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Day-ahead electricity price prediction applying hybrid models of LSTM-based deep learning methods and feature selection algorithms under consideration of market coupling

Wei Li, Denis Mike Becker

arXiv:2101.05249v2q-fin.CP

TL;DR

The paper addresses the challenge of forecasting Nordic day-ahead electricity prices as market coupling expands the relevant feature space. It combines LSTM-based deep-learning models with feature-selection algorithms and finds accurate forecasting performance, with feature selection essential and German-market features particularly influential.

  • Problem

    Accurate day-ahead electricity-price forecasting becomes harder as European market coupling requires information from an expanding set of integrated markets.

  • Method

    The study evaluates three hybrid LSTM-based architectures combined with five feature-selection algorithms for Nord Pool electricity-price forecasting.

  • Results

    LSTM-based hybrid models achieve considerably accurate electricity-price predictions, with feature selection affecting performance and German-market features playing a salient role in Nordic price generation.

  • Takeaways & Limitations

    Cross-border information and feature selection are important considerations for Nordic electricity-price forecasting under market coupling.

  • Takeaways & Limitations

    The study did not conduct an extensive hyperparameter grid search, so more comprehensive LSTM architectures might achieve better forecasting performance.

Abstract

from arXiv · show

The availability of accurate day-ahead electricity price forecasts is pivotal for electricity market participants. In the context of trade liberalisation and market harmonisation in the European markets, accurate price forecasting becomes difficult for electricity market participants to obtain because electricity forecasting requires the consideration of features from ever-growing coupling markets. This study provides a method of exploring the influence of market coupling on electricity price prediction. We apply state-of-the-art long short-term memory (LSTM) deep neural networks combined with feature selection algorithms for electricity price prediction under the consideration of market coupling. LSTM models have a good performance in handling nonlinear and complex problems and processing time series data. In our empirical study of the Nordic market, the proposed models obtain considerably accurate results. The results show that feature selection is essential to achieving accurate prediction, and features from integrated markets have an impact on prediction. The feature importance analysis implies that the German market has a salient role in the price generation of Nord Pool.

1. Introduction

European electricity-market liberalisation and coupling increase the need for accurate day-ahead price forecasts while expanding the feature space and prediction challenge. The study addresses this gap with hybrid LSTM models and feature-selection methods, examining cross-border information in Nordic price forecasting.

  • Accurate day-ahead electricity price forecasts support production planning, bidding strategies, margin improvement, and risk reduction for market participants.
  • LSTM models handle time-series information, but high-dimensional market-coupling inputs can cause overfitting and degrade performance.
  • Existing research had not applied state-of-the-art LSTM deep networks to integrated-market electricity-price forecasting or efficiently exploited expanding Nordic market-integration information.
  • The study proposes three hybrid LSTM architectures and evaluates five feature-selection algorithms using Nord Pool and six integrated markets with 62 features.
  • Feature-selection algorithms produce different feature subsets that affect LSTM prediction accuracy, while cross-border markets influence Nordic electricity-price formation.
  • SHAP analysis examines cross-border-feature relevance to improve prediction transparency and provide advice for policymakers and market participants.

2. Data description

The study forecasts Nordic hourly and daily system prices using local-market variables together with cross-border exchanges and flow deviation. Its dataset combines Nord Pool and external sources, covering interconnected European markets and converting hourly observations into daily features.

  • The Nordic system price is Nord Pool’s central reference and settlement price, calculated from bidding-zone bids and offers without congestion restrictions; daily price is the 24-hour arithmetic average.
  • The input design adds electricity exchange with integrated countries and a cross-border flow-deviation feature relating hourly flow to expected exchange capacity.
  • Eight input categories cover day-ahead price, production, production prognosis, consumption, consumption prognosis, currency exchange rate, cross-border flow, and flow deviation.
  • The dataset includes Nord Pool transmissions with Germany, the Netherlands, Lithuania, Estonia, Poland, and Russia, using data collected from Nord Pool, Thomson Reuters Eikon, and Entsoe.
  • Hourly data are converted to daily data through arithmetic averaging for variables such as price or aggregation for variables such as flow.

3. Methodology

The methodology combines LSTM networks with feature-selection procedures to forecast electricity prices while accounting for market coupling. It introduces multiple hybrid architectures and wrapper-based and regression-based selection methods.

  • 3.1. LSTM: LSTM networks learn long-term dependencies through gated memory, using forget, input, output, cell, and hidden-state components.The architecture uses sigmoid and hyperbolic-tangent functions, weight matrices, bias vectors, and element-wise products.
  • 3.2. Architectures of hybrid models: The two-step architecture separates data processing and feature selection from predictive-model training and price prediction.This design is represented as a first processing and selection step followed by a prediction step.
  • 3.2. Architectures of hybrid models: The autoencoder and two-stage architectures compress inputs or combine preliminary feature selection with subsequent autoencoder processing.The two-stage design selects explanatory variables first, then feeds the selected features into autoencoder models.
  • 3.3. Feature selection: Feature-selection methods include PC, PSO-ELM, GA-ELM, RFE-SVR, and Lasso regression, each using a different mechanism to identify relevant variables.PSO and GA provide feature subsets to ELM, RFE-SVR removes less important features iteratively, and Lasso shrinks some coefficients to zero.
  • 3.3. Feature selection: The selected-feature methods are designed to reduce irrelevant inputs while preserving predictive information for the LSTM-based forecasting models.Lasso explicitly removes less relevant features by shrinking their coefficients to zero.

3.4. Autoencoder Model

The autoencoder models compress input time series into intermediate representations before an LSTM decoder generates predictions. Different encoder choices provide alternative ways to filter temporal or convolutional information.

  • 3.4. Autoencoder Model: An autoencoder uses an encoder to compress input data into an intermediate vector and a decoder to produce predictions.In these models, the decoder is an LSTM network, while the encoder may use LSTM, CNN, or convolutional layers.
  • 3.4. Autoencoder Model: The LSTM-LSTM model uses an LSTM encoder to process raw time series and extract dynamic information through internal memory.The encoder transforms the raw sequence into an intermediate vector before decoding.
  • 3.4. Autoencoder Model: The CNN-LSTM model uses convolution and pooling to filter input data and produce a lower-dimensional intermediate representation for the LSTM.Convolved features are pooled and flattened into a long intermediate vector.
  • 3.4. Autoencoder Model: The ConvLSTM model differs from CNN-LSTM by having the LSTM process information extracted directly from preceding convolutional layers.In CNN-LSTM, the CNN generates the LSTM input; in ConvLSTM, convolutional extraction and recurrent processing are integrated more directly.

4. Experimental details

The study evaluates LSTM-based electricity price forecasting with feature selection, statistical significance testing, walk-forward nested validation, and SHAP-based interpretation. Models are benchmarked against NARMAX using repeated experiments and a defined experimental configuration.

  • Evaluation metrics and significance testing: Forecast accuracy is measured with MAE, RMSE, MAPE, and SMAPE, while one-sided Diebold-Mariano tests assess whether model differences are statistically significant.The loss differential is defined as dF1,F2_t = |e1t| − |e2t|.
  • Validation procedure: Walk-forward nested cross-validation uses fixed training, validation, and test proportions of 80%, 10%, and 10% to evaluate performance over time.The validation and test set magnitudes remain anchored during the walk-forward test.
  • Training procedure: Each model is trained in ten experiments and evaluated using the average performance to reduce variability caused by random LSTM weight initialization.Random initial conditions can produce different results for the same LSTM configuration.
  • Feature selection configuration: Feature-selection algorithms use specified stopping and population parameters, including 10,000 iterations for PSO and 10,000 generations as the GA maximum.PSO uses c1 = 0.5, c2 = 0.3, and ω = 0.7; GA uses population size 100, crossover probability 0.5, and mutation probability 0.2.
  • Model configuration: The LSTM configuration uses a single hidden layer with 300 units and a dense layer with 100 units, without extensive hyperparameter optimisation.The configuration was selected after an inexhaustive grid search.
  • Benchmark and interpretation: The proposed models are evaluated against a NARMAX benchmark, with SHAP used to interpret feature impacts on predicted prices.NARMAX is selected as a nonlinear statistical benchmark, while SHAP explains feature relevance relative to expected model output.

5. Results

The results show that feature-selection choices materially affect LSTM forecasting accuracy, with M4 and M5 performing best overall and cross-border information influencing Nordic price formation. Feature importance further highlights Germany’s role, while not all cross-border flows provide useful forecasting information.

  • 5.1. Analysis of empirical results: Different feature-selection mechanisms produce divergent feature subsets, with M1 selecting all day-ahead prices but introducing information redundancy.Wrapper methods M2 and M3 eliminate several price variables, while M4 selects a broader mix of feature types.
  • 5.1. Analysis of empirical results: M4 and M5 perform better than the other models, and eliminating redundant or less relevant features improves LSTM forecasting performance.The models are reported as overwhelmingly better than the statistical benchmark M0; M4 and M5 are identified as the best performers in the ten experiments.
  • 5.1. Analysis of empirical results: Applying LSTM-LSTM predictors improves M4, while the two-stage models are compared with two-step models using SMAPE and one-sided DM tests.The comparison is presented in Figure 11, and the DM-test sign convention identifies which model is better.
  • 5.1. Analysis of empirical results: Across 24 hourly system-price forecasts, feature selection affects accuracy, while M4 and M5 remain relatively stable and outperform the other models.The reported peak-hour comparisons cover H8, H12, and H18 and use SMAPE.
  • 5.2. Analysis of feature impact: Production, consumption, and their prognoses in Nordic and German markets are prioritized, with Germany having a significant impact on Nordic price prediction.The paper links Germany’s influence to its many electricity cables and high electricity exports to the Nordic market; electricity prices have greater impact than cross-border trade features.
  • 5.2. Analysis of feature impact: Not all cross-border flows and flow deviations help forecasting because flow capacity is often fully occupied and many deviations lack variability.The FI–Russia example shows that most flow deviations are zero, supporting the non-selection of these variables by M4.
  • 5.3. Discussion of practical importance: The hybrid models achieve considerably accurate price predictions, which can help participants reduce trading risk and adjust day-ahead bidding and production or consumption schedules.The paper reports that a 1% MAPE improvement within a 5% to 14% range leads to about a 0.1 - 0.35% cost reduction.

6. Conclusion

The study compares LSTM-based hybrid architectures and feature-selection methods for Nord Pool electricity price forecasting under market coupling. It finds that feature selection affects prediction performance, German-market features are especially significant, and broader testing remains necessary.

  • Different feature-selection methods select different inputs, and the resulting feature diversity affects LSTM-based forecasting performance.
  • LSTM-LSTM performs better as an autoencoder structure for electricity price forecasting than CNN-LSTM and ConvLSTM.
  • Two-stage models improve forecasting accuracy to some extent over two-step models, while the RFE-SVR model provides superior feature selection.
  • Features from the German market are more significant for Nord Pool price forecasting than other features, indicating stronger cross-border influence with more interconnections.
  • Exchange rates are relatively less important, while flow deviation contributes little because expected flow capacity is often fully occupied.
  • The study uses Nord Pool data; future work should expand hyperparameter searches, test additional feature-selection models, and evaluate other integrated markets.

Appendix A.1. PSO

The PSO procedure updates each particle using its velocity, personal best, and global best, then iterates until a stopping criterion is met.

  • Each particle updates its velocity using its current velocity, personal best solution, and the global best solution.
  • The particle position is updated by adding the new velocity to the previous position.
  • The iterative particle-update process continues until a stopping criterion is satisfied.

Appendix A.2. GA-ELM workflow

The GA-ELM workflow represents candidate feature subsets as binary individuals, evaluates them with ELM-based mean squared error, and evolves them through selection, crossover, and mutation.

  • GA-ELM encodes each candidate feature subset as an individual whose binary values indicate selected and unselected features.
  • Selected features are supplied to the ELM, and mean squared error determines each individual’s fitness.
  • The workflow retains better individuals, updates the global optimum, and generates new candidates through crossover and mutation until termination.
  • Figure A.20 presents the workflow of the PSO-ELM and GA-ELM feature-selection models.

Appendix A.3. SVR

SVR minimizes forecasting errors by maximizing the regression margin while allowing bounded errors through an epsilon tube and slack variables.

  • SVR individualizes the regression hyperplane by maximizing the margin to minimize forecasting errors.
  • The SVR formulation uses a parameter vector, kernel function, and bias within a linear estimation function, followed by convex minimization.
  • Slack variables permit regression errors beyond the epsilon constraint while preserving feasibility.
  • Only points outside the epsilon-radius contribute to the final cost, with epsilon defining the tube around the regression function.

Appendix A.4. The statistical details of the model performance

Appendix A.4 reports statistical performance results for eight models across MAD, RMSE, MAPE, and SMAPE metrics, with percentile values identified explicitly.

  • Appendix A.4. The statistical details of the model performance: Tables A.7–A.10 report MAD, RMSE, MAPE, and SMAPE results for models M1–M8.The tables cover one metric each and compare the same eight models.
  • Appendix A.4. The statistical details of the model performance: The reported metrics are expressed as percentages.
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