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At the occurrence of phase-to-ground faults, abnormal levels of thermal energy $I^{2}t$, due to the Joule effect, will be developed during the clearing time that protective devices take to operate. The $I^{2}t$, also referred to as specific energy or Joule Integral, is accumulated within the elements forming the fault loop, such as the protective conductors (also referred to as equipment grounding conductors), responsible to return ground-fault currents to the source. As a consequence, the temperature of these conductors elevates and may exceed, in the case of an incorrect design, the maximum value that their insulation can withstand. This dangerous situation can cause the failure of the conductor insulation and/or trigger fires in neighboring materials. The maximum $I^{2}t$ that protective conductors can endure is, therefore, crucial in order to guarantee the electrical safety. The parameters on which the maximum $I^{2}t$ depends are described by the factor $k^{2}$, which will be herein discussed and analytically evaluated. The intention of the authors is to provide a theoretical support to the Power Systems Grounding Working Group of the Technical Books Coordinating Committee IEEE P3003.2 "Recommended Practice for Equipment Grounding and Bonding in Industrial and Commercial Power Systems"; the working group is currently elaborating a dot standard based on IEEE Standard 142-2007, also referred to as the Green Book. To this purpose, a comparison with existing formulas, currently present in codes, standards of the International Electrotechnical Commission and of the IEEE, as well in the literature, will be also presented.

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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 48, NO. 1, JANUARY/FEBRUARY2012 211

An Analytical Evaluation of the Factor k2

for Protective Conductors

Massimo Mitolo, Senior Member, IEEE, and Michele Tartaglia, Senior Member, IEEE

Abstract—At the occurrence of phase-to-ground faults, abnor-

mal levels of thermal energy I2 t, due to the Joule effect, will be

developed during the clearing time that protective devices take

to operate. The I2 t , also referred to as specific energy or Joule

Integral, is accumulated within the elements forming the fault

loop, such as the protective conductors (also referred to as equip-

ment grounding conductors), responsible to return ground-fault

currents to the source. As a consequence, the temperature of these

conductors elevates and may exceed, in the case of an incorrect

design, the maximum value that their insulation can withstand.

This dangerous situation can cause the failure of the conductor

insulation and/or trigger fires in neighboring materials. The max-

imum I2 tthat protective conductors can endure is, therefore,

crucial in order to guarantee the electrical safety. The parameters

on which the maximum I2 t depends are described by the factor

k2 , which will be herein discussed and analytically evaluated. The

intention of the authors is to provide a theoretical support to

the Power Systems Grounding Working Group of the Technical

Books Coordinating Committee IEEE P3003.2 "Recommended

Practice for Equipment Grounding and Bonding in Industrial

and Commercial Power Systems"; the working group is currently

elaborating a dot standard based on IEEE Standard 142-2007,

also referred to as the Green Book. To this purpose, a comparison

with existing formulas, currently present in codes, standards of

the International Electrotechnical Commission and of the IEEE,

as well in the literature, will be also presented.

Index Terms—Adiabatic, ampacity, cables, equipment ground-

ing conductor (EGC), fault duration, ground faults, I2 t,Joule

integral, protective conductor, protective device.

NOMENCLATURE

iG ( t)Instantaneous ground-fault current.

θ0 Initial temperature of the protective conductor.

θf Final temperature of the protective conductor.

θM Maximum temperature that protective conductor insu-

lation can withstand without damage.

RFault-loop resistance.

EGC Equipment grounding conductor.

Manuscript received June 24, 2011; accepted October 15, 2011. Date of

publication November 14, 2011; date of current version January 20, 2012. Paper

2011-PSEC-255, presented at the 2011 IEEE Industry Applications Society

Annual Meeting, Orlando, FL, October 9–13, and approved for publication

in the IE EE TRANSACTIONS ON I NDUSTRY A PPLICATIONS by the Power

System Engineering Committee of the IEEE Industry Applications Society.

M. Mitolo is with Chu & Gassman, Middlesex, NJ 08846 USA (e-mail:

mmitolo@chugassman.com).

M. Tartaglia is with the Dipartimento Ingegneria Elettrica, Politecnico di

Torino, 10129 Turin, Italy (e-mail: michele.tartaglia@polito.it).

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIA.2011.2175886

PE Protective conductor.

ρ0 Resistivity at 0 ◦ C.

ρ20 . Resistivity at 20 ◦ C.

I. INTRODUCTION

THIS PAPER seeks to provide a theoretical validation of

existing formulas for sizing protective conductors (PE)

(alsoreferredtoasequipment grounding conductors ,EGCs)

currently in use in codes, standards of the International Elec-

trotechnical Commission (IEC) and IEEE standards, such as,

for example, [1] and [2].

Properly sizing PEs is extremely important, as at the oc-

currence of phase-to-ground faults, abnormal levels of thermal

energy I2

Gt, also referred to as Joule Integral, occur. This energy

develops during the clearing time that protective devices take to

operate and disconnect the faulty equipment. Such let-through

energy needs to be compared with the maximum thermal energy

that a given protective conductor can endure without damaging.

The evaluation of protective conductors' maximum thermal

energy is, therefore, crucial in order to guarantee the electrical

safety of persons under ground-fault conditions.

This maximum admissible thermal energy of the PE not only

depends on its cross-sectional area, but also on its constituting

material (e.g., copper), its type of insulation (e.g., PVC), its

initial temperature θ0 at the inception of the fault, and the

maximum temperature θM that the conductor insulation can

withstand without damage.

The initial temperature θ0 may be taken as the conduc-

tor maximum operating temperature in correspondence with

its current-carrying capability. This conservative assumption,

which may result in protective conductors oversizing, is indeed

more appropriate for line conductors involved in short circuits.

In these cases, in fact, the initial temperature of the conductors

at the inception of the short is the actual temperature in cor-

respondence with the prefault load current; such temperature

is conservatively assumed as that in correspondence with the

ampacity1 of the cable.

On the contrary, the PE is generally "at rest," as no current

normally circulates through it. Thus, if the protective conductor

is not incorporated in cables, and not bunched with other

cables, its initial temperature may be the ambient temperature

(conventionally 30 ◦ C).

1Ampacity is defined as the maximum amount of electrical current a con-

ductor can carry while its insulation remains within its temperature rating.

Exceeding temperature ratings shorten the useful life of conductors.

0093-9994/$26.00 © 2011 IEEE

212 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 48, NO. 1, JANUARY/FEBRUARY2012

II. K2 FACTOR FOR PROTECTIVE C ONDUCTORS

To estimate the thermal stress to which protective conductors

are subject, we can initially assume that the let-through energy

is entirely accumulated within the PEs, and that there is no heat

dissipation by convection or radiation by the conductor (i.e.,

adiabatic conditions).

As anticipated, in order for the protective conductor not to

be damaged during the ground fault, the let-through energy

must not exceed the maximum thermal energy that the PE can

withstand. In formulas, and for the adiabatic case,

tf



0

i2

Gdt ≤k 2 S 2 .(1)

The left-hand side of (1) is the let-through energy devel-

oped during the fault [3]; iG is the instantaneous ground-fault

current, S the cross-sectional area of the protective conductor

(mm2 ),tf is the clearing time of the protective device and k 2

is a factor that takes into account the resistivity, temperature

coefficient and heat capacity of the conductor material, the

initial temperature of the protective conductor at the inception

of the fault, and the maximum admissible temperature the

insulation of the PE can withstand without damage.

The factor k2 is given by

k2 = c

α0 ·ρ0

ln 1+α0 θ M

1+θ0 α0

.(2)

The Appendix provides an analytical calculation for the

above factor k2 .

If in (2), we pose β=1 /α0 and ρ0 =ρ20 ◦/ (1 + 20◦ α0 ) ,as

per (A3), we obtain

k2 = c(β +20

◦)

ρ2 0

ln  1+ θ M −θ0

β+ θ0  (3)

which is an equivalent formulation of the k2 factor that can be

found in [4].

With the same positions, we can obtain the following for-

mula, presented in [5]:

k2 = c

ρ20

(20 + 1/α0 )ln (θM +1 /α0 )

(θ0 +1 /α0 ) .(4)

Equations (3) and (4), included in different standards, do

confirm the analytical calculation of the k2 factor obtained

in (2).

Equations (2)–(4) can be applied to conductor at different

rated voltages, whose temperature limits, for various types of

insulation, can be found in [5]–[8].

Equations (2)–(4) can also be applied under nonadiabatic

conditions: differences in the calculated values of k2 are only

significant for smaller cross-sectional areas of cables (less than

10 mm2 ).

TAB L E I

VALUES OF P ARAMETERS FOR D IFFERENT

CONDUCTIVE MATERIALS OF PES

TAB L E I I

TEMPERATURE L IMITS FOR I NSULATION M ATERIALS OF P ROTECTIVE

CONDUCTORS NOT INCORPORATED IN C ABLES, AND

NOT BUNCHED WITH OTHER CABLES

III. VALUES OF P ARAMETERS FOR D IFFERENT M ATERIALS

AND INSULATIONS OF P ROTECTIVE C ONDUCTORS

Table I lists values of parameters for different conductive

materials of PEs to be used in the calculation of k2 :

The nature of adjacent insulating materials limits the maxi-

mum admissible temperatures of protective conductors.

In the following tables, temperature limits for protective

conductors for ground-fault durations not exceeding 5 s are

listed. If ground-fault clearing times exceed 5 s, maximum

temperatures must be reduced according to the manufacturer's

indications.

Temperature limits for insulation materials of protective con-

ductors not incorporated in cables, and not bunched with other

cables, are listed in Table II [4], [6].

Temperature limits for insulation materials of protective

conductors as a core incorporated in a cable or not bunched

with other cables or insulated conductors, are listed in Table III

[4], [6].

Temperature limits for bare protective conductors in contact

with cable covering, but not bunched with other cables, are

listed in Table IV [4], [6].

Temperature limits for insulation materials of protective

conductors as a metallic layer of a cable (e.g., armor, metallic

sheath, concentric conductor, etc.) are listed in Table V [4], [6].

If protective conductors are bare and exposed to touch, or

in contact with combustible materials, their superficial temper-

ature may be a reason of concern. In normal condition areas,

when there is no risk for the bare PE to cause damage to any

neighboring material, the maximum temperature to consider for

calculations should be 200 ◦ C. However, different temperature

limits can be adopted in different areas; if the bare protective

conductor is well visible and confined in restricted zones, the

maximum allowable temperature can be increased; on the other

hand, if the bare PE is in fire risk locations, its maximum

MITOLO AND TARTAGLIA: ANALYTICAL EVALUATION OF THE FACTOR k2 FOR PROTECTIVE CONDUCTORS 213

TABLE III

TEMPERATURE L IMITS FOR I NSULATION M ATERIALS OF P ROTECTIVE

CONDUCTORS AS A CORE INCORPORATED IN A CABLE OR BUNCHED

WITH OTHER CABLES OR INSULATED CONDUCTORS

TAB L E I V

TEMPERATURE L IMITS FOR B ARE P ROTECTIVE C ONDUCTORS IN

CONTACT W ITH C ABLE C OVE RIN G,B UT N OT B UNCHED

WITH OTHER CABLES

TAB L E V

TEMPERATURE L IMITS FOR I NSULATION M ATERIALS OF P ROTECTIVE

CONDUCTORS AS A METALLIC LAYER O F A CABLE

TAB L E V I

TEMPERATURE L IMITS FOR B ARE PE WHERE THERE IS NO

RISK OF DAMAGE TO ANY NEIGHBORING MATERIAL

temperature should be lowered. Table VI lists such temperature

limits as per [4] and [6].

It should be noted that due to safety considerations, such

as the risk of burns or of triggering fires or explosive at-

mospheres, the fusion temperatures of bare PEs, as maximum

allowed temperatures, are not considered in the IEC world.

Such temperatures would largely exceed the temperature limits

of Table VI; in fact [1] indicates that if fusing is a criterion,

then a final temperature of 1000 ◦ C for copper and 630 ◦ Cfor

aluminum may be used.

TAB L E V II

VALUES OF K2 FOR I NSULATED P ROTECTIVE C ONDUCTORS N OT

INCORPORATED IN CABLES ,AND NOT BUNCHED WITH OTHER CABLES

TABLE VIII

VALUES OF K2 FOR P ROTECTIVE C ONDUCTORS AS A CORE

INCORPORATED IN A CABLE OR BUNCHED WITH OTHER

CABLES OR INSULATED CONDUCTORS

TAB L E I X

VALUES OF K2 FOR B ARE P ROTECTIVE C ONDUCTORS IN C ONTACT W ITH

CABLE COVER ING ,B UT N OT B UNCHED W ITH O THER C ABLES

TAB L E X

VALUES OF K2 FOR P ROTECTIVE C ONDUCTORS

AS A METALLIC LAYER O F A CABLE

TAB L E X I

VALUES OF K2 FOR B ARE PE W HERE THERE I SN OR ISK OF

DAMAGE TO ANY NEIGHBORING MATERIAL

214 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 48, NO. 1, JANUARY/FEBRUARY2012

TAB L E X II

AC

OMPARISON OF V ALUES OF K2 O BTAINED W ITH D IFFERENT

FORMULAE (P ROTECTIVE CONDUCTORS NOT INCORPORATED IN

CABLES ,AND NOT BUNCHED WITH OTHER CABLES —XLPE)

IV. V ALUES OF K2 FOR P ROTECTIVE C ONDUCTORS

Based on initial and limit temperatures listed in Tables I–VI,

the k2 factor can be calculated according to (2) for any given

insulation and conductive material of wires. Tables VII–XI

report the results of the calculation:

The value of the k2 factor can also be determined through

formulas currently present in literature; however, attention must

be paid to such formulas and on the assumptions on which they

are based.

In fact, the expression of the k2 factor in [1] for copper and

aluminum are both incorrect. The Power Systems Grounding

Working Group of the Technical Books Coordinating Com-

mittee IEEE P3003.2 "Recommended Practice for Equipment

Grounding and Bonding in Industrial and Commercial Power

Systems" is aware of this issue. Thus, in the new dot standard

P3003.2, based on [1, Ch. 2], the following formulas, found

in [9] and [10] are being proposed for equipment grounding

conductors, respectively, in copper and aluminum:

I 2

A2  t=0 .0297 log10  T M + 234

Ti + 234  (5)

I 2

A2  t=0 .0125 log10  T M + 228

Ti + 228  (6)

where I is the fault current through the conductor in amperes,

Ais the protective conductor cross-sectional area in circular

mils, t is the fault clearing time in seconds, Ti is the initial

operating temperature in degrees Celsius and TM is the maxi-

mum temperature for no damage in degrees Celsius. Reference

[1] also indicates that Ti is often taken as the conductor maxi-

mum operating temperature in correspondence with its current-

carrying capability rather than its initial temperature. This is a

conservative approach of which the designer should be aware,

as may result in protective conductor oversizing.

A comparison between the values of k2 obtained with (2) and

k2

30 (T i =30◦ C and k2

90 (T i =90◦ C) calculated with (5) and

(6) is shown in Table XII for the case of protective conductors

not incorporated in cables, and not bunched with other cables

insulated in XLPE (θM = 250 ◦ C ).

It can be seen that (2), (5), and (6) substantially provide the

same results for θ0 =30 ◦ C (columns 2 and 3 of Table XII).

However, if θ0 =90◦ C is employed in (5) and (6), as implicitly

allowed in [1], the values of k2

90 (column 4 of Table XII) are

more than 30% lesser than the values k2 calculated with (2)

(column 2 of Table XII). It is important to note that reduced

values of k2 determine larger cross-sectional areas for the PE in

correspondence with the same ground-fault current and clearing

time of protective devices. This conservative approach does

compound with the assumption of adiabatic conditions during

ground faults in determining larger PE.

V. M INIMUM C ROSS -S ECTIONAL A REAS OF P ROTECTIVE

CONDUCTORS IN ADIABATIC C ONDITIONS

The analytical evaluation of the integral of the left-hand side

of (1) is rather complex, as the ground-fault current is asym-

metrical due to the development of a transient dc component

[11], [12]. The method proposed in [11] allows the evaluation

of the maximum possible thermal stress to conductors involved

in faults by taking into account the worst possible asymmetries

of fault currents due to both the making angle and the short-

circuit phase angle.

However, [4] and [13] indicate that if protective devices

can clear the ground fault within 5 s from its inception, the

following simplified formula may be used to determine the

minimum and safe cross-sectional area S( mm2 )of PEs in

adiabatic conditions:

S≥ IG

k t f .(7)

IG is the r.m.s. value of the prospective ground-fault current

circulating through the PE for a fault of negligible impedance,

and tf is the operating time of the protective device in corre-

spondence with the ground-fault current. In reality, when the

ground-fault current is not constant, the error caused by the

simplification shown in (7) is acceptable provided that either

the dc transient component of the ground-fault current quickly

expires or protective devices do not clear the ground fault within

the first cycle. As per the above simplification, the method used

in [11] is not herein used.

It is important to note, though, that the optimum wire size of

the PE is not per se a guarantee of electrical safety for persons.

In fact, also, terminations, joints, bonding jumpers, etc., in-

cluded within the ground-fault path must have equal, or greater,

thermal capabilities than that of the protective conductor.

If (7) produces nonstandard sizes, protective conductors of a

higher standard cross-sectional area must be used. In addition,

as anticipated in the previous section, the choice of using the

maximum operating temperature as θ0 may lead to further

oversizing.

To better understand this issue, let us consider the Time-

Inverse trip curve as a function of the prospective ground-fault

current for a 20-A molded case circuit breaker (Fig. 1).

The trip curve of Fig. 1 provides the clearing times tf of the

circuit breaker in correspondence with any given value of the

ground-fault current. Such values are listed in Table XIII.

Table XIII also shows the values of calculated and trade

sizes of cross-sectional areas (columns 4 and 5) as per (7).

The two initial temperatures used in the calculation of k2 are

θ0 =30◦ C and θ0 =90◦ C, in the case of a copper protective

conductor not incorporated in cables, and not bunched with

other cables, insulated in XLPE (k= 175 .55 and k90 = 141 .99 ,

as per Table XII).

It can be clearly seen that the adoption of the operating

temperature of 90 ◦ C results in some cases in protective

MITOLO AND TARTAGLIA: ANALYTICAL EVALUATION OF THE FACTOR k2 FOR PROTECTIVE CONDUCTORS 215

Fig. 1. Time-Inverse trip curve for a 20-A molded case circuit breaker.

TABLE XIII

CALCULATED AND T RADE S IZES OF C ROSS -SECTIONAL AREAS OF

PROTECTIVE CONDUCTORS NOT INCORPORATED IN C ABLES,

AND N OT B UNCHED W ITH O THER C ABLES—XLPE

conductor oversizing by one trade size, in the presence of the

same ground-fault current and clearing time.

Equation (7) can of course be used for protective conductors

such as armors, metallic sheaths, tapes, etc. In these cases,

we consider an equivalent cross-sectional area SE ( mm2 ) given

by [5]:

SE = ρ 20 · P

R20 · γ(8)

where ρ20 is the resistivity at 20 ◦ C(Ω mm ) ,R20 is the

resistance per kilometer at 20 ◦ C(Ωkm−1 ) ,P is the mass per

kilometer ( kg km−1 ),γ is the specific mass (kg mm−3 ).

TAB L E X IV

CONSTANTS X AND Y

VI. NONADIABATIC METHOD

The assumption that under ground-fault conditions all the

thermal energy is accumulated within the protective conductors

may be in some case pessimistic, as heat transfer into the

neighboring environment does occur.

References [14] and [15] provide details for the nonadiabatic

method, which is valid for all ground-fault durations, and is

based on an empirical approach.

If part of the heat is dispersed toward adjacent bodies, the

permissible ground-fault current can increase, without risk of

damaging the protective conductor of a given cross-sectional

area. Alternatively, the wire size of the PE can be safely de-

creased with respect to the value determined with the adiabatic

method.

The permissible nonadiabatic ground-fault current INAD is

given by

INAD =εIG (9)

where ε is the nonadiabatic factor, which takes into account

heat loss into the adjacent components (ε=1 in adiabatic

conditions, whereas ε is > 1 in nonadiabatic conditions); IG

is the ground-fault current calculated with (7) in adiabatic

conditions.

In the case of insulated conductors as PEs, the nonadiabatic

factor is given by the following simplified empirical formula:

ε= 1+ XZ + YZ

2.(10)

The constants X and Y for copper protective conductors are

presented in the following Table XIV, as a function of the PE

insulation, and for voltages ≤ 3kV.

Zis defined as

Z= tf /S (11)

where tf is the ground-fault duration (in s) and S is the PE

geometrical cross-sectional area (in mm2 ).

For the usual range of wire sizes encountered in the practice,

[14] indicates, as a decision-making criterion, to neglect the

improvement in the permissible ground-fault current when its

increase is less than 5%, that is, when INAD <1. 05 IG .In

this case, the nonadiabatic method is not recommended to

determine the minimum cross section of PEs.

Based on this criterion, the authors have performed compu-

tations based on (10) to determine the values of trade wire sizes

for copper conductors, for which ε≤1. 05 . These calculations

have taken into account different values of tf ,aswellas,

trade wire sizes from 1.5 to 300 mm2 . Threshold values for

S have been identified, below which the adiabatic hypothesis

is too pessimistic and protective conductors result oversized.

216 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 48, NO. 1, JANUARY/FEBRUARY2012

TAB L E X V

MAXIMUM VALUES FOR SFOR COPPER PES ,FOR WHICH ε≥ 1. 05

Table XV lists the smallest values for S for copper wires, for

which ε≤1. 05 ,fortf equal to 0.1 s, 1 s, and 10 s.

According to (10), the maximum size for protective conduc-

tors for the nonadiabatic condition to be useful in practice is

S=10 mm2 , as long as tf equals at least 0.1 s. Calculations

show, in fact, that for fault durations tf ≤0. 1s, the heat

exchange with the surrounding air or materials is negligible

(i.e., ε∼

=1) even for the smallest wire trade size of 1.5 mm2 .

VII. CONCLUSION

The authors have proposed an analytical method for the

calculation of k2 for protective conductors, which takes into

account the thermal characteristics of insulations, as well as of

neighboring materials. This has allowed the determination of

the optimum value of S.

The analytical results confirm the formulas currently present

in literature and to be adopted in P3003.2 for the adiabatic

case. However, such formulas only consider PEs as wires and

may lead to oversizing the protective conductors, of which the

engineer should be aware.

It has been substantiated, in fact, that two pessimistic choices

may be made: using the maximum operating current of protec-

tive conductors rather than the ambient temperature; consider-

ing the thermal phenomenon developing during ground faults

always adiabatic.

Compiling these two assumptions may lead to oversized

protective conductors by one or two trade sizes.

APPENDIX

In adiabatic conditions, during the ground fault, the follow-

ing thermal balance occurs:

ρ· l

S· i2

Gdt =S· l· c· dθ (A1)

where l is the length of the ECG, S its cross-sectional area

(mm2 ),ρ its resistivity (Ωmm ), and c its volumetric heat

capacity ( J/(◦ Cmm

3));i G is the instantaneous ground-fault

current.

The left-hand side of (A1) quantifies the heat developed

by the fault current during the infinitesimal time dt , while

the right-hand side is the heat accumulated in the conductor

during the same time. dθ is the difference between the initial

temperature θ0 of the conductor, at the inception of the fault,

and its temperature θf , after the fault is cleared.

The resistivity ρ of the PE is a function of the temperature θ

imposed by the ground fault and therefore does not remain con-

stant. In general, ρ( θ) is not a linear function of the temperature;

however, if we assume that the temperature varies in a small

range, we can approximate ρ( θ) with a Taylor series. The Taylor

series is a linear representation of a function as an infinite sum

of terms based on the values of its derivatives evaluated at an

initial point. It is normally acceptable for accuracy to use a finite

number of terms of the series to approximate the function.

As a consequence, the resistivity can be written as a Taylor

polynomial as a function of θ :

ρ( θ)=

∞



n=0

ρ(n) ( θ0 )

n!( θ−θ0 ) n ∼

=ρ( θ0 )+ρ (1) (θ0 )(θ− θ0 )

=ρ(θ0 )+ ρ( θ0 )αθ 0 ( θ− θ0 )= ρ( θ0 )[1+ αθ 0 ( θ− θ0 )] (A2)

where the subscript (n ) indicates the n th derivative of the

resistivity ρ with respect to the temperature θ , evaluated at the

initial temperature θ0 ;ρ( θ0 ) represents the resistivity of the PE

at the initial temperature θ0 ;αθ0 =ρ(1) ( θ0 ) /ρ ( θ0 ) is the tem-

perature coefficient of resistivity of the material of the PE at the

initial temperature θ0.

We can write

ρ( θ)= ρ(0◦ )(1 + α0 θ ) .(A3)

To calculate the heat accumulated in the protective conductor

during the ground fault, we need to integrate left- and right-

hand sides of (A1) between: the instant t=0 of the inception

of the fault, and the instant tf of disconnection of the supply;

the temperature θ0 of the PE at the inception of the fault and its

final temperature θf when the fault is cleared.

We obtain

tf



0

i2 dt = c· S 2

θf



θ0

dθ

ρ= c·S2

ρ(0◦ )

ϑf



ϑ0

dθ

(1 + α0 θ ).(A4)

Solving the above integral by substitution of the variable θ

[3], we obtain

tf



0

i2 dt = cS 2

α0ρ0

1+α0 ·θ f



1+α0 · θ 0

dx

x= cS2

α0ρ0

ln 1+α0 θ f

1+α0 θ 0

.(A5)

In order to prevent damages to the insulation of the PE, the

final temperature θf in (A5) must not exceed the maximum

temperature θM that its insulation can withstand. Hence, if we

replace θf with θM , we can define the parameter k2 as

k2 = c

α0 ·ρ0

ln 1+α0 θ M

1+θ0 α0

.(A6)

REFERENCES

[1] IEEE Recommended Practice for Grounding of Industrial and Commer-

cial Power Systems, IEEE Std. 142-2007, 2007.

[2] IEEE Recommended Practice for Protection and Coordination of Indus-

trial and Commercial Power Systems, IEEE Std. 242-2001, 2001.

[3] M. Mitolo, Electrical Safety of Low-Voltage Systems.NewYork:

McGraw-Hill, 2009.

[4] Electrical Installations of Buildings—Part 5-54: Selection and Erection of

Electrical Equipment-Earthing Arrangements, Protective Conductors and

Protective Bonding Conductors, IEC 60364-5-54, Jun. 2002.

[5] Installations of Generation, Transmission and Distribution of Electric

Energy: Electric Cables, Italian Standard CEI 11-17, Jul. 2006.

MITOLO AND TARTAGLIA: ANALYTICAL EVALUATION OF THE FACTOR k2 FOR PROTECTIVE CONDUCTORS 217

[6] Short-Circuit Temperature Limits of Electric Cables With Rated Volt-

ages of 1 kV (Um =1 .2kV) and 3 kV ( Um =3 .6kV), IEC 60724,

Oct. 2000.

[7] Short-Circuit Temperature Limits of Electric Cables With Rated Voltages

From 6 k V (Um =7 .2kV)upto30kV( Um =36 kV), IEC 60986,

Oct. 2000.

[8] Short-Circuit Temperature Limits of Electric Cables With Rated Voltages

Above 30 kV (Um =36kV), IEC 61443, Jul. 1999.

[9] D. L. Beeman, Industrial Power Systems Handbook.NewYork:

McGraw-Hill, 1955.

[10] National Electrical Code , NFPA 70, 2011.

[11] M. Tartaglia and M. Mitolo, "An analytical evaluation of the prospective

I2 tto assess short circuit capabilities of cables and busways," IEEE

Trans. Power Del., vol. 25, no. 3, pp. 1334–1339, Jul. 2010.

[12] G. Parise and M. Adduce, "Conductor protection against short circuit

current: Available I2 t evaluation," in Conf. Rec. IEEE IAS Annu. Meeting ,

1998, pp. 2336–2341.

[13] Electrical Installations of Buildings—Protection for Safety—Protection

Against Overcurrent, IEC 60364-4-43, Aug. 2001.

[14] Calculation of Thermally Permissible Short-Circuit Currents, Taking Into

Account Non-Adiabatic Heating Effects, IEC 60949, 1988.

[15] Calculation of Thermally Permissible Short-Circuit Currents, Taking Into

Account Non-Adiabatic Heating Effects, IEC 60949, Sep. 2008.

Massimo Mitolo (SM'03) was educated in Italy. He

received the Ph.D. degree in electrical engineering

from the University of Naples "Federico II," Naples,

Italy, in 1990, where his field of research was the

analysis and grounding of power systems.

He is currently the Assistant Electrical Depart-

ment Head of Chu & Gassman Inc., Middlesex,

NJ. He has authored numerous journal papers, as

well as the textbook Electrical Safety of Low-Voltage

Systems.

Dr. Mitolo is a Registered Professional Engineer

in Italy. He is very active in the IEEE Industry Applications Society Industrial

and Commercial Power Systems Department, where he currently fills the posi-

tions of Vice Chair of the Power Systems Engineering (PSE) Main Technical

Committee, Chair of the Papers Review Subcommittee, Chair of the Power

Systems Analysis Subcommittee, and Chair of the Power Systems Grounding

Subcommittee. He is also an Associate Editor of the PSE IEEE ScholarOne

Manuscripts. He was also the recipient of the Lucani Insigni Award in 2009 for

merits achieved in the scientific field.

Michele Tartaglia (SM'08) received the "Laurea"

degree in electrical engineering from the Politecnico

di Torino, Turin, Italy, in 1971.

In 1973, he joined the Istituto Elettrotecnico

Nazionale "Galileo Ferraris," Turin, Italy, where he

carried out theoretical and experimental studies on

breaking apparatus and computation of electromag-

netic fields in nonlinear devices. From 1982 to 2010,

he was an Associate Professor and he is currently a

Full Professor at the Politecnico di Torino. He has

authored more than 100 scientific papers. His main

research interests include the study of breaking apparatus, electromagnetic

fields by means of analytical and numerical methods, mitigation of low-

frequency magnetic fields, security in electrical power systems, and rational

use of energy.

Dr. Tartaglia is a member of the IEEE Industry Applications Society and

of AEIT (Federazione Italiana di Elettrotecnica, Elettronica, Automazione,

Informatica e Telecomunicazioni). He is also a member of CIGRE Task Force

Working Group 36.04.01 on "Magnetic Field Mitigation Methods" and of the

Technical Committee CT 106 on "Human Exposure to Electromagnetic Fields"

of the Comitato Elettrotecnico Italiano. He is also the scientific responsible

of research contracts with public institutions and private companies, and is

involved in European Community projects. In 1977, he was the recipient of the

Bonavera Award for Electrical Engineering from the Accademia delle Scienze

di Torino.

... For feeders, it is recommended to consider the thermal effects of the fault current they must carry [2,[4][5][6]. This thermal effect refers to the I²t heating effect and is also called the let-through energy (LTE) or Joule integral [2,[7][8][9]. and is used to evaluate the zone 2 time delay of 400 ms, the zone 3 time delay of 2.5 to 3 s and the back-up protection time delay design value of 800 ms. In general, LTE is used to determine maximum allowable fault clearing times. ...

... The assumption that is made is that during the fault the energy exchange is adiabatic. This means that there is no heat loss to the environment in the form of convection and radiation [4,6,9]. An equation (equation 1) can be created by setting the heat generated in the conductor due the flow of current equal to the heat absorbed in the conductor [5,6,[8][9][10]. ...

... This means that there is no heat loss to the environment in the form of convection and radiation [4,6,9]. An equation (equation 1) can be created by setting the heat generated in the conductor due the flow of current equal to the heat absorbed in the conductor [5,6,[8][9][10]. This is the conductor short time rating or then its LTE limit. ...

High voltage feeders are generally protected using main (e.g. impedance) and back-up protection. The main protection is generally set to operate as fast as possible for faults on the feeder. Operating time delays are introduced to ensure that selectivity is maintained between protection devices and this improves system security. The back-up protection elements are normally current based inverse definite minimum time elements (IDMT). These IDMT elements are not bound to certain reach limits like the main protection elements. Hence they are set much slower than the main protection elements. During network faults, the fault current will heat up the conductor based on the magnitude of fault current, the resistance of the conductor and the time that the conductor is exposed to this fault current. The conductor exposure energy limit can be calculated by setting the I²Rt heating effect equal to the heat gained in the conducting material and assuming it is an adiabatic process. The i²t rating of the conductor is also referred to as the let-through energy (LTE) limit or joule integral of the conductor. If the conductor is exposed to this for an extended period of time the conductor can get damaged (plastic deformation). The traditional simplified IDMT equations cannot be used in an interconnected network because the measured current can change before a trip is issued. To evaluate the LTE application on high voltage feeders, three case studies were used. The first a general evaluation of 48 actual high voltage feeders, secondly a hypothetical feeder was used and lastly a detailed evaluation on a problematic feeder. The results indicated that the probability of conductor thermal damage is unlikely when the main impedance protection is considered. However, the probability of conductor damage due to back-up protection is high. Especially for faults close to the busbar and when the back-up protection is set to initiate an auto-reclose cycle. The recommendation is to speed up the IDMT elements and apply instantaneous curves where possible. This evaluation allows the protection engineer to not only provide good selectivity, sensitivity and speed to the network protection, but that he is actually protecting the feeder and improving network availability in the long run.

... Permanent damage occurs wherever the energy exposure level or the let-through energy exceeds the short time withstand rating of the conductor [33]. The energy exposure of an overhead line is defined by Equation 1 [34,35]. ...

• R. Thomas
• Stuart van Zyl
• R. M. Naidoo
• Minnesh Bipath

The optimised placement of reclosers on a distribution network is known to improve the reliability of a power system. Furthermore, the protection settings on distribution systems rely heavily on the number and placement of such reclosers. This study examined the effect of using protection settings methodology with the placement of reclosers to ameliorate the damage sustained during faults on a distribution network. The aim of the study was to determine whether this 'damage control factor' should be a future consideration for recloser placement. It has been found that the determination of the number and placement of reclosers, which are the function of the energy exposure of feeder, helped to optimise the operation and reliability of a distribution network. This could benefit both energy users and energy suppliers, who often face different challenges during the fault levels on the network.

... As to the protection of cables against short circuits, fuses must interrupt the circuit before the let-trough energy (i.e. I 2 t) exceed the thermal withstand capacity of cables [32] [33]. When circuits are made of conductors with decreasing sections (Fig. 1), the protection performed by fuses against short circuits may be challenged. ...

... If the conductor is deformed beyond its modulus of elasticity [8], it will not return to its previous length once the conductor cooled down (can result in extra faults and contact incidents). By assuming that there is no energy transfer to the environment (adiabatic process) during a fault, (1) can be applied to calculate the letthrough energy (I²t energy) capability of the conductor [9] [10]. ...

The high current lock-out function is not a function that is commonly used when protecting medium voltage feeders for phase over current conditions. This function is beneficial in reducing the let-through energy that the power system is exposed too, to protect the feeder from unnecessary exposure to excessive high fault currents during certain network contingencies and to prolong the life expectancy of the network equipment. This function makes use of a definite time curve and a zero second time delay. It differs from the high-set function for medium voltage feeders in that it does not initiate auto reclose where the high-set function can initiate auto-reclose. Currently there is much written on the well-known high-set function and its application, but little is written on the high current lock-out function. The current practice is to apply the function during contingencies where the fault level is high, but there is no guideline on how to determine what is high. The principle setting for the function is a current pick-up. In this paper we recommend that the let-through energy capability of the protected equipment and the system fault level under maximum network conditions be used to determine the function pick-up. This function can then be used in conjunction with existing protection philosophies.

... the resistance, , R in dc of copper wires varies with its temperature, , i according to the following formula [7]: ...

In low-voltage systems, ground-faults do not necessarily involve the actual earth, but fault currents may return to the source via conductors; such conductors, which provide a clear path toward the source, are defined in applicable codes and technical standards as equipment grounding conductors or protective conductors. The above situation is for example typical of the North American low-voltage multiple grounded distribution systems, also referred to as TN-C-S, in international standards (IEC). In this paper it is substantiated that under ground-fault conditions even healthy equipment, sharing the same protective conductor with faulty equipment, as well as any metalwork in the building, acquire electrical potentials. Such potentials have a decreasing magnitude toward the electrical source and keep the same constant value on equipment located downstream the ground-fault location. The presence of non-zero potential differences between exposed-conductive-parts is a salient trait of TN systems and may constitute a hazard particularly in special locations, such as bathrooms with showers or bathtubs. In such areas, in fact, the resistance-to-ground of persons may be greatly reduced by moisture, water or the absence of clothing. This paper illustrates the advantages of supplementary equipotential bonding connections in such locations as a possible solution in the reduction of latent potential differences.

... The PE initial temperature varies according to its arrangement in raceways (e.g., bunched with other cables, incorporated [6]. The PE's final temperatures, which are the maximum temperatures that its insulation can withstand, slightly differ among standards (e.g., in [7]- [11]). ...

This paper presents the thermal sizing procedure of equipment grounding (protective) conductors and the electric shock calculations as per the standard 60364 of the International Electrotechnical Committee (IEC) that is used worldwide. The application of IEC standards in the design of low-voltage systems (i.e., not exceeding 1 kV) governed by the National Electrical Code (NEC) and the American National Standards Institute (ANSI), which are adopted in the U.S., is discussed and proposed in this paper. The authors propose an interpretation of the NEC installations in light of the earthing definitions included in the IEC realm. On this premises, IEC sizing criteria are applied to typical North American installations to prove their general applicability and their legitimacy to NEC/ANSI-based electrical systems. Examples are also presented for both thermal sizing and electric shock calculation of equipment grounding conductors.

Let-through energy (LTE) refers to the I²t or Joule energy that a conductor is exposed to during a fault on the feeder. This energy is influenced by the magnitude of the fault current and time it takes for the protection system to clear the fault. If the LTE exceeds the conductor thermal energy limit, the conductor will get damaged. This concept of LTE evaluation is applied to the inverse definite minimum time (IDMT) current based back-up protection elements on a multisource high-voltage feeder in a hypothetical and actual network. Another method to calculate the relay operating time for IDMT relays was developed based on an average disk speed of electromechanical over-current relay and the proportionality of its speed to the magnitude of the fault current. This method was incorporated into a software application to generate results. These results allow the user to evaluate the conductor LTE exposure, total fault time exposure, the effect of instantaneous fault clearing and the application of autoreclose cycles. An energy-area evaluation was applied to quantify and evaluate small protection settings changes. The conclusion is that LTE analysis on back-up protection should be considered for high-voltage feeders to ensure that the conductors are protected.

The effectiveness of phase over-current protection cannot be evaluated by using time-current graphs alone. This article develops a method to evaluate and optimize the phase over-current protection on a radial medium-voltage distribution feeder, which can reduce the risk at the point of fault and improves the overall reliability of the feeder. The method is based on the evaluation of protection operating time, let-through energy, pickup sensitivity, and extent of the voltage dips experienced at the point of common coupling due to network faults under high- and low-source impedance conditions. A Microsoft Excel-based software application (Microsoft, Redmond, WA, USA) is developed that incorporates the four evaluation measures under user-definable network criteria. The application illustrates the results in unique, special forms across the analyzed path and allows the benefits of small changes in protection settings to be quantified. The method and software applications are applied in two case studies on an actual distribution network. The results confirm that it is possible to evaluate the effectiveness of the phase over-current protection philosophy using the aforementioned criteria. 2016

In low-voltage systems, ground faults do not necessarily involve the actual earth, but fault currents may return to the source via conductors. Such conductors, which provide a clear path toward the source, are defined in the applicable codes and technical standards as equipment grounding conductors or protective conductors.

At the occurrence of three-phase or single-phase faults, abnormal levels of thermal energy are developed during the time taken by protective devices to clear them. By conservatively assuming an adiabatic process, all of the thermal let-through energy I²t, also referred to as Joule Integral, is accumulated within the components involved in short circuits; therefore, the temperature of their conductive materials is elevated. The thermal energy is proportional to the square of the short-circuit current. Evaluating the prospective I²t is, therefore, crucial in order to assess the short-circuit capability of cables and busways to withstand the thermal stress without failing or triggering fires in neighboring materials. In this paper, in the general case of resistive-inductive circuits, methods to evaluate the Joule Integral and to perform the assessment will be provided. The differences for power frequencies of 50 and 60 Hz are also shown.

• Giuseppe Parise
• M. Adduce

The admissible let-through energy I/sup 2/t allows consideration of the cable protection from overheating due to excessive short circuit current flowing in its conductors. For short circuit current duration of t<0.1 s, which is a range own for instantaneous tripping of protective devices in electric power systems below 600 V AC, the asymmetry of the current is of importance in the evaluation of the through energy. The IEC Standard suggests that the value of I/sup 2/t should be quoted by the manufacturer of the adopted protective devices. In the case of very short duration less than half-cycle, for which it is necessary to consider the protective device I/sup 2/t limiting behavior. The aim of this paper is to free the designer from the necessary knowledge of protective device behavior or from the need of prior manufacturers choice. The paper suggests evaluation of the prospective Joule integral, always referred to as RMS short circuit current by means a factor m, which considers the worst case (maximum value). Expressions and illustrations are given for m factor evaluation, which is variable with fault circuit conditions and prospective short circuit duration.

Transmission and Distribution of Electric Energy: Electric Cables, Italian Standard CEI 11-17

• Installations
• Generation

Installations of Generation, Transmission and Distribution of Electric Energy: Electric Cables, Italian Standard CEI 11-17, Jul. 2006. MITOLO AND TARTAGLIA: ANALYTICAL EVALUATION OF THE FACTOR k 2 FOR PROTECTIVE CONDUCTORS 217

Calculation of Thermally Permissible Short-Circuit Currents, Taking Into Account Non-Adiabatic Heating Effects

Calculation of Thermally Permissible Short-Circuit Currents, Taking Into Account Non-Adiabatic Heating Effects, IEC 60949, Sep. 2008.

IEEE Recommended Practice for Grounding of Industrial and Commercial Power Systems

• Std

Temperature Limits of Electric Cables With Rated Voltages of 1 kV (Um = 1.2 kV) and 3 kV (Um = 3.6 kV)

• Short-Circuit

Short-Circuit Temperature Limits of Electric Cables With Rated Voltages of 1 kV (Um = 1.2 kV) and 3 kV (Um = 3.6 kV), IEC 60724, Oct. 2000.

Temperature Limits of Electric Cables With Rated Voltages From 6 kV (Um = 7.2 kV) up to 30 kV (Um = 36 kV)

• Short-Circuit

Short-Circuit Temperature Limits of Electric Cables With Rated Voltages From 6 kV (Um = 7.2 kV) up to 30 kV (Um = 36 kV), IEC 60986, Oct. 2000.

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Source: https://www.researchgate.net/publication/239766386_An_Analytical_Evaluation_of_the_Factor_k2_for_Protective_Conductors