Natural Convection

Document Sample
Natural Convection Powered By Docstoc
					  Natural Convection




    Special Course Report
 Electronic Power Engineering
          Ørsted•DTU
Technical University of Denmark
    Submitted on 20 December 2005
1 Preface
This Report is the result of a special course at Eltek, Ørsted•DTU, The Technical University of Denmark,
carried out in autumn 2005. The Report has been inspired by previous work by the authors at Danish
Transmission System Operator (TSO) Eltra (now part of Energinet.dk).

The tutors that have been involved in the Report are Ole Tønnesen, Professor at Eltek, Ørsted•DTU and
Hans-Jørgen Høgaard Knudsen, Lektor (Senior Lecturer), Energy Engineering, Ørsted•DTU. Furthermore,
guidance and data have been received from Henning Øbro, Energinet.dk (Danish TSO) and Søren Krüger
Olsen, Energinet.dk. We also wish to mention The Valley Group, which has supplied us with additional
data and information.

The purpose of this Report has been to further investigate the usability of the natural convection in
calculation of the current-temperature relationship of an overhead line.

Lyngby, 20 December 2005




   Jan Willi Christiansen            Tue Keller                       Thomas Kjærsgaard Sørensen
   wchristiansen@gmx.de              tuekeller@gmail.com              t@ksoerensen.dk




                                                                                                      2
2 Summaries
2.1 Summary in English
The backgrounds of this Report are two earlier reports by the authors (see Literature [1] and [2]).

The purpose with this Report is to investigate the natural convection around an overhead line. It is
furthermore the purpose to investigate the potential of using the natural convection instead of the forced
convection in the calculations of the transmission capacity of an overhead line. In this Report a
mathematical model for the use of the natural convection in the determination of the current-temperature
relationship is presented. The model is examined based on both, theoretical considerations and
calculations based on tension monitoring data from multiple transmission lines.

The theories about this subject are reviewed in this Report. Both, Danish and international methods in the
form of Elsam notat TS97-438 [3] and IEEE std. 738-1993 [4] are used as a basis. Furthermore methods
or calculating the natural convection are reviewed and equations for the natural convection are constructed
from thermodynamic theories. The method of IEEE for calculating the natural convection is not
investigated in depth in this Report, because it is two to three times smaller than the method based on
thermodynamic theories.

Calculations of the current-temperature relationship based on natural convection are compared to the
previous used methods based on forced convection with a crossing wind speed of 0.6 m/s. This
comparison is conducted with several points of view.

The first view is based on a point wise analysis, which is performed by selecting ideal tension monitoring
data for the three investigated conductor types (Condor, Dove and Martin). The conclusion of this analysis
is that the calculations based on the natural convection have a bigger relative error. However the errors are
not of considerable sizes and the calculations can therefore be based either on forced convection with a 0.6
m/s crossing wind or on natural convection. There is not a noticeable difference in the result.
It was furthermore demonstrated that calculations of the forced convection, which uses the actual
measured wind from the tension monitoring system, result in large relative errors. Calculations of the
forced convection should therefore not be based on the actual measured wind.

The second view is of purely theoretical nature. The size of the natural convection is investigated with
respect to the forced convection at 0.6 m/s for different conductor types. This investigation showed that
there is a remarkable resemblance between the results. The forced convection gives approximately the
same result as the natural convection.

Summarized the Reports findings are that the natural convection can be used in the dimensioning of
overhead lines. This can be done without affecting the precision of the calculations. Furthermore it
contributes to the removal of an uncertainty factor, in this case the varying wind speed, which previously
has been calculated with a minimum value of 0.6 m/s.




                                                                                                           3
2.2 Summary in Danish
Baggrunden for denne rapport er to tidligere rapporter af forfatterne (se Literature [1] og [2]).

Formålet med rapporten er at undersøge den naturlige konvektion omkring en luftledning, samt at
undersøge potentialerne ved anvendelse af den naturlige konvektion i stedet for som hidtidigt den tvungne
konvektion. I rapporten opstilles en matematisk model til anvendelse af den naturlige konvektion i
bestemmelsen af strøm-temperaturforholdet ved dimensioneringen af en luftledning. Modellen undersøges
ud fra teoretiske betragtninger og beregninger baseret på data fra flere luftledningsspænd med
trækmålingsudstyr.

I rapporten gennemgås den hidtil anvendte teori inden for området. Der tages udgangspunkt i danske og
internationale metoder i form af Elsam notat TS97-438 [3] og IEEE std. 738-1993 [4]. Hertil gennemgås
IEEE’s beregning af naturlig konvektion, og der opstilles ligeledes formler for den naturlige konvektion
ud fra termodynamisk teori. IEEE’s metode til beregning af den naturlige konvektion undersøges ikke
nærmere i rapporten, da denne konstateres at være to til tre gange mindre end metoden opstillet ud fra
termodynamisk teori.

Beregninger af strøm-temperaturforholdet baseret på naturlig konvektion sammenlignes med den hidtidige
metode, der anvender tvungen konvektion ved 0,6 m/s. Denne sammenligning foretages ved flere
betragtninger.

Den første betragtning gennemføres ud fra en punktvis analyse, der dannes ud fra udvælgelse af data fra
trækmålingsdata for ledertyperne Condor, Dove og Martin. Konklusionen herpå er, at beregningerne ud
fra den naturlige konvektion giver en højere relativ fejl, men dette er ikke af betydelig størrelse, og
beregningerne kan således udføres med tvungen konvektion ved en vind af 0,6 m/s eller med naturlig
konvektion uden en væsentlig forskel i resultat. Yderligere blev det konstateret, at beregninger med den
tvungne konvektion med den af trækmålingsudstyret målte vind medførte store relative fejl. Beregninger
kan således ikke baseres på den reelle vind.

Den anden betragtning er ren teoretisk. Her undersøges størrelsen af den naturlige konvektion i forhold til
den tvungen konvektion ved 0,6 m/s for forskellige ledertyper. Denne undersøgelse viser, at der en
slående lighed mellem resultaterne. Den tvungne konvektion giver således samme resultat som den
naturlige konvektion.

Rapporten finder således, at naturlig konvektion kan anvendes ved dimensionering af luftledninger. Dette
kan gøres uden at påvirke præcisionen af beregningerne og medvirker endvidere til, at der ved
dimensioneringen fjernes usikkerhedsfaktorer, i dette tilfælde vinden, der hidtil har været vurderet til
minimum 0,6 m/s.




                                                                                                           4
2.3 Summary in German
Dieser Rapport basiert auf früheren Untersuchungen der Autoren (Literature [1] und [2]).

Es ist das Ziel dieser Arbeit, die natürliche Konvektion von einem Hochspannungsleiter zu untersuchen.
Die herkömmliche Art und Weise die Konvektion auszurechnen ist auf erzwungener Konvektion basiert.
In dieser Arbeit sollen die Möglichkeiten dafür analysiert werden, ob die Berechnungsmethoden der
erzwungenen Konvektion mit der natürliche Konvektion ersetzt werden können.

Es wird eine mathematische Berechnungsmethode erläutert, die die Relation zwischen der Temperatur und
der Übertragungskapazität einer Hochspannungsleitung beschreibt. Die Berechnungsmethode ist sowohl
theoretisch als auch praktisch mit Messdaten geprüft und verifiziert worden. Die Messdaten stammen von
einem „Tension Monitoring“ System, welches die Temperatur eines Hochspannungsleiter mit einer
Genauigkeit von ±3°C bestimmt.

Verschiedene Theorien bezüglich des Themas wurden in dieser Arbeit erläutert. Sowohl gängige dänische
als auch internationale Vorgangsweisen sind beschrieben ([3] und [4]). Die Methoden der natürlichen
Konvektion basieren hauptsächlich auf gängigen thermodynamischen Berechnungsmethoden.

Das Verhältnis zwischen der Temperatur und der Übertragungskapazität wurde sowohl mit der
natürlichen Konvektion als auch mit der gezwungenen Konvektion ausgerechnet. Die gezwungene
Konvektion baut dabei auf 0,6 m/s kreuzendem Wind. Ein Vergleich zwischen den Methoden wurde aus
verschiednen Perspektiven beschrieben.

Zum einem basiert der Vergleich auf einer diskreten Punktanalyse. Es wurden streng ausgewählte
Messdaten benutzt, die möglichst wenige Unsicherheitsfaktoren zuließen. Die Resultate zeigen ein relativ
ausgeglichenes Verhältnis zwischen der natürlichen und der gezwungenen Berechnungsmethode. Das
Ergebnis der Punktanalyse ist deshalb, dass beide Methoden angewendet werden können.

Der zweite und letzte Teil der Analyse ist rein theoretischer Natur. Dabei wurde nachgewiesen, dass der
Unterschied zwischen natürlicher und gezwungener Konvektion verschwindend gering ist.
Vorraussetzung hierfür ist ein Temperaturunterschied zwischen der Umgebungstemperatur und der
Leitertemperatur von mindestens 16°C. Es wurde nachgewiesen, dass für zufrieden stellende Resultate ein
Temperaturunterschied von 20°C erforderlich ist. Der Grund hierfür liegt in den gewählten Werten der
Berechnungskonstanten.

Zusammenfassend wurden in dieser Arbeit die Anwendungsmöglichkeiten der natürlichen Konvektion
nachgewiesen. Der relative Unterschied zwischen dem berechneten Zustand der Leitung und der
wirklichen Verfassung des Leiters ist ähnlich, egal ob die natürliche oder die gezwungene
Berechnungsmethode angewandt wird. Der Vorteil der natürlichen Methode ist jedoch der, dass der
Unsicherheitsfaktor, der durch den variierenden Wind in der gezwungenen Berechnungsmethode entsteht,
entfällt.




                                                                                                      5
3 Table of Contents
1      Preface................................................................................................................................................... 2
2      Summaries ............................................................................................................................................ 3
        2.1       Summary in English ..............................................................................................................................................3
        2.2       Summary in Danish ...............................................................................................................................................4
        2.3       Summary in German .............................................................................................................................................5
3      Table of Contents ................................................................................................................................. 6
4      List of symbols and expressions.......................................................................................................... 8
        4.1       Symbols ...................................................................................................................................................................8
        4.2       Expressions .............................................................................................................................................................8
5      Introduction.......................................................................................................................................... 9
        5.1       Problem.................................................................................................................................................................10
6      Calculating the temperature of an overhead line............................................................................ 11
        6.1       IEEE......................................................................................................................................................................11
        6.2       Eltra/Elsam...........................................................................................................................................................14
        6.3       Natural convection heat losses ............................................................................................................................17
7      Tension Monitoring ........................................................................................................................... 20
        7.1       Introduction to tension monitoring ....................................................................................................................20
        7.2       Accuracy of the tension monitoring equipment ................................................................................................21
        7.3       Introduction to the tension measurements ........................................................................................................21
        7.4       Theoretical expression for calculating temperature .........................................................................................22
        7.5       Tension Monitoring Data ....................................................................................................................................22
8      Data Processing .................................................................................................................................. 23
        8.1       Theoretical considerations ..................................................................................................................................23
        8.2       Parameter inaccuracy of data from Tension Monitoring Equipment.............................................................26
9      Discrete analysis ................................................................................................................................. 28
        9.1       Verification of the “natural convection calculation method”...........................................................................28
        9.2       Results of the discrete analysis............................................................................................................................29
10 Calculations with varying wind-speeds............................................................................................ 35
11 Demonstration of the mathematical model...................................................................................... 37
        11.1      Condor ..................................................................................................................................................................37
        11.2      Dove.......................................................................................................................................................................39
        11.3      Martin ...................................................................................................................................................................39
        11.4      Summation............................................................................................................................................................40
12 Upgrade potential of overhead lines................................................................................................. 42
        12.1      Condor ..................................................................................................................................................................42
        12.2      Martin ...................................................................................................................................................................44
13 Conclusion .......................................................................................................................................... 47
14 Future work........................................................................................................................................ 49


                                                                                                                                                                                            6
15 Literature............................................................................................................................................ 51
16 Appendix............................................................................................................................................. 52




                                                                                                                                                          7
4 List of symbols and expressions
4.1 Symbols
aα       = Absorption Coefficient                      R      = Electrical Resistance [Ω, Ω/km]
Alt      = Sun Altitude [°]                            RAC    = AC Electrical Resistance [Ω, Ω/km]
At       = Atmospheric Transmission Coefficient        Ra     = Rayleigh Number
Azi      = Sun Azimuth [°]                             Re     = Reynolds Number
c1       = TME Constant                                RDC    = DC Electrical Resistance [Ω, Ω/km]
c2       = TME Constant                                rs     = Distance between the Sun and Earth [m]
c3       = TME Constant                                r      = Conductor Diameter [m]
c4       = TME Constant                                Sf     = Sun Factor
c5       = TME Constant                                sq     = Sky Radiation Zone
Di       = Core Diameter [mm]                          Ta     = Air Temperature [°C]
Dy       = Conductor Diameter [mm]                     ta     = Air Temperature [K]
e        = Emission Coefficient                        taq    = Apparent Sky Temperature [K]
E        = Relative Error [%]                          Tc     = Conductor Temperature [°C]
f        = Frequency [Hz]                              tc     = Conductor Temperature [K]
g        = gravity acceleration                        Tf     = Air Film Temperature [°C]
Gr       = Grashof Number                              TNRS   = Temperature of Unloaded Conductor [°C]
He       = Elevation above Sea Level [m]               tten   = Tension [N]
I        = Current [A]                                 V      = Transversal Wind Velocity [m/s]
K        = Skin Effect Factor                          x0     = Sag at 0° C
k        = Sag-tension Factor (TME)                    αR     = Material Constant [°C-1]
k        = Thermal Conductivity [W/m·K]                ΔT     = Temperature Difference [°C]
Kangle   = Wind Direction Factor                       ϕ      = Angle between wind and conductor [°]
m        = Skin Effect Factor                          μ0     = Permeability [H/m]
n        = Skin Effect Factor                          υ      = Cinematic Viscosity [m2/s]
Nu       = Nusselt Number                              ω      = Angular Velocity [rad/s]
Pr       = Prandt Number                               ρ      = Density [kg/m3]
qc       = Convection [W/m]                            ρ0     = Specific Resistance
Qd       = Diffuse Sun Radiation [W/m2]                θ      = Effective Angle of the Sun [°]
qr       = Heat Radiation [W/m]                        θl     = Conductor Direction in Relation t. North [°]
qs       = Sun Radiation [W/m]                         θsl    = Angle Between Sun and Conductor [°]
Qs       = Direct Sun Radiation [W/m2]
Qsk      = Sun Constant



4.2 Expressions

Ampacity        =       Maximum capacity of an overhead line measured in ampere
OHL             =       Overhead line
TME             =       Tension Monitoring Equipment




                                                                                                          8
5 Introduction
Parts of the electrical grids have nowadays reached the limits of their ampacity, the limits which
originally were designed in the sixties and seventies. As a consequence to this, creative ways for
calculating higher ampacities for existing overhead lines (OHLs) have been derived. When it becomes
possible to calculate higher ampacities, then an upgrading of existing OHLs is justifiable.

The main factor which limits the ampacity is the sag of the OHL. When the line gets heated the sag
increases and visa versa. The sag is determined by the actual temperature of the line. In the
dimensioning of the OHL the maximum operating temperature of the chosen conductor type has been
used. This means that the temperature increases when the sag increases, and the maximum sag is
reached when the conductor has its operating temperature. This situation shows that the main condition
for determining the actual ampacity of an OHL is to calculate the actual temperature of an OHL. The
temperature of an OHL is dependant on the energy balance of the line, which is composed of four main
factors: The cooling of the line due to 1) radiation and due to 2) convection, the warming due to heating
of the 3) sun radiation (direct and diffuse) and due to 4) losses in the line when transmitting power.

In the dimensioning of an OHL these four factors are estimated in accordance with worst case
scenarios, which mean minimum cooling (due to 1 and 2) and maximum heating (due to 3 and 4). The
certainty of each factor varies depending on different factors and it is nearly impossible to determine
fixed values of these factors. The ratings of an OHL are therefore based on worst case assumptions and
equations. One of these assumptions is that the OHL at all times will be exposed to a minimum wind of
0.6 m/s, which leads to a minimum forced convection. This assumption ensures that the line will satisfy
declaratory requirements on OHL sag.

However, a minimum wind cannot always we expected. For example, when OHL cross forests or when
the OHL are screened from the wind by tall buildings or other obstacles. This could result in an
overheating of the OHL and thereby result in greater sag. The assumption of a 0.6 m/s crosswind is
therefore a statistical assumption. The question is how the ratings of an OHL can be improved, without
violating declaratory requirements.

One prime concern for the upgrading of OHLs is therefore the cooling effect due to convection. The
common practice is (as mentioned) to assume that a conductor will always be affected by a minimum
forced convection (wind crossing the line) of 0.6 m/s. This may be wrong and therefore another form of
convection can be utilized in the calculation of the ampacity for an OHL.

As an alternative, the natural phenomenon called natural convection or free convection, which also
cools the OHL by airflow around the conductor, can be used. Natural convection is the movement of
air around the conductor caused by the temperature difference between the overhead line (OHL) and
the surroundings. This minimum convection will be present as soon as the conductor temperature is
different to the temperature of the surroundings. Therefore, a rating based on natural convection rather
than forced convection can lead to more secure results.

However, it remains unsettled precisely how the natural convection can be calculated with concern to
OHLs. Furthermore, it is not clarified whether the use of the natural convection in the calculation of
design parameters of OHL’s will lead to safer and more precise results or not.


                                                                                                       9
This leads to a series of questions concerning natural convection. These queries will be prepared and
the results listed in this Report.

5.1 Problem
The analyses in the Report will be composed of two main subjects. The first purpose is to construct a
mathematical model for the connection between the temperature of a conductor on an OHL and the
ampacity. This connection will be considered with concern to the parameters of the conductor and the
surroundings; parameters like diameter, emission, absorption, the temperature of the surroundings and
the sun radiation. The mathematic model will be based on natural convection instead of forced
convection.

The second subject is a judgement of whether it is possible on the basis of the new model to increase
the ampacity of existing conductors or not. If it can be shown that the new model gives secure and
more precise results, then a recommendation of using the natural convection instead of forced
convection can be given.

Tension monitoring data will be used as a reference. Tension monitoring gives the actual temperature
of the conductor with a precision of ±3°C. Measured data from three different conductor types have
been provided by Energinet.dk and The valley Group.




                                                                                                  10
6 Calculating the temperature of an overhead line
The following Chapter will present the equations used in this Report. It is based on know methods, and
readers that are familiar with the dimensioning of overhead lines are advised to skip ahead to Section
6.3. In Section 6.3 natural convection is presented.

The energy balance for an overhead line (OHL) can be divided into four energy factors. The first
energy source is the heat development in the OHL due to the transmitted electric power (I2R), the
second is the solar radiation (qs), the third is the radiation from the conductor to the surroundings (qr)
and finally the fourth factor is convection around the OHL, (qc), which can be forced or natural
convection. The solar radiation and the transmitted current heats the OHL, while both the radiation to
the surroundings and the convection have a cooling effect on the OHL. This energy balance can be
expressed by following equation:
                                 I2 R + qs = qc + q r                                 (6.1)

Several methods for calculating the elements in the energy balance are available. In this Report two
methods are considered. These methods are described in IEEE std. 738-1993 [4] and in Elsam TS97-
438 [3]. IEEE std. 738-1993 is used in the USA, while the former Danish transmission system operator
(TSO), Eltra (now part of Energinet.dk), uses the method presented in Elsam TS97-438. In the
following two chapters these two methods will be explained in detail.

6.1 IEEE
The following method for calculating the current-temperature-relationship for an OHL is recommended
by IEEE. In this Report, this method will be presented in short terms. For further information the IEEE
std. 738-1993 [4] should be consulted.

The energy balance is, as mentioned, composed of four energy sources. The procedures to determine
each of them will be expressed in the following:

6.1.1 Solar heating
The determination of the solar heating, qs, is based on the absorption coefficient of the conductor, aα,
and the diameter of the conductor, Dy.
                              qs = aα ⋅ Qs sin(θ ) Dy                              (6.2)

Furthermore, (6.2) includes the effective angle of the sun, θ, and the solar radiation heat flux, Qs, which
are treated below.

θ is based on the sun altitude and azimuth, Alt and Azi, and the azimuth of the conductor, θl (conductor
direction in relation to north).
                          θ = cos −1 (cos( Alt ) cos( Azi − θ 1 ) )                 (6.3)

The sun altitude and azimuth is dependent on the latitude and on the time of the day. In Table 6.1 Alt
and Azi are given at different latitudes and times. In the calculation a northern latitude of 55° will be
used which roughly covers Denmark.




                                                                                                        11
                  Northern       10:00                 12:00                  14:00
                  latitude       Alt        Azi        Alt         Azi        Alt        Azi
                  45°            57         122        68          180        57         238
                  50°            54         128        63          180        54         232
                  60°            47         137        53          180        47         223
                  70°            40         143        43          180        40         217

               Table 6.1 Altitude, Alt, and Azimuth, Azi, in degrees of the sun at various latitudes.
                                         Northern hemisphere – June 10 and July 3

The solar radiation heat flux, Qs, is given in Table 6.2 for various solar altitudes. The influence of the
solar radiation on the OHL increases when the distance between the ground and the conductor grows.
As a consequence of this, Table 6.3 has been derived, where different multipliers for Qs are given
depending on the altitude of the conductor.

                              Solar       Clear atmosphere     Industrial atmosphere
                              altitude    (W/m2) - Qs          (W/m2) - Qs
                              5           233.6                135.6
                              10          432.7                240.0
                              15          583.4                328.3
                              20          693.2                421.9
                              25          769.6                501.6
                              30          828.8                570.5
                              35          877.3                618.9
                              40          908.5                662.0
                              45          940.8                694.3
                              50          968.8                726.6
                              60          993.5                770.7
                              70          1022.6               809.4
                              80          1031.2               833.1
                              90          1037.6               849.3
                                   Table 6.2 Heat flux at selected solar altitudes

                          Elevation above sea level             Multiplier for values in
                          He (m)                                Table 6.2
                          0                                     1.00
                          1500                                  1.15
                          3000                                  1.25
                          4600                                  1.30

                                  Table 6.3 Heat flux multiplier at high altitudes

6.1.2 Radiated heat losses
The OHL will radiate heat to the surroundings depending on the temperature difference between the
conductor and the surrounding air. This heat radiation can be described by the following equation,
(6.4).



                                                                                                        12
                                          W
                          qr = e ⋅ 5.674 ⋅ 10− 4
                                         mK2  4
                                                ⋅ π ⋅ tc − t a ⋅ Dy
                                                        4     4
                                                                   (          )        (6.4)

The radiated heat loss is calculated with the temperature of the conductor and of the surroundings in
Kelvin, tc respectively ta, the emission coefficient, e, and the diameter of the conductor, Dy.

6.1.3 Forced convection heat loss
In IEEE std. 738-1993 two ways of calculating the forced convection around a conductor, qc, are given.
The methods should be used for low and high wind-speed areas respectively. The larger calculated
value of the two heat radiations should be used.
                                                     0.52
                                          ⎛ Dy ⋅ V ⎞
                 qc1 = (0.3136 + 1.2172 ⋅ ⎜        ⎟ ) ⋅ k ⋅ (Tc − Ta )         (6.5)
                                          ⎝ ν ⎠
                                                             0.6
                                         ⎛ D ⋅V ⎞
                         qc 2 = 0.5561⋅ ⎜ y ⎟ ⋅ k ⋅ (Tc − Ta )                        (6.6)
                                         ⎝ ν ⎠
The heat loss caused by forced convection is dependent on the angle, φ, between the wind and the
OHL. When the wind-speed direction is orthogonal to the OHL, then the cooling effect is at its
maximum. A wind-speed direction in parallel with the OHL will, on the other hand, give the lowest
cooling effect. This is included in the calculation by introducing a wind direction factor, Kangle.
             K angle = 1.194 − cos ( ϕ ) + 0.194 cos ( 2ϕ ) + 0.368sin ( 2ϕ )         (6.7)

This factor K is multiplied with the convection heat loss, qc.

The forced convection heat loss is calculated from the diameter of the conductor, Dy, wind speed, V,
the conductor and air temperature, tc and ta respectively, the thermal conductivity of air, k, and the
cinematic viscosity of air, υ. An equation for the cinematic viscosity is given in (6.8).
                                                                                                             m2
 ν = (1,3272 ⋅10−5 +3,8485 ⋅10-8 ⋅ Tf -3,6592 ⋅10-11 ⋅ Tf 2 +6,2093 ⋅10-14 ⋅ Tf 3 ) ⋅ (1+3,67 ⋅10-3 ⋅ Tf )        (6.8)
                                                                                                             s

6.1.4 Conductor resistance
The resistance of the conductor is assumed to be proportional with the temperature of the conductor.
For the determination of resistance (6.9) and two resistances at known temperatures will be used. It is
common that two values of the resistance at two different temperatures are listed in the data sheets for
OHLs.
                             R (t high ) − R(t low )
                   R(t c ) =                         ⋅ (t c − t low ) + R (t low ) (6.9)
                                  t high − t low
In (6.9) a high and a low temperature, thigh and tlow respectively, and the resistances at these
temperatures, R (t high ) and R (t low ) , are used to calculate the resistance of the OHL at a certain
temperature.

This concludes the IEEE method of calculating the current-temperature relationship.




                                                                                                                          13
6.2 Eltra/Elsam
In western Denmark (the transmission area of the former TSO, named Eltra) another method then IEEE
std. 738-1993 has been used to determine the factors in the energy balance equation for an OHL. The
calculation method of Eltra [3] is in many ways similar to the method recommended by IEEE, but
simultaneously both methods lead to different results. In the following the method used by Eltra
(henceforth called Eltra’s method) will be presented (as given in Notat TS97-438, Elsam System [3]).

6.2.1 Solar heating
In Eltra’s method the solar heating qs is calculated with consideration to both the direct and the diffuse
solar heat flux, Qs and Qd respectively, see following (6.10).
                               qs = aα ⋅ ( Qs + Qd ) ⋅ Dy                            (6.10)

The direct solar heating, Qs, is calculated by (6.11).
                            Q s = Q sk ⋅ S f ⋅ At ⋅ sin(θ sl )                        (6.11)

The sun constant, Qsk, is dependent on the earth distance to the sun, rs, which varies throughout the
year. See (6.12).
                                          152.1 ⋅ 106 km
                        Qsk = 1309W m 2 ⋅                                        (6.12)
                                                rs
The sun factor, Sf, is in Eltra’s method set to one. The purpose of the sun factor is to describe how
cloudy the sky is. The factor varies between 0 and 1, where a value of 0 represents a dark cloudy sky. A
value of 1 on the other hand represents a clear sky.
Another factor in (6.11) is At, the atmosphere transmission coefficient, which is dependent on the sun
altitude as given in (6.13).
                                                 (
                                   A t = 0.73 ⋅ 1 − e −0.2⋅Alt   )                    (6.13)

The last factor in (6.11) is the angel, θsl, between the sun and the conductor. This angel can be
calculated with (6.14).
           θ sl = arccos(cosθ 1 ⋅ cos Alt ⋅ cos Azi + sin θ 1 ⋅ cos Alt ⋅ sin Azi ) (6.14)

Where θ1 is the conductor direction in relation to north, Alt is the sun altitude and Azi is the sun
azimuth.
The diffuse solar heat flux is calculated as being only dependent on the sun altitude, Alt (see (6.15)).
                                    Qd = 4.5 ⋅ Alt                                   (6.15)

(6.15) is an empirical equation and it gives an average value of the diffuse solar heat flux in relation to
the altitude of the sun.

6.2.2 Radiated heat loss
The radiated heat loss is calculated as energy losses to the surroundings in the proximity, qr1, and as
heat losses to the sky, qr2. The summation of these parts gives the total radiated heat losses as given in
following (6.16).
                                   q r = q r1 + q r 2                                (6.16)




                                                                                                        14
The radiated heat loss to the close surroundings are dependent on the emission coefficient, e, and on the
diameter of the conductor, Dy. These two factors vary for different types of conductors. Furthermore,
the temperature of the conductor, tc, and the surroundings, ta, plus the sky radiation zone, sq, are factors
which influence the heat loss to the surroundings.
                                                                               W
               qr1 = (1 − sq ) ⋅ e ⋅ Dy ⋅ π ⋅ (tc 4 − ta 4 ) ⋅ 5.66961⋅10−8 2 4       (6.17)
                                                                              m K
The radiated heat loss to the sky, qr2, indicates that not all of the heat loss goes to the surroundings.
Some of the heat loss is a consequence of radiation to the sky. This heat loss is partly dependent on the
same factors as the radiated heat loss to the surroundings, qr1. Additional factors have to be added too.
These are the atmosphere transmission coefficient, At (described by (6.13)), the sky temperature, taq,
and the sun factor, Sf. In this case At is set to 0.73 as recommended in TS97-438 (Elsam note). The sky
temperature is set to 217K. The sun factor is same as presented earlier in Section 6.2.1.
                                                                                W
             qr 2 = At ⋅ S f ⋅ sq ⋅ e ⋅ Dy ⋅ π ⋅ (tc 4 − taq 4 ) ⋅ 5.66961⋅10−8 2 4   (6.18)
                                                                               m K
                                            W
The Stefan-Boltzmann ( 5.66961 2 4 ) constant is applied in (6.17) and (6.18).
                                         m K

6.2.3 Forced convection heat loss
The forced convection can be expressed by following equation:
                            qc = Nu ⋅ k ⋅ (Tc − Ta ) ⋅ π                                                          (6.19)

(6.19) expresses the general term for the forced convection heat loss, qc, affecting a conductor. In this
equation the factor k represents the thermal conductivity of the air film around the conductor while the
air film temperature is Tf. This temperature is calculated as the mean value of the conductor and air
temperature as shown in the following (6.20).
                                          T +T
                                   Tf = c a                                          (6.20)
                                              2
The thermal conductivity of the air film is calculated with (6.21). This equation is found in IEEE std.
738-1993.
                                                                      2  W
            k = (0.0242 + 7.4767 ⋅ 10− 5 ⋅ T f − 4.4071 ⋅ 10 − 9 ⋅ T f ) 2           (6.21)
                                                                        m °C
Other temperatures which affect the heat loss due to forced convection are the temperature of the
conductor, Tc, and of the air, Ta. Furthermore the forced convection is dependent on the Nusselt
Number, Nu, which is calculated by (6.22).
                                                                                    2
                            Nu = 10( −0.070431+ 0,31526⋅log(Re)+ 0.035527⋅log(Re)       )
                                                                                                                  (6.22)

Nu is based on Reynolds Number, Re, which again can be expressed by the following equation:
                                     D ⋅V
                                Re = y                                         (6.23)
                                                          ν
Hence, the Reynolds Number is the product of the conductor diameter, Dy, and the wind velocity, V,
divided by the cinematic viscosity of the air film, ν. Furthermore IEEE [4] presents a method to
calculate the cinematic viscosity which is given in (6.24).
                                                                                                             m2
 ν = (1.3272 ⋅10−5 +3.8485 ⋅10-8 ⋅ Tf -3.6592 ⋅10-11 ⋅ Tf 2 +6.2093 ⋅10-14 ⋅ Tf 3 ) ⋅ (1+3.67 ⋅10-3 ⋅ Tf )        (6.24)
                                                                                                             s


                                                                                                                           15
The cinematic viscosity is calculated with the air film temperature, Tf.

6.2.4 Conductor resistance
In Eltra’s method the calculation of the AC resistance, RAC, of a conductor is based on the DC
resistance, RDC. It is common practice to include a DC resistance at 20ºC in the data sheet for a
conductor. This value is used to determine the DC resistance at other conductor temperatures, Tc, by
following a linear relation:
                    R DC (Tc ) = R DC (20°C ) ⋅ (1 + α R ⋅ (Tc − 20°C ))         (6.25)

For aluminium conductors the material constant αR is set to 0.00403ºC-1. Based on this an AC
resistance can be calculated by using (6.26), which adjust the DC resistance with respect to skin effect
when an AC current is applied to the conductor.
                                   ( {                     } )
                      RAC = RDC 1 + 0.00519 ⋅ (m ⋅ r ) n ⋅ K                       (6.26)

The product m·r is calculated from (6.27). ‘r’ represents the diameter of the conductor and m is
calculated by using the angular velocity, ω, the magnetic permeability, μ0, and the specific resistance,
ρ0. The expression can be rearranged in a way such that the m·r product is determined from the
frequency, f, and the DC resistance, RDC.
                               ω ⋅ μ0                   f
                      m⋅r =           ⋅ r = 0.050133 ⋅                             (6.27)
                                 ρ0                    RDC
The calculation of the AC resistance is also dependent on n, where n is defined by:
         n = 4 − 0.0616 + 0.0896 ⋅ m ⋅ r − 0.0513 ⋅ (m ⋅ r ) 2 , m ⋅ r < 2.8
                                                                                      (6.28)
         n = 4 + 0.5363 − 0.2949 ⋅ m ⋅ r + 0.0097 ⋅ (m ⋅ r ) 2 , 2.8 < m ⋅ r < 5
A correction factor K is also included in the AC resistance. This correction takes into account that the
conductor is not perfectly round, but rather a collection of smaller conductors that make up the hole of
the conductor. This leads to a reduction in the generated magnetic field and thereby a reduction in the
resistance which is corrected by the factor K.
                          π D                                     2                3
               K = (cos( ⋅ ( i )0.7 + 0.11⋅( m⋅r ) − 0.04⋅( m ⋅r ) + 0.0094⋅( m⋅r ) )) 2.35 (6.29)
                          2 Dy
The calculation of K is based on the outer and inner radius of the conductor and m·r.

Further adjustments are made due to the fact that normally conductors are composed of a conducting
aluminium part and a supporting steel core. The steel core leads to a small increase of the resistance
due to eddy currents. The enlargement of the resistance is dependent on how many layers of aluminium
the conductor is composed of. The more layers the less the core will affect the resistance. In Table 6.4
the enlargement of the resistance is listed for one, two, three and four layers. The magnification listed
in the table has to be added to the parentheses in (6.26).
                              1-layer conductor              15 %       0.15 p.u.
                              3-layer conductor              3%         0.03 p.u.
                              2- and 4-layer conductor       0,3 %      0.003 p.u.

                                   Table 6.4 Addition to the resistance relation

The Eltra method has hereby been presented.


                                                                                                      16
6.3 Natural convection heat losses
It has been common practice in the calculation of the current-temperature relationship of an OHL to use
forced convection. This practice was independent of actual applied wind-speed. However, an OHL is
also affected by another cooling factor due to wind movement, which is the natural convection. Natural
convection is caused by the temperature difference between the OHL and the surroundings. This
temperature difference causes an upwards turned movement of the air around the conductor, which has
a cooling affect. In this Section, different methods for calculating the natural convection heat loss will
be presented.

Natural convection has often been ignored in the calculation of the current-temperature relationship for
an OHL. The reason for this is partly caused by the low contribution to the heat loss by the natural
convection compared to the forced convection at high wind-speeds. It has long been the practice in
many countries to use only the forced convection heat loss (with a wind of 0.6 m/s) while rating the
ampacity of OHLs. The ratings have often been based on statistical approaches from which the general
conclusion is that an OHL will at least always be affected perpendicularly by a transversal wind of
minimum 0.6 m/s. However, many load situations can be imagined where an OHL will not be affected
by forced convection (no wind). Recent studies [2] have indicated that natural convection gives better
ratings for OHL compared to ratings based on forced convection.
Before the natural convection calculation methods will be presented it has to be pointed out that the
natural and the forced convection cannot be super-positioned. This would not give the correct resulting
wind, due to the highly complex nature of the wind. IEEE has also recommended that the largest value
of either the forced or the natural convection heat loss should be used in the calculation of the heat loss,
though IEEE does recommend the actual use of natural convection.

6.3.1 IEEE natural convection heat loss
In IEEE std. 738-1993 the following method for calculating the heat loss due to natural convection is
described. IEEE recommends that the forced and the natural convection are not used simultaneously in
the calculations (see above). The natural convection heat loss can be calculated by (6.30).
                         qc = 0.0721 ⋅ ρ 0.5 Dy (Tc − Ta )
                                               0.75       1.25
                                                                                     (6.30)

The heat loss is therefore dependent on the density, ρ, of the air film at a temperature Tf, the conductor
diameter, Dy, and the temperature difference between the conductor and the surroundings.

6.3.2 Proposed method for calculating natural convection heat loss
In [2] the authors present another method for calculating the natural convection heat loss of an OHL.
This method is based on known thermodynamic equations [5] with a correction to the Nusselt Number
after M. W. Davis [6]. The following Section is only a brief summation, for a more detailed version,
see [2].

Following (6.31) is the general equation for calculating convection heat losses. The equation has
previously been presented in Section 6.2.3.
                            q c = Nu ⋅ k ⋅ (Tc − Ta ) ⋅ π                      (6.31)

The natural convection is dependent on the temperature difference between the conductor, Tc, and the
surroundings, Ta, the thermal conductivity of the air film, k, and the Nusselt Number, Nu. In this case


                                                                                                         17
with natural convection, the Nusselt Number is calculated from the Rayleigh Number, Ra, instead of
the Reynolds Number, Re as described previously.
 Nu = 10(0.12724+ 0.02238⋅log( Ra ) + 0.043043⋅(log( Ra ))^2−0.0025973⋅(log( Ra ))^3) , 103 < Ra < 109 [6] (6.32)

The Rayleigh Number, Ra, is the product of the Grashof Number, Gr, and the Prandtl Number, Pr, as
given in (6.33).
                                Ra = Gr ⋅ Pr                                  (6.33)

Gr and Pr are described by
                 Pr = 0.715 + 10−4 ⋅ T f − (2.22 ⋅10−4 ⋅ Tc − 2.3 ⋅10−2 )                 (6.34)

    and
                                         g ⋅ Dy 3 ⋅ (Tc − Ta )
                                  Gr =                                                    (6.35)
                                               Tf ⋅ v2
.
                   Pr = 0.715 + 10−4 ⋅ T f − (2.22 ⋅10−4 ⋅ Tc − 2.3 ⋅10−2 )               (6.34)

The Prandtl Number is calculated from the air film temperature, Tf, and the conductor temperature, Tc.
                                   g ⋅ Dy 3 ⋅ (Tc − Ta )
                            Gr =                                                   (6.35)
                                        Tf ⋅ v2
The Grashof Number is calculated with the gravitational acceleration, g, the conductor diameter, Dy,
the temperature difference between the conductor, Tc, and the surroundings, Ta, the air film
temperature, Tf, and the cinematic viscosity, υ. An expression for υ is given in (6.24).

6.3.3 Summary on natural convection
A short comparison of the two natural convection calculation methods will be made in this sub-Section.

Calculation based on a Condor conductor is presented in Figure 6-1. From the figure it can be observed
that the natural convection calculated with the IEEE method is two to three times lower then the Eltra
natural convection.




                                                                                                              18
                                                                       Difference Between IEEE and Eltra Natural Convection (Condor)
                                                         55

                                                                                                                             Natural qc Eltra
                                                         50
                                                                                                                             Natural qc IEEE

                                                         45
                    Natural Convection Heat Flux [W/m]




                                                         40


                                                         35


                                                         30


                                                         25


                                                         20


                                                         15


                                                         10


                                                         5
                                                              0   20      40        60        80       100       120       140         160      180

                             Figure 6-1, IEEE and Eltra natural convection for a Condor conductor

It should be noticed that the IEEE natural convection equation cannot by the authors be recognised in
the thermodynamic literature used [5]. The proposed natural convection calculation does, however,
exist in the thermodynamic literature. It originates from the calculation of the natural convection
affecting a smooth pipe with adjustments to the Nusselt Number, which takes the increased convection
caused by the conductor geometry into account.

IEEE recommends that the natural convection is calculated by a simple equation, which takes the
conductor diameter, the air density and the temperature difference ∆T between the conductor and the
surrounding air into account (see Section 6.1). The other method presented in this Report (Eltra),
however, also takes other various factors into account. Factors such as the thermal conductivity of the
air film around the conductor and a changed dependability on temperature difference and conductor
diameter. This changed dependability includes the introduction of the Nusselt Number calculated with
the Rayleigh Number (which is calculated as the product of the Prandtl Number and the Grashof
Number). It should be noticed regarding the differences between IEEE and Eltra’s methods that
different values result from the calculations. Eltra’s method uses the Nusselt Number in contrast to the
IEEE method.

This difference between both natural convection calculations have led to rejection of the IEEE method
in this Report. This is because Eltra’s method is based on natural convection for a smoothed round pipe
with adjustments to the Nusselt Number. This method takes the increased convection caused by the
geometry of an OHL into account.

The Nusselt Number was presented by M. W. Davis [6] and it has apparently not been investigated
further. It is therefore necessary either to confirm Davis’ findings or to determine a new value of the
Nusselt Number for OHLs in the calculation of the natural convection. This investigation is not done in
this Report.


                                                                                                                                                      19
7 Tension Monitoring
The aim of this Chapter is to describe how the temperature of conductors from overhead lines can be
determined. This description will be divided into a theoretical explanation of the tension monitoring
method and a summary of the determination of the conductor temperatures of those types of
conductors, which are used in the analyses in this Report.

7.1 Introduction to tension monitoring
The basic idea with tension monitoring on OHLs is to measure the change of the length of the
conductor due to temperature changes. When the wired conductor is heated the length of the conductor
will increase and visa versa. The increase in length will result in a lower horizontal tension.
Simultaneously, the sag of the conductor increases as a consequence of the increasing length of the
conductor.
The sag and the tension are inversely proportional with a factor k. This can be verified easily by
imagining a tightrope walker. The tightrope has to be tight; it won’t be possible for the walker to walk
the tightrope when there is sag. This is the same with the sag and tension of an OHL; they are inversely
proportional.
It can be summarized that the sag is an expression for the temperature of the conductor. By measuring
the tension it becomes possible to determine the temperature of the conductor. This is utilized with the
tension monitoring equipment.

The following Figure shows two load conditions of an OHL during two different temperatures (0°C
and 80°C) of the conductor:




                            Figure 7-1, Span during two different load conditions

A significant advantage in using tension monitoring for determining the load condition of an OHL is
that the method is independent of the climatic parameters. The tension gives precise load information
of the OHL.
The influence of trees on the cooling effect could be mentioned as an example. If an OHL is passing a
forest, then the OHL is cooled less inside compared to outside the forest due to different wind-speeds
and directions. It is therefore nearly impossible to find definite climatic parameters that define the
cooling conditions of OHLs. Contrary the tension monitoring equipment; it is possible by using this




                                                                                                     20
equipment to determine the exact temperature of the conductor. By measuring the horizontal tension
between two anchor towers, the temperature within the line section of the OHL can be determined.

7.2 Accuracy of the tension monitoring equipment
Eltra has in earlier studies [7] demonstrated that the temperature of a conductor with tension
monitoring can be calculated with an accuracy of ±3°C. The provisions for this are corresponding
calculating methods and validated measurements.
The importance of the ±3°C is depended on the type of conductor chosen. A Condor conductor, for
example, has a maximum temperature of 80°C. This means that an uncertainty of ±3°C is insignificant
for Condor conductors (3°C << 80°C).

In the following it is therefore assumed that the temperature of a conductor determined with tension
monitoring represents the best available and determinable conductor temperature.

7.3 Introduction to the tension measurements
The following Figure shows an example of tension measurements from a Martin conductor on a DC
OHL in the north-western part of Denmark. The maximum temperature of this conductor is 50°C.




              Figure 7-2, showing the correlation between NRS-temperature and measured tension

The x-axis on Figure 7-2 shows the NRS temperature, which is the corresponding temperature of the
conductor when it is only influenced by the weather parameters and not heated by the current. The y-
axis shows the measured tension in N.
The red line on Figure 7-2 indicates the lowest tension (36,000 N), where the sag is at its maximum and
where the conductor temperature is at its limit. The black line (line of action) indicates load conditions,
where the OHL is in its initial conditions. This means that the influences of external parameters (load
current, weather conditions) are negligible. There are some tension measurements above the line of



                                                                                                        21
action. These measurements indicate high tension either due to strong cross-wind or due to ice weight
around 0°C.
All the measurements between the line of action and the minimum tension line indicate load conditions,
where the OHL is loaded and transmits power. [1]

7.4 Theoretical expression for calculating temperature
Before the temperature can be expressed as a function of the measured tension, calculation factors for
the line section, where the tension monitoring equipment is installed, have to be determined.
Figure 7-2 showed the line of action, where the conductor is not influenced significantly by any
external parameter. This means that the line of action indicates how the conductor behaves during its
initial conditions. The calculation parameters are determined with measurements, which correspond to
the line of action.
The calculation parameters consist of the ruling span of that line section where the measurements are
made. The ruling span is used to determine coherent values between temperature and tension.
Furthermore, the factor k, which represents the inverse reversibility between sag and tension, has to be
determined. k is used to recalculate the coherent values between tension and temperature to express the
coherent values between sag and temperature of the conductor. Those coherent values can be expressed
by a linear expression of the third or fourth order (conformity between the coherent values and the
linear expression depends on which order is used).
TC = c1 ⋅ (k / t ten − x 0 )4 + c2 ⋅ (k / t ten − x 0 )3 + c3 ⋅ (k / t ten − x 0 )2 + c 4 ⋅ (k / t ten − x 0 ) + c5 (7.1)
Where Tc and tten is the tension; x0 is the sag at 0°C; k is inverse reversibility factor between sag and
tension; and c1, c2, c3, c4 and c5 are constants defined by the linear expression.

(7.1) can now be used to determine the temperature of the conductor as a function of the measured
tension.
For a more precise technical preparation of the theoretical correlation between temperature, sag and
tension, see [1] and [8].

7.5 Tension Monitoring Data
Measurements from four OHLs will be used for further investigations. The first OHL is the DC-link
between Denmark and Norway (The Skagerak Connection). The line consists of a Martin conductor
and it is designed for a temperature of 50°C. 65,000 tension measurements have been prepared and the
correlating conductor temperatures have determined and provided for further studies in this Report.
The second and the third OHL consist of two 220kV systems, which are composed of Condor
conductors. The designed maximum temperature for these lines is 80°C. Likewise as for the Martin
conductor 65,000 measurements have been prepared and provided for further studies. This data has
been provided by Energinet.dk. One more dataset is used in the Report. This data is for a Dove
conductor and has been provided by The Valley Group, Inc.




                                                                                                                      22
8 Data Processing
8.1 Theoretical considerations
The first part of Data Processing will include a Section on theoretical considerations. These
considerations will help to understand which heat transfer contributors in the energy balance equation
can be neglected during the sorting of the data. It is very important to be able to differentiate the heat
transfer coefficients from each other. Otherwise it is not possible to get a clear result from the data that
can be interpreted by our model.

First of all it should be made clear which of the measured data represents variables in the mathematical
model and most of all which of the parameters are set to a definite absolute maximum value.
Unknown parameters are set to a definite maximum in the mathematical model. This is to make sure
that the calculated heat contribution will never exceed the real heat transfer.

The theoretical considerations stem from the energy balance equations described in Chapter 6.
                                I2R = qc + qr − qs                                (8.1)

In Section 6.3 qc was defined as one of the two types of convection qforced or qnaturally. In reality qc is a
summation of the two convections. It may therefore give an erroneous result if one of the two
contributions is neglected.

8.1.1 Neglecting wind-speed
The forced contribution is due to the wind, which is a stochastic parameter that is very difficult to
determine. This is because the wind might not have the same angle or speed along the line. If the wind
(speed and angle) is measured at one point of the line it might not be the same for the whole line. It is
therefore preferable to be able to neglect the cooling effect from the wind and thereby achieve a good
coherence between the model and measured data. It is therefore preferred to delete all data where wind
above 0 m/s is measured. But this is not possible due to the drastically-reduced amount of measured
data. It is very rare to have a case with maximum line load and no wind. Again, even if the wind is
measured to 0 m/s there might be a slight wind along the line.

This small contribution from the wind that is not taken into account will contribute positively to the
OHL’s ampacity. Therefore, it can be expected that the calculated ampacity for the OHL is always
lower compared to the measured. In other words, a stationary positive error for our calculation can be
expected when the wind is neglected.

To be able to neglect the contribution from the wind one has to look into the fact that the cooling effect
for the conductor is dependent on the temperature difference between the conductor and ambient
temperature. This applies for both convection contributions qforced and qnaturally. This means that the
calculated convection will be higher for higher temperatures but also for higher wind speed1. This
means that if the temperature difference is high enough with a low wind the contribution can be
neglected. This is shown in the following Figure.


1
    Wind angle is neglected and all calculations are done with a transversal wind.


                                                                                                          23
    Figure 8-1, Forced and natural convection as function of ΔT for a Martin, a Condor and a Dove conductor

As seen above, the convection due to natural and standard forced (0.6 m/s) is almost the same for all
three conductors as long as the temperature difference does not exceed 60ºC. If the wind is halved to
0.3 m/s, natural convection will become much greater at a temperature difference of 30ºC.
At a temperature difference of 20ºC natural convection will contribute with an observable higher heat
flux compared to forced convection. Doing the calculation of the forced wind convection it was
assumed that the wind direction was traverse compared to the conductor and the calculations are
therefore worst-case. This will probably never be the case, because the angle of the wind will change
along the line. If the wind angle is different from traverse its cooling effect on the line will decrease.
Precisely how much depends on the angle.

It will therefore be assumed that a wind-speed under 0.3 m/s and a temperature difference higher than
20ºC is enough to secure a good calculated approximation of the measured data.

8.1.2 Sun radiation
Another consideration is that the model is normally used with a fixed value for sun radiation. The
problem using a fixed value for sun radiation is that its contribution in the energy balance equation is
relatively significant. The fixed number is set as an absolute worst-case. This value may be correct
during a few days in the summer, but for the rest of the year this value may assume an unrealistically


                                                                                                              24
large value. It is not the purpose of this Report to look into the assumed value of the sun radiation. In
this sub-Section it will instead be tried to relatively minimize the contribution from the sun radiation.
Or in other words find the situations where the contribution from the known term of the energy balance
is large compared to the sun radiation.

To minimize the contribution from the sun radiation, it is important to see that all terms in the energy
balance equation except sun radiation are dependent on the temperature difference ΔT.
This means that all energy terms except sun radiation in the energy balance equation will increase with
the temperature. This means that the relative contribution from the sun radiation will decrease with
increasing ΔT. This phenomenon can be illustrated by the following figures.




     Figure 8-2, Flux contribution as a function of the temperature difference between conductor and ambient
                                                          temperature.

It can be seen for the two figures to the left in Figure 8-2 that the sun radiation is calculated positively.
This is strictly not correct because radiation, natural convection and sun radiation are calculated with an
opposite sign. This is due to the sun radiation in the equation of energy balance having the opposite
sign compared to the other elements. The reason why the terms are calculated in absolute values is to
illustrate the connection between the terms in the energy balance equation.

The two figures to the left illustrate that the sun radiation is large compared to the other terms. The
temperature difference between conductor and the ambient air has to be more than 25ºC before the two
terms of the natural convection and sun radiation are numerically equal.


                                                                                                               25
This means that that the sun radiation is very difficult to neglect even if the temperature difference is
very large. It is noticed that the size of the sun radiation is almost half compared to natural convection
at a temperature difference at 60ºC.

At the two figures to the right where all the terms are summated it is seen that the energy balance is
calculated as being below zero for temperature a difference below 15ºC. More precisely the
temperature is 16.6ºC and 17.2ºC for Condor and Martin, respectively. This means that the sun
radiation below this temperature is higher than the summation of radiation and natural convection. Or
in other words, the temperature of the line will always be 17ºC higher than the ambient temperature
even if the load is zero.

The contribution from the sun radiation is very difficult to neglect and the line’s ampacity cannot be
calculated if the temperature difference is below 18ºC. This means that the temperature at least has to
be more than 18ºC in the following analysis, and preferably more than 60ºC. Due to the fact that the
temperature difference for most measuring points was below 10ºC. This fixed value for the sun
radiation seems unrealistically large, but this will not be analysed further in this Report.

The conclusion of this sub-Section is that the contribution from the natural convection has to be
relatively large compared to radiation and wind-speed. This is accomplished by choosing ΔT as large
as possible. The reason why this is valid is due to the fact that qc becomes relatively large compared to
the other terms in the energy balance equation when ΔT increases. This means that if a small ΔT is
used, the calculation will be more unreliable compared to a situation where a large ΔT is used. To
minimize the contribution from the forced convection that also increases with ΔT it is chosen that the
wind should not exceed 0.3 m/s.

8.2 Parameter inaccuracy of data from Tension Monitoring Equipment
The data from the Tension Monitor Equipment (TME) is very useful for verifying our model because
the data from the TME can be used almost immediately. To do so it is first necessary to establish rules
for the selection of the data.

Following four parameters are used in the data processing:
    - Ambient temperature
    - Conductor temperature
    - Current
    - Wind speed

The first 3 parameters are used as calculation parameters in our theoretical model. The last parameter is
primarily used to detect data with a high wind. A filter is set up to detect and delete data with
measuring errors. Following errors have been deleted:
    - ambient temperatures above 50ºC and below -40ºC
    - conductor temperatures above 85ºC and below -40ºC
    - all negative currents
    - all negative wind-speeds and wind-speeds above 40 [m/s]

The filtered data is used in the following data processing.




                                                                                                       26
The Tension Monitor Equipment (TME) is not absolutely accurate for all parameters. There is an
uncertainty in the measured wind speed data due to the fact that the TME is placed a few meters below
the conductor. This uncertainty has not been taken into account in the following calculation. Another
problem is the temperature measurement (in reality a tension measurement), which has an inaccuracy
of ±3ºC of the conductor temperature [7]. When the temperature difference is high this error will have a
small importance. In the following calculation this error might play a role because the maximum
temperature difference is only 20ºC. It is unknown what the uncertainty of the ambient temperature
measurement is. It is assumed that it can be neglected.




                                                                                                     27
9 Discrete analysis
9.1 Verification of the “natural convection calculation method”
The aim of this Chapter is to verify a method for calculating the convection by using natural convection
instead of forced convection. The calculation method has been presented in Section 6.3.2. The
verification will be based on an investigation of the correlation between the ampacity of an OHL
calculated with the natural convection method and measured reference values. The reference values are
determined with measurements from the tension monitoring equipment, which have an accuracy of
±3°C.

The investigation of the correlation will be made with a pointwise analysis. This has the advantage that
only those measurement points where the different terms of the energy balance equation are known
exactly will be chosen. This procedure can be compared with laboratory experiments, where all the
important factors of a process can be regulated. In this case the process is the energy balance of the
conductor. The terms in the energy balance equation will in the following be represented by justifiable
assumptions.

The reason why a pointwise analysis is made instead of a general investigation of all the measurements
is that the quality of the calculation will be increased because fewer uncertainties play a role. In the
pointwise analysis, 20 selected measurements from the database for each conductor (Dove, Martin and
Condor) will be used.

9.1.1 Neglecting the solar radiation qs
It was shown in the previous Chapter (8) that the solar radiation could cause complex values as a result
from the energy balance equation. This was due to the fact that qs > qc + qr in some cases. It has
therefore in this Chapter been decided to use measurements, where the energy contribution from the
sun, qs, can be ignored.
This is satisfied when TNRS is approximately equal to the temperature of the surroundings. TNRS is the
temperature of the conductor when it is only influenced by the weather conditions and therefore not
heated by current losses (see Chapter 7).

The chosen condition is that the absolute difference between TNRS and the air temperature, Ta, must be
below 1° Celsius:
                               abs TNRS − Ta < 1°C                                (9.1)

Simultaneously, the wind-speed has to be below 0.6 m/s. When the wind is severely limited and the
temperature difference in (9.1) is very small, hence giving small qr and qc, the solar radiation can be
neglected because otherwise the value of the TNRS would be high due to the heating from the sun.

Now the energy balance equation is composed only of the transmitted current I, the resistance R, the
radiation of the conductor qr and finally the convection around the conductor qc.
                                         qc + qR
                                   I=                                             (9.2)
                                            R




                                                                                                     28
9.1.2    The uncertainty of qr
It is not possible to ignore that the energy contribution of the radiation qr. qr among other things is
composed of the temperature difference Tc3 – Ta3. Furthermore, the determination of qr requires an
emission factor, which has an uncertain value between 0.7 and 0.9 depending on the surface condition
of the conductor. This uncertainty cannot be neglected. Is has been chosen to use a value of 0.8 for the
emission coefficient for Condor and Martin conductors and 0.9 for the Dove conductor.
The calculation of qr is different between the calculation method recommended by IEEE and Eltra´s
method. The correlation between qc natural and qc forced will therefore be analyzed with both
calculation methods. For further information of the calculation methods, see Chapter 6.

9.1.3 Pointwise analysis
It must be pointed out that the reference ampacity of the conductor, the determination of which is based
on the tension monitoring equipment, has an uncertainty of ±3°C. This uncertainty will be taken into
account in the following pointwise analysis.

The criteria for the pointwise analysis have been derived above. Each measurement set has been
worked through carefully with these conditions. From each of the sets of measurements (2x Condor, 1x
Dove and 1x Martin) the data with the twenty highest temperature differences (Tc – Ta) will be used.
The ampacity is calculated for each of these sets of twenty measurements with the different calculation
methods. The correlation between the results from the calculating methods and the reference can then
be analyzed. Furthermore, the correlation will be analyzed as a function of the conductor type.

To make the analysis by the different methods more comparable it has been chosen to use the same
calculation method for the conductor resistance, R. The Eltra method of calculating R has been chosen.
The IEEE method is therefore changed with respect to this.

9.2 Results of the discrete analysis
The analysis has been carried out for both Eltra´s calculation method and for IEEE´s calculation
method. The differences between the methods have been presented earlier in Chapter 6.

In the analysis the relative error, E, is calculated with respect to the reference current, Iref, the current
measured by tension monitoring equipment. This is done as stated in the equation below.
                                      I −I
                                  E = ref     ⋅100%                                   (9.3)
                                        I ref

9.2.1 Eltra's calculation method
The following Figure shows the relative error in % between the reference and the natural/forced
convection. The forced convection in this case is based on Eltra´s calculation method.




                                                                                                          29
                                                                   Relative Error in Current by Different Convection Calculations
                                     25
                                                                                                                                     Natural Convection
                                     20                                                                                              Forced Convection


                                     15
        Current Relative Error [%]




                                     10


                                      5


                                      0


                                      -5


                                     -10


                                     -15


                                     -20
                                           0          2        4         6          8         10          12         14             16        18          20
                                                                        Data Point Number - sorted after dT (Descending)

                                               Figure 9-1, Relative error in current for Dove conductor – Eltra calculation method

It can be seen on Figure 9-1 that the relative error between the ampacity of the reference and the
ampacity calculated with the natural convection method is within +10 % and -7 %. The relative error
between forced convection and the reference is within +14 % and -3 %. The red and the blue dot-dash
lines indicate the uncertainty of the calculated current due to the ±3 °C of the tension monitoring
equipment.

Due to +10 % and -7 % the natural convection generates both to high and to low ampacities. Forced
convection on the other side generates higher ampacities (+14 % and -3 %). The mean value (and
variance) of the natural convection is -2.97 ± 24.7 %. In comparison, the mean value of the forced
convection is 1.29 ± 20.6 %.
It can therefore be summarized from Figure 9-1 that the natural convection is always below the line of
the forced convection method (blue is lower than red) and that the natural convection is relatively well-
correlating for the dove conductor.




                                                                                                                                                               30
                                                              Relative Error in Current by Different Convection Calculations
                                      50
                                                                                                                               Natural Convection
                                                                                                                               Forced Convection
                                      40



                                      30
         Current Relative Error [%]




                                      20



                                      10



                                       0



                                      -10



                                      -20
                                            0    2        4          6          8        10          12        14              16       18          20
                                                                   Data Point Number - sorted after dT (Descending)

                                        Figure 9-2, Relative error in current for Martin conductor – Eltra calculation method

In Figure 9-2 the relative error for the Martin conductor is shown. The error bandwidth of the natural
convection calculation is within +36 % to 0 %, whereas the relative error of the forced convection
calculation lies within +35 % to -5 %. It can therefore be observed from Figure 9-2 that the points for
the forced convection generally lie below the natural convection. Simultaneously, almost all data points
are above 0 %. The means for the two methods are 19.7 ± 69.5 % for the natural convection and 14.2 ±
75 % for the forced convection.
As a consequence of this, none of the methods have a good correlation with the reference current. It is
on basis of this not possible to conclude which method is the best or whether the natural convection
method is usable or not.

The computations are likewise conducted for the double Condor system (Condor 1 and Condor 2). In
Figure 9-3 and on Figure 9-4 the calculations for Condor 1 and Condor 2 are presented, respectively.




                                                                                                                                                         31
                                                                                          Relative Error in Current by Different Convection Calculations
                                                                  50
                                                                                                                                                            Natural Convection
                                                                                                                                                            Forced Convection
                                                                  40
                                 Current Relative Error [%]




                                                                  30



                                                                  20



                                                                  10



                                                                      0



                                                              -10
                                                                          0       2       4        6        8        10         12       14                 16         18          20
                                                                                               Data Point Number - sorted after dT (Descending)

                                               Figure 9-3, Relative error in current for Condor 1 conductor – Eltra calculation method

For Condor 1 the natural convection calculation generally lies higher (+40 % and +10 %) than the
forced convection. The forced convection lies within the bandwidth of +5 % to +38 %. The means of
the two methods are 22.5 ± 56.2 % for natural convection and 20.1 ± 52.1 % for the forced convection.
This indicates that the forced convection method is slightly better then the natural convection method
in this case. But again it is not clearly definable whether the natural convection method is usable or not.
                                                                                           Relative Error in Current by Different Convection Calculations
                                                    60
                                                                                                                                                              Natural Convection
                                                                                                                                                              Forced Convection
                                                    50
           Current Relative Error [%]




                                                    40



                                                    30



                                                    20



                                                    10



                                                              0
                                                                  0           2       4           6         8         10          12        14              16         18          20
                                                                                                Data Point Number - sorted after dT (Descending)

                                               Figure 9-4, Relative error in current for Condor 2 conductor – Eltra calculation method



                                                                                                                                                                                        32
Figure 9-4 shows the results of the calculations on Condor 2, which are the most inconsistent results
compared with the analysis of the other conductors. The relative error of the natural convection covers
the interval between 13 % and 57 % with a mean value of 27.7 ± 113 %. The forced convection has a
bandwidth which is between 9 % and 40 % with a mean value of 24.8 ± 97.4 %. Neither of the
calculation methods shows a good correlation and again it is difficult to analyse whether the natural
convection method is usable or not. The forced convection calculation does, however, seems to be
more precise.

9.2.2 IEEE’s calculation method
When the radiation qr is calculated on IEEE´s method, approximately the same results as in the
previous Chapter are reached. An explanation for this is the relatively small contribution of qr to the
total energy balance equation. It does therefore not give any meaning to analyse each graph
individually. Instead, the results of the IEEE method investigations are compared with the Eltra method
results in Section 9.2.3. The resulting graphs for the IEEE method can be found in Appendix 16.1.

9.2.3 Summary
The Table below shows the mean and the variance depending on whether natural or forced convection
is used and depending on whether qr is calculated based on Eltra´s method or on IEEE´s method.
         Conductor Type Mean             Variance               Convection Type
              Dove     -2.97               24.70                 Natural Eltra
              Dove      1.29               20.60                 Forced Eltra
             Martin    19.65               69.53                 Natural Eltra
             Martin    14.17               74.97                 Forced Eltra
             Condor1   22.53               56.25                 Natural Eltra
             Condor1   20.12               52.07                 Forced Eltra
             Condor2   27.72              113.19                 Natural Eltra
             Condor2   24.76               97.33                 Forced Eltra

               Dove        -7.43            24.89                Natural IEEE
               Dove        -6.66            18.78                Forced IEEE
              Martin       16.30            69.60                Natural IEEE
              Martin        6.28            68.92                Forced IEEE
              Condor1      17.49            61.15                Natural IEEE
              Condor1      10.46            50.05                Forced IEEE
              Condor2      23.35           118.70                Natural IEEE
              Condor2      15.57            91.70                Forced IEEE
                               Table 9.1, Relative Error Means and Variances

Table 9.1 shows that the results from the IEEE method generally give lower mean values while the
variance is approximately the same. But the tendency is the same; whether the mean value of the
natural convection calculation error or the mean value of the forced convection calculation error is
used, the result is dependent on the conductor type. The convection around the Dove conductor type is
clearly the best correlated. The correlation is almost independent of the used method (natural or
forced). In contrast to this are the other three OHL’s, where the forced convection error is always
smaller than the error for the natural convection calculations. This indicates that the forced convection
method provides more accurate results for the convection around a Martin and a Condor conductor.
The question that has to be investigated in this Paper is whether the natural convection method can be
used for calculating the convection around a conductor. The answer to this question depends on the
acceptable error range.


                                                                                                      33
Figure 9-1 to Figure 9-4 show that the ±3°C uncertainty of the Tension Monitoring Equipment results
in an error range of 17%. This means that the values of the relative error on Table 9.1 can vary
significantly.
On the other hand it should be noticed that the natural convection method neglects one of the
uncertainty variables in the determination of the energy equation (see 6 Calculating the temperature of
an overhead line).

Summarizing it can be condensed that the results on Table 9.1 show a bigger uncertainty when the
natural convection method is used instead of the forced convection method. This uncertainty has to be
weighted against the advantage in neglecting the uncertainty factor of the wind in the energy equation.
In following chapter the consequence of using the varying wind speed in the calculation will be
demonstrated. It shows the advantage in using the natural convection, which is independent of the
present wind speed. Thereby it gives an indication of how the weighting can be carried out.




                                                                                                    34
10 Calculations with varying wind-speeds
In the calculation of the current based on the natural convection the measured wind-speed is not taken
into account. This is a big advantage using natural convection in the determination of the ampacity. The
contrast to this is in the calculation of forced convection, where the wind-speed is taken into account.
The aim of this Chapter is to analyse the influence of the measured wind-speed on the forced
convection. As mentioned before the forced convection is calculated either with a minimum of 0.6 m/s
transverse wind or with the actual measured wind speed.
The forced convection will be determined for two scenarios. The calculations will firstly be based on
the actual wind speed and secondly on the minimum wind speed. The calculations of the forced
convection will be made based on the two methods, which earlier were introduced as Eltra’s and
IEEE’s calculation method.

All results will be compared to the reference ampacity value. The reference is as in the previous
Section based on measurements from the tension monitoring equipment. The comparisons will be
presented as a relative error between the calculated ampacity and the reference ampacity. The
definition of the relative error is presented in an earlier Chapter as (9.3). The reference current has an
uncertainty of ±3ºC. The current is based on the two forced convection calculating methods (Eltra and
IEEE) and also on the natural convection calculating method. The following Figure shows the
comparison for the Condor conductor.
                                       Condor - Relative error in current by different convection calculations, variable wind
                                   400                                                  Forced convec. Eltra, dyn. wind
                                                                                        Forced convec. IEEE, dyn. wind
                                   300
         Relative error [%]




                                                                                        Natural convec.
                                   200

                                   100

                                     0

                                   -100
                                          0            20          40          60         80            100         120             140
                                                                             Data point number
                                              Condor - Relative error in current by different convection calculations, fixed wind
                                   150                                                         Forced convec. Eltra, 0.6m/s
                                                                                               Forced convec. IEEE, 0.6m/s
              Relative error [%]




                                   100                                                         Natural convec.

                                    50


                                     0


                                    -50
                                          0            20          40          60         80            100         120             140
                                                                             Data point number
                              Figure 10-1, the relative error either based on dynamic wind speeds or on fixed wind speeds



                                                                                                                                          35
In Figure 10-1 two graphs are presented. While the blue and the green lines are based on forced
convection, the red lines are based on natural convection, i.e. independent of the measured wind speed.
The red lines on both graphs are actually the same (note the range on the y-axis).

The first graph shows the relative error when using the actual measured wind speed in the calculation
of the forced convection. The graph indicates that the relative error is high. This means that the
calculation of the ampacity should not be based on the forced convection with the actual measured
wind speed. The natural convection in contrast is within an acceptable bandwidth of the relative error.

The second graph shows an acceptable relative error for all three curves. The wind speed used in these
calculations of the forced convection is fixed to 0.6m/s. It is observable from Figure 10-1 that the
natural convection error is in between both forced convection calculating methods.

The mean values of the curves on Figure 10-1 are following:
                                    Dynamic wind speed                   Fixed wind speed
    Forced convection Eltra         71 %                                 9%
    Forced convection IEEE          39 %                                 -17 %
    Natural convection                                                 2%
                      Table 10-1, mean values of the relative errors shown on Figure 10-1

It can be seen on Table 10-1 that the natural convection actually has the lowest absolute relative error.
This indicates that a determination based on the natural convection is the best available method to
achieve the ampacity of the reference. Furthermore, the same indication as in Figure 10-1 is obtained.
The error between the reference and the calculated ampacity is much lower, if the forced convection is
based on a fixed wind speed of 0.6 m/s.

Summarizing this Chapter has shown that it does not make sense to use the actual wind measurements
for determining the ampacity of an OHL. This result could have been expected due to the fact that the
measured wind speed is not representative for the resulting effects on the line. If this result is viewed in
a wider perspective it indicates that it does not give more accurate results to use the actual wind speed
in the calculation of the ampacity. Even though the wind speed locally can reach a very high value, the
general ampacity of the OHL can be determined most correctly with a fixed wind speed of 0.6 m/s.

The sub-conclusion of this Chapter is therefore that the ampacity should be calculated with a
convection which is either based on forced convection with a fixed wind speed (0.6 m/s) or on natural
convection.




                                                                                                         36
11 Demonstration of the mathematical model
In the following Chapter the mathematical methods are used to calculate the ampacity for the three
conductors, Condor, Dove and Martin over a wider range of data than in the discrete analysis. The
calculated ampacity is then compared with the reference ampacity founded by the tension monitor
equipment. This is performed separately for each conductor.

In the following calculations it should be observed that the measured data has very low temperature
differences ∆T (ambient- conductor). The guidelines in 8, which were made to ensure a good
correlation between the reference and the calculated ampacity, have been relaxed. There was not
enough measured data which fulfils the requirements. Firstly, the requirements for the maximum wind
0.3 m/s have been neglected. Secondly, the defined temperature difference ∆T of 30ºC and preferably
40ºC cannot be fulfilled. Because the temperature difference ∆T for the three conductors have never
exceeded 30ºC and for most of the measured data the temperature difference is below 15ºC. On the
other hand, it is known that the mathematical models cannot calculate real numbers if the temperature
difference is below 17-19ºC. This is due to the applied fixed value for the sun radiation (see Section
8.1.2). The used minimum temperature for each conductor is:

   -   Martin         19ºC    number of left measurements 186
   -   Condor         18ºC    number of left measurements 210
   -   Dove           17ºC    number of left measurements 30

The number of remaining measurements is relatively low. This is due to the fact that the OHL’s have
been operated on a low load during the measurement period. This low load means that the following
calculated ampacity will be erroneous. This is caused by the fixed values (qs and qr) in the energy
balance equation, which are weighted relatively high when the temperature difference is small. This
matter is described in Section 8.1.2.

The methods used to calculate the ampacity are:
       - IEEE
       - Eltra forced (0.6 m/s)
       - Eltra natural
It has been shown in Chapter 10 that the calculation of the contribution from the wind gives bad
matches to the reference value. This is the reason why fluctuating wind is not taken into account for the
models.

11.1 Condor
The following figure shows the results for the Condor conductor:




                                                                                                      37
                       Figure 11-1, Current calculated by all three models for Condor

It can be seen in Figure 11-1 that all three models almost calculate the same ampacity, as long as the
current is above 600 A. The models’ predictions are different if the measured ampacity is below 400 A.
The difference between the model and the reference cannot be explained easily, due to uncertainties in
the calculation factors.

The model using natural convection calculates a positive relative error twice. This means that the
calculated current is larger compared to the reference. The size of this current is only around 15 A
higher than the reference and therefore this non-conformance can be neglected. The Eltra forced model
also calculates too high an ampacity compared to the reference. These calculated values can eventually
be explained by the temperature bandwidth error of ±3°C. It must also be observed that these non-
conformance values, where the calculated value is higher than the reference, only appears for small ΔT.

It can be summarized that the three methods calculate ampacities which lie far under the reference
values. This statement is valid except for a few places where the calculated values are above the
reference. An explanation for these places could be that the temperature differences are small. The
three models do also calculate the same ampacity when the load is above 600 A, even though the
calculations are based on different theoretical methods. The differences between the natural method
and the forced methods are presented in Chapter 6.




                                                                                                    38
11.2 Dove
In the following Figure, the same calculations have been made for the Dove conductor. The
calculations are made with 30 measurements:




                       Figure 11-2, Current ampacity calculated by all models for Dave

It can be seen in Figure 11-2 that the correlation between the reference and the calculated ampacity is
non-existent. This is the case for all three methods. It must be observed that the number of measuring
points are 30, which may be too low. On the other hand, this conductor type showed a very good
correlation between the calculated ampacities and the reference values in Chapter 9. The differences
between the discrete and the full model are that the discrete model does not take sun radiation into
account.

It is difficult to explain the lack of correlation between the models and the measured ampacity in Figure
11-2. This could be explained by the fixed number of both the sun radiation and the radiation of the
conductor. Even though the models do not correlate with the reference it is conspicuous that there is a
good correlation between the three models. The relative error for natural convection seems to be a bit
higher compared to the two forced convection models. But generally, all the models calculate a wrong
ampacity. Before the models can be used for the Dove conductor, new fixed values must be found.


11.3 Martin
The following figure shows the results for the Martin conductor:




                                                                                                      39
                         Figure 11-3, Ampacity calculated by all models for Martin

For the last conductor analysed in this Report, the three models calculate identical ampacities. There
are some places though where the models calculate different values, but the differences never exceed
10 %. It seems that the natural method calculates an ampacity, which is a bit closer to the reference
compared to the other two methods. This is the opposite of what was shown in Chapter 9, where the
natural method gave the worst predicted values compared to the reference. It can be seen in Figure
11-3 that the calculated ampacity exceeds the reference at one place. The non-conformance occurs only
when the load of the OHL is low and therefore the calculated values are erroneous. This is due to the
fixed numbers’ relative importance when ΔT is small.

11.4 Summation
It has been shown for all conductors that the correlation between the calculated ampacities and the
reference is not ideal. This was expected due to the low temperature differences ∆T and due to the
fixed values of the sun radiation qs and the conductor radiation qr. Before a better correlation can be
achieved the fixed values must be changed. This can be done in two ways: Either a better method has to
be found for the calculation of the conductor radiation and the sun radiation, or the measured data
should contain values with higher temperature difference ∆T compared to the ones used in this Report.

For the Condor and the Martin conductor it has also been shown that the calculated ampacity is below
the reference values as long as the ampacity was above 600 A and 900 A, respectively.




                                                                                                    40
The last thing which must be observed for all the conductors is that the three models “almost” calculate
the same ampacity for each of the conductors. This is quite conspicuous because the three models are
based on different theories. It was shown in Section 8.1.1 that natural and forced (0.6 m/s) convection
predict the same ampacity. This also showed that it would be reasonable to substitute the stochastic
wind with natural convection.




                                                                                                     41
12 Upgrade potential of overhead lines
Until now the natural convection models have been evaluated by comparing them with the other
existing methods and the Tension Monitoring Equipment (TME) data. This has been done for both
discrete analyses and analyses for the whole model. In this Chapter it is assumed that the natural
method is as accurate as the other existing methods. The significant difference is that the natural
method does not use the statistical parameter of the wind.

The maximum ampacity will be determined for the conductors. The determination will be based on the
design temperature of the conductor and on the natural convection caused by the temperature difference
between the ambient temperature and the conductor temperature. The values of the ampacities will be
compared to the measured reference ampacity. The difference between the maximum ampacity and the
reference illustrates the prospect of using the natural convection model.

The following calculation is performed only for the Martin and the Condor conductors because it is
shown in Chapter 11 that the full model is not consistent with the Dove conductor’s data.

12.1 Condor
The design temperature for the Condor conductor analysed in this Report is 80ºC. This temperature is
the maximum temperature the conductor is allowed. If the design temperature is exceeded the sag of
the line will become too great. It has therefore been decided to calculate with two design temperatures
in determining the ampacity of the line. The used design temperatures are 70ºC and 80ºC. The design
temperature of 70ºC gives a secure margin compared to the conductor’s real design temperature (80ºC).
The following calculation shows the line’s ampacity during the measured period compared to the actual
load:




                                                                                                    42
     Figure 12-1, ampacity calculated by all models for Condor with conductor temperature at 80ºC and 70ºC

It can be seen in Figure 12-1 that the Condor conductor has never been fully-loaded during the
measured period. At the same time, the conductor temperature during this period has never exceeded
48ºC. This is valid even though the design ampacity (800A) is reached during the measuring period.
The size of the maximum ampacity during this period depends on the used design temperature of the
conductor. If the design temperature is 80ºC it can be observed on the second graph on Figure 12-1 that
the line could at least be loaded with an additional 300 A. If a more conservative conductor temperature
of 70ºC is chosen the conductor could be loaded with 200 A compared to the design ampacity.

Furthermore, it has been shown that the calculated ampacities for the three methods are almost equal.
Due to the relative high temperature difference between the conductor and the surroundings, the
method using natural convection predicts a higher ampacity compared to the other models. The reason
for this is that natural convection is more dependent on the temperature difference compared to the
forced convection.
It can therefore be concluded that the Condor conductor during the measurement period has never been
fully-loaded. There is therefore a potential for upgrading the line using on of the new methods.

The answer to the question regarding how much additional ampacity can be utilized, depends of the
actual ambient temperature. A fixed value of the unused capacity is therefore difficult to determine
without taking the actual ambient temperature into account. This can e.g. be done by recalculating the
design ampacity year-round for each of the conductors.




                                                                                                             43
12.2 Martin
The design temperature of the Martin conductor is 50ºC. The following calculations are equivalent to
those performed for the Condor conductor. The chosen design temperatures for the Martin conductor
are 40ºC and 50ºC. Due to the small temperature differences between the design temperature of the
conductor and the ambient temperature it is expected that the relative load can become close to its
limits for this conductor. This means that the temperature of the conductor, which is determined with
the TME, is close to the design temperature during the periods, where the ambient temperature is high.
This could happen in load situations where the ambient temperature is 25ºC, while 20ºC is used as the
design ambient temperature.
Following figure shows the results for Martin:




    Figure 12-2, Ampacity calculated by all models for martin with a conductor temperature = 70ºC and 80ºC

It can be observed on the second graph (50°C) in Figure 12-2 that the calculated maximum ampacity of
the first 30 measurement points is below the reference ampacity. This means that if the determined
maximum ampacities are calculated correctly, then the conductor has been overloaded during this
period. For measurement point 20 to 30 the eventual overload is more than 300 A.

During the measured period from point 30 to point 60 on Figure 12-2 it seems as if the line has been
almost fully-loaded. For the last 60 measurement points the calculated ampacities are higher than the
reference. Even if the loads are the same (1000 A) for the first 30 and the last 60 measurement points
the line has been loaded differently. For the first 40 measurement points it seems overloaded and for
the last 60 measurement points it seems as if there is unused capacity. This difference is due to the
different ambient temperature in the two periods. In the last period the natural convection method


                                                                                                             44
predicts the highest unused capacity of the three methods. This capacity is calculated to be
approximately 250 A.

When the maximum temperature is assumed to be 40ºC (first graph in Figure 12-2), then the calculated
ampacities are even lower. For instance, the reference ampacity and the calculated values are
approximately equal for the last 70 measurement points.

It seems as if the Martin conductor has been fully-loaded during part of the measurement period. For
the first 30 measurement points the reference load is 300 A higher compared to that calculated
maximum ampacities. This could indicate that the conductor temperature has exceeded the design
temperature of the OHL of 50ºC. When the actual temperature is above the design temperature the sag
of the OHL exceeds its maximum value, and the regulatory requirements are violated.

With the purpose of analysing the eventual overload of the Martin conductor in detail, the ambient and
conductor temperatures are shown in the following figure:




                        Figure 12-3, Ambient and conductor temperatures for Martin

It should be observed from Figure 12-3 that the conductor temperature reaches 48.2ºC. This
temperature is close to the design temperature of the line of 50ºC. The reason why the conductor
temperature is close to its design temperature is due to the high ambient temperature and due to the
high load (1000 A). In the dimensioning of the line the ambient temperature was set to 20ºC. This
means that if the ambient temperature exceeds 20ºC, then there is a risk of overloading the OHL as
soon as the OHL is loaded with its design ampacity. This situation can be observed in Figure 12-2 and
Figure 12-3.


                                                                                                   45
The temperature of the Martin conductor may actually have exceeded 50ºC due to the error bar of ±
3ºC of the TME. Therefore, the actual sag of the line may have been too great during this period; it is
just not possible to observe due to the lack of precision of the equipment. If the ambient temperature
would have been 27ºC instead of 24.1ºC the conductor could also have been even warmer. Again, this
would result in a sag above the regulatory requirements.

The reason why the maximum calculated ampacity is below the reference on Figure 12-2 is unknown.
One explanation could be the fixed values of the radiation and of the sun radiation. This is uncertain
and is not investigated further in this study. Another explanation could also be the cooling effect from
the high wind during this period. But the stochastic contribution of the wind should, as it is shown in
Chapter 10, not be taken into account in the calculation of the ampacity.

This last data set for the Martin conductor shows how important it is to have an accurate method for
determining the ampacity of an OHL. If one fixed value for the design ampacity is used all year this
may cause problems during the periods where this value is exceeded. An example for this could be high
ambient temperatures during the summer. On the other hand, this fixed value may be the reason for a
design ampacity, which is set far too low compared to the actual ampacity in other periods of the year.
If the calculation methods presented in this Report are used to recalculate the ampacity for the line
more often, then this could result in a more effective utilization coefficient of the OHL. A frequently
recalculation could ensure that the temperature of the conductor never exceeds its design temperature.

As a final comment is should be observed that the natural convection in the calculations in Figure 12-1
and in Figure 12-2 predicts a higher ampacity compared to the two other forced convection methods.
This is due to the different dependency on the high temperature for the three models, shown in Figure
8-1. This means that natural convection will predict a higher maximum ampacity at the design
temperature compared to the other two methods.




                                                                                                     46
13 Conclusion
In this Report the determination of the ampacity (based on natural convection) of three different
conductor types has been analyzed. Measurements have been provided by the industry.

A derivation of four mathematical models calculating the convection of OHLs has been presented,
wherefrom three have been further analysed. These three models are based on the same equation,
which describes the energy balance of a conductor. The equation consists of four elements: the cooling
of the conductor due to convection (qc), the cooling due to radiation of the conductor, (qr), the heating
of the conductor due to solar radiation (qs) and finally the heating of the conductor due to current
losses, I2R. The first three elements represent three uncertain variables. Each of the three models
describes a separate method, where two of them are based on forced convection (Eltra and IEEE). The
third model for determining the convection, which was examined for its reliability and credibility, is
based on natural convection.

The second step in the Report is a presentation of tension monitoring equipment. The technical issues
of this equipment are introduced. Furthermore, measured data from the tension monitoring system is
introduced. The data consists of current, weather conditions and tension measurement. The measured
current data is used as reference values in the Report. The tension monitoring data is used to determine
the temperature of the conductor. This temperature information and the weather measurements are used
to calculate the ampacity of the conductor.

Three ampacity values can be calculated with the methods. Prior to comparisons of the reference values
and the determined ampacities, criteria for the measured data have been established. The criteria are
based on theoretical considerations of necessary conditions, which have to be fulfilled to guarantee a
meaningful comparison. It is shown that the temperature difference ∆T between the surroundings and
the conductor should be a minimum 20ºC, preferably 30ºC or 40ºC. This would increase the accuracy
of the calculations with the natural convection method considerably. Furthermore, it is shown that the
wind speed should be insignificant (less than 0.3 m/s) when the accuracy of the natural convection
method is investigated. If the wind speed is below 0.3 m/s the cooling effect of the conductor due to air
flow is caused mostly by natural convection. If the wind speed is above this value then the aggregated
convection is a combination between both forced and natural convection. As a matter of practical
correctness the theoretical value of 0.3 m/s has been increased to 0.6 m/s. Wind speed measurements
below 0.6 m/s do not seem measurable. This means that the convection is either calculated with 0.6 m/s
forced wind speed or with natural convection.

The next part of the Report is a discrete analysis of the relative error between the reference value and
the calculated ampacity. The ampacity is calculated with the natural convection method and the two
forced convection methods, Eltra and IEEE. The criteria for the data used in this analysis are very
stringent. This permits the omission of one of the uncertain elements in the energy equation, the solar
radiation qs from consideration. Nevertheless, the relative errors in the discrete analysis are much too
high. More important though is the fact that all three models for calculating the convection give
approximately the same results.

Neither the natural nor the forced convection shows 100% gratifying results. This can partly be
explained by two reasons. Firstly, the assumption of eliminating the solar radiation can cause relative


                                                                                                      47
errors. The elimination results in calculations, which are based on an incomplete energy equation
model. Secondly, the left data consisted of very few measurements due to the high criteria. These few
measurements can cause statistically-invaluable results. Summarizing, the discrete analysis showed that
the natural convection method gives neither better nor poorer results than the forced convection
methods. It is therefore clear that the natural convection method and the forced convection method give
approximately the same results.

A brief investigation of the forced convection methods is presented with the purpose of determining
how the wind speed (forced airflow) can be handled in the forced convection methods. It is shown that
it does not make sense to use the actual fluctuating wind speed in the calculations. The fixed wind
speed of 0.6 m/s should rather be used because this gives smaller relative errors.

The conclusion that 0.6 m/s should be used in the forced convection method is interesting. It explains
why the natural convection model gives similar results to the forced convection. It was shown earlier
that for reliable results for the natural convection the wind speed should be less than 0.3 m/s. This
number was due to practical measurement queries increased to 0.6 m/s, i.e. the same as forced
convection. Both methods give approximately the same results with this condition. An explanation can
be that the 0.6 m/s forced wind speed is a relative and well-approximated statistical value for the real
minimum convection. This real minimum convection is precisely the air flow caused by the natural
convection.

Finalizing some calculations of the complete energy equation method are presented for each conductor.
In these calculations it is shown that all the three models give approximately the same calculated
ampacity of an OHL. It is simultaneously shown that the calculated ampacities are below the reference
values, as long as the temperature difference ∆T between the ambient and the conductor temperature is
above 20°C. This means that the models are conservative.

Furthermore calculations have shown that there is a potential in investigating the upgrading of OHLs.
Suggestions to which further investigations could be performed to allow a secure upgrading of existing
lines are given. The main issue is to reinvestigate in the calculation of the ampacity during the
upgrading process.

The significant positive statement of this Report is that the natural convection gives the same results as
the forced convection. The natural convection is independent from the wind speed variable. The
complete mathematical model of the energy equation consists therefore of one less uncertainty when
using the natural convection method. This means that the natural convection gives a possibility for
reducing the number of variables in the calculation of the ampacity of an OHL because the wind speed
can be omitted.

Summarizing the Report has shown that the natural convection gives the same results as the forced
convection. The advantage is that the natural convection is independent of the variable wind speed, and
only dependent of the temperature difference ∆T. The calculation of the ampacity of an OHL could
therefore be made with natural instead of forced convection. A reinvestigation of the ampacity of
overhead lines would furthermore result in higher ampacities and higher utilisation coefficients of the
OHL.




                                                                                                       48
14 Future work
In this project it has been shown that the method using the natural convection calculates an almost
identical ampacity compared to the methods using the forced convection. It has also been shown that
there is a significant potential for upgrading the conductors analysed in this Report. It has on the other
hand also been shown that the design ampacity for the Martin conductor may have been set too high
during the summer. This shows the difficulties between obtaining a higher utilization and
simultaneously omitting overloading existing OHLs.

There remains work to do before a better utilization of the OHLs in the grid can be achieved. Firstly it
would be preferable, if the natural convection method described in this Report becomes more accurate.
This would require an investigation of the uncertainties of the stochastic values of the solar and
conductor radiation. In spite of this it has been shown that the existing model for natural convection can
be used advantageously. If the temperature difference ∆T is above 20º C then the natural convection
method is conservative and calculates a lower ampacity compared to the actual. The natural convection
method could therefore be used to recalculate the ampacities for high loaded OHLs in the grid.

A method how this could accomplish is by using different ampacities during the year (season ampacity
or a dynamic ampacity). If an accurate forecast of the ambient temperature can be performed, then a
dynamic ampacity table would be reasonable. If it is possible to forecast the ambient temperature with
a decent error bar, then the ampacity of OHLs could be upgraded considerably. A dynamic ampacity
table could also prevent overloading.

The following ordered list gives a proposed procedure, how an improved utilization coefficient of the
OHLs in the grid can be achieved:

1. Monitoring of the ambient temperature around the OHLs in the grid.
2. Definition of fixed ambient temperatures dependent on the time of the year. These fixed
   temperatures can be used to calculate the maximum ampacity.
3. Use of the ambient temperature to recalculate the ampacity of high-loaded OHLs.
4. Collecting data with high temperature difference from the high-loaded OHLs.

The following gives a brief suggestion as to how the ordered list could be carried out in practice.

1) Firstly, the ambient temperature in the grid has to be determinable. This is important due to fact that
the ambient temperature directly is a factor in the calculation of the ampacity of OHLs. In this research
it should be determined how precise the ambient temperature can be forecasted.

2) When the ambient temperature is determined this information could be used to define annual seasons
(summer, autumn, winter and spring). These seasons should use different ambient temperatures. The
division into seasons could have a big effect especially during the winter, where the ambient
temperature is much lower compared to the fixed design temperature, which is used nowadays (20°C).
During the summer this could result in a decrease of the ampacity of some OHLs.

3) The method in 2) should be used to calculate more precise ampacities for high-loaded OHLs.



                                                                                                       49
The precision of the calculations increases when the temperature difference ΔT is high (see chapter
8.1). The tension monitoring equipment should be installed at these selected OHLs with the purpose to
collect the data.

4) The collected data from these selected OHLs could be analysed the same way as it was demonstrated
in this Report. These data, which contains measurements with high temperature differences, would
make it possible to determine whether the mathematical models are acceptable or not. At the same
time, the two applied fixed values for the solar and the conductor radiation could be determined more
exactly. This could be time varying values and not as nowadays fixed worst-case values. The following
articles [6], [9] and [10] could probably give an alternative way as the one used today.

The upgrade of the ampacity for the OHL should be performed in stages. For each of the stages the
maximum allowed conductor temperature could be increased step by step until the design temperature
is reached. This means that step 3 and 4 would be repeated and an analysis must be performed for each
step to ensure that the maximum ampacity of the line is not exceeded at any time. An example could be
the Condor conductor, where the maximum temperature permitted for the first stage could be 60ºC. The
next stage could be 70ºC, and finally 80ºC is reached.

With the analysed method it would become possible to perform a revision of the entire grid. This
means that every OHL in the grid would be reinvestigated new ampacities could be calculated. At this
stage a computer program could be developed. The purpose of this program would be to calculate a
daily ampacity curve for each OHL in the grid. This curve should represent the time dependent
ampacity of the OHLs. The exactness of this curve would be dependent of the forecast of the ambient
temperature and of the accuracy of the calculation methods.




                                                                                                  50
15 Literature
[1]    Christiansen, J. W., ”Vurdering af potentialer ved anvendelse af trækmålingsudstyr til
       overvågning af luftningers overføringsevne”, Danish, 2004, Eltek, Ørsted•DTU, The Technical
       University of Denmark

[2]    Keller, T. & Sørensen, T. K., ”Anvendelse af højtemperaturledere på udvalgte
       transmissionsstrækninger i Eltra’s forsyningsområde”, Danish, 2004, Eltek, Ørsted•DTU, The
       Technical University of Denmark

[3]    Elsam, ”Forbedret beregningsmetode for luftledningers strømbelastningsevne”, Elsam notat
       TS97 -438, Danish

[4]    IEEE, “Standard for Calculating the Current- Temperature Relationship of Bare Overhead
       Conductors”, IEEE std. 738-1993

[5]    Carlsen, H. & Larsen, P. S., ”Teknisk termodynamik”, Danish, 2004, MEK, The Technical
       University of Denmark

[6]    Davis, M. W., “A New Thermal Rating Approach: The Real Time Thermal Rating System for
       Strategic Overhead Line Conductor Transmission Lines – Part II – Steady State Thermal Rating
       Program”, 1977, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-96, no. 3

[7]    Elsam, Kontrolmåling af ledertemperatur, INTERNT T97/JCH-005, Danish

[8]    Elsam, Nedhængsberegning på grundlag af trækmålinger, INTERNT TL97/JCH-015, Danish

[9]    Davis, M. W., “A New Thermal Rating Approach: The Real Time Thermal Rating System for
       Strategic Overhead Line Conductor Transmission Lines – Part I – General Description and
       Justification of the Real Time Thermal Rating System”, 1977, IEEE Transactions on Power
       Apparatus and Systems, Vol. PAS-96, no. 3

[10]   Davis, M. W., “A New Thermal Rating Approach: The Real Time Thermal Rating System for
       Strategic Overhead Line Conductor Transmission Lines – Part III – Steady State Thermal
       Rating Program Continued – Solar Radiation Considerations”, 1978, IEEE Transactions on
       Power Apparatus and Systems, Vol. PAS-97, no. 2




                                                                                                51
16 Appendix

Appendix 16.1

                                                      Relative Error in Current by Different Convection Calculations
                              15
                                                                                                                            Natural Convection
                                                                                                                            Forced Convection
                              10



                               5
 Current Relative Error [%]




                               0



                               -5



                              -10



                              -15



                              -20



                              -25
                                    0   2        4         6           8         10           12         14            16           18           20
                                                           Data Point Number - sorted after dT (Descending)

                                        Figure 16-1 Relative error in current for Dove conductor – IEEE calculation method




                                                                                                                                                      52
                                                                 Relative Error in Current by Different Convection Calculations
                                        50
                                                                                                                                  Natural Convection
                                                                                                                                  Forced Convection
                                        40


                                        30
Current Relative Error [%]




                                        20


                                        10


                                         0



                                        -10


                                        -20



                                        -30
                                              0     2        4          6          8        10          12        14              16        18         20
                                                                      Data Point Number - sorted after dT (Descending)


                                                   Figure 16-2 Relative error in current for Martin conductor – IEEE calculation method


                                                             Relative Error in Current by Different Convection Calculations
                                         50
                                                                                                                            Natural Convection
                                                                                                                            Forced Convection
                                         40


                                         30
           Current Relative Error [%]




                                         20


                                         10


                                          0


                                        -10


                                        -20
                                              0     2        4         6        8        10         12       14              16        18        20
                                                                   Data Point Number - sorted after dT (Descending)


                                                  Figure 16-3 Relative error in current for Condor 1 conductor – IEEE calculation method



                                                                                                                                                            53
                                                     Relative Error in Current by Different Convection Calculations
                             60
                                                                                                                      Natural Convection
                                                                                                                      Forced Convection
                             50



                             40
Current Relative Error [%]




                             30



                             20



                             10



                              0



                             -10
                                   0    2        4          6         8         10          12        14              16       18          20
                                                          Data Point Number - sorted after dT (Descending)


                                       Figure 16-4 Relative error in current for Condor 2 conductor – IEEE calculation method




                                                                                                                                                54