## 3 Extrapolation Formula  
An extrapolation of the temperature measured when the inspection was performed must be performed to identify its real importance. It is enough to focus on the factors that have the most influence and to which the electricity companies do not give standard values for all their facilities. It should not be forgotten that electricity companies set common values for all their facilities and thus do not have to change them from one substation to another or within the same substation. Furthermore, those values and the results obtained are always on the side of security. These factors are basically two: on the one hand, the temperature detected with the current measured when carrying out the inspection must be extrapolated to what it would reach when the nominal current circulated; on the other hand, the same must be done when instead of existing wind speed at the time of the inspection, its speed was zero.  
The extrapolation to the nominal current is done by the following equation [36]:  
\[\frac{\Delta T_{R}}{\Delta T_{N}}=1.0537\frac{I_{R}}{I_{N}}-0.055 \tag{1}\]  
where \(\Delta T_{R}\) is the temperature increase of the hot spot with respect to the ambient temperature when the measurement was made; \(\Delta T_{N}\) is the temperature increase of the hot spot with respect to the ambient temperature under nominal current; \(I_{R}\) is the circulating current at the time of inspection; and \(I_{N}\) is the nominal current.  
Hence, the increase in temperature due to the fact that the nominal current circulates through the circuit is:  
\[\Delta T_{N}=\frac{\Delta T_{R}}{1.0537\frac{I_{R}}{I_{N}}-0.055} \tag{2}\]  
A theoretical measure of the temperature obtained in the field is presented in [37]:  
\[\Delta\theta=4.906\cdot 10^{-5}\cdot I^{2}-1.241\cdot X_{WS}+6.626\cdot 10^{- 3}\cdot X_{SR}-1.428\cdot X_{R}+13.079 \tag{3}\]  
where \(\Delta\theta\) is the measured temperature of the hot spot, \(I\) the current at the time of the measurement, \(X_{WS}\) the wind speed, \(X_{SR}\) the solar radiation and \(X_{R}\) the rain. In this way, the temperature of a point in the electrical circuit is calculated. However, of the weather variables present in the equation, only the wind speed is applicable in a substation because, for safety reasons, maintenance in a substation cannot be carried out in the presence of rain. Solar radiation is similar between substations in the same area and is usually defined in the standards of electric companies. Furthermore, it is the current through the circuit that primarily sets the temperature that the hot spot reaches. This can also be seen from the coefficient corresponding to solar radiation in Equation (3).  
With respect to the influence of the current, it should not be forgotten that the objective is to find the temperature that a hot spot would reach under nominal conditions, which is shown with Equation (2). However, Equation (3) shows the temperature of a point that is crossed by a current \(I\). That is why the influence of \(I\) in Equation (3) is not considered and, on the other hand, it is considered in Equation (2). Something similar happens with the added constant in Equation (3). It becomes necessary for the objective intended by that equation, but not for what is sought in this work.  
Therefore, using Equation (3), the extrapolation at zero wind speed is defined by the following equation:  
\[\Delta T_{V}=1.241\ v \tag{4}\]  
where \(\Delta T_{V}\) is the increase in temperature of the hot spot due to the absence of wind; and \(v\) is the wind speed when the inspection was carried out.  
Therefore, considering the most unfavorable load and wind conditions, the increase in the temperature of the hot spot over the ambient air temperature is obtained as follows [38]:  
\[\Delta T=\frac{\Delta T_{R}}{1.0537\frac{l_{R}}{l_{N}}-0.055}+1.241\ v \tag{5}\]  
where \(\Delta T\) is the temperature increase above the ambient air temperature.  
Obtained this increment from Equation (5), the recommended action is determined from Table 1.