2.1. Related Works on IESs

Derived from the regional microgrid framework, the IES was initially developed to explore unified regional parallel frameworks with two or more energy sources [10, 11], e.g., electricity, heat, and natural gas. The IES relies on advanced information processing and energy management technologies to implement multienergy scheduling and management, which can effectively enhance energy utilization efficiency [12], increase utilization of REEs [13], reduce operating expenses, and mitigate carbon emissions [13, 14]. It also enables organizations to optimize energy usage across diverse energy sources and devices.

This study focuses on the energy allocation optimization problems under IESs, holding significant importance for achieving energy sustainability and economic benefits. The past related studies are generally categorized as follows: system design, operation strategy, and system optimization. On system designs for IESs, Li et al. [15] and He et al. [16] examined variations in energy sources integrated with IESs and assessed how various energy conversion devices influence the overall system design. Zhang et al. [17] designed an IES with a GSHP, and showed its outperformance than those without GSHPs, in terms of lowering carbon emissions and energy use costs.

On operation strategies for different equipment in IESs, Zhang et al. [18] proposed a seasonal operation strategy for GSHPs in IESs to reduce energy cost. Xiang et al. [19] designed a price-based demand response strategy aimed at minimizing annual costs of the grid-connected IES and demonstrated significant economic and environmental benefits under various configurations and market conditions. Wei et al. [20] optimized IESs through a stochastic predictive control strategy, resulting in total cost reductions of up to 3.4%.

On system optimization for IESs, Huang et al. [21] enhanced the reliability of a multienergy IES through a resilience-oriented planning approach so that the system has a stronger capability to withstand stochastic events such as natural disasters. Capone et al. [22] considered the demand response of heat storage in an electric-thermal integrated multienergy framework to optimize energy costs. The IES proposed by Su,Zhou, and Tan [23] dynamically optimizes household energy usage by switching between gas and electricity, reducing costs by up to 23.4% compared to traditional methods. Wu et al. [11] suggested an IES containing GSHPs to effectively utilize green energy so as to optimize the energy utilization efficiency as well as the utilization of green energy. Table 1 compares the abovementioned previous works on IESs with this study, in which “CHP” represents the combined heat and power.

2.2. Island Model

When reviewing the classic genetic algorithm (GA), Whitley [25] introduced a population structuring strategy referred to as the island model, which partitions the solution population into multiple subpopulations, each of which functions as a separate duplication of the GA. Upon reaching a specific number of iterations, a predetermined number of individuals exchange between islands. This process, termed migration, is repeated across all islands to uphold population diversity and aid in problem-solving without converging towards local optimal solutions. Corcoran and Wainwright [26] solved multiprocessor scheduling problems using island model GA (IMGA), which effectively preserves population diversity, thereby improving the performance of the algorithm. This strategy has been successfully combined with various metaheuristic algorithms, e.g., island crow search algorithm [27], island cuckoo search algorithm [28], multi-island GA [29], and island HSA [30]. The island model is widely recognized as one of the most popular structural population strategies due to its effectiveness in preserving population diversity throughout the search process [31].

2.3. Simplified Harmony Search

The HSA [9] is a metaheuristic algorithm inspired by the process of musicians improvising notes. The HSA employs two parameters harmony memory consideration rate (HMCR) and pitch adjustment rate (PAR) to govern the two-stage process of selecting one of three operations (i.e., randomly generating a new note, choosing a note from the harmony memory (HM) and choosing a note from the HM and adjusting its pitch) to determine new notes, as shown in Figure 2a. The HSA considers the HM, which stores the best solutions found thus far. The HSA continually searches for optimal solutions by replacing inferior solutions in the HM with better ones.

Our previous work proposed the SHS algorithm [32] to improve the HSA. As shown in Figure 2b, the SHS adopts a single-stage judgement based on the two parameters κ₁ and κ₂ to select one of the three operations for generating new notes random number. This simplification contributes to a better performance compared to the HSA. The SHS not only accelerates the algorithm’s solution generation speed but also enhances solution stability. This improvement signifies a significant advancement in optimization algorithms, making the SHS a promising choice for various optimization tasks.

3.1. System Framework

3.2. Problem Model

This study considers an IES as shown in Figure 1, including five sources of electricity: REE, GT, ESS, ETP, and power grid; five sources of heat energy are listed as follows: PE, GSHP, GT, HSS, and ETP; four sources of cold energy are listed as follows: PE, GSHP, CSS, and CE; and one source of natural gas: natural gas grid. The excess electricity and heat energy generated by this IES can be sold to other organizations in need through the ETP.

This study establishes a MIP model for optimizing energy management decisions (including determining the timing of charging and discharging energy storage systems, electricity purchased from the power grid, electricity, and heat energy purchased from the ETP, energy stored in energy storage systems, electricity used in CE, and consumption of natural gas) in the IES outlined in the previous section. Table 2 displays the relevant notations used in the model, with all energy-related variables converted to SI units for consistency. Specifically, joules (J) are used for electricity, heat energy, and cold energy, while the energy content of natural gas is also expressed in joules.

Note that the GSHP operates in a single mode (either heating or cooling) based on seasonal climate conditions. This operational limitation is due to the design characteristics of the GSHP, which cannot simultaneously generate both heat and cold energy. For simplification, this study assumes a constant COP for the GSHP under the given mode of operation, regardless of climate variations. This assumption ensures the feasibility of the model and focus on the energy allocation problem.

4.1. Solution Encoding and Initialization

4.2. Fitness Evaluation

The fitness of a harmony is evaluated by Algorithm 4, based on Equations (2)–(9) and (45).

4.3. SHS Procedure for Generating a New Harmony

The SHS improves the HSA to generate a new note in only one stage based on two parameters κ₁ and κ₂, as illustrated in Figure 2b. This study employs the scheme to generate a new harmony for the concerned problem in Algorithm 5.

5.1. Experimental Environment and Parameter Settings

The settings of the experimental environment for this study are shown in Table 3, with all date-related settings set to April 1, 2016. The date, April 1, 2016, was chosen because it represents one of the days with the most comprehensive data from the referenced dataset. From the available days with complete data, this date was randomly selected for the experiments. The parameters related to this specific date include the hourly renewable energy generation data, electricity prices, and heat energy prices, which are all based on real data corresponding to April 1, 2016. These parameters are detailed in Table 3 and Figure 4a–c. Natural gas prices are referenced to the daily natural gas price information on the Trading Economics website [33] on the day. The electricity generation data for both solar and wind is based on renewable energy generation data from a factory in southern Germany provided by Open Power System Data [34], from which the electricity generation was extracted for each hour of the day, as illustrated in Figure 4a. On that day, the electricity price of the grid is determined by the German real-time electricity price provided by [35]. The hourly price of the ETP is derived from the energy trading price model introduced by Yaagoubi and Mouftah [36]. This model captures the dynamics of real-time electricity supply and demand as well as RTP fluctuations at the time of its publication. It is crucial to highlight that the prices used in this study are reflective of the market conditions and assumptions embedded in the 2015 model. The distribution of real-time prices on electricity of the grid, the ETP, and heat energy is shown in Figure 4b, c.

This study considers two factory examples: a baseline factory with a relatively low electricity load and a high-load factory with a significantly larger electricity demand. Electricity load data for a baseline factory is referred to Open Power System Data for southern Germany [37] (Figure 5a). To account for variations in electricity load data across different industry scales, additional data for a high-load factory is extracted from German BDEW’s normalized standard load profiles [38]. Specifically, the “G3-Gewerbe durchlaufend” category, suitable for continuous commercial operations, is selected to match the requirements of this study. Considering the time frame of April 1, 2016, data for the Übergangszeit (transitional period) and Werktag (working day) conditions are used (Figure 5b).

The cold and heat energy load data of the factories are simulated based on the electricity load profiles from the Open Power System Data for southern Germany [37]. The heat load is estimated by taking the electricity demand for air conditioning during the heating season (October to April, as defined for Europe) and adding 60% of the electricity demand of a specific machine, which is assumed to perform heating functions. This assumption allows the electricity demand for this machine to be replaced by equivalent heat energy demand. The cold load is estimated by taking the electricity demand for air conditioning during the cooling season (April to October) and combining it with the cold energy demands of the cooling pumps and cooling aggregate from the dataset. Figure 4d, e illustrate the simulated heat and cold energy load profiles for the baseline factory based on these calculations.

Additionally, the heat and cold energy recovery data of the factories is calculated from the heat and cold energy recycling rate of the energy loads (Figure 4f). However, the recycling rates vary across different industries. Since this study discusses the energy integration framework of a future smart factory, the recycling rate is set at 70%.

The parameter settings of the three energy storage systems considered in this study are given in Table 4, in which the partial parameters (including the ESS capacity for the baseline factory 1.80000 × 10⁸ J) are referred to [18]. To design an ESS for the high-load factory, the total daily electricity consumption is 4.01800 × 10¹⁰ J [38]. Since the storage system is designed to integrate with REEs, consisting of solar and wind power, it would need to store up to half a day’s electricity consumption, requiring a capacity of 2.00880 × 10¹⁰ J (Table 4). These considerations ensure the energy storage system is customized to meet the operational and sustainability goals of the factory.

As shown in Figure 4g, the COP of GSHP is based on the dataset for Germany provided by Open Power System Data [37]. Note that in southern Germany during the season of April 2016, the GSHP can only generate heat energy. All the experiments were conducted on a PC with Intel Core i7-10700 CPU @ 2.90 GHz and 16 GB of RAM.

This study compares the performance of the proposed iSHS using different island numbers (i.e., iSHS_2, iSHS_3, iSHS_4, and iSHS_5 denoting the iSHS with 2, 3, 4, and 5 islands, respectively) to SHS and HAS because iSHS is an extension of SHS and HSA. Parameter settings for these algorithms have a significant impact on their performance. Therefore, three parameter levels for each parameter of these algorithms are tested for parameter optimization, while the termination condition for each experiment is 10,000 iterations and 30 experiments are conducted for each parameter. The best parameter settings of each algorithm are set based on their performance tested using one-way ANOVA. The results of all the parameters of these algorithms are presented in Table 5.

5.2. Performance Comparison of Different Algorithms

The examples for the baseline and high-load factories both involve 15 decision variables across 24 h (15 × 24). In addition to the two examples, this study additionally considers an illustrative example for a smaller problem with 15 decision variables for hour 1 (15 × 1) to better demonstrate the advantages of the proposed algorithm. The illustrative example is designed to provide an intuitive understanding of the algorithm’s performance under simplified conditions, where only 1 h is considered. The comparison of the three experimental settings is as follows: the illustrative example (15 × 1), the baseline factory with a relatively low electricity load (15 × 24), and the high-load factory with electricity demand 100 times larger than the baseline factory (15 × 24).

The statistics of the experimental results for the six algorithms, each running five times, for the illustrative example, the baseline factory, and the high-load factory scenarios are shown in Table 6; and their box plots are visualized in Figure 6. For the illustrative example, the experimental results indicate that the performance of all algorithms shows minimal variation, with differences only observable at the third decimal place of the total cost. This suggests that the simplified nature of the example reduces the complexity of the optimization problem, leading to comparable solutions across algorithms. Among the six algorithms, iSHS_4 achieves the best average performance (−6.03159) and the smallest standard deviation (8.09 × 10⁻⁵), highlighting its stability and effectiveness. These results confirm that while the illustrative example provides a straightforward problem for evaluation, the iSHS_4 algorithm still demonstrates a consistent advantage in terms of precision and reliability.

Table 7 shows nonzero decision variable values and the total cost of the best solutions generated by different algorithms for the illustrative example. These values across different algorithms in the illustrative example show minimal variation, indicating that the algorithms produce highly consistent total cost results for this simplified scenario. Note that the differences in the values for electricity, heat, and cold energy discharged or sold are only marginal, further confirming the small-scale nature of the problem. The initial settings ensure sufficient energy availability due to precharged energy storage systems (ESS, HSS, and CSS), along with adequate purchases of electricity from the grid and natural gas. After meeting the load requirements for various devices, surplus energy is sold through the ETP. This leads to a negative total cost, indicating that the system not only meets its energy demands but also generates revenue.

From Table 6, across both baseline and high-load factory scenarios, the iSHS algorithms consistently outperform the conventional SHS and HSA algorithms in terms of solution quality and stability. Among the iSHS algorithms, iSHS_4 achieves the best performance with the smallest best values of –37.84 for the baseline factory and 148.75 for the high-load factory. Meanwhile, iSHS_5 demonstrates the smallest average values (–37.35 for the baseline factory and 149.49 for the high-load factory) and the smallest worst values (–36.65 for the baseline factory and 149.94 for the high-load factory), showing its consistency across experiments. Additionally, iSHS_3 exhibits the least standard deviation in both scenarios (0.24 for the baseline factory and 0.45 for the high-load factory), indicating its reliable performance. In comparison, the SHS and HSA algorithms show competitive but generally inferior performance across all metrics, with larger average and worst values in both factory scales. These results confirm the robustness and adaptability of the proposed iSHS algorithms for optimizing energy allocation in factories of varying scales.

To address the carbon emissions associated with the experimental results, this study calculates the average carbon emissions for each algorithm in all three scenarios in Table 6 using the carbon emission factor for electricity consumption from the German grid in 2016 at 0.513 kgCO₂eq/kWh [39] and that for natural gas at 0.370 kgCO₂eq/kWh [40]. For energy traded through the ETP, this study assumes the same emission factor as the German grid at 0.513 due to a lack of specific carbon intensity data for ETP transactions. It is important to note that the proposed model seeks to minimize total costs, rather than directly minimizing carbon emissions. As such, the differences in carbon emissions among the algorithms are relatively minor and do not serve as a primary metric for evaluating algorithm performance. For instance, in the illustrative example, all algorithms yielded nearly identical average carbon emissions, with variations only occurring at a negligible scale. In the baseline and high-load factory scenarios, while there are slight differences in emissions, these are a by-product of the cost optimization process rather than an intentional reduction of emissions. This highlights a potential area for future research: incorporating carbon emission minimization into the model objective. Such an extension would provide a more comprehensive evaluation of algorithm performance in terms of environmental impact.

For execution time (Table 6), the SHS takes the least execution time, while the iSHS_5 is the most time-consuming. It is because the SHS simplifies the way of generating new solutions for HSA, while iSHSs generally require longer execution time due to their operations of multiple islands. The convergence analysis of different algorithms for both baseline factory and high-load factory scenarios is shown in Figure 7. From Figure 7, in general, all algorithms show good convergence results, in which all the iSHS algorithms show better converge performance than the legacy SHS and HSA.

5.3. Analyzing the Effects Before and After Implementation of the Proposed IES

From Table 6, the iSHS_4 achieves the best energy cost for 1 day, with €–37.84 for the baseline factory and €148.75 for the high-load factory. This implies that the baseline factory can gain a profit of €37.84 from its energy use and trading decisions. Furthermore, the hourly electricity input and output for both baseline factory and high-load factory scenarios for the best experimental results are shown in Figure 8. The overall electricity input consists of renewable energy, electricity purchased from the power grid, electricity generated by the GT, and electricity purchased from the ETP; while the electricity output is the electricity sold to the ETP. From Figure 8, the overall electricity input to the factory is generally greater than the electricity output, and the difference between them is the electricity load of the factory operation.

In Figure 8a (baseline factory), the electricity input is relatively smaller and shows higher reliance on REEs and energy trading. The GT and grid electricity inputs are less prominent due to the smaller scale of operations. In contrast, Figure 8b (high-load factory) depicts significantly higher electricity input, with a more substantial contribution from the GT and grid electricity. The higher input and output reflect the larger energy demands of the high-load factory. These differences emphasize the scalability of the proposed energy allocation model and its adaptability to factories of varying sizes.

As depicted in Figure 9, the dotted curve represents the electricity load of the plant using only grid power (before implementation), and the solid curve represents the electricity load of the plant in the IES of this study (after implementation). For the baseline factory (Figure 9a), the total energy demand for 1 day before the implementation is 4.03423 × 10⁹ J, while the total energy consumption after the implementation is 3.41557 × 10⁹ J. The difference between the two reflects the reduction due to the proposed IES. Therefore, a total of 15.34% of the electricity load has been reduced and replaced by heat and cold energies from the IES, which are partly obtained from the GSHP and recycling of waste heat and cold from factory production, thus enhancing the factory’s overall energy utilization efficiency. For the high-load factory (Figure 9b), the total energy demand before the implementation is 4.28111 × 10¹⁰ J, and the total energy consumption after the implementation is 4.27795 × 10¹⁰ J. This results in a slight reduction in energy consumption after implementing the IES, which is a minimal decrease compared to the baseline factory. However, the high-load factory still benefits from a decrease in energy consumption by integrating heat and cold energy sources, similar to the baseline factory.

With the implementation of the proposed IES, the utilization of green energy can be evaluated across different factory scales. Table 8 presents the energy consumption and the corresponding green energy percentages for baseline and high-load factories. For the baseline factory, the total energy consumption is 5.70988 × 10⁹ J. Electricity accounts for 38% of the total load, with 7.26696 × 10⁸ J provided by REEs, representing 21% of the electricity load. Heat and cold energy account for 12% and 50% of the total load, respectively, with green energy contributions of 70% and 60%. Overall, 39% of the baseline factory’s energy consumption is met by green energy, indicating a moderate reliance on renewable sources. In contrast, the high-load factory exhibits a significant increase in total energy consumption, reaching 4.50500 × 10¹⁰ J, nearly eight times higher than the baseline factory. The electricity load is the primary driver of this increase, accounting for 88% of the overall energy consumption. Despite the green energy contribution from REE remaining constant at 7.26696 × 10⁸ J, its share of the electricity load decreases dramatically to just 2%, highlighting the challenges of scaling renewable energy supply in larger operations. Heat and cold energy loads remain constant in absolute terms, at 1.05056 × 10⁹ J and 1.24375 × 10⁹ J, respectively, contributing 12% and 50% to the total energy consumption. The green energy contributions to these loads are consistent at 70% for heat and 60% for cold energy, demonstrating the stability of these contributions regardless of factory size. Overall, while the green energy contribution for heat and cold energy remains consistent between the two factory scales, the sharp increase in electricity demand for the high-load factory leads to a notable decrease in the overall green energy share, dropping from 39% in the baseline factory to only 9% in the high-load factory. This disparity underscores the importance of expanding renewable energy capacity, particularly for electricity generation, to accommodate the increased energy demands of larger facilities.