DYNAMIC THERMAL RATING MONITORING AND ANALYSIS FOR

DYNAMIC THERMAL RATING MONITORING AND ANALYSIS FOR

OVERHEAD LINES

A thesis submitted to The University of Manchester for the degree of

Master of Philosophy

in the Faculty of Engineering and Physical Sciences

2013

Feng Xu

Electrical Energy and Power Systems Group

School of Electrical and Electronic Engineering

TableofContentsList of Figures ............................................................................................................................ I

List of Tables ............................................................................................................................ II

List of Abbreviations ............................................................................................................... III

List of Terms ............................................................................................................................ IV

Abstract ..................................................................................................................................... V

Declaration ............................................................................................................................... VI

Copyright Statement .............................................................................................................. VII

Acknowledgements ............................................................................................................... VIII

Dedication ................................................................................................................................ IX

1. Introduction ................................................................................................................... - 1 -

1.1. The need to increase transferring capacity ............................................................. - 1 -

1.2. Aims and objectives ............................................................................................... - 5 -

1.3. Dissertation outline ................................................................................................ - 5 -

2. Thermal rating ............................................................................................................... - 7 -

2.1. Static thermal rating ............................................................................................... - 7 -

2.1.1. Thermal rating definition ................................................................................ - 7 -

2.1.2. Example for conservative condition ............................................................... - 7 -

2.1.3. Seasonal static thermal rating ......................................................................... - 8 -

2.2. Probabilistic thermal rating .................................................................................... - 9 -

2.2.1. General information ........................................................................................ - 9 -

2.2.2. Example of probabilistic thermal rating ....................................................... - 11 -

2.3. Dynamic thermal rating ........................................................................................ - 14 -

2.4. Main conclusions .................................................................................................. - 15 -

3. IEEE, CIGRE and IEC’s DTR calculating models ..................................................... - 16 -

3.1. IEEE model of thermal rating calculation ............................................................ - 16 -

3.1.1. Thermal equilibrium of OHL in IEEE model ............................................... - 16 -

3.1.2. Calculation of convection heat loss in IEEE model ...................................... - 16 -

3.1.3. Calculation of radiation heat loss in IEEE model ......................................... - 17 -

3.1.4. Calculation of solar radiation rate in IEEE model ........................................ - 17 -

3.1.5. Calculation of Joule heat gain in IEEE model .............................................. - 19 -

3.1.6. Conclusion of the IEEE model calculation process ...................................... - 20 -

3.2. CIGRE model of thermal rating calculation ........................................................ - 20 -

3.2.1. Thermal equilibrium of OHL in CIGRE model ............................................ - 20 -

3.2.2. Calculation of convection heat loss in CIGRE model .................................. - 21 -

3.2.3. Calculation of radiation heat loss in CIGRE model ...................................... - 22 -

3.2.4. Calculation of solar radiation rate in CIGRE model ..................................... - 22 -

3.2.5. Calculation of Joule heat gain in CIGRE model ........................................... - 23 -

3.2.6. Conclusion of the CIGRE calculation process .............................................. - 24 -

3.3. IEC model of thermal rating calculation .............................................................. - 24 -

3.3.1. Thermal equilibrium of OHL in IEC model ................................................. - 24 -

3.3.2. Calculation of convection heat loss in IEC model ........................................ - 25 -

3.3.3. Calculation of radiation heat loss in IEC model ........................................... - 25 -

3.3.4. Calculation of solar radiation rate in IEC model .......................................... - 26 -

3.3.5. Calculation of Joule heat gain in IEC model ................................................ - 26 -

3.3.6. Conclusion of the IEC model calculation process ........................................ - 26 -

3.4. Comparison for different DTR calculation models .............................................. - 27 -

3.4.1. Difference in the calculation of convection heat loss ................................... - 27 -

3.4.2. Difference in the calculation of solar radiation rate ...................................... - 29 -

3.4.3. Difference in the calculation of AC resistance (Joule heat) ......................... - 30 -

3.4.4. Difference in the calculation of radiation heat .............................................. - 31 -

3.4.5. Calculation difference for three thermal rating models ................................ - 32 -


4. Dynamic thermal rating monitoring systems .............................................................. - 35 -

4.1. The measured parameters of DTR monitoring system ......................................... - 35 -

4.1.1. Technical requirement of DTR monitoring system ...................................... - 35 -

4.1.2. DTR systems ................................................................................................. - 35 -

4.2. CAT-1 DTR Monitoring System.......................................................................... - 37 -

4.2.1. General information of CAT-1 DTR system ................................................ - 37 -

4.2.2. Working principle of conductor temperature measurement in CAT-1 ......... - 37 -

4.2.3. Load cell........................................................................................................ - 42 -

4.2.4. Working principle of solar radiation rate measurement in CAT-1 system ... - 45 -

4.2.5. Net radiation sensor ...................................................................................... - 46 -

4.2.6. Ambient temperature sensor ......................................................................... - 49 -

4.2.7. Anemometer .................................................................................................. - 50 -

4.3. Power Donut DTR Monitoring System ................................................................ - 51 -

4.3.1. Introduction to power donut.......................................................................... - 51 -

4.3.2. Working principle of conductor temperature measurement in PD2 system . - 51 -

4.3.3. Power Donut sensor ...................................................................................... - 52 -

4.3.4. Specification of weather instrumentation of Power Donut DTR system ...... - 56 -

4.3.5. Sensitivity of weather station ........................................................................ - 56 -

4.4. Ampacimon DTR Monitoring System ................................................................. - 58 -

4.4.1. Working principle of Ampacimon ................................................................ - 58 -

4.4.2. Specification of Ampacimon DTR measurement system ............................. - 60 -

4.4.3. Sensitivity of Ampacimon DTR measurement system ................................. - 60 -


5. Theoretical error analysis and comparison of DTR systems ....................................... - 64 -

5.1. Propagation of measurement error ....................................................................... - 64 -

5.1.1. Concept of error propagation ........................................................................ - 64 -

5.1.2. Categories of error ........................................................................................ - 64 -

5.1.3. The calculation of error propagation ............................................................. - 65 -

5.2. Accuracy of CAT-1 system .................................................................................. - 65 -

5.3. Accuracy of Power Donut system ........................................................................ - 67 -

5.3.1. Component error ........................................................................................... - 67 -

5.3.2. System error .................................................................................................. - 67 -

5.4. Accuracy of Ampacimon system ......................................................................... - 69 -

5.4.1. Component error ........................................................................................... - 69 -

5.4.2. System error .................................................................................................. - 69 -


6. Methods to Choose the Critical Spans ......................................................................... - 72 -

6.1. Identification of critical span by thermal ageing of overhead conductor ............. - 72 -

6.2. Identification of critical span by statistical methodology .................................... - 73 -


7. Conclusion ................................................................................................................... - 75 -

7.1. Main conclusion ................................................................................................... - 75 -

7.2. Further work ......................................................................................................... - 75 -

8. References ................................................................................................................... - 76 -

9. Appendix ..................................................................................................................... - 82 -

9.1. Matlab code for DTR calculation ......................................................................... - 82 -

9.1.1. Sub-function .................................................................................................. - 82 -

9.1.2. Heat calculation ............................................................................................ - 84 -

9.1.3. Dynamic thermal rating calculation .............................................................. - 85 -

9.1.4. Matlab code for plotting the relation between height and wind velocity ..... - 86 -

9.2. Numeric data for thermal rating calculation ........................................................ - 86 -

9.2.1. Tables for DTR calculation models comparison .......................................... - 86 -

9.2.2. Tables for CAT-1 system .............................................................................. - 88 -

9.2.3. Tables for Power Donut system .................................................................... - 90 -

9.2.4. Tables for overloading risk analysis ............................................................. - 95 -

Word Count: 30090

List of Figures

I

ListofFigures

Figure 2-1: probabilistic thermal rating ................................................................................ - 9 - Figure 2-2: correlation term and design temperature .......................................................... - 11 - Figure 2-3: scatter figure and regression function .............................................................. - 11 - Figure 2-4: ambient temperature (c) probability distribution (2006-2008) ........................ - 13 - Figure 2-5: wind speed cumulative probability distribution (2006-2008) .......................... - 13 - Figure 2-6: overloading risk for fixed operation current .................................................... - 14 - Figure 3-1: the process of IEEE model calculation ............................................................ - 20 - Figure 3-2: the process of CIGRE model calculation ......................................................... - 24 - Figure 3-3: the process of IEC model calculation .............................................................. - 27 - Figure 3-4: the comparison of convection heat loss in three models .................................. - 28 - Figure 3-5: the comparison of IEEE and CIGRE AC resistance calculation ..................... - 31 - Figure 3-6: the comparison of thermal rating calculation of three models ......................... - 34 - Figure 4-1[48]: force balance of load cell ........................................................................... - 38 - Figure 4-2[50]: the catenary model of overhead line .......................................................... - 39 - Figure 4-3: principle of CAT-1 system's conductor temperature measurement ................. - 41 - Figure 4-4[55]: the load cell installed on overhead line ..................................................... - 42 - Figure 4-5: the complete process of DTR calculation in CAT-1 system ............................ - 43 - Figure 4-6: difference between average and highest conductor temperature ..................... - 45 - Figure 4-7[55]: net radiation sensor installed on power transmission tower ...................... - 46 - Figure 4-8: the relation between NRS temperature and solar rate ...................................... - 48 - Figure 4-9: the relation between NRS temperature and maximum current ........................ - 48 - Figure 4-10: the error because of angle between NRS and OHL ....................................... - 49 - Figure 4-11: ambient temperature and maximum current .................................................. - 50 - Figure 4-12: PD conductor temperature and maximum current ......................................... - 53 - Figure 4-13: Power Donut tilt angle and maximum current ............................................... - 53 - Figure 4-14[1]: wool pole ................................................................................................... - 54 - Figure 4-15: conductor temperature at different height on wool pole ................................ - 54 - Figure 4-16: tilt sensors’ error according to error coefficients ........................................... - 55 - Figure 4-17: sensitivity of wind speed measurement ......................................................... - 57 - Figure 4-18: sensitivity of wind direction measurement .................................................... - 57 - Figure 4-19: sensitivity of Ampacimon conductor temperature measurement ................... - 62 - Figure 5-1: tension and relative error.................................................................................. - 67 - Figure 5-2: tilt angle and relative error ............................................................................... - 69 - Figure 5-3: frequency and relative error ............................................................................. - 71 - Figure 6-1[76]: ageing condition of transmission towers ................................................... - 73 -

List of Tables

II

ListofTables

Table 1: usually used high temperature conductors [7] ........................................................ - 3 - Table 2: monthly temperature of New South Wales State in 2012 ....................................... - 8 - Table 3: maximum conductor temperature assumption and overloading risk ...................... - 9 - Table 4: overloading correlation term for CEGB’s OHL experiments ............................... - 10 - Table 5: example of DTR data ............................................................................................ - 12 - Table 6: remaining experiment data ................................................................................... - 13 - Table 7: the value of solar azimuth according to hour angle and solar variable ................ - 19 - Table 8: comparison of different DTR models ................................................................... - 34 - Table 9: the working principles used in DTR monitoring systems .................................... - 36 - Table 10: the sensitivity of load cell in CAT-1 system ...................................................... - 44 - Table 11: terrain and roughness .......................................................................................... - 53 - Table 12: specification of Power Donut’s weather station ................................................. - 56 - Table 13: the complete process of Ampacimon calculation ............................................... - 61 - Table 14: the specifications of different DTR systems ....................................................... - 63 - Table 15: error of conductor temperature measurement in CAT-1 system ........................ - 65 - Table 16: error of conductor temperature measurement in Power Donut system .............. - 68 - Table 17: error of conductor temperature measurement in Ampacimon system ................ - 70 - Table 18: high temperature frequency for top 20 spans ..................................................... - 74 - Table 19: air viscosity, density and thermal conductivity by temperature ......................... - 86 - Table 20: coefficients for the calculation of heat flux rate ................................................. - 87 - Table 21: the comparison of IEEE and CIGRE AC resistance calculation ........................ - 87 - Table 22: the relation between NRS temperature and solar rate, thermal rating ................ - 88 - Table 23: the error because of angle between NRS and OHL ............................................ - 89 - Table 24: CAT-1 ambient temperature and maximum current ........................................... - 89 - Table 25: Power Donut conductor temperature and maximum current .............................. - 90 - Table 26: the sensitivity of tilt sensor in Power Donut system ........................................... - 91 - Table 27: the relation between height, wind speed and conductor temperature ................. - 91 - Table 28: tilt sensor’s systematic error ............................................................................... - 92 - Table 29: sensitivity of wind speed measurement .............................................................. - 93 - Table 30: sensitivity of wind direction measurement ......................................................... - 93 - Table 31: sensitivity of solar radiation measurement ......................................................... - 94 - Table 32: sensitivity of ambient temperature measurement ............................................... - 95 - Table 33: the relation between STR and overloading risk in Canterbury ........................... - 95 - Table 34: the relation between STR and overloading risk in Legacy ................................. - 96 - Table 35: the relation between STR and overloading risk in Taunton ............................... - 96 -

List of Abbreviations

III

ListofAbbreviationsAAAC All Aluminium Alloy Conductor ACAR Aluminium Conductor Alloy Reinforced ACCR Aluminium Conductor Composite Reinforced CEGB Central Electricity Generating Board CIGRE International Council on Large Electric Systems DC Direct current DNO Distribution network operator DTR Dynamic thermal rating GPS The Global Positioning System IEC International Electro-technical Commission IEEE The Institute of Electrical and Electronics Engineers NRS Net radiation sensor OHL Overhead line PD2 Power Donut 2 PTR Probabilistic thermal rating STR Static thermal rating TSO Transmission system operator UK The United Kingdom USA The United States of America

List of Terms

IV

ListofTermsOHL’s sag: The vertical distance from highest point to the lowest point of OHL. Catenary: A catenary is the curve that an idealized hanging chain absolutely under

the weight of itself. Strain: The deformation of a body’s shape from an original configuration to a

current configuration. Young’s modulus:

The ratio of stress along an axis over the strain over the same axis

Ampacity: The maximum allowable current of the OHL [1] Weibull distribution:

A probability distribution which is continuous. It is named after Waloddi Weibull because he specifically described it firstly.

Inclination: The angle of deviation or the supplementary angle from the horizontal or vertical

Bluetooth: A wireless communication technology using short wavelength (2.40-2.48 GHz)

Newton’s second law:

A body’s acceleration is proportional to the net force acting on the body.

Abstract

V

Abstract

The increasing power transmission capacity through thermal ratings is currently of great interest in electricity industry. This is mainly achieved by employing probabilistic thermal rating (PTR) or dynamic thermal rating (DTR) instead of implementing traditional static thermal rating (STR) methods.

This thesis discusses the concepts of STR, DTR and PTR. The methodologies that used to calculate thermal rating of OHL in IEEE, CIGRE and IEC are compared. Furthermore, it extends to the comparison of recently developed DTR monitoring systems, CAT-1, Power Donut and Ampacimon, which are based on different operational principles.

CAT-1 system directly measures the tension of OHL system and local weather. It uses the current loading from the network operator of the specific OHL as well as physical properties of the OHL. It calculates the ampacity without installing an instrument on the high voltage components of the OHL. Power Donut is directly installed on the OHL conductor to measure its surface temperature and inclination in order to calculate the ampacity. Ampacimon measures the conductor’s vibration frequency and then implementing acoustic laws to derive the instantaneous ampacity.

The accuracies and other characteristics of these calculation methods are compared for finding the optimal DTR monitoring method. Specific error analysis of these monitoring instruments is shown.

The maximum difference between the three thermal ratings is the calculation of convection heat loss. When the wind speed is 0.5m/s, wind direction is 90°, ambient

temperature is 40 , the difference between the three thermal rating methods’

results might be as high as 50%. The main reason is that the three models employ very different formulas to calculate convection heat loss in slow wind conditions.

The errors of measurement devices used in DTR system were analysed. In normal condition and using IEEE DTR model, the total errors against the correct thermal rating value for CAT-1, Power Donut and Ampacimon are 1.842%, 0.755% and 6.6277% respectively. Tension, tilt angle and vibration measurement are the main sources for the three devices’ error separately.

Declaration

VI

Declaration

That no portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning.

Copyright Statement

VII

CopyrightStatement

i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the “Copyright”) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes.

ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made.

iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the “Intellectual Property”) and any reproductions of copyright works in the thesis, for example graphs and tables (“Reproductions”), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions.

iv. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any relevant Thesis restriction declarations deposited in the University Library, The University Library’s regulations (see http://www.manchester.ac.uk/library/aboutus/regulations) and in The University’s policy on Presentation of Theses.

Acknowledgements

VIII

Acknowledgements

Firstly, I must thank my supervisor, Dr. Konstantinos Kopsidas who has extensive research experience and knowledge in electrical engineering. He gives me the opportunity to study in The University of Manchester and provides me research topic. He told me the research methods and provided research facilities. His guidance is incredible critical in my research.

My co-supervisor, Prof Simon Rowland, has emailed me his opinions about my research.

My colleagues in Ferranti Building’s B18 office: Muhammad Buhari, Sheng Ji Tee, Jiashen Teh, and Alexandra Kapetanaki have helped me a lot in academic research. I really appreciate their help.

Dedication

IX

Dedication

to my parents: Meilong Xu and Xiuling Dai

For the joy they bring for me

Dedication

X

Introduction

- 1 -

1. Introduction

1.1. TheneedtoincreasetransferringcapacityInternational Council on Large Electric Systems (CIGRE) conducted a survey [2] about the use of distribution network operator’s overhead line in 1998. In this survey, 51% of all the respondents had set a higher maximum operation temperature for their overhead lines in the past five years. In addition, 30% of all the respondents had changed the static thermal rating conditions to increase the transmission capacity. This phenomenon implies that the power demand is increasing in most TSOs and DNOs.

The World Bank provides data about development in countries annually. From the World

Bank’s 2013 database [3]， the total electrical power consumptions around the world are 1.817 × 10 , 1.945 × 10 , 2.009 × 10 kWh in 2009, 2010, 2011 respectively. It can be found that the increasing power demand is a common phenomenon in many countries. The electrical power consumption increased by approximately 7% in China in 2009. The global electricity consumption increased by 7.042% and 3.275% in 2009 and 2010 respectively.

Nowadays, power transmission is necessary in most areas of the world. Many of these areas will have a varied demand for electricity because of the change of natural circumstance or economic growth. Only by increasing the capacity of transmission lines can the growing demand for electricity be satisfied. The change of natural circumstance and the development of economy obey the physical and social laws. They will not follow the will of people all the time. Therefore, the demand for electricity increases or decreases at times. The reasons for increasing demand of electricity can be listed as follows. For example, due to the development of economy, almost all industries need electricity, which will inevitably increase the demand for electricity power. Apart from this, the popularity of electric appliances will lead to a rise of the domestic electricity consumption, too.

In order to increase transmission capacity, the instruments currently used in power transmission should be known. There are mainly two types of lines for electrical power transmission, namely overhead line and underground cable. Scientists and engineers have developed the third way, gas insulated line (GIL). However, it needs to be developed better.

The building expense of overhead line is lower than underground cable, because the installation of OHL is easier than underground cable. The maintenance of overhead line is also easier than cable. However, underground cable is more reliable than overhead line, because OHL tends to be influenced by weather conditions easily. Apart from this, OHL will harm the landscape and environment; while underground cable is invisible for common citizens [4]. Underground cable is a transmission line covered by insulator and buried in the ground. Compared with overhead line, it is more reliable because of better insulation. Nevertheless, the cost for underground cable is higher, for both the building expense and maintenance expense. Underground cable is the luxury among transmission conductors.

Introduction

- 2 -

Overhead line is a bare conductor located in the air. It is insulated by air that is the cheapest insulator. Overhead line is a cost efficient solution for power transmission [5]. Especially in remote power transmission, if a power plant in Siberia wants to transmit its electricity to Moscow or the three-gorge power station wants to transmit its electricity to Shanghai, overhead line will be the cheapest choice. For the sake of profit, the cheaper power line, overhead line, has become popular in large power transmission companies.

As discussed above, OHL is an important format for power transmission. Many distribution network operators want to increase the capacity of their overhead lines. However, it is not easy. Sometimes, it can be extreme difficult because the capacity is determined by the inherent nature of the material used to build the tower and the lines. In the past, only entirely eliminating the old equipment could increase the capacity. Generally speaking, the replacement is able to increase the capacity to a very great extent. However, it is not always so perfect because the increasing depended on the difference of the old and the new materials. In some cases, finding a new line that brings huge increase for capacity cannot be done.

Transmission capacity of overhead lines is determined by current and voltage. Uprising current or voltage can both increase OHL’s capacity. The capacity-increase methods can be divided into three categories.

The first category is conductor-change. Overhead line and transmission tower constitute power transmission system. Changing conductor and tower is considered to be building a brand new power transmission system. Even only changing conductor is a large-scale modification for power transmission system. The increase in power capacity due to conductor change depends on the capacity difference between new and old overhead conductors. When the new conductor’s capacity is much higher than the old one, the benefit from conductor change will be attractive.

However, generally speaking, an eligible transmission system design should try to make full use of its transmission tower. Normally, it will not be easy to find a new conductor with much higher capacity and same level weight. Therefore, before changing overhead line’s conductor, the transmission tower should be strengthened.

Generally, there are four kinds of transmission structures: wood pole, concrete pole, steel pole and latticed tower. From experience of engineering, wood pole is the easiest structure to be strengthened. Next one is latticed tower. Concrete pole and steel pole are relatively difficult to be strengthened. High temperature conductor is a good choice for overhead lines’ conductor change. Table 1 listed several kinds of high temperature conductors. CIGRE provides these data [6].

Introduction

- 3 -

Table 1: usually used high temperature conductors [7]

Material ACSR ACSR/TW

ACSR/EHS ACSR/AW AAC ACAR AAAC AACSR

Core material

HS EHS Alum Clad No core No core No core HS

Surface material

Hard Drawn

1350-H19

Hard Drawn 1350-H19

Hard Drawn

1350-H19

Hard Drawn

1350-H19

Hard Drawn

1350-H19

HS Alloy 6201-T81

HS Alloy 6201-T81

Material AACSR/ EHS

ACSS/GA ACSS/TW

ACSS/ HS285

ACSS/AW ACSS/MM ACSS/TW

ACCC/TW TACSR

Core material

EHS HS EXHS Alum Clad Mish metal Carbon Fibre

HS

Surface material

HS Alloy 6201-T81

1350-O 1350-O 1350-O 1350-O 1350-O TAI

Material TACSR/ AW

KTACSR KTACSR/ AW

ZTACSR GZTACSR

ZTACIR ACCR ACCR/TW

XTACIR

Core material

Alum Clad HS Alum Clad HS Galv. Invar

Al Matrix Alum Clad

Surface material

TAI KTAI KTAI ZTAI ZTAI ZTAI XTAI

The thermal limits of the conductors stated in Table 1 are higher than 90. This temperature is higher than normal conductors’ thermal limit. Before 1970, UK National Grids’ conductor temperature for all kinds of conductor could not be higher than 50 [8]. Aluminium Conductor Steel Supported (ACSS) is a kind of classic conductor that has been used for decades. This conductor is annealed when manufacturing. Before annealing, ACSS is called ACSR. These two conductors have the same material but different thermal limits. ACSS can be continually operating in temperature as high as 250. Density is a critical indicator for core material. For HS, EHS, EXHS, Galv. Invar Alloy and Mishmetal, the density is 7.778 mg/mm3. For Alum Clad, the density is 6.588 mg/mm3. Carbon fibre and Al matrix’s densities are 1.938 and 3.322 mg/mm3 respectively.

Thermal limit is a critical indicator for aluminium material. For hard drawn, MS alloy and HS alloy, the thermal limit for constant operation is 90. For TAL and KTAL which are called thermal resistant, the limit is 150. For ultra-thermal resistant, ZTAL, the limit is 200. For extra-thermal resistant conductor, XTAL, the limit is 230. All the information is provided by CIGRE document [7].

The second category is voltage-increase. Because conductor’s capacity is determined by current and voltage, increasing conductor voltage is able to increase the capacity directly.

According to Joule heating formula, = ∙ , changing voltage does not influence the Joule heat loss. This is a useful characteristic for voltage uprising because Joule heating is not only a source for power loss but it will cause damage for conductor. However, voltage uprising still has problems.

Introduction

- 4 -

Firstly, the transformer operation is a problem. High voltage power transmission systems require larger transformer. A power transformer’s size is generally larger than normal ones. It will produce more heat during operation. Fans and pumps normally do the cooling of transformer. All these accessories for powerful transformers cost more than small transformers. So the high voltage conversion will cause higher financial cost, which will influence the related companies’ choices.

Secondly, the safety clearance requirement to the ground is a problem. Clearance requirement is the main concern for most power transmission companies [9]. Increasing voltage will violate the old transmission system’s standard. Therefore, it is necessary to raise the height of current overhead line when increasing the voltage. The height of OHL is determined by transmission tower. Changing the transmission tower is obviously expensive.

Thirdly, the clearance requirement between phase and phase is a problem. When a relative lower voltage is employed on the OHL, the transmission tower may undertake a double circuit. After the voltage is increased, it maybe could only employ signal circuit. This is obviously another loss for transmission capacity.

The third category is the method based on operation to increase thermal rating. When the OHL’s operating conditions (conductor temperature, ambient condition) are known, the real time operation current can be adjusted based on DTR. This is able to maximize the capacity of the existing line. The drawback for DTR monitoring is that it will increase the complexity of operation.

Most distribution-network operators use static thermal rating typically as the maximum operation current. To meet the requirement of increasing demand for power transmission network’s capacity, the concept of dynamic thermal rating has been developed by power system researchers.

People have long realized that the real current of overhead transmission lines is usually conservative, which would lead to the under-utilization of conductors. Theoretically, if the damage by electrical current to transmission conductor and the conductor’s ambient environment were not considered, the capacity of the overhead line would be able to reach any value. However, there is a maximum current for all kinds of conductors [10]. There are different reasons for the constraints of power transmission lines’ maximum capacity.

The first reason is the clearance requirement that means the distance from OHL’s lowest point to ground. The clearance standard varies with countries. For 400KV OHL, the minimum clearance is 7.6m in UK; the clearance requirement is 7.5m in China [11]. When OHL is installed, the clearance is definitely larger than required value. However, the clearance of OHL is not fixed. Thermal expansion will cause the elongation of OHL’s length. Sag will increase with the elongation. Because of the clearance requirement, there is a maximum operation current corresponding to the maximum conductor temperature for OHL.

Introduction

- 5 -

The second reason limiting the capacity of OHL is the damage to conductor because of annealing and ageing. Any material will suffer from annealing and ageing. When the conductor temperature is extremely high (higher than its melting point) the conductor’s microstructure will collapse immediately. Similarly, when the conductor temperature is higher than the annealing point, the conductor’s microstructure will change (the process is slower than melting but faster than ageing). When the conductor temperature is below annealing point, the thermal effect still changes the conductor’s microstructure, and even the changing speed is slowest. Because annealing and ageing will decrease the lifespan of conductor, high-temperature should be avoided for OHL. The conductor temperature limitation will restrict the current of OHL.

In the CIGRE’s survey about power transmission companies [2], 79% of all the companies set the thermal rating of their OHL according to clearance requirement, 9% of all repliers use annealing of conductor as the primary reason for setting the maximum conductor temperature. It can be found that these two reasons of the limitation for OHL’s capacity are both directly related to thermal effect. Therefore, the maximum capacity of OHL is usually referred to as thermal rating.

1.2. AimsandobjectivesCurrently there are several different monitoring instrumentation and methods for implementing dynamic thermal rating on overhead line. The main aim of the thesis is to evaluate the measurement errors of the most widely used dynamic thermal rating monitoring systems that implement instrumentation on the conductor and at very close proximity to the overhead line system. The objectives of the report to achieve the main aim are:

Understanding the different thermal rating methods reported in the different standards (IEEE, IEC, and CIGRE) and comparing those in order to identify their maximum error.

Understand the implementation of different thermal rating monitoring systems (CAT1, Power Donut and Ampacimon)

Quantify the errors of the various thermal rating monitoring systems and identify the conditions that result in large error and propose some methods for mitigation of the measurement errors.

1.3. DissertationoutlineIn the main body of this thesis, there are six chapters. The first chapter is this introduction. The second chapter introduces the concept of thermal rating. The concepts include static thermal rating, dynamic thermal rating and probabilistic thermal rating. The concept of overhead line’s thermal rating is the foundation of this thesis.

The third chapter describes the models used to calculate thermal rating. The models include IEEE, CIGRE and IEC models. These models can get the numeric value of thermal rating according to overhead line’s environmental condition. The numeric value of thermal rating can quantify the benefit of DTR methods.

Introduction

- 6 -

The fourth chapter illustrates the systems used to measure the elements of thermal rating calculation. The systems include CAT-1, Power Donut and Ampacimon. These systems provide the original data for DTR calculation. The fifth chapter compares the error levels of thermal rating measurement systems. Systematic error is inevitable in measurement systems. To know the specific error range for measurement system can help overhead line operator to choose suitable measurement system.

The sixth chapter tells the methods to choose critical span from all overhead line. A complete overhead line consists by many spans. Thermal rating is the lowest value of all the spans. Some spans have relatively high probability to be the lowest rating span. These spans are critical spans. This chapter can help to find the optical measurement location on OHL.

Thermal rating

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2. Thermalrating

2.1. Staticthermalrating

2.1.1. ThermalratingdefinitionStatic thermal rating (STR) means the thermal rating under “worst weather condition”. It is a fixed “lowest” maximum operating current for OHL. STR is the fundamental value when it comes to thermal rating. It is widely used in power transmission industry. All the capacity improvements are compared with the STR value. Even the concepts of probabilistic thermal rating and dynamic thermal rating have been developed; static thermal rating is still the most commonly used in power transmission industry.

Strictly speaking, the worst weather condition for power transmission should be zero wind, highest ambient temperature and highest solar radiation. But it is quite difficult to define the highest ambient temperature and highest solar radiation. At the same time, using zero wind speed will dramatically decrease the value of thermal rating.

For the reasons stated above, the nature of static thermal rating is the thermal rating of conservative weather condition instead of worst condition. The conservative weather condition is usually defined by historical data and statistical analysis. Typically, the conservative weather condition is worse than most conditions, e.g. 97% of all conditions. So the STR is also not always applicable for OHL’s operation. However, power transmission companies would rather be tolerant of a small amount risk to get a reasonable transmission capacity and more economic OHL structures.

2.1.2. ExampleforconservativeconditionLots of researchers tried to explore a conservative condition for the application of thermal rating. For example, the Institute of Electrical and Electronics Engineers (IEEE) has provided its conservative condition as follows [12]:

Wind speed (Vw) is 0.61 m/s and perpendicular to the conductor; Ambient air temperature is 40 °C; Solar radiation rate is 1023 W/m2. The International Council on Large Electric Systems (CIGRE) also assumes its conservative condition as follow [13]:

Wind speed (Vw) is 0.60 m/s and perpendicular to the conductor; Ambient air temperature is annual maximum temperature; Solar radiation rate is 1000 W/m2. In industry, the assumption of conservative condition varies due to differences in weather conditions with the location. The most suitable conservative condition should depend on the historical weather record. Because weather condition varies with location, the fixed condition cannot meet the requirement of all locations. The weather condition which is worse than 97% historic conditions should be chosen. Weather records are not available all the time and when

Thermal rating

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the record is not available, the IEEE conservative condition is recommended, because it is a medium condition [14].

2.1.3. SeasonalstaticthermalratingStatic thermal rating is simple and fixed value. While the deterministic static thermal ratings of overhead transmission lines are typically conservative, resulting in underutilization of their potential capacity. One strategy to overcome this limitation leading to the development of

alternative rating is the seasonal static thermal rating，which uses a probabilistic rating

approach with explicit treatment of seasonal effects on conductor temperature. Considering the weather distinction in different seasons, the seasonal static thermal rating could be employed in OHL.

Seasonal weather difference is common in many countries. UK is a typical example. According to the weather record of UK meteorological office from January 1910 to June 2013, the average temperature of Spring (Mar - May) is 7.327; the average temperature of Summer (June - Aug) is 13.90; the average temperature of Autumn (Sep - Nov) is 9.05; the average temperature of Winter (Dec - Feb) is 3.42. The average temperature difference between summer and winter is approximately 10 . In normal conditions, this ambient temperature difference will result in 73A current difference of a 400kV standard OHL structure thermal rating.

The Central Electricity Generating Board (CEGB) has recommended the following seasonal temperature: Summer (20), Normal (9), Winter (2). Normal season refers to spring and autumn seasons. As the recommendation of Central Electricity Generating Board (CEGB), the weather conditions in UK are as follow:

Season Summer Spring/Autumn Winter Temperature () 20 9 2 Wind speed (m/s) 0.5 0.5 0.5

Table 2 shows the monthly average temperature of New South Wales State in Australia given by Australian Government Bureau of Meteorology [15].

Table 2: monthly temperature of New South Wales State in 2012

Month Jan Feb Mar Apr May Jun Mean tem (c) 24.1 22.9 22.7 19.4 16.7 13.7

Month Jul Aug Sep Oct Nov Dec Mean tem (c) 13.3 14.6 17.2 18.5 20.5 22.3

From Table 2, it can be found that the coldest three months are June, July and August. The seasonal temperature difference is not as obvious as in UK. So, the seasonal conservative weather condition must base on local meteorological record. There is no fixed standard method for seasonal thermal rating all around the world. Seasonal thermal rating is the static thermal rating in one season. In some conditions, the weather conditions in different seasons

Thermal rating

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are distinct, and seasonal thermal rating is able to make more use of the capacity of overhead lines.

2.2. Probabilisticthermalrating

2.2.1. GeneralinformationThe probabilistic thermal rating is another strategy to solve the problem of the underutilization of potential capacity. Before introducing the probabilistic thermal rating, the risk of static thermal rating should be well defined. Even when conservative weather condition is employed, the overloading risk of OHL is not zero.

For a value of operation current, there is a corresponding overloading probability. Figure 2-1 can be used to explain the meaning of probabilistic thermal rating. In this figure, the curve is the probability distribution of real-time thermal rating. The data are extracted from National Grid’s Taunton OHL monitoring system. The horizontal axis means the operation current; the Y-axis is the probability of overloading under this current. The vertical black line indicates the static thermal rating under CEGB recommended assumed conservative condition. The solid part of the figure indicates the conditions that the real-time operation current is higher than designed value. The total area of the solid part of the curve is the probability of overloading [16].

Figure 2-1: probabilistic thermal rating

According to Taunton’s OHL operation data, for different maximum conductor temperature assumption, the probability is different as shown in Table 3.

Table 3: maximum conductor temperature assumption and overloading risk

Maximum conductor temperature ()

60 65 70 75 80 85 90

Overloading risk (%) 23.14 17.11 11.08 9.05 5.02 3.99 1.95

The OHL’s weather data are provided by the National Grid’s Taunton site. The dynamic thermal rating calculation is according to the IEEE 738’s DTR model.

Thermal rating

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The probabilistic thermal rating is a combination of different fixed thermal rating based on different conservative conditions. The probability of overloading risk in this fixed thermal rating condition can be calculated as shown in [17].

In some occasions, the maximum allowable overloading risk is given; the corresponding fixed thermal rating can be calculated.

CEGB has provided the overloading conditions of OHL in UK. The statistic result of overloading probability for OHL is shown in Table 4.

Table 4: overloading correlation term for CEGB’s OHL experiments

Year Design temperature

Summer Spring/Autumn Winter Te(%) Log(Te) CT Te (%) Log(Te) CT Te(%) Log(Te) CT

1975 45 4.0 0.602 1.151 2.5 0.398 1.055 12.0 1.079 1.36850 1.0 0 0.929 0.5 0.301 0.934 7.0 0.845 1.236

1976 45 49.1 1.691 2.094 39.2 1.594 1.789 37.1 1.569 1.80750 35.5 1.550 1.759 27.5 1.440 1.584 29.1 1.464 1.63255 23.1 1.363 1.519 17.5 1.242 1.423 21.5 1.332 1.49060 13.0 1.115 1.339 8.87 0.948 1.292 13.8 1.139 1.37165 5.62 0.750 1.198 3.35 0.525 1.185 8.07 0.907 1.27170 1.98 0.296 1.085 0.800 0.097 1.095 3.94 0.596 1.18575 0.403 0.395 0.992 0.0453 1.344 1.018 1.50 0.177 1.11180 0.019 1.721 0.915 0.291 0.536 1.045

1977 45 58.9 1.770 2.625 45.3 1.656 1.862 20.3 1.307 1.58050 44.6 1.649 2.206 34.7 1.540 1.649 13.7 1.138 1.42755 33.8 1.529 1.905 23.2 1.365 1.481 8.15 0.911 1.30260 24.8 1.394 1.679 12.9 1.109 1.345 3.94 0.596 1.19965 15.9 1.201 1.502 5.20 0.716 1.234 1.27 0.105 1.11170 8.32 0.920 1.361 1.47 0.167 1.140 0.352 0.453 1.03675 2.58 0.412 1.244 0.443 0.354 1.060 0.105 0.979 0.97180 0.524 0.280 1.147 0.152 0.818 0.991 0.015 1.824 0.91485 0.191 0.720 1.064 0.0132 1.879 0.930 90 0.019 1.720 0.993

The correlation table presents the relation between design conductor temperature and excursion time according to the statistic result of CEGB overhead line experiment.

where: CT is the correlation term, CT =

Te is the overloading probability

Excursion time means the time when conductor temperature is higher than designed temperature. The season will influence the value of excursion time. Employing different thermal ratings according to season is able to improve the usage rates of OHL.

Thermal rating

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The correlation term in Table 4 is calculated for the purpose of looking for the possible relation between designed temperature and excursion time. By the data in Table 4, the scatter points are plotted as Figure 2-2. The regression function according to these scatter points is shown in Figure 2-3. In the two figures, the X-axis is the correlation term; Y-axis is the logarithm of corresponding overloading probability.

Using the Logarithm format, the regression function of the scatter points in Figure 2-2 can be worked out as follows: y = 1.6422 + 1.1124 × log(x − 0.8673) For the regression function curve and scatter figure, the coefficient of determination is 0.8656. The fitness for the regression function is shown in Figure 2-3.

Figure 2-2: correlation term and design temperature

Figure 2-3: scatter figure and regression function

2.2.2. ExampleofprobabilisticthermalratingThe purpose of employing DTR in power transmission system is to increase the OHL’s capacity and decrease OHL’s risk of overloading.

The advantages of dynamic thermal rating compared with other thermal ratings have been introduced in many papers [18-20]. However, the specific capacity and risk analysis of different thermal ratings are rare.

This part will analyse the dynamic thermal rating monitoring data of Canterbury, Legacy and Taunton. Each DTR record from the three places contains approximately 50000 sets of data.

From the survey of CIGRE [2], different power transmission companies are using different conservative conditions for their OHL. So, even the static thermal rating is a fixed value in specific conditions, the conservative condition used in static thermal rating is not fixed. In order to show the relations between conservative capacity and overloading risk, different conservative capacities will be analysed. Each DTR record from the three places contains more than 300 conservative capacities analyses.

Thermal rating

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Probabilistic thermal rating can be regarded as a combination of different static thermal ratings. Static thermal rating can be seen as an example of probabilistic thermal rating. So this report does not distinguish STR and PTR on purpose and only analyse the data according to the value of conservative condition.

The fixed conditions for overhead line at Canterbury, Legacy and Taunton are shown as follows:

Location Canterbury Legacy Taunton Record start 15/11/2006 12:30 09/10/2006 12:30 09/10/2006 07:50 Record end 12/02/2008 13:00 13/02/2008 13:00 13/02/2008 13:10 Instruments CAT-1 CAT-1 CAT-1 Conductor type Sorbus Sorbus Sorbus Conductor diameter 26.94 mm 26.94 mm 26.94 mm Elevation 80 m 30 m 100 m Conductor emissivity 0.5 0.5 0.5 Solar absorptivity 0.5 0.5 0.5 AC resistance at 25 7.39 / 7.39 / 7.39 / AC resistance at 75 8.64 / 8.64 / 8.64 / Interval of record 10 min 10 min 10 min

The example record for overhead lines’ conditions is shown in Table 5. The table shows an example of the Canterbury’s DTR record. The first column is recording date and the second column is recording time. The third column is one load cell’s measured tension and the fourth column is another load cell’s measured tension. One transmission tower connects two spans. There are load cells on both spans. The fifth column is the ambient temperature measured by ambient temperature sensor located in the main panel. The next two columns are the two NRS’ temperatures. The eighth column is wind speed and ninth column is wind direction.

Table 5: example of DTR data

Cell 1 Cell 2 Ambient NRS1 NRS2 Wind

Velocity Wind

Direction

Date Time N N m/s ° 15/11/2006 12:30 26,773 26,426 14.2 14.6 14.7 8.4 234 15/11/2006 12:40 26,773 26,368 14.2 14.4 14.5 6.3 231 15/11/2006 12:50 26,773 26,368 14 14.3 14.4 6.1 225

A part of original data is obviously wrong. These data will cause error for the final result. The criteria for obviously wrong data are:

The load cell tension is less than 22250N The ambient temperature is lower than -20 The ambient temperature is higher than +40 The NRS temperature is lower than -20 The NRS temperature is higher than +50 The wind speed is higher than 50m/s

Thermal rating

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As long as one of these criteria is satisfied, the data will be regarded as obvious wrong data. The original data quantity and remaining data quantity are shown in Table 6.

Table 6: remaining experiment data

Location Canterbury Legacy Taunton Original data amount 51247 55655 58381

Error data amount 2629 0 1 Remaining data amount 48618 55655 58380

According to these DTR monitoring data, the ambient temperature and wind speed probability distribution are shown in Figure 2-4 and Figure 2-5 respectively.

Figure 2-4: ambient temperature (c) probability distribution (2006-2008) Figure 2-5: wind speed cumulative probability distribution (2006-2008)

In Figure 2-4, the X-axis is the ambient temperature; Y-axis is the corresponding probability. After data processing, the probability distributions for ambient temperature in different sites are shown in the Figure 2-4. In Canterbury, the highest temperature is 44.8; the lowest temperature is -24.8; the average temperature is 11.8. In Legacy, the highest temperature is 21.4; the lowest temperature is -2.8; the average temperature is 10.3. In Taunton, the highest temperature is 25.8; the lowest temperature is -2.8; the average temperature is 11.3.

Figure 2-5 shows the cumulative probability distribution of wind speed in Canterbury, Legacy and Taunton. The X-axis is the wind speed; Y-axis is the corresponding probability. The highest wind speed in Canterbury is 17.8 (m/s); the lowest wind speed is zero wind. The average wind speed is 3.31 m/s. The highest wind speed in Legacy is 18.2 (m/s); the lowest wind speed is zero wind. The average wind speed is 2.79 m/s. The highest wind speed in Tanuton is 22.9 (m/s); the lowest wind speed is zero wind. The average wind speed is 4.06 (m/s).

Thermal rating

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The weather conditions and conductor information can be used to calculate the thermal rating. (The specific method for thermal rating calculation will be introduced in the next chapter.) The probability distribution of thermal rating will be presented by statistic method. The data of Taunton starts at 10/09/2006 and ends at 13/02/2008. According to the statistic result, the overloading probability of different operation currents is described in Figure 2-6.

The 58380 points real-time maximum current of overhead line in Taunton were calculated. For each value between 0 – 3000A with interval of 10A, the probability of overloading points were counted. The X-axis in Figure 2-6 is the operation current of OHL, and Y-axis is the corresponding overloading probability.

Figure 2-6: overloading risk for fixed operation current

Regression function could provide the relation between continuous operation current and corresponding risk. So the next part will find a regression function for the scatter points in Figure 2-6.

Using the Weibull CDF format [21], the regression function of the scatter points in Figure 2-6 can be worked out as follows:

y = 1 − e . . For the regression function curve and scatter figure, the coefficient of determination is 0.9991. The fitness for the regression function is shown in Figure 2-6. The solid line is the regression function for the scatter points. By this regression function, the specific numeric relation between operation current of OHL and the corresponding overloading risk can be known. When the operation current is determined, the value can be plugged into the regression function. And the corresponding overloading risk can be calculated.

2.3. DynamicthermalratingDynamic thermal rating is the real time maximum allowable current of overhead lines. It also has been seen as a most important tool for planning and operation of power systems. Recently,

Thermal rating

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for smart-grid applications, it is necessary to install monitoring stations along the studied lines. As the accurate measurement of dynamic thermal rating is expensive, it is necessary to explore a cost-effective DTR system. More and more researchers pay attention to the study of the dynamic thermal rating and have got some achievements. In [22], Adapa has stated the basic monitoring and calculation methods for dynamic thermal rating. In [23], Heckenbergerova has presented the influence of wind speed for DTR. When the wind speed increases, the value of DTR will increase a lot relatively. In [24], Douglass and Edris have provided a case study for the application of DTR. These researchers’ work improves our understanding for dynamic thermal rating.

Although the details of these methods are different; the required parameters are the same. The calculation of DTR requires information of real time wind speed, wind direction, solar radiation rate, ambient temperature and conductor temperature.

A dynamic thermal rating system should be able to measure all of these parameters. At the same time, the measurement must meet the requirements of sensitivity, accuracy and calibration.

In the market, there are different kinds of DTR system instruments that can be used to monitor OHL’s weather condition. But, most of them are not good enough to be used in practice. After careful selection, three DTR systems will be specifically introduced in this thesis.

2.4. MainconclusionsThis chapter introduced the concept of overhead lines’ thermal rating. According to the environment employed in the calculation of thermal rating, there are static thermal rating, probabilistic thermal rating and dynamic thermal rating. Thermal rating is the maximum operation current of overhead line. The differences between static, probabilistic and dynamic thermal rating mainly depend on the environment condition employed.

If the environment is assumed fixed, the result is static thermal rating. If the environment value is calculated according to overloading probability, the result is probabilistic thermal rating. If the environment is real time value, the result is dynamic thermal rating.

IEEE, CIGRE and IEC’s DTR calculating models

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3. IEEE,CIGREandIEC’sDTRcalculatingmodels

3.1. IEEEmodelofthermalratingcalculation

3.1.1. ThermalequilibriumofOHLinIEEEmodelThe earliest IEEE model of dynamic thermal rating calculation was defined in 1980s. After two revisions (1993 2006), the latest IEEE model is the 2006 revision [12]. Thermal equilibrium is the main assumption of IEEE’s thermal rating calculation which can be stated as follow: during the operation of OHL, the conductor’s temperature will be higher than ambient temperature because of Joule effect [25], so the thermal transmission between conductor and ambient air will not stop during the power system’s operation. By analysing the heat gain and loss of conductor, it is able to build the relationship between weather conditions and electrical current. The equation of thermal equilibrium of conductor can be given by:

+ = + ( ) (1)

Where: = heat loss rate per unit span because of convection (w) = heat loss rate per unit span because of radiation (w) = heat gain rate per unit span from sun (w) = conductor current (A) = conductor temperature (C) ( ) = AC residence of conductor at temperature (Ω)

When the conductor temperature reached its highest value, the corresponding current is the thermal rating of OHL. There were four kinds of heat in the thermal equilibrium of overhead conductor mentioned in the IEEE’s model: convection heat loss, radiated heat loss, solar heat gain and Joule heat gain.

3.1.2. CalculationofconvectionheatlossinIEEEmodelThe convection heat loss of OHL varies with the changing of the wind speed. When the wind speed is high, the formula of convection heat loss rate of OHL is given by:

= [1.01 + 0.0372( ) . ] ( − ) (2)

When the wind speed is low, the convection heat loss rate of OHL is given by:

= 0.0119( ) . ( − ) (3)

When the wind speed is zero, the convection heat loss rate of OHL is given by:

= 0.0205 . . ( − ) . (4)

Where = conductor diameter (mm)

= air density (kg/m3)


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= speed of air stream at conductor (m/s) = air viscosity (Pa∙s) = thermal conductivity of ambient air at temperature (W/m∙ )

( ) = AC residence of conductor at temperature (Ω) = temperature of conductor () = temperature of ambient air ()

The correct value of convection heat loss is the largest result of equation (2, 3 and 4). Conductor diameter is assumed constant and known. Wind speed and direction of the OHL should be measured in dynamic thermal rating or use the conservative value in static thermal rating. Air density, viscosity and thermal conductivity can be calculated by the ambient air temperature and the elevation of OHL. In engineering, the values can be obtained by looking up table. IEEE 738 [12] has provide a data table (shown in appendix) used to find the value of air density, viscosity and thermal conductivity. Ambient temperature and conductor temperature should be measured in dynamic thermal rating, or use the forecast value in probabilistic thermal rating, or use the conservative value in static thermal rating. Wind direction factor used in the calculation of convection heat loss is a coefficient based on the angle between wind direction and OHL’s direction. The calculation of wind direction factor is given by:

= 1.194 − cos(∅) + 0.194 cos(2∅) + 0.368sin(2∅) (5)

Where: ∅ = wind angle between wind direction and OHL’s direction

3.1.3. CalculationofradiationheatlossinIEEEmodelEvery object is emitting electromagnetic waves to its ambient space [26]. The energy contained in the electromagnetic waves is determined by the temperature difference between the object and its ambient environment. The radiation heat loss rate of OHL is given by:

= 0.0178 + 273100 − + 273100 (6)

Where: = the radiation heat loss of OHL (w) = the emissivity of conductor = temperature of conductor () = temperature of ambient air ()

The emissivity of conductor is assumed fixed in the operation lifetime of OHL.

3.1.4. CalculationofsolarradiationrateinIEEEmodelThe solar radiation calculation in IEEE model depends on the solar azimuth. The nature of this result is the maximum value of solar radiation. The value of solar radiation rate is given by:

= sin( ) (7)

Where: = the solar radiation rate (w)


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= solar absorptivity = corrected total radiated heat flux rate (W/m2) = the angle between solar ray and overhead line (°) = area of conductor per meter (m2/m)

Solar absorptivity of conductor is assumed constant and provided by the manufacturer of the conductor. Area of conductor per meter can be known by the specification of conductor.

The corrected total radiated heat flux rate is given by:

= (8)

Where: = solar altitude correct coefficient = the total radiated heat flux rate (W/m2)

The solar altitude correct coefficient is given by:

= 1 + 1.148 × 10 − 1.108 × 10 (9)

Where: = the height of conductor above sea level (m)

The total radiated heat flux rate is given by:

= + + + + + + (10)

Where: = solar altitude (°) A, B, C, D, E, F, G are coefficients according to the atmosphere. The values are given by table in IEEE 738 (shown in appendix).

The calculation of solar altitude is given by:

= arcsin[cos( ) cos( ) cos( ) + sin( )sin( )] (11)

Where: = latitude of overhead line (°) = solar declination (°) = hour degree (°)

The calculation of solar declination is given by:

= 23.4583sin[284 +365 360] (12)

Where: N = day of the year Hour degree is calculated according to the sun time. Noon time’s degree is zero. Every hour past noon time increase 15°, and every hour before noon time decrease 15°. The angle between solar ray and overhead line is given by:

= arccos[cos( )cos( − ) (13)


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Where: = the angle between solar ray and overhead line = solar altitude (°) = azimuth of sun (°) = azimuth of overhead line (°)

Solar altitude’s calculation was provided above. Azimuth of overhead line is assumed fixed when the line was built.

The calculation of sun’s azimuth is given by:

= + arctan( ) (14)

Where: C = constant of solar azimuth = variable of solar azimuth

The variable of solar azimuth is given by:

= sin( )sin( ) cos( ) − cos( )tan( ) (15)

The calculation and definition have been provided in previous equation.

The constant of solar azimuth is calculated by the variable of solar azimuth and hour angle. The value of C can be given by Table 7:

Table 7: the value of solar azimuth according to hour angle and solar variable

Constant of solar azimuth (c)

χ ≥ 0 χ < 0

–180 ≤ ω < 0 0 1800 ≤ ω ≤ 180 180 360

3.1.5. CalculationofJouleheatgaininIEEEmodelThe value of Joule heat is related to the conductor’s electrical resistance and the magnitude of AC current. The calculation is given by:

= ( ) (16)

Where: = the Joule heat rate (w) = the OHL’s current (A) ( ) = the AC resistance of conductor at temperature ()

The value of conductor’s resistance will be influenced by the frequency of current, conductor temperature and length of conductor. As the frequency of current is usually fixed, the conductor temperature is the primary variable in the calculation of conductor’s AC resistance.

It is a linear relation between AC resistance and conductor temperature according to the IEEE model. It uses the conductor’s AC resistance at 25 and 75 as base value firstly. Then, it can find the equation of AC resistance at any temperature:


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( ) = − ( )− ( − ) + ( )

(17)

The values of aluminium conductor’s standard AC resistance can be found in aluminium handbook [27]. The conductor types include AAAC, ACAR, ACCR and most commonly used conductors.

3.1.6. ConclusionoftheIEEEmodelcalculationprocessThe process of IEEE thermal rating model is complicate compare with other models. In order to make it clear and simple, a tree chart of IEEE DTR model is presented as follow.

Figure 3-1: the process of IEEE model calculation

Where: = heat loss rate per unit span because of convection (w)

= heat loss rate per unit span because of radiation (w) = heat gain rate per unit span from sun (w) = conductor current (A) = conductor temperature () ( ) = AC residence of conductor at temperature (Ω)

Figure 3-1 summarises the IEEE method for thermal rating calculations and the variables that affect each element of the conductor thermal equilibrium.

3.2. CIGREmodelofthermalratingcalculation

3.2.1. ThermalequilibriumofOHLinCIGREmodelCIGRE, the council on large electric system, has introduced its model for the calculation of dynamic thermal rating in 1992 [28]. The fundamental principle in CIGRE model is also heat balance. It assumes that the overhead line is under the condition of heat balance all the time. This assumption could be expressed by the expression following:

Positive Heat=Negative Heat


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The meteorological conditions and the current of transmission lines are the parameters that may influence the heat balance. The positive heats of overhead lines include the Joule heat generated by AC current, the solar heat absorbed from sun’s ray, magnetic heating and corona heating. The negative heats include the convective heat loss because of the temperature difference between the conductor surface and ambient air, the radiation heat loss because of thermal radiation and evaporative heat loss. These parameters could be expressed by the equation as follow:

+ + + = + + (18)

Where: = joule heat rate rate(w) = magnetic heat rate rate(w) = solar heat rate rate(w) = corona heat rate rate(w) = convective heat loss rate rate(w) = radiative heat loss rate rate(w) = evaporative heat loss rate rate(w)

But, in the real calculation of heat balance, magnetic heat, corona heat, and evaporative cooling are absolutely ignored. So the real equation or calculated equation is listed as follow:

+ = + (19)

3.2.2. CalculationofconvectionheatlossinCIGREmodelConvective heat transfer of overhead line is due to the temperature difference between the conductor and ambient air. Because of Joule heat, the transmission line’s temperature is higher than ambient temperature.

In CIGRE model, the calculation of convective heat loss is given by:

= ( − ) (20)

Where: = convective heat loss rate(w) = ratio of the circumference of a circle to its diameter (Thermal conductivity of air) = 2.42 ∙ 10 + 7.2 ∙ 10 (W/m∙ ) = ()

= conductor surface temperature () = ambient temperature () = the Nusselt number

The Nusselt number is a key parameter in the calculation of convective heat loss. There are different formulas for different conditions. When there is no wind, the process is called natural convective cooling. The calculation of the Nusselt number is given by:

= ( ∙ )

(21)


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Where: (Grashof number)=( )

= 1.32 ∙ 10 + 9.5 ∙ 10 (The Prandtl number)=

∙

= the specific heat of air at constant pressure (J/kg∙k) = the dynamic viscosity of air (Pa ∙ s)

A2 and are constant according the value of Rayleigh number ( ∙ ). When there is slow wind (v<0.5m/s), the process is natural convection.

The calculation of the Nusselt number is given by:

= 0.55 ∙

(22)

Where: = the corrected Nusselt number = wind direction (°)

When there is strong wind (v≥0.5m/s), the process is called forced convection.

The calculation of the Nusselt number is given by:

= ∙ ( )

(23)

Where: B1 and n are constant depend on the value of Reynolds number and the roughness of conductor surface.

3.2.3. CalculationofradiationheatlossinCIGREmodelThe calculation of radiation heat loss in CIGRE model is given by:

= [( + 273) − ( + 273) ]

(24)

Where: = the conductor diameter (mm) = conductor emissivity = Stefan-Boltzmann constant = temperature of conductor surface () = ambient temperature ()

3.2.4. CalculationofsolarradiationrateinCIGREmodelThe calculation of solar heat in CIGRE model is given by:

= (25) Where: = solar heat rate(w)

= absorptivity of overhead line = ambient solar radiation rate (w/s∙m2) = overhead line’s diameter

When the model of the overhead line is definite, the absorptivity of overhead line and the overhead line’s diameter are fixed as the assumption of engineering requirement. The only


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variable, , is measured by radiation meters. The radiation includes solar radiation and other reflections. But the source of radiation is not important as an engineering calculation.

3.2.5. CalculationofJouleheatgaininCIGREmodelThe calculation of Joule heat for general (non-ferrous) overhead conductor is given by[28]:

= [1 + ( − 20)] (26)

Where: = joule heat rate(w) =skin effect factor = effect current (A) = DC resistance at 20 (Ω) = the temperature coefficient of resistance per degree =average conductor temperature () (The value is recommended to use 1.0123).

The AC resistance is calculated by:

= ∙ 1 + ( − 20) (27)

Where: = AC resistance (Ω) = skin effect factor = DC resistance (Ω) = temperature coefficient T = conductor temperature ()

Skin effect is the uneven distribution of current when AC current flow in conductor. Most AC current will come together on the surface of the conductor. It will influence the resistance of conductor when connecting with AC current. Skin effect factor is determined by the inherit nature of the conductor’s material and the geometric shape. The CIGRE model has recommended a fixed value for skin effect factor. For aluminium strand cable, the value of skin effect factor used in engineering is usually 1.0 [29]. The CIGRE recommended value for aluminium strand cable is more accurate than engineering level. DC resistance at 20 of conductor in CIGRE model recommends the value of IEC Publication 1089 [30].


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3.2.6. ConclusionoftheCIGREcalculationprocess

Figure 3-2: the process of CIGRE model calculation

Where: = joule heat rate(w) = magnetic heat rate(w) = solar heat rate(w) = corona heat rate(w) = convective heat loss rate(w) = radiative heat loss rate(w) = evaporative heat loss rate(w)

The process of CIGRE thermal rating model is presented by the tree chart.

3.3. IECmodelofthermalratingcalculation

3.3.1. ThermalequilibriumofOHLinIECmodelThe International Electro-technical Commission (IEC) model for OHL’s thermal rating calculation is published in IEC 61597 [31]. OHL’s heat balance is also the working principle of IEC’s thermal rating model. The thermal equilibrium can be described as follows:

+ = + (28)

Where: = the OHL’s Joule heat rate(w) = the heat generated by solar radiation rate(w) = the OHL’s convection heat loss rate(w) = the heat loss because of conductor’s radiation rate(w)


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IEC model considered the same factors with other two models. It just changes the name of different kinds of heats. The nature of each component is the same.

3.3.2. CalculationofconvectionheatlossinIECmodelThe unique condition that IEC thermal rating model considered is the forced heat loss, which will happen only when the wind speed is relatively high. So there is just one formula used to calculate the convection heat loss which is given as follow:

= ∙ ( − ) (29)

Where: = the convection heat loss rate(w) = the thermal conductivity of OHL’s ambient air (W/m∙ ) = the Nusselt number = the final equilibrium temperature () = ambient temperature ()

The thermal conductivity of OHL’s ambient air in IEC model is assumed fixed and the value is 0.02585 W/(m∙K).

The Nusselt number is calculated by formula as follow:

= 0.65 . + 0.23 . (30)

Where: Re = the Reynolds number

The value of Reynolds number is given by formula as follow:

= 1.644 × 10 [ + 0.5( − )] . (31)

Where: = the wind speed (m/s) D = the diameter of OHL (mm)

, has been defined above.

The equation (31), used to calculate the radiation heat loss, is simpler than the equations both in IEEE and CIGRE model because it uses the absolute temperature instead of Celsius temperature.

3.3.3. CalculationofradiationheatlossinIECmodelThe calculation of radiation heat loss in IEC model is given by:

= ( − ) (32)

Where: = the Stefan-Boltzmann constant = the conductor diameter (mm) = conductor emissivity coefficient , has been defined above. The value of Stefan-Boltzmann constant is 5.67× 10 /( ∙ ).

The conductor emissivity is assumed fixed and provided by the manufacturer. In recent years, there are some documents [32] pointed out that the OHL’s emissivity is varying with the


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weather condition rather than being constant. But these documents did not provide a specific method can be used in industry to calculate the value of emissivity.

3.3.4. CalculationofsolarradiationrateinIECmodelThe calculation of solar radiation heat is given as following:

= (33)

Where: = the heat generated by solar heat (w) = the absorption coefficient of solar radiation for OHL D = the conductor diameter (mm) = the solar radiation intensity (w/s∙m2)

The absorption coefficient of solar radiation for OHL is the nature of conductor which will be influenced by the roughness of conductor. The value is assumed fixed and provided by the manufacturer of conductor. The measurement of solar radiation intensity can use different methods include direct and indirect measuring method. Specific methods will be introduced in the DTR methods chapters.

3.3.5. CalculationofJouleheatgaininIECmodelJoule heat is the main heat source during OHL’s operation. The calculation of Joule heat in IEC model is given as follow:

= (34)

Where: = the Joule heat (w) = the AC resistance at temperature T (Ω) = the AC current (A)

The AC resistance of OHL is given as follow:

= [1 + ( − )] (35)

Where: = the AC resistance of conductor at temperature T2 (Ω) = the AC resistance of conductor at temperature T1 (Ω) = conductor’s electrical resistance coefficient at temperature T1

The temperature T1 is the base temperature. The value is usually 20, because this value is provided in IEC 1089 [30].

3.3.6. ConclusionoftheIECmodelcalculationprocessThe process of IEC thermal rating model can be clearly presented by the following tree chart.


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Figure 3-3: the process of IEC model calculation

Where: = the OHL’s Joule heat (w) = the heat generated by solar radiation (w) = the OHL’s convection heat loss (w) = the heat loss because of conductor’s radiation (w)

3.4. ComparisonfordifferentDTRcalculationmodels

3.4.1. DifferenceinthecalculationofconvectionheatlossIEEE model use three equations (2, 3, 4) to calculate the convection heat loss. It distinguished high, low and zero wind speed. When the wind speed is low, the natural heat convection loss still exists. Using different equation to describe zero wind condition is necessary.

CIGRE model use equation (20) to calculate the convection heat loss. But CIGRE use three equations (21, 22, 23) to explain the Nu. The three equations corresponded to zero, slow and high wind speed conditions. Comparing with IEEE equations, CIGRE calculation emphasizes the influence of conductor diameter as it is in third power instead of first power in IEEE.

IEC model uses equation (29) to calculate the convection heat loss. At the same time, IEC uses equation (30) to explain Nu. In IEC model, wind speed is the crucial element in the calculation of convection heat. It ignore the natural convection condition.

From the perspective of accuracy, the three models can be compared.

In the following condition:

Conductor temperature: 100 Ambient temperature: 40 Conductor outer diameter: 28.1mm Conductor inner diameter: 20.0mm OHL’s elevation: 100m Wind direction: 90°


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Figure 3-4: the comparison of convection heat loss in three models

When the wind speed is varying from 0 to 3 m/s with interval of 0.01m/s, the corresponding convection heat loss rate of OHL can be calculated. The result is illustrated in the Figure 3-4. The X-axis is wind speed; Y-axis is the corresponding convection heat loss in three DTR models.

When the wind speed is lower than 0.15m/s, the convection heat loss of IEEE model is not influenced by the wind speed. The reason is that IEEE model regards this level wind speed as natural convection which is not sensitive to wind speed. While CIGRE model’s convection heat loss rate will change suddenly at the point of 0.5 m/s wind speed. The reason is CIGRE set 0.5 m/s as the separation of high wind speed and low wind speed. There are different formulas for high and low wind speed. However, IEC model’s convection heat loss rate is the most continuous one because it only considers the forced convection heat loss. That is why IEC’s curve looks like a general function.

When the wind speed is higher than 0.5m/s, the three thermal rating models’ performance of convective heat loss rate get very close. But, if the wind speed is lower than 0.5m/s, the calculations of convective heat loss rates are relatively distinct. Apart from that, the curves represented of convective heat loss in IEC and CIGRE model both pass through the coordinate origin point. From the experience of reality, temperature difference will definitely result in convection heat transfer. So the real curve should not pass through the origin.

To compare the performance of the three models’ accuracy, there must be another model acknowledged by most researchers. At the moment, it is difficult to find a qualified DTR model which accepted as the optimal by most researchers. But, the working principles of these models can be used to provide the accuracy information of the models. The convection heat loss rate in IEEE model has clearly differentiated natural and forced convection. In the low speed field, IEC model do not take any action to simulate the low wind speed condition.


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CIGRE model considered the low wind speed condition, but the simulation’s performance does not improve a lot.

Secondly, from the perspective of computation’s complexity, the three models can be compared too. There are three formulas in IEEE models for the direct calculation of convection heat loss. There are other three formulas for the calculations for wind angle factor or other parameters. There is one formula for the calculation of convection heat loss in CIGRE model. But for the sake of low wind speed‘s condition, there are three other formulas used to calculate the parameter in CIGRE model. The calculation of air’s properties include viscosity, density and thermal conductivity are provide by linear equation with one unknown which is very simple when computed in computer. So the computation of CIGRE model’s convection heat loss is simpler than IEEE model. There is one formula for the calculation of convection heat loss in IEC model. There is also only one formula in the calculation of other parameters. What’s more, IEC model did not mention the concepts of air density, viscosity and thermal conductivity. It directly plugs these parameters in its final formula. So IEC model’s calculation for convective heat loss is the simplest among the three models.

From what has been discussed above, just judging by the convective heat loss, it’s easily to find that IEEE thermal rating model is the most accurate model but also the most complex method, IEC model is the least accurate model but also the simplest model, IEEE model undertakes the complexity of computation to get high accuracy of thermal rating. So, when the requirement of thermal rating’s accuracy is high and there are efficient computation tools, the IEEE thermal rating model is recommended. When the requirement of thermal rating’s accuracy is not high, and there are not very efficient computation tools to use, the IEC model is suitable to calculate the convection heat loss rate. Or when the requirement of accuracy is not very either not very low, CIGRE model is a medium level choice.

3.4.2. DifferenceinthecalculationofsolarradiationrateThe calculation of solar radiation rate is a notable difference for the three models stated above. IEEE model use equation (7) to calculate the solar heat gain. The parameters of this equation are given by equation (8, 9, 10, 11, 12, 13, 14, and 15). CIGRE model use equation (25) to calculate the solar heat gain for OHL. IEC model use equation (33) to calculate the solar heat gain for OHL.

Firstly, from the perspective of computation’s complexity, the three models can be compared. There are nine equations in IEEE’s solar radiation calculation in total. There is only one formula in the CIGRE and IEC method. IEEE method is more complex than CIGRE and IEC method in computation.

The IEEE model calculates the solar radiation rate according to the solar time. It takes full advantage of the geographical information of OHL. Solar time can be counted, and OHL’s geographical information is fixed. The result of IEEE’s solar radiation rate calculation is ideal condition for maximum solar radiation rate. It does not consider the weather condition and field condition. CIGRE and IEC model can provide the real value of solar radiation rate.


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They considered the read-time weather condition and other parameters. The physical meaning of solar radiation rate in IEEE model is different from CIGRE and IEC models.

3.4.3. DifferenceinthecalculationofACresistance(Jouleheat)AC resistance is conductor resistance influenced by skin effect. Skin effect is the uneven distribution of current density in AC circuit. The tendency in AC circuit is that more current will concentrate on the outer part which is close to conductor’s skin [33]. Skin effect decreases the conductor’s effective cross-section area which will result in the change of conductor resistance.

The skin effect is reflected in the AC resistance calculation by skin effect factor. The skin effect factor will be influenced by current frequency, conductor’s material and strands. In engineering, the skin effect factor usually uses a fixed value.

The calculation of AC resistance in IEEE model is given by equation (17). Because IEEE recommend the base AC resistance value in the “Alumimum Electrical Conductor Handbook” [27], the high temperature and low temperature are respectively 75 and 25.

The nature of IEEE model’s AC resistance calculation assumes that the AC resistance has a linear relation with temperature. Then, it solves the formula for AC resistance and temperature according to two points’ value. There is no mathematic evidence to prove the linear relation between AC resistance and temperature is right, but experiment shows that the error of IEEE model’s method is acceptable.

The calculation of AC resistance in CIGRE model is given by equation (27). CIGRE’s formula for AC resistance has clear electrical meaning. It can be used as the standard formula of AC resistance calculation. It considers skin effect and temperature effect. Because the base DC resistance’s temperature is 20, the value in bracket is 20.

The calculation of DC resistance in IEC model is given by equation (35). IEC does not provide method to calculate the AC resistance. Only the DC resistance is calculated by the equation above.

The comparison of AC resistance calculation will focus on IEEE model and CIGRE model.


Conductor: peachbell (all aluminium concentric-lay class AA) Resistance: DC 20 3.481 ohm/mile AC 25 3.551 ohm/mile AC 75 4.255 ohm/mile Temperature coefficient of resistance (aluminium) at 20 : 0.00429 IEEE model AC resistance calculation:

= 4.255 − 3.55175 − 25 = 0.0141


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= 3.551 + 0.0141 × ( − 20) CIGRE model AC resistance calculation: = ∙ 1 + ( − 20) = 1.0123 × 3.481 × 1 + 0.00429 × ( − 20) = 3.5228 + 0.0151 × ( − 20) The value of is recommend to use 1.0123 by CIGRE.

The result is shown in Figure 3-5. The X-axis is conductor temperature; Y-axis is the corresponding AC resistance rate in IEEE and CIGRE DTR models. The numeric result for the calculation can be found in appendix. From the numeric result table, it can be found that the largest error is 0.6447%. The average error is 0.0246%. This error level could meet the requirement of engineering.

Figure 3-5: the comparison of IEEE and CIGRE AC resistance calculation

Figure 3-5 shows that the difference of the results between IEEE and CIGRE model is very small. For detailed, the difference tends to the most obvious at the highest and lowest temperature and becomes zero at the middle point. From the perspective of computation’s complexity, IEEE model’s AC resistance is simpler than CIGRE’s standard.

3.4.4. DifferenceinthecalculationofradiationheatIEEE model use equation (6) to calculate the radiation heat loss. CIGRE model use equation (24) to calculate the radiation heat loss. IEC model use equation (32) to calculate the radiation heat loss.


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For the calculation of radiation heat loss, three models all use one formula. Because of the similar format of the equations in three models, the components in each equation are the same; so it can be compared one by one.

Firstly, the fourth power item in these equations bracket is the most complex one. The temperature value in this item plus 273 at first step, then divide 100. In CIGRE’s model, the fourth power item plus 273 but does not divide anything. In the IEC model, the temperature in the four power item do not plus or divide any number.

From the format of three equations, IEC model is the simplest; CIGRE model second; and IEEE model least. The equations of IEEE and CIGRE model plus 273 because that the temperature value is Celsius temperature. The IEEE and CIGRE model contains the transformation of Celsius temperature and absolute temperature. As most temperature measurement systems are not absolute temperature, the IEC model also needs to transform these two temperatures, but the process is invisible.

Analysing the items outside the bracket in equation (6, 24 and 32), CIGRE model and IEC model are the same. There are two constant items outside of the bracket. IEEE model plug the constants’ values in the equation. Because the value of Stefan-Boltzmann constant is 5.67× 10 , IEEE model plug the order of magnitude into the bracket.

In conclusion, the natures of the three models’ radiation heat loss calculation are the same. From the format of equation, IEC model is the simplest. While judging by the calculation’s complexity, IEEE model is the simplest.

3.4.5. CalculationdifferenceforthreethermalratingmodelsWhen the convection heat loss rate, radiation heat loss rate, solar radiation heat gain rate and AC resistances are given, the calculations of maximum current are the same in IEEE, CIGRE and IEC thermal rating models.

The calculation is given as follow:

= + −

(36)

Where: = the maximum allowable current (A) = convection heat loss rate (w) = radiation heat loss rate (w) = solar heat gain rate (w) = AC resistance rate (Ω)


Conductor maximum temperature: 100 Ambient temperature: 40 Conductor outer diameter: 28.1mm Conductor inner diameter: 20.0mm


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OHL’s elevation: 100m Wind direction: 90° Emissivity: 0.5 OHL’s solar absorptivity: 0.5 Solar radiation rate: 1000 W/m2 Conductor’s DC resistance rate at 20: 6.87× 10 (Ω/m) Conductor’s AC resistance rate at 25: 7.10× 10 (Ω/m) Conductor’s AC resistance rate at 75: 8.47× 10 (Ω/m)

When the wind speed varies from 0 to 3 m/s with interval of 0.01m/s, the corresponding thermal rating of OHL can be calculated using IEEE, CIGRE and IEC model respectively. The result is present in the Figure 3-6. The X-axis is wind speed; Y-axis is the corresponding thermal rating in three DTR models.

Convection heat loss is the key factor because it accounting for 70% of the heat loss in in general condition. However, wind speed is the key factor in the calculation of convection heat loss rate, so wind speed is the most important parameter compare with the results of different models. Fortunately, it is able to make the difference among the three models much clear by using wind speed as the parameter.

In IEEE model, when the wind speed is lower than 0.15m/s, the maximum current is not influenced by wind speed any more from Figure 3-6. The reason is that IEEE model regard this level wind speed as natural convection which is not sensitive to wind speed. CIGRE model’s maximum current will change suddenly at the point of 0.5 m/s wind speed. The reason is CIGRE set 0.5 m/s as the separation of high wind speed and low wind speed when calculating the convection heat loss. There are different formulas for high and low wind speed. IEC model’s thermal rating curve is the smoothest because it only considers the forced convection heat loss. That is why IEC’s curve looks like a smooth curve. Apart from this, the value of thermal rating will be very low at zero wind speed condition. The reason is that the convection heat is zero at no wind condition which is not the truth. The data used in Figure 3-6 have been double checked. The Matlab program used to calculate all the data is shown in appendix.


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When the wind speed is higher than 0.5m/s, the three thermal rating models’ results get very close. But, when the wind speed is lower than 0.5m/s, the calculations of thermal rating are relatively distinct. IEC and CIGRE model’s thermal rating curves both decrease dramatically when the wind speed is low. From the experience of reality, temperature difference will definitely result in convection heat transfer. So the real curve should not decrease too much at low wind speed.

Figure 3-6: the comparison of thermal rating calculation of three

models

Generally speaking, the thermal rating calculation in IEEE model is closest to the real thermal rating curve. In the low speed field, IEC model does not take any action to simulate the low wind speed condition. CIGRE model considers the low wind speed condition, but the simulation’s performance does not improve a lot.

However, the solar radiation rate calculation in IEEE model is not the real solar rate. The solar calculation method in IEEE has not been employed in any dynamic thermal rating systems, and would not be used in the DTR calculation Matlab code in the appendix either.

In general, IEEE thermal rating model is the most accurate model also the most complex method. IEC model is the least accurate model but also the simplest model. It means that IEEE model use the complexity of computation to get high accuracy of thermal rating. So, when the requirement of thermal rating’s accuracy is high and there are efficient computation tools, the IEEE thermal rating model is recommended. When the requirement of thermal rating’s accuracy is not high, and there are not very efficient computation tools to use, the IEC model can be used.

The difference of the three DTR models can be shown by Table 8:

Table 8: comparison of different DTR models

Item IEEE CIGRE IEC Amount of formula 8 1 1 Amount of sub-formula 0 3 0 Accuracy High Medium Low

3.5. MainconclusionsThis chapter introduced and compared three models of DTR calculation. The IEEE method has a maximum value difference with the IEC method of 420A when the wind is zero m/s. However, the error is in the range of 5%-8% for the wind values of 0.5m/s-3.0m/s. The IEEE method has a maximum value difference with the CIGRE method of 430A when the wind is zero m/s. However, the error is in the range of 8%-10% for the wind values of 0.5m/s-3.0m/s.

Dynamic thermal rating monitoring systems

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4. Dynamicthermalratingmonitoringsystems

4.1. ThemeasuredparametersofDTRmonitoringsystemThe thermal rating of overhead line can be calculated by conductor conditions (conductor temperature and current) and ambient weather conditions (ambient temperature, wind speed, wind direction and solar radiation).

4.1.1. TechnicalrequirementofDTRmonitoringsystemThe technical requirements for DTR monitoring system include sensitivity, accuracy and calibration are shown as follows[7]:

Ambient temperature measurement’s minimum accuracy is 1. Wind speed measurement’s maximum start speed is 0.5 m/s. Net radiation temperature’s maximum calibration accuracy is 4. Net radiation temperature’s maximum accuracy is 1. Sag measurement’s maximum resolution is 20 mm.

4.1.2. DTRsystemsMany kinds of DTR monitoring systems have been researched. These systems include Power Donut system [34], Darmstadt University system [35], SMT system [36], TMT system [37], RIBE-Ritherm system [38], CAT-1 system [39], Arizona differential GPS system [40], Video Sagometer system [41], Ampacimon system [42] and conductor replica system [43]. These DTR systems have been reviewed, and finally three of them have been selected for this report based on the criteria below.

(1) Has the DTR system been employed in field; e.g. is there any company using the DTR system on their overhead lines? Some systems may have a good performance in laboratory but fail a lot in field. This phenomenon is frequently happening in the transformation between research and product.

(2) Did the DTR system’s manufacturer provide the technical details? It is very common that a company will hide all the details of their products to safeguard their interest. This is absolutely legal but also result in that the research of the DTR system become impossible.

(3) Did the DTR system employ new technology? Generally, only one DTR system with similar technology will be chosen.

The currently available DTR systems utilize the working principles summarised in Table 9 [6].


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Table 9: the working principles used in DTR monitoring systems

Working principle Used range Number of required devices

Location of installation

History of usage1

Weather station Globally widely used

One device for one area

Close but off the OHL

33 years (approximate)

Tension measurement

Widely used in the USA

One device for one section

On the OHL, between insulator and tower’s arm


Conductor surface temperature measurement

Not widely used One device for one span

On the surface of the line


Distributed conductor temperature measurement

Not widely used Fibre sensor should as long as the OHL

Very close the line but not touched in most case


Conductor tracking Mostly used in the USA

One device for one span

On the lowest point of the line


Vibration measurement

Pilot study One device for one span

On the line 5 years (approximate)

From Table 9, it can be found that weather station is the oldest technology used in DTR measurement. As the research of meteorology is much older than DTR, weather station can be employed on DTR system as soon as it emerges. The example of tension measurement is CAT-1 system. Conductor surface temperature’s example is Power Donut system. Both of them have been employed in field more than 20 years. Distributed temperature measurement’s example is fibre optic sensors. This technique uses a long fibre sensor tied with the OHL to get the temperature distribution of OHL. There are advantages for distributed temperature measurement. It is able to measure the total temperature of OHL which is very useful in DTR system. But the maintenance of the sensor is a problem. Along with this, there is no specific data can be found to analyse the performance of this technique. So it is not chosen to introduce in this report.

Conductor tracking’s examples include differential GPS [44] and laser distance measurement. These techniques have been designed and tested in universities. They have not entered the DTR market yet, due to inaccuracy of the GPS technology and accumulative error. GPS can provide the information of altitude using distance which is calculated by light speed and running time. The error of clock in GPS is the main source of this system’s error.

Vibration measurement’s example is Ampacimon. Ampacimon system’s history is not long, but the product has already entered the market and there are technical details available. Apart from this, the working principle of Ampacimon is totally different from the other methods. Using acoustics method to measure conductor temperature is an original approach. 1 The time of history is counted in 2013


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Three different DTR systems are discussed and compared in the next chapters. Those systems are CAT-1 system, Power Donut system and Ampacimon system. Because the working principles of these systems are totally different, the comparison is able to provide more information. This thesis will detailed illustrate the characteristics of these DTR systems and compare the performances of them.

4.2. CAT‐1DTRMonitoringSystem

4.2.1. GeneralinformationofCAT‐1DTRsystemCAT-1 dynamic thermal rating monitoring system is a typical example of DTR system using conductor tension monitoring method. Tension monitoring DTR systems has been employed in the field approximate 23 years [6]. In the early time of CAT-1 system, it was mainly used in USA. Now, it is also used in some other countries, for example, National Grids (UK) uses CAT-1 on some of their OHLs.

For a normal DTR measurement system, the five parameters of DTR measurement can be divided into two categories: conductor conditions (temperature and current) and ambient weather condition. The two categories will be introduced separately in all the DTR systems. For CAT-1 system, the solar radiation will be separately introduced, because it employs method other than normal weather station. Wunderlich[45] has introduced the method of Nano-crystal analysis to monitor OHL’s DTR. Douglass[46] has describe the method of SMT system.

4.2.2. WorkingprincipleofconductortemperaturemeasurementinCAT‐1

4.2.2.1. RelationbetweenconductortensionandhorizontaltensionThe component used to measure conductor temperature in CAT-1 system is load cell. It is installed between the transmission tower and the tension insulator and measures conductor tension. The load cell measures the tension between insulator and overhead line. This tension is not directly related to the average temperature of the conductor since only the horizontal component is used for the sag calculation [47].

In order to derive the relation between conductor tension and horizontal component of the tension, the force balance of load cell which connect the insulator and the line can be analysed. According to the classical mechanics, the force at the point of load cell should equal the composition of horizontal force and vertical force.

The composition force is the measured value by load cell. The Horizontal force need to be calculated. The Vertical force is the half weight of overhead line in one span. Then, according to the geometric character of force’s composition, the relation between composition force and horizontal force could be calculated. The equation is given as:

= − 2

(37)

Where: = the span’s weight (N)


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Figure 4-1[48]: force balance of load cell

Figure 4-1 gives the shape of one span’s OHL which is a catenary in mathematic. This is for making the force analysis clear. In the vertical direction, conductor was affected by gravity and the supported force of tower. As conductor is still in vertical direction, the value of conductor’s gravity equals the supported force’s vertical decomposition.

4.2.2.2. RelationbetweenhorizontaltensionandsagThe relation of conductor horizontal tension and overhead line’s sag is given by [49]:

= ∙8 (38)

Where: S= the conductor sag (m) W= the conductor weight (N/m) L= the span length (m) H= horizontal tensile force (N)

The derivation of equation above is given as follows[9]:

For the overhead line’s shape is catenary, the overhead line obeys the equation of a catenary which is given by:

= ∙ ℎ (39)

Where: a= the catenary constant

The mathematic drawing of catenary is shown in Figure 4-2. The following deviation is based on this figure.

The original point in Figure 4-2 is the mathematic original point instead of the geometric ground. The solid line in Figure 4-2 is the shape of overhead line. D is sag, and S is span length. As catenary can exactly describe the shape of overhead line, the real line was abstracted out in Figure 4-2. Because the overhead line’s shape is catenary, plugging B (half

of span length) in equation (51). And it results in: = ∙ cosh( ) And S(sag) = − = ℎ − 1


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According to Taylor’s series for cosh: ℎ = 1 + 12! + 14! +∙∙∙∙∙∙

As the requirement of engineering, 2 terms are enough for the cosh formula.

So, = 1 + ! − 1

And, B= half of span length =

Where: W= the conductor density(N/m) T= tensile force (N) w=conductor weight (N)

Figure 4-2[50]: the catenary model of overhead line

: = ∙8

When plug in the equation (38), the relation between sag and tension is:

= 8 − 2 (40)

The equation (40) is the final equation between sag and tension. This equation is widely used in the calculation of sag, because many other parameters could be related to the conductor tension. The equation is derived by Taylor’s series which is used for approximate calculation, so the equation is not exactly accurate. But the accuracy is enough for engineering level.

4.2.2.3. RelationbetweensagandconductortemperatureSag of overhead line is the result of conductor’s thermal expansion. So the magnitude of sag is directly related to the conductor’s temperature.

The relation between sag and conductor temperature is different from the relation of tension and sag. There is no strict mathematic equation to describe sag and conductor temperature’s relation, because the relation is not based on pure mathematic and physic laws. The performance of conductor’s thermal expansion varies with the material. Even for the same kind of materials, the ability of conductor’s expansion is not exactly the same, because other factors, like diameter and temperature will also influence the expansion performance [17].

The models available to calculate temperature by sag are experimental formulas. Because of the complexity of kinds of materials and conditions, the formats of models are distinct each other which are available in [51-53].

According to the requirement of power transmission’s requirement, this report chooses a method named “state change equation” to calculate conductor temperature by sag.


- 40 -

The equation is given by:

83 − 8 − =

(41)

Where: S= overhead line’s sag (m) L= span length (m) W= conductor weight per meter (N/m) E=Young’s modulus = conductor cross section (m2) = thermal expansion coefficient = conductor temperature () C= state constant

The constant of equation above could only be determined by a known state. The constant is not the same in different measurements.

The derivation is given as follow [54]:

For a specific span of overhead line, the horizontal tensile force has a unique value corresponding to the conductor mean temperature. As the horizontal tensile force is linear to the sag, so sag has a similar relation with conductor temperature.

As the mean conductor temperature is difficult to measure directly, so this relation stated above became an important method for the measurement of conductor mean temperature.

The length of overhead line will expand because of thermal and elastic expansion. The expansion of length could be given by:

= (1 + )(1 + ) (42)

Where: =the length of overhead line at temperature (m) = the length of overhead line at temperature (m) = thermal strain = elastic strain

The calculation of thermal strain and elastic strain are given by:

= ∙ ∆ = ∆∙

Where: =coefficient of thermal expansion =conductor tensile force =modulus of elasticity =conductor cross section (m2)

Plug in the equation of thermal strain and elastic strain, it obtained that:

= (1 + ∙ ∆ ) 1 + ∆∙ (43)


- 41 -

As the requirement of engineering, the overhead line could be regarded as parabola. The equation of parabola yields that:

= + ( )24 (44)

= + ( )24 (45)

Where: H=the horizontal tensile force (N) =the span length (m) = the mass of conductor per meter (N/m)

In engineering, the horizontal tension is much larger than the vertical tension. With acceptable accuracy, ∆ is able to replace ∆ .

When plug equations (44, 45) in previous L2 equation, the result is:

+ ( )24 = + ( )24 (1 + ∙ ∆ ) 1 + ∆∙ (46)

Neglecting the items which are two orders of magnitude lower than the others, the state change equation is obtained:

( )24 − − = ( )24 − − (47)

In engineering field, the state of one side is a known state, so one side of equation (47) is a fixed value. Considering this condition, equation above could be simplified as:

( )24 − − = (48)

Equation (48) is practical when used in engineering measurement. It could be used to calculate the mean conductor temperature by horizontal tensile force.

As sag has a linear relation with horizontal tensile force, it could also be used in equation (49).

For = ∗ , sag is able to replace H in equation (49):

83 − 8 − = (49)

Equation (49) could be used in the calculation of conductor mean temperature by sag.

4.2.2.4. TheworkingprincipleofconductortemperaturemeasurementThe process of conductor temperature’s measurement as shown as Figure 4-3:

Figure 4-3: principle of CAT-1 system's conductor temperature measurement


- 42 -

Figure 4-3 shows the complete process of conductor temperature measurement of CAT-1 system. The conductor tension was measured firstly. Then, the horizontal tension can be calculated. Next, overhead line’s sag can be worked out. Finally, the conductor temperature was able to be known.

4.2.3. Loadcell

4.2.3.1. InstallationofloadcellLoad cell is consisted by sensor and wire. There are two connection holes and long wire in load cell. The two holes can be used to connect insulator and transmission tower’s arm. The sensor has two arms used to connect the insulator of overhead line and the arm of transmission tower. The wire is used to provide power supply to the sensor and transfer signal from sensor to main unit.

There are two load cells symmetrically located on two arms of one transmission tower. In fact, the two load cells measure one section’s tension respectively. The two load cells’ measurement will transfer to the same main panel located on the transmission tower. The exact positioning of the load cell is shown in Figure 4-4 with highlighted squares. It additionally illustrates the wiring of the load cell to transmit the data to the central unit.

Figure 4-4[55]: the load cell installed on overhead line

4.2.3.2. SpecificationofloadcellThere are three different measurement ranges for load cell’s tension measurement: 22.25 to 33.78 KN; 44.50 to 66.75 KN; 111.25 to 166.88 KN. When used in CAT-1 system, the resolution of load cell is 0.05% of the full scale. For example: when the scale is 22.25 to 33.78 KN, the resolution is 5.7 N. Zero balance of load cell is 1.0% of full cell. So calibration of the zero point is necessary before using the sensor. When used in CAT-1 system, the error of load cell is 0.05% of the full scale. This value is same as resolution’s value.

4.2.3.3. SensitivityofloadcellThe sensitivity of load cell means the change of final output (ampacity) compared to the original input (conductor tension).


- 43 -

Most part of the process of calculation from conductor tension to ampacity has been presented in Chapter 4.2.2. The whole process is shown in Figure 4-5. When the real-time conductor temperature is known, the dynamic thermal rating can be calculated.

Figure 4-5: the complete process of DTR calculation in CAT-1 system

The sensitivity of load cell depends on the weather and operating condition of overhead line, so a specific sensitivity must be based on a specific condition.

Assume that: The conductor is ACSR Cuckoo/AW Conductor diameter is 28.1 mm Weight of conductor is 14.308 N/m Span length: 200m Span weight: 3000N Young’s modulus of the conductor is 57 Gpa Thermal coefficient of the conductor is 10-6 Wind speed is 0.61 m/s perpendicular to the conductor Emissivity is 0.5 Solar absorptivity is 0.5 Ambient temperature is 40 Conductor ac resistance is: (25) = 7.10 × 10 (75) = 8.47 × 10 Solar radiation rate is 1000 W/m2 Average conductor elevation 100m Conductor cross area is 4 × 10 m2

There are three ranges of CAT-1’s measurement result. In this example, the first range 22.25 – 33.78 KN was chosen. So the resolution of load cell at this range is 5.8 N (0.05% FS). The corresponding thermal rating and intermediate variables are shown in Table 10.


- 44 -

Table 10: the sensitivity of load cell in CAT-1 system

Tension (N)

horizontal tension(N)

Sag (m)

conductor temperature

(C) DTR (A)

DTR

change(A)

30000.0 29962.5 2.3877 89.072 902.763 0.07614

30005.8 29968.3 2.3872 89.064 902.687 0.07612

30011.6 29974.1 2.3867 89.057 902.611 0.07609

30017.4 29979.9 2.3863 89.049 902.534 0.07606

30023.2 29985.7 2.3858 89.042 902.458 0.07603

30029.0 29991.5 2.3853 89.034 902.382 0.07600

30034.8 29997.3 2.3849 89.027 902.306 0.07597

30040.6 30003.1 2.3844 89.019 902.230 0.07594

30046.4 30008.9 2.3840 89.012 902.154 0.07591

30052.2 30014.7 2.3835 89.004 902.079 0.07588

30058.0 30020.5 2.3830 88.997 902.003 0.07585

30063.8 30026.4 2.3826 88.989 901.927 0.07582

30069.6 30032.2 2.3821 88.982 901.851 0.07580

30075.4 30038.0 2.3817 88.974 901.775 0.07577

30081.2 30043.8 2.3812 88.967 901.699 0.07574

30087.0 30049.6 2.3807 88.959 901.624 0.07571

30092.8 30055.4 2.3803 88.952 901.548 0.07568

30098.6 30061.2 2.3798 88.944 901.472 0.07565

30104.4 30067.0 2.3794 88.937 901.397 0.07562

30110.2 30072.8 2.3789 88.929 901.321 0.07559

30116.0 30078.6 2.3784 88.922 901.245 0.07556

30121.8 30084.4 2.3780 88.915 901.170 0.07554

30127.6 30090.2 2.3775 88.907 901.094 0.07551

In Table 10, the original tension output (in the first column) was set from 30000.0 to 30139.2

N, because the sensitivity of load cell at this measurement range is 5.8 N. The interval of

tension is for the purpose to show the corresponding maximum current. From the last column of Table 10, it can be found that the minimum detectable current change of load cell is less than 0.1 ampere which is very accurate in power transmission industry. All the data in Table 10 was calculated by the method introduced in chapter 4.2.2.

4.2.3.4. SystematicerrorofloadcellThe conductor temperature measured by CAT-1 is the average temperature. But overhead lines’ thermal rating is related with the highest conductor temperature. Considering the case of 0.61m average wind speed with 90 degree angle, 40 ambient temperature and span length of 200m, the difference between average conductor temperature and highest conductor temperature is shown in Figure 4-6. The horizontal axis illustrates the tension in newton and the vertical axis is the conductor temperature. The dotted line is the average conductor temperature derived from the conductor tension while the dashed line is the highest conductor temperature derived using wind speed difference.


- 45 -

Figure 4-6: difference between average and highest conductor temperature

Wind speed varies with height. This leads to temperature difference in different points. In Figure 4-6, the highest conductor temperature is approximately 2 higher than average value. This should be considered in error analysis of CAT-1 measurement.

4.2.4. WorkingprincipleofsolarradiationratemeasurementinCAT‐1systemCAT-1 system employs a component called net radiation sensor (NRS) to measure the solar radiation rate[56]. The net radiation sensor is a conductor replica for the overhead line in the same ambient conditions. The fundamental principle of net radiation sensor is heat balance.

The heat balance in net radiation sensor is given by:

= + (50) Where: = solar heat (w)

= convective heat loss (w) = radiative heat loss (w)

The net radiation sensor is able to measure the temperature of its replica conductor. One real CAT-1’s net radiation sensor was shown in Figure 4-7. The conductor temperature is measured by the thermal sensor directly. In order to have an accurate measurement the sensor should point to the same direction as the conductor. Therefore, the solar incident angle top the conductor is identical to the one on the sensor.


- 46 -

Figure 4-7[55]: net radiation sensor installed on power transmission tower

The convection heat loss of the conductor replica is calculated by equation (20 and 24)

The ambient temperature is provided by ambient temperature sensor. So the solar radiation rate is able to be calculated by equations (24). The inputs of Net Radiation sensor include the diameter of sensor, the conductor emissivity of the sensor, sensor surface temperature and ambient temperature. The output is the radiation heat rate. As the sensor is located close to overhead line, the sensor’s radiation rate can be regarded as overhead line’s solar radiation heat rate.

4.2.5. Netradiationsensor

4.2.5.1. CharacteristicsofsolarradiationmeasurementbyNRSNet radiation sensor uses the same material and diameter of the conductor installed on OHL. Furthermore, the direction of the NRS should be in parallel to conductor as the solar radiation changes with time and direction of the line. Typical net radiation sensor output used with DTR instrumentation is temperature. The measured range is -40°C to 60°C with interval of

0.1°C [39]. In principle the NRS is a small section of a rod that has equivalent material and

geometrical properties with the conductor and therefore it absorbs the same solar energy as the conductor used. Consequently, NRS have to be selected based on the conductor size used to monitor. NRS uses Semiconductor Temperature Sensor to measure the surface temperature of it. The advantage for Semiconductor Temperature Sensor is they provide accurate result. The disadvantage is their result is analogue which cannot be understood by computer. By comparing the NRS with atypical solar radiation instrument used in regular weather stations, there are several advantages of using NRS:

The technique of temperature measurement is more mature than light intensity measurement. The output of NRS is more accurate than regular light intensity sensor.

NRS is cheaper than weather station’s light intensity sensor.


- 47 -

The disadvantages of NRS measurements utilisation instead of using the solar radiation measurement by regular nearby weather station are:

The increased complexity of the CAT-1 monitoring system since NRS is an independent component. It requires additional installation and design.

The miss-positioning of NRS’s conductor will result in additional error and therefore increases the complexity of its installation. This type of error is specifically analysed in next part.

4.2.5.2. EffectofNRSsensitivityonthermalratingcalculationConsidering an OHL system with conductor having 28.1 mm diameter and NSR sensor with 25 mm diameter and 700 mm length the effect of NRS sensitivity will be negligible. Weather conditions were assumed as follows.

Assume that: OHL’s diameter is 28.1mm NRS’s diameter is 25.0mm NRS’s length is 0.7m Ambient temperature is 40 Maximum conductor temperature is 100 Wind speed is 0.61m/s Wind direction is 90° Elevation is 100m Emissivity is 0.5 Conductor ac resistance is: (25) = 7.10 × 10 (75) = 8.47 × 10

Figure 4-8 illustrates how the solar power rate affects NRS temperature. The X-axis is NRS temperature; Y-axis is the corresponding solar radiation heat. Figure 4-9 indicates the relation of NRS temperature with the current. The X-axis is NRS temperature; Y-axis is the corresponding maximum current. The numeric data can be found in appendix.

From Figure 4-8 and Figure 4-9, it can be seen that there is a linear relationship between solar radiation rate and current with NRS temperature. The minimum change are 0.26 (W/m2) and 1.40 A for solar radiation and current respectively.


- 48 -

Figure 4-8: the relation between NRS temperature and solar

rate Figure 4-9: the relation between NRS temperature and maximum

current

Another source of error for the NRS is the improper alignment of the sensor with the conductor axis. When the radiation sensor and overhead line are not parallel, the measured solar radiation rate is different from the real rate [57].

Figure 4-10 illustrates the error that is introduced due to misplacement of the NRS for different times of the day. The X-axis is angle between NRS and OHL; Y-axis is the corresponding relative systematic error. The OHL assumed to be located at 30º latitude and the measurements considered to be taken on the 161st day of the year. The error is increasing with the increase of the angle between NRS and OHL as it was expected. However, the error is reduced when at noon and increases at early and late times of the day.

The table of error due to angle between NRS and OHL (Table 23) is given in the appendix. The time in the first line of NRS angle error table is the solar time instead of clock time. It should be noted that in solar time, AM 12:00 is equal to 90 degree of solar angle. The altitude of sun (Hc), solar declination (δ) and azimuth of sun (ZC) could be calculated according to the solar time. The value of error is calculated by the difference between measured value and real value over the real value.

= 23.4583sin[284 +365 360] = arcsin[cos( ) cos( ) cos( ) + sin( ) ( )] = + ( )


- 49 -

Figure 4-10: the error because of angle between NRS and OHL

4.2.6. AmbienttemperaturesensorAmbient temperature is a critical parameter in the measurement of dynamic thermal rating. The values of viscosity, air density and thermal conductivity all depend on ambient temperature. So the value of ambient temperature could influence the accuracy of result to a great extent. CAT-1 system uses temperature sensor to measure ambient temperature.

4.2.6.1. SpecificationofambienttemperaturesensorCAT-1’s ambient temperature sensor is enclosed in the main unit which is located on the transmission tower. The height of main unit is approximately the same level with the lowest arm of the tower.

The main unit is a synthesis of signal processing unit and measurement unit. It can also transfer digital signal to remote CAT-PAC unit via radio.

Measured temperature range of ambient temperature sensor is -40 to 60 [39]. The resolution of ambient temperature sensor is 0.1.

4.2.6.2. SensitivityofCAT‐1ambienttemperaturesensorThe output of ambient temperature sensor will directly influence the value of ambient temperature. Ambient temperature’s change would influence most parameters’ values.

Assume that: OHL’s diameter is 28.1mm NRS’s diameter is 25.0mm NRS’s length is 0.7m Maximum conductor temperature is 100 Wind speed is 0.61m/s Wind direction is 90° Elevation is 100m Emissivity is 0.5 AC-resistance: (25) = 7.10 × 10 (75) = 8.47 × 10


- 50 -

When the ambient temperature is varying from 20 to 40 with interval of 0.1, the corresponding thermal rating was calculated. The numeric table for the sensitivity of ambient temperature sensor can be found in appendix. The information in the table was shown in Figure 4-11. The X-axis is ambient temperature and Y-axis is corresponding maximum current. From Figure 4-11, it can be found that the maximum current is decreasing with the increasing of ambient temperature. The reason for this is that the increasing of ambient temperature cut the convective heat loss of overhead line.

When the ambient temperature increase 0.1 , the OHL’s maximum current will decrease less than 0.9A. This accuracy could meet the requirement of most engineering condition.

Figure 4-11: ambient temperature and maximum current

4.2.7. AnemometerThe CAT-1 system takes advantage of an anemometer to measure the wind information includes wind speed and direction [58].

4.2.7.1. SpecificationsofanemometerWind direction Range: 0 to 360° Resolution: 1° Accuracy: ±3° Wind speed Range: 0.5 to 89.0 m/s Resolution: 0.1 m/s Accuracy: 5%


- 51 -

4.3. PowerDonutDTRMonitoringSystem

4.3.1. IntroductiontopowerdonutPower donut is another dynamic thermal rating system. It includes donut sensor, weather station and operating software [59]. The donut sensor is able to directly measure overhead line’s current (A), line-ground voltage (kV), conductor temperature (degree C) and conductor inclination. The sensor is powered by the E-H field of the line. The threshold current for the operation of power donut sensor is 50 A.

The weather station is able to directly measure ambient temperature (degree C), wind speed (ft/second), wind direction (deg), solar radiation (w/m2) and internal battery voltage (V). Pytlak[60] has introduced the method of RIBE-Ritherm to monitor OHL’s DTR. Henke[61] has describe the method of TMT system.

4.3.2. WorkingprincipleofconductortemperaturemeasurementinPD2systemThere are two ways in Power Donut system that can be used to measure the conductor temperature. One is directly conductor surface temperature measurement. There are temperature sensors located at the place where Power Donut contacts OHL’s surface.

The other way is inclination measurement. There are inclination sensors in Power Donut system.

As the shape of OHL is catenary, the mathematic description of OHL’s shape can be given by:

= × ℎ (51)

where: y= the height of the point on catenary line (m) = the horizontal coordinate value of point on catenary line = the catenary constant

In overhead line, the calculation of catenary constant is given by:

= ℎ

(52)

where: ℎ = the horizontal tension of OHL (N) = the weight of OHL per meter in Newton

Differentiating the equation of catenary: = sinh

Assuming the inclination is θ, tangent of inclination is the differentiation: sinh =

Plugging in all the parameters:

ℎ = ∙ℎ ( ) (53)


- 52 -

The deviation from horizontal tension to conductor temperature has been specifically introduced in CAT-1 system chapter.

4.3.3. PowerDonutsensor

4.3.3.1. SpecificationofPowerDonutsensorThe outputs of Power Donut sensor include overhead line’s current (ampere), line-ground voltage (kV), conductor temperature (degree C) and conductor inclination [59].

The dimensions of Power Donut sensor are 32*32*14 cm. The range of Power Donut sensor’s measured current is 0 to 3000 ampere. Accuracy of current is 0.5% of the instrument’s reading value. Resolution of current is 0.1 ampere. There are two ranges for Power Donut sensor’s measured conductor temperature. Conductor temperature range: -50 to 150 (standard temperature) and -50 to 250 (high temperature). Accuracy of conductor temperature is 0.5% of the instrument’s reading value. Resolution of conductor temperature is 0.1. The range of Power Donut sensor’s measured inclination is -15 deg tilt to +15 deg tilt. Accuracy of inclination is 0.025 deg. Resolution of inclination is 0.025 deg. The range of Power Donut sensor’s measured voltage is -1 to 500 kV. Accuracy of voltage needs field scaling. Resolution of voltage is 1 kV. The operational temperature of Power Donut ranges from -40 to 60.

4.3.3.2. SensitivityofPowerDonutsensorBecause Power Donut is able to measure the conductor temperature via two ways: surface temperature measurement and tilt sensor measurement. The sensitivities of the two ways will be both introduced in this chapter. The error because of sensor’s position will be discussed later.

Assume that: The conductor is ACSR Cuckoo/AW Conductor diameter is 28.1 mm Weight of conductor is 14.308 N/m Span length: 200m Span weight: 3000N Young’s modulus of the conductor is 57 Gpa Thermal coefficient of the conductor is 10-6 Wind speed is 0.61 m/s perpendicular to the conductor Emissivity is 0.5 Solar absorptivity is 0.5 Ambient temperature is 20 Conductor ac resistance is: (25) = 7.10 × 10 (75) =8.47 × 10 Solar radiation rate is 1000 W/m2 Average conductor elevation 100m Conductor cross area is 4 × 10 m2 The distance between PD sensor and closest tower is 10m.


- 53 -

The sensitivity of surface temperature sensor is shown in the Figure 4-12. When the conductor temperature increases 0.1 , the maximum current OHL will increase approximately 1.0 – 1.5 A.

In Figure 4-12, The X-axis is conductor temperature; Y-axis is the corresponding thermal rating. It can be found that the relation between conductor temperature and OHL’s maximum current is almost linear and monotonically increasing.

The tilt sensor is able to measure the conductor temperature too. The sensitivity of tilt sensor is shown in the Figure 4-13. The X-axis is tilt angle; Y-axis is the corresponding thermal rating.

The relation between tilt angle and OHL’s maximum current is almost linear and monotonically increasing. The resolution of Power Donut’s tilt sensor is 0.025 ° , the corresponding minimum current change is approximately 4.00 A.

Figure 4-12: PD conductor temperature and maximum current

Figure 4-13: Power Donut tilt angle and maximum current

4.3.3.3. ErrorbecauseoftheconductordifferentheightalongaspanThe conductor surface temperature is divergent at different points of OHL in windy condition. The main reason is the different wind speed at different height. Wind speed is an important parameter in forced convection heat loss which is the key heat loss in OHL’s thermal balance.

Wind speed varies with height above the ground. This effect is commonly called “wind shear”. The wind shear is influenced by the roughness of ground as shown in Table 11.

Table 11: terrain and roughness

Type of terrain Roughness length, z Water, snow or sand surfaces 0.0001m Open, flat land, mown grass, bare soil 0.01m Farmland with some vegetation 0.05m Suburbs, town, forests, many trees and bushes

0.3m


- 54 -

The relation between wind speed and height is given by:

= ln( )ln( )

(54)

where: = the wind speed at point 1 (m/s) = the wind speed at point 2 (m/s) = the height of point 1 (m) = the height of point 2(m) = the roughness length

Assume that: Single wood pole 33-kV

Current: 900A Average resistance: 9.39e-5 ohm/m Solar radiation: 14.00 w/m Span length: 200m Average wind speed: 0.61m/s Wind direction: 90 degree Roughness: 0.05

The dimension of wood pole is shown as Figure 4-14.

The difference of wind speed at different points on overhead line will influence the conductor temperature, because the heat balance has changed.

where: Average conductor temperature is 91.16 Sag is 3.66m Clearance is 6.84m So the height of average temperature point is 8.67m [62]

Figure 4-14[1]: wool pole

Figure 4-15: conductor temperature at different height on wool

pole


- 55 -

The forced convection heat loss rate is given by equation (2).

In Figure 4-15, the X-axis is height of wool pole; Y-axis is the corresponding conductor temperature. It can be found that the temperature at lowest point is 93.36, the temperature at highest point is 91.69 . The temperatures of different points are calculated by the equation of overhead line’s heat balance.

4.3.3.4. ErroroftiltsensorTilt sensor is also installed in the Power Donut. The tilt angle can help to provide more accurate result of conductor temperature. The error of tilt sensor’s measurement is not only influenced by the location of Power Donut but also by the property of tilt sensor itself. The measurement range of tilt sensor is -15° to 15°. This determines that Power Donut can be only installed on a small area of whole span which insure that tilt sensors are in the right tiling. When the sensors are installed on other place, the result is meaningless.

The systematic error can be calculated by the equation as follow [63]:

= arccos(cos − ) − (55)

where: = the error in degree = the measured angle (°) = error coefficient

The systematic error is caused by sensor’s error coefficient. Different sensors have different coefficients. Therefore, the analysis of the tilt error senor is performed for various δ values as shown in Figure 4-16. The X-axis of Figure 4-16 is the measured angle; the Y-axis is the corresponding error. From Figure 4-16, it can be found that the error coefficient is a key factor in the calculation of tilt sensors’ error. When the error coefficient’s value is small, the error is also small. The error coefficient is directly related to the quality of the sensor.

Figure 4-16: tilt sensors’ error according to error coefficients


- 56 -

4.3.4. SpecificationofweatherinstrumentationofPowerDonutDTRsystemWeather station is the main part of Power Donut DTR monitoring system. Weather station measures all the ambient conditions include solar radiation rate, wind speed, wind direction and ambient temperature in the DTR calculation [64].

Power Donut DTR system’s weather instrumentation installed on the transmission tower. The weather station is consisted by weather box and anemometer. All the measurement performed form the Power Donut sensor on the conductor will be transferred via Bluetooth wireless communication system to the weather station central unit that is installed on the tower. Because of the distance limitation of Bluetooth communication, the weather station should be installed close to the Power Donut.

The Power Donut’s weather station is installed on the closest tower to the power donut in most cases. The height of weather station should be the same level with the bottom phase conductor of the overhead line and in particular at the same height of the lowest point of the phase conductor otherwise some error on wind speed is expected. The weather station provides measurements of solar radiation (w/m2), wind speed (ft/s), wind direction (°) and ambient temperature ().The resolution and range of instrumentation of the weather station is shown in Table 12.

Table 12: specification of Power Donut’s weather station

Measurement Resolution Range Solar radiation 1 w/m2 100 to 1200w/m2 Wind speed 0.1m/s 0.5 to 80 m/s Wind direction 1° 0 to 360° Ambient temperature 0.1 -50 to 60

4.3.5. SensitivityofweatherstationBecause Power Donut’s weather station measures four different parameters, its sensitivity is more complex than other components. This chapter evaluates the sensitivity of these parameters one by one in order to make it clear.

4.3.5.1. SensitivityofwindspeedmeasurementAssume that: Diameter is 28.1 mm.

Emissivity is 0.5. Solar absorptivity is 0.5. Ambient temperature is 40. Conductor ac resistance is: R(25) = 7.238 × 10 R(75) = 8.688 × 10 Average conductor elevation 100m. Maximum conductor temperature is 100. Wind direction is 90°. Solar radiation is 1000 W/m2


- 57 -

The conditions of wind speed varying from 0.5 m/s to 2.5 m/s with interval of 0.1 m/s are calculated. The result of the calculation can be found in appendix. Figure 4-17 is plotted according to the table.

In Figure 4-17, the X-axis is wind speed; Y-axis is the corresponding thermal rating. It can be found that the maximum current is 1424.5 ampere when the wind speed is 2.5 m/s; the minimum current is 952.4 ampere when the wind speed is 0.5 m/s. When the wind speed increases 0.1 m/s, the maximum current will change 16 – 40 ampere correspondingly. Comparing with the sensitivity of other parameters, wind speed is the highest. That is why wind speed is a key factor in dynamic thermal rating measurement.

Figure 4-17: sensitivity of wind speed measurement Figure 4-18: sensitivity of wind direction measurement

4.3.5.2. SensitivityofwinddirectionmeasurementAssume that: Diameter is 28.1 mm.

Emissivity is 0.5. Solar absorptivity is 0.5. Ambient temperature is 40. Conductor ac resistance is: R(25) = 7.238 × 10 R(75) = 8.688 × 10 Average conductor elevation 100m. Maximum conductor temperature is 100. Wind speed is 0.61 m/s Solar radiation is 1000 W/m2

The conditions of wind direction varying from 0° - 90° with interval of 1° are calculated. The result was shown in Figure 4-18. In Figure 4-18, the X-axis is wind direction; Y-axis is the corresponding thermal rating. It can be found that the maximum current is 1064.8 ampere when the wind direction is 90° ; the minimum current is 842.1 ampere when the wind direction is 0°. When the wind direction changes 1°; the maximum current will change 1.01 – 1.28 ampere correspondingly.


- 58 -

4.4. AmpacimonDTRMonitoringSystem

4.4.1. WorkingprincipleofAmpacimonThe fundamental working principle of Ampacimon is based on OHL’s conductor frequency measurement. According to string wave theory, the vibration frequency of a string between two dead ends is related to the tension of the string. As sag is related to the tension, it can be used to calculate the conductor temperature. Miura[65] has introduced the method of video Sagometer to monitor OHL’s DTR. Kumano[66] has describe the method of conductor replica system.

The OHL in one span is fastened to the two dead ends of the transmission towers’ arms. Because of the flow of air stream around the OHL, the conductor vibrates with changing amplitude and frequency. The string (conductor) will produce acoustic wave with frequency the same with the string’s vibration [67]. The propagation speed of acoustic wave can be described by equation below.

= (56)

Where: = wave velocity (m/s) = wave length (m) =vibration frequency (Hz) A wave’s propagation speed of a string which is between two dead ends can be described by equation below.

∝ (57)

Where: =the tension of string (N) =linear density (kg/m)

Derivation of the equation above is given as follow:

The figure right is used to show the shape of string fasten between two dead ends. Abscissa x to x+∆x is a piece of the string.

Because the string is fastened between two dead ends, from the Newton’s second law, the horizontal tension of any points of the string should be same. Assuming the horizontal tension of the string is “T”. = cos( ) = = cos( ) =

In the y axis of this piece of string, the force analysis is given as below:


- 59 -

= + = ∆

= + = sin( ) + sin( ) = tan( ) + tan( ) The tangent items are the slope of the points. So the force analysis can be given as:

| ∆ − | = ∆ ∙

When ∆ become very small, the equation above can result to a second derivative equation as follow:

= ∙

As v=x/t: ∝

This derivation is able to explain the string wave theory on engineering level. But when the piece of string is large, ∆ is not equal the piece of string any more. The more specific explanation can be found in [68].

According to equations (56, 57):

= ∝ 1 ∙

This equation is called Mersenne’s law [69].

In the overhead line, the span length equals half wave length. Plugging in the span length, the equation is given by:

= 2

(58)

L: length of the span (not the string’s length) (m)

a: coefficient of Mersenne's law

= 4

(59)

w = linear density (N/m) g = acceleration of gravity = 8

(60)


- 60 -

s = sag (m)

= 32 (61)

The coefficient of Mersenne's law can be calculated by a known state.

Ampacimon measures the vibration of OHL conductor at a point which is same at all the points of one span of overhead line. Therefore Ampacimon can be installed at any point along the span without affecting the accuracy of the measurement.

The weather measurements of Ampacimon system are similar with other DTR systems. The fixed parameters used in DTR calculation are different from normal condition. The solar absorptivity is assumed 0.9 instead of 0.5 in IEEE standard. The emissivity is assumed 0.7 instead of 0.7 used in IEEE standard.

4.4.2. SpecificationofAmpacimonDTRmeasurementsystemAmpacimon directly measure the vibration frequency of overhead line.

The vibration frequency measured range is 0 to 100 Hz. The resolution of frequency measurement is 1 Hz. The range for conductor temperature measurement is -40 to 200. Dimensions of Ampacimon is 44*18*27 cm Weight of Ampacimon is 8kg. Location of Ampacimon could be anywhere on the line as the vibration frequency is same all over the line [70].

4.4.3. SensitivityofAmpacimonDTRmeasurementsystemAssume that: The conductor is ACSR for 400KV

Diameter is 28.1mm Weight is 10N/m. Span length: 200m Young’s modulus of the conductor is 57 Gpa Thermal coefficient of the conductor is 23 × 10 Cross section is 400 mm2 Wind speed is 0.61 m/s perpendicular to the conductor Emissivity is 0.7 Solar absorptivity is 0.9 Ambient temperature is 40 Conductor ac resistance is: (25) = 7.238 × 10 (75) = 8.688 × 10 Average conductor elevation 100m Solar radiation rate is 1000 W/m2

The values of tension, sag, conductor temperature, current and the change of current is given by Table 13 according to the vibration frequency.

Table 13 shows the process of Ampacimon conductor temperature measurement process. All the data in Table 13 was calculated by the method introduced in chapter 4.1.1.


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Table 13: the complete process of Ampacimon calculation

Frequency (Hz)

Tension (N)

Sag (m)

conductor temperature (C)

Current (A)

Change (A)

35 18218 6.1 169.9 1586 104 36 18695 5.8 157.5 1492 94 37 19184 5.4 146.6 1406 86 38 19688 5.2 137.0 1328 78 39 20204 4.9 128.4 1255 72 40 20734 4.7 120.7 1188 67 41 21277 4.4 113.7 1125 63 42 21834 4.2 107.4 1066 59 43 22405 4.0 101.7 1010 56 44 22988 3.8 96.4 956 54 45 23585 3.7 91.6 905 52 46 24196 3.5 87.1 855 50 47 24820 3.4 83.0 806 49 48 25457 3.2 79.1 758 48 49 26108 3.1 75.5 711 48 50 26772 3.0 72.0 663 48 51 27450 2.9 68.8 615 48 52 28141 2.8 65.7 566 49 53 28845 2.7 62.7 515 51 54 29563 2.6 59.9 461 54 55 30294 2.5 57.2 402 58 56 31039 2.4 54.5 337 66 57 31797 2.3 51.9 258 79 58 32568 2.2 49.4 145 113

It can be found that higher the frequency, higher the tension. But sag will decrease when the frequency is increasing. Conductor temperature has the same variation tendency with sag. The same is the current. However, the change of current shows an absolute different variation tendency. It reaches the minimum value at middle point and maximize at two ends.


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Figure 4-19 is plotted according to the data of Table 13. The X-axis is vibration frequency of overhead line; Y-axis is the corresponding thermal rating. It describes the change of OHL’s thermal rating with the varying of measured OHL vibration frequency. From Table 13, it can be found that the highest maximum allowable current of OHL is 1690 ampere when the vibration frequency is 34 Hz. The lowest maximum allowable current is 145 ampere when the vibration frequency is 58 Hz. The tendency of DTR’s change is monotone decreasing. The minimum DTR change varies from 40 to 100 ampere when the measured frequency changes 1 Hz.

Figure 4-19: sensitivity of Ampacimon conductor temperature measurement

4.5. MainconclusionsThis chapter introduces the working principles and specifications of three kinds of DTR monitoring systems: CAT-1, Power Donut and Ampacimon.

The three systems all measure the conductor temperature and ambient weather conditions. But the working principles of the three systems are totally different.

Table 14 concludes the specifications of the three DTR systems. In order to make the specifications clear, same items are arranged in the same column. This could also be used for the comparison of different DTR systems.

All the values in Table 14 are calculated according to the weather conditions stated in previous parts. The accuracies of the three systems will be specifically described in next chapter.


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Table 14: the specifications of different DTR systems

DTR system

Component Measured Item

Range Resolution Sensitivity

CAT-1 Load cell Tension 22.25 to 33.78 KN44.50 to 66.75 KN111.2 to 166.8 KN

0.05% of full scale

0.07 A per 5.8N

ATS2 Ambient Temperature

-40 to 60 0.1 9.039 A per 0.1NRS3 NRS

temperature -40 to 60 0.1 1.4 A per 0.1

Anemo-meter

Wind speed

0.5 to 89.0 m/s 0.1 m/s 25 A per 0.1 m/s

Wind direction

0 to 360° 1° 1.15 A per 1° Power Donut temperature

sensor Conductor temperature

-50 to 150 -50 to 250

0.1 1.3 A per 0.1

Tilt sensor Power Donut tilt angle

-15 to 15° 0.025° 4.0 A per 0.025° Weather station

Ambient temperature

-50 to 60 0.1 9.039 A per 0.1Solar radiation

100 to 1200 w/m2 1 w/m2 0.07 A per 1w/m2

Wind speed 0.5 to 80 m/s 0.1 m/s 25 A per 0.1 m/s

Wind direction

0 to 360° 1° 1.15 A per 1° Ampacimon On-line

sensor Vibration frequency

0 to 100 Hz 1 Hz 70 A per 1 Hz

Weather station

Ambient temperature

-50 to 60 0.1 9.039 A per 0.1Solar radiation

100 to 1200 w/m2 1 w/m2 0.07 A per 1w/m2

Wind speed 0.5 to 80 m/s 0.1 m/s 25 A per 0.1 m/s

Wind direction

0 to 360° 1° 1.15 A per 1°

In Table 14, the sensitivity is calculated according to the resolution of instruction. Because the sensitivity will change by the measured value, the sensitivity is given by the average value of all measurement result in this chapter.

2 Ambient temperature sensor 3 Net radiation sensor

Theoretical error analysis and comparison of DTR systems

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5. TheoreticalerroranalysisandcomparisonofDTRsystemsThe previous chapter has introduced the specifications include resolution, measurement range and sensitivity of three DTR. Apart from the specifications mentioned above, error level or accuracy is an important criterion for DTR system. In order to emphasise the importance of measurement system’s accuracy, an individual chapter was written to present and compare the accuracies of different DTR systems. All the equations in this chapter are referred from [71].

5.1. Propagationofmeasurementerror

5.1.1. ConceptoferrorpropagationPropagation of error or uncertainty is an important concept in statistics. It describes the relation between variables’ error and the function’s error which is consisted by the variables [72]. In order to make the definition clear, a simple example of linear error propagation is given as follow. This example is the simple linear propagation of error. In real research and engineering, the format of function is usually much more complicate than the example provided here. This example is used to help make the concept clear. Assume that = +

The error of is∆ ; the error of is ∆ . The error of y is ∆ + ∆ . The process, from∆ , ∆ to ∆ is error’s propagation.

5.1.2. CategoriesoferrorWhen measurement is undertaken in laboratory or factory, the measured value is different from the true value. The difference between them is the measurement error. The probability rule for measurement error can be described by normal distribution generally [73]. Absolute error is a mostly used term when describing error. It uses the same unit with the result of measurement to describe the uncertainty of an instruction. If the instruction’s measurement range is x, the absolute error of x can be written as∆ .

An example of absolute error can be given as follow: A ruler’s measurable length is 1 meter; its error is 0.01 meter. The error here is absolute error. Apart from absolute error, relative error is also used in many occasions. Relative error is the ratio between absolute error and the instruction’s measurement range. If the instruction’s measurement range is x, the relative error of x can be write as∆ / . It is generally written as percentage.

Using the same example with absolute error: there is a ruler; its error is 1%. The error here is relative error. From this example, it can be found that the relative error does not rely on the instruction’s measurement range. The accuracy level can be known without knowing the measurement range. In the accuracy comparison between instructions with different measurement range, relative error is an important parameter [74].


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5.1.3. ThecalculationoferrorpropagationFor the function given as follow: = ( , ⋯ ) The error of y can be calculated as follow:

∆ = ∙ ∆ (62)

Where: = measurement variable ℎ = ± ,∆ = ∆ ± ∆ .

ℎ = ∙ ,∆ = ∙ ∆ + ∙ ∆ = ∙ ∆ + ∙ ∆

ℎ = / ,∆ = ∙ ∆ + ∙ ∆ = 1 ∙ ∆ − ∙ ∆

ℎ = ∙ ,∆ = ∙ ∆ + ∙ ∆ = ∆ + ∆

5.2. AccuracyofCAT‐1systemThe manufacturer specified the expected error of CAT-1 system’s individual components. In more details, the load cell error is 0.03% at 22.25KN, the thermometer error is expected to be at a range of 0.5 °C and the wind direction can be measured with a 3° error. Wind speed and solar radiation have 5% error of the reading. When wind speed is lower than 0.5m/s; or solar radiation rate is less than 100 w/m2, the error happens.

In order to evaluate the maximum error that CAT-1 monitoring system can generate during a measurement. The example of an OHL with span length of 200 m is given. Conductor installation tension at 70 °C is considered to be at 12 KN. The conductor density is 10 N/m, modulus of elasticity is 57 Gpa and thermal coefficient of expansion 10-6m/K. Conductor cross-sectional area is 4×10-4m2. The error of conductor temperature according to the error of tension measurement is given in Table 15. It can be found that the conductor temperature measurement error will slightly decrease when the measured tension increase.

Table 15: error of conductor temperature measurement in CAT-1 system

tension (N)

error of tension

(N)

horizontal tension

(N)

error of horizontal

tension (N)

Sag (m)

error of sag (m)


(C)

error of conductor

temperature (C)

Highest conductor

temperature error (C)

37144 9.0432 37107 9.05 2.38 4.99E-04 88.89 1.53E-02 1.26

35264 8.4792 35224 8.49 2.53 5.33E-04 91.5 1.57E-02 1.45

34512 8.2536 34471 8.27 2.6 5.48E-04 92.68 1.60E-02 1.5


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33008 7.8024 32965 7.82 2.76 5.80E-04 95.31 1.65E-02 1.62

31504 7.3512 31458 7.37 2.93 6.16E-04 98.4 1.73E-02 1.74

31880 7.464 31835 7.48 2.88 6.06E-04 97.58 1.71E-02 1.71

30376 7.0128 30328 7.03 3.07 6.46E-04 101.07 1.80E-02 1.84

33384 7.9152 33341 7.93 2.72 5.71E-04 94.62 1.64E-02 1.59

36392 8.8176 36354 8.83 2.44 5.12E-04 89.88 1.55E-02 1.31

37520 9.156 37483 9.17 2.35 4.93E-04 88.41 1.53E-02 1.24

36768 8.9304 36730 8.94 2.41 5.06E-04 89.38 1.54E-02 1.28

34888 8.3664 34848 8.38 2.57 5.40E-04 92.08 1.59E-02 1.48

34136 8.1408 34095 8.15 2.64 5.55E-04 93.3 1.61E-02 1.53

30000 6.9 29951 6.91 3.12 6.56E-04 102.05 1.83E-02 1.88

32256 7.5768 32211 7.59 2.84 5.97E-04 96.79 1.69E-02 1.68

36016 8.7048 35977 8.72 2.47 5.19E-04 90.4 1.56E-02 1.33

30752 7.1256 30705 7.14 3.02 6.35E-04 100.14 1.78E-02 1.81

35640 8.592 35601 8.6 2.5 5.26E-04 90.94 1.56E-02 1.43

31128 7.2384 31081 7.25 2.97 6.25E-04 99.25 1.75E-02 1.77

33760 8.028 33718 8.04 2.68 5.63E-04 93.95 1.62E-02 1.56

37896 9.2688 37860 9.28 2.32 4.87E-04 87.95 1.52E-02 1.22

38272 9.3816 38236 9.39 2.29 4.81E-04 87.51 1.51E-02 1.2

32632 7.6896 32588 7.7 2.8 5.88E-04 96.04 1.67E-02 1.65

The average error of conductor temperature measurement is 1.18. The relation between measured tension and relative error is shown in Figure 5-1. The X-axis is tension; Y-axis is the corresponding relative error.


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Figure 5-1: tension and relative error

5.3. AccuracyofPowerDonutsystem

5.3.1. ComponenterrorError of conductor temperature: 1 (power donut system manual) Error of Power Donut angle: 0.025 ° (power donut system manual) Error of ambient temperature: 0.5 (weather station manual) Error of wind speed: 5% of the reading (weather station manual) Error of wind direction: 3° (weather station manual) Error of solar radiation: 5% of the reading (weather station manual)

5.3.2. SystemerrorThe error of Power Donut system’s direct conductor temperature is 1. The error measured by tilt sensor will change with the measured value. This chapter will mainly analysis the error of tilt sensor.

Assume that: Distance between Power Donut and the closest tower: 10 m

Conductor density: 14.308 N/m Span length: 200 m Young’s modulus of the conductor is 57 Gpa Thermal coefficient of the conductor is 10 / Conductor cross area is 4 × 10 m2

According to the equations below: ℎ = ∙( ) (Horizontal tension calculation); = ∙(sag calculation)

− − = (Conductor temperature calculation)


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The error of conductor temperature according to the error of tilt measurement is given Table 16. It can be found that the conductor temperature measurement error will slightly decrease when the measured tilt increase.

Table 16: error of conductor temperature measurement in Power Donut system

angle

error of

angle horizontal

tension

error of horizontal

tension sag

error of sag


error of conductor

temperature (deg) (deg) (N) (N) (m) (m) (C) (C)

2 0.025 36883 461 1.94 1.69E-02 82.13 0.620 2.025 0.025 36427 450 1.96 1.69E-02 82.49 0.612 2.05 0.025 35983 439 1.99 1.69E-02 82.86 0.604 2.075 0.025 35549 428 2.01 1.69E-02 83.22 0.596 2.1 0.025 35126 418 2.04 1.69E-02 83.59 0.589

2.125 0.025 34712 408 2.06 1.69E-02 83.95 0.583 2.15 0.025 34309 399 2.09 1.69E-02 84.32 0.576 2.175 0.025 33914 390 2.11 1.69E-02 84.69 0.570 2.2 0.025 33529 381 2.13 1.69E-02 85.06 0.564

2.225 0.025 33152 372 2.16 1.69E-02 85.44 0.559 2.25 0.025 32783 364 2.18 1.69E-02 85.81 0.553 2.275 0.025 32423 356 2.21 1.69E-02 86.19 0.548 2.3 0.025 32070 348 2.23 1.69E-02 86.57 0.544

2.325 0.025 31725 341 2.26 1.69E-02 86.95 0.539 2.35 0.025 31387 334 2.28 1.69E-02 87.33 0.535 2.375 0.025 31057 327 2.30 1.69E-02 87.72 0.531 2.4 0.025 30733 320 2.33 1.69E-02 88.11 0.527

2.425 0.025 30416 313 2.35 1.69E-02 88.50 0.523 2.45 0.025 30105 307 2.38 1.69E-02 88.89 0.519 2.475 0.025 29801 301 2.40 1.69E-02 89.28 0.516 2.5 0.025 29503 295 2.42 1.69E-02 89.68 0.513

2.525 0.025 29211 289 2.45 1.69E-02 90.08 0.510 2.55 0.025 28924 283 2.47 1.69E-02 90.48 0.507

The average error of conductor temperature measurement is 0.55. The relation between measured tilt angle and relative error is shown in Figure 5-2. The X-axis is tilt angle; Y-axis is the corresponding relative error.


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Figure 5-2: tilt angle and relative error

5.4. AccuracyofAmpacimonsystem

5.4.1. ComponenterrorError of sag: 0.1m (Ampacimon manual) Error of ambient temperature: 0.5 (Davis weather station) Error of wind speed: 5% of the reading (Davis weather station) Error of wind direction: 3° (Davis weather station) Error of solar radiation: 5% of the reading (Davis weather station)

5.4.2. SystemerrorAssume that: Conductor density: 10 N/m

Span length: 200 m Young’s modulus of the conductor is 57 Gpa Thermal coefficient of the conductor is 10 / Conductor cross area is 4 × 10 m2 Acceleration of gravity is 9.8 N/kg

According to the equations below: = (Horizontal tension calculation); = ∙(sag calculation)

− − = (Conductor temperature calculation)

The error of conductor temperature according to the error of frequency measurement is given in Table 17. It can be found that the conductor temperature measurement error will decrease when the measured frequency increase.


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Table 17: error of conductor temperature measurement in Ampacimon system

Frequency error of

frequency Tension

error of tension

Sag error of

sag conductor

temperature

error of conductor

temperature (Hz) (Hz) (N) (N) (m) (m) (C) (C) 36 1 18695 482.4 5.8 0.317 157.5 11.57 37 1 19184 495.8 5.4 0.292 146.6 10.10 38 1 19688 509.2 5.2 0.270 137 9.08 39 1 20204 522.6 4.9 0.249 128.4 8.08 40 1 20734 536 4.7 0.231 120.7 7.30 41 1 21277 549.4 4.4 0.215 113.7 6.54 42 1 21834 562.8 4.2 0.200 107.4 5.94 43 1 22405 576.2 4 0.186 101.7 5.43 44 1 22988 589.6 3.8 0.174 96.4 4.98 45 1 23585 603 3.7 0.162 91.6 4.62 46 1 24196 616.4 3.5 0.152 87.1 4.27 47 1 24820 629.8 3.4 0.143 83 3.99 48 1 25457 643.2 3.2 0.134 79.1 3.73 49 1 26108 656.6 3.1 0.126 75.5 3.51 50 1 26772 670 3 0.118 72 3.31 51 1 27450 683.4 2.9 0.112 68.8 3.14 52 1 28141 696.8 2.8 0.105 65.7 2.99 53 1 28845 710.2 2.7 0.099 62.7 2.86 54 1 29563 723.6 2.6 0.094 59.9 2.74 55 1 30294 737 2.5 0.089 57.2 2.65 56 1 31039 750.4 2.4 0.084 54.5 2.57 57 1 31797 763.8 2.3 0.080 51.9 2.51 58 1 32568 777.2 2.2 0.076 49.4 2.46

The average error of conductor temperature measurement is 4.97. The relation between vibration frequency and relative error was shown in Figure 5-3. The X-axis is frequency; Y-axis is the corresponding relative error. It can be found that the relative error is 7.2% when vibration is 36 Hz and the relative error is 4.6% when vibration frequency is 52 Hz.


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Figure 5-3: frequency and relative error

5.5. MainconclusionsThis chapter calculated the systematic error of DTR systems. In normal condition, the error

for CAT-1 system is 1.842%; for Power Donut system, the error is 0.755%; for Ampacimon, the error is 6.627%.

Methods to Choose the Critical Spans

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6. MethodstoChoosetheCriticalSpansIn one span of overhead line, the conductor temperature is not the same at different points. But the difference is not significant for engineering level. Taking wind factor into consideration, the lowest point temperature is suitable to be used as the conductor temperature. When the requirement is not strict, any point’s temperature can be used to represent the span’s temperature.

The conductor temperature in one span varies not much. But the temperatures in different spans are quite different. So finding the highest conductor temperature is to find the highest span temperature. To achieve this purpose, the perfect method is to measure all spans’ temperature. Only by measuring all spans’ temperature, the highest conductor temperature which is definitely right can be known. However, measuring all spans’ temperature is not economic. To make DTR monitoring a practical method, it is necessary to solve the problem between cost and reliability.

One solution for this is measuring only a part of all spans. Because the possibility for one span to be the hottest span is different, it is possible to measure a low percent spans but get a result which has a high possibility to be right. For example, only the temperatures of 10% spans are measured and the hottest span which has 90% possibility to be right can be found.

In this solution, the key problem is to determine which spans should be measured. Because these measured spans should have relative high possibility to be the hottest span, they are critical spans. In this report, two methodologies which used to identify the critical span will be presented.

6.1. IdentificationofcriticalspanbythermalageingofoverheadconductorOverhead line will lose its tensile strength when operating in high temperature. When the operating temperature is not very high, the conductor’s tension loss is slow. This process is called ageing. When the operating temperature is extremely high, the conductor lose tension very fast. This process is called annealing.

The nature of ageing and annealing are the same. They could be described by uniform formula, remain strength rate formula of conductors.

According to [75], different materials have different formula for the calculation of remain strength rate. The formula for two conductors: SAC, AAAC are given as below. More formulas for other conductors are available in [75].

For SAC: = (134 − 0.24 ∙ ) . ∙ . 134 − 0.24 ≤ 100100 . ∙ . 134 − 0.24 > 100 For AAAC: = (176 − 0.52 ∙ ) . ∙ . 176 − 0.52 ≤ 100100 . ∙ . 176 − 0.52 > 100


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RS = remaining strength as a percentage of initial strength

T = temperature (C) t = operating time (hours) d = diameter (inch) From the formulas above, the conductor’s tension lose rate (1-RS) is positive correlation with operating time and operating temperature. It means that the conductor’s tension lose rate is positive correlation with the mean conductor temperature during the operating period.

The span with high average operating temperature tends to have a high possibility to be a hot span. So the more strength a span lose, the higher possibility it will become a hot span. So the method to choose a critical span is to choose the highest strength lose span.

Following content is a case using the method stated above [76]:

A line in Kelly Lake to Williston in Canada supported by 861 towers can be used as an example. The tension loss (ageing) of each span has been shown in Figure 6-1. Assume that only 5 spans can be chose as critical ones, the highest five tension loss spans should be chose among the 861 spans.

In Figure 6-1, X-axis is the number of transmission tower; Y-axis is the percentage of this span’s ageing. The highest five points are critical spans in this case.

Figure 6-1[76]: ageing condition of transmission towers

6.2. IdentificationofcriticalspanbystatisticalmethodologyStatistics has been widely used in different engineering fields. The meteorological conditions tend to be reliable during an operating period of overhead line. The record of different spans’ conductor temperatures can be used to find out the span with highest frequency to be the hottest span. These high frequent spans can be chose as the critical spans.


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The principle for this method is based on specific conductor temperature record. Because the parameters that can influence the weather condition are complicated, it will be simpler if considering the conductor temperature directly.

A real experiment has been employed in a line called “Nogales-Pan” in north central Chile [18]. There are 248 spans in the line. According to the 47235 times conductor temperature record in five years, the top 20 spans with high frequency to be hottest were listed below:

Table 18: high temperature frequency for top 20 spans

Span 143 148 93 202 18 205 205 129 231 22 Frequency 6.7% 6.5% 6.5% 6.3% 5.7% 5.7% 5.7% 5.4% 5.4% 5.1% Span 36 21 182 153 203 95 190 100 167 116 Frequency 5.1% 4.7% 3.2% 2.1% 2.0% 1.5% 1.4% 1.3% 1.1% 1.1%

If measuring the top ten spans, the confidence level is 59%. If measuring the all top 20 spans, the confidence level is 82.5%. it can be found that this method can provide a relative high confidence critical span with measuring low quantity devices. This method is based on historic specific DTR data record and assuming that the climate is similar in the recording period.

6.3. MainconclusionsThis chapter summarized two methods used to choose the critical spans among overhead lines’ all spans. In the example of Nogales-Pan overhead line, 8.0% of all spans are measured and the probability of choosing the right span is 82.5%.

Conclusion

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7. Conclusion

7.1. MainconclusionAs the demand of electrical power soaring, increasing the thermal rating of overhead line become more and more important. For DNO, higher thermal rating could increase the profit directly. For customers, higher thermal rating means more and cheaper access to electrical power. Increasing OHL’s thermal rating with reasonable expense is extremely difficult, sometimes even impossible. The research of dynamic thermal rating is able to provide a method to achieve this purpose. The achievement of this project is choosing the optimal DTR system for electrical power transmission. The outcome of this project is that Power Donut DTR system is the optimal DTR system.

This thesis specifically illustrates three thermal rating calculation models and three measurement systems. In the three models, IEEE model is the most accurate; IEC model is

the simplest. In the three DTR systems, Power Donut’s systematic error is 0.755% which is smallest. The result of this thesis can help distribution network operators to choose the optimal method to increase overhead lines’ thermal rating.

In static thermal rating, the relation between OHL’s operation current and its overloading rate is very abstract. By analysing historic data, this thesis provides a method to get the specific and numeric relation. This made the comparison between overloading risk and increased thermal rating possible. DNO could adjust the operation current according the maximum tolerable overloading risk using the method in this thesis.

7.2. FurtherworkTo increase thermal rating is an endless task in power transmission industry. A lot of research about dynamic thermal rating has been done already, to find new method is meaningful in order to get higher thermal rating. Even in dynamic thermal rating, more effort can be made in looking for new methods. All these effort is possible to increase overhead lines’ thermal rating.

Apart from increasing overhead lines’ thermal rating directly, the prediction of thermal rating is also practical in electrical industry. The research of prediction could make full use of the development of meteorology and statistics. It is helpful for developing new ideas for increasing thermal rating. The functional relation between dynamic thermal rating parameters is able to provide guide for overhead line operation. It is also a method to increase thermal rating. This method based on mathematic analysis is also new approach to boost overhead line’s existing thermal rating.

References

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8. References[1] K.Kopsidas, "ModellingThermalRatingofArbitraryOverheadLineSystems,"TheUniversityofManchester,2009.[2] CIGRE,"SurveyonFutureUseofConductors,"ed.Paris,1998.[3] P. Alby, et al., "Firms Operating under Electricity Constraints inDevelopingCountries,"TheWorldBankEconomicReview,vol.27,pp.109-132,2013.[4] M. Fulczyk, et al., "Measurement based fault location in three-terminal overhead line and underground cable," in ElectricalInsulation and Dielectric Phenomena (CEIDP), 2010 Annual ReportConferenceon,2010,pp.1-4.[5] M.Mukaida,etal.,"Taskplanningexperimenttowardanautonomousrobotsystemfortheconstructionofoverheaddistributionlines," inIntelligent Robots and Systems 95. 'Human Robot Interaction andCooperative Robots', Proceedings. 1995 IEEE/RSJ InternationalConferenceon,1995,pp.448-455vol.2.[6] CIGRE,"IncreasingCapacityofOverheadTransmissionLines:NeedsandSolutions,"ed,2010.[7] CIGRE,Guide forSelectionofWeatherParameters forBareOverheadConductorRatings:CIGRE,2006.[8] D.Hanson,etal., "A STATCOM-based relocatable SVC project in theUK forNational Grid," inPowerEngineeringSocietyWinterMeeting,2002.IEEE,2002,pp.532-537.[9] "IEEE Standard for Testing and Performance for All-Dielectric Self-Supporting(ADSS)FiberOpticCableforUseonElectricUtilityPowerLines,"IEEEStd1222‐2011(RevisionofIEEEStd1222‐2004),pp.1-55,2011.[10] M.W.Davis,"Anewthermalratingapproach:Therealtimethermalrating system for strategic overhead conductor transmission lines--Part I:Generaldescriptionand justificationof thereal timethermalrating system,"PowerApparatusandSystems, IEEETransactionson,vol.96,pp.803-809,1977.[11] D. T. Radmer,etal., "Predicting vegetation-related failure rates foroverheaddistributionfeeders,"PowerDelivery,IEEETransactionson,vol.17,pp.1170-1175,2002.[12] IEEE, "IEEE Standard for Calculation of Bare Overhead ConductorTemperature and Ampacity Under Steady-State Conditions," inANSI/IEEEStd738‐1993,ed,2006.[13] CIGRE, "Guide for the selection of weather parameters for bareoverheadconductorratings,"2006.

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Appendix

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9. Appendix

9.1. MatlabcodeforDTRcalculation

9.1.1. Sub‐function

9.1.1.1. Sub‐function:loadingtheIEEEairpropertytableSave the air property table (3-1) in Matlab file path as “air_property.txt”.

9.1.1.2. Sub‐function:calculatingtheairviscosity1) function airviscosity = air_viscosity (ambient_temperature) 2) load ('air_property.txt'); 3) line1 = air_property(:,1); 4) line2 = line1 + 2.5; 5) [row col]= size(line2); 6) for i=1:row 7) if ambient_temperature < line2(i) 8) airviscosity = air_property(i,2); 9) break 10) else 11) airviscosity = air_property(21,2); 12) end 13) end

9.1.1.3. Sub‐function:calculatingtheairdensity1) function airdensity = air_density (ambient_temperature,elevation) 2) load ('air_property.txt'); 3) line1 = air_property(:,1); 4) line2 = line1 + 2.5; 5) [row col]= size(line2); 6) if elevation < 500 7) for i=1:row 8) if ambient_temperature < line2(i) 9) airdensity = air_property(i,3); 10) break 11) else 12) airdensity = air_property(21,3); 13) end 14) end 15) elseif elevation <1500 16) for i=1:row 17) if ambient_temperature < line2(i) 18) airdensity = air_property(i,4); 19) break 20) else 21) airdensity = air_property(21,4); 22) end

Appendix

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23) end 24) elseif elevation <2500 25) for i=1:row 26) if ambient_temperature < line2(i) 27) airdensity = air_property(i,5); 28) break 29) else 30) airdensity = air_property(21,5); 31) end 32) end 33) else 34) for i=1:row 35) if ambient_temperature < line2(i) 36) airdensity = air_property(i,6); 37) break 38) else 39) airdensity = air_property(21,6); 40) end 41) end 42) end

9.1.1.4. Sub‐function:calculatingtheairthermalconductivity1) function airconductivity = air_conductivity(ambient_temperature) 2) load ('air_property.txt'); 3) line1 = air_property (:,1); 4) line2 = line1+2.5; 5) [row col]= size(line2); 6) for i=1:row 7) if ambient_temperature< line2(i) 8) airconductivity = air_property (i,7); 9) break 10) else 11) airconductivity = air_property (21,7); 12) end 13) end

Sub-function: calculating the wind direction factor

1) function wind_direction_factor = kangle(wind_conductor_angle_degree) 2) wind_direction_factor = 1.194 -

cos(wind_conductor_angle_degree*0.0174532925)+0.194*cos(2*wind_conductor_angle_degree*0.0174532925)+0.368*sin(2*wind_conductor_angle_degree*0.0174532925);

Sub-function: calculating the AC resistance

Appendix

- 84 -

1) function r=ieeeac(r25,r75,temperature) 2) r=((r75-r25)/50)*(temperature-25)+r25;

9.1.2. Heatcalculation

9.1.2.1. ConvectionheatlossofIEEEmodel1) function qc = convection_heat (diameter,conductor_temperature,

ambient_temperature,wind_angle, wind_speed,elevation) 2) tfilm = (ambient_temperature + conductor_temperature)/2; 3) D = diameter; 4) pf = air_density(tfilm,elevation); 5) Vw = wind_speed; 6) uf = air_viscosity(tfilm); 7) kf = air_conductivity(tfilm); 8) ka = kangle(wind_angle); 9) qc1 = (1.01+0.0372*(D * pf *Vw / uf)^0.52)* kf *ka

*(conductor_temperature - ambient_temperature); 10) qc2 = (0.0119*(D * pf *Vw / uf)^0.6)* kf *ka *(conductor_temperature -

ambient_temperature); 11) qc3 = 0.0205*pf^0.5*D^0.75* (conductor_temperature -

ambient_temperature)^1.25; 12) qc4 = max(qc1,qc2); 13) qc = max(qc4,qc3);

9.1.2.2. ConvectionheatlossofCIGREmodel1) function [qc] = cigre_convection(inner_diameter,outer_diameter,... 2) ambient_temperature,condcutor_temperature,wind_speed,wind_direction,... 3) elevation) 4) d = inner_diameter/1000; 5) D = outer_diameter/1000; 6) ta = ambient_temperature; 7) tc = condcutor_temperature; 8) v = wind_speed; 9) dir = wind_direction; 10) h = elevation; 11) tf = (ta+tc)/2; 12) rf = d/(2*(D-2*d)); 13) pf = exp(-1.16e-4 * h); 14) vis = 1.32e-5+9.5e-8*tf; 15) con = 2.42e-2+7.2e-5*tf; 16) re = pf * v * D / vis; 17) if re < 2650 18) b1 = 0.641; 19) n = 0.471; 20) else 21) if rf > 0.05

Appendix

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22) b1 = 0.048; 23) n = 0.800; 24) else 25) b1 = 0.178; 26) n = 0.633; 27) end 28) end 29) nu90 = b1 * re^n; 30) if v<0.5 31) nu = 0.55 * nu90; 32) else 33) if dir<24 34) nu = nu90 * (0.42+0.68*sin(dir*pi/180)^1.08); 35) else 36) nu = nu90 * (0.42+0.58*sin(dir*pi/180)^0.9); 37) end 38) end 39) qc = pi * con * (tc-ta) * nu;

9.1.2.3. Radiationheatloss1) function qr = radiated_heat (diameter,emisstivity, conductor_temperature,

ambient_temperature) 2) D = diameter; 3) e = emisstivity; 4) tc = conductor_temperature; 5) ta = ambient_temperature; 6) qr = 0.0178 * D * e * ( ((tc+273)/100)^4 - ((ta+273)/100)^4);

9.1.2.4. Solarheatgain1) function qs = solar_heat (diameter, solar_density, absorptivity) 2) qs = diameter * solar_density * absorptivity

9.1.3. Dynamicthermalratingcalculation1) function dynamic_thermal_rating = dtr (diameter,conductor_temperature,

ambient_temperature,wind_angle,wind_speed,elevation,emisstivity,solar_rate, r25,r75, absorptivity)

2) qc = convection_heat(diameter,conductor_temperature, ambient_temperature, wind_angle,wind_speed,elevation);

3) % calculate the convection heat 4) qr = radiated_heat (diameter,emisstivity,conductor_temperature,

ambient_temperature); 5) % calculate the radiated heat 6) qs = solar_heat(diameter, solar_rate, absorptivity; 7) % because the solar heat is measured directly in DTR monitoring, this value 8) % is not calculated ad the method in IEEE738

Appendix

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9) r = ((r75-r25)/50)*(conductor_temperature-25)+r25; 10) % the calculation of AC resistance in the conductor temperature as given 11) dynamic_thermal_rating = ((qc+qr-qs)/r)^0.5;

9.1.4. Matlabcodeforplottingtherelationbetweenheightandwindvelocity1) syms t 2) h = (6:0.1:10); 3) [row col] = size(h); 4) answer_final = []; 5) for i = 1:col 6) answer = solve((1.01+19.8*(log(20*h(i)))^0.52)*0.0295*(t-

40)+0.25*(((t+273)/100)^4-3.13^4)==74.1); 7) answer_final = cat(2,answer_final,answer); 8) end 9) answer_final_double = double(answer_final); 10) y_one = answer_final_double(1,:); 11) y_two = answer_final_double(2,:); 12) figure; 13) scatter(h,y_one,10);axis([6 10 0 100]); 14) grid on 15) hold all 16) scatter(h,y_two,10);axis([6 10 81 83]);

9.2. Numericdataforthermalratingcalculation

9.2.1. TablesforDTRcalculationmodelscomparisonTable 19: air viscosity, density and thermal conductivity by temperature

Temperature

( )

Air viscosity

Air density Kg/m3

Air thermal conductivity pa∙s 0m 1000m 2000m 4000m W/(m*c)

0 1.72E-05 1.293 1.147 1.014 0.785 0.0242 5 1.74E-05 1.270 1.126 0.995 0.771 0.0246 10 1.76E-05 1.247 1.106 0.978 0.757 0.0250 15 1.79E-05 1.226 1.087 0.961 0.744 0.0254 20 1.81E-05 1.205 1.068 0.944 0.731 0.0257 25 1.84E-05 1.184 1.051 0.928 0.719 0.0261 30 1.86E-05 1.165 1.033 0.913 0.707 0.0265 35 1.88E-05 1.146 1.016 0.898 0.696 0.0269 40 1.91E-05 1.127 1.000 0.884 0.685 0.0272 45 1.93E-05 1.110 0.984 0.870 0.674 0.0276 50 1.95E-05 1.093 0.969 0.856 0.663 0.0280 55 1.98E-05 1.076 0.954 0.843 0.653 0.0283 60 2.00E-05 1.060 0.940 0.831 0.643 0.0287

Appendix

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65 2.02E-05 1.044 0.926 0.818 0.634 0.0291 70 2.04E-05 1.029 0.912 0.806 0.625 0.0295 75 2.07E-05 1.014 0.899 0.795 0.616 0.0298 80 2.09E-05 1.000 0.887 0.783 0.607 0.0302 85 2.11E-05 0.986 0.874 0.773 0.598 0.0306 90 2.13E-05 0.972 0.862 0.762 0.590 0.0309 95 2.15E-05 0.959 0.850 0.752 0.582 0.0313 100 2.17E-05 0.946 0.839 0.741 0.574 0.0317

Table 20: coefficients for the calculation of heat flux rate

Clear atmosphere Unclear atmosphere A –42.2391 A 53.1821

B 63.8044 B 14.2110

C –1.9220 C 6.6138 × 10-1

D 3.46921 × 10-2 D –3.1658 × 10-2

E –3.61118 × 10-4 E 5.4654 × 10-4

F 1.94318 × 10-6 F –4.3446 × 10-6

G –4.07608 × 10-9 G 1.3236 × 10-8

Table 21: the comparison of IEEE and CIGRE AC resistance calculation

TEMPERATURE (IEEE) (CIGRE) DIFFERENCE PERCENT

25 3.6215 3.5983 ‐0.0232 ‐0.6447%

26 3.6356 3.6134 ‐0.0222 ‐0.6144%

27 3.6497 3.6285 ‐0.0212 ‐0.5843%

28 3.6638 3.6436 ‐0.0202 ‐0.5544%

29 3.6779 3.6587 ‐0.0192 ‐0.5248%

30 3.6920 3.6738 ‐0.0182 ‐0.4954%

31 3.7061 3.6889 ‐0.0172 ‐0.4663%

32 3.7202 3.7040 ‐0.0162 ‐0.4374%

33 3.7343 3.7191 ‐0.0152 ‐0.4087%

34 3.7484 3.7342 ‐0.0142 ‐0.3803%

35 3.7625 3.7493 ‐0.0132 ‐0.3521%

36 3.7766 3.7644 ‐0.0122 ‐0.3241%

37 3.7907 3.7795 ‐0.0112 ‐0.2963%

38 3.8048 3.7946 ‐0.0102 ‐0.2688%

39 3.8189 3.8097 ‐0.0092 ‐0.2415%

40 3.8330 3.8248 ‐0.0082 ‐0.2144%

41 3.8471 3.8399 ‐0.0072 ‐0.1875%

42 3.8612 3.8550 ‐0.0062 ‐0.1608%

43 3.8753 3.8701 ‐0.0052 ‐0.1344%

44 3.8894 3.8852 ‐0.0042 ‐0.1081%

45 3.9035 3.9003 ‐0.0032 ‐0.0820%

46 3.9176 3.9154 ‐0.0022 ‐0.0562%

Appendix

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47 3.9317 3.9305 ‐0.0012 ‐0.0305%

48 3.9458 3.9456 ‐0.0002 ‐0.0051%

49 3.9599 3.9607 0.0008 0.0202%

50 3.9740 3.9758 0.0018 0.0453%

51 3.9881 3.9909 0.0028 0.0702%

52 4.0022 4.0060 0.0038 0.0949%

53 4.0163 4.0211 0.0048 0.1194%

54 4.0304 4.0362 0.0058 0.1437%

55 4.0445 4.0513 0.0068 0.1678%

56 4.0586 4.0664 0.0078 0.1918%

57 4.0727 4.0815 0.0088 0.2156%

58 4.0868 4.0966 0.0098 0.2392%

59 4.1009 4.1117 0.0108 0.2627%

60 4.1150 4.1268 0.0118 0.2859%

61 4.1291 4.1419 0.0128 0.3090%

62 4.1432 4.1570 0.0138 0.3320%

63 4.1573 4.1721 0.0148 0.3547%

64 4.1714 4.1872 0.0158 0.3773%

65 4.1855 4.2023 0.0168 0.3998%

66 4.1996 4.2174 0.0178 0.4221%

67 4.2137 4.2325 0.0188 0.4442%

68 4.2278 4.2476 0.0198 0.4661%

69 4.2419 4.2627 0.0208 0.4880%

70 4.2560 4.2778 0.0218 0.5096%

71 4.2701 4.2929 0.0228 0.5311%

72 4.2842 4.3080 0.0238 0.5525%

73 4.2983 4.3231 0.0248 0.5737%

74 4.3124 4.3382 0.0258 0.5947%

75 4.3265 4.3533 0.0268 0.6156%

9.2.2. TablesforCAT‐1systemTable 22: the relation between NRS temperature and solar rate, thermal rating

NRS temperature (c)

Solar rate per meter (W/m)

Solar change (W/m)

DTR (A)

DTR change (A)

45 12.61803 0.253455 1013.569 1.366633

45.1 12.87148 0.253498 1012.202 1.368715

45.2 13.12498 0.253542 1010.833 1.370806

45.3 13.37852 0.253585 1009.462 1.372905

45.4 13.63211 0.253628 1008.09 1.375013

45.5 13.88574 0.253672 1006.715 1.37713

45.6 14.13941 0.253715 1005.337 1.379256

45.7 14.39312 0.253759 1003.958 1.38139

45.8 14.64688 0.253803 1002.577 1.383534

45.9 14.90068 0.253846 1001.193 1.385687

Appendix

- 89 -

46 15.15453 0.25389 999.8076 1.387848

46.1 15.40842 0.253933 998.4197 1.390019

46.2 15.66235 0.253977 997.0297 1.3922

46.3 15.91633 0.254021 995.6375 1.394389

46.4 16.17035 0.254065 994.2431 1.396588

46.5 16.42442 0.254108 992.8465 1.398797

46.6 16.67852 0.254152 991.4477 1.401014

46.7 16.93268 0.254196 990.0467 1.403242

46.8 17.18687 0.25424 988.6435 1.405479

46.9 17.44111 0.254284 987.238 1.407726

Table 23: the error because of angle between NRS and OHL

angle 06:00 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00

0 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

1 8% 5% 3% 2% 1% 0% 0% 0% 1% 2% 3% 5% 9%

2 16% 10% 6% 3% 2% 1% 0% 1% 2% 3% 6% 10% 18%

3 25% 15% 9% 5% 3% 2% 0% 1% 3% 5% 8% 15% 27%

4 33% 21% 11% 7% 4% 2% 0% 2% 4% 6% 11% 20% 36%

5 41% 26% 14% 9% 5% 3% 0% 2% 4% 8% 14% 25% 44%

6 50% 31% 17% 11% 6% 3% 1% 2% 5% 10% 16% 30% 53%

7 58% 36% 20% 13% 8% 4% 1% 3% 6% 11% 19% 35% 62%

8 66% 41% 23% 14% 9% 5% 1% 3% 7% 12% 21% 39% 71%

9 75% 47% 26% 16% 10% 5% 1% 3% 7% 14% 24% 44% 79%

10 83% 52% 29% 18% 11% 6% 2% 3% 8% 15% 26% 49% 88%

11 92% 57% 32% 20% 12% 7% 2% 3% 9% 17% 29% 54% 96%

12 100% 63% 35% 22% 14% 8% 2% 3% 9% 18% 31% 58% 105%

13 108% 68% 39% 24% 15% 9% 3% 3% 10% 19% 33% 63% 113%

14 117% 73% 42% 26% 16% 9% 3% 4% 10% 20% 36% 67% 121%

15 125% 79% 45% 28% 18% 10% 3% 4% 11% 22% 38% 72% 130%

16 134% 84% 48% 30% 19% 11% 4% 4% 11% 23% 40% 76% 138%

17 142% 89% 51% 33% 21% 12% 4% 3% 12% 24% 42% 81% 146%

18 150% 95% 54% 35% 22% 13% 5% 3% 12% 25% 45% 85% 154%

19 159% 100% 58% 37% 23% 14% 5% 3% 13% 26% 47% 89% 162%

20 167% 105% 61% 39% 25% 15% 6% 3% 13% 27% 49% 93% 170%

21 175% 111% 64% 41% 27% 16% 7% 3% 13% 28% 51% 97% 178%

22 184% 116% 67% 43% 28% 17% 7% 3% 13% 29% 53% 102% 185%

23 192% 121% 70% 46% 30% 18% 8% 3% 14% 30% 55% 106% 193%

Table 24: CAT-1 ambient temperature and maximum current

ambient temperature (c) DTR (A) DTR change (A)

35 1050.198 0.865441

35.1 1049.333 0.866303

Appendix

- 90 -

35.2 1048.466 0.867168

35.3 1047.599 0.868036

35.4 1046.731 0.868905

35.5 1045.862 0.869777

35.6 1044.992 0.870651

35.7 1044.122 0.871527

35.8 1043.25 0.872406

35.9 1042.378 0.873287

36 1041.505 0.87417

36.1 1040.63 0.875055

36.2 1039.755 0.875943

36.3 1038.879 0.876834

36.4 1038.003 0.877726

36.5 1037.125 0.878621

36.6 1036.246 0.879519

36.7 1035.367 0.880418

36.8 1034.486 0.88132

36.9 1033.605 0.882225

37 1032.723 0.883132

37.1 1031.84 0.884041

37.2 1030.956 0.884953

37.3 1030.071 0.885867

9.2.3. TablesforPowerDonutsystemTable 25: Power Donut conductor temperature and maximum current

conductor temperature (c) DTR (A) DTR change (A)

60 801.0108 1.175571

60.1 802.1864 1.173304

60.2 803.3597 1.171048

60.3 804.5307 1.168803

60.4 805.6995 1.166568

60.5 806.8661 1.164343

60.6 808.0304 1.162129

60.7 809.1926 1.159925

60.8 810.3525 1.157731

60.9 811.5102 1.155547

61 812.6658 1.153373

61.1 813.8191 1.151209

61.2 814.9703 1.149055

61.3 816.1194 1.146911

61.4 817.2663 1.144776

61.5 818.4111 1.142651

61.6 819.5537 1.140536

Appendix

- 91 -

61.7 820.6943 1.13843

61.8 821.8327 1.136333

61.9 822.969 1.134246

62 824.1033 1.132168

62.1 825.2354 1.130099

62.2 826.3655 1.12804

Table 26: the sensitivity of tilt sensor in Power Donut system

Tilt angle (°) horizontal

tension(N) Sag (m)


(C) DTR (A)

DTR change(A)

2 36882.97 1.939649 82.13258 827.9973 4.086254

2.025 36427.44 1.963904 82.49389 832.0836 4.074972

2.05 35983.01 1.98816 82.8565 836.1586 4.064674

2.075 35549.3 2.012417 83.22048 840.2232 4.055295

2.1 35125.9 2.036674 83.58591 844.2785 4.046774

2.125 34712.47 2.060931 83.95285 848.3253 4.039057

2.15 34308.65 2.085188 84.32138 852.3644 4.03209

2.175 33914.11 2.109446 84.69155 856.3965 4.450675

2.2 33528.54 2.133705 85.06343 860.8471 4.021247

2.225 33151.63 2.157963 85.43708 864.8684 4.016251

2.25 32783.09 2.182223 85.81254 868.8846 4.011827

2.275 32422.65 2.206482 86.18988 872.8965 4.007939

2.3 32070.04 2.230742 86.56914 876.9044 4.004551

2.325 31725.02 2.255003 86.95037 880.909 4.001629

2.35 31387.33 2.279263 87.33362 884.9106 3.999144

2.375 31056.75 2.303525 87.71893 888.9097 3.997065

2.4 30733.06 2.327786 88.10635 892.9068 3.995366

2.425 30416.04 2.352049 88.49591 896.9022 3.99402

2.45 30105.48 2.376311 88.88766 900.8962 3.993003

2.475 29801.2 2.400574 89.28164 904.8892 3.992294

2.5 29503 2.424838 89.67787 908.8815 3.991869

2.525 29210.71 2.449102 90.07639 912.8733 3.99171

2.55 28924.14 2.473366 90.47724 916.8651 3.991797

2.575 28643.14 2.497631 90.88046 920.8569 3.992113

2.6 28367.54 2.521897 91.28606 924.849 3.99264

Table 27: the relation between height, wind speed and conductor temperature

Height (m) wind velocity (m/s) conductor temperature (C)

6.84 0.581860917 93.36809599

6.98 0.584257815 93.28731226

7.12 0.586607112 93.20850923

Appendix

- 92 -

7.26 0.588910662 93.13159957

7.40 0.591170212 93.05650142

7.54 0.593387412 92.98313795

7.68 0.59556382 92.911437

7.82 0.597700911 92.84133065

7.96 0.599800079 92.77275494

8.10 0.601862646 92.7056496

8.24 0.603889869 92.6399577

8.38 0.605882936 92.57562548

8.52 0.607842981 92.51260209

8.66 0.60977108 92.45083939

8.80 0.611668257 92.39029179

8.94 0.613535488 92.33091603

9.08 0.615373705 92.27267106

9.22 0.617183795 92.21551789

9.36 0.618966606 92.15941945

9.50 0.620722948 92.10434047

9.64 0.622453595 92.05024735

9.78 0.624159289 91.99710812

9.92 0.625840739 91.94489224

10.06 0.627498624 91.89357061

10.20 0.629133595 91.84311542

10.34 0.630746278 91.79350011

10.48 0.632337273 91.74469929

10.62 0.633907154 91.69668866

Table 28: tilt sensor’s systematic error

degrees radians cos() δ=0.0005 δ=0.0015 δ=0.001 δ=0.002

0 0 1 1.811927 3.138611 2.562559 3.624307

0.5 0.008727 0.999962 1.37966 2.678208 2.110898 3.158657

1 0.017453 0.999848 1.069601 2.294144 1.750825 2.759823

1.5 0.02618 0.999657 0.85233 1.978794 1.46942 2.42264

2 0.034907 0.999391 0.698842 1.721946 1.250852 2.13984

2.5 0.043633 0.999048 0.587737 1.51298 1.080334 1.903382

3 0.05236 0.99863 0.504937 1.342279 0.945847 1.705487

3.5 0.061087 0.998135 0.441462 1.20181 0.838294 1.539224

4 0.069813 0.997564 0.391554 1.085164 0.751005 1.39873

4.5 0.07854 0.996917 0.351439 0.987351 0.679137 1.279198

5 0.087266 0.996195 0.318576 0.904524 0.619168 1.176755

5.5 0.095993 0.995396 0.291211 0.833722 0.568509 1.088309

6 0.10472 0.994522 0.268101 0.772659 0.525238 1.011395

6.5 0.113446 0.993572 0.248343 0.71956 0.487905 0.944047

7 0.122173 0.992546 0.23127 0.673033 0.455406 0.884691

Appendix

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7.5 0.1309 0.991445 0.216377 0.631979 0.426885 0.832059

8 0.139626 0.990268 0.203278 0.595521 0.401672 0.785124

8.5 0.148353 0.989016 0.191672 0.562953 0.379238 0.743048

9 0.15708 0.987688 0.181319 0.533705 0.359156 0.705143

9.5 0.165806 0.986286 0.17203 0.507306 0.341082 0.670841

10 0.174533 0.984808 0.163651 0.483372 0.324736 0.639669

10.5 0.18326 0.983255 0.156056 0.461581 0.309885 0.611231

11 0.191986 0.981627 0.149141 0.441664 0.296337 0.585192

11.5 0.200713 0.979925 0.142819 0.423395 0.28393 0.561271

12 0.20944 0.978148 0.137018 0.406583 0.272529 0.539225

12.5 0.218166 0.976296 0.131677 0.391062 0.262018 0.518849

13 0.226893 0.97437 0.126745 0.376694 0.252298 0.499965

13.5 0.235619 0.97237 0.122175 0.363357 0.243285 0.482417

14 0.244346 0.970296 0.117931 0.350945 0.234905 0.466074

14.5 0.253073 0.968148 0.113979 0.339369 0.227096 0.450817

15 0.261799 0.965926 0.110291 0.328547 0.219801 0.436545

Table 29: sensitivity of wind speed measurement

wind speed (m/s) maximum current (A) change (A)

0.5 952.467 39.74169

0.6 992.2087 35.30659

0.7 1027.515 31.89581

0.8 1059.411 29.17856

0.9 1088.59 26.95495

1 1115.545 25.09645

1.1 1140.641 23.51639

1.2 1164.157 22.15401

1.3 1186.311 24.30321

1.4 1210.615 23.42054

1.5 1234.035 22.38429

1.6 1256.419 21.45356

1.7 1277.873 20.61219

1.8 1298.485 19.84723

1.9 1318.332 19.14819

2 1337.481 18.50646

2.1 1355.987 17.91489

2.2 1373.902 17.36751

2.3 1391.269 16.85928

Table 30: sensitivity of wind direction measurement

wind direction (degree) maximum current (A) change (A)

60 957.9078 1.273065

Appendix

- 94 -

61 959.1808 1.22217

62 960.403 1.176402

63 961.5794 1.135844

64 962.7152 1.10057

65 963.8158 1.070649

66 964.8865 1.046143

67 965.9326 1.027105

68 966.9597 1.013584

69 967.9733 1.005617

70 968.9789 1.003234

71 969.9821 1.006458

72 970.9886 1.015302

73 972.0039 1.029767

74 973.0337 1.049848

75 974.0835 1.075529

76 975.159 1.106783

77 976.2658 1.143573

78 977.4094 1.185852

Table 31: sensitivity of solar radiation measurement

Solar radiation (W/m2) maximum current (A) change (A)

900 1003.444 0.074975

901 1003.369 0.07498

902 1003.294 0.074986

903 1003.219 0.074992

904 1003.144 0.074997

905 1003.069 0.075003

906 1002.994 0.075008

907 1002.919 0.075014

908 1002.844 0.07502

909 1002.769 0.075025

910 1002.694 0.075031

911 1002.619 0.075037

912 1002.544 0.075042

913 1002.469 0.075048

914 1002.394 0.075053

915 1002.319 0.075059

916 1002.244 0.075065

917 1002.169 0.07507

918 1002.094 0.075076

Appendix

- 95 -

Table 32: sensitivity of ambient temperature measurement

Ambient temperature (c) maximum current (A) change (A)

35 1039.857 0.856918

35.1 1039 0.857773

35.2 1038.142 0.858629

35.3 1037.283 0.859488

35.4 1036.424 0.860349

35.5 1035.563 0.861212

35.6 1034.702 0.862077

35.7 1033.84 0.862945

35.8 1032.977 0.863815

35.9 1032.113 0.864687

36 1031.249 0.865562

36.1 1030.383 0.866439

36.2 1029.517 0.867318

36.3 1028.649 0.868199

36.4 1027.781 0.869083

36.5 1026.912 0.869969

36.6 1026.042 0.870858

36.7 1025.171 0.871749

36.8 1024.3 0.872642

9.2.4. TablesforoverloadingriskanalysisTable 33: the relation between STR and overloading risk in Canterbury

STR (A) frequency

Accumulation frequency

Percentage frequency ratio STR/DTR ratio DTR/STR

120 0 0 0.0000% 0.086239155 11.59566091 220 0 0 0.0000% 0.158105118 6.324905951 320 2 2 0.0041% 0.22997108 4.348372842 420 2259 2261 4.6505% 0.301837043 3.313045975 520 70 2331 4.7945% 0.373703006 2.675921749 620 81 2412 4.9611% 0.445568968 2.244321467 720 82 2494 5.1298% 0.517434931 1.932610152 820 93 2587 5.3211% 0.589300893 1.696925987 920 103 2690 5.5329% 0.661166856 1.51247751 1020 121 2811 5.7818% 0.733032819 1.364195401 1120 140 2951 6.0698% 0.804898781 1.24239224 1220 205 3156 6.4914% 0.876764744 1.140556811 1320 179 3335 6.8596% 0.948630706 1.054150992 1420 256 3591 7.3862% 1.020496669 0.979915007 1520 208 3799 7.8140% 1.092362632 0.915446914 1620 243 4042 8.3138% 1.164228594 0.858937845 1720 273 4315 8.8753% 1.236094557 0.808999598 1820 285 4600 9.4615% 1.307960519 0.764549071 1920 243 4843 9.9613% 1.379826482 0.724728807 2020 232 5075 10.4385% 1.451692444 0.688851143 2120 207 5282 10.8643% 1.523558407 0.656358165 2220 182 5464 11.2386% 1.59542437 0.626792482

Appendix

- 96 -

2320 137 5601 11.5204% 1.667290332 0.599775564 2420 92 5693 11.7097% 1.739156295 0.57499145 2520 96 5789 11.9071% 1.811022257 0.552174329

Table 34: the relation between STR and overloading risk in Legacy

STR (A) frequency



390 1 1 0.0018% 0.24642045 4.058105 480 348 349 0.6271% 0.303292593 3.297146 570 160 509 0.9146% 0.360165841 2.776499 660 203 712 1.2793% 0.417043577 2.397831 750 226 938 1.6854% 0.473923291 2.110046 840 215 1153 2.0717% 0.530803677 1.883936 930 202 1355 2.4346% 0.587688029 1.701583 1020 205 1560 2.8030% 0.644565742 1.551432 1110 250 1810 3.2522% 0.701446106 1.425626 1200 285 2095 3.7643% 0.758332241 1.318683 1290 279 2374 4.2656% 0.81521881 1.226665 1380 304 2678 4.8118% 0.872114718 1.146638 1470 286 2964 5.3257% 0.929012292 1.076412 1560 274 3238 5.8180% 0.985914561 1.014287 1650 240 3478 6.2492% 1.042799256 0.958957 1740 245 3723 6.6894% 1.099691248 0.909346 1830 254 3977 7.1458% 1.15659674 0.864606 1920 288 4265 7.6633% 1.21349933 0.824063 2010 270 4535 8.1484% 1.270408876 0.787148 2100 252 4787 8.6012% 1.327312026 0.753402 2190 270 5057 9.0863% 1.384218457 0.722429 2280 251 5308 9.5373% 1.441126085 0.693902 2370 256 5564 9.9973% 1.498035187 0.667541 2460 184 5748 10.3279% 1.554944952 0.64311 2550 147 5895 10.5920% 1.61185411 0.620404 2640 117 6012 10.8023% 1.66876054 0.599247 2730 75 6087 10.9370% 1.725672036 4.058105

Table 35: the relation between STR and overloading risk in Taunton

STR (A) frequency



400 2 2 0.0034% 0.251709855 400 490 62 64 0.1096% 0.308349051 490 580 150 214 0.3666% 0.364989388 580 670 128 342 0.5858% 0.421633789 670 760 134 476 0.8153% 0.478281004 760 850 149 625 1.0706% 0.53493043 850 940 195 820 1.4046% 0.591581028 940 1030 203 1023 1.7523% 0.648233383 1030 1120 275 1298 2.2233% 0.704887343 1120 1210 318 1616 2.7680% 0.761543113 1210 1300 357 1973 3.3795% 0.818199289 1300 1390 453 2426 4.1555% 0.874855068 1390

Appendix

- 97 -

1480 440 2866 4.9091% 0.93150809 1480 1570 524 3390 5.8067% 0.9881583 1570 1660 466 3856 6.6049% 1.044813273 1660 1750 436 4292 7.3517% 1.101469313 1750 1840 441 4733 8.1071% 1.158127786 1840 1930 393 5126 8.7803% 1.214781437 1930 2020 261 5387 9.2273% 1.271440048 2020 2110 231 5618 9.6230% 1.32810411 2110 2200 167 5785 9.9090% 1.384762739 2200 2290 155 5940 10.1745% 1.441421804 2290 2380 136 6076 10.4075% 1.498086333 2380 2470 86 6162 10.5548% 1.554750055 2470 2560 91 6253 10.7107% 1.611411778 2560 2650 65 6318 10.8220% 1.668086377 2650 2740 55 6373 10.9162% 1.724759974 2740

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