Next Article in Journal
Model Predictive Controlled Parallel Photovoltaic-Battery Inverters Supporting Weak Grid Environment
Previous Article in Journal
Study on the Theme Evolution and Synergy Assessment of China’s New Energy Vehicle Policy Texts
Previous Article in Special Issue
Long-Term Forecast of Energy Demand towards a Sustainable Future in Renewable Energies Focused on Geothermal Energy in Peru (2020–2050): A LEAP Model Application
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Predicting Energy Consumption for Hybrid Energy Systems toward Sustainable Manufacturing: A Physics-Informed Approach Using Pi-MMoE

1
Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 100458, China
2
University of Chinese Academy of Sciences, Beijing 100049, China
3
Guangzhou Institute of Industrial Intelligence, Guangzhou 511458, China
*
Author to whom correspondence should be addressed.
Sustainability 2024, 16(17), 7259; https://doi.org/10.3390/su16177259 (registering DOI)
Submission received: 2 July 2024 / Revised: 2 August 2024 / Accepted: 21 August 2024 / Published: 23 August 2024
(This article belongs to the Special Issue Energy Management System and Sustainability)

Abstract

:
Hybrid energy supply systems are widely utilized in modern manufacturing processes, where accurately predicting energy consumption is essential not only for managing productivity but also for driving sustainable development. Effective energy management is a cornerstone of sustainable manufacturing, reducing waste and enhancing efficiency. However, conventional studies often focus solely on predicting single types of energy consumption and overlook the integration of physical laws and information, which are essential for a comprehensive understanding of energy dynamics. In this context, this paper introduces a multi-task physics-informed multi-gate mixture-of-experts (pi-MMoE) model that not only considers multiple forms of energy consumption but also incorporates physical principles through the integration of physical information and multi-task modeling. Specifically, a detailed analysis of manufacturing processes and energy patterns is first conducted to study various energy types and extract relevant physical laws. Next, using industry insights and thermodynamic principles, key equations for energy balance and conversion are derived to create a physics-based loss function for model training. Finally, the pi-MMoE model framework is constructed, featuring multi-expert networks and gating mechanisms to balance cross-task knowledge sharing and expert learning. In a case study of a textile factory, the pi-MMoE model reduced electricity and steam prediction errors by 14.28% and 27.27%, respectively, outperforming traditional deep learning methods. This demonstrates that the model can improve prediction performance, providing a novel approach to intelligent energy management and promoting sustainable development in manufacturing.

1. Introduction

In manufacturing with a hybrid energy supply, multiple energy types are utilized concurrently in production across industries such as minerals, chemicals, and food. Compared to conventional manufacturing that relies on a single energy source, this approach draws from a diverse energy mix, including electricity, thermal energy, fossil fuels, and natural gas. Given the volatility of energy prices, complex production demands, and the close relationship between energy consumption and issues such as environmental pollution and climate change, prioritizing energy efficiency and resource optimization is crucial. This strategy not only helps reduce operational costs for enterprises but also decreases greenhouse gas emissions, which is significant for environmental protection and is central to achieving sustainable manufacturing. Accurate energy consumption forecasting is vital to addressing these challenges and promoting sustainable development. By analyzing energy usage across equipment and production lines, strategies can be developed to reduce consumption, increase efficiency, and decrease carbon and other harmful emissions, thus promoting environmental sustainability. Such forecasting provides insights into the energy behaviors of various production tools and methods. By examining these insights, irregularities in equipment operation can be detected early, potential malfunctions can be identified, and targeted maintenance strategies can be created. This approach not only prevents significant financial losses from equipment failures but also refines production line operations, enhancing sustainability. Identifying stages with high energy consumption guides production strategy adjustments and equipment upgrades, all targeting broad energy savings and further promoting sustainable manufacturing. Moreover, advanced energy forecasting ensures more precise production cost calculations, enhancing the data foundation and thus supporting enterprises on the path to sustainable development [1].
The interplay and transformation of various energy types during production introduce complexities that pose significant challenges in energy management and optimization. Reliable energy forecasting is an urgent technical challenge, especially given the strong link between production quality and energy consumption [2]. In the heat treatment process of the steel industry, models have been developed to measure energy consumption from both natural gas and electricity. By integrating with multi-objective heuristic methods, the dual goals of minimizing total energy cost (TEC) and total tardiness (TT) are targeted. However, the omission of the interaction between natural gas and electricity in their development has led to reduced predictive accuracy. The FDEACM algorithm was designed for energy efficiency analysis in ethylene plants, encompassing crude oil, steam, fuel, water, and electricity. However, it does not adequately factor in the intricate processes of production, which limits the model’s clarity [3]. To refine the models, there is often a need to expand the model’s dimensionality. Furthermore, integrating time series data to enhance their forecasting accuracy is also a common approach [4,5]. However, with the sidelining of the interconnectedness of forecasting variables and the omission of detailed processes of production, these efforts inadvertently hike up computational and training expenses. Furthermore, the commonly employed metric in assessing industrial energy consumption is specific energy consumption (SEC), which measures total energy use against product output [6]. However, its scope is limited as it does not illuminate the interaction between energy usage and production techniques, which leaves a gap in the technical direction for enhancing energy efficiency management.
Industrial energy consumption prediction models are mainly divided into two categories: conventional and artificial intelligence (AI) models. Conventional models, such as time series, regression, and gray models, can establish clear relationships between energy consumption and its influencing factors. They achieve this without requiring extensive historical data [7]. Conversely, AI-based models learn from vast historical data, not depending on predefined relationships, and can effectively handle nonlinear problems [8]. Both data-driven modeling approaches typically concentrate solely on single energy consumption, making it challenging to discover correlations in hybrid-energy manufacturing industries. In manufacturing, the interactions and conversions among various energy types are intricately coupled. Conventional single-task models struggle to harness this intertwined knowledge, limiting optimal learning efficiency. Therefore, based on the shared input feature information, this study focuses on exploring the correlational information and distinctive features between different prediction targets. To enhance the learning of shared and unique knowledge, expert layers are introduced, forming the multi-gate mixture-of-experts (MMoE) model. Physical information is extracted by leveraging the fundamental principles of energy conversion during the energy consumption process, as well as the mechanistic processes involved in materials and production techniques [9]. By integrating both, the physics-informed multi-gate mixture-of-experts model (pi-MMoE) is formed to enhance its interpretability.
The creation of an energy consumption prediction model tailored to specific industry characteristics is the primary focus of this paper. The contributions of it are as follows:
  • An in-depth analysis of the production mechanisms in the manufacturing, such as mineral, chemical, and textile industries, has been conducted. A methodology for extracting physical information from a process-oriented perspective has been proposed.
  • To address the energy consumption prediction management needs of this manufacturing industry, an MMoE model incorporating gated mechanisms and multi-expert network mechanisms has been introduced. This facilitates the balance between knowledge sharing and the uniqueness of different prediction objectives.
  • Building upon a specific textile dyeing and finishing enterprise, physical information was distilled by analyzing its production process. Subsequently, a pi-MMoE model was constructed to achieve multi-objective predictions targeting electricity and steam consumption.
This paper is structured as follows: Section 2 provides a review of the relevant literature. Section 3 introduces the methodology framework. Section 4 introduces the application in the case study. Section 5 offers a summary and outlook.

2. Literature Review

2.1. Energy Consumption Forecasting in Manufacturing with Hybrid Energy Supply

Fossil fuels, initially utilized by humans as a heat source in industrial production, trace their origins to the 1760s during the first Industrial Revolution. Given the accompanying greenhouse gas emissions during its usage and its non-renewable nature, the development of renewable energy became a theme of the times. Many countries began to champion renewable energy development and application through policy measures, leading to the emergence of integrated energy industries [10,11,12]. Due to factors like human subjective agency, complexity of mechanistic processes, and variability in production environments, these industries exhibit significant randomness in their manufacturing processes [13]. The research focused on energy consumption in these sectors primarily remains at the electricity prediction level, including the application of artificial intelligence algorithms such as residual networks (ResNet) and multilayer perceptron (MLP) [14]. Additionally, decision trees classify production equipment like simple machines, adjustable machines, single-purpose complex machines, and multi-purpose complex machines, then compute energy consumption based on the characteristics of each type using respective formulas [15]. Employing methods like Dynamic Modeling with Memory (DMWM) based on production data features helps mitigate the effects of uncertainties in datasets, improving prediction accuracy [16,17]. Influenced by industry characteristics, fossil fuels still dominate resource consumption in the production process, whether as production materials or as sources of power and heat [18]. Relying solely on electricity for energy consumption modeling is insufficient to provide adequate technical support for energy measurement and management.
As previously mentioned, manufacturing mainly employs methods such as combustion and heat exchange to utilize other energy sources during the production process. Different energy utilization strategies are chosen based on their specific production requirements [19]. Looking at the mineral, chemical, and textile industries, which are typical examples of hybrid-energy manufacturing industries, we observe the following: The mineral industry requires a continuous high-temperature production environment, predominantly relying on the combustion of fuel for its inherent energy; the textile industry requires high-temperature and high-pressure production conditions, primarily driven by the use of medium-/high-pressure steam for heat exchange; and the chemical industry requires electrical energy as a power source for electrolytic reactions, making the physical modeling of electricity utilization indispensable, as shown in Figure 1.

2.2. Development and Application of Multi-Task Learning

In our ongoing exploration of energy consumption forecasting and the diverse needs of various industries, it is evident that traditional single-task forecasting methods can sometimes fall short. They are too rigid to adapt to the intricacies of different tasks or too resource-intensive when applied to numerous parallel forecasting scenarios [20]. Additionally, the challenge of capturing interrelated patterns across diverse datasets calls for a more integrated approach. This brings us to the concept of multi-task learning (MTL)—a paradigm that offers a solution to these complexities.
MTL is an approach that achieves simultaneous learning for related tasks through knowledge sharing between them. It aims to enhance the model’s generalization capabilities and has emerged as an effective method to address the diverse energy consumption forecasting needs of various industries [21]. Compared to single-task learning, MTL reduces the model size and computational complexity through parameter sharing. It capitalizes on the complementary features between tasks to enhance generalization performance, with increasing robustness under uncertain inputs [20,22]. At the same time, a significant challenge for MTL is striking a balance between joint task learning and ensuring that one or multiple tasks do not overshadow others. Solutions such as dynamic task prioritization optimization methods have been proposed. These methods allocate greater weight to ‘difficult’ tasks, thereby directing the network to expend more effort on these challenging areas. Essentially, this approach seeks to optimize the network’s loss function to improve multi-task learning performance [23,24]. In recent years, several MTL framework models targeting energy consumption prediction have been proposed [25,26]. These models, built upon electricity consumption forecasting tasks, incorporate other prediction tasks not directly related to energy consumption. They utilize black-box models to capture correlations between production, environmental information, and electricity consumption. However, the weak interpretability of black-box models significantly impacts their ability to grasp such correlational information. By directly extracting this information to reinforce model training, not only is the model’s learning performance enhanced, but its interpretability is also improved to a certain extent [27].

2.3. Application of Physical Information in Machine Learning

Conventional AI algorithms are predominantly data-driven models. Given a set of input and output pairs, these algorithms can extract intricate relationships by learning and fitting from vast amounts of data. However, when pursuing the optimal solution for a model, prior knowledge is indispensable. Integrating the knowledge not only unveils the inherent connections between data distribution and task features but also enhances the utilization efficiency of data and bolsters the model’s generalization and interpretability [28]. In industrial production, it can be distilled through the induction of human experience and modeling of production process mechanisms. The primary manifestations include knowledge graphs, logical rules, human feedback, and physical knowledge [29,30]. Extracting knowledge graphs, logical rules, and human feedback demands extensive expert intervention, making the process high-cost and highly subjective. Such methods struggle to capture the entirety of the production process, which introduces bias. In contrast, prior knowledge based on physical information has become a current research hotspot [31], which offers broader applicability and lesser subjectivity. The integration of physical information with deep learning algorithms has seen profound applications in architectural energy consumption prediction, fluid prediction, and fatigue life prediction [32,33]. However, there is still a research gap in forecasting energy consumption in manufacturing. This is primarily due to the intricate mechanistic processes of its production design, making it challenging to precisely characterize the knowledge. Studies have shown that foundational physical rules can serve as valuable insights. Rules like the conservation of mass/energy and heat transfer equations can effectively mitigate issues like incomplete data. When used as a supplement to prior knowledge, these foundational rules provide insights that help address information gaps. The underlying physical principles act as guides when details are lacking. By leveraging basic laws, we can make progress despite missing data [34].
The demand for hybrid energy in manufacturing is growing daily, especially given the environmental challenges and resource limitations of fossil fuels in today’s world. Accurate forecasting and management of energy consumption are highly important. However, current research often focuses solely on predicting single energy consumption types, neglecting other significant energy factors in the production process. MTL, which enables predictions for multiple types of energy consumption through knowledge sharing between tasks, offers a solution. But to genuinely enhance the accuracy and interpretability of models, relying solely on data-driven is insufficient. Integrating physical information not only endows the model with rich prior knowledge but also enhances its generalization performance and interpretability. It effectively mitigates the subjectivity and one-sidedness of conventional methods, providing a more comprehensive and objective perspective. In summary, by extracting physical information based on production mechanism processes and establishing a multi-objective prediction model, we can provide more accurate and comprehensive energy consumption forecasting for production. Moreover, these efforts help drive the industry toward more sustainable and efficient directions.

3. Methodology

3.1. Overview of the Physics-Informed Multi-Task Approach

As discussed in the Introduction section, issues involve energy consumption forecasting based on multi-task models for manufacturing. To address the forecasting challenges of hybrid energy supplied in manufacturing, we propose an advanced multi-task learning model. This model integrates physical information to guide process optimization, production recommendations, and efficiency analysis. By summarizing key factors like target energy use, production processes, intrinsic mechanisms, and product data, we refine and integrate physical insights. As outlined in Figure 2, our methodology has four main parts: (1) analyzing manufacturing processes and extracting physical equations; (2) constructing the model by integrating MMoE and physical information; and (3) exploring the predictive and application methods of the model. Given the versatility of our approach, we opt for a broader range of supplementary physical knowledge for modeling. This strategy effectively bridges the gaps in data and mechanisms.

3.2. Extraction of Physical Equation

The fundamental process of industrial production involves the transformation of raw materials and energy into their final products through physicochemical processes. In this process, the law of energy conservation is a fundamental principle, indicating that the energy input Q i n into the system is equal to the sum of the energy Q o u t consumed by the system and the energy Δ Q lost:
Q i n = Q o u t + Δ Q
In this form of manufacturing, the sources of system energy input typically include steam, fuel, and electricity, each involving different utilization processes. Steam is commonly used to provide a high-temperature and high-pressure environment, generally utilizing its internal energy Q s through heat exchange processes. Fuel, often referred to as natural gas, diesel, and other fossil fuels, usually utilizes its internal energy Q f through processes like combustion to produce heat. Electricity is mainly used to drive mechanical forces or serve as an energy source for electrolysis processes, with its consumption Q e generally only involving unit conversion, where 1 kW·h is approximately equal to 3600 kJ of energy.
Q s = m s Δ h s
Q f = Σ m f q f
m s and Δ h s represent the amount of steam used and the change in its enthalpy, respectively. m f and q f represent the mass and calorific value of the fuel used, respectively.
The basic physicochemical processes involved in production include endothermic and exothermic processes and chemical reactions. Endothermic and exothermic processes refer to substances absorbing or releasing heat to change their temperature, while chemical reactions refer to substances utilizing energy to complete their chemical reactions. The energy changes, Q p and Q c , involved in these processes are as follows:
Q p = Σ c p m p Δ T p
Q c = Σ m c Δ H c
where c p , m p , and Δ T p represent the specific heat capacity, mass, and temperature rise of the object, respectively, while m c and Δ H c represent the mass of the reactants and the enthalpy value of the reaction process, respectively.
Σ m f q f + η Q e + m s Δ h s = Σ c p m p Δ T p + Σ m c Δ H c + Δ Q
This leads to the derivation of a physical information model based on energy conservation in the production process of multi-source energy consumption manufacturing, in which Δ Q represents the system’s energy loss.
The integration of physical information and models is evident in data preprocessing and loss function enhancement. This approach enhances the model’s predictive performance and interpretability, assisting in addressing uncertainty and complexity in practical applications. Utilizing the knowledge for data preprocessing helps address uncertainty, missing values, and noise issues in real-world problems. For instance, it can be employed to fill in missing data, correct errors in the data, or extract helpful features from the raw data and used to design effective data augmentation strategies, thereby expanding the training dataset and enhancing the model’s generalization ability. This often requires specific analysis based on the processes. More broadly, by incorporating physical equations into the loss function, the model adheres to certain physical principles during training to improve the model’s predictive accuracy and reliability. For example, it can be formalized into mathematical expressions, which are added as regularization terms to the loss function to ensure the model’s output meets physical constraints and used to design new loss functions to better reflect the actual needs and characteristics of the problem.
Building on the physical knowledge discussed in the previous section regarding the process, the model’s predictive output is often the energy input in the production process. A possible loss function L p i based on physical information can be established:
L p i = Σ μ 1 h y ^ f μ 2 Q o u t + Δ Q
where h y ^ f represents the heat conversion value of the input energy (generally converted to kJ); μ 1 and μ 2 represent the correction weights. Q o u t and Δ Q represent the energy consumed by the system and the energy lost, respectively. More broadly, the Mean Squared Error (MSE) as a commonly used metric for evaluating model predictive accuracy is a conventional method for constructing loss functions in regression prediction models, denoted as L N :
L N = 1 N Σ y f y ^ f 2
where N represents the total number of model outputs. In summary, the loss function L integrating physical information can be expressed as follows:
L = L N + L p i = 1 N Σ λ 1 y f y ^ f 2 + Σ λ 2 μ 1 h y ^ f μ 2 Q o u t + Δ Q
where λ 1 and λ 2 represent the loss weights, with a corresponding λ 1 + λ 2 = 1 for each y ^ f .

3.3. The Multi-Gate Mixture-of-Experts Model

In conventional MTL models, there is a widespread issue of a “one-size-fits-all” approach, where a single strategy is used to process all data. This neglects the multimodal features within the data and struggles to incorporate prior knowledge effectively when faced with complex modeling tasks. The MMoE model introduces gating mechanisms and expert network mechanisms to provide different weight coefficients to combine the outputs of the expert networks for each learning task, enhancing the model’s adaptability for each task [35]. Its structure, as shown in Figure 3, primarily consists of an input layer, expert layer, tower layer, output layer, and gating units. The expert layer is utilized for extracting shared knowledge from input features. After passing through the gating unit, which assigns weights, the data feeds into the tower layer, eventually leading to the prediction results. The tower layer is crucial for managing task-specific information and generating intermediate feature representations for each task. It extracts and processes features using various neural network layers, such as fully connected and convolutional layers, specifically tailored to the requirements of each task. Integrating knowledge from multiple expert networks creates richer and more accurate feature representations, thereby enhancing the model’s flexibility and adaptability. Structurally speaking, if two tasks are not closely related, their weight coefficients after passing through the gate would be quite different. This enables them to utilize information from some expert network outputs, approximating multiple single-task learning models. However, if the two tasks are closely related, the weight distribution acquired from the gate should be similar, resembling a general multi-object learning framework. Compared to conventional methods, it boasts benefits such as task-specificity, better information sharing, and fewer task conflicts. It offers a more flexible approach to balancing information sharing and specificity between tasks. The model can be represented as follows:
y k = h k f k x
f k x = Σ i = 1 n g k x i f i x
where y k denotes the output of each tower; h k represents the k t h tower network; f k x signifies the shared bottom used for y k predictions; f i x is the output of the i-th expert network; g k ( x ) stands for the k -th gating unit; and g k x i represents the weight value under the gating unit for the i -th expert network. Additionally, we have the following:
g k x = s o f t m a x W g k x
where W g k R n × K ; K denotes the number of expert networks; and n represents the feature dimension of the model input.

3.4. Model Validation and Application

In industrial production, optimizing process parameters is crucial because it directly affects product quality, efficiency, and cost. Conventional optimization methods often rely on experience and fixed rules, but these methods cannot be flexible enough in rapidly changing production environments. The advantage of heuristic learning lies in its flexibility and adaptability, which allow for the optimization of complex problems without requiring a precise mathematical model. Heuristic learning algorithms can identify the optimal combination of process parameters through simulation and prediction, thereby maximizing the energy efficiency of the production process. At the same time, the introduction of the pi-MMoE model provides a solid foundation for heuristic learning. This model incorporates material data, process data, and knowledge of the energy consumption process. These provide context for the algorithm, allowing for more accurate predictions and optimizations. Additionally, this combination of data-driven and physical information can help uncover potential trends and patterns, further improving production efficiency.
Particle Swarm Optimization (PSO) is a typical heuristic learning algorithm inspired by natural phenomena like bird flocking and fish schooling. In PSO, each “particle” in the solution space has a position and velocity. They dynamically update based on particle best (Pbest) and global best (Gbest) to find the global optimal solution in multi-dimensional spaces. PSO is known for its simplicity, efficiency, and robustness. It is especially well-suited for various types of optimization problems, including nonlinear, non-convex, high-dimensional, and those with multiple local optima. Its basic process is as follows:
V i t + 1 = w V i t + c 1 X p b e s t , i X i t + c 2 X g b e s t X i t
X i t + 1 = X i t + V i t + 1
The position X i ( t + 1 ) of particle i at time t + 1 is determined by its position X i ( t ) at time t and its velocity V i ( t + 1 ) at time t + 1 . In this equation, w represents the inertia weight. c 1 and c 2 denote learning factors. X p b e s t , i and X g b e s t represent the particle historical best and global historical best solutions of the particle swarm, respectively.

4. Case Study

To investigate the performance of the pi-MMoE model for multi-energy consumption prediction proposed in this paper, a case study will be conducted targeting an actual textile dyeing and finishing factory, specifically for its electricity and steam consumption prediction needs.

4.1. Background and Data Status

4.1.1. Industrial Background

In the textile dyeing and finishing industry, the stentering machine is key production equipment. As shown in Figure 4, it includes several essential components: the feeding section, the weft straightener, the chain, the drying oven, and the cloth drop and roll-up device. The feeding section mainly consists of a trough and rollers, which ensure that the cloth is evenly coated with chemicals. The weft straightener primarily detects the skewness of the fabric through a photoelectric sensor and corrects it using a hydraulic system. The chain is driven by a motor, and the cloth is stretched by the pin plates on the chain. The drying oven comprises ten drying chambers, including steam delivery pipes and blowers, to provide the necessary temperature and humidity for the fabric setting process. The cloth drop and roll-up device mainly uses a motor-driven transmission chain to handle the subsequent processing of the fabric. The stentering process can be viewed as a heat exchange process involving steam, fabric, and the workshop environment. It relies on the collaborative operation of steam, the warp straightener, and the blower to enhance fabric quality and prolong its lifespan, making it the cornerstone for improving fabric quality. The fabric, once soaked with chemicals in the trough, is evenly pressed by rollers before entering the oven. It is then dried and set under the influence of hot air to reduce the inherent residual stress in the fabric, ensuring its dimensional consistency and achieving a good hand feel. This process is influenced by fabric properties, operational parameters, ambient temperature, and manual operations and can account for more than 25% of the total production energy consumption [36].
A textile dyeing and finishing company produced 11,738 tons of knitted dyed fabric in 2021. The stentering workshop has a total of thirteen production machines, including ten stentering machines, one auxiliary conveyor system, and two preshrinking machines. The equipment network connection rate has reached 85%, and all stretching machines are fully integrated into a networked monitoring system. The factory’s stentering process primarily uses medium-pressure steam with a pressure of 3.2 MPa and a temperature of 380 °C. Data on the electric consumption of the stretching machines and steam supply are collected every minute, while production information is manually recorded by workers using card swiping. It is worth noting that, due to the presence of current transformers, the actual energy consumed by the machine is 80 times the meter reading. Steam consumption is measured in tons.

4.1.2. Data Description

The case data were obtained from the Manufacturing Execution System (MES) of the company. The production data structure is shown in Table 1. The sample dimensions are 1673 × 33, which mainly includes information like date, production machine, process, customer product name, piece number, weight, and process number. The process number is used to record the parameters of the setting machine and is managed in another table in the database. For the convenience of the study, this paper takes the 3# stentering machine as the research object. Data recorded under the process number include the oven temperature, spindle width, fan speed for each fan, machine speed, upper super-feed, and lower super-feed, among other process parameters. Additionally, the average workshop temperature has been introduced as a supplementary input for prediction. For ease of display, the following parameters will be denoted as p1~p33:
-
Fabric count (pieces): Represents the number of fabric pieces processed during production, helping to measure the scale and efficiency of the production task.
-
Weight (kg): The weight of the fabric is an important production parameter that affects the heating and stretching force during the setting process, ensuring the quality and consistency of the final product.
-
1~10# oven temperature (°C): The oven temperatures in each section control the heating phase of the setting process. Precise temperature control is crucial to ensure the setting effect and quality of the fabric.
-
1~8# spindle width (cm): The spindle width determines the width of the fabric being stretched during the setting process, affecting the shape and size of the fabric to ensure the final product meets specifications.
-
1~10# fan speed (expressed as a percentage of the standard wind speed, with the standard wind speed being 20 m/s): Fan speed controls the air circulation and cooling efficiency within the oven, impacting the uniformity of the setting process and the cooling rate of the fabric.
-
Machine speed (m/s): The machine speed determines the dwell time of the fabric in the stentering, affecting the heating and cooling time of the fabric, thus influencing the final setting effect.
-
Upper feed (cm): The upper feed length determines the tension and position of the fabric as it enters the stentering, directly affecting the stretching and setting of the fabric.
-
Workshop temperature (°C): The ambient temperature in the workshop may influence the overall production environment.

4.2. Data Preprocessing

Under the condition of small data, the higher the data dimension, the greater the impact on the model’s learning performance. Therefore, dimensionality reduction will be carried out first, focusing on the key information in the data. Factor analysis (FA) is a statistical method designed to study relationships between variables. By examining the internal dependencies among a multitude of variables, FA extracts key latent factors that encapsulate the primary characteristic information of a vast array of observed variables. Within the realm of stentering machine process parameters, certain variables often exhibit interdependencies, such as fabric count with weight, temperatures across various ovens, and spindle widths. This inevitably introduces redundancy into the predictive model’s input. Hence, FA is employed to discern the fundamental structure of observed data, extract latent factors, reduce input dimensions, and enhance model learning performance.
FA seeks to extract latent factors from variables. These latent factors, while unobservable, are objective, influential factors that exist inherently. Each variable can be expressed as a linear function of latent factors combined with specific factors, represented as follows:
p i = Σ j = 1 m a i j F j + ε i
where F j denotes the common factor, with a count of m , and the variable dimension is n ; ε i represents the specific factor for p i . Concurrently, this model can be matrix-represented as follows:
P = A F + ε
P = p 1 , , p n T ,   A = a 11 a 1 m a n 1 a n m ,   F = F 1 , , F m T ,   ε = ε 1 , , ε n T ,   m < n
The detailed steps of factor analysis are as follows, with its pseudocode presented in Algorithm 1:
  • Standardize the data, transforming it into a distribution with a mean of zero and a standard deviation of one.
  • Compute the correlation matrix between variables.
  • Employ Principal Component Analysis to determine the factor loading matrix and extract factors.
  • Apply orthogonal rotation (Varimax) to the factors to enhance their interpretability.
  • Calculate factor scores, estimating the common factors based on original variables.
The data sample for this study has dimensions of 1673 × 33 , encompassing parameters like fabric count, weight, temperatures for ovens 1–10#, 1~8# spindle width, 1~10# fan speed, machine speed, upper feed, and workshop temperature. For ease of representation, these parameters are labeled p1 through p33. Utilizing FA, the scree plot, as shown in Figure 5a, represents the variance explained by each factor. According to the elbow method, retaining five factors is deemed optimal. The heatmap of factor loadings is illustrated in Figure 5b. Factor loadings, being the correlation coefficients between original variables and latent factors, signify the weight of original variables on latent factors. A value closer to one in absolute terms indicates a stronger correlation between the factor and the observed variable, with the sign indicating a positive/negative correlation. Conversely, a value closer to zero suggests a weaker correlation. As evident from the heatmap, the distribution of the factor loading matrix aligns with the practical implications of each process parameter.
Algorithm 1: Factor analysis data preprocessing
Input: Production data X w .
-
Workshop data X t .
-
Process data X p .
Output: Latent factor loading matrix F .
The procedure of factor analysis:
  1: Initialize the data X X w , X t , X p m × n .
  // m × n represents the size of data, with the value of 1673 and 33
  2:  While  k < n  do:
  4:     R X T X .
  5:     R V Λ V 1 .
  //     V and Λ represent the eigenvector matrix and diagonal matrix of R .
  6:     V k V [ : , a r g   s o r t ( Λ ) [ k : ] ] .
  7:     V k * V k T v a r i m a x .
  //     T v a r i m a x represents the varimax orthogonal rotation matrix.
  8:     T k X V k * .
  9:     F k T k T X .
   10:     e k R M S E ( X , T k F k T ) .
   11:  End While.
   12:   T = T k    where    k = arg min e i  for  i = 1,2 , , n .
   13:   F = F k    where    k = arg min e i  for  i = 1,2 , , n .

4.3. Evaluation Metrics

In the fields of machine learning, statistics, and other data-driven research, error metrics are key tools for evaluating the accuracy of models or research results. Selecting appropriate metrics can help judge the quality and performance of a model, compare the effects of different models or methods, assist in adjusting model parameters, and provide decision-makers with information about the reliability of the model. Commonly used error evaluation metrics include the RMSE (Root Mean Square Error), MSE (Mean Square Error), MAE (Mean Absolute Error), GM-MSE (Geometric Mean of Mean Square Error), and Cross-Entropy Loss (CE).
e R M S E = 1 n Σ y y ^ 2
e M S E = 1 n Σ y y ^ 2
e M A E = 1 n Σ y y ^
e G M M S E = i = 1 n e M S E , i n
e C E = Σ y l o g   y ^ + 1 y log 1 y ^
Among them, CE is commonly used to measure the “distance” between two probability distributions and is mainly applied to classification problems. However, energy consumption prediction is a typical regression problem. The RMSE, MSE, and MAE are commonly used for regression problems to quantify the difference between predicted and actual values. The GM-MSE is the geometric mean of the MSE values for all outputs, which is sensitive to the differences in MSE values for each output. Both the MAE and MSE are not sensitive enough to outliers and find it challenging to distinguish between different error patterns. The RMSE provides an error measurement on the same scale as the original data and will be used as the primary metric in this study to measure the error between actual and predicted values.

4.4. Construction of the Pi-MMoE Model for Textile Dyeing and Printing

As stated in Section 3, we introduced the pi-MMoE model. For the textile industry, we developed this model, specifically targeting multi-task forecasting for steam and electricity consumption. Additionally, we provided the construction of the loss function based on physical equations and detailed the process of model training.
In the analysis of complex physical and engineering systems, data acquisition costs are prohibitively high. Under this small data condition, most advanced machine learning techniques lack robustness and cannot guarantee that their predictions conform to actual production rules. Utilizing structured physical information to guide and optimize data-driven models, enhancing the model’s generalization capability, and ensuring its output conforms to physical laws is a primary method for optimizing machine learning models in small data conditions. Generally speaking, assisting data-driven modeling by integrating physical laws into the loss function can guide and constrain the model’s learning process. This ensures that model predictions not only fit the data but also align with physical principles, enhancing the model’s reliability. In summary, the physical information loss function serves as a bridge between data-driven machine learning and domain-specific knowledge, leveraging the strengths of both to produce models that are both accurate and physically consistent.
Considering that the energy consumption prediction in the textile dyeing and finishing industry is mainly oriented towards electricity and steam usage and combined with the principle of energy conservation, we derive the loss function L p i based on physical information:
L p i = i 0 y ^ s t e a m m 0 i 1 τ w 0 τ w 1 + i 2 1 τ w 0 T 1 T
L = M S E y e l e c t r i c , y ^ e l e c t r i c + λ 1 M S E y s t e a m , y ^ s t e a m + λ 2 L p i
where y e l e c t r i c , y ^ e l e c t r i c , y s t e a m , and y ^ s t e a m represent the true and predicted values for electricity and steam, respectively. M S E ( ) represents the Mean Square Error, and λ 1 and λ 2 denote the weights for the steam prediction loss, satisfying the condition λ 1 + λ 2 = 1 . The pi-MMoE model proposed in the article is shown as Algorithm 2.
Algorithm 2: Pi-MMoE multi-task modeling
Input: Temperature of the 10th drying oven T o .
-
Workshop temperature T .
-
The moisture content of the fabric m w = m × τ .
-
Loading matrix F .
-
Task number K .
-
Expert number N .
Output: Electric and Steam consumption Y .
The procedure of Pi-MMoE Modeling:
  1: Initialize the input data X [ F , T o , T , m w ] .
  2:  While training loss has not converged do:
  4:    h t a n h ( w s X + b s ) .
  5:    g k t a n h ( w g , k h + b g , k )   for   k = 1 , , K .
  6:    e i t a n h ( w e , i h + b e , i   )    for  i  = 1 , , N .
  7:    y k ^ Σ i = 1 N g k , i × e i     for   k = 1 , , K .
  //    The g k , i represents the i -th element of g k .
  8:    L M S E , k M S E ( y k , y k ^ ) .
  //    In the model, the K and N get the value of 2 and 6 .
  //      y 1 ^ and y 2 ^ represent the predicted value of electric and steam consumption.
  9:    y p i 1.31 × 10 3 m w ( T o T ) .
   10:    L p i M S E y 2 ^ , y p i .
   11:    L L M S E , 1 + 0.5 L M S E , 2 + 0.5 L p i .
   12:    w s , g , e A d a m ( w s , g , e L ) .
   13:    b s , g , e A d a m ( b s , g , e L ) .
   14:  End While.
To address the need for energy forecasting and consumption reduction in enterprises, this study introduces the MMoE model tailored for the primary energy consumption of electricity and steam during the stentering process. After obtaining the factor score matrix T through FA (factor analysis), when facing the new production data X 0 , we can derive the latent factor loading matrix F 0 .
F 0 = T T X 0
In addition, parameters such as the temperature of the 10th oven, room temperature, and fabric moisture content will be introduced to supplement physical information as model inputs. The main body of MMoE consists of the input layer, expert layer, gating unit layer, tower layer, and output layer. The expert layer is composed of six expert networks; the gating unit layer consists of two gating units; and the tower layer comprises two tower networks. Both the expert networks and tower networks use Relu as the activation function, while the gating unit employs Softmax as its activation function. Finally, the output layer simultaneously outputs the predicted values of electrical energy consumption and steam consumption of the setting machine. The model’s structure and specific parameters are shown in Table 2.

4.5. Experimental Setup

We will provide detailed descriptions of the experimental design and settings to verify the effectiveness and feasibility of the proposed method. To ensure the comprehensiveness and reliability of the experimental results, we have designed the following experimental directions:
Scenario 1—Effectiveness of Physical Information: By comparing the prediction performance of the MLP, ResNet, and MMoE before and after integrating physical information, we aim to verify the improvement effect of incorporating physical information on model performance. This is also to compare these algorithms to determine which model is optimal. The constructed MLP, ResNet, and MMoE model structures are shown in Table 3.
Scenario 2—Impact of Expert Net: The number of expert networks is the most crucial parameter in MMoE. Increasing the number of experts can enhance the model’s capacity but also lead to overfitting. To further verify its impact on the pi-MMoE model’s predictive performance, we will compare model prediction results by setting different numbers of expert networks to explore their influence on predictive performance. Generally, we have set the number of expert networks to 2, 4, 6, and 8, with each expert network having 8 neurons in the input layer, 4 neurons in the output layer, and the number of neurons in the hidden layer set to 8 and 16, respectively. These models are denoted as pi-MMoE L-N, where L represents the number of expert networks and N indicates the number of neurons in the expert network’s hidden layer. For example, pi-MMoE 2-8 would denote a pi-MMoE model that contains two expert networks with eight neurons in their hidden layers. The structures of these models are shown in Table 4.
Scenario 3—Performance of Multi-Task Modeling: The MMoE model was initially designed to handle multi-task learning problems. By ensuring that each task is handled only by specific experts, it can be equated to multiple single-task models. This change requires designing a gate network for each task. For instance, in predicting two tasks, for the gate network of Task A, Expert 1 is assigned a weight of one, and Expert 2 is assigned a weight of zero. Conversely, for the gating network of Task B, Expert 1 is assigned a weight of zero, and Expert 2 is given a weight of one. This ensures that the output of each task is the direct output of its expert network. Given the practical application scenarios of the integrated energy manufacturing energy consumption prediction model, we compare the predictive performance of single energy consumption modeling with multi-task energy consumption modeling. We aim to explore the predictive performance of the proposed method for multi-source energy consumption. Moreover, to investigate the influence of the gate unit on predictive performance, we also set up a control group with a single gate unit. We used pi-MMoE STT, pi-MMoE SGG MTT, and pi-MMoE MGG MTT to represent the single-task, single-gate multi-task, and multi-gate multi-task pi-MMoE models, respectively. Their model structures are illustrated in Table 5. We aim to explore the performance of the proposed method in predicting multi-source energy consumption.
In summary, through these experiments, we aim to demonstrate the effectiveness and robustness of the proposed method from various perspectives. We believe that with these detailed experimental settings, we can provide readers with a comprehensive and in-depth understanding and also offer valuable references for future research.
In there, we will partition the dataset into training, validation, and test sets at a ratio of 0.8:0.1:0.1, based on the 1,673 production historical data records from the textile dyeing and finishing factory. Using data preprocessing methods such as factor analysis and the addition of physical information, we form an input dataset of size 1673 × 8.

4.6. Experimental Results

4.6.1. Effectiveness of Physical Information

Table 6 presents the Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) for predicting electricity and steam consumption for each model, alongside the GM-MSE results, which reflect the overall predictive performance for both types of energy consumption. This comprehensive table allows for a direct evaluation of each model’s prediction accuracy by presenting all pertinent performance indicators together. Table 7 displays the results of the Wilcoxon signed-rank test comparing the pi-MMoE model with other models for predicting electricity and steam consumption. The p-value, a measure in statistics, quantifies the evidence against a null hypothesis, which typically represents no effect or no difference. Specifically, the p-value indicates the probability of observing the given result, or something more extreme, assuming that the null hypothesis is true. This test reveals the differences between the pi-MMoE model and other models. A p-value less than or equal to 0.05 indicates a significant difference between the models; otherwise, it suggests no significant difference.
From Table 6, we can observe that, whether it is predicting electricity or steam consumption, models integrating physical information (Pi-MLP, Pi-ResNet, and Pi-MMoE) generally perform better than their respective standard models. Particularly, Pi-MMoE, with MAE and RMSE scores of 0.10 and 0.12, respectively, for electricity consumption prediction and 0.07 and 0.08 for steam consumption prediction, outperforms other models, showcasing the best predictive performance. Furthermore, the Pi-MMoE model excels in multi-output predictions, achieving the most favorable GM-MSE score of 0.01, indicating a robust overall performance. From Table 7, it is evident that there is a significant difference in predictive ability between Pi-MMoE and most other models for electricity and steam energy predictions. However, when compared to MMoE, the differences in models for both electricity and steam energy are not significant.

4.6.2. Impact of Expert Net

For each model, Table 8 presents the MAE, RMSE, and GM-MSE of the Pi-MMoE model with different expert layers in predicting electricity and steam consumption. Based on these two indicators, we can evaluate the impact of the expert layers on the prediction accuracy of the model. Figure 6 provides a comparison of their prediction performance. Ideally, all data points should be closely distributed along the dashed line (ideal line), indicating the consistency between predicted and actual values. Table 9 performs the Wilcoxon signed-rank test to compare the pi-MMoE model, which contains a 6-layer expert network with 16 hidden neurons, with other models in predicting electricity and steam energy consumption, respectively. This reveals the differences between it and other models.
From Table 6, we can observe that, whether for electricity or steam consumption predictions, the predictive performance improves with the increase in the number of expert layers or the deepening of the expert network’s depth. The best performance is seen when the number of expert networks is 6 or 8 and the number of hidden nodes is 16, achieving MAE and RMSE scores of 0.10 and 0.11 for electricity consumption prediction, and 0.07 and 0.08 for steam consumption prediction, respectively. However, when the number of expert networks further increases to 8, the performance improvement is often not as pronounced. Furthermore, pi-MMoE 6-16 and pi-MMoE 8-16 excel in multi-output predictions, achieving the most favorable GM-MSE score of 0.01, indicating a robust overall performance. From Table 9, we can deduce that there is a significant difference in performance between pi-MMoE 6-16 and pi-MMoE 2-16, pi-MMoE 4-16, and pi-MMoE 8-16. In contrast, when compared to pi-MMoE 6-8, there is no significant difference in their performance.

4.6.3. Performance of Multi-Task Modeling

Table 10 presents the MAE, RMSE, and GM-MSE of the pi-MMoE model in single-output architecture, single-gating unit architecture, and its structure for predicting electricity and steam consumption. Based on these two indicators, we can evaluate the prediction accuracy of each model. Figure 7 provides a comparison of their predictive performance. Table 11 outlines the Wilcoxon signed-rank test comparing the model presented in this paper with other models for electricity and steam energy predictions. This reveals the differences between it and other models.
From Table 10, we can observe that the model proposed in this paper exhibits superior predictive performance compared to other models. It achieves MAE and RMSE scores of 0.10 and 0.12 for electricity consumption prediction and 0.07 and 0.08 for steam consumption prediction, respectively, which are notably lower than those of other models. The GM-MSE of the model presented in this paper is 0.01, significantly lower than the other models, further attesting to its superior performance. The RMSE box plot in Figure 7 shows that the pi-MMoE multi-gate unit architecture performs the best in predicting electricity and steam consumption, with the lowest median RMSE, the narrowest distribution range, and the fewest outliers. It significantly outperforms the single-output architecture and the single-gate unit architecture, demonstrating higher prediction accuracy and result stability. Table 11 also shows that the model in this paper demonstrates considerable differences when compared to the single-output and single-gate multi-output models for electricity and steam energy predictions.

4.7. Discussion

Based on the actual production history data of the textile dyeing and finishing factory, we further organized and processed the data by combining factor analysis and physical information supplementation, among other data preprocessing techniques. To verify the effectiveness and feasibility of the proposed method, we conducted a series of experiments.
In Scenario 1, we observed the enhancement effect of incorporating physical information on model performance. By comparing the predictive performance of MLP, ResNet, and MMoE before and after merging physical information, the results showed that physical information indeed can enhance the predictive performance of the model to some extent. This phenomenon reflects that in specific production environments, physical information contains important information not directly observed during the production process, which can assist the model in making more accurate predictions. Moreover, this result also indicates that different models have various ways of processing and utilizing physical information.
Scenario 2 explored the impact of the structure of expert networks on the predictive performance of the pi-MMoE model. The findings showed that increasing the structure of experts indeed enhances the model’s capacity, but having too complex experts will cause overfitting. This aligns with our intuition: the larger the model capacity, the stronger its representational power. However, it is also more susceptible to noise. Therefore, selecting the appropriate number of experts and ensuring the model has enough representational capacity while preventing overfitting is a significant challenge in model design.
In Scenario 3, we delved into the potential of the MMoE model in multi-task learning. The experimental results demonstrated that by setting unique gating networks for each task to ensure that each task’s output is the direct output of its expert network, simulating a single-output model structure, the multi-output prediction model can more comprehensively capture and learn the correlations between different tasks, thereby enhancing overall predictive performance. This method presents an innovative approach: by leveraging the structural advantages of multi-output models, we can address multi-task learning problems more efficiently, yielding better results in predictive tasks.
In summary, through these experiments, we can derive the following in-depth insights and patterns:
  • Physical information is highly valuable in specific production environments. Different models have varying methods and outcomes when utilizing physical information, and selecting the right model is key.
  • Choosing the appropriate expert network is a significant challenge in model design. It is vital to ensure the model has enough representational power while preventing overfitting.
  • In the field of multi-task learning, guiding model training through inter-task correlations can further enhance model predictive performance, making it a promising research direction.
Lastly, it is worth mentioning that these experiments provide us with a wealth of valuable insights, but no model or method can be perfect. In real-world applications, further optimization and adjustments to the model and methods are required based on specific production environments and needs.

4.8. Pi-MMoE-Supported Process Optimization to Energy Efficiency

4.8.1. Determinants of Energy Efficiency

The heatmap of each output’s sensitivity to input features is illustrated in Figure 8. It is evident that concerning the outputs (energy and steam), there is a significant correlation between the temperature of oven #10, workshop temperature, and fabric moisture content with steam consumption. Specifically, the temperature of oven #10 has a correlation as high as 0.74 with steam consumption, the workshop temperature has a correlation of 0.79 with steam, and the fabric moisture content’s correlation with steam is 0.60. The high correlation of these three variables demonstrates their pivotal role in predicting steam consumption. In contrast, these input features have a relatively lower correlation with electrical energy consumption. This further confirms our previous observation that considering the physical laws of heat transfer, there is a strong correlation between steam consumption and variables such as the temperature of oven #10, ambient temperature, and fabric moisture content. This pronounced correlation offers compelling evidence for the high performance of steam consumption prediction, which stands out even more when compared to the prediction of electrical energy consumption.

4.8.2. Process Optimization for Energy Efficiency

Acknowledging the uncontrollable nature of environmental and production information, the PSO optimization is confined to process parameters. This encompasses spindle widths, vehicle speeds, fan speeds for each fan, and oven speeds. By harnessing the FA filtering method, PSO seeks to optimize latent factors Factor 1, Factor 2, and Factor 3. Each particle within the swarm embodies a potential combination of these factors. The position of the particle within the swarm articulates a specific configuration of these latent factors. Through iterative updates and fitness evaluations, the objective becomes pinpointing the optimal configuration of Factor 1, Factor 2, and Factor 3 that minimizes a predefined target function. Due to the intrinsic mechanics of the setting process, constraints are set: spindle width between 150–210 cm, vehicle speed in the 18–55 m/min range, wind speed duty cycle between 55–100%, and oven temperatures spanning 130–200 °C. To reconcile the unit discrepancies between electrical and steam consumption, this study adopts kilogram standard coal (kgce) as the unified measurement standard, with a conversion factor of 29,307.6 for kgce to kJ. Typically, 1 kW·h is approximately equivalent to 0.12 kgce, and the enthalpy of 3.2 MPa, 380 °C medium-pressure steam is around 3200 kJ/kg, which is roughly 0.11 kgce in this factory setting. This study aims to predict energy consumption in kgce for both electricity and steam. The PSO oriented towards stentering machine process parameter optimization is represented as follows:
F p = λ 1 × η × k e × G e l e c t r i c p + 1000 λ 2 × k s × G s t e a m p
where G e l e c t r i c ( p ) and G s t e a m ( p ) represent the model’s predictive output in kgce for electrical and steam consumption, respectively; p represents the population; k e and k s denote the conversion coefficients for electricity and steam to kgce, valued at 0.12 and 0.11 , respectively; λ 1 and λ 2 represent decision coefficients in energy optimization management; and η denotes the inductive coefficient with a value of 80 .
In the process optimization with energy conservation as the objective, the dynamic fluctuations in the unit prices of various energy sources are closely related to their peak and off-peak usage periods. Therefore, the energy consumption optimization strategies summarized for the textile dyeing and finishing industry are as follows:
(a)
Optimal total energy consumption: This situation arises when the unit price of electricity is on par with the steam.
(b)
Prioritize reducing electricity consumption while ensuring overall energy consumption. This situation occurs when the unit price of electricity is higher than the steam.
(c)
Prioritize reducing steam consumption while ensuring overall energy consumption: This situation occurs when the unit price of electricity is lower than the steam.
To adapt to the strategies to adjust the decision coefficients for each energy source in the objective function. Taking the production data from May 6, 2023, of the enterprise as the subject of study.
Table 12 shows the results of optimizing the process under three distinct strategies: optimal total energy consumption, prioritizing electricity reduction, and prioritizing steam reduction. Across all strategies, the oven temperatures were decreased from the original 180 °C, with variations based on each strategy’s focus. The spindle width for ovens 1# to 8# was uniformly adjusted to 157 cm. As a result of these optimizations: (a) total energy consumption saw a 7.03% decrease in electricity and a 6.67% drop in steam; (b) the electricity-focused strategy achieved an 8.98% decrease in electricity and 4.67% in steam; and (c) the steam-focused strategy resulted in a 5.86% reduction in electricity and an 8.67% decrease in steam. The impact of these optimization strategies on energy consumption is mainly reflected in:
-
Oven temperature: By lowering the oven temperature, the energy required for heating is reduced, directly lowering both electricity and steam consumption.
-
Spindle width: Adjusting the spindle width ensures more uniform and efficient fabric transfer, reducing energy waste and improving production efficiency.
-
Fan speed: Optimizing fan speed significantly reduces motor power consumption while ensuring adequate ventilation and production quality.
-
Machine speed: Controlling machine speed optimizes the pace and energy consumption of fabric movement, reducing unnecessary energy waste.
-
Total energy consumption strategy: When the unit prices of electricity and steam are comparable, a comprehensive adjustment of all parameters achieves the lowest overall energy consumption.
-
Electricity priority strategy: When the unit price of electricity is higher, prioritizing the reduction of electricity consumption through optimized parameter combinations reduces total electricity usage.
-
Steam priority strategy: When the unit price of electricity is lower, prioritizing the reduction of steam consumption through precise parameter control reduces steam usage.
These adjustments demonstrate the efficacy of tailored strategies in energy conservation.

5. Conclusions

This paper proposes a physics-informed multi-gate mixture-of-experts (pi-MMoE) model for multi-task energy consumption forecasting in manufacturing with a hybrid energy supply. The model aims to enhance predictive accuracy and model interpretability by integrating physical information and leveraging a multi-task learning framework, thus promoting sustainable manufacturing. This research was motivated by the need for more comprehensive energy consumption forecasting in manufacturing industries that utilize multiple energy sources concurrently. Relying solely on data-driven approaches often overlooks the interconnectedness and conversions between different energy types during production. To address this gap, we introduce an approach that combines multi-task learning with physical insights on energy and material transformations. We first analyzed the production processes and energy usage patterns in typical hybrid energy industries like minerals, chemicals, and textiles. This provided context on where and how different energy types are utilized. Next, guided by domain knowledge and thermodynamic principles, we extracted key physical equations representing the energy balance and conversions. These equations were later incorporated into the loss function to constrain model training. The pi-MMoE model itself has a multi-gate mixture-of-experts architecture. The expert networks extract shared knowledge from inputs, while the gating networks assign task-specific weights. This setup balances inter-task knowledge sharing with specialized learning. Integrating physical insights makes the model conform to valid energy conversion rules, enhancing reliability. We demonstrated the effectiveness of the pi-MMoE model through a case study in a textile factory. The model achieved superior accuracy in predicting electricity and steam consumption compared to the MLP, ResNet, and vanilla MMoE models. Ablation studies also verified the benefits of physical information and multi-task learning. The prediction errors (e.g., the RMSE) in electricity and steam consumption decreased by 14.28% and 27.27%, compared to the conventional MMoE, pi-ResNet, and pi-MLP multi-output prediction models. Furthermore, we employed PSO guided by pi-MMoE predictions to optimize process parameters. Under various energy-saving strategy constraints, we achieved energy consumption savings of approximately 5% to 9% in electricity and 4% to 8% in steam. This not only reduced energy consumption but also lowered carbon emissions and other harmful emissions, significantly promoting sustainable development in manufacturing.
These promising results highlight the potential of combining first principles and data-driven methods. While we focused on energy forecasting, the framework could likely be extended to other aspects of production optimization using appropriate physical knowledge. As manufacturing grows more complex, such hybrid physics–AI approaches will become increasingly valuable for sustainable manufacturing. However, there remain areas for improvement. More refined parameter tuning could further boost accuracy. Extracting physical insights can also be non-trivial for intricate processes. And uncertainty representation is still limited. Future work could explore Bayesian neural networks to address stochasticity. The loss function formulation could be enhanced by adaptively weighing different terms. In closing, this research makes valuable contributions toward integrating domain knowledge into data-driven models. By merging physical and statistical insights, pi-MMoE demonstrates enhanced interpretability and accuracy for multi-objective forecasting tasks. The proposed methodology helps create more sustainable and efficient manufacturing through comprehensive energy optimization. We believe hybrid physics-AI approaches will play a major role in advancing industrial intelligence and promoting sustainability.
In future work, we will delve into the mechanistic knowledge involved in the production process, refine the rule system for physical information extraction, and establish a more realistic method for extracting physical information to support sustainable manufacturing. By integrating machine learning or deep learning technologies, we aim to capture and predict energy patterns more accurately, contributing to energy efficiency and sustainability. Additionally, establishing and improving the carbon accounting model for enterprise production, with energy consumption benchmarks at the core, and combining expert knowledge and human experience to form an energy consumption optimization decision system are key research directions. These efforts will further enhance the sustainability of manufacturing processes by reducing carbon emissions and optimizing resource utilization.

Author Contributions

Methodology, M.Y. (Mukun Yuan); Software, J.L. (Jian Liu); Validation, Z.C.; Formal analysis, Q.G. and J.L. (Jian Li); Investigation, M.Y. (Mingzhe Yuan); Resources, G.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by [China Postdoctoral Science Foundation] grant number [2023M730779], and [Guangdong Provincial Science and Technology Project] grant number [2023A0505050090].

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Katina, P.F.; Cash, C.T.; Caldwell, L.R.; Beck, C.M.; Katina, J.J. Advanced Manufacturing Management: A Systematic Literature Review. Sustainability 2023, 15, 4702. [Google Scholar] [CrossRef]
  2. Gomes, A.C.L.; Ravetti, M.G.; Carrano, E.G. Multi-objective matheuristic for minimization of total tardiness and energy costs in a steel industry heat treatment line. Comput. Ind. Eng. 2021, 151, 106929. [Google Scholar] [CrossRef]
  3. Han, Y.; Geng, Z.; Zhu, Q.; Qu, Y. Energy efficiency analysis method based on fuzzy DEA cross-model for ethylene production systems in chemical industry. Energy 2015, 83, 685–695. [Google Scholar] [CrossRef]
  4. Bai, Y.; Xie, K.; Shao, C.; Hu, B.; Yu, X.; Hu, Y. Unreliability tracing of power systems with high penetration of wind power based on a temporal decomposition model. CSEE J. Power Energy Syst. 2023, 1–14. [Google Scholar] [CrossRef]
  5. Feng, T. Do Intelligent Manufacturing Concerns Promote Corporate Sustainability? Based on the Perspective of Green Innovation. Sustainability 2023, 15, 10958. [Google Scholar] [CrossRef]
  6. Estrada, O.; López, I.D.; Hernández, A.; Ortíz, J.C. Energy gap method (EGM) to increase energy efficiency in industrial processes: Successful cases in polymer processing. J. Clean. Prod. 2018, 176, 7–25. [Google Scholar] [CrossRef]
  7. Wei, N.; Li, C.; Peng, X.; Li, Y.; Zeng, F. Daily natural gas consumption forecasting via the application of a novel hybrid model. Appl. Energy 2019, 250, 358–368. [Google Scholar] [CrossRef]
  8. Awan, M.R.; González Rojas, H.A.; Hameed, S.; Riaz, F.; Hamid, S.; Hussain, A. Machine Learning-Based Prediction of Specific Energy Consumption for Cut-Off Grinding. Sensors 2022, 22, 7152. [Google Scholar] [CrossRef]
  9. Rakib, M.I.; Saidur, R.; Mohamad, E.N.; Afifi, A.M. Waste-heat utilization–the sustainable technologies to minimize energy consumption in Bangladesh textile sector. J. Clean. Prod. 2017, 142, 1867–1876. [Google Scholar] [CrossRef]
  10. Qin, Y.; Wu, L.; Zheng, J.; Li, M.; Jing, Z.; Wu, Q.H.; Zhou, X.; Wei, F. Optimal operation of integrated energy systems subject to coupled demand constraints of electricity and natural gas. CSEE J. Power Energy Syst. 2019, 6, 444–457. [Google Scholar]
  11. Liu, Y.; Wu, L.; Li, J. A two-stage peer-to-peer energy trading model for distribution systems with participation of utility. CSEE J. Power Energy Syst. 2021, 7, 893–902. [Google Scholar]
  12. Shi, X.; Huang, G.; Hao, X.; Yang, Y.; Li, Z. Sliding window and dual-channel CNN (SWDC-CNN): A novel method for synchronous prediction of coal and electricity consumption in cement calcination process. Appl. Soft Comput. 2022, 129, 109520. [Google Scholar] [CrossRef]
  13. Huang, Z.; Yang, C.; Zhou, X.; Yang, S. Energy consumption forecasting for the nonferrous metallurgy industry using hybrid support vector regression with an adaptive state transition algorithm. Cogn. Comput. 2020, 12, 357–368. [Google Scholar] [CrossRef]
  14. Ramos, P.V.B.; Villela, S.M.; Silva, W.N.; Dias, B.H. Residential energy consumption forecasting using deep learning models. Appl. Energy 2023, 350, 121705. [Google Scholar] [CrossRef]
  15. Jana, R.K.; Ghosh, I.; Sanyal, M.K. A granular deep learning approach for predicting energy consumption. Appl. Soft Comput. 2020, 89, 106091. [Google Scholar] [CrossRef]
  16. Kahraman, A.; Kantardzic, M.; Kotan, M. Dynamic Modeling With Integrated Concept Drift Detection for Predicting Real-Time Energy Consumption of Industrial Machines. IEEE Access 2022, 10, 104622–104635. [Google Scholar] [CrossRef]
  17. He, Y.; Wu, P.; Li, Y.; Wang, Y.; Tao, F.; Wang, Y. A generic energy prediction model of machine tools using deep learning algorithms. Appl. Energy 2020, 275, 115402. [Google Scholar] [CrossRef]
  18. Giampieri, A.; Ling-Chin, J.; Ma, Z.; Smallbone, A.; Roskilly, A. A review of the current automotive manufacturing practice from an energy perspective. Appl. Energy 2020, 261, 114074. [Google Scholar] [CrossRef]
  19. Mikulčić, H.; Klemeš, J.J.; Vujanović, M.; Urbaniec, K.; Duić, N. Reducing greenhouse gasses emissions by fostering the deployment of alternative raw materials and energy sources in the cleaner cement manufacturing process. J. Clean. Prod. 2016, 136, 119–132. [Google Scholar] [CrossRef]
  20. Zhang, Y.; Yang, Q. A survey on multi-task learning. IEEE Trans. Knowl. Data Eng. 2021, 34, 5586–5609. [Google Scholar] [CrossRef]
  21. Zhang, Y.; Yang, Q. An overview of multi-task learning. Natl. Sci. Rev. 2018, 5, 30–43. [Google Scholar] [CrossRef]
  22. Vandenhende, S.; Georgoulis, S.; Van Gansbeke, W.; Proesmans, M.; Dai, D.; Van Gool, L. Multi-task learning for dense prediction tasks: A survey. IEEE Trans. Pattern Anal. Mach. Intell. 2021, 44, 3614–3633. [Google Scholar] [CrossRef]
  23. Guo, M.; Haque, A.; Huang, D.-A.; Yeung, S.; Fei-Fei, L. Dynamic task prioritization for multitask learning. In Proceedings of the European Conference on Computer Vision (ECCV), Munich, Germany, 8–14 September 2018; pp. 270–287. [Google Scholar]
  24. Shinohara, Y. Adversarial multi-task learning of deep neural networks for robust speech recognition. In Proceedings of the Interspeech, San Francisco, CA, USA, 8–12 September 2016; pp. 2369–2372. [Google Scholar]
  25. Liu, C.-L.; Tseng, C.-J.; Huang, T.-H.; Yang, J.-S.; Huang, K.-B. A multi-task learning model for building electrical load prediction. Energy Build. 2023, 278, 112601. [Google Scholar] [CrossRef]
  26. Ding, B.; Wang, F.; Chen, C.; Wang, S. Urban monthly power load forecasting based on economy-meteorology-gas demand coupling. Electr. Eng. 2022, 104, 3497–3507. [Google Scholar] [CrossRef]
  27. Zendehboudi, S.; Rezaei, N.; Lohi, A. Applications of hybrid models in chemical, petroleum, and energy systems: A systematic review. Appl. Energy 2018, 228, 2539–2566. [Google Scholar] [CrossRef]
  28. Raissi, M.; Perdikaris, P.; Karniadakis, G.E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 2019, 378, 686–707. [Google Scholar] [CrossRef]
  29. Lawal, Z.K.; Yassin, H.; Lai, D.T.C.; Che Idris, A. Physics-informed neural network (PINN) evolution and beyond: A systematic literature review and bibliometric analysis. Big Data Cogn. Comput. 2022, 6, 140. [Google Scholar] [CrossRef]
  30. Abubakr, M.; Abbas, A.T.; Tomaz, I.; Soliman, M.S.; Luqman, M.; Hegab, H. Sustainable and Smart Manufacturing: An Integrated Approach. Sustainability 2020, 12, 2280. [Google Scholar] [CrossRef]
  31. Huang, B.; Wang, J. Applications of physics-informed neural networks in power systems-a review. IEEE Trans. Power Syst. 2022, 38, 572–588. [Google Scholar] [CrossRef]
  32. Gokhale, G.; Claessens, B.; Develder, C. Physics informed neural networks for control oriented thermal modeling of buildings. Appl. Energy 2022, 314, 118852. [Google Scholar] [CrossRef]
  33. Almajid, M.M.; Abu-Al-Saud, M.O. Prediction of porous media fluid flow using physics informed neural networks. J. Pet. Sci. Eng. 2022, 208, 109205. [Google Scholar] [CrossRef]
  34. McGowan, E.; Gawade, V.; Guo, W. A physics-informed convolutional neural network with custom loss functions for porosity prediction in laser metal deposition. Sensors 2022, 22, 494. [Google Scholar] [CrossRef] [PubMed]
  35. Ma, J.; Zhao, Z.; Yi, X.; Chen, J.; Hong, L.; Chi, E.H. Modeling task relationships in multi-task learning with multi-gate mixture-of-experts. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, London, UK, 19–23 August 2018; pp. 1930–1939. [Google Scholar]
  36. Hasanbeigi, A.; Price, L. A review of energy use and energy efficiency technologies for the textile industry. Renew. Sustain. Energy Rev. 2012, 16, 3648–3665. [Google Scholar] [CrossRef]
Figure 1. Production mechanisms and energy consumption of industries: mineral, textile, and chemical industries. (a) Production process in the mineral industry; (b) production process in the textile industry; and (c) production process in the chemical industry.
Figure 1. Production mechanisms and energy consumption of industries: mineral, textile, and chemical industries. (a) Production process in the mineral industry; (b) production process in the textile industry; and (c) production process in the chemical industry.
Sustainability 16 07259 g001
Figure 2. The proposed physics-informed multi-task modeling approach.
Figure 2. The proposed physics-informed multi-task modeling approach.
Sustainability 16 07259 g002
Figure 3. The architecture of the physics-informed multi-gate mixture-of-experts model.
Figure 3. The architecture of the physics-informed multi-gate mixture-of-experts model.
Sustainability 16 07259 g003
Figure 4. Components and workflow of the stentering machine.
Figure 4. Components and workflow of the stentering machine.
Sustainability 16 07259 g004
Figure 5. Scree plot of variance by each factor and heatmap of factor loadings. (a) RMSE curve; (b) factor loadings heatmap.
Figure 5. Scree plot of variance by each factor and heatmap of factor loadings. (a) RMSE curve; (b) factor loadings heatmap.
Sustainability 16 07259 g005
Figure 6. Prediction performance for pi-MMoE with different expert networks.
Figure 6. Prediction performance for pi-MMoE with different expert networks.
Sustainability 16 07259 g006
Figure 7. Comparison of the predictive performance of the pi-MMoE model for electricity and steam consumption.
Figure 7. Comparison of the predictive performance of the pi-MMoE model for electricity and steam consumption.
Sustainability 16 07259 g007
Figure 8. Heatmap of sensitivity of each output to input features.
Figure 8. Heatmap of sensitivity of each output to input features.
Sustainability 16 07259 g008
Table 1. Data details of production records for stentering machines.
Table 1. Data details of production records for stentering machines.
Number of InstancesProduction Data DetailsProcess Data Details
1673Date, machine number, production type, quantity and weight of fabric, process number, and workshop temperature1#~10# oven temperature, 1#~8# spindle width, 1#~10# fan speed, machine speed, and upper feed
Table 2. Structure and parameters of the pi-MMoE model.
Table 2. Structure and parameters of the pi-MMoE model.
Gate UnitsInput LayerExpert LayerTower LayerOutput Layer
Total21621
Units8-488-16-48-42
Table 3. Parameter setting for verification of the effectiveness of physical information.
Table 3. Parameter setting for verification of the effectiveness of physical information.
ModelItemDetailsInput SizeOutput Size
MLPBatch size328 × 12 × 1
Hidden layer2
Hidden neurons32-32
Activation functionRelu
Pi-MLPBatch size32
Hidden layer2
Hidden neurons32-32
Activation functionRelu
ResNetBatch size32
Residual block2
Convolution kernel3 × 1
Activation functionRelu
Pi-ResNetBatch size32
Residual block2
Convolution kernel3 × 1
Activation functionRelu
MMoEBatch size32
Gate units2
Expert layers6
Expert neurons16
Activation functionRelu
Pi-MMoEBatch size32
Gate units2
Expert layers6
Expert neurons16
Activation functionRelu
Table 4. Parameter setting for verification of the impact of expert net.
Table 4. Parameter setting for verification of the impact of expert net.
ModelItemDetailsInput SizeOutput Size
Pi-MMoE
2-8
Expert layers28 × 12 × 1
Expert hidden neurons8
Pi-MMoE
4-8
Expert layers4
Expert hidden neurons8
Pi-MMoE
6-8
Expert layers6
Expert hidden neurons8
Pi-MMoE
8-8
Expert layers8
Expert hidden neurons8
Pi-MMoE
2-16
Expert layers2
Expert hidden neurons16
Pi-MMoE
4-16
Expert layers4
Expert hidden neurons16
Pi-MMoE
6-16
Expert layers6
Expert hidden neurons16
Pi-MMoE
8-16
Expert layers8
Expert hidden neurons16
Table 5. Parameter setting for verification of the performance of multi-task modeling.
Table 5. Parameter setting for verification of the performance of multi-task modeling.
ModelsItemsDetailsInput SizeOutput Size
Pi-MMoE
S.T
Gate units28 × 12 × 1
Expert layers6
Expert hidden neurons16
Activation functionRelu
Pi-MMoE
S.G M.T
Gate units1
Expert layers6
Expert hidden neurons16
Activation functionRelu
Pi-MMoE
M.G M.T
Gate units2
Expert layers6
Expert hidden neurons16
Activation functionRelu
Table 6. Evaluation metrics for validating the effectiveness of physical information.
Table 6. Evaluation metrics for validating the effectiveness of physical information.
MLPPi-MLPResNetPi-ResNetMMoEPi-MMoE
Electric energy:
MAE0.170.120.200.190.210.10
RMSE0.210.150.240.230.250.12
Steam energy:
MAE0.170.120.130.130.110.07
RMSE0.200.150.250.170.140.08
Comprehensive:
GM-MSE0.040.020.060.040.030.01
Table 7. Wilcoxon signed-rank test results for validating the effectiveness of physical information.
Table 7. Wilcoxon signed-rank test results for validating the effectiveness of physical information.
Itemp-ValueDifferent
Electric energy:
Pi-MMoE vs. MLP0.01Yes
Pi-MMoE vs. Pi-MLP0.02Yes
Pi-MMoE vs. ResNet0.02Yes
Pi-MMoE vs. Pi-ResNet0.04Yes
Pi-MMoE vs. MMoE0.39No
Steam energy:
Pi-MMoE vs. MLP0.01Yes
Pi-MMoE vs. Pi-MLP0.01Yes
Pi-MMoE vs. ResNet0.04Yes
Pi-MMoE vs. Pi-ResNet0.02Yes
Pi-MMoE vs. MMoE0.75No
Table 8. Evaluation metrics for assessing the impact of expert net.
Table 8. Evaluation metrics for assessing the impact of expert net.
Pi-MMoE
2-8
Pi-MMoE
4-8
Pi-MMoE
6-8
Pi-MMoE
8-8
Pi-MMoE
2-16
Pi-MMoE
4-16
Pi-MMoE
6-16
Pi-MMoE
8-16
Electric energy:
MAE0.120.110.160.100.120.100.100.10
RMSE0.140.130.150.120.140.120.120.11
Steam energy:
MAE0.090.090.070.070.090.070.070.07
RMSE0.110.110.080.080.110.090.080.09
Comprehensive:
GM-MSE0.0140.0140.0140.0140.010.0140.010.01
Table 9. Wilcoxon signed-rank tests for assessing the impact of expert net.
Table 9. Wilcoxon signed-rank tests for assessing the impact of expert net.
Itemsp-ValueDifferent
Electric energy:
Pi-MMoE 6-16 vs. Pi-MMoE 2-160.02Yes
Pi-MMoE 6-16 vs. Pi-MMoE 4-160.00Yes
Pi-MMoE 6-16 vs. Pi-MMoE 8-160.00Yes
Pi-MMoE 6-16 vs. Pi-MMoE 6-80.55No
Steam energy:
Pi-MMoE 6-16 vs. Pi-MMoE 2-160.00Yes
Pi-MMoE 6-16 vs. Pi-MMoE 4-160.00Yes
Pi-MMoE 6-16 vs. Pi-MMoE 8-160.00Yes
Pi-MMoE 6-16 vs. Pi-MMoE 6-80.19No
Table 10. Evaluation metrics for validating the performance of multi-task modeling.
Table 10. Evaluation metrics for validating the performance of multi-task modeling.
Pi-MMoE
S.T.
Pi-MMoE
S.G. M.T.
Pi-MMoE
M.G. M.T.
Electric energy:
MAE0.160.140.10
RMSE0.190.170.12
Steam energy:
MAE0.150.110.07
RMSE0.170.130.08
Comprehensive:
GM-MSE0.040.020.01
Table 11. Wilcoxon signed-rank test results for validating the performance of multi-task modeling.
Table 11. Wilcoxon signed-rank test results for validating the performance of multi-task modeling.
Itemsp-ValueDifferent
Electric energy:
Pi-MMoE M.G. M.T. vs. Pi-MMoE S.T.0.00Yes
Pi-MMoE M.G. M.T. vs. Pi-MMoE S.G. M.T.0.02Yes
Steam energy:
Pi-MMoE M.G. M.T. vs. Pi-MMoE S.T.0.01Yes
Pi-MMoE M.G. M.T. vs. Pi-MMoE S.G. M.T.0.00Yes
Table 12. Process optimization results based on PSO under different strategies.
Table 12. Process optimization results based on PSO under different strategies.
Production Information
Date6 May 2023
Production detailsType: Y6260; quantity: 94; weight: 2379.5 kg
Workshop temperature28 °C
Process No.06735-1-3
StrategiesTotal energyReduce electricityReduce steam
Decision coefficients λ 1 = 0.5, λ 2 = 0.5 λ 1 = 0.75, λ 2 = 0.25 λ 1 = 0.25, λ 2 = 0.75
ItemOriginal DetailsOptimized Details
1~10# Oven temperature
/°C
180173174171
1~8# spindle width
/cm
[150, 150, 150, 150, 150, 150, 150, 150][157, 157, 157, 157, 157, 157, 157, 157][157, 157, 157, 157, 157, 157, 157, 157][157, 157, 157, 157, 157, 157, 157, 157]
1~10# fan speed
/%
[98, 98, 98, 98, 98, 98, 95, 95, 95, 95][92, 92, 92, 92, 93, 92, 92, 92, 93, 98][89, 89, 89, 89, 90, 89, 89, 89, 90, 92][94, 94, 94, 94, 96, 94, 94, 94, 96, 99]
Electric consumption/kW·h256238 (↓ 1 7.03%)233 (↓8.98%)241 (↓5.86%)
Steam consumption
/t
1.501.40 (↓6.67%)1.43 (↓4.67%)1.37 (↓8.67%)
1 ↓ indicates how much the value has decreased compared to the original value.
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Yuan, M.; Liu, J.; Chen, Z.; Guo, Q.; Yuan, M.; Li, J.; Yu, G. Predicting Energy Consumption for Hybrid Energy Systems toward Sustainable Manufacturing: A Physics-Informed Approach Using Pi-MMoE. Sustainability 2024, 16, 7259. https://doi.org/10.3390/su16177259

AMA Style

Yuan M, Liu J, Chen Z, Guo Q, Yuan M, Li J, Yu G. Predicting Energy Consumption for Hybrid Energy Systems toward Sustainable Manufacturing: A Physics-Informed Approach Using Pi-MMoE. Sustainability. 2024; 16(17):7259. https://doi.org/10.3390/su16177259

Chicago/Turabian Style

Yuan, Mukun, Jian Liu, Zheyuan Chen, Qingda Guo, Mingzhe Yuan, Jian Li, and Guangping Yu. 2024. "Predicting Energy Consumption for Hybrid Energy Systems toward Sustainable Manufacturing: A Physics-Informed Approach Using Pi-MMoE" Sustainability 16, no. 17: 7259. https://doi.org/10.3390/su16177259

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Article metric data becomes available approximately 24 hours after publication online.
Back to TopTop