2. Harmonic losses in low-voltage networks

#### 2.1. Harmonic emission of LED and CFL

of solid-state energy conversion devices, industrial variable speed drive systems and cutting and welding equipment). Significant sources of harmonic currents in power systems also include electric devices such as compact fluorescent lamps (CFLs), power transformers operat-

58 Light-Emitting Diode - An Outlook On the Empirical Features and Its Recent Technological Advancements

Nowadays, with the technological advancement in semiconductors, light-emitting diode (LED) lamps are becoming a promising lighting technology due to its superior energy efficiency, longer lifetime and a better visual performance compared to most of the conventional light sources [7–9]. Due to these unique features, CFLs are now being replaced by LED, seeking a reduction in lighting costs and a lower impact on environment. In general lighting applications, a compact ac/dc (alternating current/direct current) converter should be used to supply dc current to LEDs, which introduces non-linearity to the system [10–12]. As non-linear loads, LEDs might produce highly distorted currents. Although the input power of a single LED is quite low, an incoming widespread use of them in lighting could create significant additional harmonic losses in the existing low-voltage lines [13]. Since several national standards allow for the neutral conductor reducing sizing with respect to the phase conductors, many of these existing low-voltage installations have the cross section of the neutral conductor

A large number of works were conducted on LEDs as an energy-efficient lamp, but most of them have been devoted to the internal driver circuit design [10–12, 14–17]. Several other works have concentrated on the light distribution and visual performance of LED lamps

When significant harmonic currents are present in low-voltage supply systems, additional Joule losses occur in the phase or line conductors as well as the neutral conductor. Significant harmonic currents may be present in secondary circuits of three-phase wye-connected transformers and single-phase transformers. Zero sequence harmonic currents flow in the phase conductors and are added in the neutral conductor, thus resulting in even higher harmonic current flow in the neutral conductor [25, 26]. Thus, the harmonic current flow in the neutral must be considered in the design of the supply system. The presence of harmonic currents in the supply conductors affects the ampacity of the supply system because of the additional ohmic losses [13, 26–28]. The determination of ohmic losses is complicated by the fact that the resistance of the cables is frequency dependent. Specifically, the resistance is increased with frequency because of the skin effect and proximity effect in the conductors and proximity effect from metallic conduit (if present) [26, 29]. The effects of harmonic currents in the neutral conductors can be evaluated with the same methods as for the phase or line conductors. However, the harmonic current magnitudes may be different in the neutral conductors due to non-cancelation of zero sequence harmonic currents and the cancelation of the positive and negative sequence harmonic currents. Thus, the neutral conductor becomes an additional heatgenerating conductor and must be considered in the ampacity calculation for three-phase wyeconnected and single-phase circuits [26–28]. Besides the knowledge of the increased losses due to harmonic currents, it is significant also for the economic evaluation of measures that attenuate harmonic currents. Such measures can be, for instance, passive or active harmonic filters [30, 31] or controlling the current unbalance in three-phase distribution systems by node

[18–20]. A few contributions focused on harmonic emissions of LED lamps [12, 21–24].

ing near saturation and computer system installations [4–6].

approximately equal to half of the phase conductors.

reconfiguration [32].

Figure 1 shows the current in one phase and in the neutral conductor of a four-core cable that feeds a balanced load of three identical LEDs (one per phase). Specifically, Figure 1 was obtained by testing the Philips 8 W LED. It can be seen that the LED current is highly distorted with respect to a sinusoidal waveform and that the fundamental frequency of the current in the neutral conductor is 150 Hz (i.e. the frequency of the third harmonic). Because the third-order harmonic currents (and their multiples) are zero sequence, they are added to the neutral

Figure 1. Voltage and current waveforms in one phase and current in the neutral conductor [35].

conductor. All experiments were done with an almost sinusoidal voltage waveform. The total harmonic distortion (THD) of the phase voltage waveform was relatively low (< 3%).

A frequency domain analysis of the current harmonics (Ipu(h)) produced by several commercially available LEDs is presented in Figure 2. In this figure, Ipu(h) was expressed in per unit of the fundamental current harmonic (h = 1 corresponding to a harmonic frequency of f = 50 Hz, with h being the order of the harmonic).

The experimental data can be approximately described by the power law:

I hð Þ <sup>I</sup>ð Þ<sup>1</sup> <sup>¼</sup> hm, (1)

A power quality analyzer (Fluke 435-II) was used in the measurements presented here.

Two different types of cables were examined. The first type was an arrangement of four singlecore cables in contact with each other, as they are specified by IEC 60502–1 [37]. The schematic of the used cable arrangement is shown in Figure 4, while its geometric dimensions are

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As the conductors in all cables were assumed solid in the modeling, the conductor dimensions showed in Table 1 are slightly smaller than the actual dimensions. This assumption leads to results that are on the conservative side. Cases where the conductors were copper and the cross section of the neutral conductor was approximately equal to half of phase conductors

The second type corresponded to four-core cables as they were specified by CENELEC Standard HD603 [38]. In this case, a large cross-sectional sector-shaped cable, namely, 3 � 240 + 120 mm2

In a conductor where the conductivity is sufficiently high, the displacement current density can be neglected, and the conduction current density is given by the product of the electric field and the electrical conductivity (ohm's law). With these simplifications, the Maxwell's

∇ �

B μ 

<sup>∇</sup> � <sup>E</sup> ¼ � <sup>∂</sup><sup>B</sup>

∂t

,

¼ σE, (2)

, (3)

2.2. Numerical simulation of the harmonic losses in low-voltage networks

Figure 3. Third-order harmonic current amplitude for the investigated lamps [35].

summarized in Table 1.

were modeled.

was examined.

equations are

where m = � 1.2 � 0.2. Eq. (1) (which is indicated by a solid line in Figure 2) defines the harmonic signature of the examined LED lamps. It is observed in Figure 2 that the amplitude of the harmonic currents decreases almost inversely with the order of the harmonic, thus indicating that the third-order one is usually the most significant one. Note that the data corresponded to LEDs from 3 to 120 W. FCLs also tend to present considerable amplitudes in their third-order harmonics [34]. Figure 3 shows the amplitude of the third harmonic current (expressed in per unit of the fundamental harmonic) of several commercially available LEDs and CFLs, with powers varying between 3 and 23 W. The red line (representing a constant value of 86%) indicates one of the criteria established by IEC 61000–3-2 [36] for the harmonic emission limits for lamps having an active input power < 25 W (i.e. that the third harmonic current should not exceed 86% of the fundamental one).

As it can be seen in Figure 3, the lamps tested meet the quoted emission limit imposed by IEC, except for one of them (LED 9 W Sica), which is slightly above it.

Figure 2. Frequency domain analysis of the current harmonics on several commercially available LEDs. The blue line represents the power law given by Eq. (1) [13].

Figure 3. Third-order harmonic current amplitude for the investigated lamps [35].

conductor. All experiments were done with an almost sinusoidal voltage waveform. The total

A frequency domain analysis of the current harmonics (Ipu(h)) produced by several commercially available LEDs is presented in Figure 2. In this figure, Ipu(h) was expressed in per unit of the fundamental current harmonic (h = 1 corresponding to a harmonic frequency of f = 50 Hz,

harmonic distortion (THD) of the phase voltage waveform was relatively low (< 3%).

60 Light-Emitting Diode - An Outlook On the Empirical Features and Its Recent Technological Advancements

I hð Þ

where m = � 1.2 � 0.2. Eq. (1) (which is indicated by a solid line in Figure 2) defines the harmonic signature of the examined LED lamps. It is observed in Figure 2 that the amplitude of the harmonic currents decreases almost inversely with the order of the harmonic, thus indicating that the third-order one is usually the most significant one. Note that the data corresponded to LEDs from 3 to 120 W. FCLs also tend to present considerable amplitudes in their third-order harmonics [34]. Figure 3 shows the amplitude of the third harmonic current (expressed in per unit of the fundamental harmonic) of several commercially available LEDs and CFLs, with powers varying between 3 and 23 W. The red line (representing a constant value of 86%) indicates one of the criteria established by IEC 61000–3-2 [36] for the harmonic emission limits for lamps having an active input power < 25 W (i.e. that the third harmonic

As it can be seen in Figure 3, the lamps tested meet the quoted emission limit imposed by IEC,

Figure 2. Frequency domain analysis of the current harmonics on several commercially available LEDs. The blue line

<sup>I</sup>ð Þ<sup>1</sup> <sup>¼</sup> hm, (1)

The experimental data can be approximately described by the power law:

current should not exceed 86% of the fundamental one).

represents the power law given by Eq. (1) [13].

except for one of them (LED 9 W Sica), which is slightly above it.

with h being the order of the harmonic).

A power quality analyzer (Fluke 435-II) was used in the measurements presented here.

#### 2.2. Numerical simulation of the harmonic losses in low-voltage networks

Two different types of cables were examined. The first type was an arrangement of four singlecore cables in contact with each other, as they are specified by IEC 60502–1 [37]. The schematic of the used cable arrangement is shown in Figure 4, while its geometric dimensions are summarized in Table 1.

As the conductors in all cables were assumed solid in the modeling, the conductor dimensions showed in Table 1 are slightly smaller than the actual dimensions. This assumption leads to results that are on the conservative side. Cases where the conductors were copper and the cross section of the neutral conductor was approximately equal to half of phase conductors were modeled.

The second type corresponded to four-core cables as they were specified by CENELEC Standard HD603 [38]. In this case, a large cross-sectional sector-shaped cable, namely, 3 � 240 + 120 mm2 , was examined.

In a conductor where the conductivity is sufficiently high, the displacement current density can be neglected, and the conduction current density is given by the product of the electric field and the electrical conductivity (ohm's law). With these simplifications, the Maxwell's equations are

$$\nabla \times \left(\frac{\overline{B}}{\mu}\right) \;=\; \sigma \overline{E} \; \tag{2}$$

$$
\nabla \times \overline{E} = -\frac{\partial \overline{B}}{\partial t} \,' \tag{3}
$$

Figure 4. Layout of the examined single-core arrangement (taken from [13]).


Table 1. Dimensions of the modeled cable arrangement.

where B is the magnetic field, μ is the magnetic permeability, σ is the conductor electrical conductivity and E is the electric field. Introducing the magnetic vector potential A (B � ∇ � A) in Eq. (3), E can be expressed as

<sup>E</sup> ¼ � <sup>∇</sup><sup>V</sup> � <sup>∂</sup><sup>A</sup> <sup>∂</sup><sup>t</sup> , (4)

<sup>I</sup> <sup>¼</sup> <sup>Ð</sup>

variations over the conductor's surface

corresponding to the harmonic of order h.

were assumed carrying the following currents:

tor using the integral:

L3 are the three phases.

<sup>¼</sup> <sup>V</sup> Rdc

are very small because of the large value of the thermal conductivity.

� σ

<sup>J</sup> � dS <sup>¼</sup> <sup>Ð</sup> <sup>J</sup><sup>0</sup> <sup>þ</sup> Jeddy � � � dS

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63

, (7)

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J hð Þ<sup>2</sup> dS, (8)

� �, (9)

� �, (10)

� �, (11)

<sup>3</sup> <sup>π</sup> <sup>þ</sup> <sup>ϕ</sup><sup>h</sup>

<sup>3</sup> <sup>π</sup> <sup>þ</sup> <sup>ϕ</sup><sup>h</sup>

ð ∂A <sup>∂</sup><sup>t</sup> � dS,

where I is the total current and Rdc is the dc conductor resistance (J<sup>0</sup> is the spatial average current density generated by potential electric fields, while Jeddy is the (eddy) current density induced by rotational fields). Eq. (6) assumed a uniform electrical conductivity over the conductor surface. This is justified because simple estimates showed that the temperature

> <sup>Δ</sup><sup>T</sup> <sup>≈</sup> <sup>Λ</sup><sup>2</sup> κ σE<sup>2</sup>

(where κ is the conductor thermal conductivity and Λ = R/2.4 and R is the conductor radius) due to the non-uniform distribution of the Joule heat caused by the skin and proximity effects

At each harmonic frequency, the software calculates the losses per unit length in each conduc-

σ ð

where P(h) is the harmonic losses per unit conductor length and J(h) is the current density

The cables were modeled in two dimensions assuming that at each harmonic frequency, balanced, three-phase and sinusoidal currents flow through them. The three-phase conductors

IL<sup>1</sup> ¼ Ip cos 2πhf t þ ϕ<sup>h</sup>

where Ip is the current peak value, φ<sup>h</sup> is the angle phase of the harmonic order h and L1, L2 and

For non-triplen harmonic (h 6¼ 3 N, with N = 1, 2, 3,…), the neutral conductor only carries the eddy currents calculated by the software. Notice that in this case, h = 3 N + 1 represents the direct sequence harmonics, while h = 3 N – 1 represents inverse sequence harmonics. For

IL<sup>2</sup> <sup>¼</sup> Ip cos 2πhf t � <sup>h</sup> <sup>2</sup>

IL<sup>3</sup> <sup>¼</sup> Ip cos 2πhf t <sup>þ</sup> <sup>h</sup> <sup>2</sup>

triplen harmonics the current in the neutral conductor was assumed as

P hðÞ ¼ <sup>1</sup>

(where V is the electrostatic potential), and Eq. (2) becomes

$$\nabla \times \left[\frac{\nabla \times \overline{A}}{\mu}\right] = -\sigma \nabla V \quad - \quad \sigma \frac{\partial \overline{A}}{\partial t}.\tag{5}$$

The electromagnetic software [39] solved the diffusion equation (Eq. (5)) to obtain the spatial distribution of the total current density (J) over each conductor's surface (S), having as input the measurable current in the conductor:

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$$\begin{aligned} I &=& \int \overline{f} \cdot d\overline{S} &= \int \left( \overline{f}\_0 + \overline{f}\_{\text{eddy}} \right) \cdot d\overline{S} \\ &=& \frac{V}{R\_{dc}} - \sigma \int \frac{\partial \overline{A}}{\partial t} \cdot d\overline{S} \end{aligned} \tag{6}$$

where I is the total current and Rdc is the dc conductor resistance (J<sup>0</sup> is the spatial average current density generated by potential electric fields, while Jeddy is the (eddy) current density induced by rotational fields). Eq. (6) assumed a uniform electrical conductivity over the conductor surface. This is justified because simple estimates showed that the temperature variations over the conductor's surface

$$
\Delta T\_{\perp} \approx \frac{\Lambda^2}{\kappa} \sigma E\_{\perp}^2 \tag{7}
$$

(where κ is the conductor thermal conductivity and Λ = R/2.4 and R is the conductor radius) due to the non-uniform distribution of the Joule heat caused by the skin and proximity effects are very small because of the large value of the thermal conductivity.

At each harmonic frequency, the software calculates the losses per unit length in each conductor using the integral:

$$P(h) := \frac{1}{\sigma} \int \! f(h)^2 \, dS,\tag{8}$$

where P(h) is the harmonic losses per unit conductor length and J(h) is the current density corresponding to the harmonic of order h.

The cables were modeled in two dimensions assuming that at each harmonic frequency, balanced, three-phase and sinusoidal currents flow through them. The three-phase conductors were assumed carrying the following currents:

where B is the magnetic field, μ is the magnetic permeability, σ is the conductor electrical conductivity and E is the electric field. Introducing the magnetic vector potential A

Dimensions [mm] 3 � 35 + 16 3 � 70 + 35 3 � 120 + 70 3 � 240 + 120

Phase conductor radius [Rp] 3.3 4.7 6.2 8.8 Neutral conductor radius [Rn] 2.3 3.3 4.7 6.2 Thickness of phase conductor insulation [tp] 2.6 2.8 3.1 4.0 Thickness of neutral conductor insulation [tn] 2.4 2.6 2.8 3.1 Distance [L] 8.4 10.6 13.1 18.0 Distance [Y] 6.5 8.2 10.5 12.7

62 Light-Emitting Diode - An Outlook On the Empirical Features and Its Recent Technological Advancements

Nominal cable cross section [mm<sup>2</sup>

]

<sup>E</sup> ¼ � <sup>∇</sup><sup>V</sup> � <sup>∂</sup><sup>A</sup>

The electromagnetic software [39] solved the diffusion equation (Eq. (5)) to obtain the spatial distribution of the total current density (J) over each conductor's surface (S), having as input

¼ � σ∇V � σ

∂A

<sup>∂</sup><sup>t</sup> , (4)

<sup>∂</sup> <sup>t</sup> : (5)

(B � ∇ � A) in Eq. (3), E can be expressed as

Table 1. Dimensions of the modeled cable arrangement.

the measurable current in the conductor:

(where V is the electrostatic potential), and Eq. (2) becomes

Figure 4. Layout of the examined single-core arrangement (taken from [13]).

∇ �

∇ � A μ 

$$I\_{L1} = \,^I I\_p \cos \left( 2 \pi hf \, t \, + \, \phi\_h \right) \,. \tag{9}$$

$$I\_{l2} = \,^l I\_p \cos \left( 2 \pi h f \, t \, \begin{array}{c} -h \ \frac{2}{3} \pi \end{array} \pi \right) + \,^l \phi\_h \Big) \tag{10}$$

$$I\_{l3} = \,^l I\_p \cos \left( 2 \pi h f t \, \, + \, h \, \frac{2}{3} \pi \, \, \right. \, + \, \phi\_h \Big)\_{\prime} \tag{11}$$

where Ip is the current peak value, φ<sup>h</sup> is the angle phase of the harmonic order h and L1, L2 and L3 are the three phases.

For non-triplen harmonic (h 6¼ 3 N, with N = 1, 2, 3,…), the neutral conductor only carries the eddy currents calculated by the software. Notice that in this case, h = 3 N + 1 represents the direct sequence harmonics, while h = 3 N – 1 represents inverse sequence harmonics. For triplen harmonics the current in the neutral conductor was assumed as

$$I\_N \;=\; \Im I\_p \cos(2\pi hf\,t\; +\; \pi). \tag{12}$$

To calculate the Rac conductor resistance, an ac steady-state harmonic analysis was employed. Only the odd harmonics, up to the 29th, were considered. A higher value of this upper limit did not have appreciable impact to the obtained results. Due to the geometry of the cables, the losses in the phase conductors are not identical. In fact, the losses in phase conductors L1 and L3 (Figure 4) are the same, but those in L2 are different. The losses per unit length in the threephase conductors, when a symmetrical current of rms value Irms(h) and of frequency f h flows through them, can be defined as PL1ð Þh , PL2ð Þh and PL3ð Þh , for L1, L2 and L3, respectively. The uneven heat generation inside the cable is a fact that also needs to be considered when calculating the derating of cable ampacity. According to [41], not only the average cable temperature but also the temperature at any point along the insulation of the cable should not exceed the maximum permissible one. Therefore, for derating of the cable ampacity, the maximum conductor losses should be considered and not their average. For non-triplen harmonic (h 6¼ 3 N), the neutral conductor only carries the eddy currents calculated by the software, so the maximum cable losses can be represented by an effective conductor resistance

Figure 6. Spatial distribution of the rms current density for the fifth-order (h = 5) harmonic in a sector-shaped cable of <sup>3</sup> � 240 + 120 mm2 (a) and for the third-order (<sup>h</sup> = 3) harmonic in four single-core cables of 3 � 70 + 35 mm<sup>2</sup> (b).

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per unit length Reff(h) for the harmonic order h, which was defined as

where

<sup>3</sup>PLmaxðÞ þ <sup>h</sup> PNðÞ � <sup>h</sup> <sup>3</sup>Irms<sup>2</sup>

resistance (Rac Nð Þh ) that reflects the losses of the neutral conductor were defined as

<sup>3</sup>PLmaxðÞ � <sup>h</sup> <sup>3</sup>Irms<sup>2</sup>

PNðÞ � h Irms N

When triplen harmonics are present, the neutral conductor picks up the current. An effective resistance that reflects the maximum losses of the phase conductors (R~effð Þ<sup>h</sup> ) and another

2

ð Þh Reffð Þh , (13)

ð Þ<sup>h</sup> <sup>R</sup>~effð Þ<sup>h</sup> , (15)

ð Þh Rac Nð Þh , (16)

PL maxðÞ � h maxf g PL1ð Þh ; PL2ð Þh ; PL3ð Þh : (14)

In order to obtain an accurate distribution of the current density over the conductor sections, it was checked that the size of the local numerical mesh was less than half the characteristic skin penetration length for each harmonic order.

The study domain for the cases of sector-shaped and four single-core cables, showing the nonuniform numerical grid (with up to about 4000 mesh cells), is presented in Figure 5. At the boundary of the domain (at a radius up to ten times the cable size), it was assumed that A ¼ 0 because the magnetic field vanishes at a large distance (as compared to the cable size) from the cable.

The simulation results were obtained for <sup>μ</sup> = 4 <sup>π</sup> � <sup>10</sup>�<sup>7</sup> H/m (non-magnetic material was considered). The copper electrical conductivity at 293 K was taken as <sup>σ</sup> = 5.8 � 107 <sup>Ω</sup>�<sup>1</sup> <sup>m</sup>�<sup>1</sup> according to IEC 60028 [40]. The σ value was correspondingly corrected for other cable operating temperatures.

Figure 6 illustrates the spatial distribution of the root-mean-square (rms) value of the total current density over the conductors of cables of large sections submitted to harmonic currents of different frequencies. Figure 6(a) corresponds to a 3 � 240 + 120 mm<sup>2</sup> sector-shaped cable submitted to a 15 A (peak value) fifth-order harmonic current, while Figure 6(b) corresponds to an arrangement of four single-core cables (3 � 70 + 35 mm<sup>2</sup> ) submitted to a 30 A (peak value) third-order harmonic current. The magnetic field lines produced by the current are also shown in Figure 6. A noticeable reduction in the effective area of current circulation due to the skin and proximity effects is observed in Figure 6(a), thus causing a considerable increase of the ac conductor resistance (Rac) as compared to the dc resistance, (Rdc), which in turn results in high heat losses. In Figure 6(b) the neutral conductor is assumed to carry the algebraic sum of the phase currents.

Figure 5. Non-uniform numerical grid generated by the software for the case of a four-core sector-shaped cable (a) and four single-core cables (b).

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Figure 6. Spatial distribution of the rms current density for the fifth-order (h = 5) harmonic in a sector-shaped cable of <sup>3</sup> � 240 + 120 mm2 (a) and for the third-order (<sup>h</sup> = 3) harmonic in four single-core cables of 3 � 70 + 35 mm<sup>2</sup> (b).

To calculate the Rac conductor resistance, an ac steady-state harmonic analysis was employed. Only the odd harmonics, up to the 29th, were considered. A higher value of this upper limit did not have appreciable impact to the obtained results. Due to the geometry of the cables, the losses in the phase conductors are not identical. In fact, the losses in phase conductors L1 and L3 (Figure 4) are the same, but those in L2 are different. The losses per unit length in the threephase conductors, when a symmetrical current of rms value Irms(h) and of frequency f h flows through them, can be defined as PL1ð Þh , PL2ð Þh and PL3ð Þh , for L1, L2 and L3, respectively. The uneven heat generation inside the cable is a fact that also needs to be considered when calculating the derating of cable ampacity. According to [41], not only the average cable temperature but also the temperature at any point along the insulation of the cable should not exceed the maximum permissible one. Therefore, for derating of the cable ampacity, the maximum conductor losses should be considered and not their average. For non-triplen harmonic (h 6¼ 3 N), the neutral conductor only carries the eddy currents calculated by the software, so the maximum cable losses can be represented by an effective conductor resistance per unit length Reff(h) for the harmonic order h, which was defined as

$$\Im P\_{L\text{max}}(h) \; \; \; \; \; \; P\_N(h) \; \; \equiv \; \; \Im I\_{rms} \; \; \; \; \; \; R\_{\text{eff}}(h) \; \; \; \; \; \; \; \tag{13}$$

where

IN ¼ 3Ip cos 2ð Þ πhf t þ π : (12)

) submitted to a 30 A (peak value)

In order to obtain an accurate distribution of the current density over the conductor sections, it was checked that the size of the local numerical mesh was less than half the characteristic skin

64 Light-Emitting Diode - An Outlook On the Empirical Features and Its Recent Technological Advancements

The study domain for the cases of sector-shaped and four single-core cables, showing the nonuniform numerical grid (with up to about 4000 mesh cells), is presented in Figure 5. At the boundary of the domain (at a radius up to ten times the cable size), it was assumed that A ¼ 0 because the magnetic field vanishes at a large distance (as compared to the cable size)

The simulation results were obtained for <sup>μ</sup> = 4 <sup>π</sup> � <sup>10</sup>�<sup>7</sup> H/m (non-magnetic material was considered). The copper electrical conductivity at 293 K was taken as <sup>σ</sup> = 5.8 � 107 <sup>Ω</sup>�<sup>1</sup> <sup>m</sup>�<sup>1</sup> according to IEC 60028 [40]. The σ value was correspondingly corrected for other cable

Figure 6 illustrates the spatial distribution of the root-mean-square (rms) value of the total current density over the conductors of cables of large sections submitted to harmonic currents of different frequencies. Figure 6(a) corresponds to a 3 � 240 + 120 mm<sup>2</sup> sector-shaped cable submitted to a 15 A (peak value) fifth-order harmonic current, while Figure 6(b) corresponds

third-order harmonic current. The magnetic field lines produced by the current are also shown in Figure 6. A noticeable reduction in the effective area of current circulation due to the skin and proximity effects is observed in Figure 6(a), thus causing a considerable increase of the ac conductor resistance (Rac) as compared to the dc resistance, (Rdc), which in turn results in high heat losses. In Figure 6(b) the neutral conductor is assumed to carry the algebraic sum of

Figure 5. Non-uniform numerical grid generated by the software for the case of a four-core sector-shaped cable (a) and

penetration length for each harmonic order.

to an arrangement of four single-core cables (3 � 70 + 35 mm<sup>2</sup>

from the cable.

operating temperatures.

the phase currents.

four single-core cables (b).

$$P\_{L\max}(\hbar) \equiv \max\{P\_{L1}(\hbar), P\_{L2}(\hbar), P\_{L3}(\hbar)\}.\tag{14}$$

When triplen harmonics are present, the neutral conductor picks up the current. An effective resistance that reflects the maximum losses of the phase conductors (R~effð Þ<sup>h</sup> ) and another resistance (Rac Nð Þh ) that reflects the losses of the neutral conductor were defined as

$$\Im \mathcal{B} P\_{L\text{max}}(\hbar) \equiv \Im I\_{rms}^2(\hbar) \tilde{R}\_{\text{eff}}(\hbar) \tag{15}$$

$$P\_N(h) \equiv \begin{array}{c} I\_{rms} \text{N}^2(h) R\_{\text{acN}}(h) \end{array} \tag{16}$$

where

$$I\_{rms}(h) \quad \equiv \text{ 3} \quad I\_{rms}(h), \tag{17}$$

The cable losses can be approximately calculated by the following formula:

ð Þ<sup>h</sup> RacðÞ þ <sup>h</sup> <sup>X</sup>

where the first term on the right-hand side represents the losses in the phase conductors and the second term is the losses in the neutral conductor. This second term is present only when triplen harmonics are considered (i.e. h = 3, 9, 15, 21, 29). The values of Racð Þh and Rac Nð Þh were shown in Figures 7(a) and (b), respectively. It is important to compare the above calculated cable losses (Eq. (18)) with the losses produced in an identical cable but carrying an undistorted electric current of a rms value of Irmsð Þ1 . To do this, the cable losses ratio defined as

> <sup>ξ</sup> � Ploss 3Irms<sup>2</sup>

assumption on the m value leads to results that are on the conservative side.

mainly due to the harmonic content of the distorted current.

were calculated by using the harmonic signature given by Eq. (1) for the cables described in Table 1 and for a 3 � 240 + 120 mm<sup>2</sup> four-core cable as was specified by CENELEC Standard HD603 [38]. The results obtained for the upper bound of m (= � 1.0) are shown in Table 2. The

As it is observed in Table 2, for a four-core cable with a cross section of 3 x 240 + 120 mm<sup>2</sup>

power losses reach 2.5 times the value corresponding to an undistorted current of the same rms value of the first harmonic of the LED current. Even for cables with relatively small cross

proximity effects are neglected in the cable losses given by Eq. (18) (the conductor radius is small as compared to the characteristic skin penetration length and the distances of the nearby conductors are large as compared to the conductor radius) and thus ξ is not dependent on the cable cross section, the loss ratio still reaches 2.0 for m = � 1.0. The increase in the losses is

As shown Table 2, large LED-like loads generate huge harmonic losses resulting in additional conductor heating. This heating will result in a higher-temperature rise of the cable which can

exceed its rated temperature, thus requiring the derating of the cable ampacity.

Arrangement of four single-core cables 3 � 35 + 16 2.1

Four-core cable 3 � 240 + 120 2.5

Table 2. Calculated cable loss ratio ξ of various examined PVC-insulated low-voltage cables feeding LED-type loads.

Cable type Nominal cable cross section [mm<sup>2</sup>

27

ð Þ <sup>3</sup>Irmsð Þ<sup>h</sup> <sup>2</sup>

The Impact of the Use of Large Non-Linear Lighting Loads in Low-Voltage Networks

ð Þ<sup>1</sup> Racð Þ<sup>1</sup> , (19)

] ξ

, this ratio reaches about 2.1. Furthermore, if the skin and

3 � 70 + 35 2.1 3 � 120 + 70 2.0 3 � 240 + 120 2.3

Rac Nð Þh , (18)

http://dx.doi.org/10.5772/intechopen.76752

, the

67

h¼3N

Irms<sup>2</sup>

Ploss ¼ 3

sections, such as 3 � 35 + 16 mm<sup>2</sup>

X 29

h¼1

is the neutral conductor current for the harmonic current of order h. The resistances Reffð Þh and <sup>R</sup>~effð Þ<sup>h</sup> will be referred, from now on, as Racð Þ<sup>h</sup> . The ratio Racð Þ<sup>h</sup> <sup>=</sup>Rdc for the phase and neutral conductors of the cables described in Table 1 and for a 3 � 240 + 120 mm<sup>2</sup> four-core cable are shown in Figures 7(a) and (b), respectively. As expected, the ratio Racð Þh =Rdc for the phase conductors increases with both frequency and conductor cross section due to skin and proximity effects. The curve is not smooth but presents spikes at triplen harmonics.

This is mainly due to the increased losses in conductors L1 and L3 when zero sequence currents flow in the phase conductors and thereby in the neutral conductors. The ratio Racð Þh =Rdc for the neutral conductor is shown only for triplen harmonics, because only when triplen harmonics are present the neutral conductor picks up the current (other than eddy currents). It is evident from Figures 7(a) and (b) that the ratio of the neutral conductor is much smaller than that of the respective phase conductors. This occurs because the zero sequence currents decrease the proximity effect significantly on the neutral conductor when its position, relative to the phase conductors, is as shown in Figure 4.

The simulation results corresponded to a conductor operating temperature of 343 K, which is the maximum conductor temperature for PVC-insulated cables according to IEC 60502–1 [12]. It was checked that large variations in this temperature value (in the range 283 to 343 K) only render slightly variations (less than 10%) in the conductor resistance ratio. The results of the employed electromagnetic model were validated by comparison to (a) the numerical model developed in [27] and (b) the formulae given in the standard IEC 60287–1-1 [42] for thee singlecore cable arrangements. The differences in the calculated ratios Racð Þh =Rdc were in both cases less than 3% in the whole considered frequency range.

Figure 7. Variation with the harmonic order of the ratio Racð Þh =Rdc for various examined cables, (a) for the phase conductors and (b) for the neutral conductor.

The cable losses can be approximately calculated by the following formula:

where

Irms NðÞ � h 3 Irmsð Þh , (17)

is the neutral conductor current for the harmonic current of order h. The resistances Reffð Þh and <sup>R</sup>~effð Þ<sup>h</sup> will be referred, from now on, as Racð Þ<sup>h</sup> . The ratio Racð Þ<sup>h</sup> <sup>=</sup>Rdc for the phase and neutral conductors of the cables described in Table 1 and for a 3 � 240 + 120 mm<sup>2</sup> four-core cable are shown in Figures 7(a) and (b), respectively. As expected, the ratio Racð Þh =Rdc for the phase conductors increases with both frequency and conductor cross section due to skin and prox-

This is mainly due to the increased losses in conductors L1 and L3 when zero sequence currents flow in the phase conductors and thereby in the neutral conductors. The ratio Racð Þh =Rdc for the neutral conductor is shown only for triplen harmonics, because only when triplen harmonics are present the neutral conductor picks up the current (other than eddy currents). It is evident from Figures 7(a) and (b) that the ratio of the neutral conductor is much smaller than that of the respective phase conductors. This occurs because the zero sequence currents decrease the proximity effect significantly on the neutral conductor when its position,

The simulation results corresponded to a conductor operating temperature of 343 K, which is the maximum conductor temperature for PVC-insulated cables according to IEC 60502–1 [12]. It was checked that large variations in this temperature value (in the range 283 to 343 K) only render slightly variations (less than 10%) in the conductor resistance ratio. The results of the employed electromagnetic model were validated by comparison to (a) the numerical model developed in [27] and (b) the formulae given in the standard IEC 60287–1-1 [42] for thee singlecore cable arrangements. The differences in the calculated ratios Racð Þh =Rdc were in both cases

Figure 7. Variation with the harmonic order of the ratio Racð Þh =Rdc for various examined cables, (a) for the phase

imity effects. The curve is not smooth but presents spikes at triplen harmonics.

66 Light-Emitting Diode - An Outlook On the Empirical Features and Its Recent Technological Advancements

relative to the phase conductors, is as shown in Figure 4.

less than 3% in the whole considered frequency range.

conductors and (b) for the neutral conductor.

$$P\_{\rm loss} = \Im \sum\_{h=1}^{29} I\_{\rm rms} ^2(h) R\_{\rm at}(h) \quad + \sum\_{h=3N}^{27} (\Im I\_{\rm rms}(h)) ^2 R\_{\rm at}(h) \tag{18}$$

where the first term on the right-hand side represents the losses in the phase conductors and the second term is the losses in the neutral conductor. This second term is present only when triplen harmonics are considered (i.e. h = 3, 9, 15, 21, 29). The values of Racð Þh and Rac Nð Þh were shown in Figures 7(a) and (b), respectively. It is important to compare the above calculated cable losses (Eq. (18)) with the losses produced in an identical cable but carrying an undistorted electric current of a rms value of Irmsð Þ1 . To do this, the cable losses ratio defined as

$$\xi \triangleq \frac{P\_{\text{loss}}}{\Im I\_{rms}^2(1)\mathcal{R}\_{ac}(1)},\tag{19}$$

were calculated by using the harmonic signature given by Eq. (1) for the cables described in Table 1 and for a 3 � 240 + 120 mm<sup>2</sup> four-core cable as was specified by CENELEC Standard HD603 [38]. The results obtained for the upper bound of m (= � 1.0) are shown in Table 2. The assumption on the m value leads to results that are on the conservative side.

As it is observed in Table 2, for a four-core cable with a cross section of 3 x 240 + 120 mm<sup>2</sup> , the power losses reach 2.5 times the value corresponding to an undistorted current of the same rms value of the first harmonic of the LED current. Even for cables with relatively small cross sections, such as 3 � 35 + 16 mm<sup>2</sup> , this ratio reaches about 2.1. Furthermore, if the skin and proximity effects are neglected in the cable losses given by Eq. (18) (the conductor radius is small as compared to the characteristic skin penetration length and the distances of the nearby conductors are large as compared to the conductor radius) and thus ξ is not dependent on the cable cross section, the loss ratio still reaches 2.0 for m = � 1.0. The increase in the losses is mainly due to the harmonic content of the distorted current.

As shown Table 2, large LED-like loads generate huge harmonic losses resulting in additional conductor heating. This heating will result in a higher-temperature rise of the cable which can exceed its rated temperature, thus requiring the derating of the cable ampacity.


Table 2. Calculated cable loss ratio ξ of various examined PVC-insulated low-voltage cables feeding LED-type loads.
