Solar photovoltaic generation and electrical demand forecasting using multi-objective deep learning model for smart grid systems

Abstract

The growing of the photovoltaic (PV) panel’s installation in the world and the intermittent nature of the climate conditions highlights the importance of power forecasting for smart grid integration. This work aims to study and implement existing Deep Learning (DL) methods used for PV power and electrical load forecasting. We then developed a novel hybrid model made of Feed-Forward Neural Network (FFNN), Long Short Term Memory (LSTM) and Multi-Objective Particle Swarm Optimization (MOPSO). In this work, electrical load forecasting is long-term and will consider smart meter data, socio-economic and demographic data. PV power generation forecasting is long-term by considering climatic data such as solar irradiance, temperature and humidity. Moreover, we implemented these deep learning methods on two datasets, the first one is made of electrical consumption data collected from smart meters installed at consumers in Douala. The second one is made of climate data collected at the climate management center in Douala. The performances of the models are evaluated using different error metrics such as Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) and regression (R). The proposed hybrid model gives a RMSE, MAE and R of 1.15, 0.75 and 0.999 respectively. The results obtained show that the novel deep learning model is effective in the both electrical load prediction and PV power forecasting and outperforms other models such as FFNN, Recurrent Neural Network (RNN), Decision Tree (DT), Gated Recurrent Unit (GRU) and eXtreme Gradient Boosting (XGBoost).

Keywords:

• Electrical demand prediction
• photovoltaic power forecasting
• deep learning models
• smart grid
• long short term memory
• multi-objective particle swarm optimization

REVIEWING EDITOR:

• Qingsong Ai, Wuhan University of Technology, China

SUBJECTS:

• Technology
• Computer Science
• Algorithms & Complexity
• Artificial Intelligence
• Computation
• Computer Engineering
• Computer Science (General)

1. Introduction

Nowadays, the power networks face technological developments both in the electrical power generation and on the transmission and distribution lines (Mbey et al., 2022). In addition, advances in the industrial and social sector have enormously increased electrical consumption. In this case, the International Energy Agency (IEA) projects an increase of nearly 50% in electrical consumption in 2050. The worldwide consumption of commercial and residential sectors will increase by 65% between 2018 and 2050, and the pollution will double in 2050 (Desai et al., 2022). To face these challenges, it is now necessary to use renewable sources and new information and communication technologies which make it possible to revolutionize electrical consumption using real time management techniques in the 21st century. Renewable energy sources such as solar and wind are also used today by end consumers, which leads to variability in the electrical network, with the need to balance electrical demand to electrical power generation anytime (Gupta et al., 2022). With the integration of digital technologies, a large amount of data is produced through digital equipment, sensors, Phase Measurement Units (PMU), smart meters and Advanced Metering Infrastructure (AMI). The processing and analysis of these data represent the new challenge but also an opportunity for the 21st century (Syed et al., 2021). The concept of smart grid and the application of Artificial Intelligence (AI) methods based on deep learning for data analysis, will help to face these challenges (Foba et al., 2021). The principle of smart grid is based on solving energy issues by providing a two-way flow of electrical power and information between consumers and energy producers (Ahmad et al., 2020). However, real-time data management for decision-making still represents a major challenge (Souhe et al., 2022). This is why energy distributors worked in recent years to install a large number of smart meters in order to use this data for effective demand management. To date, the data is collected monthly by the meters, but with the implementation of the AMI, the meters record the data every 15 to 30 minutes, this data can reach the terabit (Zhang et al., 2018). The smart meters being installed throughout the world in recent years will thus allow the migration to the smart grid. Compared to the conventional network, the smart grid provides several advantages, in particular: automatic restoration, better integration of renewable energies, precise knowledge of the situation of the network through the deployment of smart meters and data analysis by the deep learning and machine learning. In (Souhe et al., 2021), a roadmap for the deployment of smart meters in Cameroon has been proposed. In this article, the number of smart meters deployed is estimated at 3 million in 2040. Smart meters therefore have a particular importance in smart grid applications by providing new data analysis operations such as the analysis of consumer behavior, electricity fraud detection, outage management and automated demand response.

Although the main role of smart meters is the measurement of electrical energy parameters, they also generate an immense amount of data allowing to have a full resolution of the electrical distribution network. Thus, by analyzing this data and converting it into a control instrument, it would improve energy distribution by ensuring good energy quality for consumers. These smart meters therefore contain a memory to store the information and a communication module to transfer this information to the control center of the smart grid (Mbey et al., 2023). Similarly, smart meters now provide hourly and monthly readings of electrical energy consumption and thus collect a large amount of data. Analyzing such data can help to readjust energy consumption optimization strategies by improving the accuracy of predictive models to make them more reliable (Melzi et al., 2017). The intelligent infrastructure associated with the communication network generates a large volume of data that requires various techniques for analysis and decision-making. Machine learning and deep learning techniques play an important role in analyzing this amount of data in order to obtain suitable results (Thilakarathne et al., 2020). The use of conventional approaches for electrical operations through deterministic programming makes it difficult to control, monitor, measure and predict in the network. In addition, conventional techniques require manual monitoring and control operations leading to frequent problems especially when integrating renewable energy sources . Simple machine learning models such as Artificial Neural Network (ANN) and Support Vector Machine (SVM) have many limits, which explains their low use for complex problems in electrical power systems (Yem Souhe et al., 2022). Moreover, these models are inefficient for high-dimensional representations with high complexities. Moreover, these models cannot be improved with large amounts of data (Shamshirband et al., 2019). To face these problems, learning paradigms have migrated to deep learning to take into account this abundance of data with the extraction of hierarchical components using its strong learning potential. With the complexity of smart grids, the need for deep learning is observed in the use of important data from smart meters and Internet of Things (IoT) devices (Liang et al., 2020). These new algorithms ensuring reliable data will improve the distribution of information between machine learning and systems (Elahe et al., 2021; Martinez et al., 2021). So, the deep learning models can be classified into two categories: individual models and hybrid models.

Individual models basically include: Multilayer Perceptron (MLP), Graph Neural Network (GNN), SVM, Deep Neural Network (DNN), RNN, Convolutional Neural Network (CNN), Shallow Neural Network (SNN), FFNN, XGBoost, LSTM, Auto Encoder (AE), Generative Adversarial Network (GAN), Restricted Boltzmann Machines (RBM), Deep Reinforcement Learning (DRL), GRU, Generator Network (GN) and Capsule Networks (CN). Mashlakov et al. (2021) evaluated DL models for multivariate probabilistic energy forecasting. Similarly, Comert and Yildiz (2022) developed a novel ANN model for electrical demand forecasting enhanced by population-weighted mean temperature and unemployment rate. A linear function based on the weighted average temperature was created in order to fit a function for the monthly oscillations. For this purpose, records of average temperature values ​​for cities in Turkey were used as additional input data from January 2000 to November 2019. Mean Square Error (MSE) coefficients were calculated for training, validation and test respectively for 3.77%, 2.02%, and 1.95%. Hong et al. (2020) presented a framework for short-term residential demand forecasting which is based on Deep Neural Network and Iterative Resblock (IRBDNN). In this work, data acquisition collects measurement data from household smart meters. Data preprocessing enables data cleaning, data integration, and data transformation. At the end, the proposed model can calculate the predicted values ​​for each consumer. Real-world measurement data was used to evaluate the performance of the proposed model. Compared to existing models, the proposed approach presents a reduction of Root Mean Square Error (RMSE), Mean Absolute Error (MAE) and Mean Absolute Percentage Error (MAPE) respectively from 20.00% to 3.89%, from 22.58% to 2.18% and from 32.78% to 0.69%. Similarly, Widodo et al. (2021) developed a DNN method using LSTM as a learning model to ensure accurate prediction of renewable energy generation. The input values of the neural architecture are irradiance, air temperature, panel temperature, wind speed, wind direction and precipitation. The activation functions used in the forecasting process can be hyperbolic tangent functions and Rectified Linear Unit (ReLU) functions. The simulation results give a MAE of 0.035 and a MSE of 0.0023. Moreover, Gupta et al., (2022) compared the efficacy of novel time series estimation models such as auto-seasonal autoregressive integrated moving average (auto-SARIMA), Facebook Prophet (FBP) and Neural Prophet (NP) for predicting the future values of global horizontal irradiance. The obtained results give the RMSE of 18.69, 21.62, 14.473 and 10.70 W/m² respectively for the four datasets.

Moreover, hybrid models are based on the individual models and optimization techniques. Hafeez et al., (2020) proposed a novel hybrid model for short-term electrical demand forecasting. This hybrid model is a framework that integrates a Modified Mutual Information (MMI), the Factored Conditional Restricted Boltzmann Machine (FCRBM) and the Genetic Wind-Driven Optimization (GWDO). The accuracy of the proposed model is evaluated through historical data of hourly consumption in three electrical networks in the USA. In addition, the proposed model is compared with four other recent models including Bi-level, Mutual Information based Artificial Neural Network (MI-ANN), Accurate Fast Converging Artificial Neural Network (AFC-ANN), and LSTM. In terms of accuracy, the proposed model exceeds the MI-ANN by 31.2%, the Bi-level by 17.3%, the AFC by 4.7%. The execution time of the proposed model is 52s, on the other hand that of the AFC-ANN is 58s, the Bi-level is 102s, and the LSTM is 63s. Similarly, a combined technique using LSTM and a learning transfer approach based on XCORR has been proposed in Ozer et al. (2021) for short-term electrical demand forecasting. The XCORR is applied between the data of the buildings to be estimated and the data of each building to be transferred. The LSTM is trained with standardized data from these buildings. Thus, the performance values ​​of this model are obtained through the test data. A dataset of electricity demand of buildings of Bandırma Onyedi Eylu University with a resolution of 15 minutes was used to validate the proposed model. The accuracy of the model is evaluated using the indicators RMSE, MAE and MAPE with the respective values ​​of 736.706, 352.176 and 8.145. In addition, this model has been compared with methods such as Random Forest Algorithm (RFA), XGBoost and Light Gradient Boosting Machine (LGBM), thus presenting better results. Syed et al. (2021a, 2021b) proposed a novel cluster-based deep learning approach for short-term power consumption forecasting at distribution transformers. The performances of the proposed model are compared with the individual models. The precision evaluation coefficients are the RMSE and the MAPE. The performances of the models are evaluated using the training and execution time. For the first data set, the values of RMSE, MAPE, training time and execution time are respectively 2.6874 kWh, 15.9380%, 10.76s and 0.1070s. For the second data set, the values are respectively 21.2596 kWh, 7.2271%, 4644.82s and 4.57s. Moreover, Farsi et al. (2021) proposed an LSTM-CNN model for short-term load forecasting. LSTM and Gradient Boosting Regression (GBR) models have been adopted in Selim et al. (2021) to assess the uncertainties in the prediction of short-term electrical demand. Data collected by the Eastern Slovakia Electricity Corporation was used to validate the consumption forecasting models. RMSE and MAPE coefficients were used to verify the performance of the models. The LSTM model presents a RMSE of 18.025 and a MAPE of 0.023, the GBR model gives a RMSE of 17.42 and a MAPE of 0.023. Sengan et al. (2021) proposed a method based on deep learning for the detection of false data cyber-attacks in a smart grid. To this end, to obtain the combined Artificial Feed-forward Network (AFN) model, several techniques have been proposed, in particular: CNN, RNN and LSTM. This method has been implemented on an IEEE 14 bus network for the identification of cyber-attacks. The results obtained show an improvement in accuracy of 98.19% in the detection of false data. In same time, Siniosoglou et al. (2021) presented an intrusion detection system for smart grid environments that uses Transmission Control Protocol (TCP) and Distributed Network Protocol 3 (DNP3). The proposed method is called anoMaly dEtection aNd claSsificAtion (MENSA) adopting a new GAN encoder architecture. For the detection of cyber-attacks by the DNP3 protocol, the MENSA has an accuracy of 0.994, a True Positive Rate (TPR) of 0.983, a False Positive Rate (FPR) of 0.003 and a F1 score of 0.983. For detection in the TCP protocol, the MENSA gives an accuracy of 0.964, a TPR of 0.730, a FPR of 0.019 and a F1 score of 0.730. These results demonstrate that the MENSA model outperforms other machine learning and deep learning methods. Other authors have hybridized neural networks with fuzzy logic to improve data classification. Moreover, Pekaslan et al. (2020) developed a Mandani fuzzy logic system for predicting renewable energy production uncertainties by considering a variety of climate changes depending on the season. Yem et al. (2022) developed a hybrid model for the prediction of household electricity consumption in a smart grid system. Thus, a combined Grey-ANFIS-PSO model was built based on data from meters installed in households in order to improve the forecast of electrical energy consumption. Household electricity consumption data in Cameroon over a period of 24 years was used to validate the model. The accuracy of the model obtained gave a RMSE of 0.20158 and a MAPE of 0.6291%. These results are better in comparison with single methods such as SVM and ANN.

Table 1 summarize deep learning applications for forecasting PV power generation and electrical consumption in smart grid.

Table 1. Summary of deep learning applications for forecasting PV power generation and electrical demand.

Author	Problem	Data set	Techniques	Limit
Wen et al. (2020)	Residential building electrical consumption forecasting	Climate data in US	DRNN-GRU	Model is only validated in small dataset
Fujimoto et al. (2021)	Short-term residential load forecasting	553 consumers in Ohta, Japan	Deep reservoir architecture	Low convergence
Ren et al. (2021)	Electrical load forecasting	Households in China	GC-LSTM	Low performance
Zhang et al. (2021)	Short term electrical load forecasting	02 datasets in England	ADDPG-AEF-RIM	Performance coefficients are not effectively evaluated
Chen and Zhang (2021)	Electrical load forecasting	Historical and climate data in china	TgDLF, EnLSTM	Model complexity
Eniola et al. (2021)	Short term solar photovoltaic power generation forecasting	Climate data in US	Markov model and genetic algorithm	Low prediction
Zieba et al. (2021)	Photovoltaic power generation forecasting	Climate data in Australia	multidirectional search optimization algorithm	Abnormal climate conditions
Shaqour et al. (2022)	Short term electrical demand forecasting	479 buildings in japan	CNN, DNN, GRU, LSTM, Bi-GRU	Small aggregation
Zhao et al. (2021)	Forecasting of electrical load in microgrid	69 consumers in Australia	k-means, QRLSTM, KDE	Complexity of model
Chung et al. (2022)	Electrical load forecasting	12,000 households in korea	CNN-LSTM	Gradient problem
Haihong et al. (2022)	Electrical demand forecasting	Microgrid in china	TCN-DNN	Model complexity
Wentz et al. (2022)	Solar photovoltaic energy generation	Climate data over 3 years in California	ANN, LSTM	Complex structure
Akhter et al. (2022)	Forecasting of PV generation	03 solar fields in US over 4 years	RNN, LSTM	Model complexity
Michael et al. (2022)	Short term solar generation forecasting	Climate data in Abu Dhabi	CNN, LSTM	High simulation time
Rodriguez et al. (2022)	PV power generation forecasting	Climate data in Vitoria-Gasteiz, Spain	STFFNN	Require features adjustments
Jiandong et al. (2022)	Short-term wind power forecasting	Wind field in Dingbian, Shaanxi, China	LSTM, PSO-DBN	Increase computation complexity
Markovics and Mayer (2022)	Photovoltaic power generation forecasting	Climate data in US	MLP, SVM, LGBM, RF, XGBoost	Aggregation complexity
Salih et al. (2022)	Short term PV power generation forecasting	Climate data in Australia	LSTM, SVM, GBT, DT, ANN, GLM	Hybridization complexity
Boum et al. (2022)	Long term PV power generation forecasting	Climate data of Douala, Cameroon	ANN-SVM-PSO	Convergence speed can be affected
Rangelov et al. (2023)	Short term photovoltaic power generation forecasting	Climate data in Berlin, Germany	RF, DNN, LSTM	Comparison with existing model is not evaluated
Gaboitaolelwe et al. (2023)	Photovoltaic power generation forecasting	Climate data in Utrecht, Netherlands	Lasso, MLP, SVR, SVM, RF, RF, XGB, GB	Model is not efficiently elaborated and exploited

In this works, we proposed novel techniques based on deep learning for power generation and electrical load forecasting. The proposed deep learning models allows us to generate an output forecasting from climate data and socio-economic data.

The main contribution of this research work is developed as follows:

• We first summarized individual and hybrid deep learning models for electrical demand prediction and solar photovoltaic power generation forecasting. In addition, we highlighted the most relevant recent works for power forecasting with the highest accuracy.

• Then, we proposed a novel hybrid deep learning model combining FFNN, LSTM and MOPSO for solving classification and binary dynamic processing problems. We showed that this hybrid model ensures high classification accuracy with reduced execution time.

• We also implemented the deep learning models of our work on a Cameroon dataset for short term solar photovoltaic power generation forecasting and long term electrical demand forecasting.

• Finally, we compared the proposed deep learning models with those in the literature using accuracy coefficients such as RMSE, MSE, MAE, MAPE and regression.

The rest of the work is organized as follows: Section 2 presents the materials and the methods used in our work. Here, we first present the dataset, the software material and the computer used. Then, we developed the forecasting models based on deep learning. The results and the discussion are presented in section 3. Finally, section 4 concluded our work with future perspectives.

2. Materials and methods

2.1. Materials

2.1.1. Dataset

a) dataset for forecasting electrical consumption

The aim of this section is to make a long-term prediction of electrical demand. For this purpose, our dataset will include electrical consumption data as an output variable and socio-economic factors such as GDP, population, number of unemployed, number of subscribers (residential, commercial and industrial), the price per kilowatt of electricity as input variables. These data were obtained from the World Bank, the Ministry of Water and Energy of Cameroon, the company in charge of electricity distribution and the National Institute of Statistics from 1990 to 2020. Table 2 presents the dataset for the electrical consumption forecasting.

Table 2. Dataset for the electrical demand forecasting.

Year	GDP in billion (USD)	Population in million	Number of subscriber in thousand	Unemployment rate (%)	Electricity cost (CFA)	Real consumption (GWH)
1990	12,31	11,78	315	21.1	150.5	239
1991	11,84	12,14	329	21.6	145.55	248
1992	12,07	12,5	348	22.3	140.5	261
1993	14,52	12,86	365	24.6	135	278
1994	8,293	13,23	381	25.8	132	295
1995	10,02	13,6	401	26.3	130	289
1996	10,31	13,97	420	27.5	128	321
1997	10,04	14,34	427	28.1	125	350
1998	10,83	14,72	447	28.7	121	395
1999	10,71	15,11	451	29.3	118	400
2000	10,11	15,51	451	29.7	115.5	387
2001	10,38	15,93	452	30	112.5	388
2002	11,63	16,36	488	31.8	110.55	457
2003	14,58	16,8	504	33.4	108.15	496
2004	17,46	17,26	507	35.6	107	548
2005	17,95	17,73	527	36.4	106.25	708
2006	19,37	18,22	537	39.1	105.95	803
2007	22,39	18,73	571	42.3	104.35	828
2008	26,52	19,25	614	45.6	103.85	896
2009	26,12	19,79	660	47.5	102.25	903
2010	26,17	20,34	711	50.1	101	986
2011	29,38	20,91	707	52.3	98	1052
2012	29,1	21,49	709	57.6	96	1081
2013	32,36	22,08	852	59.1	94.5	1111
2014	34,99	22,68	887	62.9	92	1141
2015	30,93	23,3	927	63.4	90	1303
2016	32,64	23,93	969	65.1	89	1304
2017	35,01	24,57	1012	66.4	87	1305
2018	38,69	25,22	1086	67	86.2	1316
2019	39,01	25,88	1156	69	85	1325
2020	39,8	26,55	1214	70	86.95	1332

b) dataset for the photovoltaic power generation forecasting

In our work, we use a dataset consisting of daily input data including: irradiance, temperature, solar tilt angle, wind speed and relative humidity to make a short-term prediction of photovoltaic energy generation for the next day. These data were obtained from the electricity distribution company and the meteorological station of the city of Douala. Table 3 gives the dataset for solar photovoltaic generation forecasting.

Table 3. Dataset for solar photovoltaic power generation forecasting.

Hour	Temperature (Celsius degree)	Irradiance (kWh/m^2)	Wind speed (km/h)	Humidity (%)	Inclination angle (°)
0	27	3.04	3	81	15
1	27	3.1	2	83	14.5
2	27	3.4	2	84	15
3	27	3.51	3	85	14.6
4	28	3.64	2	89	14.5
5	29	3.68	2	89	15.6
6	29	3.74	3	88	15.2
7	29	4.05	4	86	14.6
8	29	4.19	5	79	13.7
9	30	4.56	5	72	13.4
10	31	4.86	6	63	14.8
11	32	5.06	6	57	15.6
12	33	5.41	7	56	15.5
13	33	5.54	10	58	14.3
14	32	5.02	13	55	13.4
15	32	4.86	12	56	13.8
16	31	4.69	11	61	14.4
17	31	3.58	11	66	13.6
18	30	3.48	7	69	12.1
19	29	3.32	6	74	13.5
20	28	3.04	5	74	14.2
21	28	3.02	4	75	13.5
22	27	3.01	4	77	14.1
23	27	3	3	79	15.4

2.1.2. Matlab

In our work, we used Matlab which is a software that allows programming for the intelligent resolution of data mining problems. Matlab allows to implement all deep learning algorithms in the smart grid applications such as LSTM and FFNN. In this paper, we used Matlab R2020b 64 bit version.

2.1.3. Computer

The hardware material is a computer with the following specifications: processor icore 5, 3.5 GHz; RAM 8 GB; hard disk 1 Terabits; Windows 10/64 bit System.

2.2. Methods

2.2.1. Feed-Forward Neural Network

The architecture of FFNN is given in Figure 1 with connection of the neuron, activation function and bias. The structure of a neuron is same with facility distribution structure from source to the final destination. The route from sources to destinations represents the interconnection of the neuron, then information flows from external environment in the neuron represent product flow from source to end destination and the summing junction is the incidence point. The neural structure under consideration is made of several input information to give an output information through bias, weight and activation function.

Figure 1. Architecture of Feed-Forward Neural Network.

From Figure 1, the linearly combination of all the inputs allow to obtain the total sum of the neuron $V_k.$ This sum is obtain by the appropriate multiplication of weights with connecting neurons so that the mathematical form of summing junction $V_k$ can be expressed in Equation 1(1) $$V_k=\sum _{j=0}^n{X}_j{W}_{\mathit{\text{kj}}}+b_k$$(1) : (1) $$V_k=\sum _{j=0}^n{X}_j{W}_{\mathit{\text{kj}}}+b_k$$(1)

The output value is expressed with equation 2(2) $$Y_k=\phi \left(t\right)V_k$$(2) . (2) $$Y_k=\phi \left(t\right)V_k$$(2)

Substituting the value of $V_k$ in Equation (1)(1) $$V_k=\sum _{j=0}^n{X}_j{W}_{\mathit{\text{kj}}}+b_k$$(1) allows to transform the output as expressed in Equation 3(3) $$Y_k=\phi \left(t\right)\sum _{j=0}^n{X}_j{W}_{\mathit{\text{kj}}}+b_k$$(3) . (3) $$Y_k=\phi \left(t\right)\sum _{j=0}^n{X}_j{W}_{\mathit{\text{kj}}}+b_k$$(3)

With $V_k$ the Linear combiner with bias to support input signal; $X_j$ the input data from signal j; $W_{\mathit{\text{kj}}}$ the Weight or strength of connection from neuron j to k; $b_k$ the bias from neuron j; $\phi (t)$ the activation function.

In this work, a Feed-Forward Neural Network Back-Propagation learning model with sigmoid transfer function [5–10–1–1] was considered using 70% of the data for training and 30% for testing and validation as shown in Figure 2.

Figure 2. FFNN back-propagation learning model.

2.2.2. Support Vector regression

The SVR was developed by Vladimir Vapnik in the late 20th century using deep learning tools for solving nonlinear problems with high dimensionality. The SVR helps to create a hyper plane for data classification to find the most appropriate for the distinction between the two classes for the separation of the data. The hyper plane separates the training data into two groups each having a label between +1 and −1 such that the distance between the hyper plane and the nearest training element is maximal with the intention of forcing generalization machine learning. Figure 3 gives the hyper plane which correctly separates the data into two spaces.

Figure 3. Hyper plane for data separation.

The linear regression function is given in Equation 4(4) $$f\left(x\right)=\omega *x+b=0$$(4) . (4) $$f\left(x\right)=\omega *x+b=0$$(4)

With $f(x)$ the forecast value; $\omega$ the weight and $b$ the bias.

The expression of hyper plane separators are given by Equations 5(5) $$P_{+1}=\omega *x+b=+1$$(5) and 6(6) $$P_{-1}=\omega *x+b=-1$$(6) respectively. (5) $$P_{+1}=\omega *x+b=+1$$(5) (6) $$P_{-1}=\omega *x+b=-1$$(6)

With $P_{+1}$ the positive region; $P_{-1}$ the negative region; $\mathit{\text{HP}}$ the hyper plane.

2.2.3. Long short term memory

LSTM was introduced in 1997 by Hochreiter and Schmidhuber. LSTM is a particular type of RNN used in deep learning to address long-term dependency problems. The major success of the LSTM model is related to its excellent ability in the extraction of temporal characteristics for data inputs. In an LSTM module, there are 3 separate memory gates, respectively the forgetting gate, the input gate and the output gate. Figure 4 gives the structure of an LSTM module.

Figure 4. Structure of an LSTM module.

The Forgetting Gate decides how much information to forget and how much will be passed on to the next step. It uses a sigmoid activation function that generates a value between 0 and 1 for this process. ‘0’ means no information should be passed, and ‘1’ means all information should be passed. It is possible to express the mathematical expression of this gate by Equation 7(7) $$f_t=\sigma \left(W_f{x}_t+U_f{h}_{t-1}+b_f\right)$$(7) . (7) $$f_t=\sigma \left(W_f{x}_t+U_f{h}_{t-1}+b_f\right)$$(7)

In the next step, it is decided what information should be stored for the input gate represented by Equation 8(8) $$i_t=\sigma \left(W_i{x}_t+U_i{h}_{t-1}+b_i\right)$$(8) . (8) $$i_t=\sigma \left(W_i{x}_t+U_i{h}_{t-1}+b_i\right)$$(8)

As a first step at this point, a 2nd sigmoid function $\sigma$ decides which values should be updated. Subsequently, a vector of new candidate values is created through a function $\mathrm{\text{tanh}},$ expressed by Equation 9(9) $$C_t=\mathrm{\text{tanh}}\left(W_c{x}_t+U_c{h}_{t-1}+b_c\right)$$(9) , and then these two processes are combined. (9) $$C_t=\mathrm{\text{tanh}}\left(W_c{x}_t+U_c{h}_{t-1}+b_c\right)$$(9)

Once the candidate values ​​are determined, the new state information of the memory cell must be calculated. The new state information calculation process is expressed in Equation 10(10) $$C_t=f_t*C_{t-1}+i_t*C_t$$(10) . (10) $$C_t=f_t*C_{t-1}+i_t*C_t$$(10)

Finally, the output gate expressed in Equation 11(11) $$o_t=\sigma \left(W_o{x}_t+U_o{h}_{t-1}+b_o\right)$$(11) can calculate the output of the system of Equation 12(12) $$h_t=o_t*\mathrm{\text{tanh}}\left(C_t\right)$$(12) : (11) $$o_t=\sigma \left(W_o{x}_t+U_o{h}_{t-1}+b_o\right)$$(11) (12) $$h_t=o_t*\mathrm{\text{tanh}}\left(C_t\right)$$(12)

With $x_t$ the input signal at time t; $\sigma$ the activation function; $C_t$ the memory unit; $f_t$ the forgetting gate at time t; $i_t$ the input gate at time t; $o_t$ the output gate at time t; $(b_f,b_i,b_o)$ and $(W_f,W_i,W_o)$ the bias and weight matrix for each gate, respectively.

2.2.4. Multi-Objective particle swarm optimization

The MOPSO is a metaheuristic technique inspired from the PSO to improve the determination of optimal values. The PSO was developed by Kennedy 1995. It is based on artificial intelligence and on the observations of fish, ants, bees, and birds in their natural environment. The PSO is an algorithm based on population that takes inspiration from the way birds fly and flocks of fish. The operating principle of the MOPSO is presented in Algorithm 1.

Algorithm 1:

MOPSO algorithm

1: Set the counter $t=0$ then randomly generate the particles

2: Update the counter $t=t+1$

3: Update the inertia weight $\omega (t)=\mathit{\beta \omega }(t-1)$ ; With $\beta$ the decrement value near to 1

4: update the velocity

$v_{\mathit{\text{ik}}}(t)=w(t)v_{\mathit{\text{ik}}}(t-1)+c_1{r}_1(x_{\mathit{\text{ik}}}^{*}(t-1)-x_{\mathit{\text{ik}}}(t-1))+c_2{r}_2(x_{\mathit{\text{ik}}}^{**}(t-1)-x_{\mathit{\text{ik}}}(t-1))$

With $x_{\mathit{\text{jk}}}^{**}$ the best global, $x_{\mathit{\text{ik}}}^{*}$ the best individual

5: Update the position of particle $x_{\mathit{\text{ik}}}(t)=v_{\mathit{\text{ik}}}(t)+x_{\mathit{\text{ik}}}(t-1)$

6: Add the new updated position of the $i^{\mathit{\text{th}}}$ particle to non-dominated local set $S_i^{*}(t).$

7: Reduce the size of $S_i^{**}(t)$ using a clustering algorithm if it exceed a given value

8: Update an external Pareto-optimal by copying the members of $S_i^{**}(t)$, then search the external Pareto set and remove all dominated solutions from the set

9: Evaluate the obtained distances between values in $S_i^{*}(t)$ and $S_i^{**}(t)$ in objective space

10: Stop the process if the numbers of iterations exceeds the limit, otherwise, go back to step 2

2.2.5. Proposed multi-objective hybrid deep learning model

In this work, we proposed a novel Multi-Objective Hybrid Model named FFNN-LSTM-MOPSO to perform power consumption prediction and renewable energy generation forecasting considering socio-economic data and climate data. In the proposed model, the FFNN is first used for feature extraction from the input data and data preprocessing, then the output data of FFNN is used as the input data of the LSTM. Then, the LSTM can process the training and validation using the obtained data from FFNN. Furthermore, the MOPSO can optimize the forecast parameters such as the bias $\beta ,$ the weight $\omega ,$ the number of particle $n$ and the objective function $f(x).$ The aim of this process is to reduce the training and processing time of the model. The proposed multi-objective hybrid deep learning model is presented in Figure 5.

Figure 5. The proposed Multi-Objective Hybrid Deep Learning model.

In the hybrid model, we use 3 layers, 90% for the learning rate and we also use the Adaptive moment estimation (ADAM) optimizer.

The proposed hybrid model is explained as follow:

Step 1: the pre-processing and feature extraction is performed by FFNN using climate data such as irradiance, temperature, wind speed and humidity; and socio-economic data such as GDP, population, number of unemployed, number of subscribers.

Step 2: Then, the training and validation is performed by LSTM using data from output of FFNN in order to realize a final and precise forecasting. The network progressively predicts and updates the trained network from the previous time during the training stage of LSTM. The LSTM network is trained separately for the forecasting of the electrical consumption and photovoltaic generation. The initial network allows the training of the data of the trained system. Then, the initial LSTM network is tested on the validation data. Subsequently, the LSTM network step-by-step can forecast the output value on the validation data.

Step 3: the accuracy coefficients are calculated after obtaining the forecasting data from the LSTM module. Consequently, a MOPSO algorithm is implemented for feature optimization to improve the accuracy of the hybrid deep learning model. In the deep learning method, the features which are optimized by MOPSO are the bias $\beta ,$ the weight $\omega ,$ the number of particle $n$ and the objective function $f(x).$

Step 4: then, the initial network relearns and readjusts the current data to the validation data until the error is definitely minimized. Finally, the final data is used to ensure the perfect forecasting with the minimal error.

2.2.6. Accuracy evaluation metrics

Several evaluation coefficients are used to compare the performance of the developed deep learning forecasting models. Table 4 shows the evaluation metrics used in this work.

Table 4. Accuracy evaluation metrics.

Coefficients	Expression	Range	Goal
Mean Square Error	$\mathit{\text{MSE}}=\frac{1}{N}{\sum }_{i=1}^N{(P_i-O_i)}^2$	$\left[0;\infty \right[$[	Minimize
Root Mean Square Error	$\mathit{\text{RMSE}}=\sqrt{\frac{1}{N}{\sum }_{i=1}^N{(P_i-O_i)}^2}$	$\left[0;\infty \right[$[	Minimize
Mean Absolute Error	$\mathit{\text{MAE}}=\frac{1}{N}{\sum }_{i=1}^N\left|P_i-O_i\right|$	$\left[0;\infty \right[$[	Minimize
Mean Absolute Percentage Error	$\mathit{\text{MAPE}}=\frac{1}{N}{\sum }_{i=1}^N\left|\frac{P_i-O_i}{O_i}\right|*100\%$	$\left[0;100\right]$	Minimize
Regression	$R=\sqrt{1-\frac{{\sum }_{i=1}^N{(P_i-O_i)}^2}{{\sum }_{i=1}^N(P_i-O_i)}}$	$\left[0;1\right]$	Maximize

3. Results and discussion

In this section, we show the results obtained from deep learning model simulations in our work on climate and socioeconomic data. The aim is to verify the performance of the proposed models in the long-term electrical demand prediction and the short-term solar photovoltaic power generation forecasting.

3.1. Experimental results on electrical demand forecasting

In this section, we have implemented power consumption forecasting using the models proposed in our work. Figure 6 gives the electrical demand forecasting using SVR model, XGBoost model, FFNN model, LSTM model, FFNN-LSTM model and FFNN-LSTM-MOPSO model.

Figure 6. Electrical demand forecasting using a) SVR b) XGBoost c) FFNN d) LSTM e) FFNN-LSTM f) FFNN-LSTM-MOPSO.

As depicted in Figure 6a, the SVR model gives a forecasting which is effective enough because its variations from the actual data between 1990 and 2020. In Figure 6b, the XGBoost model gives a better demand forecasting than the SVR model but with some fluctuations in some years. As shown in Figure 6c, the forecasting of FFNN model follows the actual data between 1995 and 2010. However, we observe that this forecasting deviates from the real consumption data between 2011 and 2020 which could be caused by its slow convergence. It can be observed in Figure 6d, that the LSTM model forecasting follows the real data between 2000 and 2020 with some variations in 2002, 2003 and 2009. However, the LSTM model is better that previous models. This result shows the effectiveness of the LSTM model in electrical demand forecasting. With the limits of individual models, we combined FFNN with LSTM model to obtain better results than single models. As shown in Figure 6e, the FFNN-LSTM model gives a forecasting essentially confused with real data between 2000 and 2020. Finally, with the aim to perfectly reach the highest precision, we optimize our FFNN-LSTM hybrid model with the MOPSO to obtain the optimal prediction as shown in Figure 6f. It can be observed that the FFNN-LSTM-MOPSO model outperforms single model with highest precision.

Finally, using the novel multi-objective hybrid model, we can predict a growing of electrical demand from 1350GWH in 2020 to 1510 GWH in 2030 as shown in Figure 7.

Figure 7. Long-term prediction of electrical demand using the proposed hybrid FFNN-LSTM-MOPSO model.

Table 5 gives a comparison of electrical demand forecasting models implemented in this study using accuracy evaluation metrics. As depicted in Table 5, the proposed multi-objective hybrid model has proven the outperformance to the baseline models as the resultant RMSE and MAE are lower than the inferred values from the baseline model for electrical demand forecasting.

Table 5. Comparison of deep learning models for forecasting electrical demand.

Model	MSE (kW)	RMSE (kW)	MAE (kW)	MAPE (%)	R
SVR	4.5	2.12	1.26	2.85	0.9852
XGBoost	0.52	0.72	1.02	1.54	0.9956
FFNN	0.26	0.51	0.45	0.84	0.9975
LSTM	0.05	0.22	0.23	0.15	0.9987
FFNN-LSTM	0.0012	0.034	0.075	0.028	0.9994
FFNN-LSTM-MOPSO	0.00014	0.011	0.005	0.013	0.9999

In Table 6, we compared our novel hybrid deep learning model with recent works in the literature on electrical demand forecasting.

Table 6. Comparison with deep learning models of the literature.

Models	MSE (kW)	RMSE (kW)	MAE (kW)	MAPE (%)	R	Authors
ADDPG-AEFRIM	2.04	1.428	2.58	0.94	0.8996	Zhang et al. (2021)
GC-LSTM	3.66	1.913	7.85	2.36	0.8795	Ren et al. (2021)
Deep reservoir architecture	2.15	1.466	5.42	0.64	0.8896	Fujimoto et al. (2021)
TgDLF, EnLSTM	1.58	1.241	1.33	0.89	0.9654	Chen and Zhang (2021)
k-means, QRLSTM, KDE	0.012	0.11	0.15	0.47	0.9979	Zhao et al. (2021)
TCN-DNN	0.0035	0.059	–	–	0.9995	Haihong,et al. (2022)
CNN-LSTM	–	–	0.04	0.38	0.9987	Chung et al. (2022)
CNN, DNN, GRU, LSTM, Bi-GRU	0.06	0.244	0.48	0.75	0.9788	Shaqour et al. (2022)
FFNN-LSTM	0.0012	0.034	0.075	0.028	0.9994	Writers
FFNN-LSTM-MOPSO	0.00014	0.011	0.005	0.013	0.9999	Writers

3.2. Experimental results on PV power generation forecasting

This section gives the results of PV power generation forecasting using the developed models. Figures 8, 9, 10, 11, 12 and 13, respectively give the daily forecasting with regression using the SVR model, XGBoost model, FFNN model, LSTM model, FFNN-LSTM model and FFNN-LSTM-MOPSO model.

Figure 8. PV power generation Forecasting using SVR model.

Figure 9. PV power generation Forecasting using XGBoost model.

Figure 10. PV power generation Forecasting using FFNN model.

Figure 11. PV power generation Forecasting using LSTM model.

Figure 12. PV power generation Forecasting using FFNN-LSTM model.

Figure 13. PV power generation Forecasting using FFNN-LSTM-MOPSO model.

As shown in Figure 8, the SVR model can effectively predict PV power generation. However, there is a variability between the forecast and the actual values between 00 a.m and 05 a.m. But the prediction is accurate between 6 a.m. and 2 p.m. The regression coefficient of this model is evaluated at 0.9824. As depicted in Figure 9, the XGBoost model gives a forecasting result close to real data. In this case, its regression coefficient is 0.9965. In Figure 10, FFNN model gives predicted values close to the observed values with some variations between 2 a.m. and 7 a.m., 5 p.m. and 6 p.m. The regression coefficient is 0.9946, this is explained by the fact that the predicted values are quite close to the observed values. In Figure 11, the LSTM presents a forecasting results close to actual data with 0.9989 regression. As observed in Figure 12, the hybrid FFNN-LSTM model can predict the PV power generation with 0.9996 regression. Finally, we improve our predictor using MOPSO to obtain a novel hybrid model named FFNN-LSTM-MOPSO model which can perfectly predict the PV power generation as shown in Figure 13 with the highest accuracy and fast convergence.

Table 7 presents a comparison of PV power generation forecasting models implemented in this work.

Table 7. Comparison of PV power generation forecasting models.

Model	MSE (kW)	RMSE (kW)	MAE (kW)	MAPE (%)	R
SVR	5.25	2.29	3.21	1.56	0.9824
XGBoost	5.02	2.24	1.85	1.23	0.9965
FFNN	4.85	2.20	1.45	0.87	0.9946
LSTM	3.54	1.88	1.26	0.75	0.9989
FFNN-LSTM	2.36	1.53	1.05	0.15	0.9996
FFNN-LSTM-MOPSO	1.15	1.05	0.75	0.05	0.9999

It can be observed in Table 7 that the multi-objective deep learning algorithm gives better forecasting results compared with other single models.

Figure 14 gives the percentage of reduction of error with respect to best algorithm found for both the dataset.

Figure 14. Reduction of error for each model.

Table 8 presents a comparison of the proposed hybrid model with those in the literature for PV power generation forecasting.

Table 8. Comparison with the literature on PV power generation forecasting.

Model	MSE (kW)	RMSE (kW)	MAE (kW)	MAPE (%)	R	Authors
STFFNN	12.86	3.58	5.75	3.07	0.9899	Rodriguez et al. (2022)
LSTM, SVM, GBT, DT, ANN, GLM	6.58	2.56	2.85	1.47	0.9965	Salih et al. (2022)
RNN, LSTM	15.26	3.90	7.85	4.59	0.9878	Akhter et al. (2022)
ANN-SVM-PSO	14.97	3.86	3.32	0.867	0.9968	Boum et al. (2022)
LSTM, PSO	10.47	3.23	4.29	2.38	0.9928	Jiandong et al. (2022)
CNN, LSTM	13.31	3.64	6.52	3.22	0.9895	Michael et al. (2022)
MLP, SVM, LGBM, RF, XGBoost	–	–	4.05	2.27	0.9945	Markovics and Mayer (2022)
Lasso, MLP, SVR, SVM, RF, RF, XGB, GB	–	–	1.58	0.82	0.9970	Gaboitaolelwe et al. (2023)
RF, DNN, LSTM	7.89	2.81	2.59	0.758	0.9969	Rangelov et al. (2023)
FFNN-LSTM	2.36	1.53	1.05	0.15	0.9996	Writers
FFNN-LSTM-MOPSO	1.15	1.05	0.75	0.05	0.9999	Writers

It can be observed in Table 8 that the proposed hybrid model is better than those in the literature with minimum error and highest regression.

4. Conclusion

This study aims to present deep learning algorithms for electrical demand prediction and solar PV power generation forecasting. Therefore, we proposed a novel multi-objective hybrid model named FFNN-LSTM-MOPSO which is efficient in data training and optimization of input parameters. These deep learning models were implemented on socio-economic, demographic and climate dataset of the city of Douala in Cameroon over the last years. The obtained results show that the proposed model is effective in electrical load prediction and PV power generation forecasting with respectively RMSE, MAE and R of 1.15, 0.75 and 0.999. In our knowledge, it is the first paper which can both forecast the electrical load and PV power generation using large amount of historical data for long term predictions. Moreover, the novel multi-objective deep learning model proposed in the paper can help power distributors for vulgarization and integration of renewable energy in the future. This paper can be optimized by considering the psychological aspect of consumer in the demand forecasting and geographical position of the city in the power generation forecasting.

Acknowledgements

The authors acknowledge the electrical engineering department of ENSET of University of Douala and the research team.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data used to support the findings of this study are included within the article.

Additional information

Notes on contributors

Camille Franklin Mbey

Dr. Camille Franklin Mbey is a lecturer at ENSET of university of Douala. He received his PhD degree in 2021 at university of Douala. His research interests include smart grid, electrical power network, renewable power generation, artificial intelligence and deep learning. He can be contacted at email: [email protected].

Vinny Junior Foba Kakeu

Mr. Vinny Junior Foba Kakeu is an assistant lecturer at ENSET of university of Douala. He received his Master 2 degree in 2023 at university of Douala. His research interests include photovoltaic power generation, artificial intelligence, smart grid and deep learning. He can be contacted at email: [email protected].

Alexandre Teplaira Boum

Prof. Alexandre Teplaira Boum is a full professor at university of Douala since 2024. He receive his PhD degree in 2014. He is currently the head of departmet of basic sciences at university of Douala. His research interests include system optimization, smart power network, deep learning and Matlab programing. He can be contacted at email: [email protected].

Felix Ghislain Yem Souhe

Dr. Felix Gislain Yem Souhe is a lecturer at IUT of university of Douala. He received his PhD degree in 2022 at university of Douala. His research interests include electrical power grid, electrical consumption forecasting, artificial intelligence and smart grid. He can be contacted at email: [email protected].