An Equivalent Heat Transfer Model Instead of Wind Speed Measuring for Dynamic Thermal Rating of Transmission Lines

: With the increase in electricity demand, the ampacity calculation based on the dynamic thermal rating (DTR) technology is increasingly signiﬁcant for assessing and improving the power transfer capacity of the existing overhead conductors. However, the DTR models now available present some inadequacies in measurement techniques related to wind speed. Therefore, it is essential to propose a new model instead of wind speed measuring in DTR technology. In this paper, the inﬂuence analysis of various weather parameters on the conductor ampacity is carried out by using the real weather data. Based on the analysis, it is conﬁrmed that the impact of wind speed is signiﬁcant, especially in the case of the low wind speed. Moreover, an equivalent heat transfer (EHT) model for DTR technology is proposed instead of wind speed measuring. For this EHT model, the calculation of conductor ampacity is realized through investigating the correlation of heat losses between the heating aluminum (Al) ball and conductor. Finally, combined with the ﬁnite element method (FEM), the EHT model proposed in this paper is veriﬁed by the Institute of Electrical and Electronic Engineers (IEEE) standard. The results indicate that the error of the EHT model is less than 6% when employing the steady thermal behavior of the Al ball to calculate the ampacity. The EHT model is useful in the real-time thermal rating of overhead conductors. It can increase the utilization of overhead conductors while also avoiding the limitation of the existing measurement techniques related to wind speed.


Introduction
With the rapid increase in electricity demand, it becomes a huge challenge for utilities to improve the power transfer capacity of transmission lines. To achieve this, the direct solutions are establishing new transmission lines or replacing overhead conductors which have greater power transfer capacity. However, there are some restrictions in these solutions, including the difficulties of land expropriation, long construction time and high costs [1,2]. Therefore, it is increasingly essential to maximize the utilization for power transfer capacity of the existing overhead conductors [3,4]. The economic benefits of power grid operation can also be increased [5]. In the field of power transmission, the power transfer capacity of overhead conductors can be characterized by the ampacity, which is the maximum current that the overhead conductor can carry yet within its allowable maximum temperature. The accurate implemented in the field and avoid the limitations of the available measurement techniques related to wind speed.

Calculation of Overhead Conductor Ampacity
For the existing DTR model, conductor ampacity is calculated based on the data collected by the sensors. IEEE standard (IEEE Std 738-2012), CIGRE standard and Morgan formula all specify the calculation methods of conductor ampacity [32][33][34]. These methods are all based on the steady-state heat balance equation of conductor. Once the maximum allowable temperature of conductor is determined, the conductor ampacity can be calculated with the steady-state heat balance equation. When the conductor temperature reaches steady state, the steady-state heat balance equation of conductor is shown in (1). q c + q r = q s + I 2 R(T c ) (1) where q c , q r and q s denote the convective heat loss rate, radiative heat loss rate and solar heat gain rate of conductor per unit length, respectively, I is the conductor current and R(T c ) is the conductor resistance at the temperature of T c . According to the IEEE standard, R(T c ) is the function related to the conductor temperature. When the maximum allowable temperature of conductor T cmax is given, the calculation formula of the conductor ampacity I amp can be derived by (1), as shown in (2).

Convective Heat Loss Rate
The convective heat loss rate of conductor per unit length q c is mainly related to the temperature difference between conductor and the external environment, wind speed and wind direction. According to the different heat convection modes, the calculation of q c is divided into two cases including natural convection and forced convection.
In the IEEE standard, natural convection is defined as the condition of zero wind speed. For the case of natural convection, the calculation of the convective heat loss rate of conductor per unit length q cn is as follows: where ρ f is the air density, D 0 is conductor diameter, T c is the temperature of conductor surface and T a is the ambient temperature. For the case of forced convection, the calculation equations of convective heat loss are also different according to the magnitude of wind speed. Under the low wind speed, the forced convective heat loss rate of conductor per unit length q c1 can be expressed as: In the case of the high wind speed, the forced convective heat loss rate of conductor per unit length q c2 can be calculated by: where k f is the thermal conductivity of air, K angle is the wind direction factor and Re c is the Reynolds number of the conductor under the forced convection. K angle is related to the included angle ϕ between the wind direction and the axial direction of conductor. The calculation formulas for K angle and Re c are shown in (6) and (7), respectively.
where V w is the wind speed and µ f is the dynamic viscosity of air at the mean volume reference temperature [32,35]. In IEEE standard, the maximum value of the three calculation results from (3), (4) and (5) is selected as the final result of the convective heat loss rate [27], as shown in (8).

Radiative Heat Loss Rate
In addition to the heat convection, the conductor also transfers energy to the surrounding environment in the form of heat radiation. The heat radiation is related to the geometric size, surface condition, temperature of conductor and ambient temperature. The calculation of radiative heat loss rate of conductor per unit length q r is as follows.
where σ and ε are the Steff-Boltzmann constant and emissivity of the conductor surface, respectively. The value of σ is 5.67 × 10 −8 W/(m 2 ·K 4 ). For the new conductor, ε is set to 0.2~0.3, while it can exceed 0.7 for the old conductor [32,36,37].

Solar Heat Gain Rate
The heat sources of conductor also involve the solar heat gain. The solar heat gain rate of conductor is related to many factors, such as the orientation of sun and the air quality. At present, it can be calculated by two methods. The first method needs to use the geographical location (i.e., latitude data) of the conductor and time information [32], and the computational procedure of this method is complex. While for the second method, a pyranometer is employed, and the solar heat gain rate of conductor per unit length q s is calculated by (10) [34].
where α is the solar absorptivity of the conductor surface and Q s is the global solar irradiance. Generally, the value of α is equal to ε. Q s can be directly measured by the pyranometer. Compared with the first method, the second method is more accurate and its calculation procedure is simpler. However, an additional sensor is required in the second method. In this paper, the second method is chosen for the calculation of q s . In the aforementioned calculation of I amp , expressions of some parameters (such as the dynamic viscosity of air µ f , the air density ρ f , and the thermal conductivity of air k f , etc.) can be found in the IEEE standard [32].

Influence of Weather Parameters on Conductor Ampacity
According to Section 2, for a certain conductor, the calculation of I amp is related to the weather parameters. In this section, an aluminum conductor steel reinforced (ACSR) 240/30 mm 2 conductor is taken as an example, and I amp is calculated by using the real weather data from a weather station in Huizhou, Guangdong Province, China. The features of the ACSR 240/30 mm 2 can be shown in Table 1.
These calculation results under different weather conditions are compared to investigate the influence of weather parameters on I amp .

Collective Influence of Multiple Weather Parameters
The weather parameters affecting I amp include the ambient temperature T a , global solar irradiance Q s , wind speed V w and wind direction ϕ. Firstly, four groups are set according to varying ambient temperature alone, varying ambient temperature and global solar irradiance, varying ambient temperature and wind, and varying all of the weather parameters. Based on the real weather data from Huizhou, I amp under the above four groups are calculated and compared to investigate the collective influence of weather parameters. As for the weather data under the four groups, the real-time varying weather parameters are obtained from the average value of data within one hour, while the weather parameters that need to be kept constant are the average value of all the data. Besides, in the calculation of I amp , the maximum allowable temperature of conductor T cmax is set to 70 • C, and the emissivity ε and the solar absorptivity α of conductor surface are both set to 0.9. Figure 1 shows the calculation results of I amp under the above four groups. The minimum, maximum, range, mean, and standard deviation of I amp results in each group are also obtained by compiling the statistics of Figure 1, as shown in Table 2. It can be found from Figure 1 and Table 2 that the weather parameters have a great impact on the calculation results of I amp . For the two groups of varying ambient temperature alone and varying ambient temperature and global solar irradiance, the curves of calculation results show a smaller fluctuation. The corresponding range and standard deviation are also smaller. These results indicate that the ambient temperature and global solar irradiance have little influence on I amp , compared with other weather parameters. While after adding the influence of wind, the fluctuation of I amp curves becomes significantly greater (as shown in Figure 1c,d). The range and standard deviation also increase correspondingly. Therefore, the wind has a significant influence on the calculation results of I amp .

Relative Importance of Single Weather Parameter
In order to highlight the relative importance of single weather parameter, the results of I amp are calculated and compared for varying ambient temperature alone, varying global solar irradiance alone, varying wind speed alone, and varying wind direction alone, respectively. The processing of weather data and the setting of conductor parameters are consistent with those in Section 3.1. The calculated results of I amp are shown in Figure 2. Table 3 further demonstrates the minimum, maximum, range, mean, and standard deviation of I amp results in Figure 2. By comparing the calculation results in Figure 2 and Table 3, when the global solar irradiance or wind direction varies alone, the change in I amp is small. This means that the global solar irradiance and wind direction have little influence on I amp . While in the case of varying wind speed alone (i.e., Figure 2d), it can be found that the variability in I amp substantially increases. The maximum range and standard deviation are also observed when the variation of wind speed is considered. As a result, the wind speed is an especially important factor in the calculation of I amp .  To further investigate the impact of wind speed on I amp , assume the ambient temperature T a is 40 • C, the global solar irradiance Q s is 1000 W/m 2 , and the wind direction ϕ is 90 • , then the I amp results are calculated under different wind speed V w . The obtained results are shown in Figure 3 (i.e., black curve). Moreover, the variation rate of I amp with wind speed is also introduced in Figure 3 (i.e., red curve) for the clear presentation of I amp variation in different wind speed. In Figure 3, as the wind speed increases, the I amp increases, while the variation rate of I amp decreases. This indicates that the impact of wind speed diminishes gradually with the increase in wind speed. In the case of low wind speed, this impact is the most significant. According to the above analysis, the variation of weather parameters has a great impact on the calculation results of I amp . Compared with other weather parameters, the impact of wind speed is the most significant. Thus, for the existing the DTR models, the larger I amp calculation error is inevitably introduced when using the wind speed sensor for direct measurement. This reinforces the need to accurately obtain wind-induced heat transfer with the external environment (i.e., heat loss) in the calculation of I amp . To accomplish this, this paper proposes a new DTR model, called the Equivalent Heat Transfer model (EHT model), which can accurately evaluate the wind-induced heat loss without using the wind speed sensor.

Model Rationale
Based on the correlation of heat losses between the heating Al ball and conductor in the same environment, an EHT model for DTR technology is proposed instead of wind speed measuring. The detailed rationale of the EHT model can be explained combined with Figure 4. In Figure 4, a device equipped with a heating Al ball is placed near the overhead conductor. Since the Al ball and conductor are in the same environment, their weather conditions including the ambient temperature, global solar irradiance, wind direction and speed are also the same. It is thought that the heat losses in the Al ball and conductor are related. Under this circumstance, for the Al ball of the device, its heat loss can be calculated first by its thermal behavior in a certain environment. According to the correlation of heat losses between the Al ball and conductor, the heat transfer between the conductor and external environment can be obtained. Then, I amp in this environment is evaluated. For the EHT model proposed in this paper, I amp can be calculated indirectly through the thermal behavior of Al ball. The model does not need the wind speed sensor. Therefore, the inaccuracy of I amp calculation results arising from the measurement error of wind speed sensor can be avoided in the EHT model. Moreover, since the device equipped with the Al ball is not in contact with the conductor, difficulties of the device maintenance can be reduced effectively. The corresponding maintenance costs can also decrease.

Model Device
The biggest challenge for establishment of the EHT model is to obtain the correlation of heat losses between the heating Al ball and conductor. Figure 4 indicates that before determining this correlation, the thermal behavior of the Al ball needs to be obtained first, which comes from the device of the EHT model. Thus, the hardware facility of EHT model (i.e., model device) is introduced in this section.
The basic components of the model device include power supply, Environmental Monitoring System (EMS), Data Transmission System (DTS) and a solid Al ball, as shown in Figure 5. The power supply provides electrical energy to all modules and systems of the device. EMS can collect weather data in real time including ambient temperature, global solar irradiance and wind direction. The collected weather data are transmitted to remote servers through DTS for analysis and processing. A resistance wire and a contact temperature sensor are placed inside the solid Al ball. The purpose of the resistance wire is to heat the Al ball. To evenly heat the Al ball, the position of the resistance wire is set in the center of the Al ball. In addition, the power of the resistance wire will be constrained to ensure that the steady-state temperature of the Al ball does not exceed 50 • C under extreme environmental conditions. This can effectively avoid the damage of the device due to high temperature. The temperature sensor is used to collect the real-time temperature data of the Al ball. Based on the obtained real-time temperature data, the thermal behavior of Al ball can be analyzed.

Calculation for Heat Loss of Al Ball
In order to investigate the correlation of heat losses between the heating Al ball and conductor, the heat loss of Al ball needs to be calculated based on its thermal behavior. The thermal behavior can be expressed by the heat balance equation, as shown in (11). q cs + q rs + m s C ps dT s dt = q ss + q gs (11) where q cs , q rs , q ss , q gs , m s , C ps and T s are convective heat loss rate, radiative heat loss rate, solar heat gain rate, internal heat source power, mass, specific heat capacity and temperature of the Al ball, respectively, and t is time. When the differential term is equal to 0 (i.e., dT s /dt = 0), the temperature of the Al ball is thought to reach the steady state.
In (11), the calculation formulas for the radiative heat loss rate and solar heat gain rate of the Al ball are similar to those of conductor in Section 2. The only distinction in the calculation formulas is that the geometric parameters of Al ball are used. For the EHT model proposed in this paper, the radiative heat loss rate and solar heat gain rate of the Al ball and conductor can be calculated directly by the collected weather data from the model device. The heat losses of the Al ball and conductor both include radiative heat loss and convective heat loss. Therefore, the correlation analysis of heat losses between the Al ball and conductor is turned into that of convective heat losses between the Al ball and conductor. In the heat transfer theory, the convective heat loss of the Al ball can be represented by the convective heat transfer coefficient h. According to the Newton's Law of Cooling, h can be calculated by (12).
where l is the diameter of the Al ball. According to the collected real-time weather data from model device (including ambient temperature, global solar irradiance and Al ball temperature) and (11)-(12), the convective heat transfer coefficient h of the Al ball can be calculated.

Correlation Analysis of Convective Heat Losses between Al Ball and Conductor
After the convective heat transfer coefficient h of the Al ball is determined, the correlation of convective heat losses between the Al ball and conductor is established based on the characteristic numbers in the experimental correlations of heat convection. In the experimental correlations, the convective heat transfer coefficient h of the Al ball is related to a series of characteristic numbers (including Nusselt number Nu, Reynolds number Re s , Prandtl number Pr, and Grashof number Gr) [35,38]. Specifically, the relationship between Nusselt number Nu and the convective heat transfer coefficient h of the Al ball is as follows: Similarly, the heat convection of the Al ball can also be divided into the forced convection and natural convection. For these two cases of heat convection, there are differences in the expression of the Nusselt number Nu. In the case of forced convection, the Nusselt number Nu is related to Reynolds number Re s and Prandtl number Pr, and their relationship can be shown by (14) [35].
where µ w denotes the dynamic viscosity of air at average surface temperature. The Reynolds number Re s is a function of wind speed, as shown in (15). The Prandtl number Pr can be calculated by (16).
where C p is the specific heat capacity of air, which is set at 1.005 kJ/(kg·K) [39].
In the case of natural convection, the Nusselt number Nu is related to the Grashof number Gr and Prandtl number Pr, and their relationship can be expressed by (17).
The calculation formula of the Grashof number Gr is as follow: where g is the acceleration of gravity, ∆t is the temperature difference between the Al ball and environment, and t m is the characteristic temperature of the Al ball. The value of t m is the average of the Al ball temperature and ambient temperature. It can be found from the above formulas that the calculation processes for characteristic numbers of the Al ball by using laboratory correlations are different in the case of forced convection and natural convection. However, in the EHT model, the mode of heat convection (natural convection or forced convection) cannot be determined according to the wind speed. Because the wind speed sensor is not applied. In this case, based on the determined convective heat transfer coefficient h of the Al ball, the calculation result of the characteristic numbers of the Al ball is not unique. To solve this problem, the convection heat transfer coefficients of the Al ball under natural convection and forced convection are compared and analyzed in the following.
Under the conditions of the same ambient temperature and Al ball temperature, the forced convection heat transfer coefficients h f with different wind speeds and the natural convection heat transfer coefficient h 0 are firstly calculated according to (13), (14) and (17). Meanwhile, a variable err c is introduced to compare the difference between h 0 and h f , which is calculated by (19). The calculated err c results are shown in Figure 6. In Figure 6, the err c results decrease first and then increase with the increase in wind speed. For any ambient temperature and Al ball temperature, there is a minimum value without exceeding 1.5% in the err c results. This indicates that there is always a specific wind speed V wf at any ambient temperature and Al ball temperature, which can keep the difference between h f and h 0 less than 1.5%. Additionally, all of the calculated specific wind speeds V wf in Figure 6 do not exceed 0.2 m/s. Combined with the above analysis, the natural convection of the Al ball can be equivalent to the forced convection with the wind speed of V wf . Therefore, in the case of any modes of thermal convection, the expressions of the forced convection based on laboratory correlations (i.e., (14)-(16)) can be used to solve the characteristic numbers of Al ball.
Since the natural convection of the Al ball can be equivalent to the forced convection with a wind speed of V wf , only the characteristic numbers involved in the expressions of the forced convection need to be considered, including the Nusselt number Nu, Reynolds number Re s and Prandtl number Pr. Moreover, it can be found from (7) and (15) that the wind speed is correlated with Reynolds number. As a result, the correlation function for the Reynolds numbers of conductor and Al ball can be established through (7) and (15), as shown in (20).
According to (20), the Reynolds number of conductor Re c can be calculated through the Reynolds number of the Al ball Re s . Additionally, when combined with the wind direction data of the model device, the convective heat loss rate of conductor can also be determined.

Ampacity Calculation Based on EHT Model
For the EHT model proposed in this paper, the calculation flowchart of I amp is shown in Figure 7, and the specific implementation procedures are as follows: (1) Calculation of convective heat transfer coefficient of Al ball. The calculation formula for convective heat transfer coefficient of Al ball h is derived according to (11) and (12), as shown in (21). Combined with (21) and the collected weather data collected from the model device, h is calculated.
(2) Calculation of the Reynolds number of the Al ball. After determining the convective heat transfer coefficient of the Al ball, the Reynolds number of the Al ball Re s is obtained according to (13)- (14).

Validation of EHT Model Based on FEM
In this section, a three-dimensional FEM of the Al ball in the model device is established by COMSOL to simulate the temperature rise characteristics of the Al ball with different wind speeds [40]. Based on the simulation results of temperature rise in the Al ball and the calculation process in Figure 7, the I amp of the EHT model I EHT is obtained. Meanwhile, the I amp of the IEEE standard I IEEE under the same conditions is also calculated. The EHT model is verified by comparing the results of I EHT and I IEEE . Figure 8 shows the specific validation process for the EHT model.

Setting and Results of FEM
Before applying the EHT model proposed in this paper, a thermal-fluid coupling FE simulation model of the Al ball in the model device should be established to obtain the thermal behavior of the Al ball. The established geometric model is shown in Figure 9. The radius of the Al ball should not be set too large, otherwise an overlarge radial temperature difference will be caused. In the case of large radial temperature difference, the heat conduction inside the Al ball cannot be ignored. However, the heat conduction is not taken into account in the heat balance equation of the Al ball (i.e., (11)) in the EHT model. Therefore, the error is inevitably introduced in the calculation of the convective heat transfer coefficient h (i.e., (21)) with an overlarge radius of the Al ball. In this paper, according to the above analysis, the radius of the Al ball is set to 1.25 cm. In addition, a concentric sphere is also established as the air domain. To ensure the outer boundary temperature of the air domain is consistent with the ambient temperature, the radius of spherical air domain is set as 25 cm [41]. The ambient temperature is set to 25 • C. The internal heat source power q gs of (11) is 0.7 W to ensure that the steady-state temperature of the Al ball does not exceed 50 • C. As the global solar irradiance has little impact on I amp , the solar heat gain rate is not considered in the FE model. The boundary conditions of the thermal field contain the heat radiation and heat convection on the surface of Al ball. The simulation of heat radiation is achieved by setting the emissivity of the Al. In this section, the emissivity is set as 0.2 for calculation. The simulation of heat convection is realized by thermal-fluid coupling. For the boundary conditions of the fluid field, the boundary S1 of the air domain (as shown in Figure 9) is set as the air inlet, while the boundary S2 (as shown in Figure 9) is set as the air outlet. Then, the standard k-ε model of the turbulence model is adopted to solve the fluid field. Because it cannot only guarantee the accuracy of the calculation results, but also avoid a long computation time [42]. Moreover, it is noted that the effect of air gravity is also considered during the solution of fluid field.
The transient calculation time is set to 80 min, and the transient and steady temperature distribution results of the Al ball can be obtained. Figure 10 shows the steady-state simulation results with a wind speed of 1 m/s, including the temperature distribution of the Al ball and the wind speed distribution in the surrounding air domain. The results in Figure 10 indicate that, due to the blocking effect of the Al ball on the wind, the temperature on the leeward side of the Al ball is higher than that on the windward side. There is still a temperature difference of 3.1 • C in the Al ball. In order to obtain a conservative I amp result, the highest temperature on the leeward side is selected as the temperature of Al ball.

Validation of EHT Model
For the validation of the EHT model in this section, the wind direction is set to 90 • , and an ACSR 240/30 mm 2 overhead conductor is selected to calculate the I amp . Based on the output thermal behavior curve (including transient and steady state) of the FE model in Section 5.1, the I amp results of the proposed EHT model I EHT at different wind speeds are calculated. Moreover, I amp results of IEEE standard I IEEE are calculated under the same conditions. Figure 11 shows the results of I EHT and I IEEE . In addition, the relative errors err I of these two I amp results are also introduced, as shown by the red curves in Figure 11. According to the results in Figure 11, when the weather conditions remain constant, I amp calculated by the IEEE standard I IEEE is a constant value, as shown by the blue curves in Figure 11. However, I amp calculated by the EHT model I EHT is variable, as shown by the green curve in Figure 11. Moreover, the I EHT results decrease and tend to be stable gradually with the increase in time. The error between I IEEE and I EHT also decreases gradually. When the I EHT results are stable, the error does not exceed 5%.
This phenomenon can be explained by analyzing the equivalent wind speed V w.calc calculated by the EHT model. In the EHT model, the equivalent wind speed V w.calc can be calculated according to (15) and the calculated Reynolds number of the Al ball. Figure 12 shows the V w.calc results corresponding to Figure 11 and the temperature rise results of the Al ball. It is indicated from Figure 12 that the V w.calc results decrease with the increase in time. When the Al ball temperature reaches steady-state, V w.calc is also stable and almost equal to the input wind speed V w . The error between V w.calc and V w is mainly in the transient temperature rise of the Al ball. This is because in the initial phase of the transient state, the temperature difference between the Al ball and the environment is small. According to (21), although there is a differential term in the formula, the small denominator value results in a large calculated convective heat transfer coefficient h. This also causes the calculation results of V w.calc to increase in the initial phase of the transient state. As a result, according to Figures 11 and 12, the reason for the large error of I EHT in the initial phase is that the calculated V w.calc is much larger than the V w . In order to more comprehensively verify the accuracy of the EHT model proposed in this paper, the calculation results of I EHT based on the Al ball steady-state thermal behavior and I IEEE at different wind speeds are obtained, as shown in Figure 13. In Figure 13, the results of I EHT and I IEEE both increase with the increase in wind speed. Compared with the I IEEE results, the relative errors of I EHT results are less than 6%. Thus, the EHT model proposed in this paper can accurately evaluate I amp in the case of using the steady-state thermal behavior of the Al ball.

Conclusions
The DTR technology can maximize the utilization for power transfer capacity of the existing overhead conductors. However, the existing models based on this technology have some deficiencies in measurement techniques related to wind speed. Therefore, in this paper an EHT model for DTR technology was proposed instead of wind speed measuring. In this model, the correlation of heat losses between the heating Al ball and conductor was established to calculate the conductor ampacity. Moreover, a thermal-fluid coupling FE model of the Al ball was established to simulate the implementation of the EHT model. Combined with the established FE model, the EHT model was validated by the IEEE standard. The results demonstrated that the EHT model showed a good agreement with the IEEE standard in the case of employing the steady thermal behavior of the Al ball. Since the EHT model can achieve desirable performance in evaluating the real-time ampacity of overhead conductors, it can facilitate a high utilization of overhead conductor, while also avoiding the restrictions of measuring technology related to wind speed.
The future work of this investigation will focus on the comparative analysis between the EHT model and the existing models based on experiments. In addition, the feasibility of using the transient thermal behavior of the Al ball to assess the conductor ampacity will be also explored.