Abstract
Accurate short-term prediction of overhead line (OHL) transmission ampacity can directly affect the efficiency of power system operation and planning. Any overestimation of the dynamic thermal line rating (DTLR) can lead to the lifetime degradation and failure of OHLs, safety hazards, etc. This paper presents a secure yet sharp probabilistic model for the hour-ahead prediction of the DTLR. The security of the proposed DTLR limits the frequency of DTLR prediction exceeding the actual DTLR. The model is based on an augmented deep learning architecture that makes use of a wide range of predictors, including historical climatology data and latent variables obtained during DTLR calculation. Furthermore, by introducing a customized cost function, the deep neural network is trained to consider the DTLR security based on the required probability of exceedance while minimizing the deviations of the predicted DTLRs from the actual values. The proposed probabilistic DTLR is developed and verified using recorded experimental data. The simulation results validate the superiority of the proposed DTLR compared with the state-of-the-art prediction models using well-known evaluation metrics.
THERMAL line rating (TLR) is the primary culprit limiting the current carrying capability of the overhead line (OHL) [
To overcome the shortcomings of the STLR, dynamic TLR (DTLR) is proposed in which the thermal condition of OHL can be monitored. Thus, DTLR unlocks the additional capacity headroom of current OHLs in a secure way, thereby addressing network congestion and postponing/eliminating the need for transmission expansion [
DTLR is a function of several climatology variables such as wind speed, wind direction, etc. [
Relying on the literature, the DTLR prediction has been interpreted with two different viewpoints. Some researchers predict the maximum allowable current at the OHL thermal limit [
DTLR can be broadly divided into direct [
From the prediction output, DTLR prediction models can be divided into deterministic [
In [
In [
Weather-based DTLR prediction models mainly limit their inputs to those features obtained directly from meteorological measurements. However, from the DTLR formulations proposed in [
The remarkable advancements in deep learning and its successful implementation in prediction have resulted in the enhancement of prediction accuracy for a range of power system applications [
This paper proposes a DTLR for predicting the various POEs using accessible latent variables in addition to meteorological measurements. A DTLR model is developed using the SDAE and LSTM unit in the DNN architecture. The prediction engine is trained by considering a novel cost function to meet the JTF recommendation, while the sharpness is maximized. The performance of the proposed models is compared with the state-of-the-art DTLR prediction models using publicly available data. Briefly, the main contributions of this work are three-fold.
1) For the first time, latent variables are introduced in DTLR prediction as valuable predictors.
2) A cost function is proposed to train the deep-learning model for probabilistic prediction.
3) A deep learning model is trained for the DTLR prediction.
The remainder of the paper is organized as follows. Section II presents the proposed framework for DTLR prediction. The proposed training framework for probabilistic deep-learning-based prediction of DTLR is explained in Section III. Data, evaluation metrics, and simulations results are presented in Section IV, followed by discussion in Section V and remarkable conclusions in Section VI.
Define and as TSs associated with wind speed and wind direction, respectively, where is the length of the TSs, and is the th sample of the TS. Also, the TSs of the wind speed components decomposed by a Cartesian coordinate system are denoted by and , respectively. The TSs of ambient temperature and solar irradiance are denoted by and , respectively.

Fig. 1 Overall framework of proposed DTLR prediction model.
In this study, the power system is assumed to be in normal operation. Therefore, the current fluctuations and the resulting variations in the OHL temperature can be considered negligible, provided that the system does not require any abrupt or temporary switching [

Fig. 2 A schematic diagram of DTLR.
As per IEEE Std 738-2012, in the steady state, the heat balance equation for an OHL at the th sample can be written as follows [
(1) |
where and are the convection cooling and radiated heat loss rates per unit length, respectively; is the heat gain rate from the sun; is the alternating current (AC) resistance associated with the conductor temperature ; and is the allowable conductor current at , which can be simply obtained as:
(2) |
From (1) and (2), one can observe that and are the cooling elements in the heat-balance equation, and that their increase helps obtain more OHL ampacity. is a heating component and is a culprit of ampacity reduction. is a function of , , and . is calculated as [
(3) |
(4) |
(5) |
(6) |
(7) |
where , , , and are intermediate variables; is the wind direction factor; , , are the air density, absolute air viscosity, and coefficient of thermal conductivity of air, respectively; and is the outside diameter of the conductor. As can be seen from (3)-(7), is a nonlinear and complicated function of meteorological variables while the relationship between and is simple as shown in (2). Therefore, it can be beneficial to consider as the elements of the predictor set for predicting . Moreover, in (1) and (2) can be calculated as [
(8) |
where is the emissivity and has a value between 0.23 and 0.91, which increases with the age of conductor. As can be seen from (8), is related to the fourth power of . Therefore, it may not be adequate to consider the simple vector of as the elements of the feature set to reflect the importance of ambient temperature and radiative heat loss in the DTLR prediction. It is worthful to analyze the influence of considering historical data of radiated heat loss rate. For a solar irradiance at time , the rate of solar heat gain can be estimated by a linear function of [
The calculated DTLR, , as well as the latent variables obtained during the DTLR calculation, is used in a feature reduction and feature learning stage as elucidated below.
As detailed in Section II-A, several climatic variables strongly influence the DTLR value including wind speed, wind direction, wind speed Cartesian components, ambient temperature, and solar irradiance. A tensor, formed by a series of lags associated with these variables, contains the potential informative predictors for DTLR. Moreover, the latent variables, i.e., convection cooling and radiated heat loss rate, may also contain valuable information about the complex relationship between climatic variables and DTLR. Historical DTLR values could also contain useful information.
It is a principal stage in the DTLR prediction to properly optimize the input features of the prediction engine by eliminating the non-informative and redundant features and identifying the features, which can demonstrate the DTLR pattern more efficiently. To this end, we first employ a feature reduction stage based on minimal-redundancy and maximal-relevance (mRMR) [
(9) |
where is the joint probability density function; and and are the individual probability density functions of and random variables, respectively, which are the discretized format of continuous predictor and target variables. Based on (9), the iterative mRMR algorithm is carried out by the following optimization problem [
(10) |
(11) |
where is the subset containing selected features at the th iteration; is the random variable describing the th lag of the th feature; and A is a set consisting of the random variables associated with feature candidates. In the first iteration of solving (10), , . In this paper, the optimization problem in (10) is iteratively solved until the of the last component selected from solving (10) with is negligible.
After conducting mRMR, the feature types that are not among the selected features in or constitute a negligible portion of are removed from the feature pool. The selected features, along with their corresponding lags, are then used to build the tensor input of the deep-learning-based feature extraction described in Section II-B-3).
LSTM is a unit of RNNs in which the temporal dependency among the elements of the TS can be captured. An LSTM block consists of a memory cell, an input gate, an output gate, and a forgetting gate. The memory cell stores the values for arbitrary time intervals. In LSTM, the three gates are neurons with activation functions.

Fig. 3 Schematic of LSTM unit.
AE is a type of neural network mainly employed for unsupervised feature learning in a wide range of applications [
At time , all features that remain after the feature reduction stage, which are described in Section II-B-1), are used to construct the input vector of the DAE , a three-dimensional tensor (), where is the number of feature variants. To capture the sequential correlation of the TS, LSTM is used as the building blocks of the DAEs in this paper. An RNN-based DAE can be formulated as:
(12) |
(13) |
(14) |
(15) |
where is the loss function that employs the mean-square error (MSE); () is the tensor that contains all tensors , is the number of available points in the validation set; is the tensor that consists of all tensors , , which are the outputs of a DAE corresponding to input ; and encapsulate the unknown parameters of encoder and decoder blocks, respectively; and are the outputs of encoder and decoder blocks, respectively; and are the information passed from the calculation of and , respectively, as the result of recurrent units; and and are the functions associated with encoder and decoder blocks, respectively.
In (13), some elements of input vector are destroyed in a stochastic process, i.e., , in order to form the corrupted input .
Stacking several DAE forms an SDAE, in which more informative features can be extracted. Using the ADAM stochastic optimization [
The training of the proposed DTLR is conducted in several steps according to the scheme presented in

Fig. 4 General training scheme of proposed probabilistic prediction of DTLR.
Afterwards, Models 1-3 are further trained using the proposed cost function (16)-(20). Once Models 1-3 are tuned considering the proposed cost function, SDAE layers are unfrozen to tune the model further using the proposed cost function.
Inspired by QR formulation, the proposed cost function for training DNN for a preferred POE is defined as:
(16) |
(17) |
(18) |
(19) |
(20) |
where is the sign function; and is the preferred POE. In (16), the term associated with penalizes the prediction model when the prediction model results in values above . The more value is higher than , the more the cost function penalizes the model. On the other hand, when , the term related to in the cost function becomes zero. However, the term corresponding to penalizes the prediction model if is deviated from the .
(21) |
where is a large constant number so that for , for , and for . Using results in a differentiable function that can use the off-the-shelf DNN optimizers for training the model.
The proposed cost function enables to fine-tune the DTLR model so that the DTLR prediction model provides the lower bound of DTLR values with the minimum deviations from the actual DTLR values.
We have performed the analysis based on a 5-year dataset (from January 1, 2010 to January 1, 2015), recorded from the M2 met tower at the National Wind Energy Center (NWEC) located in Denver, USA [
Based on [
The proposed model has been implemented in Python 3.7 on a Windows 10 PC with a 1.6 GHz Intel Core i5 CPU and 8 GB of memory. Based on the grid search, three DAE layers are selected as the suitable architecture for SDAE, respectively. The numbers of hidden layers for Models 1 and 2 are one and three, respectively. Model 3 is a single-layer model which provides the final prediction by using the outputs from Models 1 and 2. The training is conducted using the process described in Section III.
To evaluate the dependency of on different feature candidates, a 5-year meteorological dataset is used to generate historical , , and TS, using the procedure described in Section II-A.

Fig. 5 Dispersion of DTLR with respect to different feature types and Kendall rank coefficient and mutual information of different features and DTLR.
The Kendall rank coefficient is widely employed as a non-parametric statistical test in hypothesis testing to identify the statistical dependency between two random variables [
The detailed explanations of the Kendall rank coefficient can be found in [
Furthermore,
Three benchmark models are utilized in this paper: persistence [
Three evaluation metrics are used to appraise the performance of the different DTLR models. The POE of the prediction model is the most imperative evaluation criterion for a secure probabilistic DTLR prediction and is defined as:
(22) |
(23) |
where is the number of points in the training, validation, or test datasets. Any deviation of the from the preferred can lead to unprecedented issues. As a measure of sharpness of the predicted s, normalized mean absolute error () is used, which is defined as:
(24) |
where is the range of DTLR values. In a perfect prediction of DTLR, while . Root-mean-squared-error (RMSE), which is a valuable measure and signifies large deviations of DTLR prediction from its actual value, is also employed as another evaluation metric, and can be calculated as:
(25) |
As an instance, the performance of the prediction model on a two-year dataset (from January 1, 2010 to January 1, 2012) is used for numerical comparisons. 85% of the data are used for training and validation, while the remainder are employed for testing. In this section, the effectiveness of considering latent predictors, i.e., convection cooling and radiated heat loss rate, is first investigated. Thereafter, the proposed DTLR is compared with the benchmark models for different preferences using various evaluation metrics.
Section IV-B shows that in comparison to directly observed meteorological variables, the radiated heat loss rate has a more pretentious relation with DTLR. To empirically investigate the efficacy of considering a series of lags associated with the mentioned latent variables as predictors, a case study is conducted using the proposed DTLR prediction with and without the latent predictor.
The performance evaluations of different prediction models for , 95%, and 99% are presented in
To further investigate the performance of the proposed model compared with the benchmark models, the histograms of various prediction models of DTLR for is presented in

Fig. 6 Distribution of predicted DTLR for with predicted values equal or less than the maximum actual ampacity.
As can be observed from

Fig. 7 DTLR prediction using proposed model and QRF for .
As the case studies elucidate, the proposed prediction framework of DTLR results in a more accurate and reliable prediction compared with various well-established benchmark models. In Sections IV-E-1) and 2), the detailed analyses demonstrate that the superior performance of the proposed model results from both considerations of latent variables and the deep learning architecture. The proposed model can be used as a decision-supporting tool for system operators for unlocking the additional ampacity of OHL considering a pre-defined risk exposure acceptance, i.e., .
The hour-ahead secure prediction of DTLR can facilitate the accommodation of renewable generation, and alleviate the issues of transmission congestion. Consequently, the wind power curtailment is reduced, while the demand for transmission expansion can be postponed or even eliminated. Such a highly accurate hour-ahead prediction can be implemented in the energy management system for real-time security constraint economic dispatching [
In this study, the efficacy of the proposed model is validated for a range of values. In practice, the risk-based assessment of the transmission lines and the cost implications of the OHL annealing can be used to identify the optimal [
The feature selection and extraction play crucial roles in the prediction framework. In this paper, mRMR and SDAE, which are well-known feature selection and extraction methods, are adapted, respectively. Despite the extensive research on the development of the advanced feature selection and extraction techniques in various power-system-related prediction applications [
The proposed prediction framework of DTLR is validated for a direct prediction application of DTLR, in which only meteorological variables and the DTLR formulations are employed. However, considering the generality of the framework, other features such as OHL sag/tension as well as NWP can be incorporated in the framework to improve the performance of DTLR forecasting further. Furthermore, using the NWP can also facilitate extending the prediction horizon, which is beneficial for day-ahead unit commitment and economic dispatching.
This paper has proposed a deep-learning-based probabilistic prediction model of DTLR for hour-ahead power system operation problems. The latent variables, obtained in the process of calculating the DTLR values, are considered as new predictors of the proposed model, while SDAE is employed for feature learning and extraction. A training strategy is devised to train the DNN, and a cost function is put forward to train the prediction model in a probabilistic manner. The proposed prediction framework considers no hypothesis about the uncertainty of the DTLR. Simulation results confirm the efficacy of the proposed model and its superiority compared with the benchmark models.
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