# Determination of the Operating Temperature of the Gas-Insulated Transmission Line +
## 2 Magneto-Thermal Modeling  
Let us consider the operating temperature of the three-phase high-current busduct presented in Figure 1.  
At a steady state, the heat generated and dissipated in the three-phase high-current busduct satisfies the following equations [12; 13; 28]:  
\[P=Q_{C}+Q_{R} \tag{1}\]  
\[P+P_{e}=Q_{cC}+Q_{eR} \tag{2}\]  
where \(P\) and \(P_{e}\) are the power losses in the phase conductor and in the enclosure, respectively, \(Q_{C}\) and \(Q_{cC}\) are the convection heat transfer of the phase conductor and the enclosure, respectively, and \(Q_{R}\) and \(Q_{eR}\) are the radiation heat transfer of the phase conductor and the enclosure, respectively. For the phase conductor, the convection and radiation heat transfer are [12; 13; 28]  
\[Q_{C}=\frac{T-T_{o}}{\frac{1}{2\pi\alpha R_{2}}+\frac{1}{2\pi\lambda}\ln\frac{ R_{3}}{R_{2}}+\frac{1}{2\pi\alpha R_{3}}} \tag{3}\]  
\[Q_{R}=2\sigma_{o}\varepsilon_{n}\pi R_{2}\bigg{(}\bigg{(}\frac{T}{100}\bigg{)} ^{4}-\bigg{(}\frac{T_{e}}{100}\bigg{)}^{4}\bigg{)} \tag{4}\]  
where \(T\) and \(T_{e}\) are the temperatures of the phase conductor and the enclosure, respectively, \(T_{o}\) is the ambient temperature, \(\lambda\) is the thermal conductivity of the phase conductor, \(\alpha\) is the convective heat transfer coefficient of the phase conductor, \(\sigma_{o}=5.67\cdot 10^{-8}\bigg{[}\frac{W}{m^{2}\cdot 16^{4}}\bigg{]}\) is the Stefan-Boltzmann constant and \(\varepsilon_{n}\) is emissivity, which can be calculated from the following equation [12]:  
\[\varepsilon_{n}=\frac{1}{\frac{1}{\varepsilon}+\frac{R_{2}}{R_{3}}\Big{(} \frac{1}{\varepsilon_{e}}-1\Big{)}} \tag{5}\]  
in which \(\varepsilon\) is an emissivity of the phase conductor and \(\varepsilon_{e}\) is an emissivity of the enclosure. The convection and radiation heat transfer of the enclosure is [28]  
\[Q_{cC}=2\pi\alpha_{e}R_{4}(T_{e}-T_{o}) \tag{6}\]  
\[Q_{eR}=2\sigma_{o}\varepsilon_{e}\pi R_{4}\bigg{(}\bigg{(}\frac{T_{e}}{100} \bigg{)}^{4}-\bigg{(}\frac{T_{o}}{100}\bigg{)}^{4}\bigg{)} \tag{7}\]  
where \(\alpha_{e}\) is the convective heat transfer coefficient of the enclosure. Power losses in the conductors of the single-pole high-current busduct presented in Figure 1 are expressed by formula [29]  
\[P=\frac{\Gamma\;l\;l_{1}^{2}}{4\;\pi\;\gamma\;R_{2}}\;\;\;\frac{a}{b\;b^{*}} \tag{8}\]  
Figure 1: Three-phase high-current busduct with isolated phases. where  
\[P=Q_{C}+Q_{R} \tag{1}\]0  
\[P=Q_{C}+Q_{R} \tag{1}\]1  
and  
\[ \begin{array}{ll}a&=K_{1}\!\left(\Gamma R_{1}\right)K_{1}^{*}\!\left(\Gamma R _{1}\right)\left[I_{0}\!\left(\Gamma R_{2}\right)I_{1}^{*}\!\left(\Gamma R_{2 }\right)-\mathrm{j}\ I_{1}\!\left(\Gamma R_{2}\right)I_{0}^{*}\!\left(\Gamma R _{2}\right)\right]\\ &-I_{1}\!\left(\Gamma R_{1}\right)I_{1}^{*}\!\left(\Gamma R_{1}\right)\!\left[ K_{0}\!\left(\Gamma R_{2}\right)K_{1}^{*}\!\left(\Gamma R_{2}\right)-\mathrm{j}\ K_{1}\!\left(\Gamma R _{2}\right)K_{0}^{*}\!\left(\Gamma R_{2}\right)\right]\\ &+I_{1}\!\left(\Gamma R_{1}\right)K_{1}^{*}\!\left(\Gamma R_{1}\right)\!\left[ K_{0}\!\left(\Gamma R_{2}\right)I_{1}^{*}\!\left(\Gamma R_{2}\right)+\mathrm{j}\ I_{1}\!\left(\Gamma R_{2}\right)I_{0}^{*}\!\left( \Gamma R_{2}\right)\right]\\ &-K_{1}\!\left(\Gamma R_{1}\right)I_{1}^{*}\!\left(\Gamma R_{1}\right)\!\left[ I_{0}\!\left(\Gamma R_{2}\right)K_{1}^{*}\!\left(\Gamma R_{2}\right)+\mathrm{j}\ I_{1}\!\left(\Gamma R_{2} \right)K_{0}^{*}\!\left(\Gamma R_{2}\right)\right]\end{array}