Hybrid Model for Long-Term and Short-Term Power Demand Forecasting in HPC Datacenters ^†

Abstract

The power demand of High-Performance Computing (HPC) infrastructures exhibits both stable weekly regularities and rapid workload-driven fluctuations, which are difficult to capture reliably with a single modeling paradigm. Achieving more sustainable HPC operation requires accurate forecasts at multiple horizons: short-term predictions support operational control (e.g., proactive power capping and energy-aware scheduling), whereas long-term forecasts are essential for planning activities (e.g., capacity provisioning and energy procurement). Together, these capabilities reduce operational cost and risk while enabling more efficient and sustainable datacenter management. This paper investigates multi-horizon forecasting of aggregated active power consumption in an operational HPC datacenter utilizing a four-month dataset (five-minute time intervals) from the University of Turin. We propose a hybrid residual-learning framework that integrates a long-term structural forecaster with a short-term residual corrector utilizing a Temporal Convolutional Network (TCN) to address the simultaneous presence of weekly regularities and short-term workload-induced fluctuations. Assessment utilizing a rolling-origin protocol covers a timeframe of 15 min to 6 h and extends 1 to 3 weeks into the future. Performance of the proposed approach has been compared against the SARIMA baseline.

1. Introduction

High-Performance Computing (HPC) infrastructures are becoming increasingly constrained not just by performance and availability, but also by energy consumption and its economic and environmental consequences. Globally, datacenters are estimated to represent around 1.5% of total energy consumption, and their future development is heavily reliant on maintaining efficiency improvements despite growing demand [1]. In operational contexts, precise power models and forecasts are essential for planning and management tasks, including power provisioning, cost- and carbon-aware load control, and infrastructure optimization, encouraging forecast research based on real-time telemetry. In HPC facilities, forecasting is especially important as energy-aware control measures have to operate over different timescales. At short horizons, forecasts enable online scheduling decisions, proactive power limitation, and quality-of-service (QoS) maintenance under power restrictions, characteristics that have proven to be viable and advantageous in actual High-Performance Computing (HPC) systems. Furthermore, fine-grained telemetry (e.g., at a 5 min interval) is already common in large-scale implementations and facilitates high-resolution modeling of diurnal patterns and operational modes [2].

An important methodological challenge is that multi-horizon forecasting involves complex trade-offs across prediction intervals: strategies that perform well at short horizons may accumulate errors or fail to maintain low-frequency structure at extended horizons, whereas long-horizon models may not properly capture rapid fluctuations [3]. This study specifically targets two decision-relevant timeframes utilizing 5 min sampled data from an actual deployment: (i) a short-term horizon of 15 min to 6 h, corresponding to operational control and short-term scheduling; and (ii) a long-term horizon of 1 to 4 weeks, associated with near-term planning and capacity provisioning. To address these heterogeneous horizons within an integrated framework, a hybrid modeling approach that distinguishes between slow and fast dynamics has been adopted. Time-series decomposition separates a signal into trend, seasonal, and residual components, offering a systematic approach to distinguish long-term patterns from short-term fluctuations [4]. This decomposition-driven approach aligns with empirical evidence in the datacenter sector, indicating that hybrid statistical-neural methodologies can surpass single-model baselines. For example, SARIMA–LSTM combinations have been documented to enhance accuracy in short-term forecasting of electrical consumption and daily peak demand in Internet datacenters [5]. At the same time, modern multi-horizon architectures (e.g., attention-based models) highlight the importance of explicitly modeling interactions between variables and temporal dependencies across horizons [6].

We propose a residual-learning hybrid forecasting framework for HPC datacenter power demand, integrating a long-term module utilizing Prophet, a decomposable model capable of detecting trends and multiple seasonality while effectively managing calendar effects, together with a short-term learner based on a Temporal Convolutional Network (TCN). Prophet is employed to simulate the low-frequency structure applicable to the 1–4-week horizon, while the TCN is trained to forecast the residual component (i.e., the remaining rapid dynamics not accounted for by Prophet) over the 15 min to 6 h horizon. This architecture follows the recognized principles of hybrid residual modeling, where a first-stage model identifies the predominant patterns, and a second-stage model integrates the residual nonlinearities to enhance predictions [7]. The methodology is assessed using 5 min sampled telemetry from an actual deployment on the HPC infrastructure at the University of Turin, incorporating power usage metrics and relevant environmental signals. The primary contributions of the paper include: (i) a Prophet and TCN residual correction architecture that clearly differentiates long-term structure from short-term variability, subsequently integrating them into cohesive multi-horizon forecasts; (ii) a comprehensive multi-horizon evaluation method for high-resolution (5 min) HPC telemetry, encompassing 15 min to 6 h and 1 to 4 week horizons; and (iii) an empirical validation utilizing actual operational data from the University of Turin HPC datacenter, illustrating the practical efficacy of residual-learning hybrids for energy-aware management.

2. Related Work

The power-aware operation of large-scale computing infrastructures has led to significant research on measurement-driven modeling of energy consumption in datacenters and High-Performance Computing (HPC). In hyperscale environments, simplified statistical models have been created to correlate workload and resource usage with power demand, facilitating planning and operational decision-making at the datacenter scale [2]. In High-Performance Computing, power prediction has been examined at the levels of job granularity and scheduler decisions, where precise job power estimations are crucial for power-capping and energy-aware scheduling algorithms [8]. These studies highlight that power behavior in production systems is influenced by workload dynamics and system-level management strategies, and that forecasting approaches must remain resilient to operational non-stationarities characteristic of actual deployments.

A major field of research focuses on modular, interpretable forecasting models that represent a time series as a combination of trend, seasonality, and event-driven influences. The Prophet technique systematizes this methodology via an additive regression model designed for scaled forecasting, accommodating numerous seasonality and exhibiting resilience to missing data and outliers. The Prophet model has gained extensive practical application and has been assessed in scientific fields necessitating multi-step forecasting, such as hydrology, demonstrating its efficacy in capturing low-frequency patterns and its responsiveness to domain-specific attributes (e.g., sampling granularity and seasonal variations). Recent research has introduced modifications that integrate Prophet-style decompositions with neural components to enhance flexibility while preserving interpretability, as demonstrated by NeuralProphet. These contributions collectively support the utilization of decomposable models as robust baselines for week-scale forecasts, especially when periodicities, such as weekly cycles, are prevalent. Deep learning techniques, especially recurrent and convolutional architectures, have gained popularity in short-term load and energy forecasting due to their capacity to learn nonlinear temporal correlations from high-frequency telemetry data. Temporal Convolutional Networks (TCNs) received interest as a substitute for recurrent networks, utilizing causal and dilated convolutions alongside residual connections to effectively capture long-term memory while facilitating parallel training. Temporal Convolutional Networks (TCNs) have demonstrated superior efficacy in energy-related forecasting tasks, such as electrical demand and electric vehicle charging need, frequently surpassing recurrent baseline models in specific conditions while providing advantageous computational attributes [9]. These characteristics make TCNs particularly appropriate for sub-hour to multi-hour intervals where detailed temporal structures and fast workload-induced fluctuations are significant.

Hybrid forecasting has a long history in time series analysis, based on the observation that numerous real processes exhibit a combination of components that are better effectively described by different models. Initial hybrid methodologies integrated linear statistical models with nonlinear algorithms, particularly ARIMA and neural networks, to utilize complementary structures and enhance forecast precision [10]. A commonly employed approach is residual learning (or error-correction): a primary model describes trend/seasonality (or other patterns), while a second-stage model incorporates the residual errors to enhance predictions. Recent research has structured and assessed residual-driven hybrids and groups, demonstrating accuracy improvements across various domains when residuals encapsulate systematic nonlinearities missing in the baseline model. Residual-enhanced Prophet–LSTM frameworks have been presented within Prophet-centered hybrids for environmental forecasting, demonstrating the practical efficacy of Prophet as a low-frequency model with a trained residual corrector [11]. In the energy sector, hybrid models integrating (Neural) Prophet with deep learning techniques have similarly aimed for enhanced multi-horizon consistency, particularly when external factors like weather are included [12].

Hybrid and ensemble forecasting models have been extensively investigated in electricity-load and energy-demand forecasting. Early work on neural networks for short-term load forecasting demonstrated the potential of nonlinear learners, while subsequent surveys classified load-forecasting methods into statistical, artificial-intelligence, and hybrid families [13]. A common motivation behind hybrid models is that electric-load series contain both linear/seasonal structures and nonlinear or stochastic components; consequently, single models often fail to capture all relevant dynamics. This has led to combinations such as ARIMA–ANN [14], wavelet–ANN–ARIMA [15], ARIMA–SVM [16], and decomposition-based ensembles in which individual components are forecast separately and recombined [17]. More recent studies have further developed stacking and deep ensemble approaches, including frameworks in which multiple ARIMA forecasters are used as base learners and a neural network combines their outputs with auxiliary variables [18]. These studies provide the methodological background for our residual-learning design. Differently from general grid-load forecasting, however, the present work focuses on operational HPC datacenter telemetry at 5-min resolution and explicitly separates week-scale structural demand from short-horizon workload-induced residual fluctuations.

Multi-step, multi-horizon forecasting presents methodological challenges that differ from one-step prediction, such as error accumulation, horizon-dependent bias/variance trade-offs, and the necessity to model both short and long-term dependencies. Recent attention-based architectures, such as the Temporal Fusion Transformer, specifically address multi-horizon scenarios by incorporating diverse variables and mechanisms for modeling long-range dependencies and enhancing interpretability [19]. Although end-to-end models can be potent, hybrid residual frameworks are appealing for operational systems due to their (i) preservation of interpretability in low-frequency components, (ii) facilitation of modular substitution of sub-models, and (iii) substantial reduction in training complexity compared to monolithic architectures. With respect to previous research that concentrates on either coarse granularity workload-to-power modeling or single-horizon forecasting, we examine a multi-resolution hybrid specifically designed for production HPC telemetry taken at 5 min intervals. We utilize Prophet as a long-term model to capture 1–4-week dynamics and weekly load patterns, while employing a Temporal Convolutional Network (TCN) as a short-term learner to forecast residual variations occurring every 15 min to 6 h, integrating both through a residual learning framework. This design is directly inspired by the complementary strengths identified in decomposable forecasting (for stable trends and seasonality) and convolutional sequence modeling (for high-frequency nonlinear dynamics), while addressing the operationally relevant horizons encountered in HPC datacenter energy management.

3. Problem Definition and Data Description

3.1. Forecasting Problem Formulation

The power demand of a High-Performance Computing (HPC) datacenter is a complex, non-stationary time series characterized by a superposition of low-frequency, predictable cycles (e.g., academic calendars, day–night rhythms) and high-frequency, stochastic volatility driven by user job submissions. To accurately capture both phenomena, we formulate the forecasting task within an additive decomposition paradigm, treating the long-term baseline and the short-term burstiness as distinct but coupled predictive tasks.

Let $\left\{t_1,t_2,\dots ,t_T\right\}$ denote an ordered sequence of observation timestamps with constant sampling period $\Delta =5$ min. The primary target is the aggregate electrical power demand of the HPC infrastructure, denoted by the univariate time series:

$$\mathbf{y}=\{y_t\in {\mathbb{R}}_{\ge 0}{\}}_{t=1}^T$$

(1)

where $y_t$ represents the measured active power consumption (e.g., in kW) at time index $t$. In addition, we observe a set of exogenous covariates capturing environmental and operational conditions (e.g., inlet/outlet temperature, humidity, cooling-related measurements, and exogenous calendar variables), represented by a multivariate time series:

$$\mathbf{X}=\{{\mathbf{x}}_{\mathit{t}}\in {\mathbb{R}}^{\mathit{d}}{\}}_{\mathit{t}=1}^{\mathit{T}}.$$

(2)

The covariate vector ${\mathit{x}}_t$ is assumed to be synchronized with $y_t$ at the same 5 min cadence after standard alignment and cleaning procedures. We consider two forecasting regimes aligned with operational needs in HPC datacenters:

Short-term forecasting: Horizons from 15 min to 6 h ahead, corresponding to:

$${\mathcal{H}}_{\mathrm{S}}=\{3,4,\dots ,72\},$$

(3)

where $h$ is measured in 5 min steps (e.g., $h=3$ corresponds to 15 min; $h=72$ corresponds to 6 h).

Long-term forecasting: Horizons from 1 week to 4 weeks ahead, corresponding to

$${\mathcal{H}}_{\mathrm{L}}=\{2016,2017,\dots ,8064\}$$

(4)

since one week contains $7\times 24\times 12=2016$ samples at 5 min resolution, and four weeks contain $8064$ samples.

Given a forecasting origin time $t$, the objective is to estimate the future power demand vector:

$${\widehat{\mathbf{y}}}_t\left(\mathcal{H}\right)=\{{\widehat{y}}_{t+h}{\}}_{h\in \mathcal{H}}$$

(5)

for $\mathcal{H}\in \{{\mathcal{H}}_{\mathrm{S}},{\mathcal{H}}_{\mathrm{L}}\}$, using historical observations up to time $t$. We denote the information set available at time $t$ as:

$${\mathcal{I}}_t=\left\{\right(y_{\tau },{\mathbf{x}}_{\tau }) : \tau \le t\}$$

(6)

and define a generic forecasting function $f_{\theta }$ parameterized by $\theta$ such that:

$${\widehat{y}}_{t+h}=f_{\theta }({\mathcal{I}}_t,h),h\in \mathcal{H}$$

(7)

Model parameters are learned from a historical training segment $\left\{1,\dots ,T_{train}\right\}$ and evaluated on a chronologically subsequent test segment, consistent with the temporal ordering of operational data. Performance is assessed under a rolling-origin protocol: for multiple origins $t$ in the test interval, forecasts are generated for each horizon $h\in \mathcal{H}$ and compared against realized values $y_{t+h}$. The learning objective may be expressed as minimization of an empirical risk over origins and horizons, for example under an absolute-error loss:

$$\underset{\theta }{\min \left({\sum }_{t\in \mathcal{T}}{\sum }_{h\in \mathcal{H}}\mid y_{t+h}-{\widehat{y}}_{t+h}\mid \right)}$$

(8)

where $\mathcal{T}$ denotes the set of evaluation origins and $\theta$ represents the model parameters. While the primary focus of the paper is forecasting accuracy (e.g., MAE/RMSE across horizons), the formulation naturally accommodates probabilistic forecasts and prediction intervals when supported by the underlying models. Consistent with the proposed architecture, the forecasting problem is additionally framed as learning complementary predictors for different temporal components. Specifically, a long-term model provides a baseline forecast ${\widehat{y}}_{t+h}^{\mathrm{L}\mathrm{T}}$ that captures trend and seasonal structure, and a short-term learner estimates a corrective residual term ${\widehat{r}}_{t+h}$, yielding the final prediction:

$${\widehat{y}}_{t+h}={\widehat{y}}_{t+h}^{\mathrm{L}\mathrm{T}}+{\widehat{r}}_{t+h},h\in {\mathcal{H}}_{\mathrm{S}}$$

(9)

with long-horizon forecasts primarily governed by the long-term component for $h\in {\mathcal{H}}_{\mathrm{L}}$. This decomposition is particularly suited to HPC datacenter dataset, where persistent weekly regularities coexist with rapid workload-driven fluctuations at sub-hour scales.

3.2. HPC Data Description

The empirical analysis is conducted using data collected from the HPC infrastructure of the University of Turin under real operational conditions. The dataset comprises four months of continuous measurements of the overall active power consumption of the facility, sampled at a constant rate of 5 min. Each observation is indexed by a timestamp $t$ and associated with a scalar target value $y_t$, representing the aggregated electrical active power demand (e.g., in kW) at that time. The resulting time series forms the primary dependent variable for both the long-horizon and short-horizon forecasting tasks.

The empirical power consumption profile of the HPC datacenter over a seven-day temporal window is illustrated in Figure 1. A close inspection of the dataset reveals several distinct physical phenomena that justify the multi-scale modeling approach proposed in this study. The power demand exhibits a continuous, low-frequency wave-like pattern corresponding to daily operational cycles. Power consumption demonstrates systemic dips during the early morning hours (reaching a weekly nadir of approximately 55.5 kW on Wednesday, 15 October) before escalating as scheduled computational workloads and facility cooling demands peak during standard operating hours. During the designated weekend period (shaded region, 18–19 October), the macro-trend stabilizes into a more uniform, lower-amplitude rhythm. This drop reflects a reduction in human-triggered interactive jobs, leaving the facility to process steady-state background processes and long-running batch simulations. Superimposed upon the daily baseline trends is a dense, high-frequency oscillatory behavior characterized by rapid, sharp power spikes (such as the peak spike exceeding 71.5 kW on Tuesday evening). These non-linear spikes represent the chaotic arrival rate of massive computational workloads distributed across the cluster nodes.

Figure 1. Time-series profile of the actual High-Performance Computing (HPC) datacenter active power consumption over a continuous one-week horizon (14–20 October). The shaded region highlights the weekend period. The profile demonstrates a superposition of macro-level diurnal cyclicality and micro-level high-frequency stochastic volatility, with power demands bounded primarily between 55 kW and 72 kW.

In addition to the target series, the monitoring framework provides time-aligned exogenous measurements describing environmental and operational conditions, such as ambient/inlet temperature, humidity, and other facility-level data. These variables are denoted as ${\mathit{x}}_t\in {\mathbb{R}}^d$ and are used as candidate covariates for the forecasting models. This data, even if available, are not considered during the forecast in this preliminary study.

All raw measurements are first converted to a unified time basis with 5 min sampling, ensuring consistent temporal alignment. Timestamps are normalized to a single time zone and rounded to the nearest 5 min boundary when needed. If multiple observations occur within a given 5 min bin, aggregation is performed using the arithmetic mean for continuous variables (including power), while preserving the original physical units.

Given the operational nature of the deployment, the dataset may contain missing values due to transient sensor or logging outages. Missingness is addressed through a multi-stage strategy designed to preserve temporal structure while minimizing the risk of introducing artifacts:

Gap identification and characterization: Missing intervals are detected by verifying the continuity of the 5 min timestamp sequence. Gaps are characterized by length and frequency, distinguishing isolated missing points from extended outages.

Short-gap imputation: For isolated missing samples and short gaps (e.g., up to a small number of consecutive bins), values are imputed using linear interpolation on the target series and covariates, constrained by local trend. This choice preserves continuity without overly smoothing rapid variations. Advanced approaches can be used, such as the one described in [19].

Long-gap handling: For extended gaps, imputation can lead to unrealistic reconstructions; therefore, long missing segments are excluded from training windows that would require forecasting targets within the missing interval. In evaluation, any forecast targets corresponding to missing ground truth are omitted to avoid biased error estimates.

Outlier screening: Spurious spikes or physically implausible values are detected using robust statistics (e.g., median absolute deviation-based thresholds) and cross-validated against operational bounds. Flagged points are treated similarly to missing values, with conservative imputation for isolated anomalies and exclusion for extended corrupt segments.

4. Hybrid Forecasting Framework

We develop a hybrid residual-learning architecture composed of: (i) a long-term structural forecaster based on Prophet [20,21], which models the low-frequency components of power demand and supports week-scale forecasts; and (ii) a short-term residual learner based on a Temporal Convolutional Network (TCN), which corrects the long-term forecast over 15 min to 6 h. The resulting predictor is designed to remain coherent across the two horizon sets ${\mathcal{H}}_{\mathit{L}}$ and ${\mathcal{H}}_{\mathit{S}}$ defined in Section 3.1.

4.1. Long-Term Forecasting Model

Let $y_t$ denote the aggregated active power consumption at 5 min resolution. Prophet models $y_t$ via an additive decomposition:

$$y_t=g\left(t\right)+s\left(t\right)+{\mathit{w}}^{\top }{\mathit{z}}_t+{\epsilon }_t$$

(10)

where $g\left(t\right)$ is a non-linear trend component, $s\left(t\right)$ is a seasonal component, ${\mathit{z}}_t$ is a vector of optional regressors, $\mathit{w}$ are regression coefficients, and ${\epsilon }_t$ is a noise term. In our context, $g\left(t\right)$ captures slow changes in facility-level demand, while $s\left(t\right)$ captures recurring patterns (notably daily and weekly regularities). We adopt a piecewise linear trend with data-driven changepoints:

$$g(t)=\left(k+\mathit{a}(t{)}^{\top }\mathit{\delta }\right)t+\left(m+\mathit{a}(t{)}^{\top }\mathit{\gamma }\right)$$

(11)

where $k$ and $m$ define a base growth rate and offset, and $\mathit{a}\left(t\right)$ is an indicator vector encoding whether time $t$ occurs after each changepoint; $\mathit{\delta }$ and $\mathit{\gamma }$ represent the corresponding adjustments. The changepoint prior regularizes trend flexibility to reduce overfitting under limited dataset. Prophet represents seasonality using truncated Fourier series. For a seasonal period $P$ (in days), the seasonal term is:

$$s_P\left(t\right)={\displaystyle {\sum }_{n=1}^{N_P}}\left({\alpha }_{n}cos\left({\displaystyle \frac{2\pi nt}{P}}\right)+{\beta }_{n}sin\left({\displaystyle \frac{2\pi nt}{P}}\right)\right)$$

(12)

and $s\left(t\right)$ is the sum of all enabled seasonalities. Given the data span, we explicitly model daily and weekly seasonality. Yearly seasonality is disabled due to insufficient temporal coverage. When exogenous data are available (environmental or operational information), they can be included via ${\mathit{z}}_t$. The long-term model is trained in the chronological training segment and used to generate long-horizon forecasts:

$${\widehat{y}}_{t+h}^{LT}=f_P({\mathcal{I}}_t,h),h\in {\mathcal{H}}_L\cup {\mathcal{H}}_S$$

(13)

In our hybrid design, ${\widehat{y}}_{t+h}^{LT}$ provides the final prediction for $h\in {\mathcal{H}}_L$, and acts as a baseline for short-horizon correction when $h\in {\mathcal{H}}_S$. Prophet includes regularization parameters controlling trend and seasonality flexibility. These hyperparameters are selected via validation using time-respecting splits: Changepoint prior scale (trend flexibility): ${\lambda }_{\mathrm{cp}}$; Seasonality prior scale (seasonality strength): ${\lambda }_{\mathrm{seas}}$; Number of changepoints and changepoint range; and Fourier orders for daily and weekly seasonality: $N_{\mathrm{day}}, N_{\mathrm{week}}$. We report the final selected values in the experimental section, together with the validation protocol used to choose them.

4.2. Short-Term Forecasting Module

Prophet captures low-frequency structure but may under-represent fast variations associated with workload bursts and transient operational regimes. To address this, we model the residual process and learn a short-term corrector. For each time index $t$, we define the one-step residual:

$$e_t=y_t-{\widehat{y}}_t^{LT}$$

(14)

where ${\widehat{y}}_t^{LT}$ denotes the Prophet fitted value (or time-series cross-validated prediction) aligned with $t$. TCN is trained to forecast future residuals for the short horizon set ${\mathcal{H}}_S$, i.e.,

$${\widehat{\mathit{e}}}_t\left({\mathcal{H}}_S\right)=\{{\widehat{e}}_{t+h}{\}}_{h\in {\mathcal{H}}_S}.$$

(15)

Its inputs are lagged sequences of residuals and covariates over a history window of length $L$:

$${\mathit{u}}_t=\left[e_{t-L+1},\dots ,e_t, {\mathit{x}}_{t-L+1},\dots ,{\mathit{x}}_t, {\mathit{c}}_{t+1},\dots ,{\mathit{c}}_{t+h_{max}}\right]$$

(16)

where ${\mathit{c}}_t$ denotes known-in-advance calendar features (e.g., hour-of-day, day-of-week), and $h_{max}=72$ corresponds to 6 h.

TCN is a causal convolutional network composed of $B$ residual blocks. Each block applies dilated causal convolutions to ensure that predictions at time $t$ depend only on inputs $\le t$. Let $\varphi (\cdot )$ be an activation function (e.g., ReLU); then, a typical block can be expressed as:

$${\mathit{h}}^{\left(b\right)}={\mathit{h}}^{\left(b−1\right)}+Dropout\left({Conv}_{d_b,k}\left(\varphi \left({Conv}_{d_b,k}\right({\mathit{h}}^{\left(b−1\right)}\left)\right)\right)\right)$$

(17)

where ${Conv}_{d_b,k}$ is a causal 1D convolution with dilation $d_b$ and kernel size $k$. Dilations are typically scheduled exponentially (e.g., $d_b=2^{b-1}$), allowing the receptive field to grow rapidly and capture dependencies at multiple scales while retaining computational efficiency. TCN outputs a multi-horizon residual forecast (direct strategy) through a linear projection head:

$${\widehat{\mathit{e}}}_t\left({\mathcal{H}}_S\right)=f_{TCN}\left({\mathit{u}}_t\right)\in {\mathbb{R}}^{\mid {\mathcal{H}}_S\mid }$$

(18)

The TCN configuration is defined by the following hyperparameters, grouped according to their function in the model:

Input window length $\left(L\right)$: Number of past observations used as input to the TCN, expressed in 5 min time steps. This parameter controls how much recent residual history is available to predict future residuals.

Number of residual blocks $\left(B\right)$: Number of stacked TCN residual blocks. Increasing $B$ enlarges the effective receptive field and allows the model to learn more complex temporal dependencies.

Convolution kernel size $\left(k\right)$: Width of the one-dimensional causal convolutional filters. This parameter determines how many neighboring time steps are processed at each convolutional layer.

Dilation schedule ${\left\{d_b\right\}}_{b=1}^B$: Sequence of dilation factors used across the residual blocks. We adopt an exponentially increasing schedule, e.g., $d_b=2^{b-1}$, which enables the network to capture both short-range and longer-range temporal dependencies without using very deep architectures.

Number of channels $\left(C\right)$: Number of convolutional filters in each residual block. This parameter controls the representational capacity of the TCN.

Dropout rate $\left(p\right)$: Dropout probability applied inside the residual blocks to reduce overfitting, particularly important given the limited temporal span of the dataset.

Learning rate $\left(\eta \right)$: Step size used by the optimizer during training.

Batch size: Number of training windows processed in each optimization step.

Weight decay: $L_2$-regularization coefficient used to penalize large weights and improve generalization.

Early stopping patience: Number of validation epochs without improvement after which training is stopped. This criterion prevents overfitting and selects the model checkpoint with the best validation performance.

For short-term horizons $h\in {\mathcal{H}}_S$, the final hybrid forecast is obtained by residual correction:

$${\widehat{y}}_{t+h}={\widehat{y}}_{t+h}^{LT}+{\widehat{e}}_{t+h}$$

(19)

This design enforces a clear division of roles: Prophet provides interpretable trend/seasonality and stable week-scale projections, while the TCN specializes in high-frequency corrections relevant to control-oriented horizons. As mentioned above, we train the TCN by minimizing an empirical risk over the training origins ${\mathcal{T}}_{\mathrm{train}}$ and horizons ${\mathcal{H}}_S$. using mean absolute error (MAE):

$$\underset{{\theta }_{TCN}}{min} {\sum }_{t\in {\mathcal{T}}_{\mathrm{train}}}{\sum }_{h\in {\mathcal{H}}_S}\mid e_{t+h}-{\widehat{e}}_{t+h}\mid$$

(20)

The use of a residual target typically reduces non-stationarity and encourages the short-term model to focus on transient dynamics rather than re-learning low-frequency structure already captured by Prophet. Such a modular design is particularly suitable for HPC datacenter data consumption forecast, where pronounced weekly regularities coexist with rapid variations driven by workload bursts and transient facility-level control actions.

5. Validation

This section describes the experimental protocol adopted to validate the proposed hybrid residual-learning model against a classical statistical baseline, SARIMA, and a deep learning baseline based on Long Short-Term Memory (LSTM), on real 5 min active power data collected from the University of Turin HPC infrastructure. The validation targets both the short-term operational horizon (15 min–6 h) and the long-term planning horizon (1–3 weeks), consistent with the forecasting problem formulation in Section 3.1.

Let $y_t$ denote the measured aggregated active power at time index $t$ (sampling period $\Delta =5$ min). We employ a chronologically consistent train/validation/test procedure, avoiding any random shuffling. Model selection and performance estimation are carried out using a rolling-origin evaluation: for each forecasting origin $t\in \mathcal{T}$ within the evaluation interval, the models generate multi-step forecasts $\left\{{\widehat{y}}_{t+h}\right\}$ for the required horizons. Errors are then aggregated across origins and horizons. The short-term horizon set ${\mathcal{H}}_{\mathrm{S}}$ spans from 15 min to 6 h: ${\mathcal{H}}_{\mathrm{S}}=3,\dots ,72$, and results are grouped into four operationally meaningful buckets: 15–30 min: $h\in [3,6]$; 30–60 min: $h\in [6,12]$; 1–3 h: $h\in [12,36]$; and 3–6 h: $h\in [36,72]$. For long-term assessment, we consider week-ahead forecasts at the following horizons (in 5 min steps): $h_{1w}=2016, { h}_{2w}=4032, {\mathrm{a}\mathrm{n}\mathrm{d} h}_{3w}=6048$, corresponding to 1, 2, and 3 weeks ahead.

We use a Seasonal ARIMA model fitted on the univariate series $\left\{y_t\right\}$. The SARIMA configuration is selected on the validation interval using time-respecting model selection, and forecasts are produced in a standard multi-step manner for both short- and long-term horizons.

In the optimized baseline code provided to resolve the Auto-ARIMA memory constraints, the specific order specified for the model is: SARIMA(1, 1, 1)X(1, 0, 1)₂₈₈. The model structure is formally split into two parts: Non-Seasonal (p, d, q) and Seasonal (P, D, Q)_s. The Non-Seasonal Order: (1, 1, 1):

p = 1 (Autoregressive term): The model looks back exactly one step (the previous 5 min) to capture immediate temporal momentum.

d = 1 (Differencing): The data is differenced once (X_t − X_t−1) to remove any linear trend and make the high-frequency active power signal stationary.

q = 1 (Moving Average term): The model incorporates the residual error from the previous time step to smooth out sudden random shocks.

The Seasonal Order: (1, 0, 1)₂₈₈:

P = 1 (Seasonal Autoregressive term): The model looks back to the exact same 5 min interval from the previous day to account for recurring daily schedules.

D = 0 (Seasonal Differencing): Explicitly set to 0 to prevent the model from crashing. Differencing across a massive 288-step lag strips out too much variance and causes matrix inversion failures in the statsmodels optimizer.

Q = 1 (Seasonal Moving Average term): Smooths out the prediction based on the structural errors made at the same time yesterday.

s = 288 (Seasonal Period): Reflects 5 min data resolution over a 24 h cycle.

To establish a rigorous deep learning benchmark for evaluating the predictive performance of the proposed multi-scale hybrid architecture, a LSTM network was implemented as a traditional sequential baseline. LSTMs are a specialized variant of Recurrent Neural Networks (RNNs) designed specifically to model long-term temporal dependencies in high-frequency time-series data while successfully mitigating the structural challenges of vanishing or exploding gradients during backpropagation. The core capability of the LSTM relies on a cell state channel governed by a series of continuous gating mechanisms that adaptively control the flow of historical information. To guarantee complete parity and compatibility with the evaluation schema of the proposed model, the LSTM baseline is configured to solve a direct multi-horizon forecasting problem rather than executing a recursive, iterative one-step-ahead prediction strategy, which is known to introduce cumulative compound error over extended windows. The network processes a historical lookback vector of length L = 288 steps (corresponding to a 24 h continuous window at a 5 min sampling frequency) across a stacked configuration consisting of two recurrent LSTM layers. To map the sequential representations into a multi-step future output, a direct projection mechanism is integrated. The terminal hidden state vector outputted by the final layer at the conclusion of the lookback sequence is fed directly into a fully connected linear decoding layer.

We report three point-forecast accuracy metrics, computed over a set of evaluation origins $t\in \mathcal{T}$ and horizons $h$ (or horizon buckets):

(1)

Mean Absolute Error (MAE):

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{\mid \mathrm{\Omega }\mid }} {\sum }_{\left(t,h\right)\in \mathrm{\Omega }} |y_{t+h}-{\widehat{y}}_{t+h}|$$

(21)

(2)

Root Mean Squared Error (RMSE):

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{\mid \mathrm{\Omega }\mid }}{\sum }_{\left(t,h\right)\in \mathrm{\Omega }}(y_{t+h}-{\widehat{y}}_{t+h}{)}^2}$$

(22)

(3)

Mean Absolute Percentage Error (MAPE):

$$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}={\displaystyle \frac{100}{\mid \mathrm{\Omega }\mid }}{\sum }_{\left(t,h\right)\in \mathrm{\Omega }}\left|{\displaystyle \frac{y_{t+h}-{\widehat{y}}_{t+h}}{y_{t+h}+ϵ}}\right|$$

(23)

where $ϵ>0$ is a small constant introduced to ensure numerical stability in rare low-load intervals.

In addition, for long-term evaluation we quantify the planning-relevant discrepancy in total energy over weekly horizons through the Weekly Energy Error (WEE). For a given origin $t$ and a week-length window $W=2016$, we define the true and predicted weekly energy (up to a constant factor $\Delta$) as sums:

$$E_t={\displaystyle {\sum }_{i=1}^W}{y}_{t+i},{\widehat{E}}_t={\displaystyle {\sum }_{i=1}^W}{\widehat{y}}_{t+i}.$$

(24)

We report the average relative weekly energy error as follows:

$$\mathrm{W}\mathrm{E}\mathrm{E}={\displaystyle \frac{100}{\mid {\mathcal{T}}_w\mid }}{\sum }_{t\in {\mathcal{T}}_w}\left|{\displaystyle \frac{E_t-{\widehat{E}}_t}{E_t+ϵ}}\right|$$

(25)

This metric captures systematic over/under-estimation of energy demand, which is particularly important for week-ahead operational planning.

The deep learning component of the hybrid architecture (TCN) was trained using an enterprise-grade NVIDIA Tesla T4 Tensor Core GPU hosted within a Google Colab runtime environment (Python 3.13.15, libraries: Pandas 2.2.3, Numpy 2.1.3, Torch 2.11.0, prophet 1.4.0). Leveraging 16 GB of GDDR6 VRAM and 320 Turing Tensor Cores, the hardware environment enabled rapid parallel acceleration of the 1D dilated convolutions. The host virtual machine environment provided 16 GB of system RAM, facilitating the sequential execution of the macro-level Prophet baseline fitting before streaming the residuals directly to the GPU memory pipeline.

The computational overhead of the proposed framework reflects the structural differences between its components, with the curve-fitting optimization of Prophet converging rapidly in 21.472 s, whereas the multi-channel deep learning architecture of the TCN required 203.034 s for full network training.

Table 1 summarizes performance across the four short-term horizon buckets and the overall 0–6 h range. This presentation facilitates a direct comparison of error growth with the horizon, while keeping the evaluation aligned with operational decision windows.

Table 1. Short-term forecasting performance (5 min sampling). Each cell reports MAE/RMSE/MAPE aggregated over horizons in the bucket.

Model	Metrics	15–30 min	30–60 min	1–3 h	3–6 h
SARIMA	MAE (kW)	2.09	1.41	1.98	2.36
	RMSE (kW)	2.32	1.52	2.19	2.60
	MAPE (%)	3.70	2.45	3.49	4.20
LSTM	MAE (kW)	2.76	2.76	2.76	2.83
	RMSE (kW)	3.27	3.26	3.27	3.34
	MAPE (%)	4.68	4.76	4.66	4.72
HYBRID	MAE (kW)	1.36	1.59	1.81	1.53
	RMSE (kW)	1.43	1.79	2.09	1.81
	MAPE (%)	2.39	2.73	3.13	2.68

SARIMA might perform reasonably well here on short-term forecast, as the immediate autoregressive and moving average components carry strong recent momentum. Then, the performance of SARIMA degrades significantly faster than the Hybrid model. As the SARIMA forecast projects further into the future, it reverts to the seasonal meaning, entirely missing the stochastic job bursts that the TCN is designed to capture. The flat trend of the LSTM model is due because the baseline LSTM maps the sequence directly via a single dense projection layer to a vast future output array. Consequently, it sacrifices short-range precision (yielding higher errors in the 15–30 min window than SARIMA or the Hybrid model) and outputs a heavily smoothed trajectory that represents the macro-level average power distribution rather than localized tracking.

Long-horizon accuracy is reported in Table 2 for the 1-week-, 2-week-, and 3-week-ahead forecasts. As for the short-term analysis, each cell reports MAE/RMSE/MAPE at the corresponding horizon. In addition, we include the Weekly Energy Error (WEE) to evaluate the quality of energy aggregation over week-long intervals, which is a key planning signal in HPC facility management.

Table 2. Long-term forecasting performance of the proposed approach.

Metrics	1 Week	2 Weeks	3 Weeks
MAE (kW)	3.71	3.15	2.34
RMSE (kW)	4.15	4.04	2.85
MAPE (%)	+6.56	+5.79	+4.12
Et (kWh)	+509.40	+434.00	+186.11
WEE (%)	+5.29	+4.53	+1.93

The combined use of (i) interval-based short-term accuracy, (ii) discrete week-ahead errors, and (iii) a weekly energy aggregation metric provides a comprehensive validation that is aligned with the two primary operational objectives of the study: control-oriented forecasting (minutes to hours) and planning-oriented forecasting (weeks). In particular, the inclusion of WEE complements the basic error metrics by capturing whether the model is systematically biased when forecasts are integrated over a full operational week, which may not be apparent from MAE/RMSE/MAPE alone.

6. Conclusions

This paper investigated multi-horizon forecasting of aggregated active power demand in an operational HPC datacenter, using a real dataset collected from the University of Turin with a fixed sampling rate of 5 min over four months. We addressed the intrinsic multi-scale nature of the problem by proposing a hybrid residual-learning architecture. The resulting framework explicitly targets two decision-relevant regimes: short-term horizons from 15 min to 6 h, and long-term horizons from one to four weeks.

Experimental validation against a SARIMA baseline was conducted using a time-consistent rolling-origin protocol. Performance was assessed using MAE, RMSE, and MAPE across operationally meaningful short-term interval-based, as well as week-ahead errors for long-term prediction. In addition, we introduced a Weekly Energy Error metric to quantify aggregate deviations over weekly windows, which is particularly relevant for planning-oriented use cases. Overall, the results demonstrate that a residual-learning strategy is an effective mechanism to combine the interpretability and stability of decomposable long-term models with the flexibility of convolutional sequence learners for capturing high-frequency variations. The proposed approach has direct practical implications for HPC facility management, including improved near-term forecasting to support proactive operational control and enhanced week-ahead projections for capacity and energy planning.

Although the results obtained in this study demonstrate the feasibility and potential of the proposed machine learning-based approach, the analysis was conducted using data collected over a four-month period. This represents a limitation, as the dataset may not fully capture the variability associated with longer-term operation, seasonal changes, or different environmental and operating conditions. Future work will therefore focus on extending the data collection campaign to cover a full year. The availability of a larger and more representative dataset will allow for a more comprehensive validation of the proposed methodology, including the assessment of model robustness, generalization capability, and performance under a wider range of conditions. In addition, the framework will be generalized toward probabilistic forecasting with calibrated uncertainty estimates, incorporating additional operational covariates such as workload and cooling control signals, and investigating online adaptation mechanisms to maintain accuracy under concept drift induced by evolving workloads, policy changes, or infrastructure upgrades.