A Hierarchical Water Supply–Demand Regulation Model Coupling System Dynamics and Feedback Control Mechanisms: A Case Study in Wu’an City, China

Abstract

Water scarcity has become a critical global challenge, particularly in rapidly developing regions where water demand often exceeds sustainable supply capacities. Traditional “demand-driven” water management approaches have proven inadequate to address this imbalance, necessitating the development of more sophisticated “supply-driven” solutions. This study presents a groundbreaking System Dynamics (SD)-Feedback-Hierarchical Water Demand (SD-F-HWD) model that advances water resources management through three contributions. First, the model substantially extends conventional water demand hierarchy methods by developing a comprehensive classification framework with enhanced sector-specific criteria for industrial, agricultural, and ecological needs. Second, the innovative feedback regulation mechanism resolves persistent challenges from previous studies, including ambiguous control parameters and system instability. Third, the model establishes a unified analytical platform that effectively integrates these components for robust supply–demand equilibrium analysis. Validation in Wu’an City, Hebei Province—a representative water-stressed industrial region in northern China—demonstrated the model’s effectiveness. Under low-flow conditions (P = 75%), total water demand decreased by 11.24% while rigid demand was reduced by 8.50%. For normal flow conditions (P = 50%), corresponding reductions reached 9.88% and 6.99%, respectively. Crucially, all adjustments remained within practical policy implementation boundaries, demonstrating the model’s real-world applicability. The SD-F-HWD model offers a practical and scalable solution for sustainable water allocation in water-stressed regions through its integrated methodological framework.

1. Introduction

In recent years, many factors have contributed to the exacerbation of global water scarcity, including global climate change, drought, population growth, increasing water demands, and inadequate management practices [1]. To address this issue, numerous scholars have conducted a significant body of research on risk assessments, strategies, and water resource management models [2,3]. However, the escalating issue of water scarcity is both persistent and inevitable, with an increasing number of countries and regions being expected to face this challenge in the near future [4,5,6]. The problem of water supply–demand imbalance urgently requires innovative water resource management strategies to achieve sustainable and coordinated development of water resources in the social, economic, and environmental spheres [7]. Currently, extensive innovative research focusing on the climate [8], ecology [9], and socioeconomic [10] domains has provided critical insights and diverse frameworks for advancing water resource management methodologies. The challenges posed by water scarcity are global in nature, yet their manifestations in specific regions are particularly pronounced.

For any given region, the increase in water supply is limited by the region’s inherent water resource endowment. Given these constraints, implementing water demand management solutions proves significantly more crucial than seeking alternative water sources [11]. Consequently, it is necessary to curb excessive potential water demand [12,13] to keep water consumption within the carrying capacity of water resources. Given the inherent constraints of regional water resource endowments that limit supply expansion, it becomes imperative to shift from conventional demand-driven reactive approaches to establishing a supply-constrained management framework, thereby enabling more effective and scientific water demand management policies. Existing research has examined water supply and demand situations under policy control by modifying parameters that affect water demand levels [14,15]. However, the modification of these parameters lacks theoretical justification. Therefore, Dong [16] developed a water resources socioeconomic–environmental (WSE) system for water resource systems to provide a scientific basis for regulating water resource parameters. Similarly, Zhou [17] also established an SD-multiple objective optimization (SD-MOO) model for water resource systems, thereby enabling the scheme of policy control and related configuration results to be obtained. In addition, other studies on feedback mechanisms for water resource systems have rarely been reported. While existing feedback mechanisms offer valuable insights for promoting the transition of water resources management to a “supply-dictates-demand” model, these mechanisms still exhibit several inherent limitations in their structural design. First, the WSE system lacks defined control ranges for feedback intensity, which hinders practical management decision-making. Second, although the SD-MOO model incorporates fixed feedback intensity control ranges, this design significantly constrains the model’s feedback capability. Notably, when feedback control intensity increases further, the model demonstrates convergence failure issues. Finally, neither of these feedback mechanisms addresses the rationality of parameter adjustments post-feedback.

Moreover, current research on water-resources feedback regulation mechanisms has failed to account for the differential impacts of regulation on various hierarchical levels of water demand. Hierarchical classification of water demand is of equal importance in water demand identification and water resources management research [18]. Existing studies have explored sector-specific hierarchical classification approaches for water demand [19,20]. However, no research has yet synthesized these theories into a unified, generalizable approach.

To address this gap, to establish a water resources management system based on the “supply-driven” principle, this study developed an SD-F-HWD model that integrates hierarchical water demand partitioning and feedback regulation mechanisms. The comprehensive application of this model combines policy optimization for macro-level water supply–demand equilibrium with predictive capacity for stratified water demand classification, thereby creating an effective technical framework for precise water resources regulation and science-based policy formulation and revision.

2. Methodology

2.1. SD Model

SD is an interdisciplinary research method based on systems control theory, integrating theoretical frameworks from both social and natural sciences. This approach treats the research subject as an open system with multi-level structures, where nonlinear interactions among internal components generate complex dynamic behaviors. SD models have two key features: visualizing system structures to represent abstract concepts graphically, and simulating system behavior changes. The SD modeling process follows three key steps: first, establishing the conceptual framework and methodological foundation of the research system; second, visually representing structural relationships among components through stock-flow diagrams while defining variable types and their interaction mechanisms; finally, developing mathematical equation systems to conduct quantitative analysis. This method maintains system structural integrity while clearly demonstrating dynamic behavioral patterns.

Currently, SD has been widely applied in water resources management and related research fields due to its capability to model complex, dynamic water system processes [21]. In water resources research, it is essential to clarify the causal relationships among various elements in the water resources system, as this enables clear identification of key nodes within the system for research purposes [22]. The hierarchical classification of water demand and the feedback regulation mechanism investigated in this study are both conceptual frameworks. Therefore, by adopting the SD model as the baseline modeling framework, its unique modeling mechanisms and procedures enable the transformation of abstract concepts in hierarchical water demand classification and feedback regulation mechanisms into visual representations, thereby revealing their intrinsic characteristic mechanisms.

The SD model is a system model constructed with multiple variable types:

(i) Level variables: represent accumulated quantities that change only through inflows/outflows.

(ii) Rate variables: control flows altering levels over time.

(iii) Auxiliary variables: intermediate components for rate calculations.

(iv) Table functions: define nonlinear relationships.

(v) Constants: fixed parameter values.

The model developed in this study was constructed using the Vensim PLE software package (https://vensim.com).

2.2. Feedback Regulation Mechanism

The construction of a feedback regulation mechanism aims to integrate the system’s internal outputs (state variables) with external anthropogenic factors (feedback control intensity) in order to adjust the system’s inputs (adjustable variables). This adjustment drives the system’s internal outputs to approach or reach desired levels after regulation. The fundamental computational principle is as follows:

$$X_n^{\prime }=X_n\times {\displaystyle \prod _{m=1}^M{\alpha }_{m,n}}$$

(1)

where $X_n^{\prime }$ is the adjustable variable after feedback; $X_n$ is the adjustable variable before feedback; ${\alpha }_{m,n}$ is the feedback ratio, which is a quantification of the feedback effect; m is the number of state variables in the system; n is the number of adjustable variables in the system.

The concrete implementation of feedback functionality requires quantifying the feedback effect through the establishment of a feedback function. In this study, we propose a reconstructed arctangent-style feedback function:

$${\alpha }_{m,n}=1+arctan\left({\displaystyle \frac{S{V}_m-{\mu }_m}{{\sigma }_m}}\times \phi \times {\beta }_{m,n}\right)\times {{\displaystyle (arctan\phi )}}^{-1}$$

(2)

$${\alpha }_{m,n}=1-arctan\left({\displaystyle \frac{S{V}_m-{\mu }_m}{{\sigma }_m}}\times \phi \times {\beta }_{m,n}\right)\times {\left(arctan\phi \right)}^{-1}$$

(3)

where $S{V}_m$ is the state variable in the system; ${\mu }_m$ is the expected value of the state variable; ${\sigma }_m$ is the smallest distance between the $S{V}_m$ and ${\mu }_m$; ${\beta }_{m,n}$ is the feedback control intensity, taking a value in the range of [0, 1], which can be artificially regulated and reflects the sensitivity of the human influence on the system; $\phi$ is the regulation coefficient ($\phi$ > 0), which reflects the sensitivity of the system feedback itself.

In the feedback regulation mechanism, Equations (2) and (3) mathematically represent the positive and negative feedback operations, respectively. These equations determine ${\alpha }_{m,n}$ based on the intrinsic deviation relationship between $S{V}_m$ and ${\mu }_m$, then apply ${\alpha }_{m,n}$ to $X_n$ to achieve system regulation. The selection between positive and negative feedback formulations depends on the directional characteristic of how variations in $X_n$ influence output $S{V}_m$. For instance, in a given system, if an increase in $X_n$ drives $S{V}_m$ further away from the equilibrium state ${\mu }_m$, a negative feedback control must be activated to reduce $X_n$, thereby steering $S{V}_m$ back toward ${\mu }_m$. Conversely, if $X_n$ enhances the system’s convergence to ${\mu }_m$, positive feedback is applied. The designed feedback function aims to modulate the adjustable variables ($X_n$) through these bidirectional feedback effects. The feedback function proposed in this paper has the following characteristics:

(i) When $S{V}_m$ reaches ${\mu }_m$, ${\alpha }_{m,n}$ is 1, and the system is not regulated by feedback.

(ii) ${\alpha }_{m,n}$ varies monotonically and nonlinearly with the change in the state variable.

(iii) ${\alpha }_{m,n}$ has an upper (lower) limit of adjustment.

(iv) ${\alpha }_{m,n}$ is controlled by the intensity of the feedback control.

(v) $\phi$ controls the degree of nonlinearity of the feedback function and the degree of dynamic response of the system.

The positive and negative feedback curves under varying feedback control intensities are illustrated in Figure 1 (with parameter $\phi$ fixed at 1 as an example), while the corresponding curves under different regulation coefficients are shown in Figure 2 (with parameter ${\beta }_{m,n}$ fixed at 1 as an example).

As seen in Figure 1, when $S{V}_m$ is held constant, ${\alpha }_{m,n}$ increases with ${\beta }_{m,n}$. Figure 2 shows that for identical $S{V}_m$ and ${\beta }_{m,n}$, ${\alpha }_{m,n}$ grows as $\phi$ increases. In practical applications of the feedback function, $\phi$ governs the system’s intrinsic feedback sensitivity, while ${\beta }_{m,n}$ regulates the intensity of anthropogenic control. Their coordinated adjustment enables precise system behavior modulation.

The application of feedback regulation in water resources supply-demand systems requires clear definition of system variables $S{V}_m$ and $X_n$. The state variable $S{V}_m$ should be selected as a comprehensive indicator reflecting water supply-demand dynamics. The water scarcity rate, serving as a key metric for evaluating supply-demand equilibrium and effectively capturing supply-demand conflicts, is designated as the system’s ultimate output. Control variables $X_n$ should consist of policy-adjustable parameters capable of regulating $S{V}_m$ through feedback mechanisms, including sectoral water use efficiency, population size, and industrial development scale. Consequently, the water resources supply-demand system contains only one state variable ($m$ = 1), specifically $S{V}_1$. The optimal water scarcity rate is zero (${{\displaystyle \mu }}_1$ = 0), while the theoretical maximum shortage rate is 100% (${\sigma }_1$ = 1). Parameter ${{\displaystyle \beta }}_{1,n}$ represents the feedback control intensity for different regulatory variables, and ${{\displaystyle \alpha }}_{1,n}$ denotes the feedback ratio applied to these variables.

2.3. Method of Hierarchical Classification for Water Demand

Regarding water resource management, the regional water demand is determined by the local socioeconomic development. In the context of limited water resources, the primary focus is to meet survival and safety-related needs, followed by higher-level development needs. Based on Maslow’s hierarchy of needs theory, the water demand across industries is classified into two levels: rigid and flexible water demands [23]. The specific classification method and formulas are based on Guo’s hierarchical framework [19].

This study introduces modifications to the hierarchical classification of industrial, agricultural, and ecological water demands within Guo’s hierarchical framework. The improvements are as follows: For industrial water demand, the selection method for leading industries, the location entropy analysis method, has been added. For agricultural water demand, the forestry, fisheries, and livestock water demands have been added to agricultural water demand, with hierarchical classification implemented for these components. For ecological water demand, a hierarchical classification method and its basis for the terrestrial ecological water demand have been introduced. The enhanced hierarchical water demand classification methodology demonstrates greater comprehensiveness, representing an integrated and universally applicable approach. Furthermore, the principles and corresponding equations of hierarchical water demand classification for each sector are elaborated as follows:

(i) Domestic water demand is related to population and domestic water use quotas [24]. Under rigid conditions, residents’ water saving awareness and water use efficiency should be maximized to sustain basic survival activities. Therefore, the advanced values of domestic water use quotas serve as the basis for partitioning rigid and flexible domestic water demands. The equation for calculating the domestic water demand is as follows:

$${{\displaystyle W}}_{dt}={\displaystyle \frac{\left({{\displaystyle q}}_u\times {{\displaystyle p}}_u+{{\displaystyle q}}_r\times {{\displaystyle p}}_r\right)}{\left(1-{{\displaystyle c}}_d\right)\times 10000}}$$

(4)

where ${{\displaystyle W}}_{dt}$ is the domestic water demand, 10⁴ m³; ${{\displaystyle q}}_u$ is the urban domestic water consumption quota, m³/(person·year); ${{\displaystyle p}}_u$ is the urban population, person; ${{\displaystyle q}}_r$ is the rural domestic water consumption quota, m³/(person·year); ${{\displaystyle p}}_r$ is the rural population, person; and ${{\displaystyle c}}_d$ is the domestic water supply loss factor.

The equation for calculating the domestic rigid water demand is as follows:

$${{\displaystyle W}}_{dr}={\displaystyle \frac{\left({{\displaystyle q}}_u^{\prime }\times {{\displaystyle p}}_u+{{\displaystyle q}}_r^{\prime }\times {{\displaystyle p}}_r\right)}{\left(1-{{\displaystyle c}}_d\right)\times 10000}}$$

(5)

where ${{\displaystyle W}}_{dr}$ is the domestic water demand, 10⁴ m³; ${{\displaystyle q}}_u^{\prime }$ is the urban rigid domestic water consumption quota, m³/(person·year); ${{\displaystyle q}}_r^{\prime }$ is the rural rigid domestic water consumption quota, m³/(person·year).

(ii) Industrial water demand depends on the scale of industrial activities and industrial water use efficiency [25]. Under rigid conditions, water supply assurance must be guaranteed for leading industries. The LEAM is employed to identify leading industries in the target region. When an industry’s location entropy exceeds 2, it is identified as a leading industry in the region. The calculation formula is defined as follows:

$${{\displaystyle L}}_{ij}={\displaystyle \frac{\left({{\displaystyle v}}_{ij}/{{\displaystyle v}}_j\right)}{\left({{\displaystyle V}}_i/V\right)}}$$

(6)

where ${{\displaystyle L}}_{ij}$ is the location entropy of industry $i$ in region $j$; ${{\displaystyle v}}_{ij}$ is the output value of industry $i$ in region $j$, CNY 10⁴; ${{\displaystyle v}}_j$ is the output value of all industries in area $j$, CNY 10⁴; ${{\displaystyle V}}_i$ c is the output value of i industry nationwide, CNY 10⁴; V is the output value of all industries in the country, CNY 10⁴.

The equation for calculating the industrial water demand is as follows:

$$W_{it}={\displaystyle \frac{I\times \eta }{1-c_i}}$$

(7)

where $W_{it}$ is the industrial water demand, 10⁴ m³; $I$ is the industrial value added, CNY 10⁴; $\eta$ is the water consumption per CNY 10⁴ of industrial value added, m³/(CNY 10⁴); $c_i$ is the industrial water supply loss factor.

The equation for calculating the industrial rigid water demand is as follows:

$${{\displaystyle W}}_{ir}={\displaystyle \frac{{{\displaystyle I}}^{\prime }\times \eta }{1-c_i}}$$

(8)

where ${{\displaystyle W}}_{ir}$ is the industrial water demand, 10⁴ m³; ${{\displaystyle I}}^{\prime }$ is the industrial value added, CNY 10⁴.

(iii) The total agricultural water demand depends on the irrigated area of grain crops and forestry, the area of fisheries, the number of livestock, and their respective water quotas [26]. Under rigid conditions, cultivated land should produce a grain yield that fundamentally satisfies human needs, and the water that is required to produce this yield constitutes the rigid grain crop irrigation water demand. Additionally, as forestry, fisheries, and livestock also serve as food sources, their water demands should likewise undergo hierarchical classification. The rigid water demand for these sectors is calculated by multiplying their total water demand by the ratio of rigid grain crop irrigation water demand to grain crop irrigation water demand. The equations for calculating the total agricultural water demand are as follows:

$$W_{ut}={\displaystyle \frac{Q_u\times A_u}{E\times 10000}}$$

(9)

where $W_{ut}$ is the water demand for irrigation of grain crops, 10⁴ m³; $Q_u$ is the comprehensive irrigation quota for grain crops, m³/ha; $A_u$ is the actual irrigated area for grain crops, ha; $E$ is the effective utilization coefficient of irrigation water.

$$W_{ft}={\displaystyle \frac{{{\displaystyle Q}}_f\times {{\displaystyle A}}_f+{{\displaystyle Q}}_l\times {{\displaystyle A}}_l+{{\displaystyle Q}}_a\times {{\displaystyle A}}_a}{10000}}$$

(10)

where $W_{ft}$ is the water demand for forestry, livestock, and fisheries, 10⁴ m³; ${{\displaystyle Q}}_f$ is the water quota for forestry, m³/ha; ${{\displaystyle A}}_f$ is the area for forestry, ha; ${{\displaystyle Q}}_l$ is the water quota for livestock, m³/head; ${{\displaystyle A}}_l$ is the quantity of heads of livestock, head; ${{\displaystyle Q}}_a$ is the water quota for fisheries, m³/ha; ${{\displaystyle A}}_a$ is the area of fisheries, ha.

The equations for calculating the rigid agricultural water demand are as follows:

$$W_{ur}={\displaystyle \frac{\left(p_u+p_r\right)\times g\times \lambda \times Q_u}{C\times \theta \times \phi \times E\times 10000}}$$

(11)

where $W_{ur}$ is the rigid water demand for irrigation of grain crops, 10⁴ m³; $g$ is the per capita grain demand, kg/person; $\lambda$ is the grain self-sufficiency rate; $C$ is the grain yield per unit area of irrigated land, kg/ha; $\theta$ is the ratio of food to cash crop acreage; $\phi$ is the multiple cropping index.

$$W_{fr}={\displaystyle \frac{W_{ur}}{W_{ut}}}\times W_{ft}$$

(12)

where W_fr is the rigid water demand for forestry, livestock, and fisheries, 10⁴ m³.

(iv) Ecological water demand comprises two components: environmental flow demand and terrestrial ecological water demand [27]. The environmental flow demand refers to the water that is required to maintain the ecological security within rivers, while the terrestrial ecological water demand refers to the water that is needed to sustain the ecological security of wetlands and urban vegetation [28]. The environmental flow demand encompasses the water demand for maintaining fundamental riverine functions, supporting connected lake–wetland ecosystems, and preserving estuarine ecological environments. These water allocations are principally dedicated to conserving aquatic biodiversity, sustaining natural water purification processes, and maintaining ecological equilibrium. Current methodologies for determining environmental flow demand include hydrological approaches (e.g., Tennant and Q95) and habitat simulation techniques (PHABSIM) [29], among others. The terrestrial ecological water demand primarily includes urban ecological environmental water requirements, shelterbelt system water demands, and wetland replenishment, which are predominantly allocated for urban greening, shelterbelt construction, and wetland conservation to enhance regional ecological environment quality. Currently, the methodologies for calculating terrestrial ecological water demand mainly consist of the quota method [30] and evapotranspiration-based approaches (e.g., Penman–Monteith equation) [31], among others.

For the environmental flow demand, the current water resource management assessment requirements in China aim to ensure the most basic environmental flow demand. Therefore, this study does not divide the environmental flow demand into rigid and elastic components, and its water demand is deducted from the surface water supply. For the terrestrial ecological water demand, the amount of water depends on the number of urban population and the per capita green area. The “Code for Classification of Urban Land Use and Planning Standards of Development Land” (GB50137-2011) is a mandatory standard, which stipulates that the minimum controlled per capita green park space area should be 8 m²/person. Therefore, a value of 8 m²/person is taken as the rigid per capita green park space area and used to calculate the rigid terrestrial ecological water demand. The equation for calculating the terrestrial ecological water demand is as follows:

$$W_{zt}={\displaystyle \frac{p_u\times p_g\times q_g}{10000}}$$

(13)

where $W_{zt}$ is the terrestrial ecological water demand, 10⁴ m³; $p_g$ is the per capita green space area, m²/person; $q_g$ is the greening water consumption quota, m³/(m²·year).

$$W_{zr}={\displaystyle \frac{p_u\times p_g^{\prime }\times q_g}{10000}}$$

(14)

where $W_{zr}$ is the rigid terrestrial ecological water demand, 10⁴ m³; $p_g^{\prime }$ is the rigid per capita green space area, m²/person.

2.4. Workflow of the SD-F-HWD Model

Based on the clarification of feedback regulation mechanisms and the hierarchical calculation method for water demand across all sectors, a generalized SD-F-HWD model was developed (Figure 3). The model comprises three parts: the total supply–demand module, the rigid supply–demand module, and the feedback regulation module. As shown in Figure 3, the total supply–demand module and the rigid supply–demand module are interconnected yet independent, sharing certain common elements. Consequently, when implementing feedback regulation on the total supply–demand module, adjustments to certain adjustable variables will also affect the rigid supply–demand module. For instance, a reduction in irrigation water quotas for grain crops following regulation will simultaneously decrease agricultural rigid water demand. Therefore, based on comprehensive consideration of the intrinsic characteristics of rigid water demand and the relationship between total supply–demand and rigid supply–demand, no feedback regulation mechanism was implemented for the rigid supply–demand module, because this module is still influenced by the effects of feedback regulation from the total supply–demand module.

In the absence of regulation, all ${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,7\right)$ values in the model should be set to 0. Before implementing regulation, the adjustment coefficient $\phi$ needs to be determined. If $S{V}_1$ reaches the expected value of 0 when all ${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,7\right)$ values are set to their maximum of 1, it indicates that the $\phi$ value is reasonable. Additionally, to prevent the adjustable variables from deviating from reality, the feedback limits for each adjustable variable should be selected based on historical data from the study area and relevant policy planning documents, and these limits should be incorporated as constraints in the calculation process of each element in the model.

Prior to model application, model validation is required. Due to the distinctive nature of rigid water demand, validation cannot be performed for this component. Consequently, validation is conducted exclusively for the total water demand results. The model’s efficacy is appraised through the utilization of the mean relative error ($MRE$) and thecoefficient of determination ($R^2$). The $MRE$ indicates the overall direction of deviation between the simulated and actual values. The $R^2$ reflects whether the model captures the main trend of variation in the data. The equations for calculating the parameter is as follows:

$$MRE=\left|{\displaystyle \frac{\sum \left(V_s-V_a\right)}{\sum V_a}}\right|$$

(15)

$$R^2={\left({\displaystyle \frac{\sum \left(V_s-{\overline{V}}_s\right)\left(V_a-{\overline{V}}_a\right)}{\sqrt{\sum {\left(V_s-{\overline{V}}_s\right)}^2\sum {\left(V_a-{\overline{V}}_a\right)}^2}}}\right)}^2$$

(16)

where $V_s$ is the simulated values of the variables; $V_a$ is the actual values of the variables. The closer the $MRE$ is to 0, the better the model’s simulation performance. The closer $R^2$ is to 1, the greater the proportion of data variability captured by the model.

The workflow of the SD-F-HWD model is shown in Figure 4. The water resources supply–demand situation is closely related to potential hydrological year types, with significant differences in water supply and demand existing across different year types. Taking low-flow year and normal-flow year as examples, when feedback regulation is required, only the low-flow year supply–demand results require regulatory intervention. This is because: (i) normal-flow years have lower water demand and higher supply compared to low-flow years, thus achieving supply–demand balance in low-flow years through regulation will naturally satisfy normal-flow year conditions; (ii) each adjustable variable should yield unique post-regulation results, and separate regulation for different hydrological year types would produce multiple regulation schemes, which could compromise effective references for water resources management. Therefore, regulating only low-flow year supply–demand outcomes ensures the solution covers the most challenging scenario while avoiding management confusion from multiple schemes, maintaining decision-making consistency and operability.

3. Case Study

3.1. Study Area

Wu’an City is located northwest of Handan City in China (Figure 5), with the geographic coordinates 36°28′ to 37°01′ N and 113°45′ to 114°22′ E. As a prefecture-level city with a high water demand [32], Wu’an City has per capita water resources that are only 20% of the national average in China. Meanwhile, as one of the top 100 industrial counties in the whole of China, Wu’an City is very developed industrially, but at the same time, this places significant pressure on the regional water supply. Industrial water use has been increasing in recent years, the local water resources are insufficient, and the fierce competition for water use among industries has seriously constrained the sustainable socioeconomic development of Wu’an City.

3.2. Data Resources

This study takes 2022 as the base year and 2030 as the planning year, considering two hydrological year types (normal-flow year (P = 50%) and low-flow year (P = 75%)) to analyze the hierarchical water supply and demand situation in Wu’an City from 2023 to 2030, and to conduct feedback regulation on possible water shortage situations in 2030. Currently, Wu’an City’s water use sectors can be divided into domestic, industrial, agricultural, and ecological use sectors; the main water supply sources include surface water, groundwater, recycled water, coal mine drainage water and inter-regional water. All data and parameters used in this study come from the following materials:

For water demand: agricultural irrigation area data comes from “Wu’an Water Resources Bulletin” (2011–2022); population, industrial added value, per capita green space, grain self-sufficiency rate, multiple cropping index, ratio of food to cash crop acreage and grain yield per unit area data were obtained from “Wu’an Statistical Yearbook” (2011–2022); domestic, ecological and agricultural water quotas under different hydrological year types come from “Hebei Province Water Use Quota” (2021); water consumption per CNY 10⁴ of industrial value added was sourced from Handan Water Resources Bureau’s “Notice on Adjusting Handan City’s Water Use Efficiency Control Indicators During the 14th Five-Year Plan Period” (2022); environmental flow demand comes from “Ming River Water Allocation Plan” (2021); domestic sewage conversion factor, industrial effluent conversion coefficient, sewage collection coefficient, sewage treatment reuse rate, domestic water supply loss coefficient, industrial water supply loss coefficient and farmland irrigation water efficiency coefficient were extracted from “Wu’an Water Conservation Plan” (2022).

For water supply: long-term observed inflow data and hydraulic parameters for medium-sized reservoirs in Wu’an City, groundwater extraction control targets, and mine drainage discharge volumes reported by local mining enterprises were all provided by the Wu’an Water Resources Bureau. The active storage capacity of small reservoirs and their replenishment coefficients under different assurance probabilities were obtained from the “Wu’an Urban Water Supply Source Development and Utilization Plan (2021–2035)” (2023).

3.3. Application of the SD-F-HWD Model

3.3.1. Hierarchical Classification of Water Demand

Domestic water demand. For Wu’an City, the “Water Quotas for Domestic and Service Industries (DB 13/T 5450.1-2021)” specifies that the lower limit of the water quota for urban residents is 30 m³/(person·year), and the lower limit of the water quota for rural residents’ life is 18.5 m³/(person·year); therefore the lower limit of the quota standard is used as the threshold between rigid and flexible water demand in Wu’an. In 2022, the urban domestic water use quota in Wu’an City was 36.5 m³/(person·year), while the rural domestic water use quota was 24.6 m³/(person·year). By 2030, the urban domestic water use quota in Wu’an City will be reduced to 33.5 m³/(person·year), and the rural domestic water use quota to 22 m³/(person·year).

Industry water demand. Wu’an City has formed a three-pillared metallurgy, coking, and building materials industrial structure, with the iron and steel industry as the cornerstone of Wu’an’s economy. Using the LEAM, the location entropy values of various industrial sectors in Wu’an City over the past five years were calculated. The results show that the location entropy values for the ferrous metal smelting and rolling processing industry have consistently exceeded 2 each year (with annual values around 10, which indicates an extremely high level of specialization in this sector), while the location entropy values for all other industrial sectors remained below 2. Therefore, the ferrous metal smelting and rolling processing industry is selected as the leading industrial sector for Wu’an City. In 2022, the calculated industrial added value of Wu’an City was CNY 420.18 × 10⁹, with the leading industrial sector contributing CNY 327.31 × 10⁹.

Agricultural water demand. For Wu’an City, the parameters required for agricultural rigid water demand—including grain self-sufficiency rate, multiple cropping index, ratio of food to cash crop acreage, and grain yield per unit area—are calculated based on corresponding data sources. The projected 2030 values for these parameters are as follows: 0.9, 1.92, 1.43, and 14,932.65 kg/ha, respectively. The per capita grain demand is established as 400 kg/person, based on the baseline established by the United Nations Food and Agriculture Organization (FAO) for ensuring basic food security. The above parameters are used to calculate the minimum retained irrigated area. Considering the projected future agricultural planting structure of Wu’an City, the comprehensive irrigation water quota for grain crops in a normal-flow year is estimated to be 1392.21 m³/ha, while in a low-flow year, it is projected to be 2229.76 m³/ha. Wu’an’s livestock and fisheries have accounted for about 3% of the total GDP in recent years, and their water demand is minimal, so the water demands of livestock and fisheries are not considered in this study. The comprehensive irrigation water quota for forestry is estimated to be 2361.16 m³/ha in a normal-flow year and 2593.71 m³/ha in a low-flow year.

For the environmental flow demand, the value of 273 × 10⁴ m³ specified in the “Explanation for the Compilation of the Minghe River Water Allocation Plan” (2021) was adopted. In Wu’an City, the per capita green space area in 2022 was 15.20 m²/person, and based on this, the projected value for 2030 is 15.29 m²/person. The threshold for delineating terrestrial ecological rigid water demand is 8 m²/person.

3.3.2. Calculation of Water Supply

The available surface water supply in Wu’an City mainly comes from reservoirs within the region. There are four medium-sized reservoirs that are all annually regulated. Therefore, this study employs the typical year method with monthly time steps to conduct beneficial regulation calculations for each medium-sized reservoir, to obtain the maximum available water supply for both normal-flow years and low-flow years. To calculate the available supply of small reservoirs, the reoperation index method is used, which involves multiplying the beneficial reservoir capacity by the reoperation index under different hydrological year types.

The available groundwater supply in Wu’an City is determined using the groundwater withdrawal control indicators provided by the Wu’an Water Resources Bureau. Similarly, the available mine drainage water supply is based on data provided by the Wu’an Water Resources Bureau.

The main sources of recycled water in Wu’an City are wastewater generated from industrial production and urban residential activities. The calculation method follows the recycled water calculation method used by Guo [19].

3.3.3. The SD-F-HWD Model of Wu’an City

Building upon Wu’an City’s specific methodology for hierarchical water demand classification and available water supply calculations, the Wu’an City SD-F-HWD model was constructed (Figure 6) by extending the generalized SD-F-HWD framework. The model’s 10 regulatory variables were grouped correspondingly into categories G1, G2, and G3 based on classification by sectoral water use efficiency, population size, and industrial scale. The computational workflow and mathematical formulations of the model are detailed in Table S1.

Historical data from 2011 to 2019 were used for predicting 2020–2022 parameter values to evaluate model performance. This validation selected seven key variables from the model for verification: total population, industrial value added, industrial water demand, domestic water demand, agricultural water demand, ecological water demand, and total water demand. Detailed results are presented in

Table 1. Calculation results of model performance parameters.

Variable	$\mathit{M}\mathit{R}\mathit{E}$ (%)	${\mathit{R}}^2$
total population	0.24	0.999
industrial value added	3.04	0.738
industrial water demand	3.26	0.976
domestic water demand	0.50	0.778
agricultural water demand	1.18	0.654
ecological water demand	7.67	0.968
total water demand	2.17	0.902

. The actual and simulated values of key variables during the validation period (2020–2022) in Wu’an City are presented in Table S2.

The validation results demonstrate that the $MRE$ for all variables ranges between 0.50% and 7.67%, with the total water demand $MRE$ at 2.17%, indicating acceptable model simulation results. The $R^2$ for all variables falls within the range of [0.654, 0.999], with the total water demand variable achieving a high $R^2$ of 0.902, demonstrating that the model exhibits excellent trend-fitting performance across all variables and captures the actual variation patterns of water demand with remarkable accuracy.

Before conducting model simulations, all parameters were entered into the model, and feedback limits were set for each adjustable variable. The input values and feedback limits for each adjustable variable are presented in

Table 2. Feedback limits for each adjustable variable.

Adjustable Variables	Value		Upper Limit	Lower Limit
water consumption per CNY 104 of industrial value added (m3/(CNY 104))	21.6		/	17.4
rural domestic water consumption quota (m3/(person·year))	22		/	18.5
urban domestic water consumption quota (m3/(person·year))	33.5		/	30
comprehensive irrigation water quota for forestry (m3/ha)	Normal-flow year	2361.16	/	1863.12
	Low-flow year	2593.71	/	2265.18
comprehensive irrigation quota for grain crops (m3/ha)	Normal-flow year	1392.21	/	1167.17
	Low-flow year	2229.76	/	1873.46
greening water quota (m3/(m2·a))	0.22		/	/
sewage treatment reuse rate (%)	43.3		60	/
sewage collection coefficient (/)	0.9		0.95	/
industrial value added growth rate (%)	2.87		/	0.74
population growth rate (%)	0.66		/	0

. The sources of regulation limits for each adjustable variable are as follows:

(i) The lower limit for water consumption per CNY10⁴ of industrial value added should adopt the strictest water resources management target value from Wu’an City’s 15th Five-Year Plan, but since this indicator has not yet been established, the current average level of Hebei Province is selected as its adjustment lower limit.

(ii) For domestic water use quotas, the lower limits of urban and rural domestic water use quota ranges specified in the “Water Quotas for Domestic and Service Industries” (DB 13/T 5450.1-2021) were adopted as adjustment lower limits.

(iii) For agriculture, based on Wu’an City’s forest land and grain crop planting structure, types, and area, using the pipe irrigation quotas for different crops stipulated in the “Agricultural Water Use Quotas” (DB 13/T 5449.1-2021) as constraints, a comprehensive irrigation quota corresponding to the pipe irrigation standard was calculated as the adjustment lower limit.

(iv) For the ecological sector, there is no advanced benchmark for greening water use quotas, so no lower limit was set.

(v) The upper adjustment limits for sewage treatment reuse rate and sewage collection coefficient adopted the 2030 planning targets from the “Wu’an Urban Water Supply Source Development and Utilization Plan (2021–2035)” (2023).

(vi) The lower adjustment limit for industrial added value growth rate was set as the average annual growth rate of industrial added value in recent years.

(vii) The lower adjustment limit for population growth rate was set to 0.

3.4. Results

3.4.1. Water Supply and Demand Simulation

Following the SD-F-HWD model workflow, all ${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,7\right)$ values in the model were initially set to 0. Through model simulation, the water resources supply and demand situations under different hydrological year types in Wu’an City from 2023 to 2030 were obtained (Figure 7). Overall, both the rigid water demand and total water demand across various sectors in Wu’an City show an increasing trend year by year. The ranking of water demand among sectors is industrial > agricultural > domestic > ecological, with industrial water demand growing significantly faster than other sectors. The sudden increase in water supply in 2025 is due to Wu’an City’s utilization of 20 × 10⁶ m³ of water from the South-to-North Water Diversion Project starting that year, while the variation trends of water supply in other years remain relatively stable.

Water shortages only occur in the total supply–demand simulation scenario during low-flow years (as shown in Figure 7d). From 2023 to 2024, the water scarcity rate shows an upward trend; by 2025, after utilizing the South-to-North Water Diversion water, the water scarcity rate decreases significantly, then resumes a stable upward trend; until 2030 when the water scarcity rate rises to 11.15%. Therefore, regulation is required for the 2030 water resources supply and demand simulation results under low-flow year conditions.

3.4.2. Feedback Regulation

To verify the performance of the feedback regulation mechanism, 14 different regulation scenarios (S0–S13) were established by grouping various adjustable variables. Scenarios S1–S3 represent the regulation outcomes considering only water supply and usage efficiency, population size, and industrial scale, respectively. The results indicate that regulating only population and industrial scale has a limited effect on the water supply–demand situation, whereas regulating water supply and usage efficiency exerts a more substantial influence. Scenario S4 reflects the regulation outcome when water supply and usage efficiency, population size, and industrial scale are simultaneously adjusted with maximum feedback control intensity. This scenario demonstrates that when parameter $\phi$ is set to 1, the feedback regulation achieves the expected results, confirming the rationality of the chosen $\phi$ value.

To ensure the model’s practical applicability, both the effectiveness of the simulation strategy and its alignment with real-world challenges are crucial. Based on the projected water supply–demand dynamics in Wu’an City, all feasible policy measures should be considered to mitigate the acute water supply–demand imbalance. Consequently, in scenarios S5–S13, all regulatory variable groups were systematically adjusted, with the feedback factor ${\beta }_{m,n}$ initially set at 0.1 and incrementally increased by 0.1. As the feedback control intensity intensified, the system’s water scarcity rate progressively declined until supply–demand equilibrium was attained in Scenario S11. The feedback regulation results across different scenarios are detailed in

Table 3. Feedback adjustment results in different scenarios in the low-flow year 2030.

Scenario	Groups	2030 Total Supply–Demand (Low-Flow Year)
		Feedback Control Intensity	Water Scarcity Rate (%)
S0	/	${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,7\right)=0$	11.15
S1	G1	${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,5\right)=1$$,{{\displaystyle \beta }}_{1,n}\left(6,7\right)=0$	0
S2	G2	${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,5,7\right)=0$$,{{\displaystyle \beta }}_{1,n}\left(6\right)=1$	11.06
S3	G3	${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,6\right)=0$$,{{\displaystyle \beta }}_{1,n}\left(7\right)=1$	9.45
S4	G1,G2,G3	${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,7\right)=1$	0
S5	G1,G2,G3	${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,7\right)=0.1$	9.64
S6	G1,G2,G3	${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,7\right)=0.2$	8.09
S7	G1,G2,G3	${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,7\right)=0.3$	6.49
S8	G1,G2,G3	${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,7\right)=0.4$	4.85
S9	G1,G2,G3	${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,7\right)=0.5$	3.21
S9	G1,G2,G3	${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,7\right)=0.6$	1.53
S11	G1,G2,G3	${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,7\right)=0.7$	0
S12	G1,G2,G3	${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,7\right)=0.8$	0
S13	G1,G2,G3	${{\displaystyle \beta }}_{1,n}\left(n=1,\cdots ,7\right)=0.9$	0

.

When the regulation outcomes meet the expected targets, the scenario with the lowest feasible feedback control intensity is selected as the final scenario to ensure better practical applicability of the regulated variables. Therefore, Scenario S11 is chosen as the final scenario. The regulation results of each variable (excluding agricultural water use efficiency) under this scenario are applied to the hierarchical supply–demand system under normal flow year conditions for further simulation, which produces the hierarchical supply–demand regulation outcomes for the normal flow year. Figure 8 presents the hierarchical water supply–demand regulation results for Wu’an City in 2030 under different hydrological year types. The total and rigid water demands are reduced by 11.24% and 8.5%, respectively, under low-flow year conditions, and by 9.88% and 6.99%, respectively, under normal-flow year conditions. On the supply side, only the available volume of recycled water changes after the feedback regulation. Since the available volume of recycled water depends on urban domestic water use and industrial water use, both of which are reduced after feedback regulation, the increase in recycled water supply remains limited despite higher treatment reuse rate and sewage collection coefficient after the adjustment. The recycled water supply only increases by 1.85 × 10⁵ m³. (Due to the minimal magnitude of the adjusted water volume, it is not displayed in the figure.)

Table 4. Comparison of adjustable variables before and after feedback regulation in the S11 scenario.

Adjustable Variables	Before Feedback	After Feedback
per CNY 104 of industrial value added (m3/(CNY 104))	21.60	19.46
rural domestic water consumption quota (m3/(person·year))	22.00	19.82
urban domestic water consumption quota (m3/(person·year))	33.50	30.18
comprehensive irrigation water quota for orchards on low-flow year (m3/ha)	2593.71	2336.37
comprehensive irrigation quota for grain crops on low-flow year (m3/ha)	2229.76	2008.53
greening water use quota (m3/(m2·year))	0.22	0.20
industrial value added growth rate (/)	0.43	0.48
sewage collection coefficient (/)	0.90	0.95
industrial value added growth rate (%)	2.87	2.58
population growth rate (%)	0.66	0.59

presents the comparative results of adjustable variables before and after adjustment under Scenario S11. With the exception of the sewage collection coefficient, none of the adjustable variables reached their feedback limits, indicating that Wu’an City still retains certain water-saving potential after regulation. Through the application of the SD-F-HWD model, the water supply–demand situation in Wu’an City has been effectively regulated, achieving balanced hierarchical water supply–demand under different hydrological year types across the region.

4. Discussion

Since the seminal work by Hou [33] on household water demand classification methodologies, significant advancements have been made in hierarchical water demand partitioning research. The conceptual framework for hierarchical water demand has progressively matured [34]. In this study in 2030, rigid water demand represented 78.43% of the total water demand in Wu’an City during normal-flow years and 77.98% during low-flow years. After feedback implementation, the rigid water demand increased to 80.94% of the total demand in normal-flow years and 80.39% in low-flow years. Consequently, when the regional water supply and demand reached equilibrium, the rigid water demand constituted approximately four-fifths of the total water demand in the region. Wu’an City has a relatively complex industrial structure. As the lifeblood of regional economic development, its dominant industries are large-scale, and are vulnerable to significant economic impacts caused by water scarcity. Therefore, the proportion of rigid water demand is relatively high. Since there are no historical data available to validate the rigid water demand, further research is urgently needed to investigate the rationality of partitioning rigid and flexible water demands, particularly under increasingly severe water-supply–demand conflicts in the future.

In the feedback regulation mechanism, the regulation coefficient $\phi$ enables the division of feedback capacity into multiple abstract intervals. This approach ensures that while the feedback control intensity ${\beta }_{m,n}$ remains constrained within the [0, 1] range, the feedback capability can be modified by $\phi$. Besides, ${\beta }_{m,n}$ is set within the range of [0, 1]. However, as ${\beta }_{m,n}$ increases, the interval of ${\alpha }_{m,n}$ also expands continuously. When ${\beta }_{m,n}$ approaches infinity, the interval of ${\alpha }_{m,n}$ shifts from [1, 2] to (−1, 3) (Figure 9, Taking the negative feedback function curve with parameter $\phi$ fixed at 1 as an example). This change begins when exceeds 1, at which point ${\alpha }_{m,n}$ may take negative values. In the feedback mechanism, the negative values for feedback-adjusted variables are meaningless. Therefore, ${\beta }_{m,n}$ is constrained to the range of [0, 1] in this study.

Through implementing multi-scenario feedback regulation operations in the Wu’an City case study, this research has demonstrated the applicability of the SD-F-HWD model. In the model, all variables ultimately converge on the water scarcity rate as the key output. By observing changes in water scarcity rate under different feedback regulation scenarios (i.e., examining feedback effects at various intensities of ${\beta }_{m,n}$), the effectiveness of feedback regulation can be directly reflected and explained. When the feedback control intensity ${\beta }_{m,n}$ is set to 0.7, the water supply and demand achieve equilibrium. The feedback results of adjustable variables under this scenario can serve as a reference for the local water administration authorities in formulating and implementing relevant policies, albeit limited to the goal of achieving a regional water supply–demand balance. The specific implementation mechanism involves water resource management authorities establishing interdepartmental coordination protocols to incorporate adjusted parameter values (including sectoral water use efficiency, population size, and industrial scale) derived from feedback analyses into formalized planning targets for future water resource and socioeconomic development, thereby providing scientifically grounded support for water management planning and policy decision-making. In addition, depending on management needs, a larger ${\beta }_{m,n}$ value can be selected to address potential extreme water resource conditions, or a smaller ${\beta }_{m,n}$ value can be chosen for more moderate policy adjustments. These diverse control scenarios offer multiple possibilities for guiding water resource management strategies. Meanwhile, the adjustment of policy variables also has certain limitations. Under conditions that restrict the population and industrial development scale, the water scarcity situation has not been effectively improved. This may be attributed to the ongoing industrial restructuring in Wu’an City, where the growth of the population and industrial development has slowed relatively. In contrast, the adjusting water use efficiency across various sectors has a significant impact on alleviating water scarcity. However, improving efficiency requires economic and technological investments, necessitating a comprehensive evaluation of its feasibility and careful balancing of stakeholder interests [35].

The SD-F-HWD model, constructed using the standard SD model as its baseline framework, is therefore equally applicable to different study regions due to its superior extensibility. However, it should be noted that when applying the model to different regions, analysis and adjustment of the model’s constituent modules are required. For example, since water-use sectors and water supply sources may differ across study regions, certain structural modifications to the model are necessary. Furthermore, the SD-F-HWD model has certain limitations. On one hand, as an annual-scale model, it contrasts with existing refined monthly-scale (or daily-scale) studies of water demand. Theoretically, obtaining feedback regulation results at finer temporal scales would be more conducive to advancing water resources management. However, this is difficult to implement and poses significant challenges to the fundamental construction principles of all model components. On the other hand, the model’s spatial boundary delineation also has limitations, such as difficulty in capturing the spatial heterogeneity characteristics of water supply and demand. Furthermore, existing studies on feedback regulation mechanisms in water resources management have universally neglected uncertainty analysis. To enhance the reliability and applicability of the SD-F-HWD model, future iterations should systematically incorporate uncertainty quantification techniques. This is particularly critical given the model’s dependency on socio-hydrological parameters (e.g., water use efficiency adjustments) that inherently carry high epistemic uncertainty.

Water management policies alone cannot resolve all water crises, as factors such as climate change uncertainty [36] and inter-sectoral conflicts [37] pose significant challenges to water governance. Existing studies have systematically evaluated the impacts of various climate change scenarios on water demand through comprehensive modeling approaches. Notwithstanding the inherent uncertainties associated with the constructed models and methodologies, these investigations have consistently reached a robust consensus that climate change will invariably result in heightened water demand [38,39]. Building on this perspective, scholars have emphasized the necessity of combining water supply and demand management strategies to accommodate varying climatic conditions and their inherent uncertainties [40]. While the SD-F-HWD model provides valuable insights, its foundational constraints lead to limited performance in capturing spatial variations. Therefore, enhancing the model’s capacity to integrate climate change uncertainties emerges as a pivotal research priority moving forward. Furthermore, inadequate collaboration among governments, societal sectors, and regions may exacerbate water scarcity, increase agricultural economic vulnerability, and intensify social tensions regarding resource allocation [41]. Achieving balanced water management policies across these stakeholders presents considerable complexity. Consequently, building upon the SD-F-HWE model framework, future research should focus on developing sector-specific policy instruments to address these multidimensional challenges. The SD-F-HWD model, by nature an extensible framework, therefore requires continuous integration with models and tools from other domains to establish a robust approach for addressing future water scarcity.

5. Conclusions

This study developed the SD-F-HWD model to establish a systematic solution for “supply-dictates-demand” water resources management. The main innovations are reflected in three aspects: First, by refining sector-specific water demand hierarchy standards, we extended conventional water demand classification methods into a universal technical system. Second, to address issues like undefined control ranges in existing feedback mechanisms, we innovatively proposed a new regulation approach. Third, we achieved organic integration of hierarchical water demand with feedback regulation through an SD framework. The empirical study in Wu’an City demonstrated three key advantages of the model: (i) precise identification of the classification mechanism between rigid and total water demand across different sectors, (ii) achievement of regional water supply–demand balance through dynamic regulation, and (iii) assurance that all regulatory parameters remain within reasonable thresholds. The research outcomes not only provide a scientific basis for decision-making in water-scarce regions, but the modular design concept can also serve as reference for adaptive management in other areas. The primary value of this work lies in establishing a complete technical framework coupling hierarchical water demand with feedback regulation, providing a feasible technical system for advancing water resources management research. Future studies may further explore coupling and extension of the model with other climate or economic models.