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DC-Arc Models and Incident-Energy Calculations
Ravel F. Ammerman, Senior Member, IEEE, Tammy Gammon, Senior Member, IEEE,
Pankaj K. Sen, Senior Member, IEEE, and John P. Nelson, Fellow, IEEE
Abstract—There are many industrial applications of large-scale
dc power systems, but only a limited amount of scientific literature
addresses the modeling of dc arcs. Since the early dc-arc research
focused on the arc as an illuminant, most of the early data was
obtained from low-current dc systems. More recent publications
provide a better understanding of the high-current dc arc. The
dc-arc models reviewed in this paper cover a wide range of arcing
situations and test conditions. Even with the test variations, a
comparison of dc-arc resistance equations shows a fair degree of
consistency in the formulations. A method for estimating incident
energy for a dc arcing fault is developed based on a nonlinear
arc resistance. Additional dc-arc testing is needed so that more
accurate incident-energy models can be developed for dc arcs.
Index Terms—DC-arc modeling, dc-arc resistance, dc incidentenergy calculations, dc-system hazard risk category evaluation,
free-burning arcs in open air, volt–ampere (V –I) characteristics.
Fig. 1.
Series-electrode arc classification [2].
I. I NTRODUCTION
A
RC physics is complex, and the physical constants are
particularly hard to clearly define for real-world arcing
faults in power systems. Therefore, the present knowledge has
been largely developed based on the observation and analysis of
electrical measurements. The volt–ampere (V –I) characteristics of electric arcs, which are dependent on test parameters, are
essential to defining the complex arc phenomenon in power systems. Early researchers often failed to specify test conditions,
the configuration type, and if ac or dc arcs had been initiated.
Since the V –I characteristic is dependent on test conditions,
including gap width and relative current magnitude, it can be
difficult to assess the early published work for accuracy and
coherence.
High current magnitudes (on the order of kiloamperes),
typical of arcing faults in power systems, are commonly viewed
as being quasi-stationary because the large thermal inertia in
the arc discourages changes in arc temperature and conductance. Even though the dynamic nature of the arc generates
a time-varying arc length, arc voltage equations have been
Manuscript received June 22, 2009; accepted January 24, 2010. Date of
publication July 12, 2010; date of current version September 17, 2010. Paper
2009-PCIC-185, presented at the 2009 IEEE Petroleum and Chemical Industry
Technical Conference, Anaheim, CA, September 14–16, and approved for
publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by
the Petroleum and Chemical Industry Committee of the IEEE Industry Applications Society.
R. F. Ammerman and P. K. Sen are with Colorado School of Mines, Golden,
CO 80401 USA (e-mail: rammerma@mines.edu; psen@mines.edu).
T. Gammon is with John Matthews & Associates, Cookeville, TN 38502
USA (e-mail: tgammon@tds.net).
J. P. Nelson is with NEI Electric Power Engineering, Arvada, CO 80001 USA
(e-mail: jnelson@neiengineering.com).
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.2010.2057497
developed from the quasi-stationary V –I characteristics. This
paper provides an overview of the most commonly used and
published arc equations and develops dc-arc-resistance models.
A simulation study is performed to compare the formulas which
are relevant with present dc-arc research. Additionally, based
on these models, dc arcing-fault incident-energy calculations
are presented to assess the level of risk involved when working
around high-current dc apparatus. The discussion begins with a
brief summary of free-burning arcs in open air.
II. F REE -B URNING A RCS IN O PEN A IR
As Sweeting and Stokes observed, “The vast majority of
the literature deals with arcs that have been constrained or
stabilized.” They also noted that “The bulk of the arc literature
is based on single-phase opposing electrodes, where the current
comes from one side and flows across to the other side” [1].
Series electrodes have historically received the majority of
attention because this is the configuration utilized to design
power-system protective devices like circuit breakers and fuses.
In this context, arcs are often divided into two main categories: axisymmetric and nonaxisymmetric. An axisymmetric
arc burns uniformly, while nonaxisymmetric arcs are either in a
“state of dynamic equilibrium or continuous motion” [2]. Fig. 1
shows some of the commonly used arc classifications.
The “wall-stabilized” arc is constrained to a cylindrical
shape. At low currents (below 10 A), the geometry of a freeburning arc would look similar to the diagram on the right side
of the figure. As shown, the actual arc length is longer than
the electrode gap. Convective forces cause the arc plasma to
bow upward; the resulting shape helps to explain the origin
of the term “arc” used to describe this complex electrical
phenomenon.
0093-9994/$26.00 © 2010 IEEE
AMMERMAN et al.: DC-ARC MODELS AND INCIDENT-ENERGY CALCULATIONS
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TABLE I
STROM’S AVERAGE VOLTAGE GRADIENT [6]
Fig. 2.
Electric-arc characterization [3], [4].
Free-burning arcs in open air are the exclusive focus of
this paper. In industrial applications, high-current free-burning
arcing faults are extremely chaotic in nature. The arc moves
rapidly so that its length and geometry are constantly changing.
The contributing factors to the dynamic nature of high-current
free-burning arcs are the following:
1)
2)
3)
4)
5)
Fig. 3. Arcing voltage and current characteristic [5], [7].
thermal convection;
electromagnetic forces;
burn back of electrode material;
arc extinction and restriking;
plasma jets.
III. C HARACTERISTICS OF AN A RC
As shown in Fig. 2, an arc consists of three regions: the anode
region, the plasma column, and the cathode region. The electrode regions (anode and cathode) form the transition regions
between the gaseous plasma cloud and the solid conductors.
As shown in Fig. 2, an arc is also commonly associated
with a voltage profile. The voltage gradient across the arc
plasma depends on the actual arc length; the arc may deviate
from the gap width between the electrodes. Less deviation
is expected for short gap widths, series electrodes, and less
turbulent conditions.
A number of researchers have postulated that the voltage
gradient in the plasma column of an arc is nearly independent
of the arcing current. For example, Browne found that the
voltage gradient in the arc column is nearly independent of the
arc current for magnitudes above 50 A and is approximately
12 V/cm (30.5 V/in) for arcs in open air [5]. Browne’s research
investigated arc behavior in both dc and ac circuits. In 1946,
Strom published that “the voltage gradient in the arc is affected
very little by current magnitude” [6]. Strom found that, for
arc gap widths from 0.125 to 48 in (0.32 to 122 cm), the arc
voltages averaged 34 V/in (13.4 V/cm) during arc tests, which
produced peak ac currents ranging from 68 to 21 750 A. Table I
summarizes the results of Strom’s findings. These numbers are
comparable to Browne’s finding.
Fig. 4. DC-arc test circuit configuration [8].
A. Arc V –I Characteristics
Fig. 3 shows the quasi-static V –I characteristic for an arc
of “fixed” length. In the low-current region (identified by the
dotted line), the arc voltage drops as the arc current increases;
as a result, the arc power (P = V I) tends to remain relatively
constant in this region. For “larger” currents, the arc voltage
increases slightly with increasing arc current. (A transition
current, which defines the boundaries between the low- and
high-current regions, is presented later). With wall-stabilized
arcs, the arc plasma is only partially ionized in the low-current
region, whereas the plasma becomes fully ionized above some
threshold current [2]. A similar transition in the level of ionization is observed for free-burning arcs.
B. Arc Modeling Using Static V –I Characteristics
Fig. 4 shows a typical test circuit used to measure the
characteristics of a dc arc. In this diagram, the gap width,
not the actual arc length, between the electrodes is labeled as
“L.” The arc length is difficult to measure. Many equations
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Fig. 5. DC equivalent-circuit model.
use arc length. A number of experimenters probably assumed
that the arc length was equal to the gap width. The length
of the arc approximates the gap width when series electrodes,
low currents, and short gap widths are involved. Otherwise, the
arc length may be considerably longer than the gap width. In
many early papers, it is not clear when arc length is defined
as an equation parameter if the equation is based on the gap
width or an estimated arc length. Most equations are probably
based on gap width since gap width is a measurable parameter.
However, it must be remembered that the impedance of the arc
is governed by the actual arc length.
The arc’s physical processes are complex and chaotic in
nature, and it is very difficult to develop theoretical models
using arc physics. Consequently, an arc is often represented
with an equivalent electrical circuit (a “black-box” approach).
In some cases, this representation is sufficient because the
objectives are to determine arc current, power, and energy.
Fig. 5 shows the simplified dc equivalent-circuit representation
of the arc.
Fig. 6.
Sample of arc characteristic curves [8].
B. Steinmetz Equation
In 1906, Steinmetz derived a semiempirical V –I equation
based on carbon and magnetite arc experiments [10]
Varc = A +
C(L + D)
.
0.5
Iarc
(2)
In (2), A, C, and D are constants, and L is the arc length.
For a 25.4-mm (1-in) arc with carbon electrodes, the equation
is defined as
Varc = 36 +
130(1 + 0.33)
.
0.5
Iarc
(3)
C. Nottingham Equation
IV. DC-A RC V –I C HARACTERISTICS AND E QUATIONS
Much of the early arc research was focused on the use
of an arc as an illuminant. Low-current arcs were relatively
stable, while their high-current counterparts were considered
unpredictable and dangerous. This belief, coupled with the
availability of low-power dc supplies, explains why most early
arc research focused on low-current dc arcs, which exhibited
inverse V –I characteristics. This section highlights some of
the early and selected key publications; it also provides a
comparison of methods used to model a dc arc.
A. Ayrton Equation
Ayrton formulated the first known equation used to model
the electrical properties of a steady-state arc [9]. Developed in
1902, (1) was derived for arcs in air initiated between carbon
electrodes separated by a few millimeters
Varc = A + BL +
C + DL
.
Iarc
(1)
The constant A represents the electrode voltage drop, B
describes the voltage gradient, and L is the arc length; C and D
are constants, which model the arc’s nonlinear characteristic.
In the mid 1920s, Nottingham conducted arc research that
produced a similar inverse characteristic [11]
Varc = A +
B
.
n
Iarc
(4)
The constants A and B are dependent on the arc length and
the electrode material. The arc current is raised to a power n,
where n varies as a function of the electrode material. For arc
lengths ranging from 1.0 to 10.0 mm (0.0394 to 0.394 in), the
equation for copper electrodes is specified in (5). Also, note that
the exponent n is different from the previous two equations
44
(5)
Varc = 27.5 + 0.67 .
Iarc
Fig. 6 shows a sample of some typical V –I characteristics of
arcs with 6-mm (0.236-in) arc lengths and different electrodes.
For constant arc lengths, the Nottingham equation has the same
general structure as the Ayrton and Steinmetz formulas.
The early arc formulas are based on a limited number of
low-current test results. The empirical constants were actually
dependent on electrode materials, gap lengths, and gaseous
mediums. No standard testing procedure had been established,
and experimental procedures did not follow consistent testing
protocols. Consequently, many of the findings have been considered inconclusive.
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D. Van and Warrington Equation
In 1931, Van and Warrington performed a series of tests on
high-voltage ac systems for arcing currents between 100 and
1000 A and electrode distances which spanned several feet [12].
The V –I characteristic of a stable arc was determined as
Varc =
8750L
.
0.4
Iarc
(6)
In (6), L is the arc length in feet. Van and Warrington confirmed the early research performed by Ayrton and Steinmetz
by showing that arc voltages are proportional to the arc length
and decrease with increasing arcing current. The inverse characteristic was probably exhibited in this current range because
of the large gap distance between the electrodes.
E. Miller and Hildenbrand
Fig. 7. DC-arc voltage versus current in 9.5-mm (3/8 in) gap [16].
In 1972, Miller and Hildenbrand published a dc-arc model
based on an energy-balance concept [13]. As a first approximation, they recommended using the empirical relationship in
(4) developed by Nottingham. They emphasized that A, B, and
n are not absolute constants but depend on the arc conditions,
specifically, electrode material, arc length, and gas species and
pressure. Furthermore, they referenced Cobine’s statement that
constants are difficult to accurately determine even for a given
set of conditions [14]. Ignatko conducted a series of ac-arc
tests which generated arc currents ranging from 5 to 150 kA
for arc gaps between 5 and 200 mm (0.197 and 7.87 in).
Ignatko determined that the total electrode (cathode and anode)
drop remained practically constant and measured 23.5 V for
copper, 26.5 V for steel, and 36 V for tungsten [15]. Ignatko’s
results confirm earlier work reporting a 20- to 40-V drop at the
electrodes [5].
F. Hall, Myers, and Vilicheck
In 1978, a group of researchers conducted tests to evaluate
faults on dc trolley systems [16]. Over 100 dc-arc tests were
conducted using a 300-V dc supply. Arcing currents ranged
from 300 to 2400 A, and electrode gap widths ranged from
4.8 to 152 mm (3/16 to 6 in). The relationship between the
arc voltage and the arc current, shown in Fig. 7, is based on a
number of arc tests with a 9.5-mm (3/8 in) gap. The relationship
between the arc current and the arc voltage in a dc trolley
system was determined to match the form defined in (4).
G. Stokes and Oppenlander Model
Stokes and Oppenlander performed perhaps the most exhaustive study of free-burning vertical and horizontal arcs between
series electrodes in open air [17]. “Current and voltage signals have been recorded for arcs burning with exponentially
decaying currents from 1000 to 0.1 A, and 50-Hz arcs for
sinusoidal currents with amplitudes decaying from 20 kA to
30 A [17].” Figs. 8(a) and (b) and 9 show that the minimum
voltage needed to maintain an arc depends on current magnitude, gap width, and orientation of the electrodes. Stokes and
Fig. 8. (a) Minimum arc voltage for horizontal arcs [17]. Minimum voltage
characteristics for copper electrodes. Continuous lines are measured. Broken
lines are calculated based on power characteristics. Gap widths for curves
from bottom to top: 5, 20, 100, and 500 mm (0.20, 0.79, 3.94, and 19.7 in).
(b) Minimum arc voltage for horizontal arcs [17]. Stokes and Oppenlander data
presented on a linear scale (500-mm (19.7-in) gap).
Oppenlander formulated the minimum arc voltages for series
electrodes. DC arcs in an industrial setting are likely to be
initiated between parallel electrodes, which are characterized
by longer arc lengths and higher arc voltages.
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Fig. 10.
Paukert’s compilation of arcing-fault data [18].
Fig. 9. Minimum arc voltage for vertical arcs [17]. Minimum voltage characteristics for aluminum electrodes. Continuous lines are measured. Broken
lines are calculated based on power characteristics. Gap widths for curves from
bottom to top: 5, 20, 100, and 500 mm (0.20, 0.79, 3.94, and 19.7 in).
TABLE II
EMPIRICAL ARC FORMULAS FOR Iarc < 100 A [18]
The current associated with the transition point for each
gap width is clearly marked on the figures by the solid line
with dots. The transition current is defined as It = 10 + 0.2zg ,
where the length of the gap zg is expressed in millimeters [17].
The curves show the inverse V –I characteristic for an arc with
a current that is lower than the transition point. For currents
above the transition point, the arc voltage shows a very slow
rise in voltage values. Stokes and Oppenlander modeled the arc
voltage for arcing currents above a transition point. “This set
of data, totaling some two million current and voltage points,
was reassembled to current–voltage characteristics [17].” The
result is
0.12
.
Varc = (20 + 0.534zg )Iarc
(7)
TABLE III
EMPIRICAL ARC FORMULAS FOR 100 A < Iarc < 100 kA [18]
Equation (7), written in terms of arc resistance, becomes
Rarc =
20 + 0.534zg
.
0.88
Iarc
(8)
H. Paukert’s Compilation of LV Arcing-Fault Data
Paukert compiled published arcing-fault data from seven
researchers who conducted a wide range of arc tests. Some tests
were dc, and some tests were ac. Some configurations were
vertical, and others were horizontal. Arcing currents ranged
from 0.3 A to 100 kA, and electrode gaps ranged from 1 to
200 mm (0.039 to 8 in) [18]. The survey data are summarized
in Fig. 10.
Based on the collected data, Paukert formulated arc-voltage
and arc-resistance equations for various electrode gap widths;
these equations are listed in Tables II and III. Table II presents
an inverse V –I characteristic for low-current arcs, and Table III
presents positive V –I characteristic for currents above 100 A.
Good agreement was found between the measurements of
Stokes and Oppenlander and the test results compiled by
Paukert, as shown in Fig. 11(a) and (b). The best agreement is
found in the higher current range, which is greater than 100 A.
Paukert concludes his analysis with the following words: “Al-
though the author’s approximation formulas for minimal arc
voltage and minimal arc resistance have been found to be in
good agreement with other authors’ results, the uncertainty
connected with the determination of actual arc length will
hamper their successful application for exact calculation [18].”
I. Sölver
Like earlier researchers, Sölver recognized the complexity of the relationship between the arc current, arc voltage,
AMMERMAN et al.: DC-ARC MODELS AND INCIDENT-ENERGY CALCULATIONS
1815
arc voltage is primarily determined from the electrode voltage
drops, which is around 20 V. “When the arcs are long and the
current is not too low, the arc voltage tends to be on the order
of 10 V/cm [20].”
The dc-arc models presented in this section share the following characteristics.
1) Arc resistance is nonlinear.
2) Arc resistance is dependent on multiple factors:
a) gap length;
b) electrode material;
c) arc-current magnitude;
d) electrode configuration.
The need for additional testing is evident. More testing would
lead to the development of better equations for dc arcing and dcarc resistance. Section V provides a comparative analysis of the
existing arc-resistance equations.
V. A RC -R ESISTANCE M ODEL C OMPARATIVE S TUDY
Fig. 11. (a) Comparison of V –I characteristic formulas for vertical arcs
[18]: full lines—measurements of Stokes and Oppenlander [17], very full thick
lines—Paukert [18], and broken lines–theory of Lowke [19]. (b) Comparison of
V –I characteristic formulas for horizontal arcs [18]: full lines—measurements
of Stokes and Oppenlander [17], very full thick lines—Paukert [18], and broken
lines—theory of Lowke [19].
Fig. 12. Current–voltage characteristics for dc arcs in air, with copper
electrodes [20].
and arc length. Fig. 12 shows experimental results for dc
arcs between copper electrodes separated by widths of up to
200 mm (7.87 in). For lower current values, the arc voltage has
an “inverse” relationship with the arc current; as the arc currents
increase, the arc voltages tend to flatten and become relatively
constant (independent of the current). When the arc is short, the
The following models are used in the comparative study to
calculate arc resistance. The models developed by Paukert and
by Stokes and Oppenlander are included because they represent
a large number of test data, including dc arcs.
1) Nottingham: Equation (5), based on test data from dc
arcs, with lengths ranging from 1.0 to 10.0 mm (0.0394
to 0.394 in). His sample curves show arc currents up to
10 A.
2) Stokes and Oppenlander: Equation (8), based on exponentially decaying dc currents from 1000 to 0.1 A and
decaying single-phase 50-Hz amplitudes from 20 kA to
30 A. The gap widths ranged from 5 to 500 mm (0.20
to 19.7 in) between series electrodes. Copper electrodes
were tested in a horizontal configuration, and aluminum
electrodes were tested in a vertical configuration.
3) Paukert: (Table III), based on test data from dc and singlephase ac arcs. Based on readings of Fig. 10, the dc-arc
tests were conducted for arc currents of up to approximately 50 A (covered in Table II only). Rieder initiated
dc arcs between copper electrodes spaced between 1 and
160 mm (0.0394 and 6.30 in).
Figs. 13 and 14 show comparisons between the arc-resistance
formulas. Fig. 13 shows a comparison of the three approaches
for a gap length of 10 mm (0.394 in).
The Nottingham formula described in (5) is only applicable
to electrode gaps in the range of 1–10 mm (0.0394–0.394 in),
so it is not included in the sensitivity study shown in Fig. 14.
Fig. 15 shows the relationship between arc resistance, gap
length, and sensitivity to arc current.
Fig. 13 shows that the three models are somewhat consistent.
Fig. 14 shows that the V –I relationships developed by Paukert
and by Stokes and Oppenlander exhibit more deviation with
large gap widths. Some additional observations include the
following.
1) Arc resistance is nonlinear.
2) Arc resistance decreases with increasing arc current.
3) Arc-resistance drop approaches a constant value at high
current magnitudes.
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4) Arc resistance changes rapidly at low current magnitudes
(< 1 kA).
5) Paukert predicts larger arc resistances (almost by a factor
of 1.5) than what Stokes and Oppenlander predict.
6) For a given arc current, the arc resistance increases linearly with the electrode gap.
VI. A RC E NERGY
The law of “conservation of energy” principle requires that
energy is conserved during an arcing fault; therefore, the electrical energy input is equal to the energy released in the form
of heat, pressure, sound, light, and electromagnetic radiation.
Arc-resistance models may be a convenient way to estimate the
electrical energy delivered during an arcing fault.
A. Theoretical Arc-Energy Fundamentals
Fig. 13. DC-arc resistance comparative study (10-mm electrode gap).
For steady-state dc systems, power is determined as follows:
Power = Vdc Idc .
(9)
Generally speaking, the power for dc or single-phase ac arcs
can be expressed as
2
Parc = Varc Iarc = Iarc
Rarc .
(10)
Since energy is a function of time, the energy associated with
an arc is approximated as
2
Rarc tarc .
Earc ≈ Iarc
(11)
The arc duration (tarc ) is measured in seconds. It should
be noted that dc arcs do not pass through current zero every
half cycle, which makes low voltage (LV) single-phase ac arcs
susceptible to self-extinction [21].
B. DC-Arc Incident-Energy Estimates
Fig. 14. DC-arc resistance comparative study (sensitivity to electrode gap).
Since electric arcs involve extremely complex processes,
modeling electric arcs with theoretical physics is difficult, and
model parameters depend on test and environmental conditions.
Arcing faults in an industrial workplace may be initiated under
a wide range of conditions. Arcing, by nature, is a dynamic
process, and industrial arcing faults are much more dynamic,
random, and turbulent than constrained arcs initiated in a controlled environment (for example, a laboratory setup). Consequently, semiempirical models are an effective way of modeling
arcing faults in power systems and calculating incident energy.
1) Open-Air Arc Exposures: A large battery-bank installation in a nuclear power plant is an example of an open-air dc-arc
flash hazard. For this type of exposure, the heat transfer depends
on the spherical energy density, as described in (12), where d
represents the distance from the arc (in millimeters).
Es =
Fig. 15. DC-arc resistance comparative study (Stokes and Oppenlander/
Paukert formula comparison).
Earc
.
4πd2
(12)
This formula is based on radiant-heat transfer, and not all arc
energy will be transferred as radiant heat. In (12), the energy
density varies with the inverse square of the distance from the
arc source.
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1817
TABLE IV
OPTIMUM VALUES OF a AND k [22]
Fig. 17. Incident energy (arc-in-a-box) versus arc duration for 32-mm
(1.25-in) gap and 457-mm (18-in) working distance.
Fig. 16. Incident energy (open air) versus arc duration for 32-mm (1.25-in)
gap and 457-mm (18-in) working distance.
2) Arc-in-a-Box Exposures: When a dc arc is initiated
within a piece of switchgear, the enclosure tends to have a
focusing effect on the incident energy. Wilkins proposed an
approach for three-phase ac arcs where the spherical energydensity component is replaced by a value E1 that takes into
account the focusing effect of an enclosure [22]. In other words,
the term E1 also represents the additional energy reflected by
the back and sides of the enclosure
E1 = k
Earc
.
a2 + d2
(13)
Table IV lists Wilkins’ optimum values of a and k for the
three equipment classes described in the IEEE 1584 guide [23].
The use of (12) and (13) to compare the arcs initiated in
enclosures with those in open-air arc exposures shows that
the arc-in-a-box case results in an increase of incident energy
directed toward a worker.
C. DC-Arc Incident-Energy Release
Figs. 16 and 17 approximate the incident energies associated
with dc arcing faults of 2, 6, and 10 kA across a gap of 32 mm
(1.25 in). The arc power was calculated from the arc-resistance
equation (8). Incident energy at 457 mm (18 in) was determined
by (12) and (13). The LV switchgear values for a and k in
Table IV were used to calculate the incident energies associated
with an enclosure. The resulting incident-energy levels are
compared with the hazard risk categories defined in National
Fig. 18. DC system one-line diagram.
Fire Protection Association 70E [24]. For the selected enclosure type and test distance, the incident energies calculated
for enclosures are 2.2 times larger than the incident energies
calculated for open air.
VII. DC D ISTRIBUTION S YSTEM : C ASE S TUDIES
Two case studies of a large power plant illustrate a method
for estimating the potential dc-arc flash hazard associated with
high-current batteries. The one-line diagram in Fig. 18 shows
operational units feeding a 250-V dc bus through rectifiers.
The bus is backed up with 258-V battery banks. The dc-supply
sources include batteries, rectifiers, and battery chargers; any
of these sources might sustain a dc arcing fault, depending on
system operating conditions and the fault location. The dc bus
serves a variety of loads, such as motors, inverters, relay coils,
and lamps.
For the fault calculations presented in this section, it is
assumed that a fault occurs on the dc bus while being supplied
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Fig. 19. Double-string battery circuit model.
TABLE V
DC SYSTEM SPECIFICATIONS AND PARAMETER VALUES
Fig. 20. Incident energy (open air) versus arc duration for 20-mm (0.79 in)
gap and 457-mm (18 in) working distance.
TABLE VI
ITERATIVE SOLUTION RESULTS
by the battery bank. It is further assumed that the fault-current
contribution from any dc motors is negligible. The dc steadystate circuit model for the double-string battery system is
shown in Fig. 19. For the single-string system, one battery is
removed. Table V lists the system specifications and circuitmodel parameters for the case studies. The reactive (inductive
and capacitive) dynamic response of the batteries lasts approximately 15 ms after the fault occurs and is neglected [25]. The
effect of the battery charger is also transitory and neglected in
the calculations. Furthermore, any nonlinear battery-discharge
characteristics are not considered in this work.
The bolted-fault currents, listed in Table VI and associated
with the double-string and single-string battery banks, were
calculated using the nominal battery voltage and the total
system resistance. The arcing-fault current for each case was
determined using an iterative solution of (8) and the circuit
model shown in Fig. 19. As an initial guess, the arc current was
set to be equal to 50% of the bolted-fault current and converged
rapidly. The arc gap width was defined as 20 mm since a 250-V
source has limited voltage potential to sustain arcs across large
gap widths. The arc current and arc resistance for each system
are provided in Table VI.
For the single-string and double-string systems, the battery
banks lack upstream overcurrent circuit protective devices, so
immediate dc-arc interruption is not likely for sustainable gap
widths. Equations (11) and (12) were used to calculate the
incident energies at 457 mm (18 in). The incident energies,
plotted as a function of time and shown in Fig. 20, merit
concern. In particular, the magnitude of the incident energy for
the double-string battery bank increases quickly as a function of
time and reaches Hazard Category 4 soon after 1.1 s. A higher
risk of serious burn is certainly associated with the doublestring battery bank. These cases were calculated for a dc arcing
fault which occurs at the 250-V bus. However, if an arcing fault
initiates between battery terminals, chemical burns present an
additional hazard.
VIII. C ONCLUSION
The models presented in this paper have been based on tests
conducted over more than a century by different researchers
in different countries and under very different protocol. Considering the wide range of testing methods and conditions, the
results are remarkably similar. At low current levels, the V –I
characteristic is inversely proportional and nonlinear. At high
arcing-current levels, the analysis in this paper has shown that
the arc-resistance voltage-drop approaches a constant value. In
an effort to quantify the risks associated with high-current dc
systems, a method has been presented to estimate the incidentenergy levels possible during an arcing fault. Results from a
case study demonstrated that the risks associated with highcurrent dc systems may be significant.
Arcing behavior is highly variable, and the existing dcarc models cannot accurately and reliably assess all the
AMMERMAN et al.: DC-ARC MODELS AND INCIDENT-ENERGY CALCULATIONS
characteristics of dc arcs. Additional arc testing is needed to
develop more accurate V –I characteristics and better dc-arc resistance models. Extensive testing in a controlled environment
is needed to study the incident-energy levels associated with
dc arcing faults. A hazard risk assessment is needed to identify
where dc arcing faults might be initiated in industrial power
systems. The relative severity of the arc flash hazard posed by
different types of dc power equipment must be identified.
ACKNOWLEDGMENT
The authors would like to thank G. Leask of Bruce Power,
Ontario, Canada, for his assistance in providing the dc system
information presented in the case studies, and to the many
reviewers of this paper for their detailed and constructive
criticism.
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Document # RS1468.
1819
Ravel F. Ammerman (SM’09) received the B.S. degree in engineering (electric power/instrumentation)
from the Colorado School of Mines, Golden, in
1981, the M.S. degree in electrical engineering
(power/control) from the University of Colorado,
Denver, in 1987, and the Ph.D. degree in engineering
systems (electrical specialty—power systems) from
the Colorado School of Mines, in 2008.
He has over 28 years of combined teaching, research, and industrial experience. He is currently
with the Colorado School of Mines. He has coauthored and published a number of award winning technical articles, published in
archival journals. His research interests include arc flash hazard analysis, electrical safety, computer applications in power system analysis, and engineering
education.
Dr. Ammerman is a member of the IEEE/NFPA Arc Flash Collaborative
Research and Testing Project Team.
Tammy Gammon (SM’06) received the Ph.D.
degree from the Georgia Institute of Technology,
Atlanta, in 1999.
She was an Assistant Professor with the North
Carolina State Engineering Program, University of
North Carolina at Asheville, from 1999 to 2003. She
has been with John Matthews & Associates, Inc.,
Cookeville, TN, as a Senior Electrical Engineer since
2003. The firm specializes in forensic engineering
(fires of electrical origin, electrical shock, and arc
flash burns) and evaluates the safety of electrical
products and equipment. The firm is experienced in utility and distribution
power issues and in designing electrical and lighting systems for buildings.
She is also currently the Research Manager for the IEEE/NFPA Arc Flash
Collaborative Research Project. She has taught a wide range of power and
mechatronic courses.
Dr. Gammon is a Registered Professional Engineer in the State of North
Carolina.
Pankaj K. (P. K.) Sen (SM’90) received the Ph.D.
degree from the Technical University of Nova Scotia
(Dalhousie University), Halifax, NS, Canada, in 1974.
He has over 44 years of combined teaching, research, and consulting experience. Currently, he is
a Professor of engineering and the Site Director
for the NSF Power Systems Engineering Research
Center (www.PSerc.org), Colorado School of Mines,
Golden. He has published over 120 papers on a
variety of subjects related to power systems engineering, electric machines and renewable energy,
protection, grounding, and safety and has supervised over 120 graduate students. His current research interests include application problems in power
system engineering, renewable energy and distributed generation, arc flash
hazard, electrical safety, and power engineering education. He is a member of
the IEEE/NFPA Arc Flash Collaborative Research and Testing Project Team.
Dr. Sen is a Registered Professional Engineer in the State of Colorado.
John P. Nelson (S’73–M’76–SM’82–F’98) received
the B.S.E.E. degree from the University of Illinois,
Urbana, in 1970, and the M.S.E.E. degree from the
University of Colorado, Boulder, in 1975.
He is the Founder/CEO of NEI Electric Power
Engineering, Inc., Arvada, CO. He spent ten years
in the electric utility industry and the last 29 years
as an electrical power consultant. He has authored
numerous papers (over 30) involving electric power
systems, grounding and protection, and protection of
electrical equipment and personnel safety. Many of
those papers are also published in the IEEE TRANSACTIONS ON INDUSTRY
APPLICATIONS and IEEE Industry Applications Magazine. He has taught
graduate and undergraduate classes at the University of Colorado, Denver, and
Colorado School of Mines, Golden, along with a number of IEEE tutorials and
seminars.
Mr. Nelson has been active in IEEE Industry Applications Society/
Petroleum and Chemical Industry Committee for 27 years. He is a Registered
Professional Engineer in the State of Colorado and numerous other states.