Electricity is an essential requirement for all aspects of modern life. However, large-scale energy storage remains a problem, complicating operating plans in the electricity sector. As a result, energy pricing in supply and demand is an issue since it is directly related to the quantity of consumed load. This connection serves as the foundation for all utility economic planning and administration. Furthermore, trustworthy and committed policies in development plans are essential. For example, generation units must prioritize reliability alongside other policies, such as environmental protection and renewable energy infrastructure development. These policies significantly affect the quality of delivered energy (Lago et al., 2021).

Efficient planning in reorganized power systems is separated into four time periods: long-term (more than 10 years), medium-term (1 year), short-term (1 week to 1 month), and immediate planning. These plans must consider load usage, electricity pricing, economic conditions, and yearly electricity prices. Furthermore, the demand load quantity, a time-dependent nonlinear function, must be evaluated daily (day or night), seasonally (hot or cold), and considering local weather conditions (Inglada-Pérez and Gil, 2024; Iftikhar et al., 2023a).

As a result, differing electricity demand patterns and levels vary significantly over time (at least hourly). As mentioned earlier, each phase has significant characteristics that improve managerial performance. During these periods, 1-day forecasting has the most significant impact on power prices. The most crucial advantage of short-term forecasting is that it enhances supply and demand. Therefore, selecting the most accurate and rapid approach for estimating electric power costs is critical and vital (Gonzales et al., 2024; Iftikhar et al., 2023b).

To obtain efficient and accurate one-day-ahead electric power demand and prices, many studies have used classical time series forecasting models, including exponential smoothing, regression analysis, and various machine-learning models such as neural networks, fuzzy neural networks, and support vector machines (Pinhão et al., 2022; Shah et al., 2019). For example, Dudek (2015) employed stochastic models for short-term price forecasting. They utilized the regression trees method to forecast electricity prices and compared the results with those of alternative approaches such as ARIMA, exponential smoothing, and neural networks. Empirical findings indicated that their proposed method outperformed other benchmark techniques regarding predictive accuracy. Li et al. (2020) proposed a new hybrid short-term load forecasting method that combined multivariate linear regression (MLR) and long-short-term memory (LSTM) neural networks. Taheri et al. (2021) suggested forecasting the electricity demand time series employing a hybrid prediction model that combined LSTM with empirical mode decomposition (EMD). Recurrent neural networks (RNNs) can be preferable as they implicitly capture the temporal context available in the time series or sequential data (Gers et al., 2001; Karim et al., 2019; Talebjedi et al., 2024; Baruah and Organero, 2023). A gated recurrent unit (GRU) and LSTM networks have demonstrated efficiency in modeling and forecasting time series data (Coelho et al., 2024). GRU networks are mainly used in classification and are seldom applied in regression problems (Liu et al., 2017; Ugurlu et al., 2018). Fargalla et al. (2024) employed combined neural networks (GRU + MLP, CNN + bidirectional GRU (BiGRU)) for predicting gas production in various reservoirs, showcasing remarkable prediction capabilities. Hippert et al. (2001) conducted a comprehensive review of short-term demand prediction and compared the performance of artificial neural networks with classical TS methods. Their results indicate that artificial neural networks outperform classical techniques. With increasing uncertainty in electrical systems, anticipating electricity demand, generation, and pricing becomes challenging for actors in the electrical market (Rajabi and Estebsari, 2019). Numerous strategies and solutions have been proposed in response to these challenges. However, each method entails its own advantages and disadvantages. As the electricity market integrates more renewable resources with intermittent behavior, the system experiences additional fluctuations and volatility, making the development of superior and efficient forecasting models complex (Chan et al., 2012). To effectively handle complex TS data, it is essential to develop state-of-the-art visualization, modeling, and forecasting techniques to accommodate high- and infinite-dimensional data. Classical methods for multivariate data encounter difficulties when dealing with such complex data, leading to ineffective results. Therefore, alternative approaches, such as functional data analysis (FDA) methods, have been less explored in the literature.

One-day-ahead price forecasting is of utmost importance for market agents and system operators. Overproduction incurs penalties due to the inability to store electricity and the need to instantly meet demand. Price forecasting also facilitates efficient resource management, optimal scheduling, and production planning to minimize generation costs. Retailers heavily rely on price forecasts to secure the best deals for their energy requirements. Hence, predicting the price for the next day is a significant concern for electricity businesses. The advent of smart metering has increased the frequency of electricity consumption data, with measurements now available at intervals as short as every half-hour or even a few minutes. Consequently, forecasting prices for the following day require predicting many price values beyond the traditional, highlighting the need for innovative methodologies such as utilizing functional data methods (Chaouch, 2013; Paparoditis and Sapatinas, 2013; Shah et al., 2022b). FDA is a statistical framework that analyzes and models observed data as curves or functions. Functional time series (FTS) extends this framework to handle time-dependent functional data, enabling the analysis and forecasting of functional observations over time.

FTS techniques have found numerous applications in electricity markets. From a functional perspective, it is possible to generate long-term estimates of electricity demand or day-ahead clearing prices. The challenge of forecasting over longer horizons can be addressed by transforming it into a series of one-step-ahead functional forecasts. This can be achieved by segmenting the time series into segments of equal length corresponding to the desired forecast horizon. While various approaches exist for dealing with discrete TS, limited contributions specifically focus on FTS (Horváth and Kokoszka, 2012). From a univariate FTS modeling perspective, the theory of linear FTS in Hilbert space was proposed by Bosq (2000) and Bosq and Blanke (2008) by introducing the functional autoregressive ( F AR) model. Notably, the pioneering studies in this research area include functional Yule–Walker (YW) estimation and Sieve estimation. Subsequently, F AR (1) was extended to order P through sequential testing hypothesis methods (Kokoszka and Reimherr, 2013). Chen and Li (2017) investigated an adaptive functional autoregressive (AfAR) model for forecasting electricity price curves. By employing California electricity daily price data, their results demonstrated that the proposed AfAR model exhibits superior forecasting accuracy compared to several alternative approaches. Lisi and Shah (2020) forecast electricity demand and price series by dividing the TS into deterministic and stochastic components. The deterministic component was modeled and forecast using the conventional time series method, and the stochastic component was modeled and projected using the functional models. Both parametric and nonparametric functional models forecast electricity demand and prices. The results of the proposed method were compared with classical models. The results suggested that the proposed method performs relatively better than the competitors.

As seen in the above studies, the functional time series approach enhances the forecasting accuracy and efficacy of short-term electric power prices better than other methods and models. To test these studies, this research examines the application of functional data analysis to forecasting day-ahead electric power prices. Within the functional data analysis, this study proposes a functional autoregressive ( F AR) and a functional autoregressive with exogenous variables ( F AR X ) model to predict next-day electric power prices. The F AR X model includes exogenous variables describing the complex features of electricity power prices (a long-term secular trend, daily, weekly, and annual periodicity, bank holidays). In the past, some research captured these complex features by components-wise estimation (Lisi and Shah, 2020; Chen et al., 2018; Aue et al., 2015). In contrast, our research directly uses these features in a single model ( F AR X ) and checks the proposal’s performance in different ways. For instance, standard time series forecasting models, including the autoregressive (AR), the AR X , the autoregressive integrated moving average (ARIMA), and the ARIMA X , are also utilized for comparison. The model’s prediction performance was evaluated using data on electricity prices from the British electricity market, considering forecast error indicators and the same forecast statistical test. Thus, the key contributions of this study are the following.

• To improve the efficiency and accuracy of day-ahead electricity price forecasting, this work proposes novel functional autoregressive ( F AR) and functional autoregressive with exogenous variables ( F AR X ) models. The F AR X model includes exogenous variables that describe the complex features of electricity power prices (a long-run secular trend, daily, weekly, and annual periodicity, and bank holidays).

The rest of this study is structured thus. Section 2 gives an overview of the basics of the FDA. Section 3 presents an overview of the British electricity market and the out-of-sample forecasting results. Section 4 compares the current study’s proposed best model and the literature’s proposed best models. Section 5 summarizes the key conclusions drawn from our study and outlines potential future research directions.

This section addresses certain fundamental concepts that are required for developing functioning models. The TS of electricity prices ( Y i ) is converted into functional data using specific basis functions. The daily electricity price for the i t h day can be stated as follows: Y i τ = ∑ j = 1 J α j β i , j τ τ ∈ K , (1) where α j is the value of the constant parameters, and β i , j ( τ ) are the Fourier basis functions. Figure 4 shows FTS curves for 1826 days, with each functional curve representing a daily price trend. Consider e j , j ∈ J . The inner product of < ⋅ , ⋅ > generates the norm ‖ ⋅ ‖ . Define a set of random curves Y i ( τ ) , i ∈ N in H with time domain i ∈ T where H is a real and separable Hilbert space with a countable and orthonormal basis. We assume L 2 as a separable Hilbert space, with Y i ( τ ) ∈ L 2 ( [ 0,1 ] ) . Assume Y i ( τ ) has a mean curve E Y i ( τ ) = μ ( τ ) and the covariance operator C ( π , τ ) : L 2 ( [ 0,1 ] ) → R , π , t ∈ [ 0,1 ] , supplied by

C ( π , τ ) = Cov Y i ( π ) , Y i ( τ ) = E Y i ( π ) − μ ( π ) Y i ( τ ) − μ ( τ )

The eigenfunctions of C ( π , τ ) are known as functional principle components (FPCs), which are calculated by deconstructing the covariance function of the curves with interdependence. We establish the covariance function ϒ : L 2 [ 0,1 ] → L 2 ( [ 0,1 ] ) with the covariance function C and extract the FPCs that correspond to ϕ k k ≥ 1 . Thus, the mathematical form is given in Equation 2 ϒ ϕ k τ = γ k ϕ k τ (2)

The combination eigenvalue of the covariance function υ is denoted by γ k . For every m ≠ k , the eigenfunctions ϕ k are assumed to be orthonormal and normalized to the unit norm with ∫ ϕ k ( τ ) ϕ m ( τ ) d τ = 0 . Using the FPC scores at time i as the specification, the statistical method is to find the orthonormal functions ϕ 1 , ϕ 2 , … with the most considerable variances of the leading scores Var λ k . However, the λk formula is explained in Equation 3. λ k , i = ∫ 0,1 Y i π − μ π ϕ k π d π (3)

It follows that E λ k λ m = 0 for k ≠ m since ϕ k and ϕ m are orthogonal for k ≠ l . The values of E λ k 2 = γ k and E λ k = 0 are as follows. Subsequently, the Karhunen–Loève expansion of the function Y i ( τ ) is defined in Equation 4. Y i τ = μ τ + ∑ k = 1 ∞ λ k , i ϕ k τ , (4)

The FPCs at time i are represented by the coefficients λ k , i , which inherit the serial dependency from Y i ( τ ) .

The liberalization of the United Kingdom’s electricity industry can be attributed to the regulatory and structural changes implemented in the late 1980s. These reforms sought to eliminate the state-owned monopoly and establish a competitive wholesale energy market. Recognizing that transmission and distribution are natural monopolies, the main focus of the reforms was the deregulation of the generation and supply sectors. Consequently, in 1990 the UK electricity market underwent reorganization, establishing the England and Wales power pool and dividing the government monopoly into three entities: National Power, Powergen, and Nuclear Energy.

The power pool served as a trading platform for half-hourly transactions. However, market power in the generation industry became a significant concern due to National Power and Powergen holding respective market shares of 50% and 30%. Meanwhile, Nuclear Electric, responsible for supplying the base load of nuclear power, essentially acted as a price taker. Market manipulation by these two corporations led to reduced competition, resulting in the average price remaining approximately 24 per megawatt-hour between 1994 and 1996 (Karakatsani and Bunn, 2008).

A fully liberalized bilateral contracting system replaced the England and Wales power pool with the introduction of New Energy Trading Arrangements (NETA) in 2001 (becoming the British Electricity Trading Transmission Arrangements—BETTA—in 2005). This transition established a more stable market share for electricity companies in the generation and retail sectors. Additionally, the reforms resulted in the creation of three distinct power exchanges: the United Kingdom Power Exchange (UKPX), the UK Automated Power Exchange (APX UK), and the International Exchange (IE, formerly known as the International Petroleum Exchange—IPE). Subsequently, the amalgamation of APX and UKPX into APX Group in 2004, followed by the inclusion of Scotland in the UK power market in 2005, further enhanced market dynamics. This highly competitive and well-developed market exhibits a robust correlation between market price and market fundamentals (Lisi and Pelagatti, 2018).

This section compares the best proposed functional ( F AR X ( P )) forecasting model in this study with the proposed best models in the literature. A detailed comparison of this study’s best forecasting functional model with the literature’s best models is listed in Table 6. This confirms that the proposed (Stochastic-AR) best model from Lisi and Pelagatti (2018) has been applied to the database utilized in the current work, and the average accuracy metrics have been calculated. The average accuracy metrics values reported for the proposed (Stochastic-AR) best model of Lisi and Pelagatti (2018) are MAPE = 9.81 (65.01%), MAE = 11.29 (170.61%), and RMSE = 18.12 (40.00%), which are significantly higher than our accuracy mean error values: MAPE = 5.95, MAE = 4.17, and RMSE = 12.94. The proposed (the FAR(P) model) best model in Jan et al. (2022) has been compared to the current research dataset and produced performance metrics comparatively higher than our proposal (the F AR X ( P ) model). For example, the computed forecasting average errors using the proposed (the FAR(P) model) best model in Jan et al. (2022) are 6.50, 5.18, and 14.55—significantly higher than our proposed ( F AR X ( P )) best model forecasting average accuracy errors. Additionally, in Iftikhar et al. (2023c), the proposed (the best model ⁴ RSD 3 4 ) was used in our dataset obtained mean accuracy metrics also comparatively greater than the F AR X ( P ) model at 4.46%, 20.81%, and 10.02% for the MAPE, MAE, and RMSE, respectively. Furthermore, the developed best model (Ensemble) of Bibi et al. (2021) was used in the current research database and the average metric errors computed, such as MAPE, MAE, and RMSE—comparatively higher (5.30%, 23.20%, and 11.10%) than our best functional model. In Shah et al. (2022a), the best forecasting model (^NP VAR E ) was applied to our database and the average mean errors computed. However, in comparison, the computed mean errors in the NPVAR E forecasting model were 10.68%, 27.52%, and 11.18% for the MAPE, MAE, and RMSE, respectively—higher than our best functional model. In summary, it is evident that the best functional model of this work achieved higher accuracy than the best models found in the literature.

The proposed functional time series model used in this study has proven to be very accurate and effective in predicting the next-day power price in the British electricity market. These precise forecasts can help sellers (buyers and suppliers) optimize their bid planning, maximize revenue, and better use the resources required to generate electricity. Offering a stable and cost-effective energy system benefits end users. Furthermore, the accurate projection and understanding of energy prices in developing countries can help traders make more effective commercial and trade plans and better asset allocations. Finally, sellers (buyers and suppliers) can create more robust trading strategies based on the proposed F AR X ( P ) model and choose models with the most remarkable risk reward ration.

The statistical computing language programming environment R-studio implements analysis models. In comparison, all computations were performed with Intel(R) Core(TM) i7-6600 @ 2.80 GHz CPU. R uses GAM, tsDyn, and forecast libraries to model and predict standard time series forecast models. We wrote our own source code to model and forecast F AR X and F AR X ( P ) models, which also use R’s fda library. The documentation of the packages provides detailed information on the algorithms used in the estimation.

Forecasting electricity prices in a deregulated power market is complex and crucial. The price time series of electric power exhibits features such as high frequency, long-term trends, multiple seasonal patterns, spikes or jumps, and the effects of bank holidays. Addressing these features is essential for accurate price forecasting. This study has focused on the problem of day-ahead electricity price forecasting using functional data analysis. The electricity prices were modeled and forecast through two functional models, employing four classical time series models (AR, AR X , ARIMA, and ARIMA X ) for this purpose. The price series was transformed into a functional form, and F AR ( P ) was utilized for modeling and forecasting. The functional model was further modified by introducing some essential exogenous information (specific features): F AR X ( P ). Meanwhile, the AR, AR X , ARIMA, and ARIMA X models employed conventional time series models for modeling and forecasting electricity prices. In comparison to the benchmark models (AR, AR X , ARIMA, ARIMA X , and the proposed F AR model), the F AR X model reduces the day-ahead forecasting average MAPE by a range of 5.02%–45.77%, 4.07%–40.63%, 3.80%–38.99%, 1.90%–24.22%, and 0.95%–13.78%; MAE by a range of 9.43%–69.32%, 5.17%–65.48%, 6.04%–59.16%, 3.02%–42.01%, and 1.51%–26.59%; RMSE by a range of 8.98%–40.97%, 6.68%–34.03%, 4.22%–24.58%, 3.91%–23.20%, and 2.30%–15.11%. Furthermore, compared with the literature’s proposed best models, the F AR X model produces a significantly higher accuracy and efficient day-ahead forecast based on forecasting error indicators and an equal forecast statistical test. Compared with the best models proposed in the literature, the F AR X model demonstrates significantly higher accuracy and efficiency in day-ahead forecasting, as evidenced by forecasting error indicators and an equal forecast statistical test.

This study focuses entirely on the British electricity market. The idea should include additional electrical markets, such as the European, American, and Chinese markets. Other energy market factors are electricity demand, production, consumption, curves, natural gas pricing, wind speed, and crude oil prices. This will enable a more in-depth assessment of the upgraded functional model’s performance. There are various approaches to broaden the field of future study. Other functional factors, such as temperature, consumer load, fuel, carbon dioxide emission costs, average solar radiation, and wind speed, can be included in the model to improve electricity price forecasting accuracy.