Nowadays, traditional fossil fuels are being exhausted. Meanwhile, the excessive use of fossil fuels has caused serious environmental pollution. Developing renewable energy is one of the important ways to solve environmental problems and the energy crisis [1]. Solar energy is a clean, low-carbon and inexhaustible energy source. As one of the important ways to utilise solar energy, photovoltaic power generation has developed rapidly in recent years [2, 3]. Photovoltaic power generation is affected by sun's radiation, temperature, clouds, weather and other factors [4]. These factors cause frequent fluctuation of photovoltaic power, affect the safe and stable operation of the power system and restrict the further application of the photovoltaic system [5, 6]. Accurate prediction of photovoltaic power is of great significance to ensure the stability of the power grid and promote the large-scale utilisation of photovoltaic power.
At present, photovoltaic power forecasting methods are mainly divided into two categories: methods based on physical modelling and statistical methods based on historical data. The method based on physical modelling relies on the dynamic relationship between solar radiation and physical laws, but there is a general problem of low levels of accuracy and robustness [7, 8, 9]. Compared with physical modelling, the statistical method based on historical data only depends on historical data and uses data-driven feature extraction to predict the trend of photovoltaic power generation, without much physical knowledge. Therefore, the statistical method based on historical data is becoming one of the most popular photovoltaic power forecasting methods. Wang
Compared with the traditional machine learning model, the deep learning model has better feature learning ability and big data learning ability; moreover, it can deliver more accurate prediction results. Gao
Compared with machine learning models, the above-mentioned deep learning models have achieved certain results in prediction accuracy and prediction efficiency. However, in the actual prediction process, the deep learning models also have some problems, such as difficulty in convergence and poor robustness, and most models take little account of sudden weather changes, while the weather conditions have great influence on photovoltaic prediction. In order to further improve the accuracy of the ultra-short-term forecast of photovoltaic power stations, this paper divides the weather conditions into two categories according to whether there is a sudden change in the weather. The sparrow search algorithm (SSA) is introduced to optimise the parameters of the variational mode decomposition (VMD) method, and the optimised VMD method is used to decompose the photovoltaic sequence and send it to an LSTM network. The final forecast result is obtained by superimposing the forecast results of each modal component. The errors of back propagation (BP), artificial neural network (ANN), LSTM and the VMD–LSTM models are compared with that of the proposed model, and the results show the accuracy of the proposed model.
Photovoltaic output is affected by many factors, such as location, environment and meteorology. In this paper, the historical power data and meteorological data of a 50 MW photovoltaic power generation system for the year 2020 are selected as the samples, and the photovoltaic output period in a day is predicted and analysed. The sampling period of this power station is 15 minutes, and there are 52 sampling points. Figure 1 shows the photovoltaic output in different weather conditions. It can be clearly observed from the figure that the photovoltaic output in stable weather is relatively stable, and the curve is similar to a parabola. When the weather suddenly changes, the photovoltaic output fluctuates greatly, which affects the stable operation of the power grid. The output of a photovoltaic power station is quite different under stable and abrupt weather conditions, so it is necessary to separately forecast abrupt weather (abrupt weather) and smooth weather (non-abrupt weather).
VMD adopts a completely non-recursive modal decomposition method, which decomposes data sequence
The formula in Eq. (1) is solved by using the augmented Lagrange function to obtain the following expression:
The SSA is a bionic intelligent algorithm, and its behaviour characteristics are modelled after sparrows’ foraging and anti-predation behaviours. The sparrow population is composed of two sub-populations: joiner and discoverer. The search and foraging activities of the whole sparrow population is based on the change in the discoverer's position. In order to increase the predation rate, participants follow the discoverer to get food while constantly monitoring the discoverer. Sparrows in different positions will choose different escape strategies after realising that the population is in danger. The comprehensive performance of common algorithms, such as moth–flame optimisation algorithm, grey wolf optimisation algorithm and PSO algorithm, is inferior to hat of the SSA in terms of accuracy and stability; so, SSA with better searching ability is used to optimise VMD.
The VMD decomposition method decomposes data series by setting variables, such as decomposition number, fidelity coefficient and convergence conditions. The decomposition quantity
The aim of optimising VMD using the SSA algorithm is to decompose sequence
When the actual decomposition components of the signal are all orthogonal, the energy of the original data sequence
When the BIMF is not completely orthogonal, then
The orthogonality of the BIMF component is determined by the value of
To sum up, the solution process of [
Setting the initial population and related parameters of the SSA and selecting
After decomposing the data sequence by the VMD method, the fitness value of the sparrows is obtained using Eq. (8).
Updating the position of sparrows through the optimisation mechanism of the SSA and comparing the energy differences in different positions to constantly find the minimum fitness value.
The minimum fitness value is obtained through cyclic iteration of Steps (2)–(4), at which time point, the sparrow position is the best position and [
VMD decomposition is carried out on the data sequence by using the optimal component number
For selecting a photovoltaic power sequence to optimise the VMD decomposition method by SSA, the population number of SSA is set to 20; the number of iterations is set to 30; the search range of the decomposition quantity
When a recurrent neural network (RNN) processes long-time series data, the gradient is easy to explode, resulting in poor processing effect. The LSTM network solves the gradient problem when an RNN network processes long sequence data by setting the cell state and three-door structure in the hidden layer, so that the memory information can be saved for a long time. The internal structure of an LSTM network is shown in Figure 3.
The inputs of the LSTM network are the sequence
Both LSTM network and BP neural network use gradient optimisation algorithm to transform weights. In order to solve the problem of large parameter dimension, Adam algorithm is used to optimise the random objective function.
The parameters of the LSTM network model of the first and second layers are as follows: dropout value is 0.2; the function is optimised to Adam; the activation function takes tan
Because the weather type has a great influence on the photovoltaic output, in order to improve the prediction accuracy, this paper separately predicts abrupt weather (sunny, cloudy, rainy and snowy) and the non-abrupt weather (sunny to cloudy, sunny to cloudy, etc.). In order to test the accuracy of the VMD–LSTM model optimised by SSA, four models, namely BP, ANN, LSTM and VMD–LSTM, were established and compared with the proposed model. When evaluating the accuracy of the model, the average absolute percentage error
The photovoltaic output of different weather types is predicted separately, and the prediction process is shown in Figure 4. In non-abrupt weather, the historical PV power station output data in sunny, rainy, snowy, cloudy or cloudy weather is decomposed by SSA–VMD, and all sub-components are added to meteorological conditions and sent to the LSTM network for prediction. Photovoltaic output fluctuates greatly in abrupt weather, so in abrupt weather forecast, the time of maximum output power in a day (2:00 pm) is selected for decomposition, so that the original complex power sequence becomes a number of stable data sequences, and then meteorological factors are added and sent to the LSTM network.
To prove the accuracy of the VMD–LSTM model optimised by SSA, the 366-day historical power data of a power station with an installed capacity of 50 MW for the period from 1 January 2020 to 31 December 2020 is selected as the experimental sample. Sunny, cloudy, rainy, snowy and abrupt weather in 2020 has been recorded for 142 days, 46 days, 37 days, 39 days and 102 days, respectively. Taking sunny weather, rain and snow weather in non-abrupt weather, and sunny-to-cloudy weather in abrupt weather as examples, the training sample days of cloudy weather, rain and snow weather and sunny to cloudy weather are 120 days, 32 days and 24 days, respectively, and the test sample days are 22 days, 7 days and 5 days, respectively.
The forecast results of each forecast model for sunny weather are shown in Figure 5. In sunny weather, the fluctuation of photovoltaic power is small. It can be clearly seen in the figure that there is a big deviation between the predicted value and the real value of the BP neural network. When the power value is larger, the error is more obvious. Statistical errors of the five models are plotted in Table 1. Comparing the prediction errors of BP, ANN and LSTM models, the
Prediction error of photovoltaic power in clear weather
7 November 2020 | BP | 0.411 | 192.574 | 0.191 |
ANN | 0.397 | 125.084 | 0.147 | |
LSTM | 0.315 | 102.218 | 0.123 | |
VMD–LSTM | 0.239 | 75.886 | 0.096 | |
SSA–VMD–LSTM | 0.179 | 56.833 | 0.076 | |
6 October 2020 | BP | 0.456 | 183.681 | 0.212 |
ANN | 0.527 | 172.535 | 0.128 | |
LSTM | 0.327 | 90.399 | 0.125 | |
VMD–LSTM | 0.287 | 66.757 | 0.074 | |
SSA–VMD–LSTM | 0.222 | 72.644 | 0.073 |
ANN, artificial neural network; BP, back propagation;
Figure 6 shows the photovoltaic output forecast of each forecast model in rain and snow weather, and Table 2 shows the model errors. Compared with the photovoltaic power in cloudy weather, the photovoltaic power in rainy and snowy weather decreases and fluctuates obviously. At this time, the prediction accuracy of each model has decreased, and there are obvious errors between the predicted power and the actual power of the BP and ANN models, which are no longer suitable for the prediction of photovoltaic power. On June 5, compared with BP, ANN and LSTM, the
Forecast error of photovoltaic power in rain and snow
5 June 2020 | BP | 0.694 | 305.577 | 0.335 |
ANN | 0.745 | 209.348 | 0.238 | |
LSTM | 0.517 | 151.906 | 0.188 | |
VMD–LSTM | 0.369 | 109.707 | 0.148 | |
SSA–VMD–LSTM | 0.294 | 101.777 | 0.117 | |
16 August 2020 | BP | 0.871 | 287.903 | 0.312 |
ANN | 0.803 | 233.221 | 0.210 | |
LSTM | 0.632 | 140.579 | 0.186 | |
VMD–LSTM | 0.426 | 117.406 | 0.134 | |
SSA–VMD–LSTM | 0.317 | 105.887 | 0.124 |
ANN, artificial neural network; BP, back propagation;
When the weather changes from sunny to cloudy, the photovoltaic power prediction results of each model are shown in Figure 7, and the errors are shown in Table 3. The complexity of photovoltaic power data series in abrupt weather is higher than that in non-abrupt weather, and the deviation between the predicted power of BP and ANN models and the actual power is greater, which obviously reduces the prediction accuracy. As can be seen from Table 3, on January 8, the
Prediction error of photovoltaic power in abrupt weather
8 January 2020 | BP | 0.708 | 251.918 | 0.287 |
ANN | 0.676 | 252.247 | 0.287 | |
LSTM | 0.600 | 218.179 | 0.268 | |
VMD–LSTM | 0.425 | 183.933 | 0.217 | |
SSA–VMD–LSTM | 0.358 | 154.512 | 0.200 | |
13 September 2020 | BP | 0.700 | 268.636 | 0.339 |
ANN | 0.728 | 276.354 | 0.254 | |
LSTM | 0.577 | 247.426 | 0.270 | |
VMD–LSTM | 0.473 | 229.952 | 0.239 | |
SSA–VMD–LSTM | 0.381 | 150.918 | 0.229 |
ANN, artificial neural network; BP, back propagation;
In order to further highlight the accuracy of the SSA optimisation method, this paper counts the test results of all test samples of the power generation system within 1 year and obtains the average errors of the VMD–LSTM and SSA–VMD–LSTM prediction models, as shown in Table 4.
Comparison of prediction errors between VMD–LSTM and SSA–VMD–LSTM
VMD–LSTM | 0.327 | 91.150 | 0.106 |
SSA–VMD–LSTM | 0.207 | 85.673 | 0.058 |
Compared with the
In order to improve the accuracy of photovoltaic power forecasting, this paper puts forward a combined forecasting model, which combines the VMD decomposition method optimised by the sparrow intelligent algorithm with the deep learning algorithm, builds four models (namely BP, ANN, LSTM and VMD–LSTM) and compares their errors with the proposed SSA–VMD–LSTM coupling model. The main conclusions are as follows.
In data processing, SSA-optimised VMD method combined with LSTM network can describe the long memory characteristics of a time series, which greatly improves the prediction accuracy compared with the traditional non-combination prediction model.
SSA can optimise the decomposition quantity and the penalty factor in VMD, which gives lower
The model proposed in this paper can decompose high-complexity photovoltaic data into more regular sub-components, and its coupling model has obvious advantages, which renders it suitable for ultra-short-term power forecasting of photovoltaic power stations.
Forecast error of photovoltaic power in rain and snow
5 June 2020 | BP | 0.694 | 305.577 | 0.335 |
ANN | 0.745 | 209.348 | 0.238 | |
LSTM | 0.517 | 151.906 | 0.188 | |
VMD–LSTM | 0.369 | 109.707 | 0.148 | |
SSA–VMD–LSTM | 0.294 | 101.777 | 0.117 | |
16 August 2020 | BP | 0.871 | 287.903 | 0.312 |
ANN | 0.803 | 233.221 | 0.210 | |
LSTM | 0.632 | 140.579 | 0.186 | |
VMD–LSTM | 0.426 | 117.406 | 0.134 | |
SSA–VMD–LSTM | 0.317 | 105.887 | 0.124 |
Prediction error of photovoltaic power in abrupt weather
8 January 2020 | BP | 0.708 | 251.918 | 0.287 |
ANN | 0.676 | 252.247 | 0.287 | |
LSTM | 0.600 | 218.179 | 0.268 | |
VMD–LSTM | 0.425 | 183.933 | 0.217 | |
SSA–VMD–LSTM | 0.358 | 154.512 | 0.200 | |
13 September 2020 | BP | 0.700 | 268.636 | 0.339 |
ANN | 0.728 | 276.354 | 0.254 | |
LSTM | 0.577 | 247.426 | 0.270 | |
VMD–LSTM | 0.473 | 229.952 | 0.239 | |
SSA–VMD–LSTM | 0.381 | 150.918 | 0.229 |
Prediction error of photovoltaic power in clear weather
7 November 2020 | BP | 0.411 | 192.574 | 0.191 |
ANN | 0.397 | 125.084 | 0.147 | |
LSTM | 0.315 | 102.218 | 0.123 | |
VMD–LSTM | 0.239 | 75.886 | 0.096 | |
SSA–VMD–LSTM | 0.179 | 56.833 | 0.076 | |
6 October 2020 | BP | 0.456 | 183.681 | 0.212 |
ANN | 0.527 | 172.535 | 0.128 | |
LSTM | 0.327 | 90.399 | 0.125 | |
VMD–LSTM | 0.287 | 66.757 | 0.074 | |
SSA–VMD–LSTM | 0.222 | 72.644 | 0.073 |
Comparison of prediction errors between VMD–LSTM and SSA–VMD–LSTM
VMD–LSTM | 0.327 | 91.150 | 0.106 |
SSA–VMD–LSTM | 0.207 | 85.673 | 0.058 |
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Differential Equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Animation VR scene mosaic modeling based on generalized Laplacian equation Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations The influence of accounting computer information processing technology on enterprise internal control under panel data simultaneous equation Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Construction of comprehensive evaluation index system of water-saving irrigation project integrating penman Montei the quation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression Analysis of technical statistical indexes of college tennis players under the win-lose regression function equation Automatic extraction and discrimination of vocal main melody based on quadratic wave equation Analysis of wireless English multimedia communication based on spatial state model equation Optimization of Linear Algebra Core Function Framework on Multicore Processors Research on classification of e-commerce customers based on BP neural network The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Fractional Differential Equations in Electronic Information Models BIM Engineering Management Oriented to Curve Equation Model Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Design of language assisted learning model and online learning system under the background of artificial intelligence Study on the influence of adolescent smoking on physical training vital capacity in eastern coastal areas Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering Action of Aut( G ) on the set of maximal subgroups ofp -groupsThe internal mechanism of corporate social responsibility fulfillment affecting debt risk in China: analysis of intermediary transmission effect based on degree of debt concentration and product market competitive advantage Study on transmission characteristics in three kinds of deformed finlines based on edge-based finite element method Asymptotic stability problem of predator–prey system with linear diffusion Research on loyalty prediction of e-commerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long short-term memory networks A study of local smoothness-informed convolutional neural network models for image inpainting Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of vertical video creation under the background of mobile communication Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithm-based congestion control optimisation for mobile data network