KINGDOM OF SAUDI ARABIA SASO................../٢٠٠٩

SAUDI STANDARD DRAFT NO. ١٢٤٨٢

CALCULATION OF THE CYCLIC AND EMERGENCY CURRENT RATING OF CABLES

PART ٢: CYCLIC RATING OF CABLES GREATER THAN (٣٦) ١٨/٣٠ kV AND EMERGENCY RATINGS FOR

CABLES OF ALL VOLTAGES

SAUDI ARABIAN STANDARDS ORGANIZATION

Dell-٣١٨(Sef-٣)

CONTENTS

Page

Foreword................................................................................................................................ ٥

SECTION ONE – BACKGROUND Clause

١. Introduction and scope ................................................................................................. ٦

١٫١ General................................................................................................................ ٦

١٫٢ Inclusion of cable thermal capacitance............................................................... ٧

١٫٣ Conditions where thermal capacitance can be neglected ................................... ٨

١٫٤ Guide to clauses applicable to particular types of cable..................................... ٩

١٫٤٫١ Transient calculation and emergency rating ........................................... ٩

١٫٤٫٢ Cyclic rating............................................................................................ ٩

٢. Symbols...................................................................................................................... ١٠

٣. Layout of this standard............................................................................................... ١٢

SECTION TWO – TRANSIENT TEMPERATURE RESPONSE

٤. Transient temperature response to a step function of current .................................... ١٣

٤٫١ General aspects ................................................................................................. ١٣

٤٫١٫١ Background........................................................................................... ١٤

٤٫١٫٢ Criteria for the attainment factor to be unity ........................................ ١٤

٤٫١.٣ Representation of the cable................................................................... ١٥

٤٫١٫٤ Criteria for selecting the thermal circuit ............................................... ١٦

٤٫٢ Calculation of partial transients for long durations (> ١/٣ T.Q)

and cyclic rating ............................................................................................... ١٦

٤٫٢٫١ Representation of the dielectric ............................................................ ١٦

٤٫٢.٢ Representation of the cable................................................................... ١٨

٤٫٢٫٣ Calculation of cable partial transient .................................................... ٢٢

٤٫٢٫٤ Calculation of cable environment partial transient ……………………..٢٣

٤٫٣ Calculation of partial transients for short duration (≤ ١/٣ T.Q) ......................... ٢٤

٤٫٣٫١ Representation of the dielectric (single and ٣-core cables) .................. ٢٤

٢

٤٫٣.٢ Representation of the cable................................................................... ٢٥

٤٫٣٫٣ Calculation of cable partial transient .................................................... ٢٨

٤٫٣٫٤ Calculation of cable environment partial transient ............................... ٢٩

٤٫٤ Calculation of the complete temperature transient ........................................... ٣٠

٤٫٤٫١ Transient temperature response ............................................................ ٣٠

٤٫٤٫٢ Correction to transient temperature response for variation in

conductor losses with temperature (emergency ratings only) .............. ٣٢

٤٫٤٫٣ Transient temperature response caused by sudden application

of voltage (transient due to dielectric loss)........................................... ٣٢

SECTION THREE – CYCLIC RATINGS

٥. Computation of the cyclic rating factor (M) .............................................................. ٣٣

٥٫١ General.............................................................................................................. ٣٣

٥٫٢ Calculation of cyclic rating factor (including cable thermal capacitance) ....... ٣٤

٥٫٢٫١ Any load cycle of known shape............................................................ ٣٤

٥٫٢٫٢ Flat top load cycle................................................................................. ٣٥

٥٫٢٫٣ Load cycle of unknown shape but with a known loss-load

factor (μ) ............................................................................................... ٣٦

٦. Calculation of loss-load factor (μ) ............................................................................. ٣٦

٧. Calculation of ................................................................................................. ٣٦ )()(

xi

R

R

θθ

٧٫١ Single isolated ٣-core cable.............................................................................. ٣٦

٧٫٢ Single isolated circuit ....................................................................................... ٣٧

٧٫٣ Group of “N” cables with equal losses; cables or ducts not touching .............. ٣٨

٧٫٤ Group of “N” circuits, each of three single-core identical touching

cables or ducts, all cables having equal losses ................................................. ٣٩

SECTION FOUR – EMERGENCY RATING

٨. Calculation of emergency ratings .............................................................................. ٤٠

٨.١ Thermally isolated circuits ............................................................................... ٤٠

٨٫٢ Groups of circuits ............................................................................................. ٤٢

٨٫٣ Correction to transient temperature response to take account of variation in

Conductor for losses with temperature ………………………………………...٤٣

٣

٩. Bibliography............................................................................................................... ٤٣

٤

APPENDIX B – Transient average temperature variation through dielectric or

through oil filling in pipes....................................................................... ٤٥

APPENDIX C – Method for dealing with different soil resistivities, thermal

diffusivities and depths of laying ............................................................ ٤٧

APPENDIX D – Diffusivity of soil ..................................................................................... ٤٨

APPENDIX E – Physical constants of materials ................................................................ ٤٩

APPENDIX F – Examples .................................................................................................. ٥١

FIGURES ٥ & ٤ ,٣ ,٢ ,١ ...................................................................................................... ٦٣

FIGURE ٦٤..…………………………………………………………………………………٦

FIGURE ٦٥..…………………………………………………………………………………٧

NOTES TO FIGURES ٦ AND ٧ “SCALES FOR EXPONENTIAL INTEGRAL”……….٦٦

FIGURE ٦٧..…………………………………………………………………………………٨

FIGURE ٦٨..…………………………………………………………………………………٩

FIGURE ٦٩…………………………………………………………………………………١٠

٥

FOREWORD

The Saudi Arabian Standards Organization (SASO) has adopted the international standard IEC ٢/١٩٨٩-٦٠٨٥٣ “Calculation of the Cyclic and Emergency Current Rating of Cables – Part ٢: Cyclic Rating of Cables Greater than (٣٦) ١٨/٣٠ kV and Emergency Ratings for Cables of all Voltages" including its Amend. ١/٢٠٠٨, issued by the International Electrotechnical Commission (IEC). The text of this international standard has been translated into Arabic to be approved as a Saudi standard without introducing any alterations.

٦

CALCULATION OF THE CYCLIC AND EMERGENCY CURRENT RATING OF CABLES

PART ٢: CYCLIC RATING OF CABLES GREATER THAN

(٣٦) ١٨/٣٠ kV AND EMERGENCY RATINGS FOR CABLES OF ALL VOLTAGES

SECTION ONE – BACKGROUND

١ Introduction and scope

١٫١ General

This standard gives manual methods for calculating cyclic rating factors for cables whose internal thermal capacitance cannot be neglected; in general this applies to cables for voltages greater than (٣٦) ١٨/٣٠ kV. It also gives a method for calculating the emergency rating for cables of any voltage. The standard to be approved by SASO concerned with "Calculation of the cyclic and emergency current rating of cables - Part ١: Cyclic rating factor for cables up to and including (٣٦) ١٨/٣٠ kV* deals with cyclic rating factors for cables of voltages not greater than (٣٦) ١٨/٣٠ kV where the internal thermal capacitance could be neglected. The standard to be approved by SASO concerned with "Calculation of the cyclic and emergency current rating of cables - Part ٣: Cyclic rating factor for cables of all voltages, with partial drying of the soil"** deals with cyclic rating factors for cables of all voltages when partial drying of the soil occurs.

To determine the cyclic rating factors where the internal thermal capacitance cannot be neglected it is necessary to calculate the transient temperature response of the cable and its environment.

The formulae recommended in this standard contain quantities which vary with cable design and materials used. The values given in the tables are either internationally agreed, for example, resistance temperature coefficients, or are those which are generally accepted in practice, for example, thermal resistivities and volumetric specific heats of materials. In order that uniform and comparable results may be obtained, the cyclic and/or emergency current ratings should be calculated with the values given in this standard. However, where it is known with certainty that other values are more appropriate to the materials and design, then these may be used, and the corresponding cyclic/emergency current rating declared in addition, provided that the different values are quoted.

_____________ * This standard will be based on IEC ١-٦٠٨٥٣.

** This standard will be based on IEC ٣-٦٠٨٥٣.

٧

A mean of incorporating the effect of change in conductor resistance with temperature is included.

When the load current changes is discrete and multiple steps the method set out in Clause ٤ can be used to determine the temperature response of the cable.

This method shall be used carefully where a large number of steps is considered, or long duration steps are dealt with, or large variations of the load current occur.

As the method involves an iterative process, the convergence criterion shall be adapted to the different situations.

١٫٢ Inclusion of cable thermal capacitance

When a cable thermal capacitance cannot be neglected, it is necessary to calculate the internal transient temperature response of the cable(s). The manual methods recommended for this computation are:

a) For the computation of cyclic rating factors:

The use of analogous lumped thermal constant circuits to represent the cables, applicable only where time periods greater than about ١ h are involved [١]. (For the calculation of ratings for daily load cycles such a limitation is not important.) This method can be used manually or on a computer and is capable of dealing with all types of cable.

Note. – The cable thermal transient can also be calculated by a numerical method, where the division of cable components into numerous concentric cylindrical elements can be arranged so as to yield adequate accuracy for any time period [٢]*. This method applies to single-core cables only and is outside the scope of this standard.

The methods for cyclic ratings in this standard apply to cables buried in the ground, either directly or in ducts, when carrying a load which varies cyclically over a ٢٤ h period, the shape of each daily cycle being substantially the same. For these installation methods it has been assumed that the voltage had been applied for a sufficiently long time for the conductor temperature rise due to dielectric loss to have reached a steady state. The total temperature rise of the conductor above ambient is then the sum of the steady state temperature rise due to the dielectric loss [as given by the standard to be approved by SASO concerned with “Electric cables - Calculation of the current rating - Part ١-١: Current rating equations (١٠٠٪ load factor) and calculation of losses - General", "Electric cables - Calculation of the current rating - Part ١-٢: Thermal resistance - Calculation of thermal resistance" and " Electric cables - Calculation of the current rating - Part ١-٣: Sections on operating conditions - Reference operating conditions and selection of cable type"** and the transient temperature variations due to change of current. For cyclic rating factor purposes it is sufficient to independently calculate the transient temperature variations due to changes in current.

_____________________

٨

* The figures in square brackets refer to "Bibliography", clause ٩, page ٤٣.

** This standard will be based on IEC ٦٠٢٨٧ series.

٩

For cables in air, the conductor temperature follows changes in load current sufficiently rapidly so that the usual daily cycles do not permit peak loads greater than the steady-state value.

Numerical data are provided for the evaluation of cyclic rating factors for cables buried at ١ m depth in soil having a diffusivity of ٠٫٥ X ٦-١٠ m٢/s, but methods for dealing with other conditions are included. It is assumed that the soil properties are constant in both time and space.

b) For the computation of emergency rating factors:

i) For time periods longer than about ١ h, use the same method as for cyclic rating factor purposes (see Item a) above).

ii) For time periods less than ١ h down to ١٠ min, a modified version of Item a) is used having a more detailed thermal circuit [٣]*.

Note. – See also note under Item a).

The methods for calculating the emergency ratings apply to cables buried in the ground, either directly or in ducts, and to cables in air. Provision is made for incorporating the transient caused by a sudden application of voltage (i.e. the transient due to dielectric loss).

Methods for calculating emergency rating factors are intended for emergency loads not greater than about ٢٫٥ times rated full load current (١٠٠٪ load factor).

١٫٣ Conditions where thermal capacitance can be neglected

If the time periods being considered are sufficiently long compared with the thermal time constant of the cable, then the cable thermal capacitance may be neglected. In practice this applies to all cables up to and including (٣٦) ١٨/٣٠ kV and allows a simplified procedure to be used. Full details of this simplified procedure are given in the standard to be approved by SASO concerned with "Calculation of the cyclic and emergency current rating of cables - Part ١: Cyclic rating factor for cables up to and including (٣٦) ١٨/٣٠ kV.

It has been estimated that cable thermal capacitance can be neglected for:

a) all conductor cross-sections and types of cable for nominal voltages up to and including (٣٦) ١٨/٣٠ kV, for any shape of load cycle;

b) all cables where the following conditions apply together:

i) the loss-load factor μ is not less than ٠٫٦٥;

ii) the average of the ordinates Y٠, Y١ and Y٢ is not less than ٠٫٩ (see Clause ٦);

iii) the average of Y٣, Y٤ and Y٥ is not less than ٠٫٧ (see Clause ٦).

_____________________ * The figures in square brackets refer to “Bibliography”, Clause ٩, page ٤٣.

١٠

١١

c) self-contained cables where the following conditions apply together:

i) the loss-load factor μ is not less than ٠٫٤;

ii) the average of the ordinates Y٠, Y١ and Y٢ is not less than ٠٫٩ (see Clause ٦);

iii) the average of Y٣, Y٤ and Y٥ is not less than ٠٫٧ (see Clause ٦).

١٫٤ Guide to clauses applicable to particular types of cable

١٫٤٫١ Transient calculation and emergency rating Transient calculation Emergency rating

Cable type Duration of transient

Sub-clauses Duration of transient

Clauses and Sub-clauses

All types of single-core cables >

٣١ T.Q

≤ ٣١ T.Q

> ٣١ T.Q

≤ ٣١ T.Q

٤٫٤ ,٤٫٢ ,٤٫١ ,٨ ٤٫٤ ,٤٫٢ ,٤٫١

٤٫٤ ,٤٫٣ ,٤٫١ ,٨ ٤٫٤ ,٤٫٣ ,٤٫١

Pipe-type cables > ٣١ T.Q

≤ ٣١ T.Q

> ٣١ T.Q

≤ ٣١ T.Q

٤٫٤ ,٤٫٢ ,٤٫١ ,٨ ٤٫٤ ,٤٫٢ ,٤٫١

٤٫٤ ,٤٫٣ ,٤٫١ ,٨ ٤٫٤ ,٤٫٣ ,٤٫١

Three-core cables (except pipe type) >

٢١ T.Q

(whole cable) ≤ T.Q

one core (intermediate)

> ٢١ T.Q

(whole cable) ≤ T.Q

one core (intermediate)

٤٫٤ ,٤٫٢ ,٤٫١ ٤٫٤ ,٤٫٢ ,٤٫١ ,٨

٤٫٤ ,٤٫٣ ,٤٫١ ,٨ ٤.٤ ,٤٫٣ ,٤٫١

٤٫٤ ,٤٫٢،٤٫٣ ,٤٫١ ,٨ ٤٫٤ ,٤٫٣ ,٤٫٢ ,٤٫١ Note. – T = thermal resistance of cable or one core as appropriate. Q = thermal capacitance of cable or one core as appropriate.

١٫٤٫٢ Cyclic rating

The applicable clauses and Sub-clauses are ٦ ,٥ ,٤٫٤ ,٤٫٢ ,٤٫١ and ٧ and as noted below.

Cable type Load cycle type Sub-clauses

All types

Any known shape

Flat top

Where only the loss-load factor is known

٥٫٢٫١

٥٫٢٫٢

٥٫٢٫٣

١٢

٠٢ Symbols Symbol Definition Unit De = external diameter of cable covering m Di = external diameter of dielectric m Ds = internal diameter of cable covering m – Ei (-x)

= exponential integral function

F = mutual heating coefficient for a group of cables I = permissible current for given (standard) condition A I١ = constant current applied to cable prior to emergency load A I٢ = emergency load current which may subsequently be applied for times t so that the

conductor temperature rise above ambient at the end of the period of emergency load is θmax

A Imax = highest current of daily load cycle: used as denominator when determining loss-load

ordinates

A IR = sustained (١٠٠٪ load factor) rated current to attain, but not exceed, the permissible

maximum conductor temperature

A L = axial depth of burial of a cable or duct m M = cyclic rating factor N = number of cables in a group, or, for touching cables, the number of circuits, see ٧٫٤ b) Mo, No = coefficients used for calculating cable partial transient temperature rise (see Sub-

clauses

s, s٤٫٢٫٣ ٢ and ٤٫٣٫٣) Q = total thermal capacitance of a cable J/Km QA, QB = elements of two part thermal circuit (see Sub-clauses ٤٫٢٫٢ and ٤٫٣٫٢) J/Km Qa = thermal capacitance of armour J/Km Qc = thermal capacitance of conductor J/Km Qd = thermal capacitance of a duct J/Km Qf = thermal capacitance of filling in S.L. type cables, or of filling between cores of a gas-

pressure pipe-type cable

J/Km Qi = thermal capacitance of dielectric per conductor J/Km Qi١ = thermal capacitance of first portion of insulation J/Km Qi٢ = thermal capacitance of second portion of insulation J/Km Qj = thermal capacitance of outer covering of cable J/Km Qo = thermal capacitance of oil in a pipe type cable J/Km Qp = thermal capacitance of a pipe J/Km Qs = thermal capacitance of metallic sheath or screen and reinforcement J/Km R١ = A.C. resistance of conductor before application of emergency current Ω/m RR = A.C. resistance of conductor with sustained application of rated current IR, i.e. at

standard maximum permissible temperature

Ω/m Rmax = A.C. resistance of conductor at end of period of emergency loading Ω/m T = total thermal capacitance of a cable from conductor to outer surface Km/W TA, TB, TC

= elements of equivalent thermal circuit Km/W

T١ = thermal resistance of dielectric per conductor Km/W T٢ = thermal resistance of gas in a gas-pressure pipe-type cable Km/W T٣ = thermal resistance of outer covering of a cable Km/W T٤ = external thermal resistance of a cable or a duct Km/W T`٤ = thermal resistance of air space in a duct Km/W T"٤ = thermal resistance of a duct Km/W ΔT٤ = additional external thermal resistance caused by heating from other cables in a group

Km/W Ta, Tb = apparent thermal resistance used to calculate cable partial transient temperature rise

(see Sub-clauses ٤٫٢٫٣ and ٤٫٣٫٣)

Km/W Tf = thermal resistance of filling between cores of SL type cable and of gas-pressure pipe-

type cable

Km/W To = thermal resistance of the oil in a pipe-type cable Km/W

١٣

Wc = power loss in a conductor or equivalent conductor W/m W١ = total I٢R power loss of each cable W/m Wd = power loss in cable dielectric W/m Yo… Y

= scaled ordinate in (load)٢ graph, used in calculation of cyclic rating factor ٢٣

a, b = coefficients used for calculating cable partial transient temperature rise ١/s

d = Dry density of soil kg/m٣ dc = external diameter of conductor m df = spacing from centre of hottest cable for a single line source representing the heating

effect of all other cables in a group

m dpk = distance from centre of pth cable, where rating is being determined to an adjacent cable

k

m d’pk = distance from centre of pth cable, whose rating is being determined, to the image of an

adjacent cable k (see Figure ٩)

m h = heat emission coefficient for a cable in air W/m٢.K٥/٤ h١ = ratio I١/IR I = Time h k = ratio of cable, pipe or duct external-surface temperature rise above ambient to

conductor temperature rise above ambient under steady conditions

K١ = value of k for a cable in a group n = number of cores in a cable p = factor for apportioning the thermal capacitance of a dielectric (durations greater than ١/٣

T.Q)

p* = factor for apportioning the thermal capacitance of a dielectric (durations less than or equal to ١/٣ T.Q)

p’ = factor for apportioning the thermal capacitance of cable coverings pd = factor for apportioning thermal capacitance of dielectric when calculating transient

caused by dielectric loss

qa

= ratio :

conductorsinlossesarmourscreensorsheathsmetallicconductorsinlosses )( ++

qc

= ratio :

conductorsinlosses)pipescreensconductors(inlosses ++

qj

=

ratio :

conductorsinlossespipearmourscreensorsheathsmetallicconductorsinlosses )( +++

qs

= ratio :

conductorsinlossscreenorscreenorsheathmetallicconductorinlosses )]([ +

R = ratio θmax/θR(∞) T = time from start of application of heating, a general symbol for time, usually in seconds

(t = ٦٠٠ ٣i)

s a = temperature coefficient of electrical resistivity of conductor material ١/K a(t), β(t)

= attainment factors for the conductor to cable surface and cable surface to ambient temperature rises respectively

β = reciprocal of temperature coefficient of electrical resistivity at ٠oC K γ(t) = function used in calculating cyclic rating factor for groups of cables

δ = soil thermal diffusivity m٢/s η = moisture content of soil in per cent of dry weight μ = loss-load factor of a load cycle ρT = soil thermal resistivity Km/W ρi = dielectric thermal resistivity Km/W

١٤

θi = conductor temperature at commencement of transient oC θ(t) = transient temperature rise of conductor above ambient, without correction for variation

in conductor loss

K θc(t) = transient temperature rise of conductor above the outer surface of a cable

K θd(t) = transient spatial average temperature rise of dielectric above the outer surface of a cable

K θe(t) = transient temperature rise of outer surface of a cable (or hottest cable in a group of

similarly loaded cables) above ambient temperature

K θo(t) = transient average temperature rise of oil in a pipe-type cable above

the outer surface of the pipe

K θR(t) = conductor temperature rise above ambient at time t after application

of current IR, neglecting variation in conductor resistance, θR(t) is calculated by the methods given in Clause ٨

K θR(i) = θ(t), when the magnitude of the step function current is the sustained

(١٠٠٪ load factor) rated current, and i is expressed in hours

K θR(x) = value of θR in the steady state, i.e. the standard maximum permissible

temperature rise

K θa(t) = conductor transient temperature rise above ambient corrected for

variation in conductor loss with temperature

K θ(x) = conductor steady state temperature rise above ambient K θmax = maximum permissible temperature rise above ambient a) for a daily

cyclic load, b) at end of period of emergency loading

K σ = volumetric specific heat of insulation (see Appendix E) J/K.m٣ Δθd = Steady-state conductor temperature rise above ambient due to

dielectric losses K

٣ Layout of this standard

Section Two gives the manual methods for calculating the transient temperature response of a cable to a step function of current. Methods are set out for durations of:

a) from ١٠ min up to about ١ h;

b) for durations of about ١ h or greater. This is also required for use in Section Three.

A computer method for calculating the transient response of single-core cables has been published by CIGRE [٢]*.

Section Three sets out the general method of calculating the cyclic rating factor based on the calculated transient response from Section Two.

Section Four sets out the method of calculating the emergency rating of a cable based on the calculated transient responses from Section Two.

_____________________ * The figures in square brackets refer to “Bibliography”, Clause ٩, page ____.

١٥

Appendix A gives a detailed method for reducing a multiple thermal circuit to one having two sections. This technique is required for cables not covered in detail in Clause ٤.

Appendix B gives a method for calculating the transient average temperature variation through the dielectric or through oil filling in pipes. This is required when calculating transient oil-pressure variations in oil-filled cables.

Appendix C sets out methods of modifying some of the transient factors used in the cyclic rating calculation when the soil properties and depth of laying are non-standard.

Appendix D gives methods for estimating the diffusivity of the soil from other soil parameters when the actual value is unknown.

Appendix E gives values for thermal resistivity of materials [reproduced from the standard to be approved by SASO concerned with “Electric cables - Calculation of the current rating - Part ١-١: Current rating equations (١٠٠٪ load factor) and calculation of losses - General", "Electric cables - Calculation of the current rating - Part ١-٢: Thermal resistance - Calculation of thermal resistance" and " Electric cables - Calculation of the current rating - Part ١-٣: Sections on operating conditions - Reference operating conditions and selection of cable type"* and suggested values for the volumetric specific heat of materials, gleaned from various sources.

Appendix F gives sample applications of the methods enounced in this standard. Detailed calculations are given for a cable transient, a cyclic rating and an emergency rating.

SECTION TWO – TRANSIENT TEMPERATURE RESPONSE

٤- Transient temperature response to a step function of current

٤٫١ General aspects

Supporting formulae are given in the standard to be approved by SASO concerned with ““Electric cables - Calculation of the current rating - Part ١-١: Current rating equations (١٠٠٪ load factor) and calculation of losses - General", "Electric cables - Calculation of the current rating - Part ١-٢: Thermal resistance - Calculation of thermal resistance" and " Electric cables - Calculation of the current rating - Part ١-٣: Sections on operating conditions - Reference operating conditions and selection of cable type"*.

_____________________ * This standard will be based on IEC ٦٠٢٨٧ series.

١٦

٤٫١٫١ Background

The transient temperature response of a cable to a step-function of current in its conductor (or conductors) depends on the combination of thermal capacitances and resistances formed by the constituent parts of the cable itself and its surroundings. As an example, in the important case of cables laid directly in the ground, if the response is required for short times the thermal capacitances of the cable, and the way in which these are taken into account, are important. At the same time the contribution of the surrounding soil is negligible. On the other hand, when the response for long times is required, the thermal capacitances associated with the cable may be negligible and the most important factor is the thermal transient in the surrounding soil.

The method for calculating the temperature response of a cable to a suddenly applied constant value of conductor current is to consider that the whole thermal circuit is divisible into two independent parts. One part is made up of the cable components out to the outer surface of the cable, the second part is the environment of the cable. The individual responses of these two parts are partial transients, with which the total transient for the complete system can be built up.

The method for determining these two parts of the thermal circuit and the temperature transients across them are given separately in Sub-clauses ٤٫٢ and ٤٫٣ and the method for combining the transients by means of the attainment factor a (t) is described in Sub-clause ٤٫٤.

٤٫١٫٢ Criteria for the attainment factor to be unity

When the calculations are required to give the conductor temperature occurring after a period long enough to complete the transient in the first part of the thermal circuit, then the attainment factor a (t) may be assumed to be equal to ١. In practice this applies when the period from the initiation of the thermal transient is longer than:

a) ١٢ h for all cables;

b) the product T.Q: when dealing with oil-pressure pipe-types cables and all types of self-contained cables where the product T.Q ≤ ٢ h, and

c) the product ٢.T.Q: when dealing with gas-pressure pipe-type cables and all types of self-contained cables where the product T.Q > ٢ h.

where T is the total thermal resistance and Q is the total thermal capacitance of the cable.

The table below, based on design values at present commonly used to determine dimensions of cables, shows when cases b) and c) apply.

١٧

Type of cable Case b) Case c)

Oil-filled cables ٢٢٠ (١ kV : sections > ١٥٠ mm١ ٢) All voltages < ٢٢٠ kV

٢٢٠ (٢ kV : sections ≤ ١٥٠ mm٢ ٢) All voltages < ٢٢٠ kV

٢٢٠ (١ kV : sections > ٨٠٠ mm٢ Pipe-type, oil-pressure cables ١) All voltages ≤ ٢٢٠ kV

٢٢٠ (٢ kV : sections ≤ ٨٠٠ mm٢ ٢) All voltages < ٢٢٠ kV

٢٢٠ ≥ (١ kV Pipe-type, gas-pressure cables

٢) Sections ≤ ٠٠٠ ١ mm٢

٦٠ (١ kV : sections > ١٥٠ mm٢ Cables with extruded insulation ١) All voltages < ٦٠ kV

٦٠ (٢ kV : sections ≤ ١٥٠ mm٢ ٢) All voltages > ٦٠ kV

٤٫١٫٣ Representation of the cable

For the purpose of this standard, the cable or first part of the thermal circuit shall be defined for each type of installation to include the following:

a) Cables buried directly in soil

The complete cable including its outermost serving or anticorrosion protection. For pipe-type cables this includes the pipe and its protective wrappings.

b) Cables in ducts

The cable, the duct and whatever occupies the duct-space. For ducts embedded in concrete, the cable circuit includes the duct itself. (The concrete is to be considered, with the soil, as belonging to the second part of the thermal circuit.) For ducts formed directly by casting concrete, the treatment is the same, but the thermal capacity of the duct, Qd, is zero.

c) Cables in air

The complete cable including its outermost covering.

The temperature transients calculated for the first part of the thermal circuit will give the temperature rise of the conductor with respect to the outermost surface of the cable as defined above.

The cable is represented by a circuit with lumped constants, the derivation of which is important and the methods given in this standard must be followed if uniform results between users are to be obtained. This applies particularly to cables having complex constructions and where, in order to make calculation reasonably convenient, the cable circuit has been simplified to as few a number of elements as possible.

It should be noted that, in a cable, all thermal resistances, capacitances and losses are to be calculated as losses per conductor, or per equivalent conductor in the case of three-core cables (see Sub-clause ٤٫٢٫١٫٢).

١٨

٤٫١٫٤ Criteria for selecting the thermal circuit

The criteria for selecting the thermal circuits to be used for calculating the individual transients depends on the duration of the transient and should be selected from the table below:

Selection of thermal circuit Cable type Duration of transient Sub-clause

Single-core cables, all types

> ٣١ T.Q (and cyclic ratings)

٤٫٢

٦٠٠ s (١٠ min) to ≤ T.Q

٣١

٤٫٣

Pipe-type cables > ١٣

T.Q (and cyclic ratings)

٤٫٢

٦٠٠ s (١٠ min) to ≤ T.Q

٣١

٤٫٣

Three-core cables > ٢١ T.Q (total cable) (and cyclic ratings)

٤٫٢

٦٠٠ s (١٠ min) to ≤ T.Q (one core)*

٤٫٣

* In this case T and Q are the thermal resistance and thermal capacitance of one core only.

Note. – T and Q are the total thermal resistance and thermal capacitance of the cable except as marked*. T.Q is the thermal time constant of the cable.

For three-core cables the transient temperature response for durations between the two limits, T.Q for the core and

٢١ T.Q for the whole cable, is assumed to

be given by a straight line interpolation on linear temperature rise/logarithmic time axes.

In general Sub-clause ٤٫٢ can be used for durations greater than about ١ h (and for cyclic ratings) while Sub-clause ٤٫٣ can be used for durations of about ١ h down to ١٠ min.

٤٫٢ Calculation of partial transients for long durations (> T.Q) and cyclic

rating

١٣

These formulae are usually suitable for time durations greater than about ١ h: see Sub-clause ٤٫١٫٤ for exact durations.

٤٫٢٫١ Representation of the dielectric

٤٫٢٫١٫١ Single-core cables

١٩

The dielectric is represented by lumped thermal constants. The total thermal capacity of the dielectric (Qi) is divided between the conductor and the metallic sheath or screen, so that the total heat stored in the dielectric is unchanged.

The dielectric is then represented by the components shown in full lines in Figure ١,

where

p =

⎟⎟⎠

⎞⎜⎜⎝

⎛

c

i

dD

n١٢

١ –

١

١٢

T١ = total

−⎟⎟⎠

⎞⎜⎜⎝

⎛

c

i

dD

(١-٤)

thermal resistance of dielectric per conductor (or equivalent

Qi =

Qc = or

Di =

for thermal calculations metallic tapes are

٤٫٢٫١٫٢ Three-co

le is replaced by an equivalent single-core construction

-

single-core conductor of a three-core cable, see Sub-clause ٤٫٢٫١٫٢)

total thermal capacitance of dielectric per conductor (or equivalent single-core conductor of a three-core cable, see Sub-clause ٤٫٢٫١٫٢)

thermal capacitance of conductor (or equivalent single-core conductof a three-core cable, see Sub-clause ٤٫٢٫١٫٢)

external diameter of dielectric

dc = external diameter of conductor Note. – Where screening layers are present:

considered to be part of the conductor or sheath while semi-conducting layers (including metallized carbon paper tapes) are considered as part of the insulation. The appropriate component dimensions should be modified accordingly.

re cables

The three-core cabdissipating the same total conductor losses. The equivalent single-core conductor has a diameter:

⎟⎟⎠

⎞⎜⎜⎝

ie

⎛

i

Tρπ ١٢

where Di is the same value of diameter over dielectric (under the metallic

l capacitances are calculated on the following assumptions:

dc = D (٢-٤)

sheath or screen) as for the three-core cable and T١ is the thermal resistance of the equivalent single-core cable. T١ is one-third of the value for one of the cores of the three-core cable as given in the standard to be approved by SASO concerned with ““Electric cables - Calculation of the current rating - Part ١-١: Current rating equations (١٠٠٪ load factor) and calculation of losses - General", "Electric cables - Calculation of the current rating - Part ١-٢: Thermal resistance - Calculation of thermal resistance" and " Electric cables - Calculation of the current rating - Part ١-٣: Sections on operating conditions - Reference operating conditions and selection of cable type"*, ρi, is the thermal resistivity of the dielectric.

Therma

٢٠

_____________________ * This standard will be based on IEC ٦٠٢٨٧ series.

a) The actual conductors are considered to be completely inside the

b) ctor and the metallic

The f equivalent single-

٫٢٫٢ Representation of the cable

part of the thermal circuit simulates the cable, and is always

the constants in Figure ٢ for the more common types of cables

nd T٣ are adjusted by the

٫٢٫٢٫٢ Representation of common types of cable irst used.

insulated cables, extruded

(٣-٤)

TB = qs T(٤-٤) ٣

diameter of the equivalent single-core conductor, the remainder of the equivalent conductor being occupied by insulation.

The space between the equivalent single-core condusheath or screen is considered to be completely occupied by insulation (for oil-filled cables this space is filled partly by the total volume of oil in the ducts and the remainder is oil-impregnated paper).

actor ρ is then calculated using the dimensions of the core cable and is applied to the thermal capacitance of the insulation based on assumption b) above.

٤

٤٫٢٫٢٫١ General

The firstrepresented by a two-section network (see Figure ٢). The first section includes the thermal capacitance of the conductor and the inner portion of the dielectric along with the thermal resistance of the dielectric while the second section includes thermal capacitance and thermal resistance of the remainder of the cable. A typical example, based on the cables in Item a) of Sub-clause ٤٫٢٫٢٫٢ below is given in Figure ٣. Note that any thermal capacitance outside TB is omitted, as the right-hand pair of terminals is short-circuited when calculating the transient.

Formulae for and installations are given in the following sub-clause. For single-core cables in trefoil formation, the values of T١ aappropriate factors given in the standard to be approved by SASO concerned with "Electric cables - Calculation of the current rating - Part ١-٢: Thermal resistance - Calculation of thermal resistance"*

٤ Note. – Symbols are defined only where they are f

a) Solid and self-contained oil-filled paper cables and thermally similar constructions

TA = T١

٢١

_____________________ on IEC ١-٢-٦٠٢٨٧.

QA = Qc + pQi (٥-٤)

QB = (١ – p)Qi +

* This standard will be based

s

js

qQpQ '+

(٦-٤)

where:

i and p are defined in Sub-clause ٤٫٢٫١٫١.

or equivalent single-core

Qs = apacitance of metallic sheath or screen and

Qj = ance of outer covering

s

T١, Q

T٣ = thermal resistance of outer covering

Qc = thermal capacitance of conductor conductor

thermal creinforcement

thermal capacit

conductorinlossesscreenorsheathq = ratio: metallicconductorinlosses ( + :)

In three-core cables, the conductor is the equivalent single

tor used to allocate the thermal capacitance of the outer

-core

and is used to take account of the extra losses occurring in the metallic sheath or screen

conductor.

p' is a faccovering in a similar manner to that used for the dielectric.

p' =

⎟⎟⎠

⎞⎜⎜⎝

⎛

s

e

DD

n١٢

١ – ١

١٢

−⎟⎟⎠

⎞⎜⎜⎝

⎛

s

e

SD

(٧-٤)

De and Ds are the outer and inner diameters of the covering. itted because it

the conductor

b) Pipe-ty ssure oil-filled)

Note. – The outer part of the thermal capacitance of the covering is omhas been found to play only an insignificant part in determining temperature transient.

pe cables (high pre

٢٢

TA = T١ + ١ qs To ٢

(٨-٤)

TB = ١٢

qs To + qe T(٩-٤) ٣

B i

QA = Qc + pQi (١٠-٤)

١ – p) Q + oQ (١١-٤) Q = (

sq

where:

re defined in Sub-clause ٤٫٢٫١٫١. T١, Qi and p a

qs = ratio: conductorsinlosses

:)sheathconductor(inlosses +

To = thermal resistance of the oil in the pipe

e

e

T٣ = thermal resistance of covering on pipe

Qo = thermal capacitance of the oil in the pip

q = ratio: conductorsinlosses

pipescreensconductorsinlosses ( ++ )

c) Pipe-type cables (gas pressure, cables laid up, filled and armoured)

)

T

QB = (١ – p Qi +

TA = T١٢-٤) ١

B = qs Tf + qa T٢ + qj T(١٣-٤) ٣

QA = Qc + pQi (١٤-٤)

fs QQ)

sq+

+a

aQq

+ j

pQq

(١٥-٤)

where:

a

٥.٠

conductorsinlossesarmoursheathsconductorsinlosses )( ++ q = ratio:

conductorsinlossespipearmoursheathsconductorsinlosses )( +++ qj = ratio:

T٢ = thermal resistance of gas in the pipe

r

d) ing material or armour)

Tf = thermal resistance of filling

Qa = thermal capacitance of armou

Qf = thermal capacitance of filling

Qp = thermal capacitance of pipe

Pipe-type cables (gas pressure, no fill

٢٣

TA = T(١٦-٤) ١

QB = (١ – p) Qi +

TB = qs T٢ + qe T(١٧-٤) ٣

QA = Qc + pQi (١٨-٤)

pQ

s

sQq

+ eq

(١٩-٤)

where:

e

٥.٠

conductorsinlossespipesheathsconductorsinlosses )( ++ q = ratio:

e) Cables in ducts

TA = T(٢٠-٤) ١

T

B i

B = qs (T٣ + T´٤ + T´´٤) (٢١-٤)

QA = Qc + pQi (٢٢-٤)

١ – p) Q + djs QQQ ٥.٠++ (٢٣-٤) Q = (

sq

where:

al resistance of the air in the duct

t

f) e te lead sheath (SL type)

٤)

T

QB = (١ – p) Qi +

T´٤ = therm

T´´٤ = thermal resistance of the duct

Qd = thermal capacitance of the duc

Armour d cables with each core in a separa

TA = T٢-٤) ١

B = qs Tf + qa T(٢٥-٤) ٣

QA = Qc + pQi (٢٦-٤)

+ ٢

٣٢

٣

⎟⎟⎠

⎞⎜⎜⎝

⎛

+ TqTqTq

as

a⎟⎟⎠

⎞⎜⎜⎝

⎛ ++

a

ja

s

f

qQQ

qQ٥.٠

s

fs

qQQ ٥.٠+

(٢٧-٤)

٫٢٫٢٫٣ Representation of cables and installations of types not specified in previous

s sub-clause provides circuits for most cable types. However, to

resenting the thermal resistances and

٤sub-clause

The previoucover contingencies, the following method is recommended for cases not specifically dealt with in this standard.

a) Build up a ladder network repcapacitances of the cable. For this purpose the cable should be

٢٤

considered to extend as far as the inner surface of the soil for buried cables, and to free air for cables in air.

Thermal resistances are calculated by the methods used to determine the steady-state ratings. (See the standard mentioned in Sub-clause ٤٫١).

Thermal capacitances of metallic parts are placed as lumped quantities corresponding to their physical position in the cable. The thermal capacitance of materials with a high thermal resistivity (e.g. insulation and coverings) is allocated by the technique described in Sub-clause ٤٫٢٫١ (e.g. the covering in Item a) of Sub-clause ٤٫٢٫٢٫٢). It is important that the dielectric be treated exactly as specified in Sub-clause ٤٫٢٫١.

b) Sometimes this network may contain more than two sections and to allow the solution to be found shall be reduced to the two section form shown in Figure ٢. Where the thermal circuit of a cable is not easily reduced to two sections, then the second and subsequent sections are combined into one, using the method given in Appendix A. It is more important that the first section is left in the original form as derived in Sub-clause ٤٫٢٫١.

٤٫٢٫٣ Calculation of cable partial transient

The transient response of a cable circuit to a step function of load current, considered in isolation, that is with the right-hand pair of terminals in Figure ٢ short-circuited, is found as follows:

Mo = ١٢

QA TA + TB + QB TB (٤٫٢٨)

No = QA TA QB TB (٤٫٢٩)

a = o

oo

NNMM −+ ٢

٠ (٣٠-٤)

b = o

oo

NNMM −− ٢

٠ (٣١-٤)

Ta = ba −

١⎥⎦

⎤⎢⎣

⎡+− )(١

BAA

TTbQ

(٣٢-٤)

Tb = TA + TB – Ta (٣٣-٤)

٢٥

and the transient temperature rise, θc(t), of the conductor above the outer surface of the cable is:

θc(t) = ١٢

Wc Ta (١ – e-at) + Tb (١ – e-bt) (٣٤-٤)

where Wc is the power loss per unit length in a conductor or an equivalent conductor based on the maximum conductor temperature attained. The power loss is assumed to be constant during the transient.

The conductor to cable surface attainment factor a(t) is then obtained from:

a(t) = θc(t) Wc (TA + TB) (٣٥-٤)

٤٫٢٫٤ Calculation of cable environment partial transient

The cable environment is the second part of the thermal circuit. The methods for calculating the partial thermal transient are set out in the following sub-clauses.

٤٫٢٫٤٫١ Buried cables (directly or in ducts)

The transient response of the cable environment is calculated by an exponential integral formula which is the direct transient equivalent of the steady-state procedure adopted in the standard mentioned in Sub-clause ٤٫١, for groups of separate cables.

The transient temperature rise, θe(t), above ambient of the outer surface of the hottest cable of a group of cables, and similarly loaded, is:

θe(t) = π

ρ٤

١WT

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ′−−⎟

⎟⎠

⎞⎜⎜⎝

⎛ −−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −− ∑

−=

= δδδδ Td

Eit

dEi

tLEi

tD

Ei pkpkNk

k

c

٤)(

٤)(

١٦

٢٢١

١

٢٢

(٤-

٣٦)

where:

W١ = the total power loss per unit length of each cable in the group

- Ei(-x) = the exponential integral function. Evaluation of this function is facilitated by using either the exponential integral scales provided in Figures ٦ and ٧ or the formulae in the notes to those figures

ρT = soil thermal resistivity

De = external surface diameter of cable

٢٦

δ = soil thermal diffusivity

t = time from moment of application of heating

L = axial depth of burial of hottest cable

dpk = distance from centre of cable k to centre of hottest cable p

d´pk = distance from image of centre of cable k to centre of hottest cable p

N = number of cables

The summation extends over all the cables in the group except the hottest, and this summation term is not needed if only a single cable is being considered.

Equation (٣٦-٤) is the general equation for the transient response of buried cables and its application for cyclic rating purposes is given in Clause ٧. It is noted that this equation cannot be used directly for cables un unfilled troughs.

٤٫٢٫٤٫٢ Cables in air

For cables in air it is unnecessary to calculate a separate response for the cable environment. The complete transient θ(t) is obtained by replacing TB by (TB + TC) in the constituent terms of the formula for θc(t) in Sub-clause ٤٫٢٫٣. Full details are set out in Sub-clause ٤٫٤٫١٫٢

٤٫٣ Calculation of partial transients for short duration (≤ T.Q) ١٣

These formulae are usually suitable for times from ١٠ min to about ١ h; see Sub-clause ٤٫١٫٤ for exact durations.

٤٫٣٫١ Representation of the dielectric (single and three-core cables)

The method is the same as in Sub-clause ٤٫٢٫١ except that the cable insulation is divided at a diameter of dx = giving two portions having equal thermal resistance, as shown in Figure ٤.

ci dD ⋅

In this figure:

T١ = thermal resistance of insulation, given in the standard to be approved by SASO concerned with “Electric cables - Calculation of the current rating - Part ١-١: Current rating equations (١٠٠٪ load factor) and calculation of losses - General", "Electric cables - Calculation of the current rating - Part ١-٢: Thermal resistance - Calculation of thermal resistance" and " Electric cables - Calculation of the current rating - Part ١-٣: Sections on operating conditions - Reference operating conditions and selection of cable type"+.

p* = factor for each portion of the divided insulation

٢٧

p* =

⎟⎟⎠

⎞⎜⎜⎝

⎛

c

i

dD

n١

١ –

١

١

−⎟⎟⎠

⎞⎜⎜⎝

⎛

c

i

dD

(٣٧-٤)

Note. – This formula differs from that in Sub-clause ٤٫٢٫١٫١ because of the substitution of dx = . ci dD ⋅

D١ = external diameter of insulation

_____________________ + This standard will be based on IEC ٦٠٢٨٧ series.

dc = external diameter of conductor

Qc = thermal capacitance of conductor or equivalent single-core conductor

١iQ = σ

٤π dc (Di – dc) = thermal capacitance of first portion of insulation

٢iQ = σ

٤π Di (Di – dc)= thermal capacitance of second portion of insulation

σ = volumetric specific heat of insulation

Thermal resistance and capacitance are reckoned for one conductor or equivalent single-core conductor as in Sub-clause ٤٫٢٫١.

٤٫٣٫٢ Representation of the cable

٤٫٣٫٢٫١ General

The first part of the thermal circuit simulates the cable and is always represented by a two-section network (see Figure ٢). The first section includes the thermal capacitance of the conductor and the inner part of the inside portion of the dielectric plus half the thermal resistance of the dielectric. The second section contains the remaining π section along with the remainder of the cable represented as in Sub-clause ٤٫٢٫٢٫١. In general this will give a representation of a cable which consists of more than two sections. A typical example of this, based on the cables in Item a) of Sub-clause ٤٫٣٫٢٫٢ below, is given in Figure ٥.

These sections are then reduced to two sections as shown in Sub-clause ٤٫٢٫٢٫٣ and Appendix A, being careful to keep the first section unaltered.

Formulae for the constants in Figure ٢ for the more common types of cables and installations are given in the following Sub-clauses.

For single-core cables in trefoil formation, the values of T١ and T٣ are adjusted by the appropriate factors given in the standard to be approved by SASO concerned with "Electric cables - Calculation of the current rating - Part ١-٢: Thermal resistance - Calculation of thermal resistance"*

٢٨

٤٫٣٫٢٫٢ Representation of common type of cable Note. – Symbols are defined only where they are first used.

a) Solid and self-contained oil-filled paper insulated cables, extruded cables and thermally similar constructions

TA = ١٢

T(٣٨-٤) ١

_____________________ * This standard will be based on IEC ١-٢-٦٠٢٨٧.

TB = ١٢

T١ + qs T٣

(٣٩-٤)

QA = Qc + p*Qi(٤٠-٤) ١

QB = p*Qi٢ + (١ – p*)Qi(٤٫٤١) +١ ⎥⎦

⎤⎢⎣

⎡ ++−

s

jsi q

QpQQp

'*)١(

٢

٢

٣١

٣

٢١

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+ TqT

Tq

s

s

The symbols are defined in Item a) of Sub-clause ٤٫٢٫٢٫٢ and Sub-clause

٤٫٣٫١.

The final element (١ – p′ )Qj/qs (see Figure ٥) is omitted because the transient for the cable response is calculated on the assumption that the output terminals on the right-hand side are short-circuited.

b) Pipe-type cables (high pressure oil-filled)

TA = ١٢

T(٤٢-٤) ١

TB = ١٢

T١ + qs To + qe T٣

(٤٣-٤)

QA = Qc + p* Qi(٤٤-٤) ١

٢٩

QB = (١ – p*) Qi١ + Qi(٤٥-٤) ٢

The symbols are defined in Item b) of Sub-clause ٤٫٢٫٢٫٢ and Sub-clause ٤٫٣٫١.

c) Pipe-type cables (gas pressure, cables laid up, filled and armoured)

TA = ١٢

T(٤٦-٤) ١

TB = ١٢

T١ + qs Tf + qa T٢ + qj T(٤٧-٤) ٣

QA = Qc + p*Qi١ (٤٨-٤)

QB = (١ – p*)Qi١ + Qi(٤٩-٤) ٢

The symbols are defined in Item c) of Sub-clause ٤٫٢٫٢٫٢ and Sub-clause ٤٫٣٫١.

d) Pipe-type cables (gas pressure, no filling material or armour)

TA = ١٢

T(٥٠-٤) ١

TB = ١٢

T١ + qs T٢ + qe T(٥١-٤) ٣

QA = Qc + p*Qi١ (٥٢-٤)

QB = (١ – p*)Qi٠٫٣ + ١ Qi(٥٣-٤) ٢

The symbols are defined in Items b), c) and d) of Sub-clause ٤٫٢٫٢٫٢ and Sub-clause ٤٫٣٫١.

e) Cables in ducts

٣٠

TA = ١٢

T(٥٤-٤) ١

TB = ١٢

T١ + qs (T٣ + T´٤ + T´´٤) (٥٥-٤)

QA = Qc + p*Qi١ (٥٦-٤)

QB = (١ – p*)Qi٠٫٣ + ١ Qi(٥٧-٤) ٢

The symbols are defined in Items b), c) and e) of Sub-clause ٤٫٢٫٢٫٢ and Sub-clause ٤٫٣٫١

٤٫٣٫٢٫٣ Three-core cables of all types, in ducts or buried directly in the ground

a) For the very early part of the transient, i.e. for durations up to the value of the product T.Q (where T is the thermal resistance of a core and Q is its thermal capacitance) the method is to calculate the temperature difference across the core insulation only. The quantities to be entered into the equations of Sub-clause ٤٫٣٫٣ are:

TA = ١٢

T(٥٨-٤) ١

TB = ١٢

T(٥٩-٤) ١

QA = Qc + p*Qi١ (٦٠-٤)

QB = (١ – p*)Qi١ + Qi(٦١-٤) ٢

where the quantities T١, Qc, Qi١ and Qi٢ refer to one core. Mutual heating between the cores can be ignored.

b) For durations of about ١٢

T.Q (the quantities T and Q now refer to the

whole cable) and greater, the transient is calculated according to Sub-clause ٤٫٢ including if appropriate the influence of the ground given in Sub-clause ٤٫٢٫٤.

c) For durations between these two limits, T.Q for one core and ١٢

T.Q for

the whole cable, the transient is assumed to be given by straight line interpolation on linear temperature rise/logarithmic time axes.

٤٫٣٫٢٫٤ Representation of cables and installations of types not specified in previous Sub-clause

٣١

The previous Sub-clause provides circuits for most cable types. However, to cover contingencies, the method of Sub-clause ٤٫٢٫٢٫٣ is recommended for cases not specifically dealt with in this standard.

٤٫٣٫٣ Calculation of cable partial transient

The transient temperature response of a thermal circuit of Sub-clause ٤٫٣٫٢, to a step function of load current, considered in isolation, that is with the right-hand pair of terminals in Figure ٢ short-circuited, is found as follows:

Mo = ١٢

(⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

⎭⎬⎫

⎩⎨⎧

+ BBBAA TQTTQ ٦٢-٤)

No = QA TA QB TB (٦٣-٤)

a = o

oo

NNMM −+ ٢

٠ (٦٤-٤)

b = o

oo

NNMM −− ٢

٠ (٦٥-٤)

Ta = ba −

١⎥⎦

⎤⎢⎣

⎡+− )(١

BAA

TTbQ

(٦٦-٤)

Tb = (TA + TB) – Ta (٤٫٦٧)

and the transient temperature rise θc(t) of the conductor above the surface of the cable is:

θc(t) = Wc (⎥⎦

⎤⎢⎣

⎡−+− −− )١()١( bt

bat

a eTeT ٦٨-٤)

where Wc is the power loss per unit length in a conductor, based on the maximum conductor temperature. The power loss is assumed to be constant during the transient.

The conductor to cable surface attainment factor a(t) for the transient temperature rise between the conductor and outside surface of the cable is then obtained from:

a (t) = θc(t) (⎥⎦

⎤⎢⎣

⎡+ )( BAc TTW ٦٩-٤)

٣٢

٤٫٣٫٤ Calculation of cable environment partial transient

The cable environment is the second part of the thermal circuit. The methods for calculating the partial thermal transient are set out in the following Sub-clauses.

٤٫٣٫٤٫١ Directly buried cables

a) Single-core cables

For single-core directly buried cables the influence of the ground around the cable increases with time and it may be significant even within the time period ١

٣T.Q.

With the present method of calculation the contribution of the ground follows the same lines as for the longer durations dealt with in Sub-clause ٤٫٢٫٤ but some simplification is possible since the image terms are negligible.

The transient temperature rise θe(t) of the outer surface of the hottest cable of a group of cables loaded in the same manner is:

θe(t) = π٤

١WPT

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −− ∑

−=

= δδ td

Eit

DEi pk

Nk

k

c

٤)(

١٦

٢١

١

٢

(٧٠-٤)

Unless cables are touching or are very closely spaced the term inside the summation sign is likely to be negligible for the short periods of time involved.

b) Three-core cables

For three-core directly buried cables the contribution of the ground is negligible for durations up to ١

٣T.Q.

٤٫٣٫٤٫٢ Cables in ducts

For all types of cables installed in ducts the contribution of the ground is negligible for durations up to ١

٣T.Q.

٤٫٣٫٤٫٣ Cables in air

For all types of cables installed in air, use the method set out in Sub-clause ٤٫٢٫٤٫٢.

٤٫٤ Calculation of the complete temperature transient

٤٫٤٫١ Transient temperature response

٣٣

After calculating separately the two partial transients and the conductor to cable surface attainment factor (see Sub-clause ٤٫٢ or ٤٫٣) the total transient rise (θ (t)) above ambient temperature is obtained:

a) for buried configurations by simple addition of the cable and modified environment partial transients;

b) for cables in air by modification of the cable partial transient.

When the temperature response to a current that changes in discrete and multiple steps is calculated, the calculations described in this clause should be repeated for every partial steps of the current. In the calculation of these partial transients, the current heat source (the heat gained or lost in the partial current step) and the correct temperatures (for the calculation of electrical resistances) shall be used in the formulas above. One way of accomplishing this is to perform iterative calculations for each load steps as follows:

١) Start with the temperature achieved at the end of the previous load step.

٢) Calculate the electrical resistances corresponding to this temperature and obtain power losses.

٣) With these losses calculate the temperature at the end of the time step.

٤) Use this temperature to calculate the electrical resistances and power losses and go to step ٣.

٥) Repeat steps ٣ and ٤ until convergence is achieved and go to the next load step.

The total temperature response per partial step will be found by using the appropriate formulae in this clause. The complete total temperature response of the cable circuit will be found by adding all the partial transients, thereby taking the time differences between the partial steps into account.

٤٫٤٫١٫١ Buried cables (directly or in ducts)

θ (t) = θc(t) + a(t) · θe(t) (٧١-٤)

where:

θ (t) = transient temperature rise of conductor above ambient

θc(t) = transient temperature rise of conductor above cable surface

θe(t) = transient temperature rise of cable surface above ambient from t = ٠, assuming total losses (W١) from cable surface

a(t) = attainment factor for the transient temperature rise between the conductor and the outside surface of the cable

٤٫٤٫١٫٢ Cables in air

For this case:

٣٤

θ (t) = θc(t) (٧٢-٤)

and θc(t) is obtained by replacing TB by (TB + TC) in the formula of Sub-clause ٤٫٣٫٣.

where:

TC = qs,T٤ for single-core cables, solid type and oil-filled paper insulated cables

TC = qe, T٤ for pipe-type oil-filled and gas pressure cables

TC = qj, T٤ for pipe-type gas-filled cables

TC = qa, T٤ for three-core cables: for three-core cables, when the duration of the transient is less than the product T.Q of one core (see Sub-clauses ٤٫١٫٤ and ٤٫٣٫٢٫٣) then TC = ٠

qs, qe, qj and qa are defined in the list of symbols

T٤ is the external thermal resistance due to air in the steady-state, calculated according to the standard to be approved by SASO concerned with "Electric cables - Calculation of the current rating - Part ١-٢: Thermal resistance - Calculation of thermal resistance"*, and shall be expressed as a quantity per conductor or equivalent conductor.

٤٫٤٫٢ Not used.

٤٫٤٫٣ Transient temperature response caused by sudden application of voltage

(transient due to dielectric loss)

In the preceding Sub-clauses it has been assumed that the temperature rise of the conductor due to the dielectric loss has reached its steady state, and that the total temperature at any time during the transient can be obtained simply by adding the constant rise due to the dielectric loss to the transient value due to the load current.

If the application of load current and system voltage occur at the same time, then an additional transient temperature rise due to the dielectric loss has to be calculated.

٢٧٥ ٤٫٤٫٣٫١Cables at voltages up to and including kV

a) Long durations (> ١٣

T.Q)

٣٥

In this case, it is sufficient to assume that half the dielectric loss is produced at the conductor and the other half at the core-screen or sheath. The cable thermal circuit is derived by the method given in Sub-clause ٤٫٢, i.e. the coefficient p which apportions to the conductor a fraction of the dielectric thermal capacitance is the same as that used when calculating transient due to joule losses. In comparison with the transients due to joule loss, the only differences are that a dielectric loss of Wd starts from the conductor and that the coefficients qs, qe, qa and

qj are always equal to ٢.

١٢

b) Short durations (≤ ١٣

T.Q)

The approach is the same as in Item a) above, but the coefficient p apportioning the different fractions of the dielectric thermal capacitance shall be that given in Sub-clause ٤٫٣.

_____________________ * This standard will be based on IEC ١-٢-٦٠٢٨٧.

٢٧٥ ٤٫٤٫٣٫٢Cables at voltages higher than kV

In this case, the dielectric loss can be an important fraction of the total loss (i.e. paper insulated cables) and the coefficient p in Sub-clauses ٤٫٢٫١ and ٤٫٣٫١, used to apportion the thermal capacitance of the dielectric to the conductor and metallic sheath or screen, shall be replaced by:

pd = ٢٢

٢٢٢

١١

١٢١١١

⎥⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎥

⎦

⎤⎢⎣

⎡−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

c

i

c

i

c

i

c

i

c

i

c

i

dD

ndD

dD

dD

ndD

ndD

(٧٤-٤)

where:

Di = external diameter of insulation

dc = external diameter of conductor

The rest of the thermal circuit, outside the metallic sheath or screen, is the same as for the case of the conductor loss.

٣٦

The fraction of the dielectric loss starting from the conductor is still reckoned as ١

٢ Wd, and the coefficients qs, qe, qa, and qj are equal to ٢.

Information on the voltages above which dielectric losses should be considered is given in the standard to be approved by SASO concerned with "Electric cables - Calculation of the current rating - Part ١-١: Current rating equations (١٠٠٪ load factor) and calculation of losses - General"*.

SECTION THREE – CYCLIC RATINGS

٥. Computation of the cyclic rating factor (M)

٥٫١ General

The cyclic rating factor is denoted by M, and is the factor by which the permissible steady-state rated current (١٠٠ % load factor) may be multiplied to obtain the permissible peak value of current during a daily (٢٤ h) cycle such that the conductor attains, but does not exceed, the standard permissible maximum

_____________________ * This standard will be based on IEC ١-١-٦٠٢٨٧.

temperature during this cycle. A factor defined in this way has the steady-state

temperature, which is usually the permitted maximum temperature, as its

reference. The cyclic rating factor depends only on the shape of the daily cycle

and is thus independent of the actual magnitudes of the current.

The loss-load factor (μ) of the daily current cycle is determined first (see Clause ٦) by decomposing the cycle into hourly rectangular pulses. The temperature responses of the cable and soil to the complete cycle of losses can be found by adding together the response to each hourly rectangular pulse, having regard to the time period between each pulse and the time of maximum temperature.

Detail of load cycle is needed over a period of only ٦ h before the time of maximum temperature, and earlier values can be represented with sufficient accuracy by using an average. The loss-load factor μ provides this average. Location of the time of maximum temperature is done by inspection, bearing in mind that although it usually occurs at the end of the period of maximum current this may not always be the case.

٣٧

Let the conductor temperature rise at time i hours after the application of a step

function of losses corresponding to the rated current IR be θR(i). Then the

maximum conductor temperature rise above ambient for a daily cyclic current

whose peak value is the rated current is:

θmax = Y٠ θR (١) + Y١ θR (٢) – θR (١) + Y٢ θR (٣) – θR (٢)

+ Y٥ θR (٦) – θR (٥) + μ θR (∞) – θR (٦) (١-٥)

M = max

)(θ

θ ∞R (٢-٥)

where Y٠, Y١… Y٥ are the ordinates of the loss-load cycle for the ٦ h before the time of maximum temperature (see Clause ٦).

Specific formulae for various forms of load cycle are given in the following Sub-clauses.

٥٫٢ Calculation of cyclic rating factor (including cable thermal capacitance)

٥٫٢٫١ Any load cycle of known shape

The cyclic rating factor is given by:

M = ٢/١

٥

٠ )()٦(

١)()(

)()١(

١

⎥⎦

⎤⎢⎣

⎡∞

−+⎥⎦

⎤⎢⎣

⎡∞

−∞+∑

= R

R

R

R

R

R

ii

iiY

θθ

μθθ

θθ

(٣-٥)

)()(∞R

R iθθ = ١ – k + k β (i) a(i) (٤-٥)

θR (o) = ٠

Here β (i) is the attainment factor for the cable or duct external surface, i.e. the ratio of the temperature rise at time i to the rise in the steady state (equation ٧-٢)), a(i) is the attainment factor for the temperature rise of the conductor above the outside surface of the cable (equation (١-٧)).

Formulae for calculating )()(∞R

R iθθ are given in Clause ٧.

٣٨

٥٫٢٫٢ Flat top load cycle

A flat top load cycle is a cycle with a sustained maximum current for a minimum duration of ٦ h, without any restriction on the shape of the remainder of the cycle except that the maximum conductor temperature rise occurs at the end of the duration of the sustained maximum current.

a) For an isolated three-core cable, the cyclic rating factor is given by:

M = { }[( )]٢/١

)١)١(١)٦()٦(١)٦١

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+−−− βμ kaa (٥-٥)

where α(٦) is given by equation (١-٧), with i expressed in hours, and ١-β(٦) may be obtained from Table I, or if necessary from Appendix C.

Formulae for calculating k are given in Clause ٧.

b) For groups of cables with equal losses, cables or ducts not touching.

The cyclic rating factor is given by:

M = ٢/١

)()١)٦

)()٦(

١

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎭⎬⎫

⎩⎨⎧

∞−+

∞ R

R

R

R

θθ

μθθ

(٦-٥)

where θR (٦)/θR (∞) is given by equation (٦-٧).

٥٫٢٫٣ Load cycle of unknown shape but with a known loss-load (μ)

This cycle may be considered as a flat top cycle and the formulae of Sub-clause ٥٫٢٫٢ used.

٦. Calculation of loss-load (μ)

a) The daily load cycle is first expressed as ٢٤ hourly values by scaling the whole cycle so that its maximum value is equal to unity (see Figure ٨a).

b) The magnitude of each hourly value is then squared to give ٢٤ values representing the cycle of cable joule losses (see Figure ٨b).

c) This curve is then expressed as a series of step function changes so as to give a stepped curve having approximately the same average value as the original one (see Figure ٨c).

٣٩

d) Denote the ordinates of the stepped curve for the six hourly periods before the occurrence of the highest temperature by Y٠, Y١, Y٢… Y٥ (see Figure ٨c). Location of the time of the highest temperature is done by inspection, bearing in mind that although it usually occurs at the end of the period of highest current, this may not always be the case. The temperature also depends on the general level of current preceding that period. Thus, in the example given in Appendix F and Figure ٨, the highest temperature occurs at ١٧٫٥ h although the highest value of current was at ٨ h. It should be noted that the particular way in which the load and loss-load curves of Figure ٨ have been drawn assumes that the measured values in Table F٤ apply to periods of time which extend half an hour before and after each observation time.

e) The remaining ordinates are then designated by Y٦, Y٧… Y٢٣ in a cyclic manner so that Y٢٣ a is adjacent to Y٠.

f) The loss-load factor (μ) is given by:

μ = ٢٤١ Yi (∑

=

٢٣

٠i١-٦)

٧. Calculation of )()(∞R

R iθθ

٧٫١ Single isolated three-core cable

One three-core cable laid directly or in a duct.

a(t) = BA

btb

ata

TTeTeT

+

−+− −− )١()١( (١-٧)

β(t) =

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

e

e

DLn

tLEi

tD

Ei

٤١٢

١٦

٢٢

δδ (٢-٧)

k = ٤١

٤١

)( TWTTWTW

BAc ++ (٣-٧)

where:

De = diameter of cable or external diameter of duct

t = ٦٠٠ ٣ i

Ta, Tb, TA, TB, a and b are calculated as in Sub-clauses ٤٫٣٫٢ and ٤٫٣٫٣ after evaluating the parameters of the equivalent single-core cable according to Sub-clause ٤٫٢٫١٫٢.

W١ = total joule power loss in the cable per unit length at rated temperature

٤٠

Wc = joule power loss in one conductor per unit length at rated temperature

T٤ = external thermal resistance of a single isolated cable or duct

= πρ٢

T ١n (٣-٧a) eDL٤

– Ei (- x) is the exponential integral function. Evaluation of this formula is facilitated by using either the exponential integral scales provided in Figures ٦ and ٧ or the formulae in the notes to those figures

Note. – For times up to and including ٦ h the "image" term (second term in the numerator of equation (٢-٧)) can be ignored.

Then:

)()(∞θ

θ t = ١ – k + k β(t) a(t) (٤-٧)

and with t expressed in hours by i:

)()(∞R

R iθθ = ١ – k + k β(i) a(i) (٥-٧)

This expression is evaluated for i = ٦ …٣ ,٢ ,١.

t = seconds

i = hours

٧٫٢ Single isolated circuit

a) Three single-core cables, laid touching, in trefoil or flat formation.

b) Three single-core cables laid in touching ducts, in trefoil or flat formation.

c) Pipe-type cable.

The method is the same as in Sub-clause ٧٫١, except that:

١) The external thermal resistance (T٤) to be used is that for cables in trefoil, or flat touching formation or in ducts calculated from the appropriate formula in the current edition of the standard mentioned in to be approved by SASO concerned with “Electric cables - Calculation of the current rating - Part ١-١: Current rating equations (١٠٠٪ load factor) and calculation of losses - General", "Electric cables - Calculation of the current rating - Part ١-٢: Thermal resistance - Calculation of thermal resistance" and " Electric cables - Calculation of the current rating - Part ١-٣: Sections on operating conditions - Reference operating conditions and selection of cable type"*.

٢) W١ is the total power loss per unit length per single cable or duct.

٤١

٣) The value of cable or duct diameter used is that of an isolated cable or duct.

٧٫٣ Group of “N” cables with equal losses; cables or ducts not touching

In this case, θR(i) is the conductor temperature rise of the hottest cable in the group.

)()(∞R

R iθθ = ١ – k١ + k١ γ(i) a(i) (٦-٧)

where:

k١ is the ratio of cable (or duct) surface temperature rise above ambient to conductor temperature rise above ambient under steady conditions, and with all cables of the group having equal losses

a(i) is given by equation (١-٧) with i expressed in hours (t = ٦٠٠ ٣ i)

γ(i) is given by the following equation with i expressed in hours Note. – For times up to and including ٦ h, the image terms (second and fourth terms in the

numerator of equation (٧-٧)) may be ignored.

γ(i) =

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟

⎟⎠

⎞⎜⎜⎝

⎛−−−+⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

e

fe

DLFn

tLEi

td

EiNtLEi

tD

Ei

٤١٢

١٦)١(

١٦

٢٢٢٢

δδδδ (٧-٧)

_____________________ * This standard will be based on IEC ٦٠٢٨٧ series.

where:

t = ٦٠٠ ٣ i

F = pNpkpp

pNpkpp

dddd

dddd

......

......

٢١

٢١

⋅

′′′⋅′ (٨-٧)

with the term d′pp/dpp excluded, giving (N-١) terms.

df = )١)/١

٤−NF

L (٩-٧)

and the distances dpk and d′pk are defined in Figure ٩. The values of De and L to be used for the calculation of a(i) refer to the hottest cable, or the duct containing the hottest cable, of a group.

٤٢

ki = )()(

(

٤٤١

٤٤١

TTWTTWTTW

BAc Δ+++Δ+ (١٠-٧)

Where T٤ is the external thermal resistance of the hottest cable of the group if isolated, T٤ + ΔT٤ is the external thermal resistance of the hottest cable of the group taking into account the losses, assumed to be equal, in all cables of the group, and the conductor losses Wc and cable joule losses W١ per unit length of cable for the hottest cable of the group correspond to all cables working at a sustained ١٠٠٪ current.

ΔT(١١-٧) = ٤ π

ρ

٢١ FnT

k١ may also be found from the formula:

k(١٢-٧) = ١ )(

( ٤٤١

∞Δ+

θTTW

The dependence of k١ on soil thermal resistivity is given in Appendix C with k replaced by k١.

٧٫٤ Group of circuits, each of three single-core identical touching cables or ducts, all cables having equal losses

The methods are the same as those given in ٧٫٣ above, except that:

a) The external thermal resistance T٤, on which the value of k١ is dependent, relates to

T(١٣-٧) = ٤ π٥.١

Tρ ⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛٦٣٠.٠٤ln

eDL

for metallic sheathed and part-metallic covered cables (where helically laid armour or screen wires cover from ٢٠٪ to ٥٠٪ of the cable circumference),

or

T(١٤-٧) = ٤ π٢١

Tρ ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

e

L

e DDL ٢

ln٢٤ln

for non-metallic sheathed cables

where

ρT is the soil thermal resistivity, in Km/W;

L is the depth of laying, in metres (m);

De is the external diameter of cable or duct, in metres (m).

٤٣

b) The symbols N, d'pk and dpk used for the calculation of γ(i), F and df in equations (٧-٧), (٨-٧) and (٩-٧) is defined as:

- N is the number of circuits (of three single-core cables per circuit);

- dpk is distance from the center of circuit k to the centre of the circuit the hottest cable;

- d'pk is the distance of the image of the centre of circuit k to the centre of the circuit containing the hottest cable.

c) Additional external thermal resistance caused by heating from other cables in a group is expressed by:

ΔT(١٥-٧) = ٤ π

ρ٢

ln٣ FT

where F is given by item b) above.

d) W١ is the total loss per unit length per single cable or duct.

e) The value of the cable or duct diameter to be assumed for the calculation of γ(i) is that of an isolated cable or duct.

SECTION FOUR – EMERGENCY RATINGS

٨. Calculation of emergency ratings

٨٫١ Thermally isolated circuits

The following procedure for calculating the short time rating of single circuits is based on a knowledge of the conductor temperature transient as derived by the methods set out in Clause ٤. The particular method from Clause ٤ to be used depends on the criteria given in Sub-clause ٤٫١٫٤.

Consider an isolated buried circuit carrying a constant current I١ applied for a sufficiently long time for steady-state conditions to be effectively reached. Subsequently, from a time defined by t = ٠, an emergency load current I٢ (greater than I١) is applied. If I٢ is applied for any given time t, the question is how large may I٢ be so that conductor temperature does not exceed a specified value, taking into account the variation of the electrical resistivity of the conductor with temperature. The effect of dielectric loss is neglected in this treatment, but is taken into account at the end of this Sub-clause.

Denote:

r = θmax/θR(x)

t = time measured from moment of application of current I٢

٤٤

IR = sustained (١٠٠٪ load factor) rated current for the conductor to attain, but not exceed, the standard maximum permissible temperature

I١ = constant current applied to cable prior to emergency loading

I٢ = emergency load current which may subsequently be applied for time i so that the conductor temperature rise above ambient at the end of the period of emergency loading is θmax

Note. – The method only holds for values of I٢٫٥ ≥ ٢ IR.

h١ = I١/ IR

R١ = a.c. resistance of conductor before application of emergency current (i.e. at the conductor temperature corresponding to I١)

Rmax = a.c. resistance of conductor at end of period of emergency loading

RR = a.c. resistance of conductor with sustained application of rated current IR, i.e. at standard maximum permissible temperature

θmax = maximum permissible temperature rise above ambient at end of period of emergency loading

θR(t) = conductor temperature rise above ambient after application of IR, neglecting variation of conductor resistance

see the calculation for θ(t) in ٤٫٤.

θR(x) = value of θR in the steady-state, i.e. the standard maximum permissible temperature rise

The emergency current I٢ is given by:

I٢ = IR (١-٨) ٢١

١٢١١

٢١

/

RR

RmaxR

max )(/)t(])R/R[hr()R/R(

RRh

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∞

−+

θθ

This equation takes account of the variation of conductor resistance with temperature. Thus Equation (٣-٨) should not be applied when Equation (٨٫١) is used. Where armour, screen or metallic sheath losses are high, it is recommended that the losses at the final armour, screen or metallic sheath temperature are used in determination of the transient temperature response.

The effect of heating due to dielectric loss is taken into account, neglecting any change in dielectric loss with temperature, by calculating the steady-state conductor temperature rise Δθd due to the dielectric loss:

Δθd = Wd [١٢

T١ + n (T٢ + T٣ + T٤)] (٢-٨)

(See the standard to be approved by SASO concerned with “Electric cables - Calculation of the current rating - Part ١-١: Current rating equations (١٠٠٪ load factor) and calculation of losses - General")*

٤٥

and subtracting this value from θmax, θR(t) and θR(∞). The calculation of I٢ then proceeds as above, using these modified values of θmax, θR(t) and θR(∞). The values of R , Rmax, R١ and R٢ are not altered. R

٨٫٢ Groups of circuits

Where groups of circuits which are not thermally independent carry loads behaving in the same way, the methods for calculating the transients and the ratings given here can be applied.

When the loads do not behave in the same way, for example, the first circuit has an increased load while the second goes off load, the matter requires careful consideration. In most cases of buried cables the time lag associated with the mutual heating between circuits is so great that any worthwhile reduction in temperature of the first circuit due to the second going off load does not occur for many hours.

Figure ١٠ shows the time taken for a ١ K reduction in temperature of the first circuit of a double circuit ٩٣٥ ١ mm٤٠٠ ,٢ kV installation when the current in the second circuit is reduced to zero. Thus any increase in load for the first circuit during this period will depend entirely on what increase in conductor temperature can be permitted above the value the first circuit had attained before the change in load.

٨٫٣ Correction to transient temperature response to take account of variation in conductor losses with temperature

The change in conductor resistance with temperature during the transient period results in conductor losses being variable with time. Allowance for the variation of conductor loss with temperature gives the corrected temperature rise:

_____________________ * This standard will be based on IEC ١-١-٦٠٢٨٧.

θα(t) = ))()((١

)(t

tθθα

θ−∞+

(٣-٨)

where

θ(t) is the conductor transient temperature rise above ambient without correction for variation in conductor loss, and is based on the conductor resistance at the end of the transient period;

θ (∞) is the conductor steady-state temperature rise above ambient;

α is the temperature coefficient of electrical resistivity of the conductor material at the start of the transient.

٤٦

NOTE α = iθβ +

١

where

β is the reciprocal of temperature coefficient at ٠oC (see Table E.٢);

θi is the conductor temperature at the start of the transient period.

٩. Bibliography

The references given below are quoted for information purposes only; they are not obtainable from the IEC Central Office.

[١] Current ratings for cyclic and emergency loads, Part ١: ELECTRA No. ٢٤, October ١٩٧٢, p. ٦٣.

[٢] Computer method for the calculation of a single-core cable to a step function thermal transient: ELECTRA No. ٨٧, March ١٩٨٣, p. ٤١.

[٣] Current ratings for cyclic and emergency loads, Part ٢: ELECTRA No. ٤٤, January ١٩٧٦, p. ٣.

[٤] M. Abramowitz and I. Stegun: Handbook of Mathematical Functions. Dover Publications, New York.

[٥] Tables of the exponential integral for complex arguments, National Bureau of Standards Applied Mathematics Series, ١٩٥٨ ,٥١.

٤٧

APPENDIX A

REDUCTION OF A MULTIPLE THERMAL CIRCUIT TO

ONE HAVING TWO SECTIONS

An equivalent thermal circuit requires many elements in order to represent parts between conductor and outer surface of cables of complex construction, for example SL type cables, and solution of the corresponding mathematical equations calls for lengthy calculation. In order to simplify the calculation, and to standardize the procedure for all cable types not dealt with in this standard, the following method for combining the several elements in such a circuit into two sections will be used.

Following the instructions given in Clause ٤, the different elements of a cable are represented by either a capacitance or a resistance or a complete resistance-capacitance section, and are associated together in a series circuit as shown in Figure A١.

FIG. A١. – General series network representing a cable.

An equivalent circuit, which represents the cable with sufficient accuracy, can now be derived having two sections TAQA and TBQB as shown in Figure ٢. The first section TAQA of the derived circuit is made up of TA = Ta = T١ and QA = Qa = (Qc + pQi) without modification, in order to maintain the correct response for relatively short durations.

The second section TBQB of the derived circuit is made up from the remaining sections Tβ Qβ up to TvQv.

The lumped constants for the derived circuit are summarized as follows:

TA = Ta = T١ (A-١)

QA = Qa = Qc + pQi (A-٢)

TB = Tβ + T γ + Tδ + Tv-١ + Tv (A-٣)

QB = Qβ + ٢

...

...⎟⎟⎠

⎞⎜⎜⎝

⎛

+++

+++

v

v

TTTTTT

γβ

δγ Qγ

+ ٢

...

...⎟⎟⎠

⎞⎜⎜⎝

⎛

++++++

v

v

TTTTTT

γβ

εδ Qδ + … + ٢

... ⎟⎟⎠

⎞⎜⎜⎝

⎛

+++ v

v

TTTT

γβ

Qv (A-٤)

٤٨

APPENDIX B

TRANSIENT AVERAGE TEMPERATURE VARIATION THROUGH

DIELECTRIC OR THROUGH OIL FILLING IN PIPES

The transient oil pressure variation of oil-filled cables may be calculated from a knowledge of the variation of both average dielectric temperature and conductor temperature, and formulae for calculating the transient average temperature rise are given below in Clause B١.

For oil-pressure pipe-type cables, it is necessary to know the variation of average temperature of the oil contained between cables and pipe. The temperature variation to be calculated is that which occurs not through the dielectric, but outside it. Clause B٢ contains formulae for calculating the average temperature of the oil.

For very short duration transients (in general ١ h or less) these methods are not suitable.

B١. Oil-filled cables

a) Due to the conductor loss, the transient spatial average temperature rise of the dielectric above the outer surface of the cable is:

١ – e-at) + T ′b (١ – e-bt)] (B-١) θd(t) = Wc [T ′a (

This formula is similar to that given in Sub-clause ٤٫٢.٣, the only difference being the use of thermal resistances T ′a and T ′b instead of Ta and Tb. The values of T ′a and T ′b are:

T ′a = ba −

١ ⎥⎦

⎤⎢⎣

⎡+− )( BA

A

TpTbQp (B-٢)

T ′b = pTA + TB – T ′a (B-٣)

b) Due to dielectric loss, the transient average temperature rise is calculated by means of a formula similar to that in Item a), but with:

Wc replaced by ½ Wd, and the coefficient qs to be used in calculating TB and QB equal to ٢.

٤٩

In the case of cables at voltages higher than ٢٧٥ kV, the coefficient p of Sub-clause ٤٫٢٫١ shall be replaced by the coefficient pd of Sub-clause ٤٫٤٫٣٫٢.

B٢. Oil-pressure pipe-type cables

a) Due to conductor loss, the transient average temperature rise of the oil above the outer surface of the pipe (including its outer covering) is:

θo(t) = Wc baTB

− (B-⎥

⎦

⎤⎢⎣

⎡−−− −− )١()١( atbt ebea ٤)

This formula is based on some simplifying assumptions, including those of Item b) of Sub-clause ٤٫٢٫٢٫٢, and it is doubtful whether the average temperature rise worked out in this way can be related with certainty to the whole volume of oil. Unfortunately no better formula can be suggested at present.

b) Due to dielectric loss, the transient average temperature rise is calculated by means of a formula similar to that in Item a) of Clause B١ but with:

Wc replaced by ١/٢ Wd, and the coefficients qs and qe used in calculating TA, TB and QB, both equal to ٢.

٥٠

APPENDIX C

METHOD FOR DEALING WITH DIFFERENT SOIL RESISTIVITIES

THERMAL DIFFUSIVITIES AND DEPTHS OF LAYING

The factor [١ - β(٦)] is dependent on soil properties and depth of laying as well as the cable or duct diameter. The values quoted in Table I are for a depth of burial of ١ m and a soil thermal diffusivity of ٠٫٥ X ٦-١٠ m٢/s. These values may be used for cable or duct depths within the range ٠٫٧٥ to ١٫٥ m. For any other depth of burial or value of soil thermal diffusivity the numerical value may be obtained from:

[١ - β(٦)] = ١ - (C-١)

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−

)/١٢)٤١٦

٢

e

e

DLnt

DEi

δ

where:

t = ٦ x ٦٠٠ ٣ s

The factors k and k١ (see equations (٣-٧), (١٠-٧) and (١٢-٧)) are dependent on depth of laying and soil resistivity. If the laying conditions "a", for which a value of k is known, change to conditions "b" then the new value of k is given by:

k(b) =

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

)()(١

'١

١

٤

٤

akak

TT

(C-٢)

Where T٤ and T'٤ are the external thermal resistances for conditions "a" and "b" respectively. If the depth is not changed then T٤/T′٤ = p(a)/p(b).

A new value of k١ is obtained in the same way.

٥١

APPENDIX D

DIFFUSIVITY OF SOIL

For calculation of cyclic rating factors it is not necessary to know accurately the value of diffusivity and it is generally satisfactory to use the tabulated functions given in this standard which are based on the value of ٠٫٥ X ٦-١٠ m٢/s, and correspond roughly to a soil having a thermal resistivity of ١ Km/W, and a moisture content of around ٧٪ of dry weight. For exceptional cases and where moisture content is likely to be lower, it is advisable to use the methods given below to calculate alternative values.

a) The diffusivity δ can be estimated if the following quantities have been measured: density, moisture content by weight and thermal resistivity.

δ = )٠٤٢.٠٨٢.٠(

١٠ ٣

ηρ +

−

dT

m٢/s (D-١)

where:

ρT = thermal resistivity, K.m/W

d = dry density, kg/m٣

η = moisture content, % of dry weight

Alternatively, there are well-known laboratory measures for measuring diffusivity.

b) If only the thermal resistivity is known, then the diffusivity can be obtained from the following table:

TABLE D١

Values of diffusivity for soil Thermal resistivity

(K.m/W) Diffusivity

(m٢/s) ٠٫٨ ٠٫٥ x ٦-١٠ ٠٫٧ ٠٫٦ x ٦-١٠ ٠٫٦ ٠٫٧ x ٦-١٠ ٠٫٦ ٠٫٨ x ٦-١٠ ٠٫٥ ٠٫٩ x ٦-١٠ ٠٫٥ ١٫٠ x ٦-١٠ ٠٫٤ ١٫٢ x ٦-١٠ ٠٫٤ ١٫٥ x ٦-١٠ ٠٫٣ ٢٫٠ x ٦-١٠ ٠٫٢ ٢٫٥ x ٦-١٠ ٠٫٢ ٣٫٠ x ٦-١٠

Note. – These values were obtained from an empirical relationship for homogeneous and moist soils.

c) If no data are available on the soil, use a value of diffusivity of ٠٫٥ x ٦-١٠ m٢/s.

٥٢

APPENDIX E

PHYSICAL CONSTANT OF MATERIALS

TABLE E١

Thermal properties of insulating materials

Material

Thermal resistivity (K.m/W)

Volumetric *** specific heat

(J/m٣.K) Insulating materials*: Paper insulation in solid type cables ٢٫٠ ٦٫٠ x ١٠٦ Paper insulation in oil-filled cables ٢٫٠ ٥٫٠ x ١٠٦ Paper insulation in cables with external gas pressure ٢٫٠ ٥٫٥ x ١٠٦ Paper insulation in cables with internal gas pressure: a) pre-impregnated ٢٫٠ ٥٫٥ x ١٠٦ b) mass-impregnated ٢٫٠ ٦٫٠ x ١٠٦ Oil ١٫٧ **٧٫٠ x ١٠٦ Nitrogen at ١٥ atmospheres ** ٠٫٠١٧ x ١٠٦ PE ٢٫٤ ٣٫٥ x ١٠٦ XLPE ٢٫٤ ٣٫٥ x ١٠٦ Polyvinyl chloride: up to and including ٣ kV cables ١٫٧ ٥٫٠ x ١٠٦ greater than ٣ kV cables ١٫٧ ٦٫٠ x ١٠٦ EPR: up to and including ٣ kV cables ٢٫٠ ٣٫٥ x ١٠٦ greater than ٣ kV cables ٢٫٠ ٥٫٠ x ١٠٦ Butyl rubber ٢٫٠ ٥٫٠ x ١٠٦ Rubber ٢٫٠ ٥٫٠ x ١٠٦ Protective coverings: Compounded jute and fibrous materials ٢٫٠ ٦٫٠ x ١٠٦ Rubber sandwich protection ٢٫٠ ٦٫٠ x ١٠٦ Polychloroprene ٢٫٠ ٥٫٥ x ١٠٦ PVC: up to and including ٣٥ kV cables ١٫٧ ٥٫٠ x ١٠٦ greater than ٣٥ kV cables ١٫٧ ٦٫٠ x ١٠٦ PVC/bitumen on corrugated aluminium sheaths ١٫٧ ٦٫٠ x ١٠٦ PE ٢٫٤ ٥٫٥ x ١٠٦ Materials for duct installations: Concrete ١٫٩ ١٫٠ x ١٠٦ Fibre ٢٫٠ ٤٫٨ x ١٠٦ Asbestos ٢٫٠ ٢٫٠ x ١٠٦ Earthenware ١٫٧ ١٫٢ x ١٠٦ PVC ١٫٧ ٦٫٠ x ١٠٦ PE ٢٫٤ ٣٫٥ x ١٠٦

* For the purposes of current rating calculations, the semi-conducting screening materials are assumed to have the same thermal properties as the adjacent dielectric materials. Where plastics or elastomeric materials are used for protective coverings, the thermal resistivities shall be taken to be the same as those for the insulating grades of the materials given in this table.

** The thermal resistance of oil or gas filling for pipe-type cables is calculated by the method given in the standard to be approved by SASO concerned with "Electric cables - Calculation of current rating - Part ١-٢: Thermal resistance - Calculation of thermal resistance"****.

*** Different compounds have different values, but the values quoted should be used for safety.

**** This standard will be based on IEC ١-٢-٦٠٢٨٧.

٥٣

TABLE E٢

Properties of conducting materials

Material Reciprocal of temperature coefficient of resistance

at

Volumetric specific heat

Resistivity at ٢٠oC

٠oC (K) (J/m٣.K) (Ω.m) a) Conductors Copper ٣٫٤٥ x ١٫٧٢٤١ ٢٣٤٫٥ ١٠٦ x ١٠٨

٢٫٥ x ١٠٦ Aluminium ٢٫٨٢٦٤ ٢٢٨ x ١٠٨ b) Sheaths, screens and armour Lead or lead alloy ١٫٤٥ x ٢١٫٤ ٢٣٠ ١٠٦ x ١٠٨

٣٫٨ x ١٠٦ Steel ١٣٫٨ ٢٠٢ x ١٠٨ ٣٫٤ x ١٠٦ Bronze ٣٫٥ ٣١٣ x ١٠٨ ٣٫٨ x ١٠٦ Stainless steel * ٧٠ x ١٠٨ ٢٫٥ x ١٠٦ Aluminium ٢٫٨٤ ٢٢٨ x ١٠٨

* Temperature coefficient of resistance is negligible; thus, no correction is necessary in Sub-clause ٤٫٤٫٢.

٥٤

APPENDIX F

EXAMPLES

F١. General

Example calculations are given for a single circuit cable system of three cables in flat formation, not touching, for the following cases:

a) the transient temperature response of the centre cable for the first ٢٤ h after the application of a step function of rated current, the cables being assumed to have been energized for a sufficient time for the temperature rise due to the dielectric losses to have reached a steady state;

b) the cyclic rating when the daily load cycle of Figure ٨ is applied on a regular daily basis;

c) the emergency current rating for an arbitrary ٦ h period after partly loaded cables have attained a steady state, the limiting temperature being the normal operating temperature.

These have been based on the simplest method appropriate for times greater than ١ h, but are considered comprehensive enough to provide guidance for the undertaking of calculations under any conditions.

F٢. Cable and installation details

The examples are based on a single circuit ٤٠٠ kV self-contained oil-filled cable system with a ٠٠٠ ٢ mm٢ copper conductor, directly buried at a depth of ٠٠٠ ١ mm with a spacing of ٣٠٠ mm between cable centers, crossbonded. The continuous rated current (١٠٠٪ load factor) is ١٥٥٠ A, operating at a conductor temperature of ٨٥oC, in a ground ambient temperature of ١٠oC.

The circuit has been energized, but unloaded, for sufficient time for the conductor temperature rise due to dielectric losses to have become stable at ١٩٫٣ K.

٥٥

Table F١

Cable details Item Outside

diameter (mm)

Thermal resistance (K.m/W)

Thermal capacitance

(J/K.m)

Losses (W/m)

Conductor: copper ٣٠٫٣ ٠٣٨ ٧ – ٥٧٫٥ oil – ٩٤٦ Conductor screen ٢٧٥ ٠٫٠٢١ ٥٩٫٠ Dielectric (paper/oil) ١٤.٧٥ ٨٥٠ ١١ ٠٫٤٥٩ ١٠٥٫٠ Core screen ٣٣١ ٠٫٠٠٨ ١٠٦ Sheath (lead) ٢٫١ ٠٠٤ ٢ – ١١٤٫٠ Serving (PE) ٥٥٩ ٣ ٠٫٠٣٨ ١٢٢٫٠

∑ ٤٧.١٥ ٠٠٣ ٢٦ ٠٫٥٢٦

Time constant = T.Q = ٠٫٥٢٦ x ٦٧٧ ١٣ = ٠٠٣ ٢٦ s = ٣٫٨٠ h

Soil properties

Thermal resistivity = ١٫٠ K.m/W

Thermal diffusivity = ٠٫٥ x ٦-١٠ m٢/s

١٠oC Temperature =

Material properties

Thermal resistivity of serving = ٣٫٥ K.m/W

Thermal resistivity of oil/paper = ٥٫٠ K.m/W

Conductor properties

A.C. resistance at ٨٥oC = ١٢٫٦١٢ μΩ/m

F٣. Deviation of thermal circuit of the cable

From Table F١ the values for representing the dielectric (see Figure ١) are obtained as follows (see Sub-clause ٤٫٢٫١٫١):

Qc = ٩٨٤ ٧ = ٩٤٦ + ٠٣٨ ٧ J/K.m

Qi = ٤٥٦ ١٢ = ٣٣١ + ٨٥٠ ١١ + ٢٧٥ J/K.m

T٠٫٤٨٨ = ٠٫٠٠٨ + ٠٫٤٥٩ + ٠٫٠٢١ = ١ J/K.m

p = ⎟⎠⎞

⎜⎝⎛

٥.٥٧١٠٦١٢

١

n –

١٥.٥٧

١٠٦١

٢

−⎟⎠⎞

⎜⎝⎛

= ٠٫٤ (from equation (١-٤))

٥٦

The values for representing the cable (see Figure ٢) are determined using the equations in Sub-clause ٤٫٢٫٢٫٢, as follows:

TA = T٠٫٤٨٨ = ١ Km/W (from equation (٣-٤))

qs = ٣.٣٠١.٢٣.٣٠ + = ١٫٠٦٩

TB = ١ ١٫٠٦٩n = ٠٫٠٤ Km/W (from equation (٤-٤)) π٢٥.٣

١١٤١٢٢

١ QA = ٧٩٨٤ + (٠٫٤ x ٤٥٦ ١٢) = ٩٦٦ ١٢ J/Km (from equation (٥-٤))

p' = ⎟⎠⎞

⎜⎝⎛١١٤١٢٢١٢

١

n–

١١١٤١٢٢

١٢

−⎟⎠⎞

⎜⎝⎛

= ٠.٤٨٩ (from equation (٧-٤))

QB = (٠٫٤ – ١) ٩٧٦ ١٠ = + ٤٥٦ ١٢ J/Km ⎥⎦

⎤⎢⎣

⎡ +

٠٦٩.١)٠٠٤٢)٥٥٩٣٤٨٩.٠ x

(from equation (٦-٤))

F٤. Example of the calculation of the transient response

The example gives the transient response of the cable system in Clause F٢ during the first ٢٤ h after the application of a step function of current equal to the rated full load current. It has been assumed that the cables were energized, but unloaded, for sufficient time for the temperature rise due to the dielectric losses to have stabilized.

In the following example, the transient response is based on the formulae in ٤٫٢ for a transient of long duration. As the first time step chosen is less than one-third of the time constant of the cable, the use of the formulae in ٤٫٣ could be justified for the first time step. However, this would lead to a discontinuity in the temperature rise curve that cannot be justified

F٤٫١ Calculation of the response of the cable circuit

The response of the cable circuit is calculated using the formulae in Sub-clause ٤٫٢٫٣, based on the thermal circuit values in Clause F٣, as follows:

Mo = ٢١ (٠٫٠٤ + ٠٫٤٨٨) ٩٦٦ ١٢ + (٩٧٦ ١٠ x ٠٫٠٤) = ٦٤٣ ٣ s (from equation (٤-

٢٨))

٩٦٦ ١٢ x ٠٫٤٨٨ x ٩٧٦ ١٠ x ٧٧٨ ٢ = ٠٫٠٤ x ١٠٦ S٢ (from equation (٢٩-٤)) No =

a = ٦

٦٢

١٠٧٧٨.٢

٦٤٣٣)٦٤٣٣(١٠٧٧٨.٢

x

x−+ = ٢٫٤٨ x ٣-١٠ (from equation (٣٠-٤))

s١

٥٧

b = ٦

٦٢

١٠٧٧٨.٢

٦٤٣٣)٦٤٣٣(١٠٧٧٨.٢

x

x−− = ١٫٤٥ x ٤-١٠ (from equation (٣١-٤))

s١

Ta = )١٠٤٥.١١٠٤٨.٢(

١٤٣ −− − xx

٩٦٦١٢١ – ١٫٤٥ x ٤-١٠ (٠٫٤٨٨ x ٠٫٠٤)

٢٫٤٢ x ٤-١٠ Km/W (from equation (٣٢-٤)) =

٢٫٤٢ – ٠٫٠٤ + ٠٫٤٨٨ x ٠٫٥٢٨ = ٤-١٠ Km/W (from equation (٣٣-٤)) Tb =

The transient temperature rise for each time step is determined from equation (٣٤-٤); the example for ١ h (٦٠٠ ٣ s) is given below, the values for this time and for the other times being set out in Table F٣.

θc (١) = ٢٫٤٢ ٣٠٫٣ x ٠٫٥٢٨ + ٤-١٠ ( ) = ٦٫٥ K

)١( ٦٠٠٣١٠٤٨.٢ ٣ xxe−−− ٦٠٠٣١٠٤٥١ ٤

١ xx.e−−−

The conductor to cable surface attainment factor is determined for each time step from equation (٣٥-٤); the example for ١ h (٦٠٠ ٣ s) is given below, the value for this time and for the other times being set out in Table F٣.

a (١) = = ٠٫٤٠٧ )٣.٣٠)٠٤.٠٤٨٨.٠

٥.٦+

F٤٫٢ Calculation of the response of the cable environment

The response of the cable environment is given by equation (٣٦-٤). An example calculation is given for the longest time period (٢٤ h) and values of the – Ei(-x) terms for this time and for all the other times are given in Table F٢.

The response for the cable environment is summarized in column ٤ of Table F٣.

De = ٠٫١٢٢ m

L = ١٫٠ m

dpk = ٠٫٣ m

d′pk = ٢٢ )٣.٠()٠.٢( + = ٢٫٠٢٢ m (see Figure ٩)

ρT = ١٫٠ K.m/W

W٣٢٫٤ = ٢٫١ + ٣٠٫٣ = ١ W/m (dielectric loss is neglected)

٥٨

δ = ٠٫٥ x ٦-١٠ m٢/s

n a term-by-term basis for ٢٤ h (٤٠٠ ٨٦ s)

a) x =

N = ٣

O

δtDe

١٦

٢

= ٦

٢

١٠٥.٠٤٠٠٨٦١٦)١٢٢.٠(

−xxx = ٠٫٠٢١٥, - Ei (-x) = ٣٫٢٨

b) x =

δtL٢

= ٦

٢

١٠٥.٠٤٠٠٨٦)٠.١(

−xx = ٢٣٫٢, - Ei (-x) = ٠

c) x = δt

d pk

٤)( ٢

= ٦

٢

١٠٥.٠٤٠٠٨٦٤)٣.٠(

−xxx = ٠٫٥٢١, - Ei (-x) = ٠٫٥٣٥

d) x = δt

d pk

٤)'( ٢

= ٦

٢

١٠٥.٠٤٠٠٨٦٤)٠٢٢.٢(

−xxx = ٢٣٫٧, - Ei (-x) = ٠

θ (٢٤) = eπ٤ ⎥

⎦⎢⎣

٤.٣٢٠.١ x ( = ١١٫٢٢ K

(from equation (٣٦-٤))

re usually negligible at normal

depths of laying and duration of less than ٢٤ h.

on ts for c le env nment partia ansien٢٤ ١٢

⎤⎡−+− )٠٢٨.٣(٢)٠٥٣٥.٠

The second and fourth terms of equation (٣٦-٤) (items b) and d) above) represent the effect of the image sources and a

TABLE F٢

Co en ab iro al l tr t mp Time (h)

٦ ٥ ٤ ٣ ٢ ١

“x” Item

x ٠٫٢٥٨ ٥ ٠٫٥١٧ ٠٫١٧٢ ٠٫١٢٩ ٠٫١٠٣ ٠٫٠٨٦ ٠٫٠٤٣ ٠٫٠٢١a)

δtDe

١٦

٢

– E x) i (- ١٫٧٩ ١٫٥٩ ١٫٣٥ ١٫٠٢ ٠٫٥٤ ٣٫٢٨ ٢٫٦١ ١٫٩٦

x ٦٫٢٥ ١٢٫٥ ٠٫٥٢١ ١٫٠٤٢ ٢٫٠٨٣ ٢٫٥ ٣٫١٢٥ ٤٫١٦٧

δtd pk

٤)( ٢

c) – Ei (-x) ٠٫٥٣٥ ٠٫٢٠٥ ٠٫٠٤٤ ٠٫٠٢٥ ٠٫٠١١ ٠٫٠٠٣ ٠ ٠

٥٩

F٤٫٣ Calculation of the complete temperature transient

The complete transient is calculated by Sub-clause ٤٫٤. An example calculation for ١ h is shown below, the values for this time and for the remainder being summarized in column ٥ of Table F٣.

θ (١) = ٠٫٤٠٧ + ٦٫٥ x ٧٫١ = ١٫٤ K (from equation (٧١-٤))

Correcting for variation of conductor losses using equation (٧٣-٤):

θa (١) = = ٦ K (from equation (٣-٨)) )١.٧٣.٣١٨٥(

٣.٣١٥.٢٣٤١١

١.٧

−−+

+

Note. – Initial temperature = ٣١٫٣oC due to ٢١٫٣ K dielectric loss contribution. Actual conductor temperature = ٣١.٣oC + ٥٫٦ oC = ٣٦.٩oC.

The corrected values of conductor temperature rise are summarized in column ٦ of Table F٣ along with the actual conductor temperatures in column ٧.

TABLE F٣

Summary of temperature transient components

Time θc(t) a(t) θe(t) θ(t) θa(t) Actual conductor

temperature (h) (K) (K) (K) (K) (oC)

(١) (٢) (٣) (٤) (٥) (٦) (٧)

٣٧٫٠ ٦٫٠ ٧٫١ ١٫٤ ٠٫٤٠٧ ٦٫٥ ١

٤١٫٨ ١٠٫٤ ١٢٫١ ٢٫٦ ٠٫٦٤٨ ١٠٫٤ ٢

٤٤٫٨ ١٣٫٥ ١٥٫٥ ٣٫٥ ٠٫٧٩٢ ١٢٫٧ ٣

٤٦٫٩ ١٥٫٦ ١٧٫٧ ٤٫٢ ٠٫٨٧٧ ١٤٫٠ ٤

٤٨٫٣ ١٧٫٠ ١٩٫٣ ٤٫٧ ٠٫٩٢٧ ١٤٫٨ ٥

٤٩٫٤ ١٨٫١ ٢٠٫٤ ٥٫٣ ٠٫٩٥٧ ١٥٫٣ ٦

٥٢٫٧ ٢١٫٣ ٢٣٫٨ ٧٫٨ ٠٫٩٩٩ ١٦٫٠ ١٢

٥٦٫١ ٢٤٫٨ ٢٧٫٢ ١١٫٢ ١٫٠٠٠ ١٦٫٠ ٢٤

٦٠

F٥. Example of the calculation of a cyclic rating

F٥٫١ Cyclic load details

The daily load cycle is given as a fraction of the maximum current in Table F٤. Figure ٨ shows the relationship between the load cycle, the loss-load graph and the equivalent step function cyclic loss-load.

The loss-load factor is determined from equation (١-٦).

μ = ٢٤

٠٥٢.٠٠٦١.٠٠٩١.٠...٣٦٠.٠ ++++ = ٠٫٥٠٤

μ

From an examination of the load cycle, Figure ٨, it has been estimated that the highest conductor temperature will occur at ١٧ h.

F٥٫٢ Calculation of the conductor to cable surface attainment factor

The conductor to cable surface attainment factor is obtained from the transient calculations (see Sub-clause F٤٫١) if, in practice, this has been undertaken, or from equation (١-٧), using the thermal circuit parameters of Clause F٣, as set out below for the ٦ h period at hourly intervals.

a (١) = = ٠٫٤٠٧ ٠٤.٠٤٨٨.٠

)١٠٤٢.٢)١(٥٢٨.٠)١ ٦٠٠٣١٠١٤٥.١٦٠٠٣١٠٤٨.٢٤ ٤٣

+

−+−−− −−− xxxx eex

This value and the other five a(t) values are as given in column ٢ of Table F٥.

٦١

TABLE F٤

Daily cyclic load example Time (h) LoadHighest

Load Value of Yi Yi

Y١٧ ٠٫٠٩١ ٠٫٣٠٢ ٠ Y١٦ ٠٫٠٦١ ٠٫٢٤٧ ١ Y١٥ ٠٫٠٥٢ ٠٫٢٢٧ ٢ Y١٤ ٠٫٠٥٤ ٠٫٢٣٢ ٣ Y١٣ ٠٫٠٥٦ ٠٫٢٣٥ ٤ Y١٢ ٠٫٠٦١ ٠٫٢٤٦ ٥ Y١١ ٠٫٠٨٤ ٠٫٢٩٠ ٦ Y١٠ ٠٫٣٦٠ ٠٫٦٠٠ ٧ Y٩ ١٫٠٠٠ ١٫٠٠٠ ٨ Y٨ ٠٫٩٠٢ ٠٫٩٥٠ ٩ Y٧ ٠٫٨٨٤ ٠٫٩٤٠ ١٠ Y٦ ٠٫٨٢٨ ٠٫٩١٠ ١١ Y٥ ٠٫٧٩٦ ٠٫٨٩٢ ١٢ Y٤ ٠٫٥٩٣ ٠٫٧٧٠ ١٣ Y٣ ٠٫٥٩٦ ٠٫٧٧٢ ١٤ Y٢ ٠٫٦٤٠ ٠٫٨٠٠ ١٥ Y١ ٠٫٧٢٨ ٠٫٨٥٣ ١٦ Y٠ ٠٫٩٩٢ ٠٫٩٩٦ ١٧ Y٢٣ ٠٫٧٢٨ ٠٫٨٥٣ ١٨ Y٢٢ ٠٫٦٢٤ ٠٫٧٩٠ ١٩ Y٢١ ٠٫٥٤٨ ٠٫٧٤٠ ٢٠ Y٢٠ ٠٫٥٤٨ ٠٫٧٤٠ ٢١ Y١٩ ٠٫٥٢١ ٠٫٧٢٢ ٢٢ Y١٨ ٠٫٣٦٠ ٠٫٦٠٠ ٢٣

F٥٫٣ Calculations of the cable surface to ambient attainment factor

The cable surface attainment factor γ(t) is calculated using equation (٧-٧) where:

De = ٠٫١٢٢ m

L = ١٫٠ m

dpk = ٠٫٣ m

d′pk = ٢٢ )٣.٠()٠.٢( + = ٢٫٠٢٢ m (see Figure ٩)

٠٫٥ x ٦-١٠ m٢/s δ =

N = ٣

F = ٣.٠٣.٠٠٢٢.٢٠٢٢.٢

xx

= ٤٥٫٤ (from equation (٨-٧))

df = ٤.٤٥(٢/١(

٠.١٤ x = ٠٫٥٩٣ (from equation (٩-٧))

٦٢

An example is given for the value at ٦ h(٦٠٠ ٢١ s) and the values for this time and for the remainder are listed in Table F٥, column ٣. The second and fourth terms in the numerator of equation (٧-٧) represent the effects of the image sources (see Figure ٩) and at normal depths of laying are negligible when i is ٦ h or less.

x = ⎟⎟⎠

⎞⎜⎜⎝

⎛δt١٦

D ٢e

= ٦

٢

١٠٥.٠٦٠٠٢١١٦)١٢٢.٠(

−xxx = ٠٫٠٨٦, – Ei (-x) = ١٫٩٦

x = ⎟⎟⎠

⎞⎜⎜⎝

⎛δt١٦

D ٢f

= ٦

٢

١٠٥.٠٦٠٠٢١١٦)٥٩٣.٠(

−xxx = ٢٫٠٣, – Ei (-x) = ٠٫٠٤٧

γ (٦) = = ٠٫١٤ (from equation (٧-٧))

⎟⎟⎠

⎞⎜⎜⎝

⎛

+

١٢٢.٠٤.٤٥٠.١٤

١٢

)٩٦.١)٠٤٧.٠٢xx

n

x

F٥٫٤ Calculations of factor k١

k١ is calculated from equation (١٠-٧) with:

T١ = ٤n = ٠٫٥٥٥ K.m/W (from equation (٣-٧a)) π٢٠.١

⎟⎟⎠

⎞⎜⎜⎝

⎛١٢٢.٠

٠.١٤ x

ΔT١ = ٤n ٠٫٦٠٧ = ٤٥٫٤ K.m/W (from equation (١١-٧)) π٢٠.١

TA and TB are taken from Clause F٣.

k٠٫٧٠٢ = = ١ (from equation (١٠-٧)) )٣.٣٠)٠٤.٠٤٨٨.٠(٤.٣٢)٦٠٧.٠٥٥٥.٠

)٤.٣٢)٦٠٧.٠٥٥٥.٠+++

+

F٥٫٥ Calculation of )()١(∞R

R

θ

θ

This is calculated from equation (٦-٧). The input data and results are summarized in Table F٥, with an example for t = ١ h set out below.

٦٣

)()١(∞R

R

θ

θ = [٠٫٧٠٢ – ١ + (٠٫٧٠٢ x ٠٫٠٣٧)] ٠٫١٣٢ = ٠٫٤٠٧ (from equation (٦-٧))

TABLE F٥

Time (h) a(i) γ(i) θR(i)/θR(x)

(١) (٢) (٣) (٤)

٠٫١٣٢ ٠٫٠٣٧ ٠٫٤٠٧ ١ ٠٫٢٢٥ ٠٫٠٧٠ ٠٫٦٤٨ ٢ ٠٫٢٨٨ ٠٫٠٩٣ ٠٫٧٩٢ ٣ ٠٫٣٢٩ ٠٫١١٠ ٠٫٨٧٧ ٤ ٠٫٣٥٨ ٠٫١٢٦ ٠٫٩٢٧ ٥ ٠٫٣٧٩ ٠٫١٤٠ ٠٫٩٥٧ ٦

F٥٫٦ Cyclic rating factor M

٢ The cyclic rating factor is calculated from equation (٣-٥), using the values set out in the following table:

Time (h)

Yi )(

)(xi

R

R

θθ

٠ ٠٫٩٩٢ ٠ ٠٫١٣٢ ٠٫٧٢٨ ١ ٠٫٢٢٥ ٠٫٦٤٠ ٢ ٠٫٢٨٨ ٠٫٥٩٦ ٣ ٠٫٣٢٩ ٠٫٥٩٣ ٤ ٠٫٣٥٨ ٠٫٧٩٦ ٥ ٠٫٣٧٩ – ٦

μ = ٠٫٥٠٤ (see Sub-clause F٥٫١)

M = [٠٫٣٢٩)٠.٥٩٦ + (٠٫٢٢٥ – ٠٫٢٨٨)٠٫٦٤ + (٠٫١٣٢ – ٠٫٢٢٥)٠٫٧٢٨ + (٠٫١٣٢)٠٫٩٢٢ – ١/٢-[(٠٫٣٧٩ – ١)٠٫٥٠٤ + (٠٫٣٥٨ – ٠٫٣٧٩)٠٫٧٩٦ + (٠٫٣٢٩ – ٠٫٣٥٨)٠٫٥٩٣ + (٠٫٢٨٨

M = ١٫٢٨

Thus the permissible peak value of the cyclic load current is:

١٫٢٨ x ١٩٨٤ = ١٥٥٠ A

Repeat calculations for other assumed times of maximum temperature show that the actual time of maximum temperature is at ١٢ h. The cyclic rating factor based on this time is ١٫٢٧.

٦٤

٦٥

٦٦

F٦. Example of an emergency rating calculation

٣F٦٫١ Emergency load details

The cable system is assumed to be operating at ١٩٥ ١ A continuously which gives a steady-state conductor temperature of ٦٠oC. It is required to determine the emergency current level that can be carried for ٦ h without exceeding the maximum operating temperature (٨٥oC).

F٦٫٢ Emergency current rating

From Sub-clause ٨٫١ the following parameters are required (note that temperature parameters have been modified to allow for temperature rise due to dielectric loss):

r = ١

t = ٦ h (٦٠٠ ٢١ s)

IR = ١٥٥٠ A

I١٩٥ ١ = ١ A

h٠٫٧٧١ = = ١ ٥٥٠١١٩٥١

R١١.٦٢٥ = ١ μΩ/m (a.c. resistance at ٦٠oC)

Rmax = ١٢٫٦١٢ μΩ/m (a.c. resistance at ٨٥oC)

RR = ١٢٫٦١٢ μΩ/m (a.c. resistance at ٨٥oC)

Temperature rise due to dielectric loss = ٢١٫٣ K

Steady-state temperature rise due to joule losses = ٥٣.٧ = ٢١٫٣ – ١٠ – ٨٥ K;

θR(٦) = θa(٦) = ١٨.١ K (see Table F٣, column ٦) for application of a step function of rated current.

Thus the emergency current I٢ is:

I٥٥٠ ١ = ٢

٢/١٢

٢

٧.٥٣١.١٨

٦١٢.١٢

١)٧٧١.٠(٦٢٥.١١

٦١٢.١٢٧٧١.٠(٦٢٥.١١(

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −

+

١٣١ ٢ A (from equation (١-٨))

١

=

Table ١

Values of ١-β(٦) for soil thermal diffusivity of ٠٫٥ X ٦-١٠ m٢/s

Cable diameter or outer duct diameter (m)

٠٫١٠٠ ٠٫٠٩٥ ٠٫٠٩٠ ٠٫٠٨٥ ٠٫٠٨٠ ٠٫٠٧٥ ٠٫٠٧٠ ٠٫٠٦٥ ٠٫٠٦٠ ٠٫٠٥٥ ٠٫٠٥٠ ٠٫٠٤٥ ٠٫٠٤٠ ٠٫٠٣٥ ٠٫٠٣٠

١ ٠٫٦٧٦ ٠٫٦٦٧ ٠.٦٥٧ ٠٫٦٤٨ ٠٫٦٣٨ ٠٫٦٢٧ ٠٫٦١٧ ٠٫٦٠٥ ٠٫٥٩٤ ٠٫٥٨١ ٠٫٥٦٨ ٠٫٥٥٣ ٠٫٥٣٨ ٠٫٥٢١-β(٦) ٠٫٦٨٤

Cable diameter or outer duct diameter (m)

٠٫١٧٥ ٠٫١٧٠ ٠٫١٦٥ ٠٫١٦٠ ٠٫١٥٥ ٠٫١٥٠ ٠٫١٤٥ ٠٫١٤٠ ٠٫١٣٥ ٠٫١٣٠ ٠٫١٢٥ ٠٫١٢٠ ٠٫١١٥ ٠٫١١٠ ٠٫١٠٥

١ ٠٫٧٨٣ ٠٫٧٧٧ ٠٫٧٧١ ٠٫٧٦٥ ٠٫٧٥٨ ٠٫٧٥١ ٠٫٧٤٥ ٠٫٧٣٨ ٠٫٧٣١ ٠٫٧٢٤ ٠٫٧١٦ ٠٫٠٧٩ ٠٫٧٠١ ٠٫٦٩٣-β(٦) ٠٫٧٨٩

Cable diameter or outer duct diameter (m)

٠٫٢٥٠ ٠٫٢٤٥ ٠٫٢٤٠ ٠٫٢٣٥ ٠٫٢٣٠ ٠٫٢٢٥ ٠٫٢٢٠ ٠٫٢١٥ ٠٫٢١٠ ٠٫٢٠٥ ٠٫٢٠٠ ٠٫١٩٥ ٠٫١٩٠ ٠٫١٨٥ ٠٫١٨٠

١ ٠٫٨٥٧ ٠٫٨٥٣ ٠٫٨٤٨ ٠٫٨٤٤ ٠٫٨٤٠ ٠٫٨٣٥ ٠٫٨٣٠ ٠٫٨٢٦ ٠٫٨٢١ ٠٫٨١٦ ٠٫٨١١ ٠٫٨٠٥ ٠٫٨٠٠ ٠٫٧٩٤-β(٦) ٠٫٨٦١

٢

ConductorOutside diameter of insulation

١. – Representation of the dielectric for times greater than T.Q ١٣

FIG.

FIG. ٢. – Equivalent cable network for transient response calculations

٣. – Representation of a typical cable type (see Sub-clause ٤٫٢٫٢٫٢, Item a)) FIG.

for times greater than ١٣

T.Q

Outside diameter of insulation

٤. – Representation of the dielectric for times less than or equal to T.Q ١٣

FIG.

٥. – Representation of a typical cable type (see Sub-clause ٤٫٣٫٢٫٢, Item a)) FIG.

for times less than or equal to ١٣

T.Q, before reduction

٢

٦. – Scales for exponential integral ٠.١ ≤ x ≤ ٠.٣ FIG.

٣

٧. – Scales for exponential integral ٠٫٣٠ ≤ x ≤ ٣٫٠ FIG.

٤

NOTES TO FIGURES ٦ AND ٧ “SCALES FOR EXPONENTIAL INTEGRAL”

The exponential integral – Ei(–x) is defined in the standard reference books, e.g. reference [٤] pp. ٢٣١-٢٢٨, from which the following rational approximations, adequate to give a result correct to four places of decimals, have been derived:

For ٠ ≤ x ≤ ١

١n(x) + aixi ∑=

٥

٠i – Ei(–x) = –

where:

ao = -٠٫٥٧٧٢

a١٫٠٠٠٠ = ١

a٠٫٢٤٩٩– = ٢

a٠٫٠٥٥٢ = ٣

a٠٫٠٠٩٨- = ٤

a٠٫٠٠١١ = ٥

For ١ < x < ∞

– Ei(–x) = xxe١

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

++

++

٢١٢

٢١٢

bxbx

axax

where:

a٢٫٣٣٤٧ = ١ b٣٫٣٣٠٧ = ١

a٠٫٢٥٠٦ = ٢ b١٫٦٨١٥ = ٢

The expressions given above are convenient for manual or machine use. However, the following single expression, based on one given in [٥] may be convenient for computer use although its execution time can be longer.

For all values of x > ٠

– Ei(–x) = – ١ – ٠٫٥٧٧٢n(x) – ∑∞

= ١n )!()١(

nnxnn−

(It is sufficient for the purposes of this standard if – Ei(–x) is put equal to ٠ for values of x ≥ ٨, and equal to ١n(x) – ٠٫٥٧٧٢ + x for x < ٠٫٠١.)

٥

Note that ١ (B) is Highest value

Time of day (h)

FIG. ٨a. – Cyclic load divided by highest load

Time of day (h)

FIG. ٨b. – Loss-load graph

Time prior to highest temperature (hours)

FIG. ٨c. – Equivalent step functions cyclic loss-load

٤FIGURE ٨

٦

Ground surface

Cable k

Cable q

Cable p

Cable being considered

FIG. ٩. – Cables, or circuits of touching cables, and their images

٧

٨

FI|G. ١٠ - Time for ١ K reduction in temperature at axis of central cable of first circuit due to current reduce to zero in second circuit (see Sub-clause ٨٫٢).

٩

Draft No. ٣٣٧٢ “Calculation of the Cyclic and Emergency Current Rating of Cables – Part ٢: Cyclic Rating of Cables Greater than (٣٦) ١٨/٣٠ kV and Emergency Ratings for Cables of all Voltages” The preliminary draft of this standard has been prepared by the work team composed of the following: Name Agency ١. Dr. Abdul-Rahman A. Al-Oreiny King Saud University, Faculty of Engineering ٢ . Eng. Youssef A. Obaya Intertek International Ltd.

The draft standard was accepted for distribution to the concerned bodies in meeting No. (٨٧) of the Technical Committee No. (٢) “Cables and Their Accessories” composed of the following. Name Agency

١. Dr. Abdul-Rahman A. Al-Oreiny King Saud University, Faculty of Engineering ٢. Dr. Mohammed Nabih Al-Saati Riyadh Cable Co. ٣. Eng. Youssef A. Obaya Intertek International Ltd.

٤. Dr. Mohammed H. Al-Shewehdi King Fahad University for Petroleum & Mineral, Engineering Science Faculty

٥. Eng. Mohammed A. Itani Jeddah Cable Co. ٦. Eng. Abdullah A. Al-Kassar Saudi Electricity Co. - Eastern Branch ٧. Eng. Mohammed N. Nagi Saudi Cable Co. ٨. Eng. Nasser Ali Al-Sobaih Ministry of Municipal and Rural Affairs ٩. Eng. Aahed M. Al-Saqa Middle East Specialized Cable Co. ١٠. Eng. Mohammed A. Farahat SASO