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Innovative Second-Generation Wavelets Construction With Recurrent Neural Networks for Solar Radiation Forecasting

Giacomo Capizzi, Christian Napoli, Francesco Bonanno

arXiv:1308.3524v1cs.NE

TL;DR

Solar-radiation forecasting matters for photovoltaic applications, but the signal has complex time-series behavior and may lack a known generating model. The paper proposes WRNNs that forecast wavelet coefficients from meteorological data and reconstruct the predicted signal, reporting very low RMS error relative to hybrid neural-network approaches.

  • Problem

    Solar-radiation prediction is important for photovoltaic applications and challenging because the signal exhibits complex time-series behavior.

  • Method

    The proposed WRNN predicts wavelet coefficients from meteorological time-series components and performs inverse wavelet reconstruction to output the forecast signal.

  • Results

    The simulations show a very low RMS error compared with solar-radiation methods based on hybrid neural networks.

  • Takeaways & Limitations

    The WRNN provides a wavelet-domain forecasting approach that integrates coefficient prediction and signal reconstruction.

Abstract

from arXiv · show

Solar radiation prediction is an important challenge for the electrical engineer because it is used to estimate the power developed by commercial photovoltaic modules. This paper deals with the problem of solar radiation prediction based on observed meteorological data. A 2-day forecast is obtained by using novel wavelet recurrent neural networks (WRNNs). In fact, these WRNNS are used to exploit the correlation between solar radiation and timescale-related variations of wind speed, humidity, and temperature. The input to the selected WRNN is provided by timescale-related bands of wavelet coefficients obtained from meteorological time series. The experimental setup available at the University of Catania, Italy, provided this information. The novelty of this approach is that the proposed WRNN performs the prediction in the wavelet domain and, in addition, also performs the inverse wavelet transform, giving the predicted signal as output. The obtained simulation results show a very low root-mean-square error compared to the results of the solar radiation prediction approaches obtained by hybrid neural networks reported in the recent literature.

UPLOADED UNDER SELF-ARCHIVING POLICIES

The paper investigates wavelet recurrent neural networks for solar-radiation forecasting, motivated by the importance and complexity of solar radiation in photovoltaic applications. Its approach decomposes data into time-frequency components, forecasts those components with recurrent networks, and reconstructs the signal.

  • Solar radiation is important for renewable-energy applications, especially sizing stand-alone photovoltaic systems.
  • The paper investigates WRNNs to model and predict daily total solar radiation.
  • The method decomposes solar-radiation samples into time-frequency components before forecasting each component with recurrent neural networks.
  • The forecasts for the decomposed components are combined through their algebraic sum to obtain a solar-irradiation forecast.
  • Second-generation wavelets are presented as a novel contribution for improving solar-radiation prediction.

II. PROBLEM OF SOLAR RADIATION FORECASTING

The paper situates solar-radiation forecasting within time-series prediction, where the signal combines periodic and random behavior and its generating model may be unknown.

  • Solar radiation forecasting is a time-series problem involving complex periodic and random behavior.
  • The forecasting literature includes statistical, neural-network, and hybrid approaches for total solar radiation.

III. BASICS OF WAVELET THEORY AND RNN ARCHITECTURES

This section introduces wavelet analysis and recurrent neural networks as the theoretical foundations for the forecasting method.

  • The paper reviews wavelet analysis and recurrent neural-network architectures before presenting the forecasting structure.

A. Wavelet Theory

The wavelet foundations include multiresolution spaces, scaling and wavelet bases, and the lifting scheme for constructing second-generation wavelets. The lifting stage is organized around splitting, prediction, and updating.

  • Wavelet Theory: Multiresolution analysis uses nested subspaces with scaling functions that form Riesz bases.
  • Wavelet Theory: Wavelet functions form bases for complementary spaces in the multiresolution construction.
  • Wavelet Theory: Wavelet coefficients are obtained through inner products with dual wavelet functions, while projections use dual scaling functions.
  • Wavelet Theory: The lifting stage consists of split, predict, and update operations.

B. RNN Architectures

The RNN architecture models current observations using preceding observations and trains recurrent weights by minimizing instantaneous output error. Its topology includes input, processing, and output layers with nonlinear neuron activations.

  • RNN structure: RNNs model current observations as functions of a certain number of preceding observations, using delayed observations to retain context.This delay-vector approach feeds past observations into the current input rather than assigning credit backward through time.
  • RNN structure: The described network has input, processing, and output layers, with weighted inputs passed through an internal activation and nonlinear function φ.For neuron i, the output is expressed as yi(k) = φ(vi(k)).
  • Training: Real-time recurrent learning trains the RNN by minimizing the instantaneous squared error at its output.The error is defined relative to the teaching signal, and weight corrections are derived from the resulting gradient.
  • Evaluation: Table I compares neuron counts, relative RMS, correlation coefficient Γ, and convergence epochs across different wavelets.The passage identifies the table’s evaluation dimensions but does not provide the individual values.
  • Training: The recurrent-learning derivation uses time-step-specific quantities indexed by k and appropriate neuron and weight indices.The cited derivation references the RTRL algorithm and its detailed formulation.

IV. PROPOSED STRUCTURE OF PREDICTION BASED ON SECOND-GENERATION WAVELET AND RNN

The proposed structure combines second-generation wavelets with recurrent neural networks to reconstruct and predict signals in the wavelet domain. Its wavelets need not be translations and dilations of one another while retaining the key properties of first-generation wavelets.

  • WRNN structure: Figure 2 presents the RBF transfer function f(x) alongside the relative wavelet function ˜f(x).The figure links the selected transfer-function shape to the wavelet-function representation used by the architecture.
  • Second-generation wavelets: Second-generation wavelets are introduced as wavelets that are not necessarily translations and dilations of each other.Despite this broader construction, they retain the powerful properties associated with first-generation wavelets.
  • WRNN structure: The proposed WRNN architecture is designed to exploit wavelet coefficients for signal reconstruction and prediction.The surrounding paper context identifies this as the basis of the proposed prediction structure.

A. Basics of Second-Generation Wavelet Theory

The lifting scheme constructs second-generation wavelets through split, prediction, and update operations. Iterating these stages produces scaling and wavelet coefficients, while the design can accommodate nonlinear or space-varying operators.

  • Lifting operations: The lifting scheme splits the signal into even and odd samples, predicts odd samples from even samples, and forms wavelet coefficients from the prediction error.The split uses xe[n] = x[2n] and xo[n] = x[2n+1].
  • Lifting operations: The update operation combines even samples and wavelet coefficients to produce scaling coefficients representing a coarse signal approximation.An updating operator U is applied to the wavelet coefficients before combining them with xe[n].
  • Lifting operations: Repeating a lifting stage on c[n] generates the complete discrete wavelet transform scaling and wavelet coefficients.The resulting lifting steps remain invertible even when P and U are nonlinear or space-varying.
  • Lifting design: A fast in-place wavelet-transform implementation avoids auxiliary memory, while reversing update and prediction can simplify nonlinear or space-varying designs.The alternative ordering performs the update step before prediction.
  • RNN integration: Figure 3 shows the selected RNN topology used for building the prediction and update operators P and U.The caption identifies the topology’s role but does not specify its internal layout.
  • Lifting design: An update filter can preserve the first ˜N low-order polynomials while allowing a space-varying or nonlinear predictor without changing the coarse approximation.The predictor can additionally be designed to optimize performance criteria.

B. Proposed Novel Buildup of P and U Based on RNN

The proposed WRNN uses recurrent networks to adapt wavelet lifting and prediction operators to signal structure across scales. It reconstructs and forecasts signals from wavelet coefficients using a specialized architecture and wavelet-like transfer functions.

  • Adaptive lifting and prediction: RNNs adapt the predictor at each lifting stage to match the signal structure at its corresponding scale.The approach realizes both lifting operators P and U through recurrent networks.
  • Adaptive lifting and prediction: The WRNN predicts wavelet coefficients and reconstructs the forecast signal from them.This behavior combines coefficient prediction with signal reconstruction in one system.
  • Network architecture: The architecture uses paired hidden-layer neurons and a double hidden layer to support coefficient exploitation followed by signal reconstruction.The paired structure supports wavelet-like activation behavior and sequential reconstruction.
  • Wavelet-like transfer functions: RBF transfer functions approximate wavelet shapes after scaling, shifting, and repetition.The selected functions are used to simulate mother wavelet filters.
  • Wavelet-like transfer functions: Symmetric periodicity and paired neurons allow the reconstruction layer to emulate a wavelet reconstruction filter.The pairing provides equal positive and negative layer-weight structure for the theoretical reconstruction scheme.
  • Timescale selection: Different decompositions locate coefficient bands at prediction-relevant timescales, while zeroing unrelated bands retains relevant coefficients and residuals.These bands were used to study reconstruction and prediction across wavelet functions and network designs.

V. EXPERIMENTAL SETUP

The University of Catania setup measured solar radiation alongside wind, temperature, and relative humidity using dedicated meteorological instrumentation. The facility also supported PV-module electrical characterization and long-term data acquisition.

  • Instrumentation: The laboratory used pyranometers to measure total solar radiation and a three-cup anemometer with wind vane to measure wind conditions.The instrumentation was integrated with a data acquisition system and electronic loads.
  • PV characterization: PV-module electrical behavior was characterized through current–voltage curves using electronic loads.The experimental facility supported measurements of assembled PV modules and strings.
  • Instrumentation: Wind measurements used polycarbonate cups with magnetic switching over a 3–241 km/h range.The listed wind-sensor accuracy was 3 km/h, with 1 km/h resolution.
  • Instrumentation: Temperature measurements used a bandgap sensor with digital output over −40°C to 123.8°C.Accuracy was specified as 1° in the 20°C to 50°C range.
  • Instrumentation: Relative humidity was measured with a capacitive polymer sensor providing digital output and 2% accuracy over 10%–90%.The long-term survey used this sensor alongside wind and temperature measurements.

VI. SIMULATION RESULTS

The simulation used synchronized meteorological time series decomposed into dyadic wavelet bands for forecasting solar radiation about two days ahead. The synthesis filter also affects the predicted variable because the network uses it during wavelet-domain prediction.

  • Data and forecasting setup: 10 min sampling intervals produced synchronized meteorological time series for the forecasting experiment.The data were automatically stored by the laboratory information infrastructure.
  • Data and forecasting setup: A dyadic wavelet scale made the first band representative of 2-day time steps for temperature, relative humidity, and wind speed.The measured horizontal-plane solar radiation ranged from 0 to 1300 Wm2.
  • Input representation: The 12-element input vectors combined four wavelet components from each of three meteorological measurements.The components were a1(t0), d1(t0), d2(t0−), and d2(t0+).
  • Input representation: The neural networks predicted one-dimensional solar-radiation output signals approximately two days in the future.The reported simulations used multiple wavelet designs summarized with the network features.
  • Wavelet-domain prediction: The synthesis filter affects the predicted variable because the network uses that filter for prediction in the wavelet domain.This links the reconstruction stage directly to the predicted signal.

VII. CONCLUSIONS

The study concludes that its WRNN directly reconstructs predicted signals from selected wavelet information and provides efficient, robust solar-radiation forecasting. Simulations report very low RMS error relative to hybrid neural-network methods.

  • Main conclusions: The WRNN reconstructs wavelet coefficients directly from sampled signals using band-selected coefficients rather than relying on intrinsic information from other coefficients.This direct reconstruction is identified as the approach’s novelty.
  • Wavelet configurations: The wavelet biorthogonal 2.8 and 3.7 decomposition sets are among the wavelet configurations used in the simulations.Their decomposition sets are shown in Figs. 6 and 7.
  • Main conclusions: Very low RMS error was obtained compared with solar-radiation prediction methods based on previously developed hybrid neural networks.The comparison is reported as a principal simulation result.
  • Efficiency and robustness: The proposed approach is described as more computationally efficient because its neural network converges quickly.The conclusion refers to mean-square-error traces in Figs. 6–14 as evidence.
  • Efficiency and robustness: The forecasting method is reported to be robust to data errors and to reduce inverse wavelet-transform calculation.These are listed as additional practical properties of the method.
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