Photovoltaic Power Generation Forecasting with Hidden Markov Model and Long Short-Term Memory in MISO and SISO Configurations

Abstract

Photovoltaic (PV) power generation forecasting is an important research topic, aiming to mitigate variability caused by weather conditions and improve power generation planning. Climate factors, including solar irradiance, temperature, and cloud cover, influence the energy conversion achieved by PV systems. Long-term weather forecasting improves PV power generation planning, while short-term forecasting enhances control methods, such as managing ramp rates. The stochastic nature of weather variables poses a challenge for linear regression methods. Consequently, advanced, state-of-the-art machine learning (ML) approaches capable of handling non-linear data, such as long short-term memory (LSTM), have emerged. This paper introduces the implementation of a multivariate machine learning model to forecast PV power generation, considering multiple weather variables. A deep learning solution was implemented to analyze weather variables in a short time horizon. Utilizing a hidden Markov model for data preprocessing, an LSTM model was trained using the Alice Spring dataset provided by DKA Solar Center. The proposed workflow demonstrated superior performance compared to the results obtained by state-of-the-art methods, including support vector machine, radiation classification coordinate with LSTM (RCC-LSTM), and ESNCNN specifically concerning the proposed multi-input single-output LSTM model. This improvement is attributed to incorporating input features such as active power, temperature, humidity, horizontal and diffuse irradiance, and wind direction, with active power serving as the output variable. The proposed workflow achieved a mean square error (MSE) of $2.17\times {10}^{-7}$, a root mean square error (RMSE) of $4.65\times {10}^{-4}$, and a mean absolute error (MAE) of $4.04\times {10}^{-4}$.

1. Introduction

The constant increase in electric energy use around the world poses an important challenge to electric power generation [1]. Transitions to renewable energy are occurring because non-renewable energy is not viable in the long term due to social and environmental concerns and conventional fuel supply issues. Different energy sources would be needed to cover fluctuations from the sun or the wind and guarantee a quality service. For example, storage technologies would provide alternative supply during energy deficiencies due to cloud cover or at night.

Solar energy is the most used renewable energy source in the world, having generated 855.7 TWh by the end of 2020 [2]. PV systems are the main technologies based on solar energy resources [2,3]. The energy transformations achieved by PV panels are affected by manufacturing factors associated with materials, maintenance, cleaning, dimensions [4], and climate factors, grouped into atmospheric parameters, solar irradiance, and geographic characteristics.

Table 1. Classified weather variables.

Classes	Input Variables
Atmospheric characteristics	Pressure, temperature, cloud abundance, rainfall, cloud formation, cloud cover in the atmosphere, radiation, humidity, density, wind energy, wind speed, wind direction, evaporation, sunshine duration, wind gust, average temperature, ambient temperature, minimum temperature, maximum temperature, sky information, temperature variation.
Solar characteristics	Solar energy, solar irradiance, zenith angle, global horizontal irradiance, diffuse horizontal irradiance, direct normal irradiance, global solar radiation, daily solar radiation, cell temperature, wavelength, precipitation, photovoltaic energy.
Geographic conditions	latitude, longitude, altitude.

presents some weather variables that affect power generation from PV systems [5]. Irradiance has the highest correlation with the power output, as proven in previous works and later in this paper, with a correlation coefficient of 0.96. Irradiance and other weather variables exhibit stochastic behavior, which makes classic statistical methods ineffective in predicting their behavior.

Weather variables can be analyzed at different time horizons, which can be grouped into very short-term (<30 min), short-term (between 30 min and 6 h), and long-term time horizons (between 4 and 6 h) [5]. Another option is to group variables as intra-hour (15 min to 2 h), intra-day (1 to 6 h), and day-ahead (1 day to 3 days) [6]. Each group has different sources of information; for example, the long-term horizon uses satellite images, and the short term uses sky images. Prediction in the medium or long term has a high error rate. On the other hand, very short-term predictions have better accuracy and are used in power generation planning [7] or applications such as ramp rate in PV power [8].

There are diverse approaches to predicting environmental characteristics, such as time horizon, machine learning, and deep learning methods and those that estimate variables. For example, when grouping by method, the proposed groups are statistical models, cloud image models, numeric models, and hybrid models. Statistical models work with historical data to predict the subsequent values, and these models can be sub-grouped into statistical or linear models and artificial intelligence or non-linear models. Some examples of linear models are stationary analysis, autoregressive integrated moving average (ARIMA), multiple regressions, and exponential smoothing. Nonlinear models include fuzzy inferences, genetic algorithms, neural networks, and machine learning methods [6]. These linear and non-linear models are used in very short-term or intra-hour groupings. Artificial intelligence methods are well-suited for managing diverse problems. For example, neural networks are a great tool for solar irradiance prediction, and the most popular method for this prediction is multilayer perceptron (MLP) [6]. Distinguishing spurious results from actual cloud dynamics is an important problem. Dips from actual cloud dynamics can be identified as outliers and eliminated in the preprocessing stage. However, an initial model outlining common behavior for PV power generation could be used as a first step, with any relevant dips modeled at a subsequent stage. The initial baseline modeling might depend on the dataset used.

1.1. Related Works

Some relevant works for predicting power output in PV systems address the issue of discontinuous or sudden fluctuation in power generation due to factors such as temperature, wind velocity, cloud cover, and other weather variables [7]. This analysis and prediction use a short time horizon (0.5 to 6 h), and prediction is achieved with machine learning methods, particularly with support vector machine (SVM). Another paper compared multiple linear regression (MLR), decision tree (DTR), and k-nearest neighbor (KNN); the author concluded that KNN is the best machine learning method because it had more accurate predictions compared to the other models [9].

In another paper, three linear methods were compared on their ability to predict PV power output: autoregressive (AR), LM model, and exogenous input (ARX). All performed well in stable conditions but yielded inaccurate results under unexpected fluctuations; the ARX model produced better results [10]. Another author used convolutional neural network (CNN) and long short-term memory (LSTM) to create a robust deep spatiotemporal model named convolutional LSTM (ConvLSTM) to work in multiples regions or PV systems that are separated from each other [11]. ConvLSTM was compared to the ARMA model and fully connected LSTM (FC-LSMT); ConvLSTM produced better results.

Recurrent neural networks (RNNs) and LSTM architectures have been used to create a new framework for sequence learning named Evolino [12]. The Evolino framework was probed with a Mackey–Glass time series prediction. In order to perform power prediction, a path with three stages was proposed: data preprocessing to treat missing values, data normalization, and parameter initialization [13]. In the second stage, a radiation coordinate classification method was employed based on the correlation between different features at different times, typically in the time window 8:30–17:30. This classification method categorizes data points into specific radiation coordinates, capturing variations in solar radiation levels. Finally, the preprocessed and classified data were input into an LSTM model for power forecasting; the model was assessed using the benchmark dataset DKASC [14]. The evaluation metrics RMSE and mean absolute percentage error (MAPE) were reported to evaluate the accuracy of the model.

One study proposed an examination of the impact of the configuration and selection of hyperparameters in an LSTM architecture. The objective was to establish the different contributions of the LSTM configuration, input, forgetting, and output gates in forecasting applications [15]. Another work proposed a feature extraction method based on sentiment analysis. Concatenated time series information of prices and other variables were used as input in prediction models, including random forest, LSTM, and multilayer perceptron [16].

Another proposed approach focused on preprocessing the data to remove abnormalities by performing data normalization [17]. Subsequently, the ESN-CNN model is trained, combining the echo state network (ESN) and convolutional neural network (CNN) to extract spatial features from the data. This hybrid model aims to leverage the capabilities of both ESN and CNN for improved performance. Finally, the model is assessed using the DKASC dataset [14], and metrics such as RMSE and MAPE are reported to evaluate its accuracy.

Table 2. Related work summary.

Related Work	Advantages	Disadvantages
I-ACO-SVM [7]	SVM is a classical machine learning technique. A workflow was proposed where the radial basis function kernel parameters are fine-tuned using optimal parameters obtained from the ant colony algorithm. This approach yields better accuracies and can be implemented in an embedded system due to its computational efficiency [7].	The hyper-tuning of the parameters requires initial computational efforts due to the application of a search grid.
KNN [9]	KNN is a suitable method for conducting load profile forecasting because it is highly adaptable for analyzing the K-nearest neighbors.	The limitations of KNN methods arise from their requirement for a substantial amount of data to accurately perform similarity measurements for identifying the k-nearest neighbors. In contrast, deep learning approaches have demonstrated superior performance in terms of MSE, MAPE, and R2.
ConvLSTM [11]	The performance achieved by the convolutional LSTM outperforms the accuracy obtained for the baseline algorithms, including classical machine learning techniques. This model integrates spatial image analysis into the study of power prediction [11].	Given the intricacy of the convolutional LSTM and the need for hyperparameter tuning via an extensive grid search, it demands a robust computational infrastructure.
Evolino [12]	The Evolino framework typically avoids problems of vanishing gradients related to the RNN [12].	In the framework that combines Evolino with LSTM, the training process is computationally expensive and may encounter overfitting issues owing to the large number of parameters that must be tuned [12].
RCC-LSTM [13]	The framework proposed in [13] outperforms the results of baseline algorithms such as RCC-RBFNN and RCC-BPNN. One of the major contributions is made during the preprocessing stage, where similarity measurements are obtained using the window size.	The RCC-LSTM requires a selection of threshold values, and the adjustment of cell numbers is contingent upon specific weather conditions in accordance with a fixed window size.
ESN-CNN [17]	The pipeline proposed in [17] comprises three stages. The first stage involves preprocessing, which includes the removal of data abnormalities, followed by data normalization and the initialization of parameters for the echo state network (ESN). The output of the final ESN stage serves as the input for the convolutional neural network (CNN). This pipeline demonstrates excellent performance in power prediction for the benchmark datasets.	Echo state networks are susceptible to overfitting, primarily due to their large number of processing units. Furthermore, the convolutional operations within this framework require matrix multiplication, escalating computational complexity.

summarizes the related works and highlights their advantages and disadvantages.

1.2. Contributions

The main contributions of this paper are:

• A robust and precise workflow to power prediction is presented, leveraging hidden Markov models to effectively identify outliers within the raw weather data. Moreover, a deep learning LSTM architecture is designed to enhance PV prediction performance, surpassing the results reported in [7,13,17].
• PV prediction is performed at short time horizons (five minutes ahead) with time steps of five minutes using the ambient weather dataset on Puerto Rico. The Caribbean location introduces challenges, as historically, weather predictions have been unreliable due to the high variability of winds and the complicated dynamics of the heat patterns throughout the day from both sea and land.

This paper presents an implementation of a machine learning model to predict the power output of a PV system consisting of two subprocesses: the preprocessing stage and the trained model for power prediction. The initial preprocessing involves the application of a hidden Markov model (HMM) to automatically detect and eliminate outliers. Subsequently, the LSTM model is trained to predict the common behavior of power generation, establishing a baseline model as the initial step toward future work specifically addressing outlier prediction in the context of Puerto Rico. The two variations of the LSTM are presented: one uses a single input to obtain a single output (SISO), and the other uses multiple inputs to obtain a single output (MISO). The prediction horizon was set at a short time (five minutes ahead) to assess the effectiveness of HMM outlier detection and elimination. This specific time horizon captures the dynamic behavior of clouds and their impact on PV systems.

1.3. Outline

To outline the proposed method, this paper is separated into six sections. The introduction sets the stage by presenting the motivation, context, related works on PV systems and weather forecasting, and the primary contributions of this study. The proposed workflow details the stages for constructing the proposed machine learning model, with each step described in subsections. The third section describes the two datasets employed. The results outlined in section four showcase outcomes across six experiments and employ metrics for comparison with state-of-the-art solutions. The fifth section presents a discussion and interprets the results in the context of enhancing PV systems. Lastly, the sixth section provides a concise summary and conclusions, encapsulating the essential findings and contributions of this paper.

2. Proposed Workflow

Power output prediction in PV systems needs multiple steps to prepare the data, train the model, and compute the results. Figure 1 presents a workflow used to build a machine learning model.

In addition,

Table 3. Notation used in the article.

Symbol	Description
$Diff$	Magnitude difference between two consecutive points
$x$	Variable to represent some feature
$x\left[t\right]$	Magnitude of feature $x$ in the time $t$
$x[t-1]$	Magnitude of feature $x$ in the time $t-1$
$s=\left\{s_1, s_2, \dots , s_T\right\}$	Hidden state vector of HMM
$o=\left\{o_1, o_2, \dots , o_T\right\}$	Observation variable vector of HMM
$p\left(o_t\right| s_t)$	Prior probability of HMM
$O\left[i\right]$	Training data
$i$	Position of the measurement
$x_H\left[i\right]$	Present measure, used as input in the training process
$y_H\left[i\right]$	Future desired measures, used as a reference in the training process
${\widehat{x}}_H\left[t\right]$	Predicted measure. This is the output of the trained model
$c_t$	Visible state in time $t$ of LSTM cell
$h_t$	Hidden state in time $t$ of LSTM cell
$x_t=x_H\left[t\right]$	Present measure used as input to LSTM cell
$f_t$	Forget state of LSTM cell
$i_t$	Input state of LSTM cell
$o_t$	Output state of LSTM cell
${\widehat{c}}_t$	Predicted visible state of LSTM cell
$\sigma$	Sigmoid function
$\cdot$	Dot operator
$W, U,$ and $b$	Configuration parameter of LSTM cell

presents a detailed summary of the variables.

2.1. Input Variables

The first step in the workflow is related to the dataset and variables used. The dataset has the following variables: active energy delivered/received, active power, current phase average, wind speed, temperature, humidity, horizontal and diffuse irradiance, wind direction, weather daily rainfall, radiation global tilted, and radiation diffuse tilted.

The input data were normalized using min–max scaling; this process transforms the amplitude of each variable to values between 0 and 1. Normalization allows the comparison of different variables in the same space while simplifying amplitude information. To perform the model generalization and avoid overfitting, the NaN values are removed from the data and replaced by the mean values from a neighborhood.

2.2. Outlier Detection and Removal

The power output and other variables typically have outliers generated by cloud cover. In [18], the authors mention that shadows are the most negative environmental factor for PV power systems.

In the outlier detection phase, the difference between points in the signal is computed as follows.

$$Diff=x\left[\mathrm{t}\right]-x[t-1]$$

(1)

where $x\left[t\right]$ is the magnitude of feature $x$ in the time $t$. A large value represents an unusual value or outlier, and a small value represents a normal value. The Gaussian HMM classifies each point using the computed feature Diff. Three classes configured in the HMM model were: outlier, inlier, and constant.

HMM is a process involving the evolution of hidden states over time. In other words, it describes the transitions between hidden states. These states are associated with observations corrupted by noise. If the state variables are $S_1,S_2,\dots , S_T$ and the corresponding observation variables are $0_1, 0_2, \dots , 0_T$, an HMM can be parametrized by an initial distribution $p\left(S_1\right)$, transition probabilities $p\left(S_{t+1}\right| S_t)$, and observation probabilities $p\left(O_t\right| S_t)$ [19]. The joint distribution of a length $T$ is factorized as:

$$p\left(\mathit{s},\mathit{o}\right)=p\left(s_1\right)\prod _{t=1}^{T-1}p\left(s_{t+1}\right|s_t)\prod _{t=1}^{T}p\left(o_t\right|s_t)$$

(2)

where $\mathit{s}=\left\{s_1, s_2, \dots , s_T\right\}$ represents a particular assignment of hidden state and $\mathit{o}=\left\{o_1, o_2, \dots , o_T\right\}$ represents an observation variable [19]. The hidden states used in this implementation are 0 and 2 for inliers or normal values and 1 for outliers.

According to the Gaussian HMM approach, the prior probability $p\left(o_t\right| s_t)$ is obtained from the probability distribution function (PDF). The PDF provides approximations of Gaussian densities [19,20], which are obtained by:

$$p\left(o_t\right| s_t)~\mathcal{N}\left(o_t | {\widehat{\mu }}_t,{\widehat{\mathsf{\Sigma }}}_t\right)$$

(3)

where ${\widehat{\mu }}_t$ and ${\widehat{\mathsf{\Sigma }}}_t$ are estimated from each observation.

The output from the outlier detection step is a classification of measures; this classification is used in the outlier removal step to smooth the measures with higher differences. Finally, the output of the outlier removal step is the time series signal without outliers. The best performance is obtained when three classes are configured to classify the measures. These classes can be described as follows: zero or minimal differences (indicated in Figure 2 by blue dots), small differences (orange dots), and higher differences (green dots). For example, in Figure 2, the power output in one day is presented, the signal has outliers, and the outlier detection step classifies the points as outliers with green dots or normal values with orange and blue dots.

Figure 3 shows the signal output of the outlier removal step, where all outliers have been eliminated. It is observed that the difference in amplitude is less than 0.06 due to imperfections in the signal.

2.3. Split Data

The available measurements are divided into three groups: testing, training, and validation. The percentage is chosen heuristically as follows: the testing group has 30% of the measurements, the training group has 40%, and the validation group has 30%. Each of these groups is used in different steps in the next stages.

2.4. Training Data

Since statistical methods are ineffective in the prediction of irradiance and other weather variables [5], artificial intelligence methods have gained acceptance and are well-suited for PV power prediction. For example, neural networks are a great tool for solar irradiance prediction, and the most popular method for this prediction is multilayer perceptron (MLP) [6]. Deep learning is a subcategory of artificial intelligence where methods are used to resolve complex non-linear problems [21]. Some of the main methods are: convolutional neural network (CNN), support vector machine (SVN), long short-term memory (LSTM), and recurrent neural networks (RNN). LSTM is typically used for time series variables such as irradiance and other weather features.

Training data are represented by $O\left[i\right]$, where $i$ is the position of the measurement, and $i=[\mathrm{0,1},2,\dots , n]$, where $n$ is the number of measures in the training data. Using $O\left[i\right]$ generates two vectors, $x\left[i\right]$ and $y\left[i\right]$, where:

$$x_H\left[i\right]=O\left[i\right],\forall i=[\mathrm{0,1},2,\dots ,n-1]$$

(4)

$$y_H\left[i\right]=O\left[i+1\right], \forall i=[\mathrm{0,1},2,\dots , n-1]$$

(5)

Equations (4) and (5) generate two vectors with length n − 1; both vectors fit the model in the training block. The $x_H\left[i\right]$ vector represents the present measure, and the $y_H\left[i\right]$ vector represents the future desired value. Both vectors are given after the application of the outlier detection and removal steps using HMM.

2.5. Testing Data

The testing data have two vectors, $x_H\left[i\right]$ and $y_H\left[i\right]$, computed from Equations (4) and (5), and n is the number of measures in the testing data. Both vectors are used to test the built model in the training block. The $x_H\left[i\right]$ vector represents the present measure and input to the model; the output model is the predicted vector ${\widehat{x}}_H\left[t\right]$. The $y_H\left[i\right]$ vector is used to compute metrics and is explained in Section 2.7 (Metrics).

2.6. Training Model

The power prediction methods applied to PV systems have been extensively addressed in [6,7,9,13,17]. In recent years, deep learning methods, particularly those utilizing long short-term memory (LSTM) networks, have gained significant attention for time series prediction problems [19], outperforming traditional machine learning techniques in terms of MSE and RMSE [7]. This paper presents an approach based on SISO and MISO LSTM models, wherein a hidden Markov model removes outliers for power prediction.

2.6.1. Long Short-Term Memory (LSTM)

LSTM is a deep learning method used for time series prediction. A key feature of LSTM is the possibility of protecting the memory cells using sigmoid gates [12]. With this, the LSTM method can update the output based on previous outputs and present input data [22]. Figure 4 presents the cell components in LSTM.

Each LSTM cell receives the previous state $c_{t-1}$, the previous hidden state $h_{t-1}$, and the actual measure $x_t=x_H\left[t\right]$. Forget gate, input gate, and output gate use the sigmoid function to protect the memory cell [11].

For a single LSTM cell, as presented in Figure 4, it is possible to calculate the values using the equations in (6).

$$\begin{gathered} f_t=\sigma \left(W_f\times x_t+U_f\times h_{t-1}+b_f\right) \\ {i}_t=\sigma (W_i\times x_t+U_i\times h_{t-1}+b_i) \\ {o}_t=\sigma (W_o\times x_t+U_o\times h_{t-1}+b_o) \\ {\widehat{c}}_t=\tanh (W_c\times x_t+U_c\times h_{t-1}+b_c) \\ {c}_t=f_t\cdot c_{t-1}+i_t\cdot {\widehat{c}}_t \\ {h}_t=o_t\cdot \mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(c_t\right) \end{gathered}$$

(6)

where $\sigma$ is the sigmoid function used as activation function, and $‘\cdot ’$ is the dot operator or dot product operation. Finally, $W, U,$ and $b$ are configuration parameters [21].

The LSTM implemented uses $c_{t-1}$ and $h_{t-1}$ initial values in zeros. The output is taken as $h_t$.

2.6.2. Implementation

The proposed SISO and MISO LSTM models were implemented using TensorFlow version 2.11.0 [23]. For the SISO LSTM model, the network hyperparameters were configured with four neurons in the hidden layer and trained for ten epochs, with a batch size of 1. The learning rate was set to $1\times {10}^{-4}$, the optimizer used was Adam, and the optimization metric was mean squared error (MSE).

The MISO LSTM model was configured with three hidden layers, followed by a dense operation, and trained for 200 epochs. The optimizer used was Adam, and the loss function employed was MSE. The batch size was determined by grouping the following features: active power, temperature, humidity, horizontal irradiance, diffuse irradiance, and wind direction. These features were set as the input for the MISO LSTM model to perform power prediction as the target.

2.7. Metrics

In order to evaluate the performance of the proposed MISO and SISO LSTM models, the following metrics were employed: mean squared error (MSE), root mean squared error (RMSE), and mean absolute error (MAE). These models were tested using the benchmark dataset DKA Solar and Ambient Weather dataset.

3. Dataset Description

In order to evaluate the proposed SISO and MISO LSTM models, experiments were conducted using two distinct benchmark datasets: DKA Solar and Ambient Weather. The details of each dataset are described as follows:

• DKA Solar [14]: This online hub provides a platform for sharing variable weather measurements obtained from PV farms located in Australia. The dataset used to evaluate the proposed models spans from 2013 to 2020, with 1,281,324 measurements taken every five minutes. The first 12,950 dataset values were not used because there are no active power values; however, the number of samples is high, with 1,268,374 measurements. In addition, DKA has 13 numeric features such as active power, wind speed, weather temperature, relative humidity, global horizontal irradiance, and others. The dataset was accessed on 22 June 2022.
  The first task involved in using the DKA dataset was to find the place and time selected by previous state-of-the-art papers; this was conducted in order to enable a comparison of the results of the proposed method. Next, the time spans were selected where all features were available because some features presented zero or empty information for some periods. Finally, step “Section 2.1” from the proposed workflow was executed.
• Ambient Weather [24]: This is a platform used to register, share, and download weather measurements. The “UPRM CID Sustainable Energy Center, Mayagüez” device was selected in Puerto Rico. The data used span from 20 February 2022 to 6 September 2022, with 56,340 measures taken every five minutes and 18 features.
  Ambient Weather is a community where owners of meteorological sensors can seamlessly share measurements from Puerto Rico in real time. For instance, the Sustainable Energy Center (SEC) laboratory operates two meteorological stations, contributing real-time data such as irradiance, temperature, humidity, and more. However, it is important to note that the Ambient Weather dataset currently only offers records dating back one year, resulting in a relatively short time span for analysis. Furthermore, instances of electrical interruption have occasionally led to gaps in the data from certain meteorological stations. To address this, the first task involved downloading data from various stations and selecting a time span with minimal gaps. These gaps were replaced using the ffill (forward fill) interpolation technique available in Python 3.10.7 [25]. Subsequently, step “Section 2.1” from the proposed workflow was executed.

4. Results

Correlation analysis of the DKA Solar Center dataset determined which features provided more information to train the ML model. The Pearson correlation method was used to calculate the correlation between features. The results are presented in Figure 5.

Based on Figure 5, the most correlated features to active power are horizontal and diffuse irradiance, with correlation coefficients of 0.96 and 0.55, respectively. The features used in the experiments presented below were selected based on this correlation analysis.

The ML model proposed was trained with the DKA Solar Center dataset [14]. Experiment 1 used only the active power feature as input and output. Experiment 2 used active power, wind speed, temperature, humidity, horizontal and diffuse irradiance, wind direction, and weather daily rainfall as input, and active power as output. Experiment 3 used active power, temperature, humidity, horizontal and diffuse irradiance, and wind direction as input, and active power as output.

Experiment 3 used temperature, humidity, global horizontal radiation, diffuse horizontal radiation, and wind direction, based on the features presented in [7]. The results obtained with the proposed LSMT model are better than the three models (support vector machine, RCC-LSTM, and ESNCNN) tested in [7,13,17]. Another difference is that the LSTM model was trained and tested with three years of data; on the other hand, the SVM model in [7] was trained and tested with one year of data.

The optimization metric MSE along each epoch was analyzed in the training model for Experiment 3, and the results are presented in Figure 6. The MSE computed by subsequent epochs shows lesser values after the first epoch. Therefore, the MSE in epoch 10 is smaller than MSE in epoch 1. To show the behavior of the optimization metric MSE in detail, Figure 7 presents a zoomed-in look at MSE vs. each epoch for the training model in Experiment 3.

A comparison of the original signal $x\left[t\right]$, the signal after outlier removal $y_H\left[t\right]$, and the predicted signal ${\widehat{x}}_H\left[t\right]$ is presented in Figure 8. Because the signals $y_H\left[t\right]$ and ${\widehat{x}}_H\left[t\right]$ are future values, the signal $x\left[t\right]$ must be moved one time step into the future.

An additional test was executed using the Ambient Weather Network dataset [24]. The objective was to assess the proposed ML model’s performance using a dataset with more outliers than DKA and build a baseline model as the first step towards future work focusing on outlier prediction in Puerto Rico. The “UPRM CID Sustainable Energy Center, Mayagüez” device was selected in Puerto Rico, and the records from 20 February 2022 to 6 September 2022 were downloaded. The dataset has 56,340 measures and 18 features. However, Experiment 4 used only solar radiation as input and output. Experiment 4 had an MSE of $3.49\times {10}^{-1}$ KW, an RMSE of $5.91\times {10}^{-1}$ KW, and an MAE of $9.18\times {10}^{-3}$ KW.

The results of Experiment 4 are less accurate than those of the other tests because the Ambient Weather Network dataset has more outliers than the DKA Solar Center dataset. For example, in Figure 9 and Figure 10, both datasets are compared over one day.

Experiment 5 work was trained with the DKA Solar Center dataset [14], similar to Test 1, but without the outlier detection and removal steps from the proposed workflow, so the input data to train the ML model includes outliers. Experiment 5 had an MSE of $8.97\times {10}^{-2}$ KW, RMSE of $2.99\times {10}^{-1}$ KW, and MAE of $9.98\times {10}^{-2}$ KW.

Experiment 6 used the same subset selected in [7]: data from the DKA Solar Center dataset [14] from January 2018 to December 2019. The subset has 199,097 measures. All of the steps on the model were used, and the results have an MSE of $1.72\times {10}^{-3}$ KW, RMSE of $4.15\times {10}^{-2}$ KW, and MAE of $2.79\times {10}^{-2}$ KW.

Experiment 7 included the DKA Solar Center dataset [14] using all of the available data but only the two features most correlated to active power, which were horizontal and diffuse irradiance, with correlation coefficients of 0.96 and 0.55, respectively. All of the steps in the model were used, and the results have an MSE of $5.55\times {10}^{-7}$ KW, an RMSE of $7.45\times {10}^{-4}$ KW, and an MAE of $1.21\times {10}^{-4}$ KW.

The results of all seven experiments and the previous results reported by researchers who worked with the DKA Solar Center dataset [14] are presented in

Table 4. LSTM model proposed accuracy with the DKA solar center dataset.

Name	Dataset	Features	Outlier Delete	MSE (KW)	RMSE (KW)	MAE (KW)
Experiment 1	DKA	Only active power	Yes	$3.05\times {10}^{-3}$	$5.54\times {10}^{-2}$	$3.09\times {10}^{-2}$
Experiment 2	DKA	8 features	Yes	$4.55\times {10}^{-6}$	$2.13\times {10}^{-3}$	$1.30\times {10}^{-3}$
Experiment 3	DKA	5 features as SVM [7]	Yes	$2.17\times {10}^{-7}$	$4.65\times {10}^{-4}$	$4.04\times {10}^{-4}$
Experiment 4	Ambient Weather	Only solar radiation	Yes	$3.49\times {10}^{-1}$	$5.91\times {10}^{-1}$	$9.18\times {10}^{-3}$
Experiment 5	DKA	Only active power	No	$8.97\times {10}^{-2}$	$2.99\times {10}^{-1}$	$9.98\times {10}^{-2}$
Experiment 6	DKA subset selected in SVM [7]	5 features as SVM [7]	Yes	$1.72\times {10}^{-3}$	$4.15\times {10}^{-2}$	$2.79\times {10}^{-2}$
Experiment 7	DKA subset selected in SVM [7]	2 features	Yes	$5.55\times {10}^{-7}$	$7.45\times {10}^{-4}$	$1.21\times {10}^{-4}$
SVM [7]	DKA	5 features	Yes	$3.49\times {10}^{-2}$	$1.86\times {10}^{-1}$	$1.15\times {10}^{-1}$
RCC-LSTM [13]	DKA	-	-	$8.84\times {10}^{-1}$	$9.4\times {10}^{-1}$	$5.87\times {10}^{-1}$
ESNCNN [17]	DKA	-	-	$3.09\times {10}^{-2}$	$1.73\times {10}^{-1}$	$9.71\times {10}^{-2}$

, highlighting the best result in bold.

5. Discussion

A discussion of the results obtained from the application of the models MISO and SISO LSTM is presented below:

• The MSE metric in Experiment 1, Experiment 2, and Experiment 3 have orders that range from ${10}^{-3}$ KW to ${10}^{-7}$ KW. These MSE values are smaller than those presented in [7,13,17], where the MSE value had an order of ${10}^{-2}$ KW.
• Experiment 1 presented good results using only one input, which reduced the model’s complexity and training time. Typically, the predicted values are only differ slightly from the real values; this error is smaller than 4 Watts in most cases.
• The trained models show good performance regarding signals with low outliers. This model can predict the common behavior of irradiance or power signals in PV systems. However, the proposed model does not produce good predictions regarding measures with sudden fluctuations, because the datasets do not have information on cloud movements to let the ML model anticipate a sudden fluctuation or outlier. Future work will include cloud movement information to train the prediction model to account for these abrupt variations.
• Experiment 3 showed the best results because it used more information than other experiments, but the model used here was more complex than in other experiments.
• Experiment 6 used one year of data, less than other experiments. The objective was to replicate the same conditions as reference [7]. The results of the proposed model are better than those presented in [7,13,17].
• The results in Table 4 show that adding more features to the ML model does not guarantee better results. For example, Experiment 3, with five features, produced better results with its fewer features than Experiment 2, which has eight features.
• Another example of the previous discussion can be found in Experiment 6 and Experiment 7. Both of these Experiments use the same subset of the dataset, but Experiment 7 has fewer features (only two) than Experiment 6 (five features). However, Experiment 7 achieved better results than Experiment 6. One explanation for this is that more features can introduce noise instead of relevant information; because of this, feature analysis, such as correlation analysis, is an essential part of building ML models.
• The LSTM method outperforms the traditional ML methods used in forecasting problems because LSTM can save and remove information. This was confirmed with Experiment 3, which applied the LSTM method and obtained better results than [7], where SVM was utilized.
• In accordance with what was expected, the outlier removal step improved the model’s performance. Comparing Experiment 1, where outliers were removed, to Experiment 5, in which outliers were kept, the results obtained in Experiment 1 were more accurate.

The model developed in this paper will be used to forecast solar irradiance in Puerto Rico, which has a goal of achieving 100% renewable energy by 2050 and where the main renewable energy source is the sun [26]. Such an aggressive goal requires not only the massive deployment of PV systems but also new ways to plan and operate the local power system. A key aspect to securing the stability of the local grid is to have as clear a prediction as possible of the amount of power that will be available in the near future from the thousands of PV systems spread around the archipelago. The problem becomes more complicated due to Puerto Rico’s location in the Caribbean, where long-term weather predictions have historically been unreliable due to the high variability of winds, the central mountainous region of the main island, the relatively small size of the jurisdiction, and the complicated dynamics of the heat patterns throughout the day from both sea and land.

6. Conclusions

• The preprocessing step and specific feature selection used data correlation analysis. Horizontal irradiance and active power were the most correlated variables, with a correlation coefficient of 0.96. Therefore, horizontal irradiance and active power are the most important features for active power prediction.
• The evaluation of the proposed ML model was conducted via five experiments. The best results were obtained in Experiment 3 with an MSE of $2.17\times {10}^{-7}$ KW, an RMSE of $4.65\times {10}^{-4}$ KW, and an MAE of $4.04\times {10}^{-4}$ KW. These results are better than those presented in [7,13,17].
• Applying HMM for outlier detection and elimination enables the classification of measures without the need for a predefined threshold setup. Outlier detection and elimination improved the results compared to the original signal. This is evident when comparing Experiment 1 to Experiment 5, where Experiment 1 used a signal without outliers as input, whereas Experiment 5 used the original signal as input. The results of Experiment 1 were better.
• The Puerto Rico dataset [24] has more outliers than the Australian dataset [14]. Because of this, the proposed ML model trained with the Australian dataset produced better results than the Puerto Rico dataset. Although this model does not consider outliers, we understand that cloud dynamics can cause dips identified as outliers. Future work will improve the ML model prediction by including outliers.
• The proposed ML method is an excellent tool for reducing the photovoltaic generation planning error implicit in medium- or long-term prediction by updating the generation planning at regular intervals. This enables an energy management system (EMS) to execute necessary actions such as battery charging or utilizing grid energy to maintain high-quality service. However, the proposed ML model does not capture outliers because it requires additional information about cloud movements, which is currently unavailable in the datasets used for this study.
• The outlier detection and elimination strategy using HMM can be used in preprocessing steps for weather datasets and other datasets with time series variables. In addition, the LSTM model can be used in short-term generation planning to complement the information used in EMS and enable proactive response to specific situations, for example, activating batteries when the irradiance decreases below the defined limit.