Forecasting United Kingdom's energy consumption using machine learning and hybrid approaches

Abstract

Investigating the current and future dynamics of energy consumption in modern economies such as the UK is crucial. This paper predicts the UK's energy consumption using data spanning January 1995 to March 2022 by comparing and evaluating the forecast performance of machine learning, dynamic regression, time series and combination modelling techniques. The analysis reveals that the seasonal ARIMA and TBATS hybrid models yield the lowest forecast errors in predicting the UK's electricity and gas consumption. Although the combination forecasts performed poorly relative to other models, machine learning techniques such as neural network and support vector regression produced better results compared to the dynamic regression models, whereas the seasonal hybrid model performed better than the machine learning and time series models. The results indicate that the UK's electricity consumption would either stabilise or decline over the forecast horizon, suggesting that it will take some years for electricity consumption to attain pre-2019 levels. For gas consumption, the results indicate that consumption would either maintain current levels or increase over the forecast period. We also show that combination forecasts do not often generate the best predictions, and therefore, choice of methodology matters in energy consumption forecasting. Overall, changing seasonal patterns, energy efficiency improvements, shift to renewable sources and uncertainties due to the COVID-19 pandemic, Brexit, and the Russia–Ukraine crisis appear to be significant drivers of energy consumption in the UK in recent times. These findings are expected to help in designing more effective energy policies as well as guide investor decisions in the energy sector.

Keywords

Energy consumption forecasting machine learning combination forecasts hybrid techniques

Introduction

Energy consumption forecasting particularly in the aftermath of the COVID-19 pandemic and the current geopolitical uncertainties has become an even more complex endeavour. This is largely due to the inherent asymmetry and seasonality associated with energy consumption and its complex interaction with macroeconomic fundamentals. Several factors drive energy consumption such as temperature and climate, socio-economic factors, adoption of efficient appliances and emerging technologies, population growth as well as changes in industrial and economic structures, amongst others. The combination of these factors makes the prediction of energy consumption intricate and warranted especially in the context of emerging challenges. The key to reliable forecasts is to develop models that reflect the energy consumption of economic agents and their determinants. Due to the United Kingdom's (UK's) limited energy resources, careful energy planning and accurate consumption forecasts are necessary for ensuring sufficient energy supply and sustainability to enhance the country's energy security. The UK's target of cutting CO₂ emissions by 80% by 2050 makes it imperative to generate forecasts that can guide policymakers towards achieving the target.¹

Figure 1 presents the UK's energy consumption and temperature trends from January 1995 to March 2022. We observe that electricity consumption has been declining since the early 2010s, while gas consumption has been on the increase since 2016. This indicates a shift towards cleaner energy sources as electricity in the UK used to be mainly generated through coal and non-renewable fuels. Both electricity and gas consumption are characterised by seasonal variation suggesting the influence of climatic factors. Recently, the UK's overall energy consumption has started to decline. Plausible reasons for this trend include improved energy efficiency, the decline of energy-intensive industries, milder winters, government energy policies and tougher macroeconomic conditions, amongst others.²

Figure 1.

Energy consumption and temperature in the UK (1995–2022).

The UK serves as an important case study because recent geopolitical developments such as Brexit and the Russia–Ukraine war could have severe implications for energy consumption in the country. Also, recent advances in new technologies such as artificial intelligence, machine learning and advanced forecasting methods can help improve energy consumption forecasts. These issues make it imperative to investigate the current and future dynamics of energy consumption. This is because the responses of economic agents to external shocks, risks and uncertainty may differ from forecasts predicated on classical rational optimising behaviour.¹ The outcome of this paper would help shape the design and implementation of more effective energy policies.^3–5 In addition, energy consumption forecasts can provide crucial signals useful for trading in energy markets.

Some other factors have also affected the UK's energy consumption pattern. For instance, the country's lockdown measures and travel restrictions due to the COVID-19 pandemic led to a decline in energy consumption. The total final energy consumption (excluding non-energy use) was 13% lower in 2020 compared to 2019 due to the impact of the pandemic.^6,a The decline in the services and industry sectors was due to lockdown measures taken, whilst the large fall in the transport sector was due to the travel restrictions imposed with air travel declining by 60% due to the closure of international travel corridors and road consumption down by 18% (ibid.). Industrial electricity consumption fell by 5% between January and March 2020, reflecting a similar decline in the manufacturing output index. The consumption by all other sectors dropped by 1.9%.⁷

In addition to the Russia–Ukraine war which has significantly distorted global demand and supply, particularly in the energy sector, these emerging challenges underscore the importance of improved forecasts of energy consumption in the UK. Barassi and Zhao³ opine that ‘accurate and rigorous electricity demand modelling and forecasting are extremely important for energy suppliers, independent system operators, financial institutions, and other participants in electric energy generation, transmission, distribution, and markets’. Furthermore, energy consumption prediction is important because it has significant implications for global warming and climate change.⁸ Since electricity generated cannot be readily stored, forecasting is critical as it could substantially reduce waste and ensure optimal energy resource allocation.⁹

The literature on energy consumption forecasting is dominated by four main approaches: (i) classical methods such as regression (Amber et al.,² Baker and Rylatt,¹⁰ etc.), (ii) Time series approaches, such as seasonal autoregressive integrated moving average (SARIMA) models,^3,11,12 exponential-smoothing models such as Holt-Winters (HW) and HW-Seasonal (HWS) (see the study by Williams and Short¹³), (iii) Machine-learning methods such as Artificial Neural Network (ANN) and Support Vector Regression (SVR)^4,14 and (iv) hybrid/combination techniques (González-Romera et al.¹⁵ and Berriel et al.¹⁶ However, machine-learning techniques have increasingly been adopted to improve forecast accuracy.¹⁷

Despite the proliferation of forecasting methods, they have limited capacity to account for heterogeneous consumption patterns observed from UK data and no single technique outperforms others in all scenarios. The deregulation of the UK's energy market in the early-1990s made energy consumption forecasting more complex and challenging. Thus, recent studies such as Barassi and Zhao³ propose hybrid and combination techniques but this has not been adequately researched especially in the context of recent developments in the energy literature. Moreover, extant studies have been dominated by variants of time series econometric techniques which are characterised by endogeneity bias. Thus, the empirical literature remains largely inconclusive.

Using monthly energy consumption, temperature and energy prices (crude oil price) data (January 1995–March 2022), we investigate and predict the UK's energy consumption. Specifically, we evaluate the performance of benchmark energy consumption forecasts generated using SARIMA, Seasonal and Trend-Decomposition using Loess (STL), Trigonometric Box-Cox transform ARMA-errors Trend Seasonal (TBATS), Dynamic Regression, Error-Trend-Seasonal (ETS), Neural Network Autoregression (NNAR) and SVR models. Digging further, a forecast of the UK's energy consumption is obtained by averaging five selected forecast models (SARIMA, ETS, NNAR, STL and TBATS) using simple model averaging (SMA). This is a clear departure from extant studies that have combined relatively different and less diverse techniques. The forecasts are produced from these models and within each class are compared and the optimal specification is selected based on various forecast-error measures. Thereafter, the identified optimal models are then combined using SMA to generate out-of-sample energy consumption predictions.

On this backdrop, the objectives of this paper are: (i) to examine energy consumption patterns and dynamics in the UK; (ii) to forecast the UK's energy consumption; (iii) to identify and evaluate suitable models or a combination of models for predicting UK's energy consumption; (iv) compare and contrast competing approaches; and (v) provide policy implications.^b This paper is structured as follows: Section ‘Introduction’ presents the background and contexts while Section ‘Energy Consumption Developments in the UK’ discusses energy consumption developments in the UK. Section ‘Literature Review’ reviews prior studies. Section ‘Methodology and Estimation Strategy’ presents the methodology and empirical strategy, while Section ‘Data Description and Preliminary Diagnostics’ examines the data characteristics. Section ‘Results and Discussion’ presents the empirical findings and discussion. Section ‘Conclusion and Policy Implications’ concludes and presents some policy implications.

Energy consumption developments in the UK

This section highlights the patterns and main drivers of the UK's energy consumption in the last three decades. The issues around energy consumption have always been at the heart of energy planning and policy frameworks because of their importance to the economy. Illustratively, between 2017 and 2018, energy accounted for 3.9% of the UK households’ total expenditure, while households and businesses spend about GBP£55 billion on energy yearly (The Office of Gas and Electricity Markets).²⁰ Electricity which constitutes a major component is generated using gas, coal, renewable and nuclear energy. Due to the recent decline in coal usage and other fuels in electricity generation, gas has become the main energy source for electricity generation and household use, accounting for 77% of the total in 2018.²⁰, p. 120.

Weather and climatic factors have always had a significant impact on energy consumption in the UK, while new phenomena are beginning to influence the UK's energy consumption trajectory. The announcement of the COVID-19 lockdown on 23 March 2020 led to a significant fall in energy consumption. This measure led to a reduction in consumption of diesel, aviation fuel and gasoline due to flight cancellations, reduced public transport usage and fewer vehicular movements. This was worsened by the closure of businesses, factories, shops/restaurants and offices, which also affected electricity and gas consumption. From January to March 2020, total electricity consumption fell by 1.5% compared to March 2019, while domestic electricity consumption increased by 1.4% in 2020Q1, compared to 2019Q1.⁷

Renewable energy is playing an increasingly germane role in the UK's energy space. Power generation from wind, bioenergy and solar accounted for 33% of the overall electricity supply in 2018.²⁰^{, p. 119} Conversely, there was a decline in coal use with its contribution to energy production declining from 7% to 5% in 2018 which is in line with the government's commitment to eliminate coal power generation by 2025.²⁰^{, p. 119} Consequently, the UK's CO₂ emission from electricity generation has declined by more than 50% since 2012 with emissions in 2017 hovering at 42% lower than the values recorded in 1990.²⁰^{, p.113–115} According to OFGEM,²⁰^{, p.128} the carbon price and renewable subsidies are the main drivers of the reduction of electricity-related emissions in the UK.

Figure 2(a) presents the relationship between electricity and gas prices (monthly day-ahead contracts) in the UK. We observe a strong relationship between electricity and gas prices for the periods 2014–2019 and 2020–2021. OFGEM²⁰ highlights that ‘this is consistent with competition driving electricity prices to reflect changes in input cost’. Figure 2(b) presents the price volatilities of day-ahead gas and electricity from January 2003 to March 2022. This indicator is important for energy companies and other stakeholders in the sector. It is discernible that energy-price volatility has been decreasing from 2011 to 2014 before it increased in 2016–2018 followed by a large spike in 2019–2022. A significant increase in gas price occurred in 2021 due to high gas consumption and recently due to the Russia–Ukraine crisis, while electricity recorded a significant increase in 2020Q2 due to the COVID-19 pandemic.

Figure 2.

Electricity and gas prices and their volatilities in the UK (2009–2020).

It is pertinent to note that energy price is an important determinant of consumption because industries and households might use less energy if energy cost is high. Thus, incorporating energy prices into the modelling process is expected to improve forecast accuracy. However, taking into account the effects of energy prices is not straightforward as both electricity and gas prices are influenced by external factors such as fluctuating exchange rates, changing weather, together with geopolitical and economic shocks. They are also closely linked because gas prices influence electricity prices²⁰ due to intersectoral linkages.

Figure 3 shows scatterplots of electricity and gas consumption against temperature between 1995 and 2022. Both figures show a negative relationship between energy consumption and temperature. Temperature is an important predictor of energy consumption particularly during summer periods when there is a high demand for air conditioning and during winter when there is a need for heating. Figure 4 shows a Treemap indicating the share of electricity demand in the UK based on sectors in 2020. Observably, domestic demand accounted for about 33% which is greater than its corresponding share of 30% in 2019. Industrial electricity consumption represented 25%, while commercial consumption's share of electricity demand stood at 19%. Overall, electricity demand in 2020 declined by 4.6% in comparison to 2019.

Figure 3.

Scatterplots of electricity and gas consumption (1995–2022).

Figure 4.

Share of total electricity demand by sector, 2020.

In 2021, electricity and gas consumption remained low due to the effect of COVID-19 pandemic on the economy. The key driving factor behind this phenomenon is the movement restriction imposed due to the pandemic. This resulted in a significant decline in both commercial and industrial electricity consumption in the UK. However, domestic consumption recorded a significant increase. This is attributed to the fact that during the pandemic, UK residents spent a significant amount of time at home, hence, the increase in their energy consumption. Energy consumption is expected to regain its pre-pandemic level with the easing of restrictions; however, new challenges have emerged due to the Russia–Ukraine crisis.

Figure 5 depicts monthly gas demand and supply sources in the UK showing that gas consumption continues to drop partly due to lower demand from energy companies and increased efficiency of power equipment, amongst others. According to OFGEM,²⁰ more than 70% of the UK's gas supply is from Norway and the UK's Continental Shelf (UKCS) with the rest sourced from Europe and the global Liquefied Natural Gas (LNG) markets.

Figure 5.

Gas demand and supply sources (2009–2022).

Note that in Figure 5, positive values represent gas supplies to UK's energy network comprising domestic production from UKCS, gas imports at LNG terminals, including withdrawals from imports and storage through pipelines. The negative estimates denote supplies from the system that include injections to storage and exports through interconnectors. Overall, the UK's gas production and consumption have witnessed a long-term decline; but have since 2014 stabilised.²¹ The EIA estimates that from 2014 to 2016, gas production grew at an average rate of 5% per annum while consumption grew by 7%, suggesting a rise in gas consumption. A similar trend is observed for electricity generation and consumption. However, it is the electricity generated from renewable sources, such as wind that continues to grow.²¹ From a global perspective, net UK electricity generation in 2016 was 324 billion kWh of electricity, while consumption was 342 billion kWh, down 14% and 12%, respectively, from 2006.²¹

Literature review

Despite the proliferation of machine learning methods in the analysis of energy consumption recently, the most common forecasting methods are econometric models. The classical methods serve as benchmarks that are used to compare the performance of more advanced forecast models. Hor et al.²² examine the impact of weather and climatic variables on electricity consumption in England and Wales using rainfall, temperature, relative humidity, socio-economic factors and wind speed as variables, amongst others. They find that the nexus between electricity consumption and temperature extremes remains weak. A plausible explanation of this trend could be explained by the fact that the UK uses more gas for domestic heating than electricity.

Baker and Rylatt¹⁰ employ annual gas and electricity meter data to forecast the UK's energy consumption. Considering the total floor area, built-form type, occupancy, number of (bed)rooms, main heating, regular home working and numbers of TVs, PCs and electric heaters as variables, they find significant relationships between energy consumption and number of bedrooms. They argue forcefully that changes in household designs are expected to drive future energy consumption. Amber et al.² employ multiple linear regression and genetic programming models to predict the daily electricity consumption of buildings in London. They use five variables as follows: solar radiation, temperature, humidity, weekday index and wind speed. The findings show that the genetic programming technique outperforms the linear regression model employed in forecasting electricity consumption.

Hor et al.¹¹ employ ARIMA/GARCH models to predict daily electricity consumption patterns and to investigate the impact of climate change and extreme weather conditions on electricity consumption. They find that apart from climatic factors, GDP and population growth affect long-term electricity consumption. Cancelo et al.¹² examine models employed by Red Eléctrica de España in forecasting short-term electricity consumption. They assess the model's forecast accuracy through the prediction errors for daily and hourly data for 2006 and examine the dependence of electricity consumption on temperature and special days. They find that the forecast performance of the model on normal weekdays is better than on special weekdays while concluding that the model has been largely successful in capturing the dynamics as a result of the occurrence of special days.

Ardakanian et al.²³ compute electricity consumption profiles for residential consumers. Using hourly smart-meter readings from 1000 Canadian homes measured between April 2011 and October 2012. The model employed isolates the effect of temperature on electricity consumption before employing an AR model for the remaining component and this improved the root mean squared error (RMSE) compared to existing approaches. Recently, numerous techniques have been proposed to model intra-week and intra-day seasonal cycles in intra-day electricity consumption data. These models include double-SARMA and modified-HWS for double seasonality. Taylor²⁴ extends the double-SARMA to account for the intra-year seasonal cycle. Using the UK and French data, Taylor shows that for a day ahead forecast, the triple-seasonal model outperforms both double-seasonal and neural network models.

Barassi and Zhao³ perform out-of-sample combination forecasting of energy consumption in the UK using 30-min to one-day data to obtain improved short-term forecast averaging using ARIMA, neural network, vector autoregression, HWS, Bayesian-VAR and Factor-Augmented-VAR. Their results reveal that model averaging always provides a lower mean squared error (MSE) than the optimal model within each class regardless of how it is selected. Recently, Williams and Short¹³ propose a data-driven model for predicting electricity consumption and find that the out-of-sample forecast accuracy of HWS outperforms their proposed model but their model outperforms the benchmark (naive) method.

The advantage of machine-learning methods such as ANN over classical forecasting models according to Abdel-Aal and Al-Garni²⁵ is their ‘increased tolerance to noise, uncertainty and missing data, reduced need for theoretical knowledge on the modelled phenomenon, freedom from assumptions on the probability distributions of input variables, and the fact that no programming is required’. Despite its advantages, particularly its ability to identify and capture asymmetries, ANN has several limitations such as difficulty in identifying the optimal network architecture, difficulty in determining training and design parameters,^c training stopping criteria, black-box nature of the models, etc.²⁵

Since the 1990s, substantial research has been conducted using neural networks, see Gassar et al.,²⁶ Barassi and Zhao³ and Rana et al.,¹⁴ amongst others. The development of a support vector machine (SVM) represents the most significant attempt at overcoming the drawbacks of ANN. SVM provides innovative analysis on both classification and regression problems. The SVMs were employed to address classification, regression, pattern recognition and novelty detection tasks.^18,27

Oğcu et al.²⁸ forecast Turkey's electricity consumption using ANN and SVR and conclude that SVR performs slightly better than ANN. Rana et al.¹⁴ introduce an interval forecasting method based on ANN for computing forecasts and applied it to predicting energy consumption from half-hourly time series of electricity consumption for the UK and Australia. Their method produces valid forecast intervals. De Felice et al.²⁹ use temperature forecasts to perform medium-term electricity consumption predictions for Italy (1990–2007). They discover a relationship between Europe's summer mean temperatures and Italy's electricity consumption using regression and SVR. They demonstrate that using numerical weather predictions will improve the accuracy of energy consumption forecasts.

Bonetto and Rossi¹⁷ forecast energy consumption in residential micro-grids in Italy using SVM, ANN, long short-term memory (LSTM) and ARMA models. They find that machine-learning techniques outperform time-series models but with no clear model to choose from, amongst the machine learning models. Gassar et al.²⁶ propose a data-driven model to predict electricity and gas consumption in London's residential buildings using multilayer neural network (MNN), regression and machine-learning algorithms with economic, socio-demographic and building features as explanatory variables. Gassar et al. find that building characteristics, household income and sizes are the most crucial predictors of energy consumption and that MNN outperforms other competing models. Similarly, Beyca et al.⁴ employ MLR, ANN and SVR to predict gas consumption in Turkey. Their results indicate that SVR outperforms ANN by providing more accurate and reliable results and a lower gas consumption forecast error.

Despite the development of machine learning methods that represent a significant attempt at overcoming the drawbacks of classical models, studies are still ongoing to improve forecasts from these methods. Izidio et al.⁸ observe that machine learning techniques ‘do not possess established methods for feature selection and are sensitive to hyperparameter misspecification, which can degenerate their performance’. Consequently, numerous articles have proposed hybrid specifications to address these issues. These models aim to address the shortcomings of conventional models, including machine learning tools to improve energy consumption forecast accuracy.

González-Romera et al.¹⁵ propose a hybrid approach to predict Spain's monthly electricity demand. A mean absolute percentage error (MAPE) of 1.74% for the hybrid model was obtained which is an improvement compared to using ARIMA or ANNs. Their method hinges on separating trends and irregular components before conducting forecasts on them. Wang et al.⁹ combine the moving average and SVR to forecast electricity consumption in China. Their model, known as TF-ε-SVR-SA performs satisfactorily as forecast error reductions were achieved and significantly outperforms ARIMA(1,1,1) model. Similarly, Wang et al.³⁰ propose FS-SARIMA residual modification model to enhance the forecast accuracy of SARIMA models for electricity consumption in China. They apply particle swarm optimisation, SARIMA and FS-SARIMA to correct SARIMA's forecast results. Their approach enhances the forecast accuracy of SARIMA models.

Yan et al.³¹ propose a modified ANN that combines LSTM-NN with a wavelet transform technique. Using UK's household data, the model outperforms SVR, LSTM-NN and convolutional-NN-LSTM techniques. Jiang et al.³² introduce an electricity consumption prediction framework that accounts for noise and seasonality and integrates Fourier transform, variational mode decomposition and SVM techniques. They find that the hybrid model employed generates a superior forecast compared to ARIMA, SVR and other variants. Neto et al.³³ proffer a data-driven ensemble that combines five forecast methods by employing extreme learning machines (ELM) as the combination model. They find that the ELM-based ensemble is superior to the single forecast approaches. Izidio et al.⁸ combine machine learning with the SARIMA method to derive a hybrid specification that can help improve energy consumption forecasting. Their framework, known as the Evolutionary Hybrid System (EvoHyS) was implemented via the modelling of the error series while employing GA in the optimisation of hyperparameters and feature selection. However, their framework did not consider the influence of key energy variables as well as temperature which affect energy consumption. The approach employed in this paper differs from Izidio et al.⁸ because we account for these important variables to predict energy consumption. Table A2 (Appendix) provides a summary of selected empirical studies on energy consumption forecasting in the UK.

Methodology and estimation strategy

This section explores the models employed for the forecasting exercise and describes the estimation strategy and criteria for selecting the optimal forecast model(s). To determine the stationarity of relevant data, we conduct unit roots and stationarity tests: Augmented Dickey–Fuller (ADF) and Kwiatkowski, Phillips, Schmidt and Shin (KPSS) tests. Next, we explore statistical methods starting with four benchmark models (average, naive, seasonal naive and random-walk with drift methods) to compare with other models considered. For the average method, the predictions are equal to the mean of a given variable's historical data. Given all data denoted by y₁,…,y_T, the forecast can be computed in equation (1) by

{\hat{y}}_{T + h | T} = (y_{1} + y_{2} + y_{3} + \dots + y_{T}) / T = \frac{1}{T} \sum_{t = 1}^{T} y_{t},

(1)

where integer h and

{\hat{y}}_{T + h | T}

denote forecast duration and estimate of

y_{T + h}

from historical data

y_{1}, \dots, y_{T}

, for

h

= 1,2,….¹⁹ For naive forecasts, all predictions are equal to the value of the last observations. This is derived using equation (2):

{\hat{y}}_{T + h | T} = y_{T},

(2)

for

h = 1, 2, \dots .

The seasonal-naive forecast model is used when historical data exhibits strong seasonality. Using the seasonal-naive method, each forecast is equal to the last observation from the same month of a given year.¹⁹ For

T + h

predictions,

{\hat{y}}_{T + h | T} = y_{T + h - m (k + 1)},

(3)

where m and k are the seasonal period and integer component of

(h - 1) / m

. The random-walk with drift method allows naive forecasts to decrease or increase over time. Forecasts for time

T + h

are derived in equation (4):

{\hat{y}}_{T + h | T} = y_{T} + h (\frac{y_{T} - y_{1}}{T - 1}) = y_{T} + \frac{h}{T - 1} \sum_{t = 2}^{T} (y_{t} - y_{t - 1}) .

(4)

We further subject the data to time series methods using exponential smoothing and SARIMA models. These models were in turn extended to enable the regression errors to be serially correlated. The modified specifications are known as dynamic regressions. Enders³⁴ and Hyndman and Athanasopoulos¹⁹ provide a lucid exposition of the various time series forecasting approaches. A dynamic harmonic regression model is given by

y_{t} = b t + \sum_{j = 1}^{T} [α_{j} \sin (\frac{2 π j t}{m}) + β_{j} \cos (\frac{2 π j t}{m})] + η_{t},

(5)

where

η_{t}

and m denote an ARIMA process and a seasonality term, respectively. The decomposed time series is assumed to be represented as:

y_{t} = f (S_{t}, T_{t}, R_{t}),

(6)

where

y_{t}

S_{t}, T_{t}

and

R_{t}

denote time series at time t, seasonal, trend and irregular/error components at period t. We consider STL decomposition due to its advantages of robustness and versatility, ability to capture non-linearities and seasonality, robustness to outliers, assumption of a dynamic

S_{t}

and a flexible trend-cycle smoothness parameter.^19,35 However, STL only handles additive decompositions and it cannot automatically incorporate trading day or calendar effects.

Seasonal hybrid models

De Livera et al.³⁶ propose the TBATS that combines state-space exponential smoothing with Fourier terms and a Box-Cox transformation for forecasting complex seasonality. The TBATS specification is given in equations (7)–(11):

y_{t}^{(ω)} = {\begin{matrix} (y_{t}^{ω} - 1) / ω & if ω \neq 0, \\ \log y_{t} & if ω = 0 \end{matrix} .

(7)

y_{t}^{(ω)} = ℓ_{t - 1} + ϕ b_{t - 1} + \sum_{i = 1}^{T} s_{t - 1}^{(i)} + d_{t},

(8a)

ℓ_{t} = ℓ_{t - 1} + ϕ b_{t - 1} + α d_{t},

(8b)

b_{t} = (1 - ϕ) b + ϕ b_{t - 1} + β d_{t},

(8c)

d_{t} = \sum_{i = 1}^{p} ϕ d_{t - i} + \sum_{j = 1}^{q} θ ε_{t - j} + ε_{t},

(8d)

where

y_{t}^{(ω)}

is the Box-Cox transformed value with transformation parameter

ω

ℓ_{t}

denotes the local level in time t, b represents the long-run trend, b_t denotes the short-run trend in time t,

ε

represents the white-noise process, d_t is ARMA(p, q) process and y_t is observation at period t.³⁶ The trigonometric specification based on Fourier terms is given in equations (9)–(11) as follows:

s_{t}^{(i)} = \sum_{j = 1}^{k_{i}} s_{j, t}^{(i)}

(9)

s_{t}^{(i)} = s_{j, t - 1}^{(i)} \cos λ_{j}^{(i)} + s_{j, t - 1}^{* (i)} \sin λ_{j}^{(i)} + γ_{1}^{(i)} d_{t}

(10)

s_{j, t}^{* (i)} = - s_{j, t - 1}^{(i)} \sin λ_{j}^{(i)} + s_{j, t - 1}^{* (i)} \cos λ_{j}^{(i)} + γ_{2}^{(i)} d_{t}

(11)

where

s_{t}^{(i)}

denotes i^th seasonal component at period t,

s_{j, t}^{(i)}

is the stochastic level of the i^th seasonal component,

s_{j, t}^{* (i)}

is stochastic growth level of the i^th seasonal component and

γ_{1}^{(i)}

and

γ_{2}^{(i)}

denote smoothing parameters, while

λ_{j}^{(i)} = 2 π j / m_{i}

.³⁶ The TBATS model accounts for both linear and non-linear series with single, multiple, high-period and non-integer seasonalities along with dual calendar effects (ibid.).

Machine learning methods

Artificial neural network (ANN) models

ANNs are non-linear models designed to avoid restrictions that characterise time-series models. Despite criticisms of being rather complex and lacking strong theoretical foundations, the NNAR and other ANN variants often outperform regression and time-series models in terms of forecast accuracy. The NNAR model is a dynamic model with feedback on several layers.³ The NNAR equation regresses the current response output signal on previous output signals, defined in equation (12) as,

y_{t + h - 1} = f (y_{t - 1} + y_{t - 2} + y_{t - 3} + \dots + y_{t - p}),

(12)

where f is a non-linear function with p denoting the oldest value of signals. The forecast is computed using equation (13):

{\hat{y}}_{t + h - 1} = \hat{f} (y_{t} + y_{t - 1} + y_{t - 2} + \dots + y_{t - p + 1}) .

(13)

In this framework, the time series’ lagged values are used as input to a neural network. For data exhibiting seasonal variations, the NNAR(p,P,k)_m model where k, p and P denote neurons in the hidden layer, non-seasonal AR order and seasonal-AR order, respectively, is specified as follows:

y_{t + h - 1} = f (y_{t - 1} + y_{t - 2} + \dots + y_{t - p} + y_{t - m} + y_{t - 2 m} + y_{t - P m} + ε_{t}),

(14)

{\hat{y}}_{t + h - 1} = \hat{f} (y_{t} + y_{t - 1} + \dots + y_{t - p + 1} + y_{t - m + 1} + y_{t - 2 m + 1} + y_{t - P m + 1} + ε_{t}),

(15)

where

\hat{f}

denotes a neural network with hidden nodes and layer(s) and

ε_{t}

is an error term. We consider a multilayer feed-forward network (MLFFN) with several neurons in one hidden layer. Hyndman and Athanasopoulos¹⁹ note that in an MLFFN, layers of nodes obtain inputs from prior layers and that outputs of nodes in a given layer become inputs to the next layer. The outcome is then adjusted by a non-linear function before being output.³⁷ Furthermore, the input into one of the hidden layers can be linearly combined to yield the specification in equation (16):

z_{j} = b_{j} + \sum_{i = 1}^{4} w_{i, j} x_{i},

(16)

where z_j denotes activations, b_j's are parameters while w₁,₁,…, w_4,3 are the weight coefficients. Equation (16) is then modified in the hidden layer using a non-linear activation function.^27,37 Figure 6 presents an architecture of a basic neural network model known as MLFFN, where the arrows show the direction of relationships within the network. In the time-series domain, the output is the prediction at time t.^19,37

Figure 6.

Basic neural network framework.

Support vector regression models

SVM was proposed by Vapnik³⁸ to address problems associated with ANN models such as overfitting, identification of optimal network architecture and stopping criteria issues, amongst others (see Beyca et al.⁴ and Bishop²⁷). SVM technique's variant for regression is known as support vector regression (SVR). Equations (17)–(21) describe the SVR technique. Assuming a functional form as follows:

f (x) = w^{T} ϕ (x) + b,

(17)

where f( x ), w ,

ϕ (x)

and b denote a non-linear function, weight vector, feature space transformation function and bias, respectively.^4,27 To compute the coefficients, the regularised-error function given in equation (17) is minimised,

R (f) = C \frac{1}{T} \sum_{i = 1}^{T} L_{ϵ} (y_{i}, f (x_{i})) + \frac{1}{2} | | w | |^{2},

(18)

where C,

L_{ϵ}

(

y_{i}

f (x_{i})

) and

| | w | |

represent inverse-regularisation parameter,

ϵ

-insensitive error function and weights vector norm. Following Beyca et al.,⁴ a simple

ϵ

-insensitive error function with associated linear cost and errors outside the insensitive region can be specified as follows:

L_{ϵ} (y_{i}, f (x_{i})) = {\begin{matrix} 0 & if | y - f (x) < ϵ | \\ | y - f (x) | - ϵ & if | y - f (x) \geq ϵ | \end{matrix} .

The error function gives 0 when the absolute difference between the prediction and the target is less than

ϵ

(

ϵ

> 0).^4,27 Introducing slack variables

ξ_{i}

and

ξ_{i} *

denoting negative and positive deviation from the ε region, equation (18) can be re-specified in constraint form as follows:

Minimise C \sum_{i = 1}^{T} (ξ_{i} + ξ_{i} *) + \frac{1}{2} | | w | |^{2},

(19)

subject to:

y_{i} - (w^{T} ϕ (x) + b) \leq ϵ + ξ_{i},

(20a)

(w^{T} ϕ (x) + b) - y_{i} \leq ϵ + ξ_{i} *,

(20b)

ξ_{i}, ξ_{i} * \geq 0.

(20c)

Note that points above the

ϵ

-tube indicate

ξ_{i} > 0

and

ξ_{i} * = 0

, points below the

ϵ

-tube indicate

ξ_{i} = 0

and

ξ_{i} * > 0

while points inside the

ϵ

-tube indicate

ξ_{i} = ξ_{i} * = 0

.²⁷ Forecasts for new inputs are computed using,

f (x) = \sum_{i = 1}^{T} (α_{i} + α_{i} *) k (x_{i}, x_{j}) + b .

(21)

Equation (21) is known as the SVR expansion where w is a linear combination of the training patterns, x _i .²⁷ Despite some of SVR's advantages ranging from its non-parametric nature to flexibility; it also suffers some limitations such as scaling inefficiently with the data size arising from the use of quadratic optimisation and kernel-transformation techniques.

Forecast error metrics

To assess the forecast performance of the models, the following standard measures were considered. The mean absolute error (MAE) is defined as follows:

M A E = \frac{1}{T} \sum_{t = 1}^{T} | y_{t} - {\hat{y}}_{t} |,

(22)

where

y_{t}

is the actual value at time t and

{\hat{y}}_{t}

is the predicted value.³⁹ The mean squared error (MSE) is given as follows:

M S E = \frac{1}{T} \sum_{t = 1}^{T} (y_{t} - {\hat{y}}_{t})^{2} .

(23)

The root mean squared error (RMSE) is defined as follows:

R M S E = \frac{1}{T} \sqrt{\sum_{t = 1}^{T} {(y_{t} - {\hat{y}}_{t})}^{2}} .

(24)

The RMSE is a scale-dependent measure on the same scale as the data but is relatively more sensitive to outliers compared with MAE.³⁹ The mean absolute percentage error (MAPE) is given by:

M A P E = \frac{1}{T} \sum_{t = 1}^{T} | \frac{y_{t} - {\hat{y}}_{t}}{y_{t}} | 100 % .

(25)

The mean absolute scaled error (MASE) is defined by:

M A S E = m e a n (| e_{j} / \frac{1}{T - m} \sum_{t = m + 1}^{T} | y_{t} - y_{t - m} | |),

(26)

where

e_{j}

means forecast error. Hyndman and Koehler³⁹ provide a lucid exposition of forecast error measurements. The models were estimated using R statistical software and some estimation carried out using E-views statistical package.

Data description and preliminary diagnostics

The energy consumption, temperature and energy price data spanning January 1995–March 2022 were obtained from the Digest of United Kingdom Energy Statistics (DUKES) (www.gov.uk). We predict monthly energy consumption as this data frequency can provide information on seasonal consumption patterns, which are not easily observed using annual or quarterly data. Moreover, monthly data ensure that abrupt changes in weather and energy consumption are smoothed-out and problems associated with higher frequency data are minimised. In performing the forecasts, the obtained time series is divided into two samples: training and test sets. Training data for electricity consumption is from January 1995 to March 2019, while the test data cover the period April 2019 to March 2022. For gas consumption, the training period starts from January 1996 and ends in April 2019, while the span of the test data is same as that of electricity consumption. Next, we examine the statistical characteristics of the variables of interest.

Table 1 presents descriptive statistics of the variables. The mean, maximum and minimum values of electricity consumption are 27.177, 36.450 and 20.360 TWh, respectively. For gas consumption, the corresponding values are 76,485, 119,087 and 37,666 GWh. The highest average temperature from 1995 to March 2022 was 19.3°C, while the lowest was −0.3°C. The standard deviations indicate that Brent crude oil price fluctuates more than other energy variables. In terms of the distributional properties of the variables, all are positively skewed. The kurtosis is equally positive for the temperature as well as the other energy variables. To further characterise the distribution of the variables, we perform the Shapiro–Wilk normality test. Results reveal that all p-values are less than 0.05 suggesting a rejection of the null hypothesis. This indicates that the variables are not normally distributed.

Table 1.

Descriptive statistics of relevant variables (1995–2022).

Statistic	Electricity consumption (TWh)	Gas consumption (GWh)	Temperature (°C)	Oil price (US$/Barrel)
Mean	27.1772	76,485.21	10.3416	55.6333
Median	26.8200	74,777.00	9.9000	53.1000
Maximum	36.4500	119,087.00	19.3000	132.7200
Minimum	20.3600	37,666.00	−0.3000	9.8200
Standard Dev.	3.3712	21,680.41	4.4278	32.0917
Skewness	0.4766	0.1422	0.0971	0.4917
Kurtosis	2.6938	1.9066	1.7753	2.1698
Shapiro-Wilk	0.981**	0.9623**	0.951**	0.933**
Observations	327	315	327	327

Note: Computations of the descriptive statistics were done with E-views Statistical Software.

** Significance at a 1% level.

To examine the seasonal patterns and changes in energy consumption, we present seasonal plots for electricity and gas consumption in Figures 7 and 8, where y-axis represents electricity and gas consumption and x-axis indicates the 12-month horizon. There is strong seasonality in energy consumption where the Winter months recorded higher energy consumption, while Spring and Autumn recorded lower consumption levels.

Figure 7.

Seasonal plots: electricity consumption by month (1995–2022).

Figure 8.

Seasonal plot: gas consumption by month (1995–2022).

A similar trend is noticed for gas in Figure 8 except that the consumption in Summer does not rise as much as the consumption of electricity in the corresponding period. This means that seasonal effects have a significant influence on energy consumption in the UK and that overall consumption is influenced by changes in seasons, amongst other factors. The analysis above highlights the strong seasonality that is present in these datasets and any forecasting endeavour of the UK's energy consumption needs to take seasonality into account. Based on the identified data characteristics, we conduct unit roots and stationarity tests.

Table 2 presents stationarity and unit-root test results. The Augmented Dickey–Fuller (ADF) test indicates that energy consumption and energy price series are non-stationary, while the test for temperature reveals stationarity. Thus, the null hypothesis of unit root is rejected implying that there are no unit roots in the temperature series. The Kwiatkowski–Phillips–Schmidt–Shin (KPSS) tests for temperature, electricity and gas consumption indicate that the time series is stationary. Finally, for energy price (crude oil), the test statistic is 2.7433 indicating non-stationarity in the series.

Table 2.

Stationarity and unit-root test result for selected variables.

Test	Electricity Consumption (TWh)	Gas Consumption (GWh)	Temperature (°C)	Oil price (US$/Barrel)
ADF	Unit Root Test Statistics
With Intercept	−0.3238	−0.5511	−3.9352	−1.9097
p-value	(0.9182)	(0.8776)	(0.0020)**	(0.3277)
With Intercept/Trend	−2.0498	−2.0882	−3.9854	−2.7127
p-value	(0.5712)	(0.5499)	(0.0101)**	(0.2321)
	Stationarity Test Statistics
KPSS	0.0163	0.2094	0.0163	2.7433

Note: ** and * indicate significance at 1% and 5% levels, respectively. The tests for unit root are conducted in levels and the null hypothesis for ADF test is that the time series has a unit root. The computations were done in E-views Software. For KPSS test, the computations were done in R Software and critical values for significance at 10%, 5%, 2.5% and 1% are 0.347, 0.463, 0.574 and 0.739, respectively. The KPSS test is of type mu with 5 lags.

Results and discussion

Benchmark forecasts

In Table 3, we present forecast performance of electricity and gas consumption from benchmark methods. We find that for both the training and test sets, the seasonal-naive model outperforms alternative specifications for electricity and gas consumption. This is because the model had the least forecast error based on all the considered criteria. This result is not unexpected because the electricity consumption series exhibits strong seasonality. Typically, demand is higher in winter than in summer because peak demand in summer is usually lower compared to the peak demands of the winter and the low demands in the summer period are low when compared with the low demands of the winter.^d The random walk with drift model is superior to the average method based on MAE and quite similar to the naive specification. For gas consumption, training set results indicate that naive is superior to the average method, while the seasonal naive outperforms both the average and naive forecasts. The random-walk with drift performs better than both naive and average methods for gas consumption but is comparable to the average method. These results underscore the need to account for seasonality.

Table 3.

Baseline forecasts for electricity and gas consumption.

Electricity consumption					Gas consumption
Method	MAE	MAPE	RMSE	MASE	MAE	MAPE	RMSE	MASE
Training set
Average	2.6988	9.8709	3.3137	2.7229	18.6304	26.5760	21.4986	3.2229
Naive	2.5101	9.0476	3.3638	2.5325	9.7532	12.7981	12.2476	1.6873
Seasonal Naive	0.9911	3.5347	1.2958	1.0000	5.7805	7.6733	7.5975	1.0000
RW-Drift	2.5101	9.0477	3.3639	2.5326	9.7487	12.7789	12.2472	1.6865
Test set
Average	3.2113	14.1559	3.8952	3.2400	18.5851	36.7068	22.7881	3.2152
Naive	2.7978	12.3127	3.4036	2.8228	18.1792	35.6125	22.1984	3.1449
Seasonal Naive	1.0069	4.3891	1.4053	1.0159	5.7224	8.7391	7.8593	0.9899
RW-Drift	2.7929	12.2909	3.3982	2.8178	17.8214	34.4406	21.6287	3.0830

Note: The best values are in bold.

Exponential smoothing forecasts

Since the energy consumption data exhibit seasonal patterns, the exponential smoothing method is a suitable technique. Chatfield³⁷ notes that ‘this procedure can cope with the trend and with additive or multiplicative seasonality’ and thus recommends its use for data exhibiting seasonality. We examine six exponential smoothing models namely: Holt's linear trend (HLT), Holt's Damped (HD), HWS (additive (A), damped (D)), HWS (multiplicative (M)) and ETS methods. Table 4 presents forecast results from these models for the UK's electricity and gas consumption.

Table 4.

Exponential smoothing prediction error measures (2022–2025).

Electricity consumption					Gas consumption
Method	MAE	MAPE	RMSE	MASE	MAE	MAPE	RMSE	MASE
Training set
SES	2.4700	8.9463	3.1029	2.4921	10.0162	13.1061	12.2439	1.7328
HLT	2.4700	8.9463	3.1029	2.4921	10.0162	13.1061	12.2439	1.7328
HD	2.4650	8.9219	3.1001	2.4871	9.0775	11.8521	11.4212	1.5704
HWS (A)	0.8034	2.8604	1.0929	0.8105	3.7662	5.2363	4.8481	0.6515
HWS (M)	0.8235	2.9573	1.1223	0.8308	3.8308	5.1985	5.0301	0.6627
HWS(A,D)	0.8193	2.9261	1.1010	0.8266	3.5979	4.8967	4.7769	0.6224
ETS	0.0284	0.8544	0.0382	0.8051	0.0489	1.1550	0.0655	0.6495
Test set
SES	2.7132	11.9286	3.2999	2.7375	116.7004	178.1365	138.1648	20.1888
HLT	2.7132	11.9286	3.2999	2.7375	116.7004	178.1365	138.1648	20.1888
HD	2.7215	11.9669	3.3108	2.7458	23.7539	31.1113	29.7719	4.1094
HWS (A)	0.5825	2.5135	0.8938	0.5877	8.2375	12.5273	9.9753	1.4251
HWS (M)	0.6398	2.7207	0.9222	0.6455	6.4426	9.3391	8.2306	1.1145
HWS(A,D)	0.7522	3.2717	1.0666	0.7589	6.1362	9.0125	7.7537	1.0615
ETS	0.0323	1.0314	0.0466	0.9147	0.0891	2.1562	0.1079	1.1818

Note: The best values are in bold.

Since the identified form of seasonality is additive, there is no need to transform the electricity consumption data. Besides, transformations tend to make the interpretation of results difficult. Holt's Damped method outperforms the HLT model, while the HWS (A) outperforms both HLT and HD. Surprisingly, HWS (A, D) performs worse than HWS (additive) in the training set, while ETS outperforms all exponential smoothing specifications. For gas consumption, the DH method outperforms the HLT model, while HWS (A) model performs better than both HLT and HD. The HWS (A, D) performs better than HWS (A) in both the training and test sets. The ETS outperforms all previous specifications, particularly in the test set.

Time series forecasts: SARIMA model

Table 5 presents forecast-error measure results for electricity and gas consumption from SARIMA models. In selecting the preferred SARIMA model, we used the well-known ARIMA model automatic selection procedure known as the auto.arima technique. This approach returns the ‘best’ ARIMA model based on a combination of model selection criteria (AIC, BIC and AIC), unit root tests and optimisation of the maximum likelihood estimator. However, there is a caveat to applying this technique. The researcher can only apply this technique to the time series data at levels without any differencing. This method was employed in the selection of the best SARIMA model for both electricity and gas consumption. The results show that for electricity consumption, SARIMA (3,1,3)(2,1,2)₁₂ produces the least forecast error amongst competing models in both the training and test sets. This model is superior to the benchmark and exponential-smoothing methods earlier considered. For gas consumption, the result indicates that for both the training and test sets, SARIMA(1,1,1)(0,1,1)₁₂ produces the least forecast error based on MAE, MAPE, RMSE and MASE.

Table 5.

SARIMA forecast error measures (2022–2025).

Electricity consumption
METHOD	MAE	MAPE	RMSE	MASE
Training set
SARIMA(5,1,3)(1,1,1)₁₂	0.0247	0.7462	0.0336	0.7017
SARIMA(3,1,3)(2,1,2)₁₂	0.0246	0.7402	0.0338	0.6963
Test Set
SARIMA(5,1,3)(1,1,1)₁₂	0.0282	0.9045	0.0452	0.7999
SARIMA(3,1,3)(2,1,2)₁₂	0.0274	0.8789	0.0441	0.7774
Gas consumption
Training set
SARIMA(1,1,1)(0,1,1)₁₂	0.0487	1.1456	0.0649	0.6460
SARIMA(3,0,0)(2,1,0)₁₂	0.0508	1.1934	0.0677	0.6730
Test Set
SARIMA(1,1,1)(0,1,1)₁₂	0.0489	2.1575	0.1090	1.1856
SARIMA(3,0,0)(2,1,0)₁₂	0.0854	2.0704	0.1138	1.1327

Note: The best values are in bold.

The forecast from SARIMA(3,1,3)(2,1,2)₁₂ is shown in Figure 9(a). It suggests that within the 36-month forecast horizon, electricity consumption could experience continued seasonal fluctuations and a downward trend. This could be traced to macroeconomic uncertainties induced by the COVID-19 pandemic and Brexit, amongst other drivers. The dark-blue shaded region indicates 80% prediction intervals while the light-blue region indicates 95% prediction intervals. The forecast from SARIMA(1,1,1)(0,1,1)₁₂ model related to gas consumption is shown in Figure 9(b). It suggests that gas consumption will experience stronger seasonal variation. The predictions from SARIMA(1,1,1)(0,1,1)₁₂ indicate a downward trend in gas consumption in the UK over the three-year forecast span.

Figure 9.

Electricity and gas consumption forecasts (SARIMA models).

Note that despite the SARIMA model's flexibility, its underlying assumption of linearity in the data-generating process of the series undermines its performance in dealing with non-linear processes.⁸ Therefore, we explore dynamic regression forecast methods.

Dynamic regression forecasts

We forecast energy consumption based on temperature changes and this is because it is expected to provide additional insights into the historical pattern of energy consumption and improve forecast accuracy. Within the class of dynamic regression models, three variants were estimated: regression-with-ARMA errors, quadratic regression-with-ARMA errors (QREG-ARIMA) and dynamic harmonic regression (DHR) models. Table 6 presents the forecast performance of dynamic regression models for electricity and gas consumption.

Table 6.

Forecasts using dynamic regression models.

Electricity consumption					Gas consumption
Method	MAE	MAPE	RMSE	MASE	MAE	MAPE	RMSE	MASE
Regression-ARIMA	0.6829	2.5109	0.9177	0.6863	4.0670	5.7671	5.5220	0.6739
Q-REG-ARIMA	0.6477	2.3726	0.8964	0.6508	4.0538	5.7225	5.5036	0.6717
DHR-with-ARMA-errors	0.7011	2.5665	0.9432	0.7045	3.9379	5.4584	5.1249	0.6526

Note: The best values are in bold.

For electricity consumption, the regression with ARIMA(4,1,1)(1,1,2)₁₂ error model forecast metrics are 0.683, 2.511, 0.918, and 0.686 corresponding to MAE, MAPE, RMSE and MASE, respectively. The Q-REG with ARIMA(4,1,1)(1,1,2)₁₂ error conversely outperforms the regression with ARIMA(4,1,1)(1,1,2)₁₂ error model, while the DHR performs worse than other models. For gas consumption, Q-REG with ARIMA(3,1,1)(1,0,0)₁₂ error specification outperforms the regression with ARIMA(3,1,1)(1,0,0)₁₂ error but not the DHR-with-ARMA-error models. Figure 10(a) presents forecasts using quadratic regression with ARIMA(4,1,1)(1,1,2)₁₂ error for electricity consumption in the UK. The forecast results indicate a downward trend in electricity consumption over the forecast horizon. The reasons behind the decline in electricity consumption at the end of March 2022 can be traced to the slow post-COVID-19 recovery and the Russia–Ukraine crisis which has disrupted global energy and food supply as the predictions indicate lower electricity consumption. This indicates that there could be a potential substitution of energy use between gas and electricity by economic agents due to rising costs.

Figure 10.

(a) Forecasts using quadratic regression with ARIMA (4,1,1)(1,1,2)₁₂ error (electricity consumption). (b) Forecasts using dynamic harmonic regression (gas consumption).

The out-of-sample forecasts of the three models in the case of gas consumption show that the DHR model performs better. The predictions for gas consumption based on DHR-with-ARMA-errors are shown in Figure 10(b). Furthermore, amongst competing DHR-with-ARMA-errors models for gas consumption, the regression with ARIMA(1,1,1)(1,0,0)₁₂ errors was more accurate. This model predicts that within a 36-month horizon, gas consumption will experience seasonal fluctuations and is likely to increase. This could be traced to the price substitution effect as consumers switch from electricity to gas. The result also shows that the forecast intervals for the dynamic regression model become narrow compared with the fitted SARIMA models. This could be due to the inclusion of the temperature variable which further explains the dynamics of gas consumption. This conforms to earlier findings where gas consumption plots show seasonality and trends suggesting the need to account for seasonality and asymmetries. Therefore, the DHR-with-ARMA errors incorporate Fourier terms that include seasonality with the ARIMA errors reflecting other time-series dynamics.

The gas consumption predicted trajectory is in line with the recent behaviour of gas consumption with future predictions trending upwards. To highlight the impact of K (number of Fourier terms) on seasonal patterns, we vary K from 1 to 6 following extant studies. Hyndman and Athanasopoulos¹⁹ succinctly state the merits of this technique. They include handling any length of seasonality and seasonal pattern smoothness being controlled by K. Note that a common feature of regression and ARIMA models is that stationarity conditions are crucial while for machine learning models, such assumptions are not important.

Artificial neural network forecasts

Table 7 presents forecasts error measures of energy consumption from selected ANN models that were established after extensive experiments using a plethora of learning algorithms, hidden layers, activation functions and neurons, amongst others. The forecasts indicate that for electricity consumption, NNAR(9,1,10)₁₂ produces the least forecast error amongst competing models. In terms of model structure, we use lagged values of electricity consumption as inputs to the NNAR model. The NNAR(9,1,10)₁₂ model shows nine lagged inputs including the last observation from the same month as input, 1 hidden layer and 10 nodes.

Table 7.

ANN forecasts Error Measures (2022–2025).

Method	MAE	MAPE	RMSE	MASE
Electricity consumption
NNAR(9,1,2)₁₂	0.5007	1.8245	0.6563	0.5031
NNAR(9,1,6)₁₂	0.4917	1.7946	0.6497	0.4941
NNAR(9,1,10)₁₂	0.4889	1.7842	0.6375	0.4914
Gas consumption
NNAR(13,1,2)₁₂	0.4618	0.5593	0.7609	0.0765
NNAR(13,1,7)₁₂	0.4759	0.5837	0.7477	0.0789
NNAR(13,1,10)₁₂	0.4932	0.6069	0.7938	0.0817

Note: The best values are in bold.

For gas consumption, NNAR(13,1,2)₁₂ produces the least forecast error. The ANN model is discovered to be superior to all methods earlier considered but for electricity consumption, SARIMA outperforms the ANN. The 36-month ahead out-of-sample forecasts from NNAR(13,1,2)₁₂ model for electricity and gas consumption are shown in Figure 11(a) and (b). It suggests that UK's electricity consumption is expected to experience seasonal fluctuations and a declining trend, while gas consumption may likely witness seasonal fluctuations with an increasing trend albeit heightened volatility. Several factors would determine the magnitude of fluctuations in the coming months and years. These include the conclusion of the Brexit deal, a significant recovery from the COVID-19 pandemic, as well as the resolution of the Russia–Ukraine crisis, amongst other factors.

Figure 11.

Electricity and gas consumption forecasts – NNAR(9,1,10)₁₂ and NNAR(13,1,2)₁₂.

Support vector regression forecasts

Here, we present forecast results from the fitted SVR models based on different kernels and parameters. It is important to note that the SVR technique is sensitive to the choice of parameters. Studies have shown that the performance of SVR depends on three parameters: cost of error (C), kernel and ε. Before fitting the SVR models, we use a cross-validation procedure to determine the type of kernel to use for the estimation. In terms of parameter selection, four kernel functions are considered and they are; polynomial, sigmoid, linear and Radial Basis Function (RBF). This is in line with Izidio et al.'s⁸ choices. Specifically, the parameter selection for the SVR model for electricity consumption is defined according to the following value options:

kernel: (linear, polynomial2, polynomial3, polynomial4, polynomial5, sigmoid, RBF)

gamma (γ): (0.1, 0.5, 1, 2, 3, 4, 100)

C = (0.001, 0.01, 0.1,1,5,100)

Epsilon (ε) = (0.0, 10⁻⁵, 10⁻⁴, 10⁻³, 10⁻², 10⁻¹, 10⁰, 10¹).

The kernel options considered comprise linear, polynomials (with degrees 2, 3, 4 and 5), sigmoid and RBF. The C denotes the cost of error also known as the regularisation parameter, ε is the parameter used in the computation of the error function, while γ is the kernel parameter. For gas consumption, the parameters of the SVR model are defined in line with the following:

kernel: (linear, polynomial, polynomial3, polynomial4, polynomial5, sigmoid, RBF)

gamma: (0.1, 0.5, 1, 2, 3, 4, 5)

C = (0.001, 0.01, 0.1,1,5)

Epsilon (ε) = (0.0, 10⁻⁵, 10⁻⁴, 10⁻³, 10⁻², 10⁻¹, 10⁰, 10¹).

In determining the optimum value for the parameters of interest, we lean on the literature and employ extensive experimentation by varying the parameter values to identify the best combination. For the parameter tuning, the strategy employed is as follows: we first set ε-values and then vary C for the training set. The C that yields the optimal performance is selected for each training set. Finally, forecast error metrics are computed and the combination yielding the smallest value is selected as optimal for each training set. Table 8 presents SVR forecast error measures for electricity and gas consumption models.

Table 8.

SVR model forecast error measures (2022–2025).

Method	Cost	γ	ε	Kernel	NSV	MAE	MSE	RMSE
Electricity consumption
ε-SVR	100	0.064	0.10	Radial	253	1.8022	4.9697	2.2293
	200	0.100	0.10	Radial	250	2.2772	7.3506	2.7112
	100	0.065	0.10	Polynomial	258	2.1070	8.0197	2.8319
	200	0.100	0.10	Polynomial	258	2.2489	7.7521	2.7843
	100	0.064	0.10	Linear	261	1.7893	4.7809	2.1865
	200	0.100	0.10	Linear	254	2.2701	7.6506	2.7659
	100	0.064	0.10	Sigmoid	263	2.1674	7.3091	2.7035
	200	0.100	0.10	Sigmoid	254	1.9814	5.4447	2.3334
Gas consumption
ε-SVR	5	0.10	0.10	Radial	246	8.1438	101.4811	10.0738
	10	0.01	0.10	Radial	240	9.3136	115.0005	10.7238
	5	0.10	0.10	Polynomial	250	13.0101	259.2262	16.1005
	10	0.10	0.10	Polynomial	245	11.2776	220.1894	14.8388
	5	0.10	0.10	Linear	237	9.4219	126.0333	11.2265
	10	0.01	0.10	Linear	237	9.0884	118.9622	10.9069
	5	0.10	0.10	Sigmoid	252	10.2927	141.2296	11.8840
	10	0.01	0.10	Sigmoid	239	10.8428	154.7112	12.4383

Note: Cost is a general penalising parameter, ε is a parameter used in the computation of the error function, while γ is the kernel parameter. NSV represents the number of support vectors. The parsimonious models are in bold.

The results from Table 8 indicate that for electricity consumption, ε-SVR with a linear kernel (100, 0.064 and 0.1) produces the least forecast error amongst competing specifications. The MAE, MSE and RMSE are 1.789, 4.7809 and 2.1865, respectively. For gas consumption, ε-SVR with RBF kernel (5, 0.1 and 0.1) produces the least forecast error. The forecast metrics are 8.1438, 101.4811 and 10.0738, respectively. Note that for both electricity and gas consumption, ANNs are superior to the SVR forecast results.

Combination forecast results

Combination forecast procedures have often been equal or superior to individual forecasts from single models. The combination procedure is performed by averaging five selected (preferred) forecast models, namely: SARIMA, ETS, NNAR, STL and TBATS using simple model averaging (SMA). Forecasts are generated from these models and within each class, we compare models and select the optimal specification based on model selection criteria and forecast-error measures. The identified optimal models are then combined based on simple model averaging to generate forecasts. Therefore, we compute combination forecasts before comparing results using the last 36 months. Table 9 presents the forecast-error measures associated with combination forecast results.

Table 9.

Selected models error measures (2022–2025).

Method	MAE	MAPE	RMSE	MSE
Electricity consumption
SARIMA	0.6757	2.9683	1.0414	0.4566
ETS	0.7712	3.3489	1.0762	0.5947
NNAR	0.9878	4.2439	1.2949	0.9757
STL	0.7376	3.2235	1.0706	0.5441
TBATS	1.1727	4.8891	1.3625	1.3752
COMBINATION	0.7674	3.3229	1.0659	0.5889
Gas consumption
SARIMA	5.7368	8.7194	7.3096	32.9109
ETS	7.4679	11.1889	9.1758	55.7695
NNAR	6.5850	10.7982	7.9221	43.3622
STL	6.1968	9.2305	7.7811	38.4003
TBATS	5.7062	8.8836	7.1338	32.5607
COMBINATION	5.7444	8.6781	7.1815	32.9981

Note: The best values are in bold.

For electricity consumption, forecast metrics reveal that SARIMA performs well in predicting its dynamics and is the preferred model while for gas consumption; TBATS performs better than other models including the combination forecasts. This is because seasonality and non-linearity are present in the energy consumption datasets. In addition, the ETS outperforms NNAR as its forecast errors are lower. The TBATS performs worse than SARIMA, while the STL outperforms both ETS and NNAR. The combination forecasts for electricity and gas consumption are shown in Figures 12 and 13.

Figure 12.

Forecast combination results for electricity consumption.

Figure 13.

Forecast combination results for gas consumption.

The combination forecast is outperformed by STL and SARIMA. Some of the merits of the TBATS forecasting strategy according to De Livera et al.³⁶ is that it accommodates a large parameter space, enables multiple seasonal components and effectively deals with non-linearity and serial correlation compared with employing a simple and efficient estimation procedure.

The findings on combination forecasts are contrary to the results obtained by Barassi and Zhao³ and Kourentzes et al.⁴⁰ Our results indicate that combination forecasting does not always improve forecast accuracy.

Out-of-sample forecast: SARIMA and TBATS

The out-of-sample forecast is carried out using the best-performing models: the SARIMA for electricity consumption and TBATS for gas consumption. Figure 14(a) and (b) shows the predictions. For electricity consumption, the result indicates that it will continue its decline over the forecast horizon and it is unlikely to return to its pre-COVID-19 levels. Figure 13(b) indicates that gas consumption would stabilise over the forecast horizon. The deployment of vaccines across the country and the phased removal of lockdown measures could contribute in this direction. However, the energy shortage that stemmed from the Russia–Ukraine crisis has dampened modest recovery. Notably, the high gas consumption volatility remains a source of concern over the short- to medium-term due to the substitution effect.

Figure 14.

Energy consumption forecasts: SARIMA and TBATS.

Robustness checks

Sensitivity analysis is conducted by comparing forecasts from the various models used. The six classical regression models considered are simple linear regression, multiple linear regression (MLR), MLR₁ (with seasonal dummy and trend), MLR₂ (with seasonal dummies, trend and predictor variables), harmonic regression (HR) and HR₂ (harmonic with the trend and predictor variable). The model accounts for predictor variables such as temperature, seasonal indices and trends. These considerations, in addition to the inclusion of oil prices, constitute another value addition to the paper.

Table 10 presents in-sample forecast results from these models for electricity and gas consumption. In the SLR model, electricity consumption represents the forecast variable, while temperature denotes the predictor variable. For the MLR model, oil price is an explanatory variable. The in-sample forecasts of the SLR and MLR models are similar. However, MLR₁ outperforms both models. The HR specification however performs better than MLR₁, while the MLR₂ model outperforms all benchmarks considered including HR₂. Based on cross-validation (CV) criteria, the optimal model selected is the one with the smallest CV statistic and highest R². Notably, the MLR₂ specification outperforms all other models. See Table A1 in the Appendix for the detailed results.

Table 10.

Classical regression (in-sample) prediction error measures (2022–2025).

Electricity consumption					Gas consumption
Method	MAE	MAPE	RMSE	MASE	MAE	MAPE	RMSE	MASE
SLR	1.9721	7.2510	2.4841	1.9817	9.1653	12.8491	11.0523	1.5188
MLR	1.9850	7.3090	2.4808	1.9947	8.8896	12.3182	10.7780	1.5219
MLR₁	1.5739	5.8782	1.9577	1.5817	6.1886	8.4541	7.7605	1.0255
HR	1.5009	5.5874	1.8879	1.5082	6.3773	8.7128	7.9626	1.0568
MLR₂	1.4430	5.3595	1.8299	1.4501	5.8793	8.0886	7.3701	1.0065
HR₂	1.5009	5.5874	1.8879	1.5082	6.0148	8.2735	7.5779	1.0297

Note: The best values are in bold.

Synopsis of model parameter selection

The key experimental protocols associated with some of the forecast models employed are summarised in Table 11. For the SARIMA model, optimal parameter selection was accomplished using the automatic selection procedure (auto.arima). This approach returns the optimal SARIMA model based on model selection criteria, unit root tests and maximum likelihood estimator optimisation. There are six model variants considered for the exponential smoothing approach and they are as follows: simple exponential smoothing, Holt's linear trend, Holt's Damped, Holt-Winters Seasonal (comprising both additive and damped specifications), Holt-Winters Seasonal (with multiplicative seasonality) and ETS methods. The parameters used for the estimation of the SVR models for electricity and consumption are presented in Table 11, while the parameters used for the estimation of the ANN models for electricity and gas consumption are provided in Table 12.

Table 11.

Parameter selection for SVR model.

S/N	Parameters	Electricity consumption
1	kernel	linear, polynomial2, polynomial3, polynomial4, polynomial5, sigmoid, RBF
2	gamma (γ)	0.1, 0.5, 1, 2, 3, 4, 100
3	C	0.001, 0.01, 0.1, 1, 5, 100
4	Epsilon (ε)	0.0, 10⁻⁵, 10⁻⁴, 10⁻³, 10⁻², 10⁻¹, 10⁰, 10¹
	Parameters	Gas Consumption
1	kernel	linear, polynomial2, polynomial3, polynomial4, polynomial5, sigmoid, RBF
2	gamma (γ)	0.1, 0.5, 1, 2, 3, 4, 5
3	C	0.001, 0.01, 0.1, 1, 5
4	Epsilon (ε)	0.0, 10⁻⁵, 10⁻⁴, 10⁻³, 10⁻², 10⁻¹, 10⁰, 10¹

Table 12.

Parameter selection for ANN model.

S/N	Parameters	Electricity consumption
1	Hidden Layer	1
2	Nodes	10
3	Lagged Inputs	9
	Parameters	Gas consumption
1	Hidden Layer	1
2	Nodes	2
3	Lagged Inputs	13

For electricity consumption analysis, the multilayer perceptron (MLP) used for estimating the model has 1 hidden layer with 10 nodes (neurons) and 9 lagged inputs (predictors), while for the gas consumption model, the MLP has 1 hidden layer with 2 nodes and 13 predictors. The activation function used in the estimation of the ANN models is the sigmoid after extensive experimentation with a range of alternatives.^e

Regarding the key parameter for the dynamic harmonic regression forecast model, especially the K (which denotes the number of Fourier terms) which has a significant effect on the seasonal patterns, we vary K from 1 to 6. The combination forecasts were computed by averaging five selected models: SARIMA, ETS, Neural Network, STL and TBATS using simple model averaging. The predictions are generated from each of these models before combining the estimates by computing the average as the forecasts.

Summary of major findings

This subsection summarises the most significant results from the empirical investigation. Table 13 provides the main results of the paper. Overall, for electricity consumption, SARIMA is the preferred model while for gas consumption; TBATS (a seasonal trigonometric forecast model) performs better than the alternative models considered. In view of observed seasonality and non-linearity in the datasets, electricity consumption will continue its decline over the forecast horizon and it is unlikely to return to its pre-COVID-19 levels, while gas consumption would stabilise over the forecast duration.

Table 13.

Summary of major findings.

	Model	Key findings
1	Benchmark Methods	For the benchmark forecast method, the seasonal-naive model outperforms alternative specifications for electricity and gas consumption in both the training and test sets. This result underscores the need to account for seasonality.
2	Exponential Smoothing	The Holt-Winters Seasonal (with additive seasonality) model outperforms alternative exponential smoothing models for electricity and gas consumption in the training sets while Holt-Winters Seasonal (with multiplicative seasonality) outperforms alternative specifications in the test set for electricity but not for gas consumption. The state space (ETS) model outperforms all exponential smoothing specifications for both electricity and gas consumption in both the training and test sets. In addition, it outperforms the neural network as its forecast errors are lower.
3	SARIMA	For electricity consumption, SARIMA(3,1,3)(2,1,2)₁₂ is the preferred model amongst competing models in both the training and test sets. The model is superior to the benchmark and exponential-smoothing methods. For gas consumption, the result indicates that for both the training and test sets, SARIMA(1,1,1)(0,1,1)₁₂ is the preferred specification.
4	Dynamic Regression	The preferred SARIMA model outperforms the three main dynamic regression specifications for both electricity and gas consumption. In addition, the multiple linear regression specification with trend, seasonal dummies and predictor variables (i.e., MLR2) outperforms all other models including the harmonic regression.
5	ANN	The ANN model is superior to all methods earlier considered. However, for electricity consumption, SARIMA outperforms the ANN.
6	SVR	For both electricity and gas consumption, ANNs are superior to the SVR forecast results.
7	Combination	Findings suggest that combination forecasting approach does not always improve forecast accuracy.

Conclusion and policy implications

Projecting a country's energy consumption can help to determine the requisite capacity for future energy supply. This is particularly important given the gradual post-COVID-19 recovery, post-Brexit impacts, and more recently, heightened uncertainty due to the Russia–Ukraine turmoil. The key to developing more accurate energy consumption forecasts is building dynamic models that account for seasonality, asymmetries and the underlying predictors of energy consumption. This is expected to improve energy consumption forecast accuracy which could in turn aid better energy policy planning and implementation. This would also provide useful information that can help energy stakeholders such as energy companies, households and governments make informed consumption, strategic investment and regulatory decisions. Moreover, because of the UK government's target of cutting CO₂ emissions, reliable energy consumption predictions particularly for electricity and gas could improve the understanding of consumption patterns and mitigate potential supply-side binding constraints.

In this paper, we subject the UK's energy consumption to further scrutiny using a broad range of methods to generate in-sample and out-of-sample forecasts. The empirical exercise reveals that the underlying energy consumption series exhibit seasonality and persistence. The forecast performance evaluation of the models shows that the SARIMA model outperforms other models in predicting electricity consumption, while TBATS outperforms both SARIMA and machine learning models for gas consumption forecasting. Further analysis prompted the following findings: (i) for the regression models, the MLR specification with trend, seasonal dummies and predictor variables (i.e., MLR₂) outperforms all other models including the harmonic regression; (ii) for exponential smoothing technique, the ETS specification performs better than all the other specifications within the model class; (iii) based on the three dynamic regression models considered, the QREG-ARIMA model outperforms other specifications for electricity consumption in terms of the in-sample forecasts while for gas consumption, the dynamic harmonic regression model outperforms other specifications for the out-of-sample forecasts within its model class; (iv) surprisingly, the combination predictions yield higher forecast errors; and (v) the machine learning models significantly outperform the statistical and regression models, while seasonal decomposition/hybrid models such as the TBATS outperform both statistical and regression specifications.

The quantitative analysis provides some useful insights for policy consideration. The outcome of the simulations conducted is expected to help improve the design of more effective energy policies in the UK. This is particularly important against the backdrop of gradual recovery from the COVID-19 lockdown measures and the mounting energy bills and fuel prices magnified by the Russia–Ukraine crisis. Therefore, it is imperative to consider adopting policies that support households while taxes on intermediates used in electricity and gas sectors could be reduced to sustain growth and minimise pressure occasioned by energy supply disruptions, especially in Europe. As energy costs rise and are likely to remain so over the short to medium term, intensifying energy efficiency could help mitigate the energy consumption deficit. Also, in line with the UK's net-zero target, measures that support clean energy use such as a temporary reduction of VAT on hydro turbines, solar panels and heat pumps could be considered to maintain steady-state energy consumption. This could also be complemented by tax exemptions for renewable energy firms due to the possible substitution effect.

In conclusion, the implementation of the UK energy strategy^f would be critical to maintaining a balanced interplay between electricity and gas consumption by economic agents over the forecast horizon whilst gradually shifting towards renewable energy. This draws from the finding that electricity consumption is likely to maintain its current trend or even dip in the absence of appropriate policy measures while gas consumption is likely to increase or stabilise over the forecast horizon which largely conforms with the increasing pursuit of environmental quality and sustainable green growth through the net-zero target of the UK government. This paper is however not without some limitations. For example, we use monthly energy consumption time series; yet, we acknowledge that using higher frequency data could be considered in future studies as it could shed more granular insights on energy consumption pattern and structure over time. It would also be interesting to investigate the dynamics of energy consumption in the home nations that make up the UK specifically and, in the EU due to heterogeneities and increased energy interdependence. The use of new approaches that reflect recent dynamics and that can provide more robust strategies for combining predictors could also be explored.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Mohammed Shuaibu

Notes

Dahiru A Bala currently works for the Federal Inland Revenue Service, Nigeria. He received his BSc in Economics from University of Abuja, MSc in Economics from University of Ibadan (Nigeria), MSc in Data Science and Analytics from The University of Nottingham, United Kingdom and PhD in Economics from Kyushu University, Fukuoka City, Japan (Major: Financial Econometrics and Macroeconometrics). His research interests span the fields of Financial Econometrics and Macroeconometrics, Revenue Forecasting and Tax Policy Analysis, and Data Science and Analytics, Machine Learning and Economic Forecasting, amongst others.

Mohammed Shuaibu is a Senior Lecturer in Economics at the University of Abuja. He obtained his PhD in Economics from the University of Ibadan, Nigeria. His research interests cover international and development economics. He has published extensively in peer-reviewed journals such as: International Economics, Economic Change and Restructuring, African Development Review, Journal of African Trade, Foreign Trade Review, Scientific African. He has provided research and consultancy services for various domestic and international organisations.

Appendix

See Tables A1 and A2.

References

Hallin

Hooper

Weyman-Jones

. Case study evidence and behavioural analysis of residential energy consumption in the UK. Open J Energy Effic 2017; 6: 14–40.

Amber

Aslam

Hussain

, Electricity consumption forecasting models for administration buildings of the UK higher education sector. Energy Build 2015; 90: 127–136.

Barassi

Zhao

. Combination forecasting of energy demand in the UK. Energy J 2018; 39: 209–236.

Beyca

Ervural

Tatoglu

, et al. Using machine learning tools for forecasting natural gas consumption in the province of Istanbul. Energy Econ 2019; 80: 937–949.

Granell

Axon

Kolokotroni

, et al. A data-driven approach for electricity load profile prediction of new supermarkets. Energy Procedia 2019; 161: 242–250.

UK Energy Brief. UK Energy in Brief, https://www.gov.uk/government/statistics/uk-energy-in-brief-2020. Accessed July 20, 2022. 2020.

Department for Business, Energy and Industrial Strategy, Digest of United Kingdom Energy Statistics (DUKES), 28th May, 2020.

Izidio

DMF

Neto

PSGM

Barbosa

, et al. Evolutionary hybrid system for energy consumption forecasting for smart meters. Energies 2021; 14: 1–19.

Wang

Zhu

Sun

. A trend fixed on firstly and seasonal adjustment model combined with the ε-SVR for short-term forecasting of electricity demand. Energy Policy 2009; 37: 4901–4909.

10.

Baker

Rylatt

. Improving the prediction of UK domestic energy- demand using annual consumption data. Appl Energy 2008; 85: 475–482.

11.

Hor

Watson

Majithia

. Daily load forecasting and maximum demand estimation using ARIMA and GARCH. In: 9th International conference on probabilistic methods applied to power systems, June 11–15, KTH Stockholm, Sweden, 2006.

12.

Cancelo

Espasa

Grafe

. Forecasting electricity load from one day to one week ahead for the spanish system operator. Int J Forecast 2008; 24: 588–602.

13.

Williams

Short

. Electricity demand forecasting for decentralised energy management. Energy Built Environ 2020; 1: 178–186.

14.

Rana

Koprinska

Khosravi

. Feature selection for interval forecasting of electricity demand time series data. In: Artificial neural network. Midtown Manhattan, New York City, USA: Springer, 2015, pp. 445–462. https://doi.org/10.1007/978-3-319-09903-3_22

15.

González-Romera

Jaramillo-Morán

Carmona-Fernández

. Monthly electric energy demand forecasting with neural networks and Fourier series. Energy Convers Manage 2008; 49: 3135–3142.

16.

Berriel

Lopes

Rodrigues

, et al. Monthly energy consumption forecast: a deep learning approach. In: Conference paper, International joint conference on neural networks. May, Alaska, USA, 2017.

17.

Bonetto

Rossi

. Machine learning approaches to energy consumption forecasting in households. 2017, June 6–9, 2017.

18.

Hong

W-C

. Intelligent energy demand forecasting. London/UK: Springer, 2013. https://doi.org/10.1007/978-1-4471-4968-2

19.

Hyndman

Athanasopoulos

. Forecasting: Principles and practice. 2nd edition. Melbourne, Australia: OTexts, 2018; Otexts.com/fpp2 (retrieved on 26 June 2020).

20.

OFGEM. State of the energy market report. 2019. www.ofgem.gov.uk.

21.

Energy Information Administration (EIA). Country analysis brief: United Kingdom. 2018.

22.

Hor

Watson

Majithia

. Analysing the impact of weather variables on monthly electricity demand. IEEE Trans Power Syst 2005; 20: 2078–2085.

23.

Ardakanian

Koochakzadeh

Singh

, et al. Computing electricity consumption profiles from household smart meter data. In: Workshop proceedings of the EDBT/ICDT 2014 joint conference, March 28, Athens, Greece, 2014.

24.

Taylor

. Triple seasonal methods for short-term electricity demand forecasting. Eur J Oper Res 2010; 204: 139–152.

25.

Abdel-Aal

Al-Garni

. Forecasting monthly electric energy consumption in eastern Saudi Arabia using univariate time series analysis. Energy 1997; 22: 1059–1069.

26.

Gassar

AAA

Yun

Kim

. Data-driven approach to the prediction of residential energy consumption at urban scales in London. Energy 2019; 187: 1–13.

27.

Bishop

. Pattern recognition and machine learning. New York/USA: Springer, 2006.

28.

Oğcu

Demirel

Zaim

. Forecasting electricity consumption with neural networks and support vector regression. Procedia – Social Behav Sci 2012; 58: 1576–1585.

29.

De Felice

Alessandri

Catalano

. Seasonal climate forecasts for medium-term electricity demand forecasting. Appl Energy 2015; 137: 435–444.

30.

Wang

Zhao

, et al. Application of residual modification approach in seasonal ARIMA for electricity demand forecasting: a case of China. Energy Policy 2012; 48: 284–294.

31.

Yan

, et al. A hybrid LSTM neural network for energy consumption forecasting of individual households. IEEE Access 2019; 7: 157633–157642.

32.

Jiang

Liu

, et al. A novel composite electricity demand forecasting framework by data processing and optimized support vector machine. Appl Energy 2020; 260: 114243.

33.

Neto

PSGM

Oliveira

JFL

Bassetto

, et al. Energy consumption forecasting for smart meters using extreme learning machine ensemble. Sensors 2021; 21: 1–18.

34.

Enders

. Applied econometric time series. 4th edition. USA: John Wiley & Sons Inc, 2015.

35.

Chatfield

. Time-series forecasting. USA: Chapman and Hall/CRC, 2000.

36.

De Livera

Hyndman

Snyder

. Forecasting time series with complex seasonal patterns using exponential smoothing. J Am Stat Assoc 2011; 106: 1513–1527. https://robjhyndman.com/publications/complex-seasonality/

37.

Chatfield

. The analysis of time series: An introduction. 6th edition. USA: Chapman and Hall/CRC, 2003.

38.

Vapnik

. The nature of statistical learning theory. New York, USA: Springer, 1995.

39.

Hyndman

Koehler

. Another look at measures of forecast accuracy. Int J Forecast 2006; 22: 679–688.

40.

Kourentzes

Barrow

Crone

. Neural network ensemble operators for time series forecasting. Expert Syst Appl 2014; 41: 4235–4244.

41.

Al-Qahtani

Crone

. Multivariate k-nearest neighbour regression for time series data – A novel algorithm for forecasting UK electricity demand. In: The 2013 International Joint Conference on Neural Networks (IJCNN), Dallas, TX, USA, 2013, pp. 1–8, doi: 10.1109/IJCNN.2013.6706742

42.

Kolokotroni

Davies

Croxford

, et al. A validated methodology for the prediction of heating and cooling energy demand for buildings within the urban heat island: case-study of London. Sol Energy 2010; 84: 2246–2255.