Joint Command and Control Versus Integrated Energy Systems: A Comparative Analysis Based on a Quantity–Quality–Spatiotemporal Model

Abstract

As modern energy systems become increasingly complex and multi-source integrated, efficient coordination between diverse energy carriers and dynamic demand is essential. This study identifies a structural parallel between integrated energy system (IES) scheduling and the weapon-target assignment (WTA) problem in joint command and control, and proposes a quantity–quality–spatiotemporal (QQST) framework to model multi-dimensional supply–demand matching. The QQST framework formulates scheduling as a coupled optimization problem integrating quantity balance, energy quality (exergy), spatial distribution, and temporal dynamics. A real-world industrial IES case, involving 60 textile enterprises and a 62 km steam network, is used for validation. The proposed model is benchmarked against a conventional mixed-integer linear programming-based scheduling approach under identical system configurations. Results show that QQST improves overall exergy efficiency by 8.4% and reduces energy quality mismatch by 18.2%, as measured by an exergy-based index. Sensitivity analysis under varying load conditions further confirms the robustness of the approach. These findings demonstrate that the QQST framework provides a structured and effective methodology for enhancing multi-dimensional coordination in complex energy systems.

1. Introduction

In the context of the ongoing energy transition, integrated energy systems (IESs) have become pivotal for the synergistic and efficient use of diverse regional energy resources [1]. By leveraging advanced information management technologies, an IES orchestrates the entire lifecycle of multiple energy carriers (electricity, heat, cooling, natural gas) [2], encompassing their production, transmission, conversion, storage, and consumption. The high degree of coupling between multiple energies, however, introduces significant complexity, characterized by energy coupling [3], multi-timescale dynamics [4], and the involvement of multiple agents [5]. These features give rise to three critical challenges: achieving optimal coupling, coordinating across different timescales, and facilitating collaboration among stakeholders.

Attaining supply–demand equilibrium within an IES presents a scheduling dilemma, entailing the derivation of optimal strategies under intricate constraints, which has been demonstrated to be an NP-hard problem [6]. In response, the academic community has developed a variety of optimization models. Fan et al. [7] introduced a two-stage scheduling model using distributionally robust adaptive model predictive control to reconcile the trade-off between renewable energy uncertainty and economic efficiency. Nozari et al. [8] enhanced both exergy efficiency and operating costs by integrating ammonia-based energy storage within a multi-objective optimization framework. To address energy supply uncertainty and improve performance in industrial parks, Zhu et al. [9] and Wang et al. [10] separately devised a fast, distributed multi-energy management framework using stochastic gradients and a multi-timescale scheduling framework based on a Stackelberg game. Meanwhile, Lin et al. [11] concentrated on ultra-short-term scheduling, proposing a multi-timescale dynamic operation method that leverages the inherent flexibility of fluid networks and accounts for demand elasticity. The literature on IES scheduling has also introduced novel methods and concepts from the perspectives of optimization models [12], algorithms [13], uncertainty handling [14], demand response [15], and electricity markets [16].

Despite this significant progress in cost, efficiency, and reliability, these studies typically concentrate on individual dimensions or a subset of these dimensions. A critical gap continues to be the absence of a unified framework that can comprehensively characterize the matching relationships between diversified resources and dynamic demands. Specifically, conventional models often fail in high-complexity scenarios where exergy degradation occurs, high-grade energy being wasted on low-grade demand, because they lack a rigorous “quality matching” logic. Furthermore, existing frameworks frequently decouple spatial transmission latency from dispatch signals, leading to temporal synchronization errors in large-scale networks.

The challenge of the matching relationship also occurs in a vastly different domain: modern military command and control (C2) [17]. Modern warfare, defined by its rapid tempo and high-intensity offensive–defensive interactions, requires the seamless coordination of diverse operational assets across land, sea, air, and space. The central challenge of this endeavor can be formally framed as the classic weapon-target assignment (WTA) problem [18]. As an NP-complete problem within operations research (OR), the objective of WTAs is to allocate heterogeneous weapon systems to a portfolio of targets to maximize overall combat effectiveness [19]. In response, the military community has cultivated sophisticated joint command and control theories and the C4ISR (command, control, communications, intelligence, surveillance, and reconnaissance) framework. These advancements offer a powerful theoretical foundation and a technical paradigm for the efficient coordination of information flows and firepower allocation, especially within a cyber–physical system (CPS) context [20,21].

Research into the WTA optimization problem has predominantly centered on the development of novel algorithms and methods. Kong et al. [19] introduced an improved multi-objective particle swarm optimization algorithm to tackle the dynamic weapon-target assignment (DWTA) problem. Silav et al. [22] developed a novel bi-objective DWTA model that accounts for the disruption of initial allocation plans during an engagement, caused by factors such as target destruction, system failures, or the emergence of new targets. To address communication constraints, Hendrickson et al. [23] formulated a distributed WTA method operable under asynchronous conditions. In pursuit of solutions that reflect decision-maker preferences, Wu et al. [24] designed an enhanced algorithm based on the multi-objective evolutionary algorithm in a decomposition framework, which is applied in the multi-stage WTA problem. Furthermore, Kim et al. [25] devised a WTA algorithm employing a rotation strategy and clustering to mitigate the reduction in kill probability from heading errors, which is solved using a mixed-integer linear programming framework. The applicability of reinforcement learning is explored by Liu et al. [26], who demonstrated its feasibility for time-sampled dynamic weapon allocation. Extending this to complex scenarios, Wang et al. [27] solved the multi-weapon-target assignment problem for unmanned ground combat vehicles in urban warfare by integrating a reinforcement learning framework with an improved multi-objective artificial bee colony algorithm.

While extensive research exists independently in both energy engineering and military science, the existing literature remains fragmented, with limited studies systematically addressing the integration of these two domains. This gap is not due to a lack of relevance, but rather the absence of a unified abstraction capable of bridging heterogeneous system characteristics across disciplines. Existing studies primarily rely on domain-specific formulations, which limit their transferability. Recent advances in CPS and cross-domain resource allocation provide a potential theoretical bridge. These studies highlight that structurally similar optimization problems may exist across different application domains, even when their physical interpretations differ. However, most existing studies focus on either standalone energy system design or military operational requirements, lacking a unified framework that simultaneously satisfies the dual constraints of energy efficiency and military operational reliability.

In this study, we move beyond a simple descriptive analogy between military and energy domains by identifying a fundamental structural isomorphism in their resource allocation logic. While IES focuses on matching multi-modal energy supply with heterogeneous demand, WTA is concerned with the strategic matching of weapon assets to hostile targets. We demonstrate that the effectiveness of matching in both domains is governed by a shared four-dimensional state-space: quantity, quality, space, and time. Correspondingly, we have explicitly emphasized that the mathematical framework is adapted from the WTA model’s optimization logic to avoid ambiguity about the research’s civilian focus.

Leveraging this insight, the “quantity–quality–spatiotemporal” (QQST) paradigm is systematically formalized. Unlike conventional multi-objective optimization that treats these constraints as additive or separable factors, the proposed QQST framework explicitly structures these dimensions into a unified, interdependent mathematical representation. Specifically, we define a generalized mapping function $\Phi :({\mathrm{Q}}_{\mathrm{s}},{\mathrm{Q}}_{\mathrm{q}},\mathrm{S},\mathrm{T})\to \mathcal{u}$, where the four dimensions jointly determine the system utility $\mathcal{u}$. In this paradigm, supply–demand matching is reformulated as a multi-dimensional state-space mapping problem rather than a simple constraint satisfaction task.

The core contribution of this paper is the establishment of an explicit cross-domain transformation. We define a formal mapping where military weapon effectiveness $\theta$ corresponds to thermodynamic exergy $\eta$, and target threat $U$ corresponds to demand-side quality requirements D. This mapping is formalized through equivalent constraint structures, ensuring that the military logic of preventing overkill is rigorously translated into the engineering goal of minimizing exergy destruction. The efficacy of this WTA-inspired methodology is validated using real-world industrial data from a steam-based energy network, demonstrating significant improvements in systemic exergy efficiency.

The remainder of this paper is organized as follows: Section 2 details the comparative analysis between joint command and control and IESs, establishing the theoretical basis for the isomorphism. Section 3 formalizes the proposed QQST mathematical framework and the cross-domain transformation functions. Section 4 presents the industrial case study, quantitative results, and a critical discussion on practical applicability and sensitivity. Finally, Section 5 concludes the paper.

2. Joint Command and Control Versus Integrated Energy Systems

To eliminate ambiguity and ensure reproducibility, a unified notation system is introduced for both WTA and IES formulations. All subsequent equations strictly follow this notation to avoid inconsistencies and indexing ambiguity (

Table 1. Analogous definitions of key symbols in WTA and IES scheduling.

Symbol		Description	WTA Context	IES Context
Indices	i ∈ $ℐ$	Index of supply units	Weapons/Interceptors	Energy sources/Plants
	j ∈ J	Index of demand targets	Enemy targets/Assets	Load nodes/Users
	$\mathrm{t}$	Index of time intervals	Engagement stages	Scheduling periods
Variables	xij	Allocation decision	xij ∈ [0, 1]	Continuous flow assignment
	qij	Effectiveness coefficient	Kill probability ($P_k$)	Transfer efficiency (${\eta }_{trans}$)
Parameters	ci	Unit cost of supply i	Ammunition/Asset cost	Operational/Resource cost
	dj	Requirement of demand j	Target threat level $U_j$	Load demand magnitude
	${\mathsf{\eta }}_{\mathrm{i}}$	Quality coefficient	Weapon lethality index ${\theta }_i$	Energy quality coefficient (EQC)
	${\mathrm{b}}_{\mathrm{i}\mathrm{t}}$	Capacity of source i	Total available ammunition	Maximum supply capacity

Abbreviations: QQST, quantity–quality–spatiotemporal; IES, integrated energy systems; WTA, weapon-target assignment; EQC, energy quality coefficient.

).

2.1. Joint Command and Control

The architecture of C2 is a typical “cloud–network–edge” CPS [28]. As illustrated in Figure 1, the object layer (the cloud side) encompasses diverse operational environments characterized by cloud-based technologies. The physical resource pool serves as the foundation for the integration of naval, land, and air forces. It further incorporates communication, reconnaissance, early-warning, and computational resources to enable real-time simulation of the operational environment.

The integration layer (the network side) ensures connectivity and situational awareness, forming a cloud-based resource pool. This pool supports intelligence integration and operational planning, which in turn allows for the establishment of a global C2 system. A pervasive network connects disparate military assets to a unified cloud platform. These assets include naval, air, and ground forces, as well as surveillance equipment and communication nodes. This connection is possible regardless of geographical location or operational environment. Therefore, the integration layer ensures the true aggregation, integration, and sharing of information.

The optimization layer (the edge side) supports full-spectrum joint command functions, including intelligent intelligence analysis, comprehensive situational assessment, and real-time optimization of command strategies, which are subsequently transmitted to the object layer for execution.

The interaction among these layers is highly dynamic: the object layer executes physical strikes while transmitting real-time sensor data upward, the integration layer filters and synchronizes this data across disparate networks, and the optimization layer computes the allocation strategy based on the synthesized operational picture. Similarly, in IESs, the object layer executes flow adjustments, the integration layer aggregates multi-modal energy data, and the optimization layer resolves the QQST model to issue precise set-points, closing the feedback loop.

Leveraging a “cloud–network–edge” environment, the C2 system delivers global decision-making data. It employs intelligent algorithms to accelerate the decision cycle and integrates advanced decision-support tools to enable effective human–machine collaborative decision-making. The architecture ensures comprehensive all-domain situational awareness, on-demand resource aggregation, and agile command and control.

In the C2 architecture, the entire system comprises the integration of global resources at the cloud layer, the aggregation and sharing of information at the network layer, and the real-time optimization of command decisions at the edge layer. The C2 needs to be implemented during the execution phase of specific combat missions. Among these elements, the accurate matching between weapon systems and targets serves as the core support for translating the effectiveness of command decisions into actual combat effects. Only by dynamically adjusting the allocation strategy of weapon resources according to the battlefield situation can we achieve efficient strikes against threatening targets in complex adversarial environments, reduce resource waste, and ensure the overall accomplishment of combat missions. Therefore, it is essential to construct a scientific WTA model to quantitatively provide operable mathematical support for the decision-making execution of the C2 system.

The construction of the WTA model in C2 is based on weapon performance and target attributes, formulated as a supply–demand matching problem.

Suppose that, at time t, the enemy set W_WTA consists of n threat targets, $\mathit{W}=\left\{w_1^t,w_2^t,\dots ,w_j^t,\dots ,w_n^t\right\}$, while the weapon set B_WTA includes m units across s categories, $\mathit{B}=\{b_1^t,b_2^t,\dots ,b_i^t,\dots ,b_m^t\}$, and has the following relations: $\mathit{B}={\mathit{B}}_1^t\cup {\mathit{B}}_2^t\cup \dots \cup {\mathit{B}}_k^t\cup \dots \cup {\mathit{B}}_s^t$. The value of each threat target is derived from prior battlefield knowledge and operational rules. Denoting the threat degree of the j-th enemy target as U_j, the overall threat set can be represented as $\mathit{U}=\{u_1^t,u_2^t,\dots ,u_j^t,\dots ,u_n^t\}$.

Define the weapon damage probability matrix as Q_WTA = [$q_{ij}^t$] and a strike decision matrix X_WTA = [$x_{ij}^t$]. The combat effectiveness of the allocation scheme based on X_WTA is represented by the sum of all weapon damage probabilities in Equations (1) and (2).

$$F_1=\sum _{j=1}^n{u}_j^t\left(1-\prod _{i=1}^m{\left(1-q_{ij}^t\right)}^{{\mathsf{\Lambda }}_{ij}^t}\right)$$

(1)

$${\mathsf{\Lambda }}_{ij}^t=\{\begin{array}{l}1,{ x}_{ij}^t\ne 0\\ 0,{ x}_{ij}^t=0\end{array}$$

(2)

where $q_{ij}^t$ is the probability that the i-th weapon successfully damages the j-th enemy target at time t; the binary variable $x_{ij}^t$ indicates whether the i-th weapon is assigned to target j at time t.

To ensure weapon resources are fully utilized, the cost of weapon consumption should also be considered in Equation (3).

$$\{\begin{array}{l}{F}_2={\displaystyle \sum _{i=1}^m}{\overline{V}}_i\\ {\overline{V}}_i={\displaystyle \frac{V_i-V_{\min }}{V_{\max }-V_{\min }}}\\ V_i={\displaystyle \sum _{j=1}^n}{v}_{ij}{x}_{ij}^t\end{array}$$

(3)

where V_i is the total unit cost for the i-th weapon, and v_ij is the unit cost of weapon i to target j. Since weapon costs may differ significantly [29], normalization is applied, yielding ${\overline{V}}_i$ as the normalized unit cost of ammunition; V_min and V_max are the minimum and maximum of all weapon costs, respectively.

Accordingly, the WTA problem is formulated with the objective function in Equation (4) and constraints in Equation (5).

$$F=\min (F_2-F_1)$$

(4)

$$\{\begin{array}{l}{\displaystyle \sum _{j=1}^n}{x}_{ij}^t\in \{0,1\},\\ {\overline{q}}_j\le q_j^t\le 1,\\ 0\le u_j^t\le 1,\\ 0<{\overline{V}}_i\le 1\end{array}$$

(5)

where u_j denotes the degree of threat of target j; and $q_j^t$ is the cumulative probability of damage inflicted on target j; ${\overline{q}}_j$ is the damage probability threshold of target j. The combined damage probability of assigned weapons against target j $q_j^t$ should exceed ${\overline{q}}_j$.

2.2. Integrated Energy Systems

In next-generation power systems, IESs are subject to strong fluctuations on both the supply and demand sides [30]. The variability, which can occur on the order of minutes, complicates the allocation of multiple heterogeneous energy carriers. Consequently, the fluctuating conditions within IESs resemble the WTA problem.

The scheduling model of IESs can also be structured into three hierarchical layers: object, integration, and optimization layers. Each layer has relatively independent internal functions, while interaction between layers is achieved via automatic data flow [31]. The automatic data flow includes four key stages: perception, analysis, decision-making, and execution.

At the object layer, intelligent devices collect operational status data, transforming energy flows into information flows. The integration layer processes the data using advanced control methods to provide references for decision-making. At the optimization layer, optimal strategies are generated under technical and economic constraints and transmitted to the object layer. Finally, the object layer executes the control commands, updates the system state, and ensures that the IES scheduling strategy evolves toward optimal operation.

The IES scheduling corresponds to the dynamic assignment of energy demand and supply across spatiotemporal dimensions. Specifically, Figure 2 demonstrates the hierarchical mapping mechanism inherent in the QQST framework. From a spatial perspective, the system transitions from localized device-level control (point) to regional multi-energy microgrids (surface), and eventually to city-wide integrated networks (volume), mirroring the tactical-to-strategic deployment in military C2. From a temporal perspective, the spatiotemporal dimension of the QQST paradigm is operationalized through a multi-layer feedback loop: high-frequency millisecond-level adjustments ensure quality stability, while hourly dispatch optimizes quantity distribution. This nested structure ensures that the heterogeneous resource allocation—a core tenet of WTA logic—is maintained even as the system scales. By decomposing the global optimization goal into coordinated sub-tasks across these dimensions, the framework effectively mitigates the computational burden while preserving the precision of the supply–demand matching.

A general IES model encompasses multiple dimensions, including energy carriers (electricity, heating, cooling, gas), devices, spatial distribution, and temporal variability. At time t, the demand set W_IES consists of n demands, $\mathit{W}=\{w_1^t,w_2^t,\dots ,w_j^t,\dots ,w_n^t\}$, while the supply set B_IES consists of m energy sources across s energy types, $\mathit{B}=\{b_1^t,b_2^t,\dots ,b_i^t,\dots ,b_m^t\}$, and has the following relations ${\mathit{B}=\mathit{B}}_1^t\cup {\mathit{B}}_2^t\cup \dots \cup {\mathit{B}}_k^t\cup \dots \cup {\mathit{B}}_s^t$.

Define the efficiency of energy conversion and transmission matrix as Q = [$q_{ij}^t$] and a transfer decision matrix X = [$x_{ij}^t$]. The total profit based on X and Q is represented in Equations (6)–(8). It is noted that Equations (7) and (8) are represented for the constraints of IES scheduling, the demand satisfaction constraint, and the supply capacity constraint, respectively.

$$F_1=\sum _{i=1}^m\sum _{j=1}^n{b}_i^t{q}_{ij}^t{x}_{ij}^t$$

(6)

$$\sum _{i=1}^m\left(b_i^t{q}_{ij}^t{x}_{ij}^t\right)\ge w_j^t,\forall j$$

(7)

$$\sum _{j=1}^n\left({\displaystyle \frac{w_j^t}{q_{ij}^t}}{x}_{ij}^t\right)\le b_i^t,\forall i$$

(8)

To ensure the energy consumption is minimized, the cost of energy should also be considered in Equation (9).

$$\{\begin{array}{l}{F}_2={\displaystyle \sum _{i=1}^m}{\overline{V}}_i\\ {\overline{V}}_i={\displaystyle \frac{V_i-V_{\min }}{V_{\max }-V_{\min }}}\\ V_i={\displaystyle \sum _{j=1}^n}{v}_{ij}{x}_{ij}^t\end{array}$$

(9)

Therefore, the IES scheduling can be formulated with the objective function in Equation (10) and constraints in Equation (11).

$$F=\min (F_2-F_1)$$

(10)

$$\{\begin{array}{l}{\displaystyle \sum _{j=1}^n}{x}_{ij}^t\in \{0,1\},\\ {\stackrel{-}{q}}_j\le q_j^t\le 1,\\ 0\le u_j^t\le 1,\\ 0<{\stackrel{-}{V}}_i\le 1\end{array}$$

(11)

2.3. Comparative Analysis

The architectural and operational similarities between command and control (C2) systems and integrated energy systems (IESs) suggest a shared underlying logic of resource allocation. However, to move beyond a purely conceptual metaphor, it is necessary to position the proposed QQST framework against existing state-of-the-art (SOTA) energy optimization methodologies, as shown in

Table 2. Comparison of QQST with conventional IES optimization methodologies.

Feature	Standard MILP/Energy Hub	Model Predictive Control (MPC)	Proposed QQST
Primary Goal	Economic cost minimization	Tracking error minimization	Multi-dimensional quality matching
Quality Tracking	Static or neglected	Partial (temperature only)	Dynamic EQC and exergy destruction
Spatiotemporal	Linear network flow	Temporal horizon focus	Coupled non-linear degradation
Logic Origin	Commodity supply-chain	Feedback control theory	Strategic Resource-Target Assignment

.

As summarized in

Table 3. Structural similarities between IES and C2.

Type	C2	IES
Multiple forces or energy flows	Army, Navy, Air Force, missile and special forces	Production, transmission, distribution, conversion, storage, and consumption of energy forms
Multi-domain	Land, sea, air, space, cyber, electromagnetic spectrum	Energy stations with different locations
Multi-constraint	Equipment technical performance and troop tactical deployment	Energy markets and environmental regulations
Strong coupling	Tight coupling of troop elements, types, missions, architectures and functions	Multiple energy forms are tightly interconnected
Fine granularity	Granular control of combat units; real-time monitoring and command of soldiers	Granular control of energy units; real-time monitoring and dispatch of devices

, C2 and IESs exhibit common structural characteristics, including multi-objective coordination, multi-domain coupling, multi-constraint interactions, and fine-grained decision variables.

While C2 systems and IESs operate in distinct contexts—with C2 requiring high flexibility against adversarial battlefield dynamics [32] and IESs prioritizing economic sustainability amidst market and renewable fluctuations [33]—they converge at a fundamental level of resource allocation.

Beyond a purely conceptual analogy, these systems exhibit a structural isomorphism. Thus, we introduce a transformation operator $\Phi$. This operator maps the probabilistic state-space of the weapon-target assignment (WTA) problem into the thermodynamic state-space of IES dispatch:

$$\Phi :(\mathcal{W},\mathcal{U},{\mathcal{P}}_k)\to (\mathcal{E},\mathcal{D},{\eta }_{ex})$$

Under this transformation, the following functional equivalences are established:

• Quality Equivalence: The weapon effectiveness/lethality index ${\theta }_i$ maps to the energy quality coefficient (EQC, ${\eta }_i$).
• Demand Equivalence: The target threat level ($U_j$) maps to the load’s minimum quality requirement ($D_j$).
• Operational Logic: The objective of preventing overkill is mathematically translated into preventing exergy destruction.

The fundamental justification for this mapping lies in the non-linear degradation shared by both domains. In WTAs, the probability of destruction degrades with distance and counter-measures; in IESs, exergy degrades with transmission distance and entropy generation. By adopting the WTA-inspired QQST framework, IES scheduling can transition from simple quantity balancing to a strategic quality matching paradigm, ensuring optimal systemic efficiency under complex spatiotemporal constraints.

The classical WTA problem is a well-recognized NP-complete problem, which can be polynomially reduced to the set cover problem (a canonical NP-complete problem) as follows:

(1)

Let each weapon correspond to a subset of targets that the weapon can destroy;

(2)

The goal of selecting the minimum number of weapons to ensure all targets are destroyed is equivalent to the set cover problem (selecting the minimum number of subsets to cover the universal set).

By the structural isomorphism between WTAs and IESs, the QQST-based IES scheduling problem generalizes the WTA problem. Detailed proof can be found in Supplementary Materials.

(1)

Discrete binary allocation $x_{ij}^{WTA}$ ∈ {0, 1} is extended to continuous flow allocation $x_{ij}^{IES}$ ∈ [0, 1];

(2)

Kill probability constraints $1-{\prod }_{i=1}^m{(1-p_{ij})}^{x_{ij}^{WTA}}\ge$ ${\overline{\mathrm{p}}}_j$ are extended to non-linear exergy degradation constraints ${\sum }_{\mathrm{i}=1}^m{\mathrm{x}}_{ij}^{\mathrm{I}\mathrm{E}\mathrm{S}}{q}_i{\eta }_{ij}{\mathrm{EQC}}_i\ge$ $D_j$.

Since NP-complete problems are a subset of NP-hard problems, and the generalized problem cannot be easier than the original NP-complete problem, the QQST-based IES scheduling problem is NP-hard. Notably, this is not a conversion of complexity classes but a reflection of the problem’s inherent generalization—continuous variables and non-linear constraints exclude it from the NP-complete class (restricted to discrete problems) but retain NP-hardness.

3. Quantity–Quality–Spatiotemporal Model

This section establishes the quantity–quality–spatiotemporal (QQST) framework as a structured optimization paradigm. Instead of treating quantity, quality, spatial, and temporal factors as loosely coupled constraints, the proposed approach reformulates the resource allocation problem as a unified multi-dimensional mapping, thereby explicitly capturing cross-dimensional interactions.

The QQST framework is defined over a state-space S = {Qs, Qq, S, T}, where Qs, Qq, S, and T denote quantity, quality, spatial, and temporal variables, respectively. The system objective is expressed as a non-separable mapping: U = $\Phi$(Qs, Qq, S, T), where U represents system utility. This formulation captures intrinsic coupling among dimensions, including quantity–quality compatibility, spatiotemporal interactions, and quality degradation during spatial transfer, thus transforming the problem into a structured multi-dimensional optimization model. The QQST model originates from command and control (C2) scheduling and is extended to integrated energy systems (IESs) through structural mapping, providing a unified representation of energy quantity, quality, spatial distribution, and temporal dynamics.

To ensure the mathematical tractability and structural isomorphism of the QQST framework, the following foundational assumptions are adopted for both the WTA and IES models:

• Stationarity of Coefficients: It is assumed that effectiveness coefficients (kill probability $P_k$ in WTA and transfer efficiency $\eta$ in IES) remain constant within a single 15 min scheduling interval, neglecting ultra-fast transient fluctuations.
• Deterministic Load/Threat Forecasts: The model assumes that short-term demand (energy loads) and operational environments (target threats) are known via high-fidelity forecasting systems, allowing for a deterministic optimization approach.
• Linearized Network Constraints: While exergy degradation is non-linear, the physical flow constraints within the energy pipelines are linearized to maintain computational efficiency for real-time dispatch, assuming localized laminar flow conditions.

3.1. Quantity–Quality–Spatiotemporal Model for Weapon-Target Assignment

To clarify the theoretical positioning, QQST is not merely a decomposition of existing constraints, but a structured reparameterization of the optimization problem. By embedding physical and operational relationships into the mapping function F, the framework reduces the dimensional ambiguity present in conventional formulations and enables more consistent cross-domain modeling.

The WTA problem is NP-complete in its canonical form, which limits the applicability of optimization methods [34]. To overcome this limitation and achieve higher solution precision with reduced computational time, an extension of the WTA problem into an NP-hard optimization framework is proposed. The new model is constructed upon a proposed QQST model. A key advantage of QQST is its ability to resolve the ambiguities inherent in traditional WTA models, which often oversimplify complex factors into a single probabilistic metric. By contrast, the QQST model offers a more granular and precise characterization by explicitly defining these factors across four independent dimensions. The proposed QQST model in WTAs is introduced sequentially below.

3.1.1. Quantity Model for Weapon-Target Assignment

In the quantity model of WTAs, there are two constraints, the firepower damage capability (Equation (12)) and ammunition limits for platforms (Equation (13)). Based on the supply–demand balance principles, Equation (12) considers enemy damage thresholds, while Equation (13) incorporates ammunition limitations for different weapons. Equation (12) ensures that the total expected damage inflicted on target j meets or exceeds its minimum required damage threshold. Equation (13) limits the total number of type-i weapons allocated to targets within its maximum available ammunition quantity. It can be observed that the quantity model for the WTA is derived from the original one from Section 2.1.

$$1-\prod _{i=1}^m{\left(1-q_{\mathrm{WTA}ij}^t\right)}^{{\mathsf{\Lambda }}_{ij}^t}\ge w_{\mathrm{WTA} j,\min },\forall j$$

(12)

$$\sum _{j=1}^n{\mathsf{\Lambda }}_{ij}^t=\sum _{j=1}^n{\displaystyle \frac{\ln \left(1-w_{\mathrm{WTA} j,\min }\right)-\sum _{r\ne i}{\mathsf{\Lambda }}_{rj}\ln \left(1-q_{\mathrm{WTA}rj}^t\right)}{\ln \left(1-q_{\mathrm{W}\mathrm{T}\mathrm{A}ij}^t\right)}}\le {\mathsf{\Lambda }}_{i,\max }$$

(13)

where w_WTA j,min is the damage threshold for enemy target j, and Λ_i,max is the maximum number of weapons of type i.

Compared to the original model (Equation (1)), the quantity model strictly adheres to the principle of supply–demand matching, ensuring that the probability of all weapons failing is less than the 1 minus the damage threshold for enemy target j. Equation (13) is derived from Equation (12) through a natural logarithm transformation, which represents that the quantity of the i-th type of weapon cannot exceed its maximum quantity.

3.1.2. Quality Model for Weapon-Target Assignment

In WTAs, the Lanchester equation [35,36] (Equation (14)) is used to describe parameterized firepower predictions, but it does not account for weapon quality or target quality, which can lead to overkill or underkill and inefficient firepower use. To avoid the phenomenon, it is necessary to precisely match the quality of different weapons with the quality of targets, in order to maximize strike effectiveness while reducing dependence on high-quality weapons. A composite combat power index $\theta$ is introduced based on indicators such as kill radius (θ₁), damage range (θ₂), attack frequency (θ₃), mobility (θ₄), interception rate (θ₅), kill intensity (θ₆), and hit rate (θ₇), as shown in Equation (15) [37]. A constraint model is then developed in Equation (16). This constraint ensures that the combat capability of weapon i is sufficient to satisfy the quality requirement of target j. The indicators are selected based on Table 3. These indicators comprehensively reflect the multi-dimensional combat capability of the weapon system, covering aspects such as attack, defense, and mobility, and can comprehensively evaluate weapon quality. The composite combat power index θ constructed through these indicators can accurately quantify the matching degree between weapons and targets.

$$q=1-e^{-\lambda A}$$

(14)

$$\{\begin{array}{l}\theta =\sum {\omega }_i{\overline{\theta }}_i\\ \sum {\omega }_i=1\end{array}$$

(15)

$${\theta }_i^t{x}_{\mathrm{WTA}ij}^t\ge {\theta }_j^t, \forall i=1,2,\dots ,m, \forall j=1,2,\dots ,n$$

(16)

where q, λ and A are damage probability, weapon damage rate parameter and weapon attack intensity, respectively; ω represents the weight of each quality indicator, and $\overline{\theta }$ is the normalized value of each indicator. The weights ${\mathsf{\omega }}_{\mathrm{i}}$ are determined using a hybrid Analytic Hierarchy Process (AHP) and Entropy Weight Method (EWM). AHP incorporates expert prior knowledge to establish the relative importance of operational indicators, while EWM objectively corrects these weights based on data variance.

Table 4. Explanations of each indicator for a quality model in WTA [38,39].

Indicators	Explanations	Indicators	Explanations
Kill radius (θ1)	Measures the effective damage range of the weapon	Interception rate (θ5)	Assesses the weapon’s defensive and counter-capability
Damage range (θ2)	Evaluates the coverage area and impact region of the weapon	Kill intensity (θ6)	Measures the damage effect and power of the weapon
Attack frequency (θ3)	Reflects the operational rhythm and sustained strike capability of the weapon	Hit rate (θ7)	Reflects the precision and combat effectiveness of the weapon
Mobility (θ4)	Demonstrates the deployment flexibility and response speed of the weapon

illustrates explanations of each indicator for a quality model in WTAs. A sensitivity analysis demonstrating the robustness of these assigned weights against a ±10% perturbation is detailed in Section 4.

3.1.3. Spatiotemporal Model for Weapon-Target Assignment

The spatiotemporal model for WTAs involves time window constraints (Equation (17)), trajectory collision constraints (Equation (18)), and regional weapon capacity constraints (Equation (19)). These constraints restrict the engagement time of weapon i against target j within a predefined feasible time window, maintain a safe spatial distance between trajectories of different weapons to avoid mid-air collisions, and limit the number of weapons deployed in region z to not exceed its regional capacity limit.

The time window constraints stipulate that the engagement must occur within a predefined temporal interval. The trajectory collision constraints utilize three-dimensional spatial coordinates to calculate the distance between weapon trajectories, mandating that this distance remain below a designated safety radius R to prevent potential collisions. Furthermore, the regional weapon capacity constraints impose limitations on the quantity of weapons deployed within a specific geographical area. Collectively, these constraints are employed to construct the spatiotemporal model for the WTA problem, ensuring the feasibility of weapon allocation schemes under temporal, spatial, and resource limitations.

$$t_{\mathrm{in},ij}<t_{ij}<t_{\mathrm{out},ij}$$

(17)

$$\sqrt{{\left(X_{i1}-X_{i2}\right)}^2+{\left(Y_{i1}-Y_{i2}\right)}^2+{\left(Z_{i1}-Z_{i2}\right)}^2}>R, i_1,i_2=1,2,\dots ,m, i_1\ne i_2$$

(18)

$$n_{bz}\le {\Omega }_z$$

(19)

where X, Y, Z represent spatial coordinates, R is the maximum radius of m weapon units, t_ij is the time at which weapon i strikes enemy target j, and Ω represents the regional weapon capacity. The indices in and out denote the entry and exit times of targets from the strike range, while z indicates the region.

3.1.4. Optimization Model for Weapon-Target Assignment

By considering the QQST model, the WTA optimization function also can be rewritten as Equations (1), (3) and (20), with the overall optimization model formulated in Equation (21). In the model, Equation (1) maximizes the total strike damage probability, and Equation (18) minimizes the mismatch of weapon and target quality. These three objectives are in a trade-off relationship. Quality matching may reduce damage probability and increase ammunition consumption, while increasing damage probability may lead to excessive reliance on high-quality weapons, resulting in inefficiencies.

$$F_{\mathrm{WTA}3}=\sum _i^m\sum _j^n{\theta }_i^t{x}_{ij}^t-\sum _i^m\sum _j^n{\theta }_j^t, \forall i=1,2,\dots ,m, \forall j=1,2,\dots ,n$$

(20)

$$\min F_{\mathrm{WTA}}=F_{\mathrm{WTA}2}+F_{\mathrm{WTA}3}-F_{\mathrm{WTA}1}$$

(21)

Through multi-objective optimization, the QQST model enables the balancing of multiple conflicting objectives. The QQST model can maximize damage probability while simultaneously minimizing weapon-target quality mismatch and controlling ammunition consumption. Traditional WTA models typically focus solely on maximizing damage probability; however, the QQST model can prevent the wasteful overuse of high-quality weapons, ensure proper matching between weapon quality and target threat levels, optimize overall combat effectiveness, and achieve optimal resource allocation while guaranteeing mission completion.

3.2. Quantity–Quality–Spatiotemporal Model for Supply–Demand Scheduling

On the supply side, IESs are characterized by multi-energy coupling, multi-scale temporal dynamics, and multi-agent participation. Accordingly, IES scheduling should consider the supply–demand characteristics of diverse energy carriers and the heterogeneous demands of loads, achieving supply–demand matching at the QQST level, which is inherently an NP-hard optimization problem.

Based on the aforementioned analysis of the WTA problem and the proposal of the QQST model-based WTA optimization model, the QQST model can be further extended to the IES schedule to achieve precise supply–demand matching under multi-energy coupling, multi-scale temporal dynamics, and multi-agent participation. The following sections will analyze the characteristics of QQST in the IES schedule and establish the QQST model.

3.2.1. Quantity Model for Supply–Demand Scheduling

The available weapon quantity in WTAs corresponds to the total energy supply capacity in IESs, while the damage threshold in WTAs corresponds to the load demand quantity in IESs. Furthermore, analogous to the situation in WTAs, where multiple weapons of diverse types can concurrently attack a single target, in IES scheduling, multiple energy supply devices and various forms of energy can, via conversion and transmission, concurrently supply a single demand-side user. Consequently, the two exhibit similarities in terms of quantity.

At time t, the total energy demand of the system is denoted as $w_{\mathrm{IES},\mathrm{all}}^t=\sum _{j=1}^n{w}_{\mathrm{IES}j}^t$, while the total energy supply is $b_{\mathrm{IES},\mathrm{all}}^t=\sum _{i=1}^m{b}_{\mathrm{IES}i}^t$. Accordingly, the supply–demand balance at the quantity level satisfies the constraint in Equation (22), ensuring real-time balance between supply and demand across all demand nodes.

$$\sum _{i=1}^m{b}_{\mathrm{I}\mathrm{E}\mathrm{S}i}^t{x}_{\mathrm{I}\mathrm{E}\mathrm{S}ij}^t{q}_{\mathrm{I}\mathrm{E}\mathrm{S}ij}^t\ge w_{\mathrm{I}\mathrm{E}\mathrm{S}j}^t, \forall j=1,2,\dots ,n$$

(22)

3.2.2. Quality Model for Supply–Demand Scheduling

Different types of energy in IESs have varying quality levels, expressed by exergy, which measures the maximum useful work attainable during a thermodynamic process. To avoid conceptual confusion often found in standard dispatch models, it is crucial to clarify that energy refers to the absolute quantity, whereas exergy denotes the energy quality. Electricity has the highest grade and can be converted into any form of energy; thus, its energy quality coefficient (EQC) is defined as 1 [40]. Other carriers, such as cooling, heating, and natural gas, have lower EQC values depending on their thermodynamic state, as detailed in

Table 5. Energy quality coefficients of different energy carriers.

Energy Carrier	EQC	Energy Carrier	EQC
Electricity	1	Natural gas	$1-{\displaystyle \frac{T_{\mathrm{b}}}{T_{\mathrm{b}}-T_0}}\ln \left({\displaystyle \frac{T_{\mathrm{b}}}{T_0}}\right)$
Cooling	${\displaystyle \frac{T_{\mathrm{u}}}{T_{\mathrm{c}}}}-1$	Heat	$1-{\displaystyle \frac{T_{\mathrm{u}}}{T_{\mathrm{h}}}}$
Hot water	$1-{\displaystyle \frac{T_0}{T_{\mathrm{h}}-T_{\mathrm{u}}}}\ln {\displaystyle \frac{T_{\mathrm{h}}}{T_{\mathrm{u}}}}$

T_b, T₀, T_u, T_c and T_h are burn temperature, ambient temperature, user temperature, heating temperature and cooling temperature, respectively.

.

Building upon the mathematical structure of the quantity model, we formalize the mapping between the WTA and IES quality dimensions. Rather than relying on a simple descriptive analogy, we define an explicit structural equivalence: the weapon quality index ($\theta$) in the WTA model maps directly to the comprehensive exergy coefficient of the energy supply (${\eta }_i$) in the IES. Similarly, the target threat level maps to the minimum energy quality requirement of the load (${\eta }_{jk}$). By doing so, the precise matching mechanics of WTA—which simultaneously prevent the overkill of targets and the overuse of high-value assets—are adapted to ensure that high-grade energy is not wasted on low-grade thermal demands.

During transmission, spatiotemporal latency and thermal dissipation cause additional losses, reducing the effective quality of the energy flow. The energy quality required by each demand node must not exceed the effective quality provided by its assigned supply sources, forming the structural quality constraint expressed in Equation (23). This constraint guarantees that the comprehensive energy quality provided to each load meets or exceeds its required energy quality level:

$$\sum _{i=1}^m{b}_{\mathrm{IES}i}^t{q}_{\mathrm{IES}ij}^t{x}_{\mathrm{IES}ij}^t{\eta }_i^t\ge \sum _k{\eta }_{jk}^t{w}_{\mathrm{IES}jk}^t, \forall j=1,2,\dots ,n$$

(23)

where ${\eta }_i={\eta }_i^{EQC}\cdot {\eta }_{i,o}$ is the comprehensive quality coefficient of supply source i, ${\eta }_{jk}={\eta }_{jk}^{EQC}\cdot {\eta }_{jk,o}$ is the required quality coefficient of demand node j for energy type k, and ${\eta }^{EQC}$ and ${\eta }_o$ represent the EQC and other supplementary quality coefficients. Furthermore, to quantitatively evaluate the systemic performance of this quality matching paradigm, the overall exergy efficiency ${\eta }_{ex}$ is explicitly defined. It calculates the ratio of the total exergy effectively delivered to the loads against the total exergy supplied by the sources, formulated in Equation (24):

$${\eta }_{ex}={\displaystyle \frac{{\sum }_{j=1}^n(w_{jt}^{IES}\cdot {\eta }_{EQC,j})}{{\sum }_{i=1}^m(b_{it}^{IES}\cdot {\eta }_{EQC,i})}}$$

(24)

3.2.3. Spatiotemporal Model for Supply–Demand Scheduling

The spatiotemporal model encompasses both temporal scheduling and spatial allocation. The application of the spatiotemporal model in the IES schedule bears resemblances to the spatiotemporal strategies in WTAs. In the temporal domain, WTA necessitates the consideration of weapon deployment and response at different time points. Conversely, the IES attains temporal supply–demand equilibrium via energy storage devices, absorbing surplus energy when supply surpasses demand and releasing stored energy when demand exceeds supply.

In the spatial domain, WTA requires the consideration of weapon deployment and coordination across different geographical locations. Meanwhile, the IES realizes spatial energy optimal allocation through transmission networks and distributed facilities, minimizes transportation costs, and accounts for geographical disparities.

Moreover, renewable energy sources demonstrate spatiotemporal complementarity. Consequently, optimization demands the characterization of load variations in both time and space, as well as the variability of renewable energy generation, to achieve precise supply–demand matching. Therefore, in the IES schedule, by implementing the aforementioned spatiotemporal migration in energy transmission pipelines and energy storage equipment, it becomes feasible to achieve long-term scheduling across regions and time periods. Specifically, this implies satisfying the quantity model and quality model described in Section 3.1.1 and Section 3.1.2 in both the temporal and spatial dimensions.

3.2.4. Optimization Model for Supply–Demand Scheduling

Based on the QQST constraints, the optimization objectives of the IES schedule are formulated in Equations (25)–(27). Equation (26) quantifies the matching degree of the EQC between supply and demand. Equation (27) denotes the minimization of operational costs. Moreover, constraints such as device capacity, ramping limits, operational safety, energy balance, topology limits, and coupling relations are taken into account in the IES schedule. Figure 3 illustrates the EQC matching rules in the IES schedule, which are based on energy type conversion, depicting a scenario where one supplier corresponds to one demander with different energy types. In conclusion, the QQST model enables the simultaneous optimization of quantity–quality matching and spatiotemporal coordination in the IES schedule, thereby expanding the IES schedule methodology.

$$F_{\mathrm{IES}}=\min F_{\mathrm{IES}3}-F_{\mathrm{IES}1}-F_{\mathrm{IES}2}$$

(25)

$$F_{\mathrm{IES}2}=\sum _{j=1}^n\left(k\sum _{i=1}^m{\eta }_i^t{x}_{\mathrm{IES}ij}^t-m_j^{*}\sum _k{\eta }_{jk}^t\right)$$

(26)

$$F_{\mathrm{IES}3}=\sum _{i=1}^m{\overline{V}}_{\mathrm{IES}i}$$

(27)

where $m_j^{*}$ denotes the number of supply sources serving the demand node j, with constraint $m_j^{*}$ < m, meaning not all supply units simultaneously provide energy to a single demand node.

3.3. Optimize Solution Settings

To ensure the commensurability of heterogeneous objective functions, namely the minimization of operational costs (in monetary units) and the maximization of exergy efficiency (as a ratio), a min–max normalization approach is employed. The normalized cost index ${\mathrm{V}}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ is defined as follows: ${\mathrm{V}}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}={\displaystyle \frac{{\mathrm{V}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}-{\mathrm{V}}_{\mathrm{m}\mathrm{i}\mathrm{n}}}{{\mathrm{V}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{V}}_{\mathrm{m}\mathrm{i}\mathrm{n}}}}$, where ${\mathrm{V}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${\mathrm{V}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ represent the theoretical lower and upper bounds of operational costs or ammunition expenditure in WTAs determined by single-objective optimization under extreme scenarios. This transformation maps the cost into a dimensionless interval [0, 1], preventing the objective function from being dominated by variables with larger absolute numerical scales.

The computational complexity of the underlying problems should be clarified. The classical WTA problem is NP-complete, while the IES scheduling problem with mixed-integer and multi-objective characteristics is generally NP-hard. The proposed QQST framework does not alter these complexity classes. Instead, it provides a generalized reformulation that embeds quality and spatiotemporal coupling into the model structure, resulting in a mixed-integer optimization problem.

Although the worst-case complexity remains unchanged, the structured representation improves modeling clarity and facilitates solution strategies. In this study, the QQST-based model is solved using an MILP formulation with appropriate linearization, ensuring practical computational tractability.

To address the non-linear constraints in the QQST model, the linear approximation methods are adopted. For exponential non-linearity (the Lanchester equation in Equation (14)), A 1D piecewise linear approximation is employed, where the variable is partitioned into 3–5 segments to balance approximation accuracy and computational efficiency. For product-type non-linearities, including multiplicative terms in the WTA damage constraint (Equation (12)) and bilinear couplings in the ammunition limit constraint (Equation (13)), a 2D piecewise linear approximation is utilized to explicitly preserve the coupling relationship between variables. Specifically, for the Euclidean norm in the trajectory collision constraint (Equation (18)), the norm is first squared to eliminate the square root, converting it into a sum of quadratic terms; McCormick relaxation is then applied to these quadratic/product terms, generating linear constraints.

4. Case Study Analysis

4.1. Case Analysis

The QQST characteristics within an IES are identified by analyzing operational data from a real-world industrial park case. Subsequently, the similarities between IES scheduling and the WTA problem in C2 systems are revealed.

The case study centers on an industrial park in which a steam network provides superheated steam to 60 textile enterprises. The heating capacity of the energy carriers amounts to 2500 t/h, and there is a main pipeline extending 62 km. Its infrastructure encompasses eight heat sources, consisting of eight boilers, seven steam turbines, two condensing thermal power units, and five back-pressure thermal power units. Sixty textile enterprises are distributed along five low-pressure pipelines and three medium-pressure pipelines. The topology of the IES presents two-ring networks with multiple sources. To manage thermal energy of different temperature grades, an IES scheduling platform was established (Figure 4), which executes the following hierarchical scheduling process:

• Medium-to-long-term planning: Drawing upon historical production data, the IES predicts the aggregate load and its temporal distribution for the forthcoming season or year. This prediction serves as a basis for formulating the annual generation and heating plans for each thermal power unit.
• Day-ahead scheduling: Leveraging forecasts of renewable energy output, heating demand, and electricity market price signals, the IES applies multi-objective optimization algorithms to ascertain the unit commitment and generation profiles for the subsequent day, thus achieving a balance between economic efficiency and system reliability.
• Real-time dispatch: The IES consistently gathers real-time data on heating load and production. It executes rolling optimization to alleviate the effects of renewable energy fluctuations and load variations, and dynamically modifies unit outputs to uphold the supply–demand equilibrium.

The data used in this case study is derived from the SCADA (Supervisory Control and Data Acquisition) system of the textile industrial park, which is the core data acquisition and monitoring system for the park’s IES operation. The data is measured from January to June 2024, with a high sampling frequency of 5 min to capture the dynamic spatiotemporal characteristics of the IES. There are approximately 1.08 million valid data points after preprocessing, which fully reflects the actual operational state of the IES under different working conditions (peak load, valley load, equipment maintenance, etc.). Linear interpolation is used to supplement missing data caused by short-term SCADA system downtime. Dimensionless processing is performed on indicators with different units to facilitate model calculation and parameter calibration, with the normalization range of [0, 1]. The optimization algorithm is MILP by using CPLEX 20.1 with optimality gap = 1% and time limit = 30 s. The convergence criterion is the objective function value change < 1 × 10⁻⁴ for 20 consecutive iterations. The calculation is done in Intel Xeon Gold 6326 (16 cores, 2.9 GHz).

To verify the generalizability and robustness of the QQST model, four representative scenarios are designed based on the actual operational characteristics of the industrial park IES, covering normal, extreme, and uncertain conditions (

Table 6. Comparison of optimization results between QQST and MILP under typical scenarios.

Scenario Type	Description	Key Parameters
Normal Load Scenario	Routine operation of textile enterprises (two-shift system), stable load demand	Load fluctuation range ± 10% with all equipment in normal operation
Peak Load Scenario	Simultaneous production of 60 enterprises (peak season), maximum heat demand	Load increase with 30% above normal, steam supply pressure requirement 3.5 MPa
Equipment Failure Scenario	Sudden failure of 2 boilers (12.5% of total heat supply capacity)	Heat supply capacity reduction 15% with fault duration for 2 h

).

4.2. Results

The simulation results, illustrated in Figure 5, reveal the dynamic evolution of QQST features across the demand side. The quantity (Q1) analysis shows high volatility in steam mass flow, peaking during the morning shift (08:00–11:00) at approximately 1850 t/h. More critically, the quality (Q2) dimension, characterized by the energy quality coefficient (EQC), exhibits significant spatiotemporal degradation. In the medium-pressure pipelines, the EQC remains relatively stable near 0.32, whereas the peripheral nodes of the low-pressure network experience a drop to 0.24 during peak demand periods. This indicates that, while the quantity of energy reaches the end-users, the quality is compromised due to pressure drops and thermal losses over the 62 km span. The spatiotemporal (ST) analysis confirms that the mismatch between supply-side ramping and demand-side surges leads to a “quality lag” of approximately 15–20 min, mirroring the latency issues often encountered in military C2 communication and strike coordination. The quality lag refers to the time delay between the supply-side adjusting energy quality (increasing/decreasing steam pressure/temperature) in response to demand changes and the demand-side actually perceiving stable energy quality.

Analysis of Figure 5a,b indicates that the average heating flow for 60 demand-side users and the 23 supply-side pipelines exhibits congruent fluctuation patterns, reaching a minimum at 8:00 and a maximum at 12:00. This validates the quantity model (Equation (20)), where the supply capacity $b_{\mathit{IES},i}^t$ and decision matrix $x_{\mathit{IES},\mathit{ijt}}$ dynamically satisfy the demand $w_{\mathit{IES},\mathit{jt}}$ to maintain real-time equilibrium.

Furthermore, the quality model’s application is evidenced by the temperature and pressure data in Figure 5c,d. By applying the energy quality coefficient (EQC) constraints from Equation (23) and the matching rules in Equation (25), the system ensures that high-grade energy is not wasted on low-grade thermal demands. This precise matching reflects the “precision strike” philosophy of WTAs, where weapon quality $\theta$ must match target threat $U$ to avoid resource over-consumption. Finally, the hierarchical scheduling process (I–III) embodies spatiotemporal coordination, utilizing energy storage and transmission networks to achieve equilibrium across nodes, consistent with the optimization objectives in Equation (24). The case study thus validates that IES scheduling is effectively governed by the QQST framework, aligning quantity, quality, and spatiotemporal dimensions in real-time operations.

Table 7. Comparison results of multiple scenarios.

Scenario Type	Quantity Dimension Data (Average)	Quality Dimension Data (Average)	Spatiotemporal Dimension Data (Average)
Normal Load	Demand peak: 31.8 t/h; Supply peak: 192 t/h	EQC (medium-pressure: 0.32, low-pressure: 0.24–0.28); Exergy efficiency: 29.6%; Quality mismatch rate: 4.8%	Quality lag: 11–15 min
Peak Load	Demand peak: 39.5 t/h; Supply peak: 204 t/h	EQC (medium-pressure: 0.30, low-pressure: 0.22); Exergy efficiency: 27.3%; Quality mismatch rate: 6.5%	Quality lag: 18–22 min
Equipment Failure	Demand minimum: 23.9 t/h; Supply peak: 159 t/h	Exergy efficiency: 24.8%; Quality mismatch rate: 8.3%	Quality lag: max 25 min; Recovery time: 8 min

lists the results of different scenarios in quantity, quality, and spatiotemporal dimensions. When the system operates under the peak load scenario, the surging heat demand caused by the concentrated production of textile enterprises forces the energy supply side to increase its output to meet the load demand, which intensifies the energy quality loss during pipeline transmission. This is manifested as a significant decline in the EQC of both medium- and low-pressure pipelines, a reduction in exergy efficiency and a rise in the quality mismatch rate in the quality dimension. Meanwhile, the mismatch between the peak-regulating response speed of the energy supply side and the change rhythm of the demand-side load leads to a substantial extension of the quality lag time in the spatiotemporal dimension.

In the equipment failure scenario, the sudden failure of two boilers results in a sharp reduction in the heat supply capacity of the IES, compressing the adjustment space of the energy supply side in the quantity dimension. This single-dimensional disturbance is transmitted and amplified through the strong coupling of the multi-energy system, triggering a cascading deterioration of energy quality in the quality dimension, where the system’s exergy efficiency drops to the lowest level and the quality mismatch rate rises to the peak in all tested scenarios. In the spatiotemporal dimension, the energy supply disturbance caused by equipment failure breaks the original balance of spatiotemporal scheduling, leading to the maximum quality lag time of the system; the IES requires a specific recovery time to re-adjust the supply–demand allocation strategy and restore the spatiotemporal matching of energy supply and demand.

It is worth noting that real-world industrial IESs operate with inherent uncertainties that may affect the practical application of the QQST model. These uncertainties mainly include random fluctuations in demand, parameter uncertainties, and unforeseen supply-side disturbances. Based on different scenarios, a load increase or equipment failure may lead to a 1.3–3.5% increase in quality mismatch rate and a 2.3–4.8% decrease in exergy efficiency. However, as demonstrated in Table 5, even under the 30% peak load surge and 15% supply capacity reduction, the QQST model still maintains stable operational performance, indicating its basic adaptability to common real-world uncertainties. Further enhancement of its robustness against extreme uncertainties will be the focus of future research.

As illustrated by Figure 6 and the case study, the IES effectively matches the supply–demand relationship across different temporal and spatial scales, ensuring equilibrium in both quantity and quality. A comparison with the WTA problem reveals analogous patterns. In WTA, the real-time dynamic adjustment of firepower allocation achieves an optimal alignment of firepower quantity and quality across multiple targets, thereby maximizing the probability of target damage while minimizing resource consumption.

4.3. Comparison with Traditional Optimization Methods

To quantitatively verify the optimization effect and computational efficiency of the proposed QQST model, mixed-integer linear programming (MILP) is selected as a comparison benchmark. The MILP baseline model is a standard IES scheduling model widely used in industrial practice, focusing on energy quantity balance and economic efficiency optimization. It does not consider energy quality matching (EQC constraints) or spatiotemporal latency, which are the core innovations of the QQST framework. All experiments are conducted based on the same SCADA operational dataset of the case. The quantitative comparison results of the QQST model and the MILP methods are shown in

Table 8. Performance and efficiency comparison between QQST and MILP models.

Evaluation Indicator	QQST Model	MILP
Exergy efficiency (%)	29.6	21.2
Quality mismatch rate (%)	4.8	23.0
Computation time (s)	9.2	10.5
Convergence iteration number	126	189

, in which the QQST model shows significant advantages in both optimization effect and computational efficiency. The QQST model constructs a unified optimization framework from the three dimensions of quantity, quality, and spatiotemporal, which comprehensively characterizes the energy degradation and spatiotemporal latency in the IES, and avoids the one-sided optimization of only focusing on quantity balance or economic efficiency in traditional methods. By explicitly defining the quantity, quality, and spatiotemporal constraints, the QQST model compresses the search space of the optimization problem by about 30%, avoiding the redundant search for traditional methods in the high-dimensional constraint space. In particular, compared with stochastic optimization, which needs to consider a large number of uncertainty scenarios, the QQST model integrates the spatiotemporal and quality constraints into the objective function, which greatly reduces the computational complexity.

Based on the above comparative experimental results, the QQST model achieves an 8.4% increase in overall exergy efficiency and an 18.2% reduction in quality mismatch rate for the studied industrial park IES under the normal load scenario (the typical steady-state operation condition) compared with the traditional MILP method, which are the core quantitative optimization effects of the QQST framework in realizing precise quantity–quality–spatiotemporal matching of energy supply and demand.

The case study results in Table 7 and Table 8 provide direct empirical support for the QQST model, but also reveal context-dependent characteristics of its optimization effects that require critical reflection. First, in terms of quantitative optimization outcomes, the 8.4% absolute increase in exergy efficiency and 18.2% absolute reduction in quality mismatch rate under the normal load scenario (Table 8) are closely tied to the textile park’s stable thermal load feature. Figure 5 shows that the load fluctuation range is only ±10% in steady state, which minimizes the interference of random variations on the QQST model’s quantity–quality–spatiotemporal matching logic. However, under the peak load and equipment failure scenarios (Table 7), the model’s exergy efficiency decreases by 2.3% and 4.8%, respectively, compared with the normal load scenario, and the quality mismatch rate increases by 1.7% and 3.5%. This indicates that the QQST model’s optimization performance is sensitive to system operational pressure: as the load surges or supply capacity declines, the trade-off between quantity balance, quality preservation, and spatiotemporal alignment becomes more intense, leading to a slight degradation of comprehensive effects. Second, regarding the uncertainty of the results, the 1.08 million valid SCADA data points (with 5 min sampling frequency) ensure the statistical reliability of the conclusions, but the data is limited to a single textile park’s steam network (62 km pipeline, 60 enterprises). The generalization of the QQST model to industrial parks with different pipeline lengths, enterprise types, or energy carrier combinations (e.g., electricity-heat-gas coupling) remains to be verified by more heterogeneous cases.

4.4. Discussions

4.4.1. Technological Integration and Theoretical Advancement Based on Cross-Domain Comparison

Through the comparative analysis of WTA and IES scheduling, the following suggestions for technological integration and theoretical advancement are proposed:

(1)

Establish a unified descriptive paradigm based on the QQST framework. Disparities exist in the mathematical descriptions between the WTA and IES models. This study establishes a unified paradigm that allows for a transition from simple “volume matching” to “value-based matching.” By integrating the EQC into the objective function (Equation (24)), the IES achieves refined resource distribution, similar to WTAs optimizing ammunition costs, laying the theoretical foundation for cross-domain scheduling models.

(2)

Leverage collaborative control for precise spatiotemporal energy matching. Drawing on C2’s multi-platform collaborative control for WTA plans, IESs can adopt distributed strategies to address system coupling. Aligning energy “quality” and “quantity” across spatiotemporal dimensions minimizes exergy destruction, fulfilling the multi-objective optimization in Equation (25).

(3)

Construct a refined quantity–quality matching model. Adopting WTA’s “precision strike” concept, EQC matching rules (Equation (26)) reserve high-grade energy for high-quality demand, avoiding resource over-consumption while meeting user requirements.

(4)

Incorporate dynamic network topology to enhance resilience. Learning from C2’s dynamic strike paths, IESs can adopt dynamic path generation for line failures or cyber-attacks. Research on real-time adaptive spatiotemporal matching algorithms improves system survivability under extreme events.

(5)

Develop a hierarchical scheduling framework. Following C2’s strategic–operational–tactical hierarchy, the IES scheduling center functions as a command post, using the optimization model (Equation (25)) to resolve NP-hard allocation problems.

4.4.2. Practical Implementation Challenges and Solutions

The QQST model is compatible with the SCADA system widely used in industrial parks, as it only requires real-time data on flow rate, temperature, and pressure, avoiding additional hardware investment and reducing practical application thresholds. However, the QQST model faces core challenges in real-world applications (regulatory policy constraints, insufficient data availability, and heavy computing burden). Targeted solutions are proposed as follows:

(1)

Regulatory policy adaptation. Embed EQC matching degree into energy pricing, align scheduling results with market transaction rules, and establish a parameter dynamic adjustment mechanism based on policy updates to ensure compliance.

(2)

Data availability improvement. Deploy edge computing nodes for real-time data collection and preprocessing, build an industrial internet-based data sharing platform, and adopt desensitization and access control to ensure data security.

(3)

Computing burden reduction. Design a three-level “cloud–edge–end” solving architecture. Terminal devices handle preprocessing, edge nodes perform parallel computing, and the cloud undertakes global coordination, which can reduce cloud computing burden and control solving time.

4.4.3. Extension to Broader System Contexts

The findings of this study can be extended to broader IES contexts beyond industrial park steam networks.

(1)

Multi-energy coupling systems (electricity–heat–gas–cooling). The QQST model’s quality matching logic can be extended to other energy carriers, and the spatiotemporal constraints can be adapted to multi-network coordination (power grid, gas pipeline, heating network).

(2)

Urban-level IES. For large-scale urban energy systems with more complex topology and higher uncertainty, the QQST model can be combined with distributed decomposition (as discussed in Section 4.4.2) to maintain optimization efficiency while ensuring scenario adaptability.

(3)

Renewable energy-rich systems. The model’s performance in the renewable energy integration scenario confirms its potential for application in distributed photovoltaic/wind–solar–thermal complementary systems, providing a new solution for intermittent energy absorption.

4.4.4. Research Limitations

While the QQST model shows advantages in steam-based IESs, it has limitations requiring improvement:

(1)

Insufficient coverage of extreme operating conditions. The model focuses on normal conditions but lacks consideration of extreme weather (e.g., pipe network freezing) and sudden failures. Future research will integrate emergency dispatch mechanisms to enhance robustness.

(2)

Insufficient coverage of uncertain operating conditions. The model is based on deterministic parameters for a textile industrial park with stable thermal load and no renewable energy integration, without explicit modeling of renewable energy output uncertainty and demand-side random fluctuation.

(3)

Lack of multi-stakeholder interest coordination. The model optimizes system-wide efficiency without incorporating the interests of suppliers, users, and operators. Future research will build a multi-agent collaborative framework using game theory to balance interests.

(4)

Limitations in multi-energy coupling adaptation. The case targets a single steam network, while practical IESs involve electricity–heat–gas–cooling coupling. Future research will expand the QQST model to multi-energy systems, refining quality matching and spatiotemporal coordination rules.

5. Conclusions

A comparison of the architectures of the C2 and the IES is presented. By proposing and comparing the WTA model for C2 with the IES schedule model, the inherent QQST characteristics within the WTA model are revealed. A QQST optimization model framework for WTA is established, and based on the structural isomorphism between C2 and IES, an analogous QQST optimization model for the IES schedule is formally developed. The QQST characteristics of IESs are then validated using a real-world industrial case study, with comparative experiments demonstrating that the proposed model outperforms the traditional MILP method by 8.4% in exergy efficiency and 18.2% in quality mismatch reduction, while maintaining computational tractability.

Despite the theoretical advancements proposed by the QQST framework, several practical limitations must be critically acknowledged. First, transforming the scheduling task into an MINLP problem introduces a substantial computational burden, which may challenge real-time operational feasibility under extreme or large-scale network conditions. Second, the framework relies heavily on high-fidelity, synchronous data acquisition; therefore, its effectiveness is highly sensitive to sensor latency and data availability. Finally, although the military analogy employed in this study offers robust optimization logic, our research is confined to civilian integrated energy systems (IESs) and thus cannot be directly applied to military scenarios without extensive reconfiguration and adaptation. Furthermore, the practical deployment of real-world civilian integrated energy systems is constrained by regulatory policies, economic limitations, and energy market dynamics, all of which are typically absent or differ substantially in military contexts.

Future research will focus on addressing these limitations by integrating stochastic optimization to better handle renewable uncertainty, developing distributed computing architectures to reduce computational burdens, and explicitly incorporating market mechanism constraints into the QQST multi-agent scheduling framework.