### Transient thermal model of cable installed direct in the soil

The cable layers are performed by lumped circuit thermoelectric equivalent method (TEE) according to IEC 60,853-2 [27,28,29,30,31, 30, 31]. As shown in Fig. 7a, the thermal analysis at each node of the thermal model of distribution cable directly buried in the soil.

The thermal circuit contains on three thermal capacitances. \({Q}_{1}\), \({Q}_{3}\), and \({Q}_{4}\) are calculated by using IEC 60,853-2 [27]:

$$ Q_{1} = Q_{c} + \rho .Q_{i} $$

(1)

$$ Q_{3} = \left( {1 - \rho } \right).Q_{i} + Q_{s} + Q_{j} $$

(2)

$$ Q_{4} = Q_{{{\text{soil}}}} $$

(3)

where \({Q}_{soil}\),\({Q}_{s}\),\({Q}_{j}\),\({Q}_{i}\) and \({Q}_{c}\) are the thermal capacitances of surrounding soil, screen, jacket, insulation and conductor, respectively.\({T}_{4}\), \({T}_{1}\) and \({T}_{3}\) are the thermal resistances of surrounding soil, insulation and jacket, respectively. The thermal resistances of the metallic layers are ignored.\({\theta }_{e}\), \({\theta }_{s}\) and \({\theta }_{c}\) are the jacket, screen and conductor temperatures above the surrounding temperature \({(\theta }_{a})\), respectively. Finally \(\rho \) is the coefficient of Van Wormer [27].The cable losses of the conductor (*W*_{c}) are produced by the resistance of the conductor. Screen losses (*W*_{s}) are due to circulating current flowing in the cable sheath. The insulation losses (*W*_{d1}) and (*W*_{d2}) are dependent on the insulation material type and ignored for low-voltage cables. The different losses of the cable component are calculated according to IEC 60,287-1-3 [1].

$$ W_{c} = I^{2} .R_{ac} $$

(4)

$$ W_{s} = W_{c} .\lambda_{1} $$

(5)

$$ W_{d} = 2\pi .f.C.V_{o}^{2} .\tan \left( \sigma \right) $$

(6)

where *I*, *R*_{ac} and \({\uplambda }_{1}\) are load current, conductor electrical resistance and sheath loss factor, respectively. *V*_{o}, \( f\), \(C\) and tan (\(\sigma\)) are the phase voltage, system frequency (Hz), the electrical capacitance and the insulation loss factor, respectively. The thermal capacitances and resistances of the cable in each part and the around soil are calculated as given in [1, 27].

$$ Q_{c} = C_{pc} .A_{c} $$

(7)

$$ Q_{i} = \frac{\pi }{4}\left( {D_{i}^{2} - d_{c}^{2} } \right)C_{pi} $$

(8)

$$ Q_{s} = \frac{\pi }{4}\left( {D_{s}^{2} - D_{i}^{2} } \right)C_{ps} $$

(9)

$$ Q_{j} = \frac{\pi }{4}\left( {D_{e}^{2} - D_{s}^{2} } \right)C_{pj} $$

(10)

$$ Q_{{{\text{soil}}}} = \pi \left( {L^{2} - \left( {\frac{{D_{e} }}{2}} \right)^{2} } \right)C_{{{\text{psoil}}}} $$

(11)

$$ T_{1} = \frac{{\rho_{i} }}{2\pi }\ln \left( {\frac{{D_{i} }}{{d_{c} }}} \right) $$

(12)

$$ T_{3} = \frac{{\rho_{j} }}{2\pi }\ln \left( {\frac{{D_{e} }}{{D_{s} }}} \right) $$

(13)

$$ T_{4} = \frac{{\rho_{{{\text{soil}}}} }}{2\pi }\left\{ {\ln \left( {\frac{4L}{{D_{e} }}} \right) + \ln \left( {1 + \left( \frac{2L}{S} \right)^{2} } \right)} \right\} $$

(14)

where *d*_{c}, \(D_{i}\), \(D_{{\text{s}}}\) and \(D_{{\text{e}}}\) are the external diameter of the conductor, insulation, screen and the cable surface, respectively. \(\rho_{i}\), \(\rho_{j}\) and \(\rho_{{{\text{soil}}}}\) are the thermal resistivity of the cable different parts and soil around the cable. \(C_{{{\text{pc}}}}\), \(C_{{{\text{pi}}}}\), \(C_{{{\text{ps}}}}\), \(C_{{{\text{pj}}}}\) and \(C_{{{\text{psoil}}}}\) are volumetric specific heat of each cable elements material and its surrounding soil. L, S and A_{c} are burial depth, the distance between conductor axes of the cables in case of flat formation and the area of the conductor, respectively. The current sources in the thermal model represent the heat sources in the metallic elements inside the cable.

### Transient thermal model of cable installed inside PVC duct

To study the cable installed in the PVC duct, the different components such as the air and the duct medium are added to the thermal circuit as given in Fig. 7b. The thermal analysis at each node is represented in the following equations.

$$ \theta^{\prime}_{c} = \frac{1}{{Q_{1} }}.\left( {W_{c} + W_{d1} - \frac{{\theta_{c} - \theta_{s} }}{{T_{1} }}} \right) $$

(15)

$$ \theta^{\prime}_{s} = \frac{1}{{Q_{3} }}.\left( {W_{s} + W_{d2} + \frac{{\theta_{c} - \theta_{s} }}{{T_{1} }} - \frac{{\theta_{s} - \theta_{e} }}{{T_{3} }}} \right) $$

(16)

$$ \theta^{\prime}_{j} = \frac{1}{{Q_{{{\text{air}}}} }}.\left( {\frac{{\theta_{s} - \theta_{j} }}{{T_{3} }} - \frac{{\theta_{j} - \theta_{{{\text{air}}}} }}{{T^{\prime}_{4} }}} \right) $$

(17)

$$ \theta^{\prime}_{{{\text{air}}}} = \frac{1}{{Q_{{{\text{duct}}}} }}.\left( {\frac{{\theta_{j} - \theta_{{{\text{air}}}} }}{{T^{\prime}_{4} }} - \frac{{\theta_{{{\text{air}}}} - \theta_{{{\text{duct}}}} }}{{T^{\prime\prime}_{4} }}} \right) $$

(18)

$$ \theta^{\prime}_{{{\text{duct}}}} = \frac{1}{{Q_{4} }}.\left( {\frac{{\theta_{s} - \theta_{j} }}{{T^{\prime\prime}_{4} }} - \frac{{\theta_{j} - \theta_{{{\text{air}}}} }}{{T^{\prime\prime\prime}_{4} }}} \right) $$

(19)

where \(Q_{{{\text{air}}}}\) and \(Q_{{{\text{duct}}}}\) are the thermal capacitances of air in the space between duct inner surface and outer surface of cable jacket and the cable duct, respectively. \(T_{4}^{\prime }\) is the air thermal resistance between the inner surface of the duct and the cable jacket. The thermal resistance of the soil around the duct is defined as \(T_{4}^{\prime \prime \prime }\). \(T_{4}^{\prime \prime }\) indicates to the duct thermal resistance. The thermal resistances of the metallic layers are ignored.\(\theta_{{{\text{duct}}}}\) and \(\theta_{{{\text{air}}}}\) are the duct and air temperatures \(^\circ{\rm C} \) above surrounding temperature, respectively. The additional thermal capacitances and resistances of cable in each part and the around soil are calculated as given in [1, 28].

$$ Q_{{{\text{air}}}} = \frac{\pi }{4}\left( {D_{di}^{2} - D_{e}^{2} } \right)C_{{{\text{pair}}}} $$

(20)

$$ Q_{{{\text{duct}}}} = \frac{\pi }{4}\left( {D_{{{\text{do}}}}^{2} - D_{{{\text{di}}}}^{2} } \right)C_{{{\text{pduct}}}} $$

(21)

$$ \rho = \frac{1}{{2\ln \left( {\frac{{D_{i} }}{{d_{c} }}} \right)}} - \frac{1}{{\left( {\frac{{D_{i} }}{{d_{c} }}} \right)^{2} - 1}} $$

(22)

$$ T_{4} = T^{\prime}_{4} + T^{\prime\prime}_{4} + T^{\prime\prime\prime}_{4} $$

(23)

$$ T^{\prime}_{4} = \frac{U}{{1 + 0.1\left( {V + Y\theta_{{{\text{duct}}}} } \right)D_{e} }} $$

(24)

$$ T^{\prime\prime}_{4} = \frac{{\rho_{d} }}{2\pi }\ln \left( {\frac{{D_{{{\text{do}}}} }}{{D_{{{\text{di}}}} }}} \right) $$

(25)

$$ T^{\prime\prime\prime}_{4} = \frac{{\rho_{{{\text{soil}}}} }}{2\pi }\left\{ {\ln \left( {\frac{4L}{{D_{do} }}} \right) + \ln \left( {1 + \left( \frac{2L}{S} \right)^{2} } \right)} \right\} $$

(26)

where \(D_{{{\text{do}}}}\) and \(D_{{{\text{di}}}}\) are the PVC duct and inner PVC duct, respectively \(.\) The thermal resistivity of the PVC duct is defined as\(\rho_{d}\). The V, Y and U are constants according to IEC 60,287 [1] in case of the cables installed in duct and buried in the soil.\(C_{{{\text{pduct}}}}\) and \(C_{{{\text{pair}}}}\) are the volumetric specific heat of duct and air inside the duct.

### The effect of harmonic distortion on the distribution cable losses

In this study, four-core cables with and without harmonic current effects are investigated. The fundamental frequency and harmonic order frequencies are considered in transient harmonic analysis. Taking into account harmonic current at each frequency, total harmonic distortion current losses are estimated. Odd orders of harmonic current are determined. The conductor resistance (Ω/m) without harmonic effects according to IEC-60287-1 is illustrated as follows [1].

$$ R_{ac} = R_{dc} \left( {1 + Y_{s} + Y_{p} } \right) $$

(27)

$$ Y_{s} = \frac{{x_{s}^{4} }}{{192 + 0.8 x_{s}^{4} }} $$

(28)

$$ Y_{p} = \left[ {\frac{{x_{p}^{4} }}{{192 + 0.8x_{p}^{4} }}} \right].\left( {\frac{{d_{c} }}{S}} \right)^{2} .\left[ {0.312.\left( {\frac{{d_{c} }}{S}} \right)^{2} \left( {\frac{1.18}{{\frac{{x_{p}^{4} }}{{192 + 0.8x_{p}^{4} }} + 0.27}}} \right)} \right] $$

(29)

where \(R_{dc}\) denotes the resistance of the DC conductor at maximum operating temperature (θ). \({Y}_{p}\) and \({Y}_{s}\) are the proximity and skin effect factors can be used to calculate the effects of harmonic current orders on the increase of temperature in distribution cable layers. \({x}_{p}\) and \({x}_{s}\) equations are given in [1].

Due to increasing harmonic current orders, this will cause an increase in the R_{ac} as a consequence dependent on proximity and skin factors. These factors are dependent on frequency change as follows [1, 29].

$$ x = 0.01528 \sqrt {\frac{f.\mu }{{R_{dc} }}} $$

(30)

$$ Y_{s} = 10^{ - 3} \left( {\begin{array}{*{20}c} { - 1.04 x^{5} + 8.24 x^{4} } \\ { - 3.24 x^{3} + 1.447 x^{2} } \\ { - 0.2764 x + 0.0166} \\ \end{array} } \right) , if\,\,x \le 2 $$

(31)

$$ Y_{s} = 10^{ - 3} \left( {\begin{array}{*{20}c} { - 0.2 x^{5} + 6.616 x^{4} } \\ { - 83.343 x^{3} + 500 x^{2} } \\ { - 1061.9 x + 769.63} \\ \end{array} } \right) , if \,\,2 < x \le 100 $$

(32)

$$ Y_{p} = Y_{s} .\left( {\frac{{d_{c} }}{S}} \right)^{2} .\left( {\frac{1.18}{{Y_{s} + 0.27}} + 0.312.\left( {\frac{{d_{c} }}{S}} \right)^{2} } \right) $$

(33)

where \(x \) and \(\mu\) are the skin parameters and magnetic permeability, respectively. Consequently, for a non-sinusoidal current as a result of harmonic orders, the \({\mathrm{I}}_{\mathrm{rms}}\) is root-mean-square current can be defined in relation to harmonic order (*h)* as given in (34).

The losses in distribution cables depend not only on the total harmonic distortion (THD), but also on the magnitude of each harmonic order. In [4], IEEE Standard 519 recommends a limit on both of them. The THD is used to express the effect of harmonic currents on the distribution cable. It is a percentage, as shown in (35).

$$ I_{rms} = \sqrt {I_{1}^{2} + \mathop \sum \limits_{h = 2}^{\infty } I_{h}^{2} } = I_{1} \sqrt {1 + \left( {THD} \right)^{2} } $$

(34)

$$ THD = \frac{{\sqrt {\mathop \sum \nolimits_{h = 2}^{\infty } I_{h}^{2} } }}{{I_{1} }} $$

(35)

In the case of odd harmonic current orders, the conductor power loss \(W_{c}\) can be calculated at each frequency and summed by:

$$ W_{c} = \left( {I_{1} } \right)^{2} .\left( {R_{ac\left( 1 \right)} + \mathop \sum \limits_{h = 3}^{\infty } H_{h}^{2} .R_{ac\left( h \right)} } \right) $$

(36)

where \( {\text{I}}_{1}\), \({\text{I}}_{{\text{h}}}\) and \({\text{ R}}_{{{\text{ac}}\left( 1 \right)}}\) are the fundamental current component harmonic current and AC conductor resistance at fundamental frequency in case of non-sinusoidal waveform, respectively. \({H}_{h}\) is the percentage harmonic load current. \({R}_{ac(h)}\) is the conductor resistance of harmonic frequency.

In this article, the current harmonic distortion for the distribution cable system is taken into account and all the odd order harmonic orders up to the 23^{rd} are studied. All the higher orders, even zero harmonic orders, are neglected.