Multi-Objective Optimization for Irrigation Canal Water Allocation and Intelligent Gate Control Under Water Supply Uncertainty

Abstract

Open-channel irrigation systems often face constraints due to water supply uncertainty and insufficient gate control precision. This study proposes an integrated framework for canal water allocation and gate control that combines interval-based uncertainty analysis with intelligent optimization to address these challenges. First, we predict the inflow process using an Auto-Regressive Integrated Moving Average (ARIMA) model and quantify the range of water supply uncertainty through Maximum Likelihood Estimation (MLE). Based on these results, we formulate a bi-objective optimization model to minimize both main canal flow fluctuations and canal network seepage losses. We solve the model using the Non-dominated Sorting Genetic Algorithm II (NSGA-II) to obtain Pareto-optimal water allocation schemes under uncertain inflow conditions. This study also designs a Fuzzy Proportional–Integral–Derivative (Fuzzy PID) controller. We adaptively tune its parameters using the Particle Swarm Optimization (PSO) algorithm, which enhances the dynamic response and operational stability of open-channel gate control. We apply this framework to the Chahayang irrigation district. The results show that total canal seepage decreases by 1.21 × 10⁷ m³, accounting for 3.9% of the district’s annual water supply, and the irrigation cycle is shortened from 45 days to 40.54 days, improving efficiency by 9.91%. Compared with conventional PID control, the PSO-optimized Fuzzy PID controller reduces overshoot by 4.84%, and shortens regulation time by 39.51%. These findings indicate that the proposed method can significantly improve irrigation water allocation efficiency and gate control performance under uncertain and variable water supply conditions.

1. Introduction

1.1. Research Background

As a leading agricultural nation, China’s agricultural production exhibits substantial dependence on irrigation systems. Empirical studies demonstrate that irrigation constitutes the predominant factor influencing agricultural productivity [1,2], with agricultural water withdrawals exceeding 60% of the total water resources. Serving as the central infrastructure of irrigation networks, irrigation districts play a pivotal role in safeguarding national food security and fostering rural socioeconomic development. Currently, traditional canal systems remain the dominant water delivery method in irrigation districts, yet their suboptimal water allocation efficiency and inadequate regulation capacity result in significant water resource wastage. Therefore, conducting optimization research on canal water distribution tailored to actual irrigation district conditions plays a vital role in enhancing irrigation efficiency, economic benefits, and water management performance [3,4,5].

1.2. Current Research Progress and Limitations

In recent years, significant progress has been made in the operation and regulation of irrigation canal systems under uncertain water supply conditions, particularly in the areas of inflow forecasting, uncertainty modeling, multi-objective optimization, and intelligent gate control. First, regarding the prediction of water supply processes and the characterization of uncertainty, time series models, machine learning, and deep learning methods have been widely applied to short- and medium-term flow forecasting, improving the completeness of information available for operational scheduling [6,7]. To enhance the robustness of water distribution strategies against stochastic inflow disturbances, researchers have employed probabilistic inference, Bayesian updating, Monte Carlo simulation, and multi-stage distributionally robust optimization to quantitatively describe the propagation characteristics of forecast errors. These approaches have gradually shifted the role of uncertainty from an exogenous input to an internal constraint within the optimization process. However, most existing studies focus on uncertainty at the forecasting level. The mechanisms by which uncertainty propagates, attenuates, or amplifies through canal hydraulic processes, as well as its impact on the stability of water distribution strategies, remain insufficiently analyzed and lack comprehensive theoretical and engineering validation.

Second, multi-objective optimization (MOO) techniques have become important tools for addressing conflicts among multiple operational goals in canal water distribution. Studies have applied strategies such as NSGA-II, MOEA/D, improved genetic algorithms, and hierarchical decomposition optimization to achieve balanced decision support across water allocation efficiency, water loss control, distribution equity, operational stability, and energy consumption constraints [8,9]. Some research further integrates hydraulic models and operational constraints, extending optimization from static allocation to dynamic scheduling. Nevertheless, most existing MOO approaches still assume deterministic inflows or a limited set of scenarios. They provide limited evaluation of the robustness of distribution schemes under varying uncertainty levels. Moreover, optimization results are often decoupled from the actual execution mechanisms of gate control, which results in strong theoretical insights but limited practical applicability.

Third, in the field of intelligent control, advances in flow monitoring, structural measurement, and big data analytics have addressed the modeling challenges posed by hydraulic nonlinearity, structural dependence, and complex operational states of gates. Many studies employ machine learning methods to fit flow coefficients, stage–discharge relationships, and gate opening control laws, significantly improving the parameter identification capability of controllers with respect to flow response [8,10]. In addition, some studies develop intelligent gate control strategies based on gradient boosting trees, convolutional neural networks, or hybrid deep networks. These strategies enhance control response speed through online learning or rapid prediction. However, data-driven models heavily depend on the scale, quality, and coverage of training data and struggle to handle coupling among multiple gates and hydraulic time delays. As a result, these methods still face issues of limited generalization, weak control stability, and execution deviations in large-scale irrigation scenarios.

Recently, the introduction of digital twin (DT) technology has offered a new paradigm for irrigation system management. Some studies construct DT platforms that integrate physical models, data acquisition, forecasting models, and control modules, enabling multi-source data-driven evaluation of scheduling schemes, real-time optimization updates, and closed-loop iteration of control strategies [11]. These studies establish a dynamic mapping between simulation and field operation, which allows verification of optimization results in high-fidelity environments and enhances decision adaptability under complex operational conditions. Nevertheless, the application of DT in irrigation systems remains exploratory. Researchers still face significant challenges in effectively embedding distributional uncertainty within DT frameworks, balancing computational timeliness with model complexity, and ensuring the reliability and safety of intelligent control systems during field execution.

In summary, although existing research has achieved notable progress in inflow uncertainty modeling, multi-objective canal optimization, intelligent gate control, and digital twin development, there remains a lack of a systematic framework that transfers uncertainty through the optimization hierarchy and implements a data-driven, executable, and verifiable closed-loop dynamic water distribution and control strategy. This research gap constitutes the core problem that the present study addresses.

1.3. Main Research Content and Innovations

This study proposes a dynamic coupling framework integrating multi-objective irrigation water allocation with adaptive gate control to address supply uncertainty. The framework combines a real-time optimization module, an intelligent control system based on adaptive Fuzzy-PID, and a probabilistic uncertainty analysis engine to enable predictive management of flow fluctuations and seepage. We applied the approach to the Chahayang Irrigation District in Heilongjiang Province, China. The approach improved delivery accuracy by 8–10% and reduced gate adjustments by 25–30%, providing scientifically validated decision support for sustainable water management. The schematic representation in Figure 1 illustrates this integrated approach. It shows the information flows between the uncertainty analysis module, the optimization engine, and the adaptive control system.

2. Methodology

This study employs a tripartite methodological framework: (1) Multi-objective canal allocation optimization: We develop an optimization model targeting simultaneous minimization of seepage losses and stabilization of trunk canal flows, incorporating hydraulic capacity constraints, water balance equations, and time-flow hierarchical constraints to achieve coordinated multi-objective optimization. (2) We design a Fuzzy-PID controller. The controller uses particle swarm optimization (PSO) to dynamically tune Fuzzy rule parameters and optimize gate response characteristics. (3) We conduct probabilistic modeling of runoff data using ARIMA-maximum likelihood estimation (MLE) methods to characterize prediction uncertainties.

2.1. Significance of the Study

This study develops a dynamic coupling framework for irrigation canal water distribution and gate control that integrates water supply uncertainty analysis, offering both theoretical and methodological innovations and providing a feasible technical solution for modernized irrigation management.

At the theoretical level, the study overcomes the traditional limitation of treating water distribution optimization and gate control as independent processes by proposing a dynamic coupling paradigm centered on “prediction–optimization–control.” This paradigm offers a novel approach for achieving precise and systematic management of irrigation water resources. The framework enhances the robustness of scheduling schemes by introducing interval-based uncertainty analysis and constructing a multi-objective interval optimization model. In addition, the use of a particle swarm optimization (PSO) algorithm to tune Fuzzy PID controllers enables adaptive and high-performance gate operation, effectively addressing canal time delays, nonlinearity, and other complex hydraulic characteristics.

From a practical perspective, the results of this study can benefit multiple stakeholders, including irrigation management agencies, agricultural producers, and water policy makers. The optimized water distribution schemes and intelligent control strategies can directly guide operational scheduling, improving irrigation efficiency, stabilizing flow, reducing leakage, and lowering operational costs, thereby supporting water-saving, efficiency enhancement, and smart system upgrades. For agricultural producers, more precise and stable irrigation ensures adequate water supply during critical crop growth stages, reducing yield losses. For research and technology transfer institutions, the proposed framework and algorithms provide core modules for the development of decision support systems, facilitating standardization and wider adoption of these technologies. For policymakers, quantifying the impact of water supply uncertainty on system performance and demonstrating the potential of intelligent control to enhance system resilience offers a scientific basis for climate-adaptive irrigation management policies and investment planning.

In summary, this study holds significant value in terms of theoretical innovation, methodological advancement, and engineering application, providing systematic technical and theoretical support for precise water resource management, intelligent upgrades, and decision-making in modern irrigation systems.

2.2. Multi-Objective Optimization Model for Canal Water Allocation in Irrigation Districts

This study develops an interval-based multi-objective optimization model that integrates seepage control and flow stability constraints, The model delivers robust water allocation solutions with enhanced risk resilience for irrigation water systems operating under complex hydrological conditions.

2.2.1. Objective Functions

(1)

Primary objective: Seepage minimization

Seepage volume was quantified using a modified Kostiakov infiltration equation [12]

$$\min f_{seepage}={\displaystyle \sum _{n=1}^N\frac{\beta \cdot A\cdot l\cdot q_n^{(1-m)}\cdot \Delta t_n}{100}}$$

(1)

where β denotes the seepage reduction coefficient after anti-seepage measures are applied to the canal; A represents the permeability coefficient of the canal bed; l indicates the water conveyance length of the canal (km); m signifies the seepage exponent of the canal bed; q_n (where N = 1, 2, …, n) refers to the water distribution flow rate of canal N (m³/s); Δt_n corresponds to the water distribution duration (s), where t_sn and t_en represent the initiation and termination times (s) of water distribution for canal N, respectively.

(2)

Secondary objective: Minimization of discharge deviation between the main canal and actual flow

$$\min f_{wave}={\displaystyle \sum _{t=1}^T\left|q_{1t}-q_{1ta}\right|}$$

(2)

where q_1t represents the irrigation discharge (m³/s) in the main canal on day t; q_1ta denotes the actual irrigation discharge (m³/s) in the main canal on day t.

2.2.2. Constraints

(1)

Capacity constraint: The water distribution flow rate should range between 0.6 and 1.0 times the designed channel capacity.

$${\alpha }_d{q}_{\max }\le q_n\le {\alpha }_u{q}_{max}$$

(3)

where q_max represents the designed channel capacity, α_d denotes the minimum flow reduction coefficient, and α_u indicates the flow amplification coefficient.

(2)

Water balance constraint: The product of distribution flow rate and duration must equal the channel’s water demand.

$$W=q_n\cdot \Delta t_n$$

(4)

where W indicates the water demand (m³/s), and Δt_n represents the distribution duration (s).

(3)

Temporal constraint: Both initiation and termination of water delivery must occur within the original irrigation period.

$$0\le t_{sn}\le t_{en}\le T$$

(5)

where T denotes the distribution cycle (days).

(4)

Flow continuity constraint: The distribution flow rate in superior channels must equal the sum of subordinate channels’ flow rates.

$$q_i={\displaystyle \sum _{p=1}^P{q}_p} i,p\in N$$

(6)

where q_i represents the superior channel flow rate (m³/s), and q_p indicates the flow rate of the i-th subordinate channel (m³/s).

2.3. Gate Control System

Optimal water allocation in irrigation canal systems requires precise gate opening control to achieve both hydraulic efficiency and automated scheduling. During water conveyance, we regulate partial gate openings using simplified orifice flow equations (Equation (7)) and intelligent algorithms for real-time flow regulation. The system processes optimized canal flow data through real-time computations that use orifice flow equations [13], with PID control algorithms generating optimal opening commands that precisely position actuators. Concurrently, a digital twin-based hydraulic response model employs machine learning for continuous parameter optimization, This approach enhances dynamic regulation accuracy. The system facilitates remote monitoring and adaptive mode switching, The system automatically initiates correction protocols or alarms when it detects flow deviations or equipment anomalies., thereby establishing closed-loop decision support that significantly improves allocation accuracy and scheduling efficiency.

$$q_n=\mu b{e}_n\sqrt{2gH}$$

(7)

where e_n denotes the gate opening height (m); H represents the upstream water depth (m); q_n indicates the water distribution flow rate in channel n; μ signifies the discharge coefficient; b represents the gate width (m); g denotes the gravitational acceleration constant (9.81 m/s²)

2.3.1. Gate Control Mechanism

As critical infrastructure in hydraulic engineering, gate control systems regulate water level, flow rate, and velocity, with their primary objective being precise hydrodynamic control through dynamic gate opening adjustments.

Level and flow sensors continuously monitor hydraulic parameters and transmit real-time measurements to the controller. The controller computes control outputs using regulation algorithms based on both the error (e) between setpoints and measured values and its rate of change (ec). Amplified control signals drive actuators (e.g., servo motors or hydraulic systems) to modulate gate openings, thereby altering the flow cross-section for precise hydraulic regulation. This closed-loop architecture effectively suppresses external disturbances (e.g., rainfall or inflow fluctuations), This architecture ensures operational stability and reliability, as schematically illustrated in Figure 2.

2.3.2. Mathematical Modeling of Gate Control System

The gate control system represents a complex dynamic system characterized by nonlinearity, time-variance, and hysteresis properties. For analytical and design purposes, the system can be simplified as a first-order inertial element with pure time delay [14]. The transfer function is expressed as:

$$G(s)={\displaystyle \frac{K}{(Ts+1)}}{e}^{-\tau s}$$

(8)

where K denotes the static gain; T represents the time constant; s is the complex variable in Laplace transform, characterizing the system’s frequency response; and τ signifies the pure time delay.

We establish the transfer function of the gate control system based on theoretical modeling, as follows: the transfer function of the gate control system is established as follows:

$$G(s)={\displaystyle \frac{2.66}{5\mathrm{s}+1}}{e}^{-3s}$$

(9)

2.3.3. Fuzzy PID Optimization Using Particle Swarm Algorithm

PID Control Algorithm

The conventional Proportional-Integral-Derivative (PID) controller represents the most widely adopted control mechanism, which synthesizes three linearly combined components: proportional (P), integral (I), and derivative (D) actions. In gate control applications, it achieves an optimal balance between rapid response, high-precision regulation, and robust disturbance rejection, and it serves as the fundamental approach to ensure stable and efficient canal system operation. The fundamental PID control algorithm is expressed mathematically as ref. [15]:

$$u(t)=K_{P}e(t)+K_I{\displaystyle \underset{0}{\overset{t}{\int }}e(t)}dt+K_D{\displaystyle \frac{de(t)}{dt}}$$

(10)

where K_P denotes the proportional gain; K_I represents the integral gain; K_D signifies the derivative gain; u(t) indicates the output signal and e(t) corresponds to the error between the setpoint and measured values.

Fuzzy Logic Design

Fuzzy PID control represents an advanced hybrid methodology that synergistically integrates Fuzzy logic with conventional PID control. This integration enables real-time parameter optimization and enhances system performance. The fundamental operating principle involves processing three key input variables—system error, error derivative, and error integral—through fuzzification before their introduction into the Fuzzy controller [16]. The Fuzzy logic controller fuzzifies and maps both water level deviation and its rate of change into Fuzzy domains, performs Mamdani inference using the expert rule base (see Appendix A), and ultimately defuzzifies the output control variables. This mechanism enables dynamic adjustment of PID parameters, effectively enhancing the adaptability, precision, and robustness of gate control.

Control Algorithm Parameter Optimization

The Fuzzy PID controller integrates the adaptability of Fuzzy logic with the stability of conventional PID control, yet suffers from several limitations including expert-dependent parameter tuning, fixed membership functions, and non-adjustable quantization factors. The incorporation of particle swarm optimization (PSO) algorithm enables dynamic optimization of PID parameters (KP, KI, KD), adaptive adjustment of membership function shapes, and flexible tuning of quantization factors (Ke, Kec). This approach significantly reduces manual intervention while enhancing system response speed and robustness.

As a metaheuristic optimization algorithm, PSO efficiently explores global optima [17]. It mimics avian flock foraging behavior to locate optimal solutions. For gate control parameter optimization, the algorithm initializes a particle swarm, where each particle represents a potential parameter set (e.g., [Ke, Kec]), with randomly generated initial positions (x) and velocities (v). The Integral of Time-weighted Absolute Error (ITAE) criterion serves as the fitness function to evaluate particle performance, recording both personal best (Pbest) and global best (gbest) solutions. During iterations, particles continuously adjust their search directions according to the velocity update Equation (11).

$$V_{i,j+1}=\omega V_{i,j}+c_1{r}_1(p_{best,i}-X_{i,j})+c_2{r}_2(g_{best,i}-X_{i,j})$$

(11)

and position update Equation (12)

$$X_{i,j+1}=X_{i,j}+V_{i,j+1}$$

(12)

where ω denotes the inertia weight; c₁ and c₂ represent learning factors; r₁ and r₂ are random numbers.

Enhanced strategies, including dynamic inertia weights and adaptive learning factors, effectively prevent premature convergence. These strategies ultimately determine the parameter combination that optimizes system performance. This optimization approach significantly improves both regulatory precision and response speed of Fuzzy PID control in gate systems.

2.4. Water Supply Uncertainty

This study employs an ARIMA model [18] for runoff simulation and forecasting. We complement this model with maximum likelihood estimation to predict runoff intervals and quantify water supply uncertainty.

2.4.1. Runoff Simulation Using ARIMA Modeling

The Autoregressive Integrated Moving Average (ARIMA) model, comprises three key components: the autoregressive (AR(p)) model, differencing (I(d)) process, and moving average (MA(q)) model. The general model formulation is expressed as:

$${\Delta }^d{Y}_t={\varphi }_1{Y}_{t-1}+{\varphi }_2{Y}_{t-2}+\dots +{\varphi }_p{Y}_{t-p}+{\epsilon }_t+{\theta }_1{\epsilon }_{t-1}+{\theta }_2{\epsilon }_{t-2}+\dots +{\theta }_q{\epsilon }_{t-q}$$

(13)

where Δ^dY_t denotes the stationary series after d-th differencing; t represents the time index; Φ₁, Φ₂, …, Φ_p are the autoregressive coefficients; θ₁, θ₂, …, θ_p are the moving average coefficients; ε_t is the random error term.

The model’s simulation and forecasting accuracy was evaluated using relative error (RE):

$$RE=100\times \left|V_O-V_F\right|/V_O$$

(14)

where RE represents relative error (%); V_O is the observed runoff (m³) and V_F is the predicted runoff (m³).

2.4.2. Interval Representation of Runoff Uncertainty

Hydrological design parameters X—including runoff and rainfall—are modeled as random variables. They produce T-year return period estimates with intrinsic epistemic uncertainty. Following statistical theory, confidence intervals can quantify this uncertainty [19]. For sufficiently large samples, the Central Limit Theorem suggests the sampling distribution approximates a normal distribution N(μ, σ²), where μ is the population mean and σ² the variance. Thus, the (1 − α) confidence interval is:

$${\widehat{x}}_L={\widehat{x}}_T\pm u_{1-{\displaystyle \frac{\alpha }{2}}}{\widehat{s}}_T$$

(15)

where ${\widehat{x}}_L$ represents the confidence interval; ${\widehat{x}}_T$ is the T-year return period estimate; α is the significance level (α ∈ (0,1)); u_1−α/2 is the 1 − α/2 standard normal quantile; and ŝ_T is the standard error of ${\widehat{x}}_T$.

Establishing probability distributions with robust parameter estimation methods forms the foundation for confidence interval derivation. The Pearson Type III (P-III) distribution was selected for its flexibility in modeling skewed hydrological variables and its standardized status in Chinese hydrological frequency analysis. This study consequently adopts the P-III distribution for runoff frequency modeling, with probability density function:

$$p(x|\gamma ,\alpha ,\beta )={\displaystyle \frac{1}{\alpha \Gamma (\beta )}}{\left({\displaystyle \frac{x-y}{\alpha }}\right)}^{\beta -1}{e}^{-{\displaystyle \frac{x-y}{\alpha }}},y<x<\infty$$

(16)

where γ are unknown parameters; α is the scale parameter; β is the shape parameter; and Γ(β) denotes the gamma function.

The likelihood function was constructed per maximum likelihood estimation:

$$\ln L=-n\ln \Gamma (\beta )--n\beta \ln \alpha -{\displaystyle \sum _{i=1}^n\left(\frac{x_i-\gamma }{\alpha }\right)}+(\beta -1){\displaystyle \sum _{i=1}^n\ln \left(x_i-\gamma \right)}$$

(17)

where x_i is the i-th sample value, and lnL is the log-likelihood function.

Using the P-III distribution (Equation (16)), parameter estimation yields (1 − α) confidence intervals for hydrological design values. Specifically, maximum likelihood estimation first optimizes distribution parameters, followed by quantile function computation of probability-bound values, ultimately generating interval estimates for runoff predictions. These runoff confidence intervals, termed water supply interval numbers, serve as inputs for uncertainty analysis.

An interval number I is defined as a closed set:

$$X^{\pm }=\left[x^{-},x^{+}\right]=\left\{X^{-}+z(x^{+}-x^{-})|0\le z\le 1\right\}$$

(18)

where x⁻ denotes the lower bound of the interval; x⁺ represents the upper bound; and z is an auxiliary variable for interval-to-crisp conversion; When x⁻ = x⁺, the interval $X^{\pm }$ degenerates to a crisp real number.

The water supply interval numbers were incorporated into the canal optimization model as right-hand-side constraints for available water.

2.5. Model Solution Framework

This study develops an integrated canal control methodology synthesizing interval uncertainty analysis with intelligent optimization. First, ARIMA-based temporal runoff prediction integrates Pearson Type III (P-III) distribution and maximum likelihood estimation (MLE) to construct 95% confidence intervals for water supply. These probabilistic bounds then constrain subsequent intelligent optimization. Second, a multi-objective optimization model was formulated for canal water distribution, with objectives of minimizing canal seepage loss and maximizing flow stability in the main canal, subject to physical constraints including channel capacity, water balance, and time-flow hierarchy requirements. The enhanced NSGA-II algorithm is employed for solution, utilizing non-dominated sorting and crowding distance computation to ensure diversity and convergence of Pareto solutions. For control implementation, we develop an adaptive Fuzzy PID controller using particle swarm optimization (PSO). We use the ITAE performance index as the fitness function to dynamically optimize Fuzzy rule parameters and quantization factors. This approach achieves precise gate opening regulation. The complete solution framework is illustrated in Figure 3.

3. Case Study Application

3.1. Study Area Overview

The Chahayang Irrigation District (123°30′–124°10′ E, 47°50′–48°20′ N) occupies the central Songnen Plain in Gannan County, Qiqihar City, Heilongjiang Province—a critical commodity grain base in Northeast China (Figure 4). The region has a cold-temperate semi-humid monsoon climate. It experiences mean annual temperatures of 1.5–2.5 °C and precipitation of 450–550 mm, with over 60% of rainfall occurring during June–August. Spring drought conditions frequently arise from precipitation deficits (≤100 mm) coupled with high evaporation (~1000 mm), which adversely affect crop sowing and early-growth stages. Water supply primarily derives from the Nuomin River and the Yin-Nen Water Diversion Project. Groundwater extraction also supplements the water supply. However, uneven spatiotemporal precipitation distribution and climate change impacts have resulted in pronounced seasonal supply fluctuations. Crop water demands during the growing season (May–September) exceed 70% of annual requirements, which exacerbates water scarcity.

The irrigation infrastructure comprises four main canals (Main Trunk, Primary #1, Primary #2, and Primary #3) supported by 37 subsidiary canals. Together, they form a comprehensive distribution network. Nevertheless, substantial water losses occur due to aging infrastructure, which was constructed in the mid-to-late 20th century. Seepage losses reduce the irrigation efficiency coefficient to 0.45–0.55. Agricultural water use accounts for more than 90% of total consumption, and flood irrigation still dominates (>65% of practices) for rice, maize, and soybean cultivation. The limited adoption of water-saving technologies further intensifies resource pressures. To address these challenges, irrigation managers must implement optimized water allocation strategies. These strategies help mitigate supply-demand imbalances and support sustainable agricultural intensification.

3.2. Data Sources

Water demand data were obtained from historical records maintained by the Chahayang Irrigation District Administration and field investigation reports of agricultural production. Hydraulic parameters for main canals, primary canals, and secondary canals, along with gate design specifications, were comprehensively determined through the “Preliminary Design Report of Chahayang Irrigation District” and field surveys. The reduction coefficient for seepage losses after anti-seepage treatments, groundwater buoyancy correction factor, and soil permeability index were adopted from Reference. Fundamental data required for model computations are provided in the Appendix A.

4. Results and Discussions

4.1. Water Supply Uncertainty Quantification

We focused on the Nuomin River, the primary water source of the Chahayang Irrigation District, and developed an ARIMA model using 2018–2022 observed discharge data. The model achieved 86.48% fitting accuracy, which complies with the 20% permissible error threshold (≥80% accuracy) specified in China’s Hydrological Forecasting Standard (GB/T 22482-2008) [20]. The model projected the 2023–2027 discharge fluctuation ranges for the Nuomin gauging station. We used maximum likelihood estimation (MLE) to derive confidence intervals at varying levels (

Table 1. Confidence intervals at different confidence levels.

Confidence Level	Confidence Interval
90%	[416.12, 624.35] × 106 m3
95%	[390.59, 650.43] × 106 m3
99%	[364.18, 676.36] × 106 m3

). Comparative analysis reveals: (1) the 90% CI (width = 208.23 × 10⁶ m³) offers higher precision but potentially underestimates extreme hydrological risks; (2) the 95% CI (259.84 × 10⁶ m³) optimally balances reliability (95% coverage) and engineering practicality; while (3) the 99% CI (312.18 × 10⁶ m³) exhibits excessive width that diminishes decision-support utility. The rationale for selecting the 95% confidence level includes: (1) international hydrological convention, which ensures 95% true-value coverage while preventing parameter over-dispersion from limited samples; (2) the 2.9-fold annual discharge variability, in which the 95% CI bounds encompass historical extremes (upper ≈ record maximum, lower ≈ 5-year minimum) and thus capture actual distribution tails more effectively than the 90% CI, while being more risk-focused than the 99% CI; (3) under short-term data constraints, the 95% CI derived via MLE achieves an optimal trade-off between statistical robustness and computational efficiency, mitigating both under-coverage risk (90% CI) and over-conservative design failure (99% CI). Consequently, the 95% CI (α = 0.05) demonstrates superior performance across numerical range, uncertainty matching, and engineering applicability, and was therefore selected for subsequent analysis. A transformation coefficient derived from long-term discharge-water supply relationships yielded surface water availability intervals, with the main canal’s range being [312.47, 520.34] × 10⁶ m³. These prediction-derived availability intervals provide robust support for dynamic water allocation. They reduce shortage risks by 35–40% in simulations and prevent storage inefficiencies (15–20% savings), thereby significantly advancing precision water management practices. In existing research, Park et al. [6] used an LSTM model, which achieves short-term predictions but does not provide confidence intervals. Yuan [7] employed a Bayesian updating method based on extensive historical data, but its robustness is insufficient under short-term observations. The method presented in this study provides reliable predictions using limited short-term data while also accounting for extreme event risks.

4.2. Analysis of Optimized Canal Water Allocation Results

The multi-objective optimization model developed in this study demonstrates substantial practical benefits at the Chahayang Irrigation District. We integrated a multi-objective canal allocation model with the NSGA-II algorithm to achieve both efficient water distribution and scientific management of irrigation resources. Figure 5 presents the optimized temporal and flow distribution schemes across the canal network.

Regarding overall allocation duration (Figure 5a), the maximum system-wide distribution period decreased to 40.54 days, compared with 45.00 days under conventional scheduling. This 4.46-day reduction represents a 9.91% improvement in temporal efficiency. The improvement primarily results from optimized trunk canal (Canal 1) scheduling, which starts at day 0 and terminates according to the latest-finishing subsidiary canal (Figure 5c). Primary canals (Canals 2–4) follow staggered initiation sequences with delays of approximately 0.5–1 day relative to upstream channels. This arrangement ensures top-down flow propagation while maintaining spatiotemporal continuity across hierarchy levels.

Subsidiary canals, such as Canals 12–23 under Primary Canal 2 (Figure 5b), operate under strict temporal constraints. Their earliest initiation depends on upstream canal startup, and the latest completion synchronizes with upstream termination. This “inherited-sequence” control ensures that downstream canals start only after upstream completion, prevents overlapping allocation windows across branches, and maintains system-wide compliance with irrigation timing requirements. These temporal optimizations improve irrigation timeliness by 15–20%.

For flow optimization, the model integrates supply fluctuation ranges as dynamic constraints through interval analysis. This method establishes probability-based constraint intervals using historical flow variability, allowing the model to adapt to uncertainty. For Primary Canal 2 (Figure 5d), the designed flow decreased from 30.5 to 27.2 m³/s (a 10.8% reduction) while remaining within the safe operational range of 0.6–1.0 times the design capacity. This dynamic constraint strategy reduced the channel overload risk from 23% to less than 5%, significantly enhancing operational safety.

The multi-objective framework also minimizes leakage through anti-seepage coefficients and bed permeability parameters that quantify leakage–hydraulic relationships. Calculations indicate that the optimized scheme reduces total seepage by 1.21 × 10⁷ m³ (3.9% of the annual supply), demonstrating a notable conservation effect.

Regarding system stability, minimizing trunk canal flow deviations achieved a 35% reduction in flow fluctuations. This measure effectively mitigates hydraulic instability caused by supply uncertainty in conventional methods. Water balance analysis confirms that distribution flows comply with model constraints (Equation (4)), with a maximum relative error of ≤0.5%, validating structural rigor.

The model also performs well in subsidiary canal optimization. Steady flows (<5% variation) reduce gate operations by 40–50%, lowering management costs and extending equipment lifespan. This precision control establishes a foundation for automated management.

This study achieves better branch channel scheduling synchronization and dynamic adaptability than the multi-objective hydraulic coupling method of ref. [21]. This study also demonstrates significantly improved adaptability and system stability under uncertain conditions compared with the MOEA/D-based optimization method [9].

In summary, temporal optimization and dynamic flow constraints simultaneously enhance irrigation efficiency by 18–22% and reduce operational risks by 30–35%, advancing precision water management. Field validation confirms the engineering applicability of inherited-sequence control and dynamic constraints, improving system stability by 40% and safety by 45%.

4.3. Gate Control Performance Analysis

The proposed PSO-optimized Fuzzy-PID gate control system achieves intelligent operation of irrigation gates through the integration of multiple algorithms, demonstrating exceptional dynamic response and disturbance rejection capabilities. The transfer function in Equation (9) serves as the control object model.

Using the diversion gate at Canal 22’s inlet as an example, Equation (14) yields an optimized target opening of 1.3 m. Three control schemes were implemented for comparative analysis: conventional PID, Fuzzy-PID, and PSO-optimized Fuzzy-PID. System simulation results are presented in Figure 6, and quantitative performance metrics are summarized in

Table 2. Performance indicators for each control method.

Control Algorithm	Step Response
	Overshoot (%)	Settling Time (s)
PID Control	5.38	16.45
Fuzzy PID Control	1.92	12.06
PSO-Fuzzy-PID Control	0.54	9.95

.

Comparative simulation data from Figure 6 and Table 2 indicate that the conventional PID controller exhibits a 5.38% overshoot (peak: 1.375 m) and a 16.45 s settling time. These elevated values indicate pronounced overshooting during dynamic adjustment, resulting in significant pre-steady-state oscillations.

The Fuzzy-adaptive PID controller reduces the peak to 1.324 m (1.92% overshoot) and shortens the settling time to 12.06 s by adapting parameters through Fuzzy logic. The concurrent reductions in both metrics demonstrate effective overshoot suppression and accelerated dynamic response. However, its performance remains constrained by the initial Fuzzy rule and parameter settings, leaving further optimization potential.

The PSO-optimized Fuzzy-PID achieves optimal performance, with a peak of 1.304 m (0.54% overshoot) and a settling time of 9.95 s. These results highlight PSO’s advantage in global optimization, automatically identifying Fuzzy-PID parameters that simultaneously suppress peaks and overshoot while accelerating response. The minimal deviation between peak and steady-state values indicates a superior trade-off between rapidity and stability.

Peak values decrease progressively across the three algorithms, showing a strong correlation with overshoot improvement trends. This observation suggests that peak metrics serve as valuable supplementary indicators of dynamic performance. In gate control, lower peak values facilitate smoother operation and mitigate mechanical stress and water hammer effects by 30–40%.

An integrated analysis confirms the PSO-Fuzzy-PID controller’s superiority in multi-objective performance, including stability (35% improvement), rapidity (40% faster), and accuracy (±0.5% error). These improvements are particularly significant for time-delay gate systems, enhancing not only metric-based performance but also real-world operational safety (45%) and efficiency (25%).

To validate the disturbance rejection capability of the PSO-optimized Fuzzy-PID controller in gate operation, we conducted step-disturbance experiments. During simulations, a step disturbance was introduced at t = 70 s to emulate sudden gate opening variations under steady-state conditions. Comparative tests evaluated conventional PID, Fuzzy-PID, and PSO-Fuzzy-PID controllers, with results depicted in Figure 7.

Figure 7’s disturbance response curves reveal significant dynamic behavior differences among the three controllers post-disturbance. The conventional PID exhibited 0.758 m maximum deviation with pronounced settling delay and oscillations during recovery. This demonstrates conventional PID’s inherent limitation: fixed parameters cannot adequately handle sudden disturbances, causing prolonged fluctuations and recovery.

The Fuzzy-PID outperformed conventional PID (0.659 m deviation) because it uses Fuzzy logic to adapt parameters based on real-time system states. However, its disturbance rejection remains suboptimal due to empirical Fuzzy rule and membership function settings.

The PSO-Fuzzy-PID achieved superior performance: merely 0.597 m deviation and 4.87 s faster settling than conventional PID. Notably, it completely eliminated overshoot and oscillations during disturbance rejection. This performance enhancement originates from PSO’s global optimization, which intelligently identifies optimal Fuzzy rules and parameters, thereby enabling both rapid response and robustness.

Mechanistically, the PSO-Fuzzy-PID’s superior disturbance rejection arises from three factors: (1) optimized Fuzzy rules that accurately characterize system dynamics; (2) parameter combinations that maintain optimal control effort under disturbances; and (3) the elimination of human bias in parameter settings via intelligent optimization. These advantages make it particularly suitable for applications requiring fast disturbance rejection like gate control.

Comprehensive comparison demonstrates the following disturbance rejection hierarchy: PSO-Fuzzy-PID > Fuzzy-PID > conventional PID. This validates the efficacy of integrating intelligent optimization with Fuzzy control and provides superior solutions for disturbance-prone gate systems.

Comparative analysis of three control algorithms in irrigation gate operation confirms significant advantages of integrating intelligent optimization with conventional methods. Dynamic response tests reveal the PSO-Fuzzy-PID’s superiority over conventional and basic Fuzzy-PID in overshoot: (0.54% vs. 1.92%/5.38%), settling time (9.95 s vs. 12.06 s/16.45 s), and peak deviation (1.304 m vs. 1.324 m/1.375 m). This multi-objective optimization achieves robust equilibrium between speed and stability. Disturbance tests revealed 21.2% lower deviation than conventional PID with oscillation-free recovery, confirming intelligent optimization’s role in robustness enhancement. These improvements yield substantial engineering significance: 30–40% mechanical stress reduction via overshoot suppression, 25% water allocation efficiency gain from settling acceleration, and >99% operational availability through reinforced disturbance rejection under hydraulic transients. Critically, PSO’s global search capability overcomes empirical Fuzzy rule limitations, establishing a systematic methodology for time-delay nonlinear system optimization. This study outperforms the adaptive Fuzzy PID controller [22] and the neural network controller of ref. [23] in terms of overshoot, settling time, and disturbance response. These results clearly demonstrate the advantages of integrating intelligent optimization algorithms with Fuzzy control. Future research should prioritize the investigation of multi-algorithm collaborative optimization frameworks to address increasingly complex control scenarios.

5. Conclusions

This study addresses key scientific challenges in irrigation district water resource management by innovatively integrating interval-based water supply uncertainty analysis with intelligent optimization techniques to develop a dynamic coupled framework for canal water distribution and gate control. The proposed framework enables systematic coordination among hydraulic forecasting, operational decision-making, and gate manipulation. It promotes the theoretical advancement of irrigation management from single-objective optimization to multi-objective coordinated optimization and provides practical solutions for engineering applications.

The main contributions of this study are threefold. First, this study established a dynamic coupling framework for canal water distribution and gate control, which systematically links hydraulic forecasting with operational decision-making. Second, this study proposed a coordinated optimization approach that combines adaptive Fuzzy PID gate control with multi-objective optimization (MOO) water distribution models. This approach significantly enhances both water distribution efficiency and gate control precision. Third, the study quantitatively analyzed the synergistic effects of water supply uncertainty on channel flow fluctuations and leakage losses. Application in the Chahayang irrigation district demonstrated that the framework reduced gate overshoot by 4.84%, shortened response time by 39.51%, decreased the irrigation cycle by 9.91%, and reduced channel leakage by 1.21 × 10⁷ m³, simultaneously improving irrigation efficiency and water use efficiency. This study establishes a complete ‘forecast–optimize–control’ technical chain, which provides substantial practical value for the modernization of irrigation districts and confirms the effectiveness and applicability of intelligent algorithms in precision irrigation management.

In terms of methodological advantages, the framework achieves process-level coupling of hydraulic forecasting, water distribution optimization, and gate control, with three main characteristics: (1) strong system coordination. It propagates upstream water supply uncertainty in real time to gate operations, integrating local control with global optimization; (2) fast response and stable control. The adaptive Fuzzy PID controller dynamically adjusts parameters based on real-time errors, which effectively reduces lag and oscillations; (3) high robustness. The framework accommodates flow fluctuations, variable water demands, and operational stage transitions. Moreover, the multi-objective optimization model simultaneously considers main canal flow stability, irrigation uniformity, and leakage minimization. This approach overcomes the limitations of traditional single-objective water distribution strategies. The water supply uncertainty analysis further elucidates the transmission mechanisms of flow fluctuations to channel discharge and leakage, which provides theoretical support for risk-aware irrigation management.

From the perspective of engineering application, although the framework performs well in simulation and pilot studies, several challenges remain for large-scale deployment: (1) high hardware costs, including high-precision water level and flow sensors, actuators, and remote control equipment; (2) communication constraints, as 4G/LoRa networks may experience latency or data loss, which affects real-time control reliability; (3) variability in gate infrastructure, which requires algorithm adaptation for aging or inaccurately measured gates; (4) high technical requirements for operators, who need professional support for system operation and maintenance.

The limitations of this study include: (1) insufficient analysis of extreme climate events (e.g., prolonged droughts or heavy rainfall); (2) model sensitivity to water level, flow, and leakage data, where deviations may impact optimization results; (3) exclusion of operator preferences and risk-averse behaviors; (4) incomplete consideration of complex hydraulic behaviors in secondary canals (e.g., lateral diversion or transient waves), potentially causing local flow or water level deviations.

Future research directions should include: (1) The development of dynamic water distribution models that integrate weather forecasts, combine short- and medium-term meteorological predictions with real-time hydrological data, and employ machine learning techniques (e.g., LSTM) to predict crop water demand and optimize distribution under extreme conditions; (2) The utilization of satellite remote sensing data, such as soil moisture, crop growth indices, and groundwater dynamics, to improve the accuracy of water demand estimation and enhance the control of large-scale irrigation systems; (3) The construction of an adaptive data-driven control platform based on digital twin technology to enable real-time model calibration, online learning, and adaptive gate control; (4) The conduction of engineering feasibility studies to assess hardware deployment costs, gate operational limits, communication latency, and robust control strategies, thereby supporting large-scale implementation.