Platform-Independent Web Application for Short-Term Electric Power Load Forecasting on 33/11 kV Substation Using Regression Tree

Veeramsetty, Venkataramana; Sai Pavan Kumar, Modem; Salkuti, Surender Reddy

doi:10.3390/computers11080119

Open AccessArticle

Platform-Independent Web Application for Short-Term Electric Power Load Forecasting on 33/11 kV Substation Using Regression Tree

by

Venkataramana Veeramsetty

^1,†

,

Modem Sai Pavan Kumar

^2,† and

Surender Reddy Salkuti

^3,*,†

¹

Center for AI and Deep Learning, Department of Electrical and Electronics Engineering, SR University, Warangal 506371, India

²

Department of Electrical and Electronics Engineering, SR Engineering College, Warangal 506371, India

³

Department of Railroad and Electrical Engineering, Woosong University, Daejeon 34606, Korea

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Computers 2022, 11(8), 119; https://doi.org/10.3390/computers11080119

Submission received: 18 June 2022 / Revised: 19 July 2022 / Accepted: 25 July 2022 / Published: 29 July 2022

(This article belongs to the Special Issue Computing, Electrical and Industrial Systems 2022)

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Abstract

:

Short-term electric power load forecasting is a critical and essential task for utilities in the electric power industry for proper energy trading, which enables the independent system operator to operate the network without any technical and economical issues. From an electric power distribution system point of view, accurate load forecasting is essential for proper planning and operation. In order to build most robust machine learning model to forecast the load with a good accuracy irrespective of weather condition and type of day, features such as the season, temperature, humidity and day-status are incorporated into the data. In this paper, a machine learning model, namely a regression tree, is used to forecast the active power load an hour and one day ahead. Real-time active power load data to train and test the machine learning models are collected from a 33/11 kV substation located in Telangana State, India. Based on the simulation results, it is observed that the regression tree model is able to forecast the load with less error.

Keywords:

load forecasting; regression tree; hour-ahead market; day-ahead market; machine learning

1. Introduction

An electric power distribution substation takes the power from one or more transmission or subtransmission lines and delivers this power to residential, commercial, and industrial customers through multiple feeders. Short-term load forecasting at the distribution level estimates the active power load on a substation in a time horizon ranging from 30 min to 1 week [1]. The load forecasting of a distribution system gives advance alarms to the operator about the overloading of feeders and substations. Load forecasting helps the distribution substation operator to schedule and dispatch the storage batteries to shave the peak load in a smart grid environment [2]. Electrical power load forecasting is classified as very short term, short term, medium term and long term based on the length of the prediction horizon [3,4,5]. Due to deregulated power system structure and more liberalization in energy markets, electric power load forecasting has become more essential [6]. Long-term load forecasting is generally used for planning and investment profitability analysis, determining upcoming sites, or acquiring fuel sources for production plants. Medium-term load forecasting is usually preferred for risk management, balance sheet calculations, and derivatives pricing [7]. An accurate short-term load forecasting will help an electric power distribution utility to optimize the power grid load and strengthen reliability, reduce the electricity consumption cost and emphasize electric energy trading possibilities.

Forecasting the distribution-level load is far more challenging than forecasting the system-level load, such as the Telangana State’s electric power demand, due to the intricate load characteristics, huge number of nodes, and probable switching actions in distribution systems. Since the end-user behaviour has a far greater influence on distribution systems than it does on transmission systems, the load profiles of distribution systems will have more stochastically abrupt departures. Operating an independent distribution system successfully necessitates significantly more precise and high-resolution load forecasting than today’s approach can deliver [8]. Load estimates over a vast region are highly accurate because the aggregated load is steady and consistent. The distribution-level load, on the other hand, may be dominated by a few major clients, such as industrial businesses or schools, and the load pattern may not be as regular as that of a vast region. Furthermore, due to reconfigurations caused by switching activities, the load may be temporarily moved from one feeder to another, causing significant changes in distribution-level load profiles and affecting the trend at a given time. The main challenge in electric power load forecasting is data loss. Some works are available in the literature to deal with data loss, e.g., by designing a voltage hierarchical controller against communication delay and data loss [9]. The communication delay was treated using delay-tolerant power compensation control (DTPCC) in [10]. PCC uses normal PCC for effective operation when the communication delay is within the maximum tolerable communication delay, or switches to predictive PCC under abnormal communication delay conditions. However, in that paper, the authors collected historical data from the distribution company not through any communication channels.

Four general categories are identified for short-term load forecasting, i.e., similar day, variable selection, hierarchical forecasting, and weather station selection [11]. The similar-day technique identifies the load data as a set of related daily load profiles, whereas the variable selection method assumes that the load data act as a series of variables that are either correlated or independent of one another. The hierarchical technique, on the other hand, treats data as an aggregated load that is extremely variable due to changes in the load at the lower levels of the hierarchy. Finally, weather station selection is a strategy for determining which weather data are best fitted into the load model [12]. Load forecasting is regarded as one of the most critical duties for power system operators in the demand management system (DMS) as shown in Figure 1.

Many researchers have been working on short-term load forecasting of distribution systems. An ANN-based methodology was developed in [13] to forecast the load on a 33/11 kV substation near Kakatiya University in Warangal, Telangana State. In that study, the authors used the load from the previous three hours and the load at the same time but in the previous four days as input features. Load forecasting on an electric power distribution system using various regression models was proposed in [14] by considering the load from the previous three hours and the load at the same time but on the previous day as input features. Short-term load forecasting on an electric power distribution system using factor analysis and long short-term memory was proposed in [15] by considering the load from the previous three hours, the load at the same time but in the previous three days, and the load at the same time but in the previous three weeks as input features. Electric power load forecasting at the distribution level using a random forest and gated recurrent unit was proposed in [16], by considering previous three hours load, load at same time but previous three days and load at same time but previous three weeks as input features. Electric power load forecasting at the distribution level using a correlation concept and an ANN was proposed in [17] by considering the load from the previous two hours and the load at the same time but in the previous three days as input features. Similarly, the active power demand on a 33/11 kV electric power distribution system using principal component analysis and a recurrent neural network was proposed in [18], by considering the load from the previous three hours, the load at the same time but in previous three days and the load at the same time but in previous three weeks as input features.

Electric power load forecasting on a medium voltage level based on regression models and ANN was proposed in [19] using time series DSO telemetry data and weather records from the Portuguese Institute of Sea and Atmosphere, and applied to the urban area of Évora, one of Portugal’s first smart cities. A new top-down algorithm based on a similar day-type method to compute an accurate short-term distribution loads forecast, using only SCADA data from transmission grid substations was proposed in [20]. That study was evaluated on the RBTS test system with real power consumption data to demonstrate its accuracy. A convolutional-neural-network-based load forecasting methodology was proposed in [21]. Electric demand forecasting with a jellyfish search extreme learning machine, a Harris hawks extreme learning machine, and a flower pollination extreme learning machine was discussed in [22]. Electric power load forecasting using gated recurrent units with multisource data was discussed in [23]. Short-term load forecasting using a niche immunity lion algorithm and convolutional neural network was studied in [24]. Electricity demand forecasting using a dynamic adaptive entropy-based weighting was discussed in [25]. A demand-side management technique by identifying and mitigating the peak load of a building was studied in [26]. Electric power demand forecasting using a vector autoregressive state-space model was discussed in [27].

Electric power load forecasting using a random forest model was discussed in [28]. In that study, authors considered wind speed, wind direction, humidity, temperature, air pressure, and irradiance as input features. Electric power load forecasting using a group method of data handling and support vector regression was discussed in [29]. The electric power load prediction at the building and district levels for day-ahead energy management using a genetic algorithm (GA) and artificial neural network (ANN) power predictions was discussed in [30]. Short-term electric power load forecasting using feature engineering, Bayesian optimization algorithms with a Bayesian neural network was discussed in [31]. Active power load forecasting using a sparrow search algorithm (ISSA), Cauchy mutation, and opposition-based learning (OBL) and the long short-term memory (LSTM) network was studied in [32]. A new hybrid model was proposed in [33] based on CNN, LSTM, CNN_LSTM, and MLP for electric power load forecasting. All these methodologies provided valuable contributions towards the load forecasting problem, but these studies did not include the weather impact, season and day status in load forecasting.

The main contributions of this paper are as follows:

A new active power load dataset is developed to work on load forecasting problem by collecting the data from a 33/11 kV distribution substation in Godishala (village), Telangana State, India and available at https://data.mendeley.com/datasets/tj54nv46hj/1, accessed on 1 May 2022.
A machine learning model, i.e., a regression tree model is used to forecast the load on a 33/11 kV distribution substation in Godishala.
The active power load on a 33/11 kV substation is forecast one hour ahead based on input features L(T-1), L(T-2), L(T-24), L(T-48), day, season, temperature, and humidity.
The active power load on a 33/11 kV substation is forecast one day ahead based on input features L(T-24), L(T-48), day, season, temperature, and humidity.
A web application is developed based on a regression tree model to forecast the load on a 33/11 kV distribution substation in Godishala.
The impact of weather and days on short-term load forecasting is analysed by incorporating the season and day-status (weekday/weekend) in the data.
A practical implementation of the system in a prototype web application, where a regression tree model is deployed and execute the forecasts on a daily and hourly basis.

2. Methodology

This section presents the active power load data that are used to train and test the machine learning models. Furthermore, we discuss about the regression tree model that is used for electric power load forecasting on a 33/11 kV distribution substation in Godishala. This substation has four feeders: the first feeder (F1) supplies load to Godishala (town), the second feeder supplies load to Bommakal, the third feeder supplies load to the Godishala (village), and the fourth feeder (F4) supplies load to Raikal. The complete pipeline to develop the web application for short-term load forecasting using a regression model is presented in Figure 2.

2.1. Active Power Load Data Analysis

To train and test the machine learning model, active power load data are required. Hourly data consisting of voltage (V), current (I) and power factor (cos(

ϕ

)) from a 33/11 kV distribution substation in Godishala were collected from 1 January 2021 to 31 December 2021. Based on these data, the hourly active power load was calculated using Equation (1) and the sample load data are presented Table 1.

P = \sqrt{(} 3) V I c o s (ϕ)

(1)

2.2. Features Information and Data Preparation

In this paper, the load at a particular time of the day “L(T)” was predicted based on the last two hours of load data, i.e., L(T-1), L(T-2), the load at the same time but in the last two days, i.e., L(T-24), L(T-48), the temperature, the humidity, the season, and the day. Hence, data that were prepared based on collected information from the 33/11 kV substation were rearranged as shown in Table 2. This was the approach used for hour-ahead forecasting, whereas for day-ahead forecasting, the load at a particular time of the day “L(T)” was predicted based on the load at the same time but in the last two days, i.e., L(T-24), L(T-48), the temperature, the humidity, the season, and the day as presented in Table 3. The dataset for hour-ahead forecasting had 8712 samples, 8 input features and 1 output feature. Similarly, the dataset for day-ahead forecasting had 8712 samples, 6 input features and one output feature.

2.3. Machine Learning Models

In this paper, a regression tree model was used to forecast the load on a 33/11 kV substation one hour ahead and one day ahead. The problem discussed here is a regression problem. Models need to predict the load on the substation based on input features such as L(T-1), L(T-2), L(T-24), L(T-48), day, season, temperature, and humidity in the case of hour-ahead forecasting, and based on input features such as L(T-24), L(T-48), day, season, temperature, and humidity in the case of day-ahead forecasting. The performance of each machine learning model for electric power load forecasting on a 33/11 kV substation was observed in terms of the MSE as shown in Equation (2)

\begin{matrix} T r a i n i n g M S E = \frac{1}{n_{s}} \sum_{1}^{n_{s}} {(L^{a} (T) - L (T))}^{2} \\ T e s t i n g M S E = \frac{1}{n_{t}} \sum_{1}^{n_{t}} {(L^{a} (T) - L (T))}^{2} \end{matrix}

(2)

Regression Tree

A regression tree is basically a decision tree model that is used for the task of regression, which can be used to predict continuous valued output. In this paper, a regression tree was used to forecast the load. For a regression problem, a tree is constructed by splitting the input features such that the mean squared error shown in Equation (3) is minimum. A step-by-step procedure to construct the regression tree with sample data is explained in Appendix B, as mentioned in Algorithm 1. The performance of the regression tree model on short-term load forecasting problem for Godishala substation was measured in terms of error metrics such as the MSE [34], RMSE [35,36,37], and MAE [38]. A decision tree can also used for classification problems [39,40,41].

M S E = \frac{1}{n_{s}} {(L^{a} (T) - L^{p} (T))}^{2}

(3)

Algorithm 1 Regression tree model formulation

1:: Read data, initialize max-depth.
2:: for Depth $ϵ$ range(max-depth) do
3:: for Feature $ϵ$ data table do
4:: for Value $ϵ$ feature do
5:: Find the MSE for each unique split of the feature
6:: Find the best split among all feature values based on the minimum MSE
7:: Split the data table based on the feature corresponding to the best split
8:: Start building tree by identifying the root/decision node among all features
9:: end for
10:: end for
11:: end for

Figure 2. Workflow to develop web application.

3. Result Analysis

All the machine learning models were developed based on the data available in [42] using Google Colab. this section presents the data analysis, training and testing performance of machine learning models, and the web application developed to predict the load. Out of 8712 samples, 95% of samples were used for training and the remaining 5% of samples were used for testing. The data processing techniques for observing the data distribution and outliers and for data normalization were used before using these data to train and test the regression model. A stochastic gradient descent optimizer was used to train the regression models.

3.1. Regression Tree Model

The performance of the regression tree model that was developed to forecast the load L(T) based on features L(T-1), L(T-2), L(T-24), L(T-48), day and season status, temperature, and humidity was observed based on training and testing errors for hour-ahead load forecasting (HALF). The training and testing error metrics of the regression tree model are presented in Table 4 for HALF. From Table 4, it is observed that the regression model with a depth “5” had lowest testing MSE, i.e., 0.005 and was also well fitted without much difference between training and testing errors. Hence, the regression tree with a depth “5” was considered as the optimal model to deploy in a web application for hour-ahead forecasting. The complete architecture of the regression tree with a depth “5” for day-ahead load forecasting is shown in Figure 3.

Similarly, for day-ahead forecasting (DALF), the performance of the regression tree model that was developed to forecast the load L(T) based on features L(T-24), L(T-48), day and season status, temperature, and humidity was observed based on training and testing errors. The training and testing error metrics of the regression tree model are presented in Table 5 for DALF. From Table 5, it is observed that regression model with a depth “6” had lowest testing MSE, i.e., 0.00869 and was also well fitted without much difference between training and testing errors. Hence, the regression tree with a depth “6” was considered as the optimal model to deploy in a web application for day-ahead forecasting. The complete architecture of the regression tree with a depth “6” for day-ahead load forecasting is shown in Figure 4.

The distribution of the predicted load with the regression tree model having a training MSE of 0.004 and a testing MSE of 0.005 was compared with actual load samples for the training and testing data for HALF and is presented in Figure 5. Similarly, the distribution of the predicted load with the regression tree model having a training MSE of 0.0065 and a testing MSE of 0.00869 was compared with actual load samples for the training and testing data for DALF and is presented in Figure 6. From Figure 5 and Figure 6, it is observed that most of the predicted and actual load samples are overlapping each other.

The predicted load using the regression tree model was compared with the actual load on 31 December 2021 and presented in Figure 7. From Figure 7, it is observed that the predicted load using the regression tree model is almost following the actual load curve during night time but has more differences during day time, and also that the predicted load is slightly further away from the actual load curve in the case of DALF in comparison with HALF. As in the first case, the load was predicted one day earlier, i.e., a 24 h time horizon.

A web application was developed using the optimal regression tree models to predict the load one hour ahead and one day ahead for a real-time usage as a prototype and is shown in Figure 8. This web application is accessible through the link https://loadforecasting-godishala-rt.herokuapp.com/, accessed on 1 May 2022 or through the QR code shown in Figure 8.

3.2. Impact of Season and Day on Regression Model Prediction

The forecasting performance of the trained machine learning model for HALF on various seasons, i.e., rainy, winter, and summer, is presented in Figure 9. From Figure 9, it is observed that the developed regression tree model is able to forecast the load with almost the same level of error (with very minor changes in error) for all seasons.

The forecasting performance of the trained machine learning model for HALF on various seasons, i.e., rainy, winter, and summer, is presented in Figure 10. From Figure 10, it is observed that the developed regression tree model is able to forecast the load with almost the same level of error irrespective of whether it is weekday or weekend.

3.3. Comparative Analysis

The performance of the machine learning model, i.e., the regression tree model, was compared with a linear regression model in terms of training and testing mean squared error for both HALF and DALF and presented in Figure 11. From Figure 11, it is observed that the regression tree model is forecasting the load with less error in comparison with the linear regression model for both HALF and DALF. In case of DALF, the model is forecasting the load with more error than HALF, as the latter one forecast the load just one hour ahead. The regression tree model is well fitted without much difference between training and testing errors.

4. Conclusions

Electric power load forecasting one hour ahead and one day ahead are required for utilities to place a bid successfully in hour-ahead energy markets and day-ahead energy markets. In this paper, the active power on a 33/11 kV substation was predicted one hour ahead based on the load available in the last two hours and last two days at the time of prediction, and the day status, season status, temperature, and humidity. Similarly, the load was predicted one day ahead based on the load available in the last two days at the time of prediction, and the day status, season status, temperature, and humidity. A robust machine learning model was developed to forecast the load with good accuracy irrespective of the weather conditions and types of the day by incorporating features such as season, temperature, humidity, and day-status.

In this work, a machine learning model, i.e., a regression tree model was developed to predict the active power load on a 33/11 kV substation located in the Godishala village in Telangana State, India. Based on the results, it was observed that the regression tree model predicted the load one hour and one day ahead with less mean squared error in comparison with the linear regression model.

This work can be further extended by considering deep neural networks, sequence models, and conventional time series data prediction models. In this paper, the temperature and humidity data at the time of prediction were considered from an open-source website. However, we are currently further extending the model by integrating temperature and humidity forecasting models with the current load forecasting models.

Author Contributions

Conceptualization, V.V.; methodology, V.V.; software, V.V. and M.S.P.K.; validation, V.V.; formal analysis, V.V.; investigation, V.V.; resources, M.S.P.K.; data curation, M.S.P.K.; writing—original draft preparation, V.V.; writing—review and editing, V.V., M.S.P.K. and S.R.S.; visualization, V.V.; supervision, V.V.; project administration, V.V. All authors have read and agreed to the published version of the manuscript.

Funding

Woosong University’s Academic Research Funding—2022.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Active power load data used to train and test machine learning models are available at https://data.mendeley.com/datasets/tj54nv46hj/1, accessed on 7 February 2022.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

L(T-1)	Active power load one hour before the time of prediction
L(T-2)	Active power load two hours before the time of prediction
L(T-24)	Active power load one day before the time of prediction
L(T-48)	Active power load two days before the time of prediction
L(T)	Active power load at hour “T”
MSE	Mean squared error
HALF	Hour-ahead load forecasting
DALF	Day-ahead load forecasting
RMSE	Root-mean-square error
MSE	Mean absolute error
$L^{a} (T)$	Actual load at hour “T”
$L^{p} (T)$	Predicted load at hour “T”
$n_{s}$	Number of training samples
$n_{t}$	Number of testing samples

Appendix A. Conversion of Continuous Data into Categorical Data

In this section, the step-by-step procedure that was used to build the regression tree model is presented. For this purpose, a sample dataset that was built from a few samples of the original dataset is shown in Table A1.

Table A1. Sample data to build the regression tree.

L(T-24)	L(T-48)	DAY	SEASON	TEMP	HUMIDITY	L(T)
2176	412	1	1	67	88	432
2354	1829	0	1	68	88	2260
2777	2647	0	2	70	83	2681
3112	3203	1	2	75	67	3343
1663	1549	1	0	75	73	1579
1010	1027	0	0	71	93	1018

To convert continuous features into categorical features, multiple subtables were formed from Table A1. Each subtable consisted of one input feature and one target feature. We sorted each subtable in ascending order of input feature in that table. We calculated the average between every two continuous input feature values. We converted continuous input features into categorical features based on their average value. We found the mean squared value for each average and the average value that gave lowest “MSE” were treated as threshold for that feature.

Prepare the subtable for input feature L(T-24) and output feature L(T) as shown in Table A2.

Table A2. Sorted subtable—L(T-24) vs. L(T).

L(T-24)	L(T)
1010	1018
1663	1579
2176	432
2354	2260
2777	2681
3112	3343

Calculate the average between every two continuous input feature values for L(T-24) and the average values are [1336, 1919, 2265, 2566, 2945].
Convert the continuous input feature L(T-24) into a categorical feature based on the average value 1336 and Table A2, as shown in Table A3. The predicted value against each category of input feature is the average of all output variables for that category.

Table A3. Categorical subtable—L(T-24) vs. L(T).

L(T-24)	L(T)	$L^{P}$ (T)
<1336	1018	1018
≥1336	1579	$\frac{1579 + 432 + 2260 + 2681 + 3343}{5}$ = 2059
≥1336	432	2059
≥1336	2260	2059
≥1336	2681	2059
≥1336	3343	2059

Calculate the mean squared error based on the actual and predicted load values shown in Table A3 and presented below

\frac{{(1018 - 1018)}^{2} + {(1579 - 2059)}^{2} + {(432 - 2059)}^{2} + {(2260 - 2059)}^{2} + {(2681 - 2059)}^{2} + {(3343 - 2059)}^{2}}{6}

= 825,622.

Convert the continuous input feature L(T-24) into a categorical feature based on the average value 1919 and Table A2, as shown in Table A4. The predicted value against each category of input feature is the average of all output variables for that category.

Table A4. Categorical subtable—L(T-24) vs. L(T) with average value 1919.

L(T-24)	L(T)	$L^{P}$ (T)
<1919	1018	1298
<1919	1579	1298
≥1919	432	$\frac{432 + 2260 + 2681 + 3343}{4}$ = 2179
≥1919	2260	2179
≥1919	2681	2179
≥1919	3343	2179

Calculate the mean squared error based on the actual and predicted load values shown in Table A4 and presented below

\frac{{(1018 - 1298)}^{2} + {(1579 - 1298)}^{2} + {(432 - 2179)}^{2} + {(2260 - 2179)}^{2} + {(2681 - 2179)}^{2} + {(3343 - 2179)}^{2}}{6}

= 803,903.

Convert the continuous input feature L(T-24) into a categorical feature based on the average value 2265 and Table A2, as shown in Table A5. The predicted value against each category of input feature is the average of all output variables for that category.

Table A5. Categorical subtable—L(T-24) vs. L(T) with average value 2265.

L(T-24)	L(T)	$L^{P}$ (T)
<2265	1018	$\frac{1018 + 1579 + 432}{3}$ = 1010
<2265	1579	1010
<2265	432	1010
≥2265	2260	$\frac{2260 + 2681 + 3343}{3}$ = 2761
≥2265	2681	2761
≥2265	3343	2761

Calculate the mean squared error based on the actual and predicted load values shown in Table A5 and presented below

\frac{{(1018 - 1010)}^{2} + {(1579 - 1010)}^{2} + {(432 - 1010)}^{2} + {(2260 - 2761)}^{2} + {(2681 - 2761)}^{2} + {(3343 - 2761)}^{2}}{6}

= 209,064.

Convert the continuous input feature L(T-24) into a categorical feature based on the average value 2566 and Table A2, as shown in Table A6. The predicted value against each category of input feature is the average of all output variables for that category.

Table A6. Categorical subtable—L(T-24) vs. L(T) with average value 2566.

L(T-24)	L(T)	$L^{P}$ (T)
<2566	1018	$\frac{1018 + 1579 + 432 + 2260}{4}$ = 1322
<2566	1579	1322
<2566	432	1322
<2566	2260	1322
≥2566	2681	$\frac{2681 + 3343}{2}$ = 3012
≥2566	3343	3012

Calculate the mean squared error based on the actual and predicted load values shown in Table A6 and presented below

\frac{{(1018 - 1322)}^{2} + {(1579 - 1322)}^{2} + {(432 - 1322)}^{2} + {(2260 - 1322)}^{2} + {(2681 - 3012)}^{2} + {(3343 - 3012)}^{2}}{6}

= 341,675.

Convert the continuous input feature L(T-24) into a categorical feature based on the average value 2945 and Table A2, as shown in Table A7. The predicted value against each category of input feature is the average of all output variables for that category.

Table A7. Categorical subtable—L(T-24) vs. L(T) with average value 2945.

L(T-24)	L(T)	$L^{P}$ (T)
<2945	1018	$\frac{1018 + 1579 + 432 + 2260 + 2681}{5}$ = 1594
<2945	1579	1594
<2945	432	1594
<2945	2260	1594
<2945	2681	1594
≥2945	3343	3343

Calculate the mean squared error based on the actual and predicted load values shown in Table A7 and presented below

\frac{{(1018 - 1594)}^{2} + {(1579 - 1594)}^{2} + {(432 - 1594)}^{2} + {(2260 - 1594)}^{2} + {(2681 - 1594)}^{2} + {(3343 - 3343)}^{2}}{6}

= 551,591.

Prepare subtable for input feature L(T-48) and output feature L(T) as shown in Table A8.

Table A8. Sorted subtable—L(T-48) vs. L(T).

L(T-48)	L(T)
412	432
1027	1018
1549	1579
1829	2260
2647	2681
3203	3343

Calculate the average between every two continuous input feature values for L(T-48) and the average values are [720, 1288, 1689, 2238, 2925].
Convert the continuous input feature L(T-48) into a categorical feature based on the average value 720 and Table A8, as shown in Table A9. The predicted value against each category of input feature is the average of all output variables for that category.

Table A9. Categorical subtable—L(T-48) vs. L(T).

L(T-48)	L(T)	$L^{P}$ (T)
<720	432	432
≥720	1018	$\frac{1018 + 1579 + 2260 + 2681 + 3343}{5}$ = 2176.2
≥720	1579	2176.2
≥720	2260	2176.2
≥720	2681	2176.2
≥720	3343	2176.2

Calculate the mean squared error based on the actual and predicted load values shown in Table A9 and presented below

\frac{{(432 - 432)}^{2} + {(1018 - 2176.2)}^{2} + {(1579 - 2176.2)}^{2} + {(2260 - 2176.2)}^{2} + {(2681 - 2176.2)}^{2} + {(3343 - 2176.2)}^{2}}{6}

= 553,557.1333.

Convert the continuous input feature L(T-48) into a categorical feature based on the average value 1288 and Table A8, shown in Table A10. The predicted value against each category of input feature is the average of all output variables for that category.

Table A10. Categorical subtable—L(T-48) vs. L(T).

L(T-48)	L(T)	$L^{P}$ (T)
<1288	432	$\frac{432 + 1018}{2}$ = 725
<1288	1018	725
≥1288	1579	$\frac{1579 + 2260 + 2681 + 3343}{4}$ = 2465.75
≥1288	2260	2465.75
≥1288	2681	2465.75
≥1288	3343	2465.75

Calculate the mean squared error based on the actual and predicted load values shown in Table A10 and presented below

\frac{{(432 - 725)}^{2} + {(1018 - 725)}^{2} + {(1579 - 2465.75)}^{2} + {(2260 - 2465.75)}^{2} + {(2681 - 2465.75)}^{2} + {(3343 - 2465.75)}^{2}}{6}

= 302,709.4583.

Convert the continuous input feature L(T-48) into a categorical feature based on the average value 1689 and Table A8, as shown in Table A11. The predicted value against each category of input feature is the average of all output variables for that category.

Table A11. Categorical subtable—L(T-48) vs. L(T).

L(T-48)	L(T)	$L^{P}$ (T)
<1689	432	$\frac{432 + 1018 + 1579}{3}$ = 1009.666667
<1689	1018	1009.666667
<1689	1579	1009.666667
≥1689	2260	$\frac{2260 + 2681 + 3343}{3}$ = 2761.333333
≥1689	2681	2761.333333
≥1689	3343	2761.333333

Calculate the mean squared error based on the actual and predicted load values shown in Table A11 and presented below

\frac{{(432 - 1009.67)}^{2} + {(1018 - 1009.67)}^{2} + {(1579 - 1009.67)}^{2} + {(2260 - 2761.33)}^{2} + {(2681 - 2761.33)}^{2} + {(3343 - 2761.33)}^{2}}{6}

= 209,005.5556.

Convert the continuous input feature L(T-48) into a categorical feature based on the average value 2238 and Table A8, as shown in Table A12. The predicted value against each category of input feature is the average of all output variables for that category.

Table A12. Categorical subtable—L(T-48) vs. L(T).

L(T-48)	L(T)	$L^{P}$ (T)
<2238	432	$\frac{432 + 1018 + 1579 + 2260}{4}$ = 1322.25
<2238	1018	1322.257
<2238	1579	1322.25
<2238	2260	1322.25
≥2238	2681	$\frac{2681 + 3343}{2}$ = 3012
≥2238	3343	3012

Calculate the mean squared error based on the actual and predicted load values shown in Table A12 and presented below

\frac{{(432 - 1322.25)}^{2} + {(1018 - 1322.25)}^{2} + {(1579 - 1322.25)}^{2} + {(2260 - 1322.25)}^{2} + {(2681 - 3012)}^{2} + {(3343 - 3012)}^{2}}{6}

= 341,588.4583.

Convert the continuous input feature L(T-48) into a categorical feature based on the average value 2925 and Table A8, as shown in Table A13. The predicted value against each category of input feature is the average of all output variables for that category.

Table A13. Categorical subtable—L(T-48) vs. L(T).

L(T-48)	L(T)	$L^{P}$ (T)
<2925	432	$\frac{432 + 1018 + 1579 + 2260 + 2681}{5}$ = 1594
<2925	1018	1594
<2925	1579	1594
<2925	2260	1594
<2925	2681	1594
≥2925	3343	3343

Calculate the mean squared error based on the actual and predicted load values shown in Table A13 and presented below

\frac{{(432 - 1594)}^{2} + {(1018 - 1594)}^{2} + {(1579 - 1594)}^{2} + {(2260 - 1594)}^{2} + {(2681 - 1594)}^{2} + {(3343 - 3343)}^{2}}{6}

= 551,228.3333.

Prepare subtable for input feature L(TEMP) and output feature L(T) and shown in Table A14

Table A14. Sorted subtable—L(TEMP) vs. L(T).

L(TEMP)	L(T)
67	432
68	2260
70	2681
71	1018
75	3343
75	1579

Calculate average between every two continuous input feature values for L(TEMP) and the average values are [67.5, 69, 70.5, 73, 75].
Convert the continuous input feature L(TEMP) into a categorical feature based on the average value 67.5 and Table A14, as shown in Table A15. The predicted value against each category of input feature is the average of all output variables for that category.

Table A15. Categorical subtable—L(TEMP) vs. L(T).

L(TEMP)	L(T)	$L^{P}$ (T)
<67.5	432	432
≥67.5	2260	$\frac{2260 + 2681 + 1018 + 3343 + 1579}{5}$ = 2176.2
≥67.5	2681	2176.2
≥67.5	1018	2176.2
≥67.5	3343	2176.2
≥67.5	1579	2176.2

Calculate the mean squared error based on the actual and predicted load values shown in Table A15 and presented below

\frac{{(432 - 432)}^{2} + {(2260 - 2176.2)}^{2} + {(2681 - 2176.2)}^{2} + {(1018 - 2176.2)}^{2} + {(3343 - 2176.2)}^{2} + {(1579 - 2176.2)}^{2}}{6}

= 553,557.

Convert the continuous input feature L(TEMP) into a categorical feature based on the average value 69 and Table A14, as shown in Table A16. The predicted value against each category of input feature is the average of all output variables for that category.

Table A16. Categorical subtable—L(TEMP) vs. L(T).

L(TEMP)	L(T)	$L^{P}$ (T)
<69	432	$\frac{432 + 2260}{2}$ = 1346
<69	2260	1346
≥69	2681	$\frac{2681 + 1018 + 3343 + 1579}{4}$ = 2155.25
≥69	1018	2155.25
≥69	3343	2155.25
≥69	1579	2155.25

Calculate the mean squared error based on the actual and predicted load values shown in Table A16 and presented below

\frac{{(432 - 1346)}^{2} + {(2260 - 1346)}^{2} + {(2681 - 2155.25)}^{2} + {(1018 - 2155.25)}^{2} + {(3343 - 2155.25)}^{2} + {(1579 - 2155.25)}^{2}}{6}

= 830,559.

Convert the continuous input feature L(TEMP) into a categorical feature based on the average value 70.5 and Table A14, as shown in Table A17. The predicted value against each category of input feature is the average of all output variables for that category.

Table A17. Categorical subtable—L(TEMP) vs. L(T).

L(TEMP)	L(T)	$L^{P}$ (T)
<70.5	432	$\frac{432 + 2260 + 2681}{3}$ = 1791
<70.5	2260	1791
<70.5	2681	1791
≥70.5	1018	$\frac{1018 + 3343 + 1579}{3}$ = 1980
≥70.5	3343	1980
≥70.5	1579	1980

Calculate the mean squared error based on the actual and predicted load values shown in Table A17 and presented below

\frac{{(432 - 1791)}^{2} + {(2260 - 1791)}^{2} + {(2681 - 1791)}^{2} + {(1018 - 1980)}^{2} + {(3343 - 1980)}^{2} + {(1579 - 1980)}^{2}}{6}

= 967,159.

Convert the continuous input feature L(TEMP) into a categorical feature based on the average value 73 and Table A14, as shown in Table A18. The predicted value against each category of input feature is the average of all output variables for that category.

Table A18. Categorical subtable—L(TEMP) vs. L(T).

L(TEMP)	L(T)	$L^{P}$ (T)
<73	432	$\frac{432 + 2260 + 2681 + 1018}{4}$ = 1597.75
<73	2260	1597.75
<73	2681	1597.75
<73	1018	1597.75
≥73	3343	$\frac{3343 + 1579}{2}$ = 2461
≥73	1579	2461

Calculate the mean squared error based on the actual and predicted load values shown in Table A18 and presented below

\frac{{(432 - 1597.75)}^{2} + {(2260 - 1597.75)}^{2} + {(2681 - 1597.75)}^{2} + {(1018 - 1597.75)}^{2} + {(3343 - 2461)}^{2} + {(1579 - 2461)}^{2}}{6}

= 810,489.

Convert the continuous input feature L(TEMP) into a categorical feature based on the average value 75 and Table A14, as shown in Table A19. The predicted value against each category of input feature is the average of all output variables for that category.

Table A19. Categorical subtable—L(TEMP) vs. L(T).

L(TEMP)	L(T)	$L^{P}$ (T)
<75	432	$\frac{432 + 2260 + 2681 + 1018 + 3343}{5}$ = 1946.8
<75	2260	1946.8
<75	2681	1946.8
<75	1018	1946.8
<75	3343	1946.8
≥75	1579	1579

Calculate the mean squared error based on the actual and predicted load values shown in Table A19 and presented below

\frac{{(432 - 1946.8)}^{2} + {(2260 - 1946.8)}^{2} + {(2681 - 1946.8)}^{2} + {(1018 - 1946.8)}^{2} + {(3343 - 1946.8)}^{2} + {(1579 - 1579)}^{2}}{6}

= 957,301.

Prepare subtable for input feature Humidity and output feature L(T) and shown in Table A20

Table A20. Sorted subtable—Humidity vs. L(T).

Humidity	L(T)
67	3343
73	1579
83	2681
88	432
88	2260
93	1018

Calculate average between every two continuous input feature values for Humidity and the average values are [70, 78, 85.5, 88, 90.5].
Convert the continuous input feature Humidity into a categorical feature based on the average value 70 and Table A20, as shown in Table A21. The predicted value against each category of input feature is the average of all output variables for that category.

Table A21. Categorical subtable—Humidity vs. L(T).

Humidity	L(T)	$L^{P}$ (T)
<70	3343	3343
≥70	1579	$\frac{432 + 2260 + 1018 + 3343 + 1579}{5}$ = 1594
≥70	2681	1594
≥70	432	1594
≥70	2260	1594
≥70	1018	1594

Calculate the mean squared error based on the actual and predicted load values shown in Table A21 and presented below

\frac{{(3343 - 3343)}^{2} + {(1579 - 1594)}^{2} + {(2681 - 1594)}^{2} + {(432 - 1594)}^{2} + {(2260 - 1594)}^{2} + {(1018 - 1594)}^{2}}{6}

= 551,228.

Convert the continuous input feature Humidity into a categorical feature based on the average value 78 and Table A20, as shown in Table A22. The predicted value against each category of input feature is the average of all output variables for that category.

Table A22. Categorical subtable—Humidity vs. L(T).

Humidity	L(T)	$L^{P}$ (T)
<78	3343	$\frac{3343 + 1579}{2}$ = 2461
<78	1579	2461
≥78	2681	$\frac{2681 + 432 + 2260 + 1018}{4}$ = 1597.75
≥78	432	1597.75
≥78	2260	1597.75
≥78	1018	1597.75

Calculate the mean squared error based on the actual and predicted load values shown in Table A22 and presented below

\frac{{(3343 - 2461)}^{2} + {(1579 - 2461)}^{2} + {(2681 - 1597.75)}^{2} + {(432 - 1597.75)}^{2} + {(2260 - 1597.75)}^{2} + {(1018 - 1597.75)}^{2}}{6}

= 810,489.

Convert the continuous input feature Humidity into a categorical feature based on the average value 85.5 and Table A20, as shown in Table A23. The predicted value against each category of input feature is the average of all output variables for that category.

Table A23. Categorical subtable—Humidity vs. L(T).

Humidity	L(T)	$L^{P}$ (T)
<85.5	3343	$\frac{3343 + 1579 + 2681}{3}$ = 2534.33
<85.5	1579	2534.33
<85.5	2681	2534.33
≥85.5	432	$\frac{432 + 2260 + 1018}{3}$ = 1236.67
≥85.5	2260	1236.67
≥85.5	1018	1236.67

Calculate the mean squared error based on the actual and predicted load values shown in Table A23 and presented below

\frac{{(3343 - 2534.33)}^{2} + {(1579 - 2534.33)}^{2} + {(2681 - 2534.33)}^{2} + {(432 - 1236.67)}^{2} + {(2260 - 1236.67)}^{2} + {(1018 - 1236.67)}^{2}}{6}

= 555,105

Convert the continuous input feature Humidity into a categorical feature based on the average value 88 and Table A20, as shown in Table A24. The predicted value against each category of input feature is the average of all output variables for that category.

Table A24. Categorical subtable—Humidity vs. L(T).

Humidity	L(T)	$L^{P}$ (T)
<88	3343	$\frac{3343 + 1579 + 2681 + 432}{4}$ = 2008.75
<88	1579	2008.75
<88	2681	2008.75
<88	432	2008.75
≥88	2260	$\frac{2260 + 1018}{2}$ = 1639
≥88	1018	1639

Calculate the mean squared error based on the actual and predicted load values shown in Table A24 and presented below

\frac{{(3343 - 2008.75)}^{2} + {(1579 - 2008.75)}^{2} + {(2681 - 2008.75)}^{2} + {(432 - 2008.75)}^{2} + {(2260 - 1639)}^{2} + {(1018 - 1639)}^{2}}{6}

= 945,708

Convert the continuous input feature Humidity into a categorical feature based on the average value 90.5 and Table A20, as shown in Table A25. The predicted value against each category of input feature is the average of all output variables for that category.

Table A25. Categorical subtable—Humidity vs. L(T).

Humidity	L(T)	$L^{P}$ (T)
<90.5	3343	$\frac{3343 + 1579 + 2681 + 432 + 2260}{5}$ = 2059
<90.5	1579	2059
<90.5	2681	2059
<90.5	432	2059
<90.5	2260	2059
≥90.5	1018	1018

Calculate the mean squared error based on the actual and predicted load values shown in Table A25 and presented below

\frac{{(3343 - 2059)}^{2} + {(1579 - 2059)}^{2} + {(2681 - 2059)}^{2} + {(432 - 2059)}^{2} + {(2260 - 2059)}^{2} + {(1018 - 1018)}^{2}}{6}

= 825,578

From all the above calculations, the minimum MSE value for the feature “T-24” is 209,064 for the split ≥2265, the minimum MSE value for the feature “T-48” is 209,006 for the split ≥1689, the minimum MSE value for the feature “Temperature” is 553,557 for the split ≥67.5, the minimum MSE value for the feature “Humidity” is 551,228 for the split ≥70. Hence, these splits against each feature were used to convert the continuous data shown in Table A1 into categorical data, as shown in Table A26. Furthermore, the MSE value for the day with categories (1 and 0) is 965,922 and the MSE value for the season with categories (0, 1, and 2) is 747157. All the MSE values are presented in Table A1 with bold font.

Table A26. Sample categorical data to build regression tree.

L(T-24)	L(T-48)	DAY	SEASON	TEMP	HUMIDITY	L(T)
<2265	<1689	1	1	<67.5	≥70	432
≥2265	≥1689	0	1	≥67.5	≥70	2260
≥2265	≥1689	0	2	≥67.5	≥70	2681
≥2265	≥1689	1	2	≥67.5	<70	3343
<2265	<1689	1	0	≥67.5	≥70	1579
<2265	<1689	0	0	≥67.5	≥70	1018
209,064	209,006	965,922	747,157	553,557	551,228	–

Appendix B. Regression Tree Model Formulation

From Table A26, we observe that L(T-48) has a minimum MSE value, i.e., 209,006 in comparison with all the remaining features. Hence, the input feature L(T-48) is considered as a root node for the regression tree and that node has two branches ≥1689 and <1689. In order to identify the decision node under each branch, Table A26 is divided into two subtables, presented in Table A27 and Table A28.

Table A27. Subtable: L(T-48) < 1689.

L(T-24)	DAY	SEASON	TEMP	HUMIDITY	T
<2265	1	1	<67.5	≥70	432
<2265	1	0	≥67.5	≥70	1579
<2265	0	0	≥67.5	≥70	1018

Table A28. Subtable: L(T-48) ≥ 1689.

L(T-24)	DAY	SEASON	TEMP	HUMIDITY	T
≥2265	0	1	≥67.5	≥70	2260
≥2265	0	2	≥67.5	≥70	2681
≥2265	1	2	≥67.5	<70	3343

In order to identify the decision node among L(T-24), day, season, temperature, and humidity under branch <1689 Table A28 is further divided into multiple subtables based on each input feature.
A subtable based on input feature L(T-24) and target variable L(T) is presented in Table A29. From Table A29, it is observed that input feature L(T-24) has an MSE value of 219,303.

Table A29. L(T-24) vs. L(T) for L(T-48) < 1689.

L(T-24)	L(T) Prediction	Squared Error	MSE
<2265	432	1010	333,699
<2265	1579	1010	324,140	219,303
<2265	1018	1010	69

A subtable based on input feature day and target variable L(T) is presented in Table A30. From Table A30, it is observed that input feature day has an MSE value of 219,268.

Table A30. Day vs. L(T) for L(T-48) < 1689.

Day	L(T) Prediction	Squared Error	MSE
1	432	1005.5	328,902.25
1	1579	1005.5	328,902.25	219,268
0	1018	1018	0

A subtable based on input feature season and target variable L(T) is presented in Table A31. From Table A31, it is observed that input feature season has an MSE value of 52,454.

Table A31. Season vs. L(T) for L(T-48) < 1689.

Season	L(T) Prediction	Squared Error	MSE
1	432	432	0
0	1579	1298.5	78,680.25	52,454
0	1018	1298.5	78,680.25

A subtable based on input feature temperature and target variable L(T) is presented in Table A32. From Table A32, it is observed that input feature temperature has an MSE value of 52,454.

Table A32. Temperature vs. L(T) for L(T-48) < 1689.

Temperature	L(T) Prediction	Squared Error	MSE
<67.5	432	432	0
≥67.5	1579	1298.5	78,680.25	52454
≥67.5	1018	1298.5	78,680.25

A subtable based on input feature humidity and target variable L(T) is presented in Table A33. From Table A33, it is observed that input feature humidity has an MSE value of 219,303.

Table A33. Humidity vs. L(T) for L(T-48) < 1689.

Humidity	L(T) Prediction	Squared Error	MSE
≥70	432	1009.67	333,698.78
≥70	1579	1009.67	324,140.44	219,303
≥70	1018	1009.67	69.44

It is observed from the above calculations that season and temperature have a minimum MSE, i.e., 52,454. Here, season is considered as a decision node under branch L(T-48) < 1689. Now, the node season has two branches, i.e., season “1” and “0”. In order to identify the decision/leaf node under each branch, Table A27 is divided into two subtables, presented in Table A34 and Table A35. From Table A34, it is observed that the branch corresponding to season “1” has a leaf node with value 432.

Table A34. Subtable: L(T-48) < 1689 and season = “1”.

T-24	DAY	TEMP	HUMIDITY	T
<2265	1	<67.5	≥70	432

Table A35. Subtable: L(T-48) < 1689 and season = “0”.

T-24	DAY	TEMP	HUMIDITY	T
<2265	1	≥67.5	≥70	1579
<2265	0	≥67.5	≥70	1018

In order to identify the decision node among L(T-24), day, temperature, and humidity under branch season “0”, Table A35 is divided into multiple subtables with respect to each feature.

A subtable based on input feature L(T-24) and target variable L(T) is presented in Table A36. From Table A36, it is observed that input feature L(T-24) has an MSE value of 78,680.

Table A36. L(T-24) vs. L(T) for L(T-48) < 1689 and season “0”.

L(T-24)	L(T)	Prediction	Squared Error	MSE
<2265	1579	1298.5	78,680	78,680
<2265	1018	1298.5	78,680

A subtable based on input feature day and target variable L(T) is presented in Table A37. From Table A37, it is observed that input feature day has an MSE value of 0.

Table A37. Day vs. L(T) for L(T-48) < 1689 and season “0”.

Day	L(T)	Prediction	Squared Error	MSE
1	1579	1579	0	0
0	1018	1018	0

A subtable based on input feature temperature and target variable L(T) is presented in Table A38. From Table A38, it is observed that input feature temperature has an MSE value of 78,680.

Table A38. Temperature vs. L(T) for L(T-48) < 1689 and season “0”.

Temperature	L(T)	Prediction	Squared Error	MSE
≥67.5	1579	1298.5	78,680	78,680
≥67.5	1018	1298.5	78,680

A subtable based on input feature humidity and target variable L(T) is presented in Table A39. From Table A39, it is observed that input feature humidity has an MSE value of 78,680.

Table A39. Humidity vs. L(T) for L(T-48) < 1689 and season “0”.

Humidity	L(T)	Prediction	Squared Error	MSE
≥70	1579	1298.5	78,680	78,680
≥70	1018	1298.5	78,680

It is observed from the above calculations that feature “Day” has a minimum MSE, i.e., 0. Here, “Day” is considered as decision node under the season “0” branch. Now, node “Day” has two branches, i.e., day “0” and “1” as presented in Table A37. From Table A37, it is observed that the branch corresponding to day “1” has a leaf node with value 1579 and day “0” has a leaf node with value 1018.

In order to identify the decision node among L(T-24), day, season, temperature, and humidity under branch ≥1689, Table A40 is further divided into multiple subtables based on each input feature.
A subtable based on input feature L(T-24) and target variable L(T) is presented in Table A40. From Table A40, it is observed that input feature L(T-24) has an MSE value of 198,708.

Table A40. L(T-24) vs. L(T) for L(T-48) ≥ 1689.

L(T-24)	L(T)	Prediction	Squared Error	MSE
≥2265	2260	2761.33	251,335
≥2265	2681	2761.33	6453	198,708
≥2265	3343	2761.33	338,336

A subtable based on input feature day and target variable L(T) is presented in Table A41. From Table A41, it is observed that input feature day has an MSE value of 29,540.

Table A41. Day vs. L(T) for L(T-48) ≥ 1689.

Day	L(T) Prediction	Squared Error	MSE
0	2260	2470.5	44310
0	2681	2470.5	44310	29,540
1	3343	3343	0

A subtable based on input feature season and target variable L(T) is presented in Table A42. From Table A42, it is observed that input feature season has an MSE value of 73,041.

Table A42. Season vs. L(T) for L(T-48) ≥ 1689.

Season	L(T) Prediction	Squared Error	MSE
1	2260	2260	0
2	2681	3012	109,561	73,041
2	3343	3012	109,561

A subtable based on input feature temperature and target variable L(T) is presented in Table A43. From Table A43, it is observed that input feature temperature has an MSE value of 198,708.

Table A43. Temperature vs. L(T) for L(T-48) ≥ 1689.

Temperature	L(T) Prediction	Squared Error	MSE
≥67.5	2260	2761.33	251,335
≥67.5	2681	2761.33	6453	198,708
≥67.5	3343	2761.33	338,336

A subtable based on input feature humidity and target variable L(T) is presented in Table A44. From Table A44, it is observed that input feature humidity has an MSE value of 219,303.

Table A44. Humidity vs. L(T) for L(T-48) ≥ 1689.

Humidity	L(T) Prediction	Squared Error	MSE
≥70	2260	2470.5	44310
≥70	2681	2470.5	44310	29,540
<67.5	3343	3343	0

It is observed from the above calculations that day and humidity have a minimum MSE, i.e., 29,540. Here, day is considered as a decision node under branch L(T-48) ≥ 1689. Now, the node day has two branches, i.e., day “1” and “0”. In order to identify the decision/leaf node under each branch, Table A28 is divided into two subtables, as presented in Table A45 and in Table A46. From Table A45, it is observed that the branch corresponding to day “1” has a leaf node with value 3343.

Table A45. Subtable: L(T-48) ≥ 1689 and day = “1”.

L(T-24)	SEASON	TEMP	HUMIDITY	L(T)
≥2265	2	≥67.5	<70	3343

Table A46. Subtable: L(T-48) ≥ 1689 and Day = “0”.

L(T-24)	SEASON	TEMP	HUMIDITY	L(T)
≥2265	1	≥67.5	≥70	2260
≥2265	2	≥67.5	≥70	2681

In order to identify the decision node among L(T-24), season, temperature, and humidity under the branch day “0”, Table A46 is divided into multiple subtables with respect to each feature.

A subtable based on input feature L(T-24) and target variable L(T) is presented in Table A47. From Table A47, it is observed that input feature L(T-24) has an MSE value of 44,310.25.

Table A47. L(T-24) vs. L(T) for L(T-48) ≥ 1689 and day “0”.

L(T-24)	L(T)	Prediction	Squared Error	MSE
≥2265	2260	2470.5	44,310.25	44,310.25
≥2265	2681	2470.5	44,310.25

A subtable based on input feature deason and target variable L(T) is presented in Table A48. From Table A48, it is observed that input feature season has an MSE value of 0.

Table A48. Season vs. L(T) for L(T-48) ≥ 1689 and day “0”.

Season	L(T)	Prediction	Squared Error	MSE
1	2260	2260	0	0
2	2681	2681	0

A subtable based on input feature temperature and target variable L(T) is presented in Table A49. From Table A49, it is observed that input feature temperature has an MSE value of 44,310.25.

Table A49. Temperature vs. L(T) for L(T-48) ≥ 1689 and day “0”.

Temperature	L(T)	Prediction	Squared Error	MSE
≥67.5	2260	2470.5	44,310.25	44,310.25
≥67.5	2681	2470.5	44,310.25

A subtable based on input feature humidity and target variable L(T) is presented in Table A50. From Table A50, it is observed that input feature humidity has an MSE value of 44,310.25.

Table A50. Humidity vs. L(T) for L(T-48) ≥ 1689 and day “0”.

Humidity	L(T)	Prediction	Squared Error	MSE
≥70	2260	2470.5	44,310.25	44,310.25
≥70	2681	2470.5	44,310.25

It is observed from the above calculations that feature “Season” has a minimum MSE, i.e., 0. Here, “Season” is considered as a decision node under branch day “0”. Now node “Season” has two branches, i.e., season “1” and “2” as presented in Table A48. From Table A48, it is observed that the branch corresponding to season “1” has a leaf node with value 2260 and season “2” has a leaf node with value 2681. Finally, the complete decision tree to predict load L(T) based on the features, i.e., L(T-24), L(T-48), day, season, temperature, and humidity is shown in Figure A1. The decision tree shown in Figure A1 was used to predict the load shown in Table A1 and the predicted load is shown in Table A51. From Table A51, it is observed that both actual and predicted load values are equal.

Figure A1. Regression tree architecture with sample data.

Table A51. Predicted load from sample data using regression tree.

L(T-24)	L(T-48)	DAY	SEASON	TEMP	HUMIDITY	L(T)	$L^{p}$ (T)
2176	412	1	1	67	88	432	432
2354	1829	0	1	68	88	2260	2260
2777	2647	0	2	70	83	2681	2681
3112	3203	1	2	75	67	3343	3343
1663	1549	1	0	75	73	1579	1579
1010	1027	0	0	71	93	1018	1018

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Figure 1. Main functions of a distribution management system.

Figure 3. Regression tree architecture for HALF.

Figure 4. Regression tree architecture for DALF.

Figure 5. Distribution of predicted and actual load samples with regression tree model for HALF. (a) Hour-ahead load forecasting: predicted load vs. actual load for training data. (b) Hour-ahead load forecasting: predicted load vs. actual load for testing data.

Figure 6. Distribution of predicted and actual load samples with regression tree model for DALF. (a) Day-ahead load forecasting: predicted load vs. actual load for training data, (b) Day-ahead load forecasting: predicted load vs. actual load for testing data.

Figure 7. Distribution of predicted and actual load samples with regression tree model on 31 December 2021.

Figure 8. Web application to predict active power load on a 33/11 kV substation in Godishala, Telangana State, India, developed using a regression tree model.

Figure 9. Regression tree performance with respect to various seasons for HALF.

Figure 10. Regression tree performance with respect to various day status for HALF.

Figure 11. Machine learning models performance comparison in terms of training and testing mean squared errors.

Table 1. Sample load data for first 5 h on 1 January 2021 at the 33/11 kV substation in Godishala.

TIME	VOLTAGE (kV)	CURRENT (A)	cos( $ϕ$ )	POWER (kW)
01-00	11.6	102	0.96	1967
02-00	11.6	102	0.96	1967
03-00	11.6	102	0.96	1967
04-00	11.3	130	0.96	2443
05-00	11.2	148	0.96	2756

Table 2. First 6 samples from the dataset which was used to train and test the machine learning models for hour-ahead forecasting.

Sample	L(T-1)	L(T-2)	L(T-24)	L(T-48)	DAY	SEASON	Temperature	Humidity	L(T)
0	2175.941	446.5747	1828.916	1967.388	1	1	65	92	2236.757
1	2236.757	2175.941	1828.916	1967.388	1	1	65	92	2236.757
2	2236.757	2236.757	1828.916	1967.388	1	1	65	92	2354.481
3	2354.481	2236.757	1892.266	2442.607	1	1	77	52	2511.446
4	2511.446	2354.481	2532.345	2756.206	1	1	77	52	2805.756
5	2805.756	2511.446	3012.158	3203.158	1	1	77	52	3212.469

Table 3. First 6 samples from the dataset which was used to train and test the machine learning models for day-ahead forecasting.

Sample	L(T-24)	L(T-48)	DAY	SEASON	Temperature	Humidity	L(T)
0	1828.916	1967.388	1	1	65	92	2236.757
1	1828.916	1967.388	1	1	65	92	2236.757
2	1828.916	1967.388	1	1	65	92	2354.481
3	1892.266	2442.607	1	1	77	52	2511.446
4	2532.345	2756.206	1	1	77	52	2805.756
5	3012.158	3203.158	1	1	77	52	3212.469

Table 4. Training and testing errors of regression tree model for HALF.

Depth	Error Metrics	Training	Testing	Depth	Error Metrics	Training	Testing
5	MSE	0.004	0.005	25	MSE	0.000	0.008
	RMSE	0.066	0.072		RMSE	0.001	0.090
	MAE	0.039	0.044		MAE	0.000	0.049
10	MSE	0.001	0.006	30	MSE	0.000	0.008
	RMSE	0.037	0.080		RMSE	0.001	0.089
	MAE	0.021	0.043		MAE	0.000	0.048
15	MSE	0.000	0.008	34	MSE	0.000	0.008
	RMSE	0.011	0.087		RMSE	0.001	0.087
	MAE	0.006	0.047		MAE	0.000	0.048

Table 5. Training and testing errors of regression tree model for DALF.

Depth	Error Metrics	Training	Testing	Depth	Error Metrics	Training	Testing
6	MSE	0.00650	0.00869	24	MSE	0.00001	0.01419
	RMSE	0.08063	0.09322		RMSE	0.00436	0.11911
	MAE	0.04931	0.05647		MAE	0.00068	0.06984
12	MSE	0.00209	0.01201	30	MSE	0.00001	0.01367
	RMSE	0.04569	0.10959		RMSE	0.00320	0.11692
	MAE	0.02621	0.06301		MAE	0.00025	0.06930
18	MSE	0.00019	0.01358	36	MSE	0.00001	0.01387
	RMSE	0.01390	0.11655		RMSE	0.00319	0.11776
	MAE	0.00547	0.06793		MAE	0.00024	0.06933

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Veeramsetty, V.; Sai Pavan Kumar, M.; Salkuti, S.R. Platform-Independent Web Application for Short-Term Electric Power Load Forecasting on 33/11 kV Substation Using Regression Tree. Computers 2022, 11, 119. https://doi.org/10.3390/computers11080119

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Veeramsetty V, Sai Pavan Kumar M, Salkuti SR. Platform-Independent Web Application for Short-Term Electric Power Load Forecasting on 33/11 kV Substation Using Regression Tree. Computers. 2022; 11(8):119. https://doi.org/10.3390/computers11080119

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Veeramsetty, Venkataramana, Modem Sai Pavan Kumar, and Surender Reddy Salkuti. 2022. "Platform-Independent Web Application for Short-Term Electric Power Load Forecasting on 33/11 kV Substation Using Regression Tree" Computers 11, no. 8: 119. https://doi.org/10.3390/computers11080119

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Platform-Independent Web Application for Short-Term Electric Power Load Forecasting on 33/11 kV Substation Using Regression Tree

Abstract

1. Introduction

2. Methodology

2.1. Active Power Load Data Analysis

2.2. Features Information and Data Preparation

2.3. Machine Learning Models

Regression Tree

3. Result Analysis

3.1. Regression Tree Model

3.2. Impact of Season and Day on Regression Model Prediction

3.3. Comparative Analysis

4. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

Abbreviations

Appendix A. Conversion of Continuous Data into Categorical Data

Appendix B. Regression Tree Model Formulation

References

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