# Data-driven hybrid SARIMAX-MLP framework for energy consumption prediction in residential micro-grid

Ibrahim Ali Kachalla, Christian Ghiaus, Adeniran Ademuwagun, Olufemi

Babajide Odeyinde, Majid Baseer

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Ibrahim Ali Kachalla, Christian Ghiaus, Adeniran Ademuwagun, Olufemi Babajide Odeyinde, Majid Baseer. Data-driven hybrid SARIMAX-MLP framework for energy consumption prediction in residential micro-grid. Results in engineering, 2025, 26, pp.105336. 10.1016/j.rineng.2025.105336. hal-05084514

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Submitted on 26 May 2025

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Research paper

# Data-driven hybrid SARIMAX-MLP framework for energy consumption prediction in residential micro-grid

Ibrahim Ali Kachalla<sup>a, ID, *</sup>, Christian Ghiaus<sup>a</sup>, Adeniran Ademuwagun<sup>b</sup>, Olufemi Babajide Odeyinde<sup>c</sup>, Majid Baseer<sup>a</sup>

## ARTICLE INFO

### Keywords:

Endogenous and exogenous factors

Energy consumption

Hot water consumption

Hybrid SARIMAX-MLP

Explainable Artificial Intelligence (XAI)

Model Predictive Control (MPC)

## ABSTRACT

With electric water boilers (EWBs) energy consumption increasing by over 30% due to global warming, an accurate load demand prediction model incorporating exogenous factors, such as hot water consumption, seasonal variations, and time of day, can mitigate energy efficiency, energy resilience, and optimisation challenges in micro-grid management. This, in turn, reduces dependence on carbon-intensive grids and enhances model predictive control (MPC) strategies for EWBs scheduling in residential applications. This study proposes a hybrid SARIMAX-MLP model that integrates exogenous variables to improve energy consumption forecasting. While SARIMAX captures linear dependencies and seasonal trends, MLP model's complex nonlinear relationships that SARIMAX cannot estimate. A case study of two residential blocks, with one year six months (18 months) of energy consumption data, was utilised to evaluate the model's prediction accuracy using performance metrics (RMSE, MAE, R²) and computational cost. The proposed hybrid model was compared with statistical learning models (ARIMA, ARIMAX, SARIMAX), machine learning models (MLP, SVR), and other hybrid approaches (ARIMA-MLP, ARIMA-SVR, SVR-MLP). Results show that SARIMAX-MLP outperforms both single and hybrid models, offering higher accuracy and robustness across different time horizons. SHapley additive exPlanations (SHAP) analysis depicts hot water consumption as the most influential variable, subsequently hour of the day. Regardless of its higher computational cost, the SARIMAX-MLP model achieves a balanced trade-off between accuracy and computation efficiency, making it suitable for real-time energy management and MPC applications residential micro-grid. This study highlights the importance of exogenous variables in predictive modelling and suggests future exploration of deep learning architectures, such as 1D CNNs and Transformers, to further enhance forecasting performance and computation efficiency.

## 1. Introduction

The building sector accounts for 36% of total global energy consumption and 40% of associated carbon dioxide emissions. Therefore, accurately predicting energy consumption plays a key role in energy management applications, such as implementing intelligent control strategies and demand-side management, while promoting energy efficiency measures. According to the International Energy Agency (IEA), the global energy demand in residential dwellings is expected to increase more than 40% by 2024 [1,2]. However, water heating remains a significant contributor to residential energy consumption, accounting for 25% of total residential energy use in 2022. Furthermore, Fig. 1 illustrates

that energy consumption of electric water boilers (EWBs) in residential dwellings is projected to increase by 2% annually, reaching approximately 30% of total residential energy consumption by 2030. The 30% rise in EWB energy consumption is primarily attributed to increased ambient temperature and humidity. Higher temperatures reduce the cooling effect of incoming water, requiring more energy to reach the desired heating setpoints, while also contributing to increased hot water demand by occupants.

Integrating classical regression methods with machine learning has led to more accurate hybrid load demand models, particularly in residential applications for grid demand response management and control scheduling [3–6]. These models combine the strengths of both

<footnote><sup>a</sup> National Institute of Applied Sciences of Lyon, CETHIL UMR5008, INSA Lyon, University of Lyon, F-69621 Villeurbanne, Lyon, France</footnote>

<footnote><sup>b</sup> Electrical and Electronics Engineering Department, Air Force Institute of Technology Kaduna, Air Force Base, Kaduna, Nigeria</footnote>

<footnote><sup>c</sup> School of Computer Information Sciences, University of the Cumberland, 6178 College Station Drive, Williamsburg, 40769, KY, USA</footnote>

<footnote>* Corresponding author.
E-mail address: ibrahim-ali.kachalla@insa-lyon.fr (I.A. Kachalla).</footnote>
Nomenclature and Abbreviations

Nomenclature

B Backshift operator (e.g., By_t = y_{t-1})

H Forecast horizon

L Number of layers in MLP model

N Total number of input features

T Length of the time series

X_{t-k} Exogenous variable at lag k

X_{\text{hotwater}} Hot water consumption feature

X_{\text{hours}} Hour of the day feature

X_{\text{days}} Day of the week feature

e_t Residual error at time t, e_t = y_t - \hat{y}_t

f(x) Predictive model function

h^{(l)} Activation output of layer l in MLP

W^{(l)} Weight matrix of layer l in MLP

b^{(l)} Bias term of layer l in MLP

y_t Actual observed energy consumption at time t

$\hat{y}_t$ Predicted energy consumption at time t

$\hat{y}_{\text{Hybrid},t}$ Prediction from hybrid model at time t

$\hat{y}_{\text{MLP},t}$ Prediction from MLP at time t

$\hat{y}_{s,t}$ Prediction from SARIMAX at time t

$\alpha$ Weight assigned to SARIMAX in hybrid model

$\beta$ Weight assigned to MLP in hybrid model

$\alpha_t$ Coefficient of AR (autoregressive) term

$\beta_j$ Coefficient of MA (moving average) term

$\theta_k$ Coefficient of exogenous variable k

$\epsilon_t$ White noise or error term

Abbreviations

AIC Akaike Information Criterion

ANN Artificial Neural Network

ARIMA AutoRegressive Integrated Moving Average

ARIMAX ARIMA with Exogenous Inputs

BIC Bayesian Information Criterion

DoW Days of the Week

EWBs Electric Water Boilers

HoD Hour of the Day

MA Moving Average

MAE Mean Absolute Error

MLP Multi-Layer Perceptron

MPC Model Predictive Control

PACF Partial Autocorrelation Function

RMSE Root Mean Squared Error

SARIMA Seasonal AutoRegressive Integrated Moving Average

SARIMAX SARIMA with Exogenous Inputs

SHAP Shapley Additive Explanations

SVR Support Vector Regression

WHW Work Hour Weekend

<figure><img src="image_2.png" /><figcaption>Fig. 1. Energy consumption projection of EWBs in residential dwellings from IEA data.</figcaption></figure>

approaches to enhance hot water consumption predictions [6–8]. However, they can be complex to implement and interpret, often facing overfitting issues, where the model performs well on training data but poorly on test data [9]. Research over the years has shown that exogenous variables can improve accuracy and mitigate overfitting. Elias et al. [10] demonstrated that incorporating exogenous variables into ARIMA and NAR models significantly improved predictions, particularly for NARX, which responded faster to operational disturbances. Additionally, additive and multivariate hybrid models captured both linear and nonlinear variations, enhancing model stability during full-load operation [10].

Manar Abu et al. [7], analysed the impact of COVID-19's on hot water prediction demand, considering lockdown effects and exogenous factors

such as previous consumption (PC), socio-demographic (SDF), seasonal (SF), and climatic (CF) features. Using metering data from 128,000 consumers in Dubai, they tested multiple ensemble models (GBM, XGB, Light GBM, RF, STK) against traditional MLR, decision trees, and neural networks. Feature importance analysis showed PC (0.95) and SDF (0.74) had the highest impact, while SF and CF had negligible effects. Before COVID, RF and STK performed best ($R^2$ = 0.928, RMSE = 0.039), while during COVID, MLR was most accurate ($R^2$ = 0.90, RMSE = 0.05). An ensemble-based error prediction model further improved prediction by 9.2%.

Hyeongseok et al. [11] developed linear-form models, including ARX for indoor temperature prediction and threshold piecewise models for return and supply water temperatures. Evaluating MPC performance un-
<figure><img src="image_3.png" /><figcaption>Fig. 2. Integration of EWB prediction model into the MPC controller framework.</figcaption></figure>

der different prediction horizons and weightings, a residential case study showed the models achieved goodness-of-fit 0.9. The resulting MPC framework reduced the use of water heating energy use by 12% compared to basic thermostats and comfort violation magnitude by less than 0.5 °C [11]. However, most study did not consider hot water usage patterns as an exogenous factor, which could significantly influence energy consumption. This research aims to address this gap by incorporating exogenous variables and evaluating model accuracy and computational efficiency using performance metrics such as RMSE, MAE, and the coefficient of determination.

The proposed model aims to accurately predict and forecast residential energy consumption patterns based on occupants behaviour to hot water consumption over time, considering seasonal variations, time of day, and days of the week. This will improve energy efficiency management through model predictive control (MPC) strategies, optimising energy distribution between the PV-grid and the national grid, with priority given to the PV-grid based on the EWB model's energy consumption predictions. Furthermore, the novel prediction model will forecast EWB consumption based on user behaviour, determining when energy is needed by occupants. When energy is not required, excess energy from the PV-grid will be injected into the national grid [11–14]. These control strategy requires a accuracy and robust model, which is crucial for enhancing effective MPC strategies for the management and scheduling of response on the demand side, ultimately reducing the dependency on the national grid, which is associated with higher carbon emissions [15–18], as illustrated in Fig. 2.

Fig. 2 depicts the framework application, integrating a model predictive controller (MPC) with a weather forecast model, Photovoltaic grid, and utility grid supply (carbon-based grid) to optimise electric water boiler (EWB) operations. It leverages the EWBs prediction/demand model and energy, and hot water data repository to enhance energy efficiency and demand forecasting in a microgrid, with a priority on reducing dependency on the carbon-based grid supply.

This research aims to address the gap in accurately predicting energy consumption for domestic hot water boilers using a novel hybridised machine learning technique. The proposed model will contribute to the holistic objectives of energy demand and management by developing robust prediction models that incorporate exogenous factors to analyse their impact on EWB consumption.

The remainder of this article is structured as follows, from the subsection for review of related work on the energy consumption prediction model using endogenous and exogenous inputs; the methodology section outlines the development of the novel hybrid model; the results and discussion section presents a comparative analysis of the proposed model against single and hybrid models; the research further explores opportunities for further advancements in predictive modelling and energy management application; and finally, the conclusion summarises the key findings and implications of this study.

## 1.1. Review of related work

Over the years, research on modelling and forecasting the energy consumption of hot water boilers has primarily focused on endogenous variables (dependent variables) that represent the target energy consumption to be predicted [4,9,19]. However, limited or no attention has been given to exogenous variables (independent variables) such as temperature, house size, climate, hot water consumption, region, time and day of use, seasonality, and demographics in the prediction of electric water boiler (EWB) energy consumption [6,14,20]. These external factors significantly impact the accuracy and robustness of the prediction models energy consumption. Traditional and statistical prediction techniques, including Monte Carlo simulations [21,22], probabilistic methods [23], and statistical learning and regression models (e.g., ARIMA, ARMA, exponential smoothing) [9,24,25], are commonly used. However, these methods only estimate or predict endogenous variables and have significant limitations in integrating external feature labels as exogenous inputs, making them less accurate as compared to machine learning models.

In recent years, machine learning models have emerged as more advanced alternatives, not only predicting endogenous variables but also effectively capturing exogenous factors as feature labels, leading to improved predictive accuracy and robustness. Kachalla et al. [9] compared single models, specifically two supervised learning algorithms, support vector regression (SVR) and multi-layer perceptron (MLP) artificial neural network (ANN), with a regression-based forecasting technique (ARIMA) for predicting energy consumption in electric water boilers. Their research also evaluates five different endogenous input forecast scenarios, with time steps ranging from 10-minute intervals to hourly, daily, and seasonal energy consumption over six months. Results indi-
<table><caption>**Table 1**
Comparison of approaches for EWBs energy prediction.</caption><thead><tr><th>Approaches</th><th>Hybrid</th><th>Exogenous</th><th>Endogenous</th><th>Authors</th></tr></thead><tbody><tr><td>Single</td><td></td><td></td><td></td><td></td></tr><tr><td>ARX</td><td></td><td>IT</td><td>Energy</td><td>Hyeonseok Lee & Heo., [11]</td></tr><tr><td>ARIMA, ANN, SVR</td><td></td><td></td><td>Energy</td><td>Kachalla et al., [9]</td></tr><tr><td>ANN, ARIMA, CNN, LR, LSTM, MLR, SVM</td><td>SARIMA-ANN, SARIMA-SVR</td><td></td><td>Energy</td><td>Mohamad Khalil et al., [26]</td></tr><tr><td>MLR</td><td></td><td>TEC, HW, AT, RH, SI</td><td>Energy factor</td><td>Kachalla & Ghiaus, [4]</td></tr><tr><td>LSTM</td><td></td><td></td><td>Energy</td><td>Stephen Tangwe et al., [20]</td></tr><tr><td>XGBoost</td><td>Prophet-GRU</td><td>HW, EC</td><td>Energy</td><td>Weicong Kong et al., [27]</td></tr><tr><td>MLR, STK, MED, XGB, GBM</td><td>SVR-MLP, ARIMA-MLP</td><td>ODT, HCF, CHD, HoD, WHW</td><td>Energy</td><td>Chunyan Wang et al., [8]</td></tr><tr><td></td><td>ANN-LSTM</td><td>SDF, SF, CF</td><td>Energy</td><td>Etienne Saloux et al., [28]</td></tr><tr><td>UV-LSTM</td><td></td><td></td><td>Energy</td><td>Manar Abu Talib et al., [7]</td></tr><tr><td>ANN, ARIMA, MLP, SVR, SARIMAX</td><td>SARIMAX-MLP, ARIMA-MLP, ARIMA-SVR, SVR-MLP</td><td>CF</td><td>Energy</td><td>Kachalla et al., [25]</td></tr><tr><td></td><td></td><td>HoD, HW, DoW</td><td>Energy</td><td>Xin Zhou et al., [29]</td></tr><tr><td></td><td></td><td></td><td></td><td>Azar Niknam et al., [30]</td></tr><tr><td></td><td></td><td></td><td></td><td><strong>Proposed method</strong></td></tr></tbody></table>

IT: Indoor temperature, TEC: Total energy, HW: Hot water, AT: Ambient temperature, RH: Relative humidity, SI: Solar irradiance, MLR: Multiple linear regression, EC: Energy consumption, GRU: Gated recurrent unit, ODT: Outdoor dry air temperature, HCF: Heating cosine function, CHD: Cosine hour of the day, HoD: Hour of the day, DoW: Days of the week, WHW: Work hour weekend, SDF: Socio-demographic factors, SF: Seasonal factors, CF: Climatic factors, MEM: Multiple ensemble models, GBM: Gradient boosting machines, XGB: Extreme gradient boosting, RF: Random forest, STK: Stack regressor.

cated that SVR demonstrated moderate overall performance compared to ANN and ARIMA, optimal performance was achieved in the hourly scenario. This study highlighted the limitations of single models, emphasizing the need for hybridisation to leverage their combined strengths [9].

Building on this, Kachalla and Christian [25] investigated hybridised models by integrating SVR and ARIMA with MLP. Using more scalable data for one year and five months of domestic hot water consumption data, the study found that SVR-MLP achieved 94% accuracy in winter, effectively capturing irregularities and nonlinear dependencies in hot water demand, though at a higher computational cost of 3 minutes and 35 seconds. Meanwhile, ARIMA-MLP performed optimally in summer, achieving accuracy 98% with higher computational efficiency, making it ideal for stable daily seasonal variations and suitable for real-time grid control and scheduling strategies. Similarly, Kachalla et al. [4] further developed hybrid forecasting models by combining a seasonal auto-regressive integrated moving average (SARIMA) model with machine learning techniques, specifically SVR and ANN. Their findings indicated that SARIMA-ANN outperformed SARIMA-SVR in both predictive accuracy and computational efficiency. SARIMA-ANN's superior feature mapping capability across both short and long-term made it more suitable for model predictive control (MPC) applications and real-time implementations, particularly in renewable energy grid integration. SARIMA-ANN offered a more scalable and generalised solution than the hyperplane-dependent SARIMA-SVR model. However, these studies focused primarily on endogenous factors, limiting the generalisation and robustness of the models. These limitations underscore the need to incorporate exogenous variables to enhance predictive performance and address existing research gaps optimise model robustness.

Table 1 presents an overview of related works on energy consumption prediction models, highlighting the use of endogenous and exogenous variables in electric water boiler (EWB) applications. In addition, various modelling approaches used to predict energy consumption have been presented in the literature, and the paradigm shift presents a change in predictive modelling, emphasizing the integration of exogenous variables to address the limitations of traditional forecasting techniques. The proposed SARIMAX-MLP model stands out by incorporating multiple exogenous factors such as hour of the day, hot water use, and days of the week as input features to the hybrid machine learning and statistical model. This further highlights the contribution to the advancement of predictive modelling for energy optimisation and grid management.

## 2. Novel hybridised SARIMAX-MLP model design

Research has shown that SARIMAX, a class of autoregressive models, is the most robust for handling seasonality and exogenous inputs compared to other linear regression techniques such as ARIMA, ARIMAX, and exponential smoothing. Meanwhile, the MLP model is well-suited for capturing non-linear relationships, as it can process high-dimensional feature labels as exogenous variables and offers better generalisation compared to models like LSTM and RNN. Furthermore, Table 2 highlights the strengths and weaknesses of these models.

The proposed hybridisation approach integrates both sequential and parallel hybrid modelling strategies by combining the strengths of the SARIMAX and MLP models. As shown in Fig. 3, the SARIMAX model is first trained on endogenous and exogenous input variables to generate baseline forecasts ($\hat{y}_{f,t}$) and corresponding residuals ($f_t = y_t - \hat{y}_{f,t}$). These residuals capture complex, nonlinear patterns and stochastic behaviours in energy consumption that are not fully captured by SARIMAX alone. To model these residuals, the MLP is employed using a set of exogenous inputs, including the SARIMAX residuals, hours of the day, days of the week, and hot water consumption. These inputs serve as feature labels to enable the MLP to learn and forecast the non-linear components of the energy consumption signal, producing its own prediction output $\hat{z}_{\text{fore},t}$. Finally, the outputs of both models are integrated using a weighted sum:

$$ \hat{y}_{\text{adj},t} = w_1 \cdot \hat{y}_{f,t} + w_2 \cdot \hat{z}_{\text{fore},t} \quad (1) $$

This fusion of SARIMAX's linear predictions with MLP's non-linear learning capabilities enhances the overall prediction accuracy and robustness. The hybrid SARIMAX-MLP framework effectively captures both structured trends and behavioural variabilities in Electric Water Boiler (EWB) consumption, making it well-suited for predictive control applications.

### 2.1. SARIMAX model development

The SARIMAX configuration effectively models both short-term fluctuations and daily seasonality, ensuring accurate forecasting while maintaining computational efficiency. The optimal SARIMAX configuration (1, 0, 1), (1, 0, 1, 24), derived from empirical analysis, balances model complexity and accuracy while capturing both short-term dependencies and daily seasonality. The non-seasonal parameters (1, 0, 1) consist of
<figure><img src="image_4.png" /><figcaption>Fig. 3. SARIMAX-MLP hybrid methodology framework.</figcaption></figure>
<table><caption>**Table 2**
Comparison of different prediction models and justification for SARIMAX-MLP hybridisation.</caption><thead><tr><th>Model</th><th>Strengths</th><th>Weaknesses</th><th>Justification for hybridisation</th></tr></thead><tbody><tr><td>ARIMA</td><td>Effective for linear time-series data.<br>Works well with stationary data.<br>Provides interpretable results.</td><td>Struggles with non-linearity.<br>Cannot incorporate exogenous variables.<br>Limited in capturing seasonal variations.</td><td>SARIMAX improves upon ARIMA by handling seasonality and exogenous inputs, while MLP compensates for ARIMA's inability to model non-linearity.</td></tr><tr><td>SARIMAX</td><td>Extends ARIMA by integrating exogenous variables.<br>Incorporates exogenous variables.</td><td>Limited in handling complex non-linear relationships.<br>Sensitive to parameter selection.</td><td>SARIMAX further refines ARIMAX by explicitly predicting seasonal variations, while MLP captures complex, non-linear dependencies.</td></tr><tr><td>SVR</td><td>Effective for small datasets.<br>Good generalization for certain applications.</td><td>Sensitive to hyperparameter selection.</td><td>SVR lacks seasonality predicting, which SARIMAX provides, and MLP enhances non-linear pattern recognition.</td></tr><tr><td>MLP</td><td>Capable of predicting complex, non-linear relationships.<br>Handles high-dimensional feature spaces.</td><td>Requires large training datasets.<br>Computationally intensive.</td><td>MLP alone does not explicitly model temporal dependencies or seasonality, making SARIMAX a suitable complementary model.</td></tr><tr><td>LSTM</td><td>Excellent for sequential data.<br>Captures long-term dependencies.</td><td>Computationally expensive.<br>Requires large datasets.</td><td>SARIMAX is computationally more efficient and interpretable for seasonality, while MLP offers flexibility in non-linear learning.</td></tr><tr><td>ARIMA-MLP</td><td>Combines statistical forecasting with non-linear learning.<br>Addresses ARIMA's linearity limitation.</td><td>ARIMA component still struggles with seasonal variations.<br>MLP requires careful tuning to avoid overfitting.</td><td>SARIMAX enhances ARIMA by explicitly capturing seasonality, while MLP retains the benefits of non-linear predicting.</td></tr><tr><td>ARIMA-SVR</td><td>Improves ARIMA's predictive performance with SVR's non-linear modelling.<br>SVR's performance highly depends on kernel selection.</td><td>ARIMA still lacks seasonal adaptation.</td><td>SARIMAX provides a more robust seasonal component than ARIMA, making it a better hybrid choice with MLP.</td></tr><tr><td>SVR-MLP</td><td>Strong for non-linear relationships.<br>SVR prevents overfitting common in neural networks.</td><td>Lacks explicit time-series seasonality predicting.<br>High computational cost.</td><td>SARIMAX-MLP provides explicit seasonality predicting, which SVR-MLP lacks.</td></tr><tr><td>SARIMAX-MLP</td><td>Captures seasonality and exogenous inputs (SARIMAX).<br>Handles non-linearity and high-dimensional features (MLP).<br>Provides improved forecasting accuracy.</td><td>Computationally expensive.<br>Requires extensive hyperparameter tuning.</td><td>The combination of SARIMAX and MLP offers a balanced model that handles seasonality, exogenous variables, and non-linear patterns, making it ideal for energy consumption forecasting.</td></tr></tbody></table>

one autoregressive term ($p = 1$), no differencing ($d = 0$), and one moving average term ($q = 1$), selected based on empirical analysis of the autocorrelation function (ACF) and partial autocorrelation function (PACF), which guide the identification of appropriate autoregressive (AR) and moving average (MA) terms [4]. The seasonal parameters (1, 0, 1, 24) include one seasonal autoregressive term ($P = 1$), no seasonal differencing ($D = 0$), one seasonal moving average term ($Q = 1$), and a periodicity of 24, capturing daily cycles in hourly data. This configuration ensures the model effectively incorporates both short-term and long-term seasonal trends for improved forecasting accuracy. The models implementation was carried out using python programming and the SARIMAX model is mathematically formulated as:

$$y_t = \alpha_0 + \sum_{i=1}^{p} \alpha_i y_{t-i} + \sum_{j=1}^{q} \beta_j \epsilon_{t-j} + \sum_{k=1}^{r} \theta_k X_{t-k} + e_t \quad (2)$$

where $y_t$ represents the energy consumption at time $t$, $p$ is the autoregressive (AR) order, $q$ is the moving average (MA) order, $r$ represents the

number of exogenous variables included in the SARIMAX model, $\alpha_i$, $\beta_j$, and $\theta_k$ are model coefficients, $X_{t-k}$ represents the exogenous variables (hour of the day, day of the week, hot water consumption), and $e_t$ is the white noise error term at time $t$.

The SARIMAX model not only captures linear dependencies in time-series data but also integrates the influence of exogenous variables. The residual error $e_t$ in a SARIMAX model is defined as:

$$e_t = y_t - \hat{y}_t \quad (3)$$

where $\hat{y}_t$ is the predicted value at time $t$, incorporating contributions from autoregressive terms $\phi(B)y_{t-1}$, moving average terms $\theta(B)\epsilon_t$, seasonal components $\Phi(B^s)y_{t-s}$, and exogenous variables $\beta X_t$. Seasonal components $\Phi(B^s)y_{t-s}$ capture repeating patterns in time-series data by incorporating seasonal autoregressive terms at specified lags. Grid search and information criteria, such as Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), are used to optimise these parameters by selecting the best-fitting model while minimising
complexity and avoiding over-fitting. Using the backshift operator B and seasonal period s = 24, the residual expression is:

$$e_t = y_t - (\phi(B)y_t + \theta(B)\epsilon_t + \Phi(B^S)y_{t-s} + \beta X_t) \quad (4)$$

This formulation ensures the SARIMAX model effectively captures both temporal dependencies and external influences, making it well-suited for energy consumption forecasting in residential applications [4,7].

## 2.2. Multi-layer perceptron (MLP) model development

The multilayer perceptron (MLP) operates on the principle of forward propagation, where input data is passed through multiple layers of interconnected neurons, and back propagation, which adjusts weights based on error minimisation to improve predictions. Its ability to capture complex, non-linear relationships in data makes it particularly effective for modelling patterns that are not easily represented by traditional statistical methods. The MLP is significant as it complements traditional models by enhancing the prediction accuracy of energy consumption in electric water boilers, addressing non-linear dependencies and improving energy management strategies in residential applications [4].

The MLP design adopted, after fitting the SARIMAX model, takes the residuals as input to the MLP model. This allows the MLP to learn and predict nonlinear patterns that the SARIMAX model may not capture. Furthermore, the MLP model takes energy consumed as the endogenous (target) variable, and the exogenous inputs are the SARIMAX residual, hours of the day, days of the week, and hot water consumption. The MLP models the non-linear relationships between these exogenous variables and the residual errors from SARIMAX to improve the accuracy of energy consumption predictions. The inputs to the MLP can be represented as the vector:

$$X_{\text{MLP}} = [e_{\text{SARIMAX}}, X_{\text{hours}}, X_{\text{days}}, X_{\text{hotwater}}] \quad (5)$$

where $X_{\text{MLP}}$ represents the predicted energy consumption from the MLP model, $e_{\text{SARIMAX}}$ is the residual from the SARIMAX model, $X_{\text{hours}}$ is the “hours of the day” feature, $X_{\text{days}}$ is the “days of the week” feature, and $X_{\text{hotwater}}$ is the “hot water consumption” feature. Additionally, $W^{(l)}$ and $b^{(l)}$ represent the weights and biases for the l-th layer of the MLP.

The architecture of the MLP is:

$$h^{(l)} = f(W^{(l)}h^{(l-1)} + b^{(l)}) \quad (6)$$

where $h^{(l)}$ represents the activations in the l-th layer, $W^{(l)}$ is the weight matrix for layer l, and $b^{(l)}$ is the bias term for layer l. The first hidden layer is connected to the input vector $X_{\text{MLP}}$:

$$h^{(1)} = f(W^{(1)}X_{\text{MLP}} + b^{(1)}) \quad (7)$$

The output of the MLP is the predicted energy consumption, expressed as:

$$y_{\text{MLP}} = W^{(L)}h^{(L-1)} + b^{(L)} \quad (8)$$

where $y_{\text{MLP}}$ represents the predicted energy consumption produced by the final layer (L) of a Multilayer Perceptron (MLP). Here, $W^{(L)}$ and $b^{(L)}$ are the weight matrix and bias vector of the final layer, while $h^{(L-1)}$ denotes the activation output from the previous layer. These components collectively form a weighted linear combination adjusted by biases to generate the prediction. The overall MLP prediction is expressed as:

$$y_{\text{MLP}} = W^{(L)} \cdot f(W^{(L-1)} \cdot f(\dots f(W^{(1)}X_{\text{MLP}} + b^{(1)})) + b^{(L-1)}) + b^{(L)} \quad (9)$$

where, $y_{\text{MLP}}$ is the predicted energy consumption as a function of the SARIMAX residuals and the exogenous inputs (hours of the day, days of the week, and hot water consumption). The weights $W$ and biases $b$ are

learned during training through back propagation and gradient descent, capturing the remaining non-linear dynamics in the data.

## 3. Hybridisation of SARIMAX and MLP prediction model

To enhance predictive performance, the residuals from the SARIMAX model were used as input features for the MLP. These residuals represent the portion of the time series not captured by the SARIMAX component, often due to nonlinear dynamics or stochastic variations in user behaviour. By incorporating them as inputs to the MLP, the architecture is explicitly guided to learn from and correct these systematic errors. This sequential hybridisation strengthens the model's ability to represent complex consumption dynamics and improves the accuracy of final predictions.

The hybridisation of SARIMAX and MLP for prediction was implemented as the weighted sum by combining the predictions of both models to capture both linear and nonlinear variables. Capturing both linear and non-linear variables through the hybridisation of SARIMAX and MLP enhances predictive accuracy by leveraging SARIMAX for temporal dependencies and seasonality while utilising MLP to model complex, nonlinear relationships in the data. The adoption of a weighted sum in the hybrid SARIMAX-MLP model is significant because it allows the integration of SARIMAX strength in modelling linear and seasonal patterns with MLP's ability to capture complex non-linear relationships. This approach ensures a more comprehensive prediction framework, leveraging the complementary capabilities of both models to improve accuracy and robustness in energy consumption forecasting. The prediction $\hat{y}_{\text{Hybrid},t}$ from the hybrid SARIMAX-MLP model at time t is presented as the weighted sum of both models' predictions:

$$\hat{y}_{\text{Hybrid},t} = \alpha \cdot \hat{y}_{s,t} + \beta \cdot \hat{y}_{\text{MLP},t} \quad (10)$$

where $\alpha$ and $\beta$ are the weights of the SARIMAX and MLP models, respectively. These weights sum to 1, i.e., $\alpha + \beta = 1$, and are proportional to the final prediction. For a full time series of length T, the hybrid predictions are expressed as:

$$\hat{Y}_{\text{Hybrid}} = \alpha \cdot \hat{Y}_s + \beta \cdot \hat{Y}_{\text{MLP}} \quad (11)$$

where:

$$\hat{Y}_{\text{Hybrid}} = \begin{bmatrix} \hat{y}_{h,1} \\ \hat{y}_{h,2} \\ \vdots \\ \hat{y}_{h,T} \end{bmatrix}, \quad \hat{Y}_s = \begin{bmatrix} \hat{y}_{s,1} \\ \hat{y}_{s,2} \\ \vdots \\ \hat{y}_{s,T} \end{bmatrix}, \quad \hat{Y}_{\text{MLP}} = \begin{bmatrix} \hat{y}_{m,1} \\ \hat{y}_{m,2} \\ \vdots \\ \hat{y}_{m,T} \end{bmatrix}$$

where, $\hat{y}_{s,t}$ represents the prediction from the SARIMAX model at time t, and $\hat{y}_{m,t}$ represents the prediction from the MLP model at time t. The parameters $\alpha$ and $\beta$ are the respective weights of the SARIMAX and MLP predictions.

The weighted hybridisation allows the model to effectively combine the linear dependencies captured by SARIMAX with the nonlinear patterns learned by MLP, resulting in improved predictive performance. The weights $\alpha$ and $\beta$ are adjusted to reflect the proportional contributions of each model based on their individual predictive capabilities.

### 3.1. Hybridising the SARIMAX and MLP forecasting

Similar to the prediction implementation, the forecast $\hat{y}_{\text{Hybrid},t+h}$ of the SARIMAX-MLP model at forecast horizon h is determined by the weighted sum of the SARIMAX and MLP forecasts:

$$\hat{y}_{\text{Hybrid},t+h} = \alpha \cdot \hat{y}_{s,t+h} + \beta \cdot \hat{y}_{\text{MLP},t+h} \quad (12)$$

The forecast for the full horizon of length H is presented as:

$$\hat{Y}_{\text{Hybrid}} = \alpha \cdot \hat{Y}_s + \beta \cdot \hat{Y}_{\text{MLP}} \quad (13)$$

where:
$$ \hat{Y}_{\text{Hybrid}} = \begin{bmatrix} \hat{y}_{h,t+1} \\ \hat{y}_{h,t+2} \\ \vdots \\ \hat{y}_{h,t+H} \end{bmatrix}, \quad \hat{Y}_s = \begin{bmatrix} \hat{y}_{s,t+1} \\ \hat{y}_{s,t+2} \\ \vdots \\ \hat{y}_{s,t+H} \end{bmatrix}, \quad \hat{Y}_{\text{MLP}} = \begin{bmatrix} \hat{y}_{\text{MLP},t+1} \\ \hat{y}_{\text{MLP},t+2} \\ \vdots \\ \hat{y}_{\text{MLP},t+H} \end{bmatrix} $$

where, $\hat{y}_{s,t+h}$ and $\hat{y}_{\text{MLP},t+h}$ represent the SARIMAX and MLP model forecasts at time $t+h$, and $\alpha$ and $\beta$ are the weights of the SARIMAX and MLP forecasts respectively.

Furthermore, the weights $\alpha$ and $\beta$ are cross-validated to optimise and find the best weight balance between SARIMAX and MLP, depending on the respective forecasting and prediction accuracy of the models.

## 3.2. Performance metrics

Statistical learning evaluation techniques are used to assess the performance, accuracy, and reliability of models or predictions based on statistical measures and tests. The methods adopted in this article are root mean squared error (RMSE), mean absolute error (MAE), coefficient of determination ($R^2$), and shapley additive explanations (SHAP). These methods provide insights into the statistical significance of the results and estimate the influence of exogenous factors to understand the robustness of the prediction models. The measures of technique fitness estimation are as follows:

### 3.2.1. Root mean squared error (RMSE)

The RMSE is computed by taking the square root of the average of squared differences between predicted values $P_i$ and actual values $A_i$:

$$ \text{RMSE} = \sqrt{\frac{\sum_{i=1}^{n} (A_i - P_i)^2}{n}} \qquad (14) $$

where $n$ is the number of observations, and the squared differences $(A_i - P_i)^2$ emphasize larger errors, making RMSE valuable when large errors should be penalised more [5,31,32].

### 3.2.2. Mean absolute error (MAE)

The MAE represents the average absolute difference between predicted values $Y_i$ and observed values $\tilde{Y}_i$:

$$ \text{MAE} = \frac{1}{N} \sum_{i=1}^{N} |Y_i - \tilde{Y}_i| \qquad (15) $$

Unlike RMSE, which is sensitive to outliers, MAE is robust in the presence of extreme values as it gives equal weight to all errors. This makes MAE suitable for datasets with outliers [32–34].

### 3.2.3. Coefficient of determination ($R^2$)

The coefficient of determination $R^2$ measures the proportion of variance in the dependent variable $Y_i$ that is predictable from the independent variable(s) $X_i$:

$$ R^2 = \left( \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sigma_X \sigma_Y} \right)^2 \qquad (16) $$

where $N$ is the number of observations, $X$ and $Y$ are the observed values, $\bar{X}$ and $\bar{Y}$ are their respective means, and $\sigma_X$ and $\sigma_Y$ are standard deviations. Higher $R^2$ values indicate better model performance, with a value of 1 representing perfect prediction [32,33].

### 3.2.4. Shapley additive exPlanations (SHAP)

SHAP is adopted as an explainable artificial intelligence (XAI) approach to estimate the impact of exogenous features. SHAP is a game-theoretic approach that explains individual predictions by assigning each feature a contribution value. It is based on Shapley values, originally derived from cooperative game theory. SHAP values quantify how much each feature contributes to a prediction, elucidating the factors influencing both individual predictions and overall model behaviour.

Additionally, SHAP ensures fair and consistent contributions across different feature sets.

For a given predictive model $f(x)$, the Shapley value for feature $i$ is given by:

$$ \phi_i = \sum_{S \subseteq N \setminus \{i\}} \frac{|S|! (|N| - |S| - 1)!}{|N|!} [f(S \cup \{i\}) - f(S)] \qquad (17) $$

where $N$ is the set of all features, $S \subseteq N \setminus \{i\}$ represents a subset of features excluding feature $i$, $f(S)$ is the model's output when using only the features in subset $S$, and $f(S \cup \{i\}) - f(S)$ measures the marginal contribution of feature $i$ when added to subset $S$. The term $\frac{|S|! (|N| - |S| - 1)!}{|N|!}$ ensures a fair contribution distribution across all feature permutations.

Furthermore, The model's output for a given input $x$, SHAP approximates the function $f(x)$ as a linear summation of feature attribution:

$$ f(x) = \phi_0 + \sum_{i=1}^{M} \phi_i x_i \qquad (18) $$

where $\phi_0$ is the expected model output (baseline prediction), $\phi_i$ is the SHAP value for feature $i$, indicating its impact on the prediction, and $M$ is the total number of features.

## 4. Results and discussion

The data input of the developed model is described and presented in Fig. 4. These figures provide a comprehensive view of energy consumption patterns over time and across the exogenous variables, particularly over a 24-hour cycle. Predicting time steps shorter than one hour are challenging. Averaging accumulated hot water and energy consumption across apartments improves model prediction accuracy and significantly reduces outliers [4,8,25].

Performance metrics, including the coefficient of determination ($R^2$), root mean square error (RMSE), and the mean absolute error (MAE), are employed to assess the accuracy and computational efficiency under the three scenarios. To determine the optimal weighting of SARIMAX and MLP predictions in the hybrid model, a grid search cross-validation method was employed. A range of weights $(\alpha, \beta)$, subject to the constraint $\alpha + \beta = 1$, was tested. For each combination, the hybrid output was computed as a weighted sum of the SARIMAX and MLP predictions, and the model's performance was evaluated using RMSE on a validation dataset; this correlates with the dropout rate values presented in Table 9. The weight pair that yielded the lowest RMSE was selected, ensuring that the final model effectively leveraged both linear and nonlinear predictive components.

### 4.1. Data description

The data used in this analysis are sourced from two blocks of residential dwellings, located in the northeastern part of the commune of the department of central France as illustrated in Fig. A.17. The data set pertains to energy and hot water consumption from electric water boilers (EWBs) across 67 apartments, spanning the period from 2020-02-14, 08:00:00, to 2021-08-02, 12:00:00. This time frame covers 1 year, 5 months, 19 days, and 4 hours, providing a comprehensive dataset for evaluating model performance.

The data description and statistical analyses is aim to validate the efficiency, accuracy, and generalisation capabilities of the developed prediction models. Figs. 5 to 7 present statistical summaries of data on energy and hot water consumption for 7 of the 67 apartments, highlighting patterns in use of Electric Water Boiler (EWB). The structured dataset is categorised into distinct scenarios based on time intervals and seasonality to capture variations in consumption patterns, enabling a more accurate and context-aware predictive modelling approach for both short-term and long-term prediction.

Table 3 outlines the three distinct time-step scenarios used for evaluating the prediction models. The training period refers to the historical
<table><caption>**Table 3**
Time-step scenarios for EWB energy consumption in model evaluation.</caption><thead><tr><td>Scenarios (time-steps)</td><td>Training period</td><td>Prediction duration</td><td>Forecast length</td></tr></thead><tbody><tr><td>Summer (1 hour)</td><td>6 months</td><td>7 days</td><td>24 hours</td></tr><tr><td>Winter (1 hour)</td><td>6 months</td><td>7 days</td><td>24 hours</td></tr><tr><td>Daily (24 hours)</td><td>17 months</td><td>180 days</td><td>30 days</td></tr></tbody></table>

data used for model training, *prediction duration* denotes the time frame for which one-time step predictions are made, and *forecast length* refers to the extended period for which the model forecasts beyond the available data. Furthermore, the endogenous variable refers to the target variable (energy consumption), while the exogenous variables include hours of the day, daily patterns, time, and hot water consumption.

## 4.2. Visualisation of the exogenous variables

In Fig. 4(a), This energy consumption plot is obtained by aggregating and averaging energy usage data over hourly intervals, visualising consumption trends throughout the day. Energy consumption remains relatively low for most of the day, with a sharp spike in the late evening (21:00 - 22:00 hours), indicating peak usage likely driven by occupant behaviours, specific activities, and lower energy tariffs for residential dwellings during off-peak hours. Fig. 4(b) illustrates the hot water consumption trend, aggregating data at hourly intervals to visualise usage patterns throughout the day. The trend steadily increases, decreases around noon, and peaks in the mid-afternoon, probably corresponding to typical household activities such as morning routines and evening use. Furthermore, the accumulation of hot water consumption, influenced by lower tariffs, aligns with the trend of energy usage observed in Fig. 4(a).

In Fig. 4(c), the peak energy consumption with higher variability and outliers during the 21:00 - 22:00 period aligns with the trend observed in Fig. 4(a). Both figures indicate that energy consumption remains relatively low and stable throughout most of the day, with a sharp rise in the evening, likely influenced by household routines and lower electricity tariffs. The distribution further reveals a significant increase in energy consumption between 20:00 and 23:00, highlighting the impact of time-of-day pricing, where lower tariffs at night and higher costs during the day shape consumer energy usage under France's energy policy.

The box plot in Fig. 4(d) visualises energy consumption patterns across different days of the week, showing notable variability and extreme outliers, which may correspond to peak tariff periods. The interquartile range (IQR) indicates that most energy usage remains within a stable range, reflecting standard residential demand, while extreme spikes likely result from increased energy use during peak tariff hours when electricity costs are elevated. This pattern underscores the impact of dynamic electricity pricing, as consumers tend to shift high-energy activities to off-peak hours to minimise costs, leading to more stable median consumption but occasional extreme variations. These insights suggest that both the time of day and specific daily patterns are critical for optimising energy efficiency.

## 4.3. Scenarios data presentation and visualisation results

The exogenous variables analysed in this article include hours of the day, days of the week, time, and hot water consumption. These variables are used to evaluate their influence on predicting the endogenous variable (energy consumption) and to analyse the generalisation and scalability of the developed prediction model. The results presented in this section provide insight into the performance of the model in different time-steps and seasonal conditions, thereby supporting their robustness and applicability in real-world residential energy management scenarios.

Fig. 5 illustrates the hourly energy and hot water consumption of electric water boilers (EWBs) during summer across different apartments. In Fig. 5a, energy consumption exhibits substantial variability and frequent outliers, with certain apartments showing significantly higher consumption, likely influenced by differences in occupancy and usage patterns.

Fig. 5b highlights the distribution of hot water consumption, which is relatively consistent across apartments, with narrower plots indicating less variability and a more predictable pattern compared to energy consumption. These predictable patterns demonstrate the importance of accounting for exogenous factors such as hot water consumption and seasonal variations when developing prediction models to improve their robustness and accuracy for residential energy management.

Fig. 6 present the hourly energy and hot water consumption of electric water boilers (EWBs) during the winter season across multiple apartments. In Fig. 6a, energy consumption remains highly variable across apartments, with significant outliers depicting instances of peak consumption driven by increased hot water demand during the colder months. Conversely, Fig. 6b shows a more concentrated distribution for hot water consumption, with limited variability and a noticeable peak in apartment 5, potentially linked to specific occupant behaviour or higher hot water demand. These findings underline the seasonality impact on energy consumption and the necessity to incorporate both endogenous (energy) and exogenous (hot water) variables to enhance prediction accuracy and capture seasonal dynamics in residential energy modelling.

Fig. 7 display the daily energy and hot water consumption patterns for electric water boilers (EWBs) across multiple apartments. In Fig. 7a, daily energy consumption reveals significant variability among apartments, with apartment 4 exhibiting the widest range of consumption, likely due to higher occupant activity or irregular usage patterns. Fig. 7b, focusing on hot water consumption, shows a more consistent and structured distribution across apartments, with apartment 5 showing an extended tail, reflecting occasional spikes in hot water usage. The distribution highlights the importance of daily time-step modelling to capture broader consumption trends and identify irregular consumption patterns to improve prediction accuracy and efficient of energy management strategies.

# 5. Comparative analysis of SARIMAX-MLP model performance

The performance of the novel SARIMAX-MLP model is compared with other models in the literature, including single models [9,26] and hybrid models with endogenous inputs [8]. The comparison focuses on prediction accuracy and computational efficiency for EWB energy consumption in residential dwellings.

The SHapley Additive exPlanations (SHAP) was adopted to interpret the model complexity and impact of exogenous factors, many machine learning models, including neural networks (MLP), and hybrid models like SARIMAX-MLP are often black-box models with complex decision-making processes. SHAP translates these decisions into interpretable feature contributions, facilitating validation and enhancing model dependability. It was also used to visualise the correlation of features, illustrating how variables interact and contribute differently across various data inputs. Furthermore, SHAP insights can help optimise Model Predictive Control (MPC) strategies by prioritising the most influential
<figure><img src="image_5.png" /><figcaption>Fig. 4. Energy consumption patterns visualisation with exogenous variables over 24-hours.</figcaption></figure>

variables (e.g., hot water consumption, time of use, etc) in predictive models.

## 5.1. Computational efficiency of prediction models

The computational efficiency of the novel hybridised model, SARIMAX-MLP was evaluated using the validation dataset across three distinct scenarios as presented in Table 3, while the results of the computation time are summarised in Table 4 and Table 5. The models were

implemented using Python, and the performance tests were conducted on a system equipped with an 8th Gen Intel(R) Core(TM) i5-8265U CPU @ 1.60 GHz, 1.80 GHz, and 16.0 GB of RAM. During simulation, computational time was recorded, with comparative results for the prediction and forecasting performance of the models.

The results in Table 4 compare the prediction computation costs of single models, hybrid models, and models with exogenous inputs across different time-step scenarios, highlighting significant variations in efficiency. Among the single models, ARIMA consistently shows the
<figure><img src="image_6.png" /><figcaption>Fig. 5. EWBs summer energy and hot water consumption at hourly time-steps.</figcaption></figure>

<table><caption>**Table 4**

Prediction computation time results for model's performance analysis.</caption><thead><tr><th>Models/Types</th><th>Hourly summer (s)</th><th>Hourly winter (s)</th><th>Daily time-steps (s)</th></tr></thead><tbody><tr><td><strong>Single models</strong></td><td></td><td></td><td></td></tr><tr><td>ARIMA</td><td>1.0</td><td>1.0</td><td>1.0</td></tr><tr><td>SVR</td><td>433.0</td><td>530.0</td><td>550.0</td></tr><tr><td>ANN (MLP)</td><td>73.0</td><td>84.0</td><td>51.0</td></tr><tr><td><strong>Hybrid models</strong></td><td></td><td></td><td></td></tr><tr><td>ARIMA-MLP</td><td>27.5</td><td>24.0</td><td>36.0</td></tr><tr><td>ARIMA-SVR</td><td>234.8</td><td>200.9</td><td>1645.0</td></tr><tr><td>SVR-MLP</td><td>257.7</td><td>214.0</td><td>1668.0</td></tr><tr><td><strong>Models with exogenous input</strong></td><td></td><td></td><td></td></tr><tr><td>SARIMAX</td><td>14.0</td><td>9.4</td><td>4.5</td></tr><tr><td>MLP</td><td>42.6</td><td>27.8</td><td>42.7</td></tr><tr><td>SARIMAX-MLP</td><td>39.1</td><td>29.1</td><td>36.5</td></tr></tbody></table>

lowest computation cost (1.0 seconds) across all scenarios, demonstrating its suitability for real-time, low-complexity predictions. In contrast, SVR incurs the highest computation cost (up to 550.0 seconds for daily time-steps), while ANN (MLP) demonstrates moderate costs, indicating a balance between complexity and efficiency.

For hybrid models, ARIMA-MLP emerges as the most efficient, with computation costs ranging from 24.0 to 36.0 seconds, making it a practical option for balancing computational demands and model complexity. ARIMA-SVR and SVR-MLP, however, exhibit significantly higher costs (up to 1668.0 seconds for daily time-steps), reflecting the computational expense of integrating SVR with other models.

Models with exogenous inputs further demonstrate the efficiency of SARIMAX, which achieves computation costs as low as 4.5 seconds for daily time-steps, making it ideal for real-time energy predictions

<figure><img src="image_7.png" /><figcaption>Fig. 6. Energy and hot water consumption during winter at hourly time-steps.</figcaption></figure>

<table><caption>**Table 5**

Computation cost results for forecast models.</caption><thead><tr><th>Models/Types</th><th>Hourly summer (s)</th><th>Hourly winter (s)</th><th>Daily time-steps (s)</th></tr></thead><tbody><tr><td><strong>Single models</strong></td><td></td><td></td><td></td></tr><tr><td>ARIMA</td><td>49</td><td>10</td><td>70</td></tr><tr><td>SVR</td><td>433</td><td>532</td><td>596</td></tr><tr><td>ANN (MLP)</td><td>73</td><td>114</td><td>59</td></tr><tr><td><strong>Hybrid models</strong></td><td></td><td></td><td></td></tr><tr><td>ARIMA-MLP</td><td>29</td><td>25</td><td>36</td></tr><tr><td>ARIMA-SVR</td><td>234</td><td>202</td><td>1679</td></tr><tr><td>SVR-MLP</td><td>258</td><td>217</td><td>1665</td></tr><tr><td><strong>Models with exogenous input</strong></td><td></td><td></td><td></td></tr><tr><td>SARIMAX</td><td>15</td><td>11</td><td>6</td></tr><tr><td>MLP</td><td>44</td><td>29</td><td>60</td></tr><tr><td>SARIMAX-MLP</td><td>39</td><td>31</td><td>38</td></tr></tbody></table>

when external variables are considered. SARIMAX-MLP offers a balance between computational efficiency and the ability to handle nonlinear patterns, with costs ranging from 29.1 to 39.1 seconds.

These results highlight the trade-offs between computation efficiency and model complexity. While ARIMA and SARIMAX are best suited for scenarios requiring high efficiency, hybrid models like ARIMA-MLP and SARIMAX-MLP are preferable for applications demanding improved accuracy without incurring excessive computational costs. This analysis underscores the need to align model selection with the specific requirements of time sensitivity and accuracy in energy consumption prediction tasks.

The forecast computation cost results presented in Table 5 highlight the efficiency of different models across various time-step scenarios. Among the single models, ARIMA demonstrates the lowest computa-
<figure><img src="image_8.png" /><figcaption>(a) EWBs energy consumption at daily time-step.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(b) EWBs hot water consumption at daily time-step.</figcaption></figure>

<caption>Fig. 7. EWBs energy and hot water consumption at daily time-steps.</caption>

tional cost (49 to 70 seconds), making it highly efficient compared to SVR and ANN (MLP), which incur significantly higher costs, particularly for hourly and daily time-steps. Hybrid models, such as ARIMA-MLP, offer a balance between computational efficiency and model complexity, outperforming ARIMA-SVR and SVR-MLP in terms of computation cost. However, ARIMA-SVR and SVR-MLP require substantially more time for daily time-steps (over 1600 seconds), indicating computational inefficiencies for long-term predictions.

For models with exogenous inputs, SARIMAX has the lowest computation cost across all scenarios (6 to 15 seconds), reflecting its efficiency in integrating exogenous variables. SARIMAX-MLP and MLP, while more computationally demanding than SARIMAX, remain moderately efficient, particularly for daily time-steps. The results suggest that while hybrid models and exogenous input models improve model computation efficiency compared to single models like SVR and MLP. Notably, computational costs must be carefully considered for real-time applications. These findings highlight the trade-off between computational cost and model complexity, guiding the selection of forecasting models based on the application scenario.

In real-time MPC applications, model latency must be low enough to enable timely control actions. Traditional MPC strategies typically rely on linear models such as ARX or state-space formulations due to their fast inference capabilities. However, these approaches often compromise predictive accuracy. The SARIMAX-MLP model, while computationally efficient, maintains forecast runtimes below 40 seconds per cycle, well within the acceptable latency range (1–5 minutes) for real-time control in residential MPC applications. This modest trade-off in speed yields significant improvements in accuracy, making the hybrid model a viable and more effective alternative for MPC-based energy scheduling in micro-grid systems.

<table><caption>Table 6
SARIMAX parameter and exogenous variables results summary.</caption><thead><tr><th>Parameters</th><th>P > z</th><th>Standard error</th></tr></thead><tbody><tr><td>Constant</td><td>0.676</td><td>1069.023</td></tr><tr><td>Time</td><td>0.037</td><td>29.853</td></tr><tr><td>Hot water consumed</td><td>0.000</td><td>1705.179</td></tr><tr><td>Hours of the day</td><td>0.037</td><td>29.653</td></tr><tr><td>Days of the week</td><td>--</td><td>--</td></tr><tr><td>MA-L1</td><td>0.000</td><td>0.033</td></tr><tr><td>MA-L1</td><td>0.809</td><td>0.037</td></tr><tr><td>AR-S-L24</td><td>0.000</td><td>0.002</td></tr><tr><td>MA-S-L24</td><td>0.000</td><td>0.007</td></tr><tr><td>σ<sup>2</sup></td><td>9.600</td><td>8066.119</td></tr></tbody></table>

## 5.2. SARIMAX model results

The SARIMAX model configuration was determined through empirical analysis, resulting in the best-fit order of (1, 0, 1), (1, 0, 1, 24). This configuration indicates one lag in both the autoregressive (AR) and moving average (MA) terms, while the seasonal component includes both AR and MA terms with a seasonal period of 24 hours. This reflects daily cycles in energy consumption, capturing both short-term dependencies and daily seasonal patterns effectively.

Table 6 presents the parameter estimates for the SARIMAX model applied to hot water consumption and energy prediction and forecasting. The SARIMAX parameters include components such as *Constant*, which represents the model's intercept; *time*, capturing the linear trend; and *hot water consumed*, an exogenous variable accounting for external factors affecting energy usage. Seasonal and non-seasonal components include terms like *MA-L1* (moving average lag-1), which represents the effect of past forecast errors on the current prediction, *AR-S-L24* (seasonal autoregressive term with a 24-hour lag), and $\sigma^2$, which denotes the model's variance, capturing the overall dispersion of prediction errors and indicating the model's uncertainty in estimating energy consumption; collectively, these components model both time-dependent and external influences on energy consumption.

The constant term has a high standard error and a non-significant p-value (0.676), indicating that it has not contributed significantly to the model's predictions. Time represents a continuous variable that captures the chronological progression of observations, enabling the model to identify temporal trends, seasonality, and long-term patterns in energy consumption. Furthermore, time and hours of the day are statistically significant ($p = 0.037$) with relatively low standard errors, highlighting their significant impact on energy consumption. The hot water consumed variable is highly significant ($p = 0.000$), confirming its strong influence on energy usage. Seasonal terms, such as AR-S-L24 and MA-S-L24, are also significant, capturing the daily seasonality (24-hour cycles). However, the day of the week has missing values, suggesting it lacks variation in this SARIMAX model.

The autocorrelation function (ACF) plot measures the correlation between a time series and its lagged values to identify patterns like seasonality, while the partial autocorrelation function (PACF) plot isolates the direct relationship between a variable and its lags by removing intermediate influences, helping to determine the appropriate lag order for ARIMA-based models, as presented in Fig. 8. Most points fall within the confidence interval (shaded blue region), and there are no significant spikes beyond lag 1. This suggests that the SARIMAX model has successfully captured the temporal structure in the data, leaving residuals that behave like white noise. The absence of significant autocorrelation in the residuals indicates that the model does not suffer from underfitting as presented in Fig. 8, making it suitable for forecasting.

The result in Fig. 9 presents the SHapley Additive exPlanations (SHAP) values for exogenous variables in the SARIMAX model, illustrating their impact on energy consumption prediction. The SHAP values on the x-axis indicate how much each feature influences the model's output, with positive values increasing the prediction and negative values
<figure><img src="image_10.png" /><figcaption>Fig. 8. SARIMAX residuals autocorrelation.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 9. Summary of exogenous features influences on SARIMAX energy consumption prediction.</figcaption></figure>

decreasing it. The feature values, represented by colour (blue for low and red for high), show that hot water consumption has the most significant impact, followed by hours of the day and time, while days of the week exhibit minimal influence.

Hot water has the most significant impact on the model, with both high and low values strongly affecting energy use, indicating that water consumption drives energy consumption substantially. *Hours of the day* and *time* show moderate influence, with energy usage following predictable daily cycles, especially during specific hours. The *days of the week* feature show limited variability, likely indicating that it has little to no impact on energy consumption in this dataset, as shown by the clustering around zero. The spread of hot water consumption SHAP values suggests its strong and consistent effect, meaning that variations in hot water usage significantly drive changes in energy consumption predictions. This analysis highlights the importance of exogenous variables, particularly hot water usage, in improving model accuracy and capturing consumption dynamics.

The percentage impact represents the relative contribution of each exogenous feature to the SARIMAX model's energy consumption prediction. Table 7 shows that hot water consumption has the most significant influence on energy prediction, contributing 59.36%, indicating a strong relationship between hot water usage and energy consumption. *Time* and *hours of the day* are also influential, contributing around 20% each, while *days of the week* have no significant effect, suggesting that energy usage patterns do not vary much across different days.

In Fig. 10, highlights the SARIMAX model's performance in both prediction and forecasting of hourly energy consumption time-steps. During the prediction period, the model effectively captures the underlying patterns in the observed data, indicating a strong fit to historical data. In the forecasting period (shaded area), the model continues to capture the general trend but shows some deviations, particularly in the magni-

<caption>Table 7</caption>

Exogenous input influence on energy consumption prediction.

<table><caption>Table 8</caption><thead><tr><td>Exogenous feature</td><td>Percentage impact (%)</td></tr></thead><tbody><tr><td>Hot water</td><td>59.36</td></tr><tr><td>Time</td><td>20.43</td></tr><tr><td>Hours of day</td><td>20.21</td></tr><tr><td>Days of the week</td><td>0.00</td></tr></tbody></table>

MLP model parameters with exogenous variables.

<table><thead><tr><td>Parameter</td><td>Details</td></tr></thead><tbody><tr><td>Input size</td><td>SARIMAX residuals + Exogenous variables (total 4 inputs)</td></tr><tr><td>Hidden Layers</td><td>- Layer 1: 32 neurons, ReLU activation<br>- Layer 2: 64 neurons, ReLU activation</td></tr><tr><td>Dropout</td><td>0.3 (30%)</td></tr><tr><td>Output Layer</td><td>1 neuron, Sigmoid activation</td></tr><tr><td>Loss Function</td><td>Huber loss</td></tr><tr><td>Optimiser</td><td>Adam</td></tr><tr><td>Metrics</td><td>Mean Squared Error (MSE)</td></tr><tr><td>Epochs</td><td>100</td></tr><tr><td>Batch Size</td><td>32</td></tr><tr><td>Validation Split</td><td>20%</td></tr></tbody></table>

tude of peaks, which suggests potential areas for improving the model's forecast accuracy over longer horizons.

### 5.3. Multilayer perceptron (MLP) model result

The parametric tuning configuration of the MLP model was determined empirically, as presented in Table 8. The simulation results of the RMSE test results for different dropout rates of the MLP model, derived from the empirical analysis presented in Table 9. The dropout rate refers to the fraction of neurons randomly deactivated during training to mitigate overfitting and enhance generalisation. Setting neurons to zero forces the model to rely on different neurons during each training iteration; however, excessively high or low dropout rates (e.g., 0.1, 0.2, 0.4, 0.5) can lead to under- or over-regularisation, increasing RMSE and reducing prediction accuracy. The lowest RMSE is achieved at a dropout rate of 0.3, indicating an optimal balance between regularisation and predictive performance by effectively preventing overfitting while maintaining model stability.

The model parameters are selected empirically through iterative experimentation, optimising for prediction accuracy while balancing regularisation and computational efficiency. Key choices include rectified linear unit (ReLU) activation for hidden layers to enhance learning, a
<figure><img src="image_12.png" /><figcaption>Fig. 10. SARIMAX model prediction and forecasting energy consumption.</figcaption></figure>

<table><caption>**Table 9**
RMSE test results for different dropout rates.</caption><thead><tr><td>Dropout rate</td><td>RMSE test</td></tr></thead><tbody><tr><td>0.1</td><td>2103.74</td></tr><tr><td>0.2</td><td>2103.09</td></tr><tr><td>0.3</td><td>2067.99</td></tr><tr><td>0.4</td><td>2103.01</td></tr><tr><td>0.5</td><td>2104.72</td></tr></tbody></table>

<table><caption>**Table 10**
Exogenous feature impact on energy consumption prediction.</caption><thead><tr><td>Exogenous feature</td><td>Percentage impact (%)</td></tr></thead><tbody><tr><td>Hot water</td><td>39.54</td></tr><tr><td>Time</td><td>25.25</td></tr><tr><td>Hours of day</td><td>35.20</td></tr><tr><td>Days of the week</td><td>0.00</td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>Fig. 11. Exogenous variables' influence on the MLP model's predictions.</figcaption></figure>

dropout rate of 0.3 to prevent over-fitting, the Adam optimiser for adaptive learning, and Huber loss to mitigate the impact of outliers, ensuring robust performance, were obtained from empirical simulation.

Fig. 11 presents the MLP model's exogenous variables, demonstrating that *Hot water* has the highest influence on model predictions, with a spread of SHAP values between -0.01 and 0.03. This indicates that changes in hot water consumption significantly impact the model's output, although the overall effect is relatively moderate. *Hours of the day* and *Time* have narrower SHAP value distributions close to zero, suggesting a more minor and less significant role in prediction accuracy. Consistent with previous findings, *Days of the week* shows no noticeable impact, likely due to a lack of distinct weekly patterns in energy consumption.

Table 10 illustrates the percentage impact of exogenous variables on the hybrid MLP model's predictions. *Hot water* (39.54%) has the highest influence on model performance, indicating that fluctuations in water consumption significantly affect energy usage predictions. *Hours of the day* (35.20%) and *Time* (25.25%) also contribute meaningfully, suggesting that the timing of energy usage patterns is important, while *Days of the week* (0.00%) show no impact, reflecting the absence of a clear weekly trend in the data.

## 5.4. Comparative analysis of models accuracy

Tables 11 and 12 present the performance metrics of various endogenous and exogenous models evaluated in three scenarios: hourly time step in summer, hourly time step in winter, and daily time steps. The metrics include the coefficient of determination ($R^2$), root mean square error (RMSE), and mean absolute error (MAE). These metrics reflect each model's prediction accuracy, robustness, and ability to generalise across different time intervals and seasons.

The prediction performance metrics in Table 12 provide a comprehensive comparison of different models across three time-step scenarios: hourly time-step summer, hourly time-step winter, and daily time-steps.

### • Single endogenous models

- ANN (MLP) consistently achieves the highest forecasting accuracy, as indicated by a significant coefficient of determination $R^2$ = 0.860, and the lowest RMSE and MAE values across all scenarios. This highlights its superior ability to capture complex relationships in the data, particularly for shorter time-steps.

- ARIMA performs well in hourly scenarios, with $R^2$ values of 0.895 (summer) and 0.909 (winter), indicating a strong prediction accuracy. However, it struggles in daily time steps ($R^2$ = 0.496), reflecting its limitation in modelling long-term dependencies.

- SVR demonstrates moderate performance across scenarios, with $R^2$ values ranging from 0.707 to 0.754. However, its higher RMSE and MAE values indicate less precision compared to ARIMA and ANN.

### • Hybrid endogenous models

- ARIMA-MLP emerges as the best hybrid model, combining ARIMA's strength in capturing temporal dynamics with MLP's ability to model nonlinear relationships. It consistently achieves high $R^2$ values (0.909 in winter and 0.838 daily), as well as low RMSE and MAE, making it ideal for both short and long-term predictions.
<table><caption>**Table 11**
Prediction results for model's performance metrics.</caption><thead><tr><th rowspan="2">Models/Types</th><th colspan="3">Hourly summer</th><th colspan="3">Hourly winter</th><th colspan="3">Daily time-steps</th></tr><tr><th>R<sup>2</sup></th><th>RMSE</th><th>MAE</th><th>R<sup>2</sup></th><th>RMSE</th><th>MAE</th><th>R<sup>2</sup></th><th>RMSE</th><th>MAE</th></tr></thead><tbody><tr><td colspan="10"><strong>Single models</strong></td></tr><tr><td>ARIMA</td><td>0.895</td><td>0.057</td><td>0.033</td><td>0.909</td><td>0.048</td><td>0.029</td><td>0.496</td><td>0.066</td><td>0.053</td></tr><tr><td>SVR</td><td>0.754</td><td>0.087</td><td>0.079</td><td>0.707</td><td>0.086</td><td>0.077</td><td>-0.049</td><td>0.096</td><td>0.092</td></tr><tr><td>ANN (MLP)</td><td>0.810</td><td>0.040</td><td>0.022</td><td>0.850</td><td>0.033</td><td>0.012</td><td>0.860</td><td>0.054</td><td>0.041</td></tr><tr><td colspan="10"><strong>Hybrid Models</strong></td></tr><tr><td>ARIMA-MLP</td><td>0.906</td><td>0.050</td><td>0.031</td><td>0.909</td><td>0.053</td><td>0.039</td><td>0.838</td><td>0.058</td><td>0.045</td></tr><tr><td>ARIMA-SVR</td><td>0.836</td><td>0.071</td><td>0.048</td><td>0.828</td><td>0.066</td><td>0.045</td><td>0.528</td><td>0.066</td><td>0.055</td></tr><tr><td>SVR-MLP</td><td>0.763</td><td>0.092</td><td>0.053</td><td>0.576</td><td>0.103</td><td>0.069</td><td>-0.818</td><td>0.572</td><td>0.565</td></tr><tr><td colspan="10"><strong>Models with Exogenous Input</strong></td></tr><tr><td>SARIMAX</td><td>0.892</td><td>0.381</td><td>0.127</td><td>0.823</td><td>0.317</td><td>0.196</td><td>0.658</td><td>0.153</td><td>0.091</td></tr><tr><td>MLP</td><td>-0.050</td><td>0.400</td><td>0.281</td><td>0.826</td><td>1.523</td><td>0.166</td><td>-0.169</td><td>0.423</td><td>0.300</td></tr><tr><td>SARIMAX-MLP</td><td>0.910</td><td>0.075</td><td>0.055</td><td>0.927</td><td>0.016</td><td>0.018</td><td>0.983</td><td>0.029</td><td>0.033</td></tr></tbody></table>

<table><caption>**Table 12**
Forecast results for model's performance metrics.</caption><thead><tr><th rowspan="2">Models/Types</th><th colspan="3">Hourly time-step summer</th><th colspan="3">Hourly time-step winter</th><th colspan="3">Daily time-steps</th></tr><tr><th>R<sup>2</sup></th><th>RMSE</th><th>MAE</th><th>R<sup>2</sup></th><th>RMSE</th><th>MAE</th><th>R<sup>2</sup></th><th>RMSE</th><th>MAE</th></tr></thead><tbody><tr><td colspan="10"><strong>Single models</strong></td></tr><tr><td>ARIMA</td><td>-0.097</td><td>0.206</td><td>0.123</td><td>-0.014</td><td>0.193</td><td>0.157</td><td>-1.021</td><td>0.168</td><td>0.150</td></tr><tr><td>SVR</td><td>0.817</td><td>0.084</td><td>0.081</td><td>0.823</td><td>0.080</td><td>0.072</td><td>-0.585</td><td>0.100</td><td>0.100</td></tr><tr><td>ANN (MLP)</td><td>0.801</td><td>0.070</td><td>0.072</td><td>0.820</td><td>0.073</td><td>0.072</td><td>0.890</td><td>0.088</td><td>0.064</td></tr><tr><td colspan="10"><strong>Hybrid models</strong></td></tr><tr><td>ARIMA-MLP</td><td>0.901</td><td>0.043</td><td>0.021</td><td>0.905</td><td>0.010</td><td>0.005</td><td>0.808</td><td>0.038</td><td>0.034</td></tr><tr><td>ARIMA-SVR</td><td>0.579</td><td>0.128</td><td>0.082</td><td>0.630</td><td>0.116</td><td>0.087</td><td>0.023</td><td>0.092</td><td>0.079</td></tr><tr><td>SVR-MLP</td><td>0.845</td><td>0.070</td><td>0.052</td><td>0.711</td><td>0.103</td><td>0.079</td><td>-0.209</td><td>0.533</td><td>0.529</td></tr><tr><td colspan="10"><strong>Models with exogenous input</strong></td></tr><tr><td>SARIMAX</td><td>-0.263</td><td>0.090</td><td>0.264</td><td>0.892</td><td>0.215</td><td>0.178</td><td>-0.480</td><td>0.913</td><td>0.210</td></tr><tr><td>MLP</td><td>-0.049</td><td>0.412</td><td>0.289</td><td>0.828</td><td>0.256</td><td>0.166</td><td>-0.169</td><td>0.422</td><td>0.301</td></tr><tr><td>SARIMAX-MLP</td><td>0.560</td><td>0.070</td><td>0.040</td><td>0.890</td><td>0.069</td><td>0.065</td><td>0.148</td><td>0.080</td><td>0.061</td></tr></tbody></table>

- **ARIMA-SVR** performs reasonably well, particularly in hourly scenarios ($R^2 = 0.836$ and $R^2 = 0.828$), but exhibits a noticeable drop in accuracy for daily time-steps ($R^2 = 0.528$).

- **SVR-MLP** shows the weakest performance among hybrid models, with lower $R^2$ values and significantly higher RMSE and MAE in daily time-steps ($R^2 = -0.818$), indicating poor generalisation in long-term predictions.

### Exogenous models

- **SARIMAX** achieves reasonable accuracy in hourly and daily time-steps, particularly for summer ($R^2 = 0.892$) and winter ($R^2 = 0.823$). However, its RMSE and MAE are relatively high in daily time-steps, suggesting that its predictive capacity diminishes with longer intervals.

- **MLP** model is deficient in effectively capturing linear and substantially influential exogenous variables, as reflected in its poor daily prediction performance ($R^2 = -0.169$). In contrast, the **SARIMAX-MLP** hybrid model exhibits optimal performance for daily predictions ($R^2 = 0.983$), underscoring the critical role of exogenous inputs in improving the accuracy of long-term forecasting.

## 5.5. Comparative analysis of model forecast accuracy

Similarly, the forecast performance metrics in Table 12 provide a comprehensive comparison of different models across three time-step scenarios: hourly time-step summer, hourly time-step winter, and daily time-steps.

### Single endogenous models

- **ARIMA** performs poorly with negative $R^2$ values across all scenarios, particularly for daily time-steps ($R^2 = -1.021$), indicating its inability to capture long-term trends. This is further supported by higher RMSE and MAE values, especially for daily time-steps.

- **SVR** exhibits moderate performance, with positive $R^2$ values for hourly predictions ($R^2 = 0.817$ in summer and $R^2 = 0.823$ in winter). However, its performance declines significantly for daily time-steps ($R^2 = -0.585$), suggesting challenges in capturing longer-term patterns.

- **ANN (MLP)** outperforms other single models, achieving $R^2$ values ($R^2 = 0.801$ to $0.890$) across all scenarios, along with the lowest RMSE and MAE values. These highlights its strength in handling both short- and long-term predictions.

### Hybrid endogenous models

- **ARIMA-MLP** demonstrates superior performance among hybrid models, with consistently high $R^2$ values ($R^2 = 0.901$ for hourly summer, $R^2 = 0.905$ for hourly winter, and $R^2 = 0.808$ for daily). Its low RMSE and MAE values further validate its accuracy and robustness.

- **ARIMA-SVR** shows moderate performance, with $R^2 = 0.579$ for hourly summer and $R^2 = 0.630$ for hourly winter. However, it struggles with daily time-steps ($R^2 = 0.023$), indicating its limitations in long-term forecasting.

- **SVR-MLP** performs well for hourly time-steps ($R^2 = 0.845$ in summer and $R^2 = 0.711$ in winter) but fails drastically for daily time-steps ($R^2 = -0.209$), likely due to over-fitting, hyperplane architecture, and insufficient generalisation.

### Exogenous models

- **SARIMAX** shows mixed results, performing moderately for hourly time-steps ($R^2 = -0.263$ in summer and $R^2 = 0.892$ in winter) but poorly for daily time-steps ($R^2 = -0.460$).

- **MLP** with exogenous inputs performs inconsistently, with poor $R^2$ values across all scenarios and relatively high RMSE and MAE.
values, suggesting that the inclusion of exogenous variables does not enhance its performance.

- SARIMAX-MLP demonstrates some improvement over SARIMAX alone, particularly for daily time-steps ($R^2 = 0.848$), but its performance remains optimal compared to hybrid models like endogenous ARIMA-MLP due to the influence of the exogenous variables.

Among the scenarios evaluated, the SARIMAX-MLP model exhibited the most pronounced performance advantage in the daily time-step forecasts and the winter hourly predictions. These scenarios are characterised by complex, nonlinear usage patterns, particularly sharp consumption spikes and seasonal shifts in occupant behaviour. The hybrid model achieved higher $R^2$ values and lower RMSE and MAE scores compared to both single (ARIMA, SVR, MLP) and hybrid (ARIMA-MLP, SVR-MLP) models. The integration of SARIMAX's capacity to model seasonal dependencies with MLP's strength to capture nonlinear residuals was particularly effective during winter, when user demand was more variable due to increased hot water usage and tariff-sensitive behaviour.

# 6. Research findings

## 6.1. Significance of models computational efficiency

The stacked bars in the Fig. 12 represent the computational cost of different prediction models across various time-step scenarios. The computation efficiency results demonstrate the significant trade-offs between model complexity and runtime across different scenarios. Models with exogenous variables (MLP-exo, SARIMAX, and SARIMAX-MLP) show higher computation costs due to the inclusion of additional external factors, which enhance their ability to capture seasonal variations and external influences like hot water usage. Among these, SARIMAX-MLP exhibits the highest cost but achieves superior performance in daily time-steps, making it effective for long-term forecasts.

In contrast, models with only endogenous variables such as ARIMA-SVR, SVR-MLP, ARIMA-MLP, ARIMA, SVR, and ANN (MLP), generally have lower computation costs but may sacrifice accuracy in scenarios requiring the incorporation of exogenous factors. ARIMA-MLP, for instance, balances computation cost and accuracy, making it suitable for hourly scenarios, while SVR-MLP incurs higher costs 1668.0 seconds due to its dual-stage nature but struggles with shorter intervals.

The influence of exogenous variables is particularly evident in SARIMAX and SARIMAX-MLP, which outperform simpler models in capturing complex relationships and seasonal dynamics, despite their computation cost. Conversely, models like ANN (MLP) and SVR, while computationally efficient, lack robustness in scenarios with significant seasonality or external dependencies. This analysis underscores the importance of hybrid models incorporating both endogenous and exogenous inputs for achieving optimal performance, especially for long-term forecasting or scenarios with seasonal variability. However, their application must consider the computation cost, especially in real-time or resource-constrained environments.

## 6.2. Analysis of the significance of performance metrics for model accuracy

Notably, sharp consumption spikes during evening hours (20:00–22:00), influenced by low tariff incentives, irregular usage patterns at weekend, and asymmetric consumption trends during seasonal variations were not effectively captured by the SARIMAX model, which relies on linear and stationary assumptions. These residual nonlinear behaviours were more accurately learned by the MLP component, which utilised the SARIMAX residuals in conjunction with key exogenous variables. This synergy significantly improves the prediction accuracy of the hybrid model, particularly under conditions characterised by fluctuating user behaviour and temporal variability. By integrating SARIMAX's

strength in modelling linear and seasonal dependencies with the capacity of the MLP model to detect complex nonlinear patterns, the hybrid framework effectively addressed both structured and irregular dynamics in residential energy consumption.

### 6.2.1. Significance of ($R^2$) in model performance

The coefficient of determination ($R^2$) serves as a critical performance metric, quantifying the proportion of variance in energy consumption explained by the models. The ($R^2$) value can be negative when the chosen model fits the data worse than an ideal horizontal line at the mean of the observed data. In practice, ($R^2$) typically ranges between 0 and 1, where 0 indicates no predictive power and 1 represents a perfect prediction. From Fig. 13, models with endogenous variables demonstrate superior performance compared to models with exogenous inputs, particularly in hourly scenarios. ANN (MLP) achieves consistently high $R^2$ approximately 0.900 all scenarios, demonstrating its exceptional ability to model nonlinear dependencies in energy consumption patterns, even without exogenous variables. This makes ANN particularly effective for both short and long-term predictions.

Hybrid models, such as ARIMA-MLP, also exhibit strong $R^2$ values, with $R^2 = 0.808$ for daily time-steps in prediction scenarios Fig. 13a, outperforming all other hybrid and single models. This highlights the advantage of combining ARIMA's linear temporal modelling with MLP's nonlinear capabilities to address residual complexities. Conversely, ARIMA-SVR performs moderately, especially in hourly scenarios, but struggles with generalisation in daily scenarios, reflecting the limitations of SVR in handling variability over longer time-steps.

Exogenous models, particularly SARIMAX and SARIMAX-MLP, show inconsistent performance. SARIMAX achieves moderate $R^2 = 0.892$ for hourly winter scenarios but fails for daily time-steps ($R^2 = -0.480$), indicating that exogenous inputs such as weather or seasonal effects may not significantly enhance model performance due to SARIMAX's linear limitations. Similarly, the SARIMAX-MLP model exhibits significant improvement in certain hourly scenarios ($R^2 = 0.890$), while the standalone MLP underperforms compared to endogenous hybrid models such as ARIMA-MLP. This suggests that the capabilities of MLP may already internalise much of the contextual information provided by exogenous variables.

The significance of these results lies in the role of time-step duration and exogenous variable integration. Hourly seasonal scenarios generally yield higher $R^2$ values across models, reflecting their ability to capture higher-level specificity in consumption patterns. However, daily time-steps introduce greater variability, leading to a decline in $R^2$, particularly for linear and hybrid models like ARIMA and ARIMA-SVR. The integration of exogenous variables shows limited added value for machine learning models like ANN or hybrid models, suggesting that the inherent nonlinear modelling capabilities may already subsume the effects of these external inputs.

### 6.2.2. RMSE model performance significance

The performance of RMSE in all scenarios in Fig. 14 highlights the significant differences between models with exogenous variables (MLP-exo, SARIMAX, SARIMAX-MLP) and those with endogenous variables (ARIMA-SVR, SVR-MLP, ARIMA-MLP, ARIMA, SVR, ANN (MLP)). Models with exogenous variables generally achieve lower RMSE values, particularly in scenarios with seasonal variations (daily time-steps), as they account for external factors like hot water consumption, which enhance predictive accuracy. Among these, SARIMAX-MLP demonstrates superior performance, achieving low RMSE across all scenarios due to its hybrid structure and the ability to capture short and long-term dependencies.

In contrast, models relying solely on endogenous variables often show higher RMSE, especially in complex scenarios like hourly winter predictions. For instance, ARIMA-SVR and SVR-MLP exhibit higher errors due to their inability to incorporate exogenous influences model
<figure><img src="image_14.png" /><figcaption>Fig. 12. (a) Computational cost for prediction and forecast models across scenarios.</figcaption></figure>

complexity, which limits their robustness in capturing seasonal and ex-
ternal consumption dynamics.

ANN (MLP) and ARIMA perform well in simpler scenarios (hourly summer) due to their lower complexity but are less reliable in more dynamic scenarios requiring detailed contextual data. This analysis underscores the significance of integrating exogenous variables in hybrid models, as it improves accuracy and robustness, particularly in long-term forecasting or seasonally influenced scenarios. However, the choice of model should balance the improved accuracy (reduced RMSE) with computational efficiency, as discussed in the previous analysis.

6.2.3. MAE model performance significance:

The mean absolute error (MAE) results in Fig. 15 emphasize the improved performance of models with exogenous variables (MLP-exo, SARIMAX, SARIMAX-MLP) compared to those relying solely on endogenous variables. Exogenous variable-based models achieve lower MAE values across scenarios, reflecting their ability to incorporate external factors, such as hot water consumption, which enhances predictive accuracy, particularly in dynamic seasonal conditions like winter or daily time-steps. Among these, SARIMAX-MLP consistently performs well, demonstrating its robustness in capturing both seasonal trends and long-term patterns.

Endogenous models, such as ARIMA-SVR, SVR-MLP, and ARIMA,
generally exhibit higher MAE, particularly in daily time-step scenar-
ios where external influences significantly impact consumption. For in-
stance, ARIMA-SVR shows high MAE in both prediction and forecasting,
indicating its limited ability to handle complex seasonal variations with-
out exogenous inputs.

Simple models like ANN (MLP) and SVR perform adequately in
hourly summer scenarios with lower variability but fail to maintain ac-
curacy under more complex conditions. The results highlight that the
integration of exogenous variables (MLP-exo and SARIMAX) is crucial
to improving model performance in scenarios with higher variability
and longer forecasting horizons. This demonstrates the importance of
hybrid models, such as SARIMAX-MLP, which leverage the strengths of
both endogenous and exogenous components to achieve enhanced ac-
curacy and scalability across time-varying and seasonal scenarios.

6.3. Significance of exogenous variables in the SARIMAX-MLP model

The exogenous factors include hot water, hours of the day, and days
of the week (daily consumption). The significance of these variables
in predicting the energy consumption of electric water boilers (EWBs)
varies across different scenarios, as shown in Table 13. Hot water con-
sumption consistently exhibits the highest impact, especially during
<figure><img src="image_15.png" /><figcaption>(a) Comparative coefficient of determination for prediction models across scenarios.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(b) Comparative coefficient of determination for forecast models across scenarios.</figcaption></figure>

<caption>**Fig. 13.** (a) Comparative coefficient of determination for prediction models across scenarios. (b) Comparative coefficient of determination for forecast models across scenarios.</caption>

<table><caption>**Table 13**
Exogenous variables influence on EWBs energy consumption.</caption><thead><tr><th rowspan="2">Scenarios</th><th colspan="4">Rate of impact of exogenous variables (%)</th></tr><tr><th>Hot water</th><th>Hours of day</th><th>Daily consumption</th><th>Time</th></tr></thead><tbody><tr><td>Summer (hourly)</td><td>36.11</td><td>31.69</td><td>0.00</td><td>32.20</td></tr><tr><td>Winter (hourly)</td><td>40.95</td><td>34.27</td><td>0.00</td><td>24.78</td></tr><tr><td>Daily over all data</td><td>100.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr></tbody></table>

winter, with a contribution of 40.95%, reflecting its strong influence on energy consumption for heating during colder periods. Similarly, hours of the day significantly affect energy use, contributing 31.69% in summer and 34.27% in winter, likely due to diurnal patterns in water consumption and household activity.

The time variable, which indirectly represents temporal trends, also shows a notable influence, contributing 32.20% in summer and 24.78% in winter, likely capturing seasonal variations in energy demand. However, daily consumption appears to have no significant impact in hourly

scenarios, as finer time steps (hourly) already capture fluctuations in usage patterns.

For aggregated data, hot water usage alone accounts for 100% of the impact as shown in Table 13, highlighting its dominance when modelling energy consumption on a broader time scale. This finding underscores the importance of incorporating exogenous variables, particularly hot water consumption and time-associated factors, to improve the predictive accuracy and robustness of EWB energy consumption models. Models that exclude these variables risk reducing the complexity of energy consumption dynamics, especially in scenarios with significant seasonal or hourly variations.

The simulation analysis in Fig. 16 illustrates the significance of exogenous variables on the SARIMAX-MLP model's prediction and forecasting of EWB energy consumption across different scenarios. Hot water consumption consistently emerges as the most influential variable, with significant values in both summer and winter scenarios (Figs. 16a and 16b), as well as in daily time-steps (Fig. 16c). This underscores its critical role in driving energy consumption, especially during periods of high demand for hot water heating in winter. These suggesting occa-
<figure><img src="image_17.png" /><figcaption>Fig. 14. (a) Prediction model performance of RMSE across scenarios. (b) Forecast model performance of RMSE across scenarios.</figcaption></figure>

sional high energy usage, potentially during off-peak tariff hours when energy costs are lower. This pattern indicates that consumers may be shifting their high energy activities to periods of reduced tariffs, optimising their energy expenditure based on cost variations.

Time and hours of the day also show notable impacts, particularly in hourly (summer & winter) scenarios, where they reflect daily patterns and time-varying trends in energy consumption. In the summer scenario (Fig. 16a), these variables display balanced contributions, capturing diurnal variations in energy demand. Similarly, during winter (Fig. 16b), the importance of time-dependent variables reflects higher energy consumption during peak hours due to hot water heating needs.

In contrast, daily consumption has negligible influence in hourly scenarios, but represents a vital contributor in daily time steps (Fig. 16c), where it dominates alongside hot water consumption. This indicates that aggregated time-step models rely more heavily on macro-level consumption trends. The results highlight the necessity of incorporating hot water consumption, time, and hours of the day as exogenous variables to enhance model performance. These variables capture both short-term

fluctuations and seasonal dynamics, providing better predictive accuracy to manage energy consumption from the EWBS.

In addition to hot water consumption, the hour of the day emerged as a strong secondary predictor of energy use. This reflects well-defined behavioural routines, such as morning and evening peaks in hot water and energy demand, driven by occupant activities like bathing and cooking. Furthermore, these peaks often coincide with off-peak electricity pricing periods, indicating that tariff structures also shape consumption behaviour. Therefore, 'hour of the day' effectively captures both temporal and economic influences on residential energy usage, making it a valuable exogenous variable in the predictive model.

## 7. Conclusion

This article examines and develops a novel energy consumption prediction algorithm for domestic electric water boilers in residential dwellings, focusing on systems that store energy during non-peak hours and deliver it during peak hours. An accurate prediction model can significantly improve renewable energy mini-grid management and load
<figure><img src="image_18.png" /><figcaption>Fig. 15. (a) Prediction model performance of MAE across scenarios. (b) Forecast model performance of MAE across scenarios.</figcaption></figure>

scheduling based on hot water usage demand, reducing costs and over-
all carbon emissions from national grid dependency.

The novel hybrid SARIMAX-MLP model demonstrates an optimal approach for predicting energy consumption, utilising the strengths of linear and nonlinear modelling techniques. The hybridised technique combines both parallel and sequential hybridisation methods and further uses the weighted sum of individual prediction model output as the final prediction of the hybrid model. The results highlight the robustness and accuracy of hybrid models, particularly SARIMAX-MLP, which outperforms other models in most scenarios by effectively integrating exogenous variables such as hot water consumption, time, and hours of the day. Although SARIMAX captures temporal dynamics across hourly, daily, and seasonal patterns, MLP effectively models non-linear dependencies in the residuals.

Significantly, results indicate that hot water consumption has the greatest impact on prediction accuracy, compared to time-based variables (e.g., time of day and hour of use), which are minimally influential. Additionally, computational analysis reveals that while the hybrid model incurs higher computational costs, particularly when incorpo-

rating hourly and daily variables, this does not always translate to
improved accuracy, suggesting that certain exogenous inputs may in-
troduce unnecessary complexity. While SARIMAX alone proves to be
faster and computationally efficient, its inability to fully model non-
linear relationships limits its predictive accuracy. Thus, although the
SARIMAX-MLP hybrid model enhances prediction accuracy, further im-
provement may be necessary to optimise the selection of exogenous
variables and balance computational efficiency. SHAP analysis confirms
that exogenous variables significantly contribute to energy consumption
predictions, emphasizing the importance of capturing temporal and be-
havioural trends in model development.

The performance metrics including RMSE, MAE, and computational
costs, indicate that models incorporating exogenous variables achieve
higher accuracy compared to purely endogenous models like ARIMA or
SVR. Hot water consumption emerges as the most influential variable in
daily and seasonal (winter) hourly time-steps, while temporal variables
(e.g., time of day and hour of use) remain critical in hourly predictions.

This study also highlights the computational trade-offs of hybrid
models, where higher hyperplane dependent and model complexity re-
<figure><img src="image_19.png" /><figcaption>Fig. 16. Analysis of the influence of exogenous variables on the SARIMAX-MLP model across various time-step scenarios.</figcaption></figure>

Results in increased computation times, particularly for daily time-step scenarios. However, the exogenous improved prediction accuracy justifies this computational cost, making hybrid models well-suited for real-world residential energy management applications. These findings emphasize the importance of incorporating exogenous factors in predictive modelling to account for seasonal and time-dependent variations, which significantly enhance both model accuracy and real-time applicability. The findings from this research lay the foundation for optimising residential energy management and advancing data-driven sustainable energy systems, including the application of model predictive control (MPC) for grid control and scheduling.

Despite the strong performance of the SARIMAX-MLP model, it has limitations in modelling long-range dependencies and dynamically prioritising features under varying conditions. Future work will explore deep learning architectures such as one-dimensional Convolutional Neural Networks (1D CNNs), which can efficiently capture localised temporal patterns, and Transformer-based models, which are well-suited for long-term sequence modelling and adaptive feature weighting learning. These approaches have the potential to enhance the model's ability to adapt to demand variability and multi-scale feature interactions in real-time energy prediction.

### CRediT authorship contribution statement

**IbrahimAli Kachalla:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation,

Funding acquisition, Formal analysis, Data collection, Conceptualization. **Christian Ghiaus:** Writing – review & editing, Supervision, Software, Resources, Project administration, Funding acquisition, Data collection, Conceptualization. **Adeniran Ademuwagun:** Writing – review & editing, Resources. **Olufemi Babajide Odeyinde:** Writing – review & editing, Resources. **Majid Baseer:** Writing – review & editing.

### Funding

This work was financed by the Petroleum Technology Development Fund (PTDF) Nigeria, and supported by Immobilier Rhône Alpes-Group 3F.

### Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Ibrahim Ali Kachalla reports financial support, administrative support, article publishing charges, statistical analysis, and writing assistance were provided by National Institute of Applied Sciences of Lyon. Ibrahim Ali Kachalla reports a relationship with Petroleum Technology Development Fund that includes: funding grants and financial support. Authors declare no conflict of interest. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
# Appendix A. Case study

<figure><img src="image_20.png" /><figcaption>(a) Front view</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>(b) Floor plan of the 7th floor section X-X</figcaption></figure>

<caption>Fig. A.17. Case study of residential dwellings.</caption>

## Data availability

Data will be made available on request.

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