Influence of the wind variability on the calculation of dynamic line rating

Electric Power Systems Research 211 (2022) 108234 Available online 5 July 2022 0378-7796/© 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr Influence of the wind variability on the calculation of dynamic line rating M.Á. González-Cagigala,∗, J.A. Rosendo-Macíasa, A. Bachiller-Solera, D. Señas-Sanvicenteb aDepartment of Electrical Engineering, University of Seville, Seville, Spain bIngelectus S.L., Seville, Spain ARTICLE INFO Keywords: Dynamic line rating Electrical clearance Monte Carlo Overhead transmission lines ABSTRACT This paper presents a novel technique to calculate the ampacity of an overhead transmission line in real time, considering the dynamic evolution of the conductor temperature. In case this method cannot be applied due to lack of adequate information, a correction is proposed for the maximum current, based on Monte Carlo simulations assuming unfavorable external conditions, and validated with real data from a weather station. This technique might be used to avoid the temporary violation of the minimum electrical clearance in transmission lines. 1. Introduction The optimization of the transmission utilities is a prevailing challenge in the operation and control of electric power systems. In the particular case of overhead transmission lines (OHTLs), and due to the increasing electricity demand and the arising of new generation plants, it is important to accurately establish the capacity of the conductor in order to reduce infrastructure expansion and the corresponding economic outlay. The ampacity (or rating) of OHTLs is defined as the maximum current that can circulate through the conductor without causing overheating and its derived problems, like an excessive sag, [1], or conductor damage, [2]. A traditional approach is the so-called static line rating (SLR), where an ampacity is calculated for long periods of time (e.g., for the whole year or seasonal) assuming the worst-case meteorological conditions, [3,4]. Given that the SLR is not the most efficient technique, since the OHTL ampacity usually gets underrated, a new framework can be used for real-time calculation of the ampacity, namely the dynamic line rating (DLR), [5], where the measured external conditions (wind, ambient temperature or solar irradiation) are taken into account to obtain the line rating. Two major standards can be highlighted for thermal rating: IEEE, [4], and CIGRE, [6]. Although these standards are not directly oriented to the DLR approach, their thermal equations are usually used for the calculation of the ampacity in OHTLs, [7,8]. In this context the CIGRE guide has been considered in this work because, as studied in [9], the IEEE std. might represent a too conservative approach, reducing excessively the DLR with high wind speeds. Regarding the measurements of the weather conditions used to obtain the DLR, information from weather stations has been considered ∗Corresponding author. E-mail address: [email protected] (M.Á. González-Cagigal). by a remarkable number of studies, like that in [10], where the method is applied for the identification of critical spans, or the work in [11], analyzing direct and indirect measurements. Finally, some lines of investigation are oriented to the prediction of the DLR using forecasting techniques for the meteorological variables, [12]. As mentioned before, the ampacity is calculated so that the conductor temperature rises to a determined limit value, which might be established using different criteria, such as conductor deterioration or regulatory requirements for minimum electrical clearance. However, the real-time variability of the external conditions, produces a dynamic evolution of the conductor temperature, leading to a possible temporary violation of the defined limit. In this research, the influence of the environmental conditions on the DLR is analyzed, and an alternative approach is proposed to obtain the ampacity of OHTLs considering the dynamic evolution of the conductor temperature. Even though the presented technique assumes a high frequency in the measurement gathering, a method is presented to correct the DLR when this information is not available. The rest of the work is organized as follows: Section 2presents the traditional calculation of the ampacity of OHTLs and the possible transient problems related to the conductor temperature. The proposed technique is presented in Section 3, while its application to a simulated scenario and a real dataset is included in Sections 4and 5respectively. The most relevant conclusions are included in Section 6. 2. Problem statement In this section, a typically used method to obtain the ampacity of an OHTL is described, jointly with its potential problems in terms of minimum electrical clearance. https://doi.org/10.1016/j.epsr.2022.108234 Received 28 March 2022; Received in revised form 25 May 2022; Accepted 25 June 2022 Electric Power Systems Research 211 (2022) 108234 2 M.Á. González-Cagigal et al. 2.1. Ampacity calculation For a given conductor, the ampacity, 𝐼𝑚𝑎𝑥, is defined as the maximum current that can circulate permanently, so that the temperature of the conductor rises to a previously established limit value, 𝑇𝑙𝑖𝑚 𝑐, under a set of external meteorological conditions, namely: - Ambient temperature, 𝑇𝑎𝑚𝑏 [◦C]. - Wind speed, 𝑣𝑤[m∕s]. - Wind direction w.r.t. the line span, characterized by the angle 𝛼𝑤 [rad], e.g., 𝜋∕2 for perpendicular wind. - Solar radiation, 𝐼𝑠[W∕m2]. In order to calculate the value of 𝐼𝑚𝑎𝑥, the thermal equilibrium equation presented in [6], 𝑃𝑟(𝑇𝑙𝑖𝑚 𝑐) + 𝑃𝑐(𝑇𝑙𝑖𝑚 𝑐) − 𝑃𝑠−𝑃𝐽(𝑇𝑙𝑖𝑚 𝑐, 𝐼𝑚𝑎𝑥)=0,(1) must be satisfied under these ambient conditions, where 𝑃𝑟and 𝑃𝑐are respectively the radiative and convective heat losses, 𝑃𝑠is the solar heating and 𝑃𝐽is the Joule heating, which can be obtained as 𝑃𝐽(𝑇𝑐, 𝐼) = 𝐼2⋅𝑅𝐴𝐶 (𝑇𝑐),(2) with 𝑅𝐴𝐶 (𝑇𝑐)being the AC resistance of the conductor at temperature 𝑇𝑐. Substituting (2) in (1), the following expression can be derived for the ampacity: 𝐼𝑚𝑎𝑥 =√𝑃𝑟(𝑇𝑙𝑖𝑚 𝑐) + 𝑃𝑐(𝑇𝑙𝑖𝑚 𝑐) − 𝑃𝑠 𝑅𝐴𝐶 (𝑇𝑙𝑖𝑚 𝑐).(3) In the previous equation, the values of 𝑃𝑟,𝑃𝑐and 𝑃𝑠for the temperature 𝑇𝑙𝑖𝑚 𝑐are calculated in this work according to the expressions provided by the CIGRE guide, [6]. 2.2. Dynamic temperature evolution The calculation of the ampacity given by Eq. (3) assumes constant external conditions during the corresponding time period, typically the mean values of the involved variables in a 10-min interval. Even though this assumption might be valid for some magnitudes, as the ambient temperature, other variables present a significant variation in this period. Take as an example the evolution of the wind speed in a 10-min interval, represented in Fig. 1, where real-time measurements from a weather station Froggit hp1000se PRO have been used. For a given conductor with a thermal capacity 𝑐𝑝, the mentioned variability in the meteorological conditions produces a transient behavior in the temperature of the conductor, 𝑇𝑐, which can be described using the equation 𝑚⋅𝑐𝑝⋅ 𝑑𝑇𝑐 𝑑𝑡 =𝑃𝑟(𝑇𝑐) + 𝑃𝑐(𝑇𝑐) − 𝑃𝑠−𝑃𝐽(𝑇𝑐, 𝐼),(4) where 𝑚is the mass per unit length of the conductor. The reader may notice that, for constant meteorological conditions, the steady-state regime (null derivative) reduces to Eq. (1). However, under timevariant conditions, the temperature of the conductor can transiently exceed the established limit (𝑇𝑐> 𝑇 𝑙𝑖𝑚 𝑐), causing a temporary increase in the conductor sag which might produce a violation of the regulatory requirements for minimum electrical clearance. 2.3. Numerical example In order to illustrate the described situation, the 10-min wind speed profile shown in Fig. 1 has been considered, while, for simplicity sake, the rest of the external conditions have been taken as constant values, summarized in Table 1. An aluminum conductor steel reinforced (ACSR) 455-54/7 (CONDOR) has been used, as a typical one in OHTLs, with an initial temperature of 50◦C. Fig. 1. Evolution of the wind speed in a 10-min period. Table 1 External conditions in the numerical example. Variable Unit Value 𝐼𝑠W∕m2700 𝑇𝑎𝑚𝑏 ◦C 25 𝛼𝑤rad 𝜋∕3 Fig. 2. Evolution of the conductor temperature in a 60-min period. Fig. 2 represents the evolution of the conductor temperature during 60 min in two situations: considering a constant wind speed over time, equal to the mean value (in this example, 𝑣𝑤= 2.79 m∕s), and assuming that the wind-speed profile presented in Fig. 1 is repeated 6 times. In both cases it is considered that the current corresponds to the value of 𝐼𝑚𝑎𝑥 calculated with Eq. (3) and a typical limit temperature 𝑇𝑙𝑖𝑚 𝑐= 85 ◦C(𝐼=𝐼𝑚𝑎𝑥 = 1541 A). It is observed how the value of 𝑇𝑐is higher than the limit in some moments during the 60-min interval, which might cause the violation of the minimum electrical clearance. In the next section, an alternative method is proposed to obtain a new value of 𝐼𝑚𝑎𝑥 for which it is satisfied that the conductor temperature never exceeds the previously defined limit. Electric Power Systems Research 211 (2022) 108234 3 M.Á. González-Cagigal et al. 3. Proposed method In this section, the new value of the ampacity will be calculated taking into account the dynamic evolution of the conductor temperature. Then a numerical example is presented to test the influence of the initial conditions in the results. Finally, a method is proposed for real-time applications. 3.1. Ampacity calculation As previously mentioned, the value of 𝐼𝑚𝑎𝑥 is typically calculated with Eq. (3), considering steady-state conditions for the variables involved. The alternative method proposed in this work takes into account the variability in the meteorological conditions and the thermal capacity of the conductor using the dynamic Eq. (4). This novel method to obtain the value of 𝐼𝑚𝑎𝑥 is described below: •First, the initial conductor temperature, 𝑇𝑐0, is selected. In the numerical example presented later in this section it will be shown how the influence of this initialization vanishes in time and does not affect the steady-state results. •Using Eq. (4), the evolution of 𝑇𝑐during 𝑁𝑐cycles is calculated for each value of the current 𝐼. The profiles of the external conditions are repeated 𝑁𝑐times, this number being chosen to ensure that the total time is high enough, compared to the time constant 𝑚⋅𝑐𝑝in the differential Eq. (4). In this manner, the periodic steady-state regime of the temperature 𝑇𝑐will be reached irrespective of the initial conditions taken, as will be proven in the subsequent numerical example, shown in Fig. 3, where the calculation process will be illustrated for a better comprehension. •Finally, the ampacity corresponds to the value of 𝐼that satisfies the equality max(𝑇𝑐) = 𝑇𝑙𝑖𝑚 𝑐. 3.2. Numerical example For the numerical example presented in the previous section, the new value of 𝐼𝑚𝑎𝑥 is calculated using the proposed method. The number of 10-min cycles simulated is 𝑁𝑐= 6. Regarding the initial conductor temperature, different values have been used in order to verify how this initial condition has no influence on the calculated value of 𝐼𝑚𝑎𝑥. In all cases, the ampacity obtained is 1427 A, independently of the value of 𝑇𝑐0.Fig. 3 represents the evolution of 𝑇𝑐for 𝐼=𝐼𝑚𝑎𝑥, and different initial conditions. 3.3. Application to reduced sampling rates In the proposed ampacity calculation, it is assumed a high sampling frequency for the measurement gathering, for this reason, the value of 𝐼𝑚𝑎𝑥 obtained through this method will be denoted as 𝐼𝑚𝑎𝑥 ℎ𝑓 in the sequel. However, in real applications, this rate is typically much lower (e.g., 10 min between two consecutive measurements) and the information provided for the different magnitudes corresponds to their mean values during this period, so that the ampacity must be calculated using Eq. (3), which will be named as 𝐼𝑚𝑎𝑥 𝑙𝑓 . However, as it has been shown in Fig. 2, the value of 𝐼𝑚𝑎𝑥 𝑙𝑓 might result in temporary violations of the conductor limit temperature and therefore the minimum electrical clearance, so that it needs to be corrected. In this work, when the sampling rate is not high enough, a method is proposed to consider the dynamic effect in the conductor temperature for the calculation of the maximum current, based on a coefficient 𝑐, which is calculated in simulated scenarios where this meteorological variability is taken into account and a high sampling frequency is considered to calculate the value of 𝐼𝑚𝑎𝑥 ℎ𝑓 . For each time period 𝑘the ratio between the high and low frequency ampacities is calculated: 𝑐𝑘= 𝐼𝑚𝑎𝑥 ℎ𝑓,𝑘 𝐼𝑚𝑎𝑥 𝑙𝑓 ,𝑘 (5) Fig. 3. Evolution of the conductor temperature using the proposed method and different initial conditions. Fig. 4. Example of the considered SDE-based model for the wind speed. In order to take an ampacity 𝐼𝑚𝑎𝑥 as a conservative estimation of 𝐼𝑚𝑎𝑥 ℎ𝑓 , a coefficient 𝑐is defined as the minimum of those coefficients for the total number of periods analyzed, 𝑁, yielding: 𝑐= min (𝑐𝑘),with 𝑘= 1,…, 𝑁 (6) and 𝐼𝑚𝑎𝑥 =𝑐⋅𝐼𝑚𝑎𝑥 𝑙𝑓 (7) In the next section, a simulated scenario is presented where the value of this coefficient is obtained with different external conditions. 4. Simulated scenario The previously defined coefficient 𝑐is calculated in a synthetic scenario where a set of independent 10-min time periods are simulated, with a sampling frequency of 0.1 Hz and the following considerations for the meteorological conditions: •Constant values are taken for the ambient temperature, 𝑇𝑎𝑚𝑏, and the solar radiation, 𝐼𝑠, since the variability of these magnitudes in a 10-min time interval can be neglected. Electric Power Systems Research 211 (2022) 108234 4 M.Á. González-Cagigal et al. Fig. 5. Ampacities obtained in the Monte Carlo simulations. •For the wind speed, a continuous-time model has been considered, based on stochastic differential equations (SDEs) and following, as proposed and validated in [13], a Weibull distribution. The mean value of this variable, 𝑣𝑤, is previously assigned in each 10-min time interval as a uniformly distributed random variable with the range shown in Table 2. An example of this evolution for 𝑣𝑤= 5 m∕s is represented in Fig. 4. •Finally, for the wind direction, the most unfavorable case has been assumed in terms of thermal refrigeration of the cable, so that a lower bound is obtained for the coefficient 𝑐and therefore, for the ampacity of the OHTL. As for the wind speed case, a SDE-based model is considered for the angle 𝛼𝑤. This variable is modeled in [14] using a Von Mises distribution, which has no analytic expression for its cumulative distribution function, so that it is not possible to obtain the corresponding SDE. Nevertheless, as the Von Mises distribution approaches to normal or uniform distributions in extreme cases, these approximations will be used in this work to look for bounds in the coefficient 𝑐for worst case scenarios: taking 𝛼𝑤∼𝐍(𝜋 2,𝜋 6)and 𝛼𝑤∼𝐔(0, 𝜋). In both cases, the mean value of 𝛼𝑤in the 10-min period will be close to 𝜋 2rad (optimal refrigeration angle), while the extreme values, 0and 𝜋 rad, are associated to a poor refrigeration. First, an analysis is carried out using the ampacity calculated with high sampling frequency, 𝐼𝑚𝑎𝑥 ℎ𝑓 , and low sampling frequency, 𝐼𝑚𝑎𝑥 𝑙𝑓 . The involved variables are randomly assigned values within the ranges shown in Table 2, and the total number of Monte Carlo-based simulations was set to 60000. For a clearer representation, the ampacities obtained for the two distributions considered for the wind angle for a sample of only 500 simulations are represented in Fig. 5, including a zoomed plot with 10 simulations. In this graph it can be noticed how the value of 𝐼𝑚𝑎𝑥 𝑙𝑓 is higher than 𝐼𝑚𝑎𝑥 ℎ𝑓 , and the ampacity calculated with uniform distribution of the wind angle is lower than that considering a normal distribution. Additionally, the quotient 𝑐𝑘=𝐼𝑚𝑎𝑥 ℎ𝑓,𝑘∕𝐼𝑚𝑎𝑥 𝑙𝑓 ,𝑘 for both distributions is presented, in descending order, in Fig. 6 for the whole set of simulations. It is observed that the normal distribution provides higher values of the quotient 𝑐𝑘, while its minimum value, for the uniform Table 2 Range of the variables in the Monte Carlo-based simulations. Variable Unit Range 𝐼𝑠W∕m2[0 800] 𝑇𝑎𝑚𝑏 ◦C [−10 50] 𝑣𝑤m∕s [0.5 10] Fig. 6. Quotient 𝑐𝑘obtained for the Monte Carlo simulations. distribution, is approximately 0.862, which would be taken as a lower bound of the coefficient 𝑐for the correction of 𝐼𝑚𝑎𝑥 𝑙𝑓 in a real application. Although the value taken in the previous scenario as low sampling frequency (one sample each 10 min) is a typical value provided by commercial weather stations (as that considered for the application included in Section 5), the Monte Carlo simulations carried out to obtain the coefficient 𝑐can consider different low sampling frequencies. Electric Power Systems Research 211 (2022) 108234 5 M.Á. González-Cagigal et al. Table 3 Variation of the coefficient 𝑐with the sampling period of the weather station. Sampling period (min) Coefficient 𝑐(pu) 10 0.862 5 0.879 2 0.901 1 0.932 Fig. 7. Quotient 𝑐𝑘with variations of (a) 𝑇𝑎𝑚𝑏 (b) 𝐼𝑠(c) 𝑣𝑤. Table 3 represents the values obtained for the coefficient 𝑐with different sampling periods. As expected, it can be noticed that, the value of 𝑐 increases (resp. decreases) with the sampling frequency (resp. period), since the variability of the weather conditions between two consecutive samples is lower. Finally, the individual influence of each variable in the coefficient 𝑐is subsequently analyzed. For a certain magnitude, it is assumed a variation within the range shown in Table 2.Fig. 7 presents the variation of 𝑐𝑘with the ambient temperature (Fig. 7a), the solar radiation (Fig. 7b) and the mean value of the wind speed (Fig. 7c). In all cases, the two distributions used for the wind direction has been compared, maintaining the assumption of the most unfavorable conditions. It can be noticed that the variations of 𝑣𝑤have a higher influence on the coefficient 𝑐than those of the ambient temperature or the solar radiation for both the normal and the uniform distributions, with approximately 0.02 pu variation of the quotient 𝑐𝑘from 𝑣𝑤= 0.5 m∕s to 𝑣𝑤= 10 m∕s. 5. Application to real data In the previous section, the value of the coefficient 𝑐was calculated in a simulated scenario where the most unfavorable conditions were assumed for the wind direction. In this section the results obtained will be validated using a real dataset from the previously mentioned weather station Froggit hp1000se (see Fig. 8). Fig. 8. Weather station Froggit hp1000se, [15]. Fig. 9. Quotient 𝑐𝑘with real data. For this weather station, the time between two consecutive measurements ranges from 20 to 25 s for each magnitude, so that the algorithm proposed in this work can be used to obtain the ampacity of the conductor, 𝐼𝑚𝑎𝑥 ℎ𝑓 . For simplicity sake, it is assumed that the weather station angle reference is aligned with the line span under study, directly providing the angle 𝛼𝑤. Additionally, the weather station used in this work also provides the mean values of the involved magnitudes in time intervals of 10 min, so that 𝐼𝑚𝑎𝑥 𝑙𝑓 can be obtained, jointly with the quotient 𝑐𝑘, at instant 𝑘. Measurements obtained from a one-year period were preprocessed in order to remove bad data, yielding approximately a total number of 42000 valid periods of 10 min. Fig. 9 represents, in descending order, the values obtained for the quotient 𝑐𝑘. Two aspects can be highlighted from the graph: •The values obtained with real data are, in most of the cases, higher than those from the simulated scenario. The reason is that the most unfavorable case was assumed in the synthetic scenario for the generation of the wind-direction profile. •The minimum value of 𝑐𝑘(0.872) is very similar to that obtained in Section 4, so that the coefficient 𝑐, as defined in Eq. (6), approximately matches in both cases, giving evidence of the accuracy of the proposed method to correct the ampacity calculated with a reduced sampling rate. Electric Power Systems Research 211 (2022) 108234 6 M.Á. González-Cagigal et al. Fig. 10. Experimental 𝑐𝑘for different values of 𝑣𝑤. In the simulated scenarios presented in Section 4it was observed that the mean value of the wind speed was the variable with the higher influence in the quotient 𝑐𝑘. In order to assess if the simulated results fit the real data in worst case scenarios, Fig. 10 shows the minimum values of 𝑐𝑘obtained for different mean values of the measured wind speed, where the synthetic curves presented in Fig. 7c are also represented as a reference. It can be noticed that most experimental values remain between the normal and the uniform distributions, so that the latter can be considered to obtain a lower bound of the coefficient 𝑐. 6. Conclusion In this work, it has been presented a method to calculate the maximum current which can permanently circulate through an overhead transmission line, taking into account the dynamic evolution of the conductor temperature, so that it never exceeds a previously established limit. Additionally, a coefficient has been defined to correct the ampacity calculated in cases where only a reduced sampling rate (e.g., every 10 min) is available for the measurement gathering. For an ACSR 455-54/7 (CONDOR), a set of simulated cases have been considered with unfavorable external conditions so that a lower bound can be obtained for the mentioned coefficient. Finally, the validity of the results obtained has been assessed using a set of real measurements from a weather station, leading to a very similar minimum value (0.872) compared to the simulated one (0.862). The use of the proposed coefficient might prevent the violation of the regulatory requirements for minimum electrical clearance in OHTLs, reducing accordingly the derived risks. CRediT authorship contribution statement M.Á. González-Cagigal: Conceptualization, Methodology, Data curation, Writing – original draft, Writing – review & editing. J.A. Rosendo-Macías: Conceptualization, Methodology, Supervision, Writing – original draft, Writing – review & editing. A. Bachiller-Soler: Conceptualization, Methodology, Supervision, Writing – original draft, Writing – review & editing. D. Señas-Sanvicente: Conceptualization, Methodology, Writing – original draft, Writing – review & editing. 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