √ 1 4− 𝑝2− 𝑝𝑘 (2.19) And: 𝑣(𝑝) = √1 2− 𝑝𝑘 − √ 1 4− 𝑝2− 𝑝𝑘 (2.20)

By using the real parts of the functions (2.19) and (2.20), the top and bottom part of the P-V curve can be constructed. The operating point of power transfer between the two areas is obtained by the scatter plot of p(t) and v(t) on this curve, as shown in Figure 6.

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Optimisation of the power transfer over a transmission line can now be based on the measured voltage and current phasors at sending- and receiving end of the line. This information can be used by the TSO to track voltage stability. By having the operating point in the stable operating region of the P-V curve, the operator can react before a voltage collapse occurs [21].

The need to know the line reactance (XL) in section 2.3, 2.4 and 2.5, illustrates the need for accurate

transmission line parameter information and a possibility exist to derive it from voltage and current synchrophasors across a transmission line. Theoretical concepts of transmission line parameter are discussed next.

2.6 Transmission line parameter theoretical principles

Monitoring power transfer- and voltage stability requires an accurate estimation of the transmission line parameters. Transmission line parameters are affected by various variables such as the geometrical construction of the line i.e. the spacing between conductors, ambient conditions and conductor temperature. These considerations must be understood when modelling a transmission line. Stranded aluminium conductors are mostly used (ACSR) in the construction of overhead lines, as they are easier to manufacture, have a lower cost and are lighter than the equivalent copper conductors [24]. Stranded conductors are simpler to handle and the steel-reinforced strands gives it a high strength to mass ratio.

Although aluminium is an excellent conductor, it is more resistive than copper and the conductor configuration leads to an inductive and capacitive effect between the conductors, conductor bundles, as well as between the conductors and ground. The impedance of the ASCR line parameters influences the power transfer capability and energy- efficiency of a transmission line [15] and is analysed next.

2.6.1 Conductor resistance

The DC resistance of a conductor is calculated as [15]:

𝑅𝑑𝑐,𝑇 =

𝜌𝑇𝑙

𝐴 Ω

(2.21)

Where 𝜌_𝑇 represents the conductor resistivity, 𝑙 the length and 𝐴 the conductor cross-section area. Conductor resistance depends on four factors:

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i. Spiralling

In stranded conductors, alternate layers of strands are arranged in opposite winding directions to ensure the strands remain together. This technique causes the conductor to be 1% to 3% longer than the estimated conductor length [15]. This is normally considered in the manufacturer’s specification of the transmission line parameters.

ii. Temperature

The resistivity of a conductor varies linearly with temperature as shown by equation (2.22) [15]:

𝜌_𝑇2 = 𝜌_𝑇1(𝑇2+ T 𝑇₁+ T)

(2.22)

Where 𝜌_𝑇2 and 𝜌_𝑇1 represents the conductor resistivity at temperatures 𝑇₂ and 𝑇₁ respectively, while T is the temperature constant of the conductor [24].

iii. Frequency

Frequency of the current flowing through a conductor increases the resistance of a conductor due to the skin effect and is referred to as AC resistance. The current in a cylindrical conductor crowd towards the conductor surface, resulting in a reduction in the effective cross-sectional area of the conductor [15]. This smaller conducting surface increases the AC resistance of the line beyond the DC resistance value.

iv. Current Magnitude

In magnetic conductors, the internal flux linkages depend on how the current flow through the conductor. In ACSR conductors the steel core has a high resistivity compared to the aluminium strands, and the effect of current magnitude on the resistance of an ACSR conductor is small [15].

2.6.2 Conductance

The active power loss between the conductor and the earth is modelled by means of conductance. In the case of overhead transmission lines, conductance losses are mainly due to corona and leakage currents through insulators [24]. Corona occurs when conductors produce a high enough electricity field to electrically ionise the air surrounding the conductors and can be increased by ambient conditions such as moisture in the air around conductors. Losses due to corona and leakage currents are small enough to be neglected in most power system studies [15].

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2.6.3 Line Inductance

Line inductance consists of self and mutual inductance as caused by the internal and external magnetic flux [24]. The line inductance of a magnetic circuit is obtained from first principles by calculating the flux linkages per ampere:

𝐿 =𝜆 𝐼

(2.23)

𝐿 represents the number of flux linkages (𝜆) produced per ampere of current flowing through the line [15]. The inductance for a single-phase line is [15]:

𝐿 =𝜇0 𝜋 ( 1 4ln ( 𝐷 𝑟)) 𝐻/𝑚 (2.24) Simplified as: 𝐿 = 4 × 10−7_{ln (}𝐷 𝑟′) 𝐻/𝑚 (2.25)

With 𝑟 being the radius of a solid conductor, 𝑟′ the geometric mean radius (GMR) of a stranded conductor (supplied by manufacturer), 𝐷 the distance between conductors and 𝜇0 the permeability of

air.

When considering a fully transposed three-phase line, equation (2.25) has to be adjusted to include the mutual inductance between the phases as well. The total inductance of a three-phase, fully transposed line is:

𝐿 = 2 × 10−7_{ln (}𝐺𝑀𝐷

𝐺𝑀𝑅) 𝐻/𝑚

(2.26)

The GMD (Geometric mean distance) is defined as:

𝐺𝑀𝐷 = √𝐷3 𝑎𝑏𝐷𝑏𝑐𝐷𝑐𝑎 m (2.27)

Where 𝐷_𝑎𝑏, 𝐷_𝑏𝑐 and 𝐷_𝑐𝑎 are the distances between the different phases. The GMR (geometric mean radius) is the physical radius of the conductor and must be adjusted when more than one conductor is used per phase (Figure 7). The different GMR calculations for more conductors per phase are shown below [15]:

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Figure 7: Conductor bundling configurations

𝐺𝑀𝑅₂ = √𝐺𝑀𝑅. 𝑑 m (Two conductor bundle (a)) (2.28) 𝐺𝑀𝑅₃ = √𝐺𝑀𝑅. 𝑑3 2_m_{(Three conductor bundle (b))}_(2.29)

𝐺𝑀𝑅4 = 1.09√𝐺𝑀𝑅. 𝑑4 3 m (Four conductor bundle (c)) (2.30)

The greater the radius of the conductor, the smaller the total inductance. In practical applications, conductors with smaller radii are rather bundled together than using heavy, inflexible conductors. Through equations (2.26) and (2.27), it can be seen that the larger the distance between the phases, the larger the total inductance will be. Since the phases of a high voltage overhead transmission line must be spaced far away from each other to ensure proper insulation, high voltage lines will have a larger line inductance than lower voltage lines [24].

2.6.4 Shunt Capacitance

A transmission line can be modelled as a voltage (V) being applied to conductors with a dielectric medium (air in this case) in-between. Charges (q) of equal magnitude but opposite charge will accumulate between conductors. Capacitance between the conductors can then be found by the ratio of charge to voltage:

𝐶 = 𝑞_𝑉 F (2.31)

The capacitance of a single-phase transmission line based on the geometry is:

𝐶 = 2𝜋𝜀

ln (𝐷 𝑟⁄ ) 𝐹/𝑚

(2.32)

With D being the distance between the conductors, 𝑟 the radius of the conductor and 𝜀 the dielectric constant of air (1.00059) [24].

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Transposition of a three-phase transmission line is done to distribute the capacitance and inductance uniformly between phase conductors [15]. A single-phase equivalent model can then be used to model the total capacitance per phase [15]:

𝐶𝑝 = 2𝜋𝜀 ln (𝐷𝑒𝑞⁄𝐷𝑆𝐶) 𝐹/𝑚 (2.33) 𝐷_𝑒𝑞 is defined as: 𝐷_𝑒𝑞 = √𝐷3 𝑎𝑏𝐷𝑏𝑐𝐷𝑐𝑎 m (2.34)

Where 𝐷_𝑎𝑏, 𝐷_𝑏𝑐 and 𝐷_𝑐𝑎 are the distances between the different phases. 𝐷_𝑆𝐶 depends on the number of conductors bundled together (refer to Figure 7):

𝐷𝑆𝐶 = √𝑟𝑑 m (Two-conductor bundle (a)) (2.35)

𝐷_𝑆𝐶 = √𝑟𝑑3 2_m(Three-conductor bundle (b)) (2.36) 𝐷𝑆𝐶 = 1.091√𝑟𝑑4 3 m (Four-conductor bundle (c)) (2.37)

Through equations (2.33) and (2.34) it can be derived that the further conductors are spaced from each other (larger value of 𝐷_𝑒𝑞), the smaller the capacitance will be. This means that higher voltage lines will have lower capacitances than low voltage lines due to sufficient spacing between phases to insure isolation [24].

Equation (2.33) also shows that larger conductor radii leads to higher capacitances. This means that the bundling of conductors will result in a larger capacitance per phase [24].

Transmission line parameters are modelled in power systems by means of equivalent transmission line models and are discussed next.

2.7 Transmission line models

A transmission line can be modelled as a 2-port equivalent network making use of ABCD parameters as shown in Figure 8. VS and IS are the sending-end voltage and current, whilst VR and IR are the

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Figure 8: Two-port network model

The relation between the sending- and receiving end values are:

𝑽_𝑆 = 𝑨𝑽_𝑅 + 𝑩𝑰_𝑅 (2.38) 𝑰_𝑆 = 𝑪𝑽_𝑅+ 𝑫𝑰_𝑅 (2.39) In matrix form: [𝑽_𝑰𝑆 𝑆] = [𝑨 𝑩𝑪 𝑫] [ 𝑽_𝑅 𝑰𝑅] (2.40)

A, B, C and D are complex line constants based on the transmission line parameters R, L, C and G (G

to be neglected in most power system studies). A and D are dimensionless, while B and C has units of Ohms and Siemens respectively [15]. Three different models are used:

 short,

 medium, and  long line models.

Short transmission lines in a 50 Hz system are classified as shorter than 100 km, medium lines between 100 km and 300 km, and above as a long line [15]. Short and medium length transmission lines simplify the distributed nature of the line parameters and is within the distance constraints, the accepted approach.

A long transmission line can be modelled with more accuracy based on the distributed nature of line parameters. ABCD parameters applied in the modelling of the distributed nature of a transmission line model is briefly analysed in section 2.7.1.

2.7.1 Distributed transmission line model

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Figure 9: Distributed nature of a transmission line

The total series impedance and admittance of a line with length l from the per unit length values and can be written as:

𝒛 = 𝑟 + 𝑗𝜔𝑙 Ω 𝑚⁄ (series impedance per unit length)

(2.41)

𝒚 = 𝑔 + 𝑗𝜔𝑐 Ω 𝑚⁄ (shunt admittance per unit length)

(2.42)

𝒁 = 𝒛𝑙 Ω (total series impedance) (2.43)

𝒀 = 𝒚𝑙 Ω (total shunt admittance) (2.44)

Voltage and current as distributed parameters relates to the circuit elements [25]:

𝑉(𝑥) = cosh(𝜸𝑥)𝑽𝑅+ 𝑍𝑐sinh(𝜸𝑥)𝑰𝑅 (2.45)

𝐼(𝑥) = 1

𝑍_𝑐sinh(𝜸𝑥)𝑽𝑅 + cosh(𝜸𝑥)𝑰𝑅

(2.46)

with 𝑍_𝑐 representing the characteristic impedance in ohms:

𝒁_𝑐 = √𝒛_𝒚 Ω (2.47)

and 𝛾 the propogation constant, whose units are m-1_:

𝜸 = √𝒛𝒚 m-1 _(2.48)

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38 [𝑽(𝑥) 𝑰(𝑥)] = [ 𝑨(𝑥) 𝑩(𝑥) 𝑪(𝑥) 𝑫(𝑥)] [ 𝑽𝑅 𝑰_𝑅] (2.49) where:

𝑨(𝑥) = 𝑫(𝑥) = cosh(𝜸𝑥) per unit (2.50)

𝑩(𝑥) = 𝒁_𝑐sinh(𝜸𝑥) Ω (2.51)

𝑪(𝑥) =_𝒁1

𝑐sinh(𝜸𝑥) S

(2.52)

Equation (2.49) can be used to calculate the current and voltage at any point along the transmission line in terms of the receiving-end.

An equivalent transmission line model can be used to represent the distributed nature of the line parameters by means of equivalent line parameters. Equivalent ABCD parameters are derived in section 2.7.2.

2.7.2 Equivalent transmission line model

Figure 10 presents an equivalent  network model for a long transmission line. This model has a similar structure to the nominal  circuit, with the exception that Z’ and Y’ are used instead of Z and Y. The objective of this section is to determine Z’ and Y’ so that the equivalent  circuit has the same

ABCD parameters as the distributed circuit model discussed in the previous section [15].

Figure 10: Equivalent  transmission line model [15]

The approximations applied in [15] to the nominal  circuit, will be used to derive the equivalent  circuit shown above, due to their similar configuration.

In order to derive the equivalent ABCD parameters, note that the current in the series branch of the circuit equals 𝑰𝑅 +𝑽𝑅₂𝒀′. Based on Kirchoff equations, the following can be written:

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39 𝑽_𝑆 = 𝑽_𝑅+ 𝒁′ (𝑰_𝑅+𝑽𝑅𝒀′ 2 ) = (1 +𝒀′𝒁′ 2 ) 𝑽𝑅 + 𝒁′𝑰𝑅 (2.53)

Similar for current at the sending-end:

𝑰_𝑆 = 𝑰_𝑅+𝑽𝑅𝒀′ 2 + 𝑽_𝑆𝒀′ 2 (2.54) By using equation (2.53) in (2.54): 𝑰_𝑆 = 𝑰_𝑅+𝑽𝑅𝒀′ 2 + [(1 + 𝒀′𝒁′ 2 ) 𝑽𝑅+ 𝒁𝑰𝑅] 𝒀′ 2 = 𝒀′ (1 +𝒀′𝒁′ 4 ) 𝑽𝑅 + (1 + 𝒀′𝒁′ 2 ) 𝑰𝑅 (2.55)

Now by writing (2.53) and (2.54) in ABCD matrix format:

[𝑽_𝑰𝑆 𝑆] = [ (1 +𝒀′𝒁′ 2 ) 𝒁′ 𝒀′ (1 +𝒀′𝒁′ 4 ) (1 + 𝒀′𝒁′ 2 )] [𝑽_𝑰𝑅 𝑅] (2.56)

The ABCD parameters are now identified as:

𝑨 = 𝑫 = 1 +𝒀′𝒁′₂ per unit (2.57)

𝑩 = 𝒁′_Ω _(2.58)

𝑪 = 𝒀′_{(1 +}𝒀′𝒁′

4 ) 𝑆

(2.59)

Based on distributed line parameters ((2.50) – (2.52)):

𝑨 = 𝑫 = 1 +𝒀′₂𝒁′= cosh(𝛾𝑙) per unit (2.60) 𝑩 = 𝒁′_{= 𝑍}

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40 𝑪 = 𝒀′_{(1 +}𝒀′𝒁′ 4 ) = 1 𝑍_𝐶sinh(𝛾𝑙) 𝑆 (2.62)

The equivalent  circuit parameters can now be used to derive the characteristic impedance (𝑍𝑐) and

the propagation constant (𝛾). It was mentioned in section 2.6, that the line parameters of a transmission line has an effect on the power transfer capability of the line. More constraining factors on transmission lines will be discussed in the following section.

2.8 Integrity and congestion of transmission lines

Transmission line congestion refers to the concept of a transmission line that cannot support the increased power transfer without risking system stability [26]. The typical power transfer constraints are thermal limitations, voltage regulation and voltage stability.

A power transfer limit is normally based on theoretical considerations of line construction [26] and a conservative approach is used. The unused capacity can be made available without compromising any stability concern, if knowledge on the state of the transmission line can be based on field measurements and not estimated values.

The power transfer constraints of a transmission line are presented next.

2.8.1 Thermal constraints

Exceeding a line’s thermal limit causes the ACSR conductors to sag due to the increase in temperature. The elasticity of the conductor can be exceeded under extreme conditions.

The decrease in distance between conductors, and between conductors and vegetation can lead to flashovers between the phases and/or phase to earth. Protective equipment should remove the line from service, but a fire hazard and possible harm to equipment, animals and even humans before the line is de-energised, is a concern. Remaining lines can then be overloaded when the load is transferred [26].

While a conductor is within its practical thermal limit, the relation between the variation in line

resistance and temperature is approximately linear [27], shown in equation (2.63):

𝑇 = (_𝑅𝑅

𝑟𝑒𝑓) − 1

𝛼 + 𝑇𝑟𝑒𝑓

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41

T and R are the calculated line temperature and positive-sequence resistance respectively. 𝑅_𝑟𝑒𝑓

represents the line resistance at a reference temperature (𝑇_𝑟𝑒𝑓) whilst α is the thermal coefficient of the conductor.

If the temperature of the conductors is known, it is possible to estimate line sag. Estimation of line sag of a long line exposed to different climatic conditions along the length of the line can be done more accurately when the line is viewed as a number of different sections, each at a different temperature. In order to derive such a complex model, either a large number of measurement units has to be installed, or each sector’s real-time ambient data has to be available [28].

Line sag can be derived from the estimated temperature of the line because the physical properties of the transmission line are known (obtained by direct measurement for example) [28]. It refers to the distance a transmission line sags below the support points of two consecutive towers.

Figure 11: Line sag representation

Considering Figure 11 above, where 𝑤 is the per unit weight of the conductor, S is the horizontal distance between the conductor supports, D the line sag distance, H the tension component and L the actual length of the line [28].

A parabolic equation is used to derive the reference line sag of a transmission line:

𝐷𝑟𝑒𝑓 =

𝑤𝑆2

8𝐻

(2.64)

The data is first used to determine the actual length (𝐿_𝑟𝑒𝑓) of the line between towers using the reference line sag provided by the utility company [28]:

On-line monitoring of a transmission line using synchrophasor measurements