Modeling Probabilistic-Based Reliability Analysis for Irrigation Water Supply Due to Uncertainties in Hydrological and Irrigation Factors

Abstract

This study aims to model a probabilistic-based reliability analysis, named the RA_IWS_Canal model, for calculating the probability of the irrigation water supply exceeding the water demand (i.e., reliability) within a multi-canal irrigation zone due to variations in hydrological and irrigation uncertainty factors. The proposed RA_IWS_Canal model is developed by coupling uncertainty and risk analysis with a logistic regression equation. The Zhudong irrigation zone, located within the Touqian River watershed in northern Taiwan, was selected as the study area, with the inflow from Shanping Weir, water supplies at 15 irrigation canals, and water intakes of two reservoirs (Baoshan and Baoshan II) and a water treatment plant (Yuandon); 1000 simulations of 10-day irrigation water allocations and resulting exceedance probabilities of the water supplies at the 15 canals were achieved using the multivariate Monte Carlo simulation and the uncertainty with the water allocation model (RIBASIM), and employed in the development of the proposed RA_IWS_Canal model. The model development and application results indicate that the uncertainty factors and the inflow from Shanping Weir markedly and positively influence the exceedance probability of the canal-based irrigation water supply to boost the corresponding reliability (about 0.8). The water intake of the Baoshan Reservoir has a lower relationship (by 0.19) than the Yuandon water treatment plant with the reliabilities of the irrigation water supplies at its downstream canals. As a result, the proposed RA_IWS_Canal model can evaluate the effect of not only the canal-based uncertainty factors, but also the regional features on the irrigation water supply reliability. In addition, using the proposed RA_IWS_Canal model, the planned irrigation water demands at various canals within a multi-canal irrigation zone could be accordingly formulated based on acceptable reliability.

1. Introduction

Irrigation water commonly comes from two sources: surface runoff through rivers, lakes, or reservoirs, and groundwater via the springs or wells. Specifically, the surface runoff has greater irrigation water use efficiency than the groundwater, due to enhanced recharge rates [1]. As for the surface runoff, the irrigation water supply system comprises reservoir-based and natural river-based systems. The reservoir-based supply system could stably supply the irrigation water according to the control operation rules. In contrast, the irrigation water from the natural river-based system might be unsteadily provided, due to the uncertainties in the hydrological variables caused by climatic change (e.g., precipitation and evapotranspiration) and variation of the topographic features, such as the riverbed and cross-section [2,3,4,5,6,7,8]. With high variations possibly existing in the natural river-related irrigation water, the practical water allocation is rarely implemented, due to the unsteady water from the river-based supply system. Additionally, within the irrigation zones, the hydraulic structures and water intake facilities (e.g., the gates, reservoirs, water-treatment factories, and reservoirs) probably impact the reliability of the irrigation water supply without the corresponding well-regulated operation rules and the initial water levels [9,10,11]. In summary, the uncertainty existing in the process of supplying irrigation water via a variety of water resources possibly includes the surface water and hydraulic facilities as well as the geometrical features [12,13]. Furthermore, despite the water allocation simulation models being widely applied in irrigation water management, the reliability of the resulting irrigation water supply from the water allocation models might be attributed to the lack of hydrological data needed in the model configuration, such as rainfall, runoff, and crop water demand in the growing seasons [14].

In general, the reliability analysis is commonly carried out to evaluate the uncertainty factors’ effect on the irrigation water supply. The resulting irrigation water supply reliability is widely represented in terms of the ratio of the water supply to the water demand (called the reliability index) [12,15,16,17,18,19]. Therefore, a number of reliability analysis models have been proposed to pay attention to the reliability index, equal to the ratio of the irrigation water supply to the corresponding water demands. The above models could be treated as the deterministic model without considering the variation in the relevant irrigation-based factors, including the hydrological, geometrical, and operating factors; namely, the shortage risk of irrigation water supply might be quantified via the deterministic models with great difficulty, due to the uncertainties in the irrigation-based factors of interest. Recently, the uncertainty quantification and the induced reliability assessment have been coupled in the probabilistic design and risk-based analysis in various hydrosystems [20]. Specifically, the relevant probabilistic-based approaches were comprehensively applied in the reliability analysis of the water resource system [4,8,11,21]. For example, Meza et al. [4] analyzed the impact of climate change on the reliability of water rights based on the exceedance probability of the monthly flow thresholds calculated via the log–normal probability distribution; the associated parameters were calibrated using the observations. Wu et al. [11] employed the advance first-order second-moment (AFOSM) method coupled with logistic regression analysis to quantify the reliability of the water supply from reservoirs in terms of the probability of the water supply exceeding the specific water demand attributed to the uncertainties in the hydrological, hydraulic, and operating rules. Guerrero-Baean [21] adopted the shift-related parameter of the probability density function regarding the irrigation water supply to quantify the shortage risk of supplying the irrigation water supply caused by the farmers’ willingness to improve the irrigation water supply.

Despite several probabilistic-based models developed for evaluating the reliability of the irrigation water supply, the resulting reliabilities are mainly quantified through a prior probability density function (PDF) under the statistical properties of the irrigation water supply provided in advance. However, the above statistical information adopted (e.g., the prior PDF) might not be available to respond to the effect of the challenging variation in the practical irrigation-related factors [22]. Therefore, this study aims to couple the uncertainty/risk analysis approaches with logistic regression analysis to develop a probabilistic-based reliability analysis model for assessing the effect of irrigation-related uncertainty factors on the reliability of the irrigation water supply within a canal-based irrigated agriculture zone (named the RA_IWS_Canal model). To enhance the model’s performance, a significant number of irrigation water supply simulation cases were reproduced via the deterministic water allocation model, with the irrigation-related uncertainty factors for model development generated by the Monte Carlos simulation method. Consequently, the reliability of the canal-based irrigation water supply could be achieved via the proposed RA_IWS_Canal model by calculating the exceedance probability of the irrigation water supply, in consideration of the planned water demand with the magnitudes of irrigation-related uncertainty factors provided.

2. Methodology

2.1. Model Concept

This study aims to develop a probabilistic-based model for assessing the reliability of the resulting irrigation water supply in a multi-canal irrigation zone with multiple irrigation canals (called the RA_IWS_Canal model) by coupling the water allocation model with the uncertainty and risk analysis considering variations in irrigation-related features. Generally speaking, in agriculture regions, irrigation water is commonly contributed to by supplies from natural sources (e.g., surface runoff) and manufactured sources, such as hydraulic structures, reservoirs, artificial channels, and associated types of equipment (e.g., water gates). The surface runoff from rivers significantly contributes to the irrigation water volume. In addition, reservoirs play a vital role in the irrigation water supply system, which can substantially influence irrigation water. Moreover, to effectively allocate water in the sub-regions within the irrigation zones, the water supply is controlled by operating the water gates based on the planned water demand for the irrigation zone of interest. In summary, the factors related to the irrigation water supply can be briefly classified into two types: the hydrological and irrigation factors. The hydrological factors include inflow from the river and outflow into hydraulic structures, whereas the planned irrigation water demand and the maximum water intakes of canal-based gates with different opening heights could be regarded as the irrigation factors [11,12].

In this study, the water supply reliability is defined as the probability of the supply of water exceeding a specific magnitude in consideration of variations in the hydrological and hydraulic factors. Thus, when developing the proposed RA_IWS_Canal model, the numerous simulations of the irrigation water supplies could be obtained by the well-known water allocation simulation model with the various combinations of simulated and hydraulic factors using the multivariate Monte Carlo simulation approach [23]. Afterward, using the irrigation water supply simulations and the corresponding responses to the uncertainties in irrigation water-related factors, the probability of the water supply exceeding demand can be estimated via the uncertainty/risk analysis. Since the reliability analysis carried out via the AFOSM approach might be complicated due to more uncertainty factors adopted, logistic analysis is applied to derive the relationship between the exceedance probability of the irrigation water supply with the uncertainty factors to efficiently quantify the irrigation-related supply reliability in the multi-canal irrigation zones.

To sum up the above model concept, the development of the proposed RA_IWS_Canal model can be classified into five steps: (1) configuration of the water allocation model for the multi-canal irrigation zone; (2) generation of hydrological and irrigation factors; (3) simulation of the irrigation water supply; (4) quantification of the reliability regarding the irrigation water supply; and (5) derivation of the exceedance probability calculation equation. Therefore, the relevant methods required are addressed as follows:

2.2. Simulations of Irrigation Water Supply and Uncertainty Factors

As mentioned above, within the proposed RA_IWS_Canal model, the irrigation water supplies at the irrigation canals could be determined using the water resource allocation simulation model under various conditions of the hydrological and irrigation factors recognized. Therefore, to enable the quantification of the irrigation water supply via the proposed RA_IWS_Canal model, a significant number of irrigation water supply simulations, with the corresponding uncertainty factors considered, could be achieved in advance via the multivariate Monte Carlo simulation approach [23] coupled with the well-known water allocation simulation model. The process of simulating the irrigation water supplies with various uncertainty factors is addressed below:

2.2.1. Identification of Uncertainty Factors

To quantify the reliability of irrigation water supply, the uncertainty factors induced should be recognized in advance. As mentioned in the Introduction section, the uncertainty factors corresponding to the irrigation water system comprise hydrological and irrigation factors. Regarding the hydrological factors, the runoff from rivers and from underground and artificial channels can make a significant contribution to irrigation water. In general, precipitation and evapotranspiration play an important role in crop production, especially during the wet and dry seasons; to be specific, evapotranspiration is probably adversely related to precipitation and rainfall-induced surface runoff [24]. Thus, the uncertainty of surface runoff could be quantified in response to the variation of the precipitation and evapotranspiration with high likelihood. As a result, in this study, the river runoff is regarded as a hydrological uncertainty factor that responds variations in the precipitation and evapotranspiration. Moreover, surface runoff of long durations, such as a 10-day and monthly duration, are frequently required in the irrigation system; however, the surface runoff in the irrigation system might be delivered as the intake water to the hydraulic facilities (e.g., water-treatment plants and reservoirs). Therefore, the surface runoff carried as the water supply and intake in the irrigation water system could be identified as the hydrological factors. Climate change and the occurrence of extreme events could possibly impact surface runoff, causing uncertainty and variability; thus, uncertainties in the hydrological factor should be considered in the development of the proposed RA_IWS_Canal model.

In the process of distributing the water volume in the irrigation system comprised of the sub-zones, including their channel-based canals, the canal-based water supplies are determined according to the corresponding planned water demands and the capacity of the water gages in terms of the maximum delivered water (i.e., water intake). Since the above planned water demand changes with the type of crop and corresponding growth seasons, it is highly likely to be uncertain; additionally, the maximum delivered water at the water gates is generally determined based on the operating rules and strategies [17]; thus, the corresponding variations might be induced by the uncertainties on the operation of intakes and offtakes [25]. By doing so, the variations regarding the maximum delivered water of the gates may affect the reliability of the irrigation water allocation. In summary, the maximum water intakes of the water gates and planned water demands at the canals in the irrigation zones play a vital role in supplying the irrigation water. Consequently, quantifying the uncertainties in the irrigation factor should be accomplished in developing the proposed RA_IWS_Canal model. The above uncertainty factors for the irrigation water supply are listed in

Table 1. Summary of the uncertainty factors for the irrigation water supply.

Type of Uncertainty Factors	Uncertainty Factors	Sources/Definitions
Hydrological factors	Surface runoff	River
		Underground
		Hydraulic structures
Irrigation factors	Planned water demand	The minimum irrigation water volume required at the canals for the specific crop
	Maximum water intake of the water gages	The maximum water volume through the water gates.

.

2.2.2. Simulation of Uncertainty Factors

As mentioned in the above sections, the hydrological and irrigation factors are needed to irrigate the crops at all canals in the irrigation zones. Since the hydrological variables (e.g., surface runoff and precipitation) are commonly treated as correlated in time and space [26], the hydrological factors concerning the surface runoff should be temporally and spatially correlated variates. Moreover, the offtakes and intakes of several hydraulic structures are mainly delivered to the irrigation zone; in addition, they significantly change with the demands of the livelihood and industry. Therefore, among the hydraulic features, it is necessary to take temporal and spatial correlations into account when generating a significant number of uncertainty factors in seeking to quantify the reliability of the irrigation water supply.

Regarding irrigation factors, the irrigation water demands in the canals are set up based on the types of crops and their growing seasons; thus, the planned water demands are regarded as correlated variates in time and space. In addition, the maximum water intake via the water gates corresponding to the irrigation canal is commonly determined based on the associated operating rules with the practical water levels; furthermore, the water intake via the water gages at the upstream canals might influence the water supplies at the remaining canals. That is, in reality, it is highly likely that the gate-related maximum water intakes in the irrigation zones contain spatial covariances.

Consequently, by referring to the model concepts addressed in Section 2.1, this study employs the multivariate Monte Carlo simulation (MMCS) method [27] to simulate a significant number of the hydrological and irrigation factors, with the correlations of various degrees in time and space. The above multivariate Monte Carlo simulation (MMCS) method is briefly addressed as:

The MMCS method is a modified Monte Carlo simulation approach to achieve the generation of more than two variables with correlations of various degrees. The MMCS method with three normalized-based algorithms, including the standardized, orthogonal, and inverse transformations, is adopted, and their correlations can be computed via the following Nataf bivariate distribution [28]:

$${\mathsf{\rho }}_{ij}=\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}\lfloor \frac{X_i-{\mu }_i}{{\sigma }_i}\rfloor \lfloor \frac{X_j-{\mu }_j}{{\sigma }_j}\rfloor {\varnothing }_{ij}(\langle Z_i,Z_j|{\rho }_{ij}^{*}\rangle ){\mathrm{dZ}}_i{\mathrm{dZ}}_j$$

(1)

$$Z_i=\frac{X_i-{\mu }_i}{{\sigma }_i};Z_j=\frac{X_j-{\mu }_j}{{\sigma }_j}$$

(2)

where X_i and X_j are the correlated variables at the points i and j, respectively, with the means ${\mu }_i$ and ${\mu }_j$, the standard deviations ${\sigma }_i$ and ${\sigma }_j$, and the correlation coefficient ${\rho }_{ij}$; _i and Z_j are corresponding bivariate standard normal variables to the variable Xi and Xj with the correlation coefficient and the joint standard normal density function ϕ_ij(·). To simplify the generation process, Chang et al. [27] proposed a series of relationships between the abnormally correlated variables and the corresponding standardized ones’ functions based on correlations computed using Equation (1).

Therefore, in this study, the simulations of the uncertainty factors related to the reliability of supplying irrigation water would be achieved via the MMCS method to derive the proposed RA_IWS_Canal model. Note that the statistical properties corresponding to the uncertainty factors can be quantified from the observations (i.e., hydrological features) and relevant historical references (irrigation factors).

2.2.3. Simulation of Irrigation Water Supply

Recently, water allocation models are mainly applied to identify optimal water supply strategies, such as MODSIM, RIBASIM, and HEC-ResSim [11]. Among these above models, RIBASIM is a water balance model for estimating the basic spatiotemporal information of the water volume and discharge at different nodes along with a river network via the following Equation [29]:

$$S_{t_1}-S_{t_2}+\theta \times \left(Q_{in,t_1}-Q_{out,t_2}\right)=0$$

where $Q_{in,t_1}$ and $Q_{out,t_2}$ serve as the inflow at the time steps t₁ and the outflow at the time step t_2, respectively; $S_{t_1}$ and $S_{t_2}$ denote the water storage at the time step t₁ and t_2, respectively; and $\theta$ is the conversion factor (efficiency coefficient) of various diversion structures. Moreover, RIBASIM can display the entire process of allocating water intake and offtake through various hydraulic structures (e.g., weirs and gates) through a GIS-oriented graphical user interface.

As a result, when developing the proposed RA_IWS_Canal model, a significant number of simulations of the irrigation water supply in a multi-canal irrigation zone could be accomplished in advance using the RIBASIM model with numerous generated hydrological and irrigation uncertainty factors.

2.3. Reliability Quantification of Reservoir Water Supply

In this study, the reliability of supplying irrigation water is defined as the probability of the irrigation water supply ($W_{IS}$) exceeding the water demand ($w_{ID}$):

$$\mathrm{Reliability}=P_r\left(W_{Is}>w_{ID}\right)$$

(3)

Since the difference between the water supply and demand could be treated as the shortage index (SI), the reliability Equation (4) can be rewritten as:

$$\mathrm{Reliability}=P_r\left(W_{IS}>w_{dID}\right)=1-Pr\left[SI\right]$$

(4)

Regarding Equation (5), the reliability of the irrigation water supply is adversely related to the probability of the water shortage $Pr\left[SI\right]$ occurring; i.e., the above water supply reliability can account for the shortage risk.

Specifically, in the proposed RA_IWS_Canal, the probabilities of irrigation water supply exceeding the demand could be calculated using the well-known uncertainty and risk analysis (i.e., advance first-order second-moment, AFOSM) with a significant number of irrigation water supply simulations; subsequently, to effectively evaluate this reliability $Pr\left[W_{ID}>w_{IS}\right]$ without carrying out the Monte Carlo simulation and uncertainty/risk analysis, the equation for calculating the exceedance probability of the irrigation water supply should be established using the multivariate regression approach (i.e., logistic regression analysis) with a significant number of the resulting exceedance probabilities of irrigation water supply. The detailed concepts regarding the quantification of irrigation water supply could be addressed as follows:

2.3.1. The Quantification of the Simulated Irrigation Water Supplies

In establishing the proposed RA_IWS_Canal model, the probability of the water supply exceeding the demand (i.e., the exceedance probability) could be computed via the existing uncertainty analysis method, i.e., the advance first-order second-moment approach (AFOSM) [30], frequently employed in hydrological-related risk analysis [9,11,31,32,33,34].

AFOSM is a well-known method used in water resources reliability analysis, originally developed to assess the safety of the structural component and structural systems by calculating the exceedance probability based on the standard normal distribution with the first two statistical moments (mean and variance). Accordingly, in this study, by employing the AFSOM method in the quantification of the irrigation water supply reliability, the resulting exceedance probability can be defined via the following equation:

$$\begin{gathered} \mathrm{Reliability}=P_r\left(W_{Is}>w_{ID}\right)=Pr\left[W_{IS}-w_{ID}>0\right]=Pr\left[Z>0\right] \\ =1-Pr\left[Z\le 0\right]=\Phi \left(-\beta \right) \end{gathered}$$

(5)

$$Z=W_{IS}-w_{ID}$$

(6)

$$\beta =\frac{E\left(z\right)}{s_z}$$

(7)

where Z denotes the performance function; E(z) and s_z are the mean and standard deviation of Z, respectively; $\Phi \left(·\right)$ stands for the standard normal distribution, and $\beta$ is the reliability index. Note that in the AFSOM method, $E\left(z\right)$ and $s_z$ could be obtained using the following equations:

$$E\left(z\right)=W_{IS,{\theta }^{*}}+{{\displaystyle \sum }}_{i=1}^m{\left|\frac{\partial W_{IS}}{\partial {\theta }_i}\right|}_{{\theta }^{*}}\left({\mu }_{{\theta }_i}-{\theta }_i^{*}\right)-w_{IS}$$

(8)

$$s_z=\sqrt{{\displaystyle \sum }_{i=1}^m{\left(\frac{\partial W_{IS}}{\partial {\theta }_i}\right)}^2{\sigma }_{{\theta }_i^{*}}^2}$$

(9)

where ${\theta }^{*}$ stands for the failure points of the ith uncertainty factor when the performance function z = 0; ${\mu }_{{\theta }_i}$ and ${\sigma }_{{\theta }_i}$ are the mean and standard deviation of the ith uncertainty factor, respectively; $W_{IS,{\theta }^{*}}$ denotes the rainfall threshold using the uncertainty factors’ failure points; and $\frac{\partial W_{IS}}{\partial {\theta }_i}$ is the sensitivity coefficient of the ith uncertainty factor. In the case of the mean and variance of the given uncertainty factors, the probability of the water supply $W_{IS}$ exceeding the demand $w_{ID}$, $\Phi \left(-\beta \right)$ can be obtained.

Note that the relationship between the model outputs and inputs should be known in advance to calculate the quantification of the resulting reliability and risk through the AFOSM; accordingly, the equation of the irrigation water supply with the corresponding uncertainty factors could be established using the multivariate regression method as:

$$W_{IS}=f\left({\theta }_{hydro},{\theta }_{IRR}\right)$$

(10)

where $W_{IS}$ represents the irrigation water supply through the branch-based canals; ${\theta }_{hydro}$ stands for the hydrological factors, and ${\theta }_{IRR}$ stands for the irrigation factors; note that the above regression coefficients could be calibrated via multi-variate regression analysis with a significant number of the simulations of the irrigation supplies with the corresponding generated uncertainty factors.

2.3.2. Establishment of the Exceedance Probability Estimation Equation

In addition to the irrigation water supply reliability carried out via the AFOSM method, to effectively evaluate the reliability $Pr\left[W_{ID}>w_{IS}\right]$ of a specific water demand under the uncertainty factors provided, this study aims to derive an equation calculating the corresponding exceedance probability of the irrigation water supply to the uncertainty factors known. In general, the relationship between the multiple hydrological variables is established using regression analysis [31]; specifically, logistic regression analysis helps derive the relationship among numerous independent variables and a dependent variate which takes only two dichotomous values [11,33,34,35], such as:

$$log\left(\frac{P}{1-P}\right)={\beta }_0+{\beta }_1{X}_1+\cdots +{\beta }_n{X}_n$$

(11)

where X_i and P represent the ith independent variables and the corresponding occurrence probability, respectively. ${\beta }_i$ (i = 1, 2, …n) stands for the coefficients and ${\beta }_0$ is the intercept. Thus, the proposed RA_IWS_Canal model is coupled with logistic regression analysis to establish the exceedance probability calculation equations for quantifying the reliability of the irrigation water supplies at the canals. In greater detail, P in Equation (9) is regarded as the exceedance probability $P_r\left(W_{IS}>w_{ID}\right)$, and this logistic regression equation is expected to robustly provide stochastic information on the water supply regarding the demand at various canals in the multi-canal irrigation zones.

2.4. Model Framework

According to the concepts and methods used in the proposed RA_IWS_Canal model previously mentioned, a detailed flowchart of model development can be introduced as follows:

• Step (1): Collect the historical data for the uncertainty factors, including the hydrological factors related to surface runoff and irrigation factors regarding the operation of delivering the runoff into the canals.
• Step (2): Configure the water supply allocation (RIBASIM) model based on the structure of the irrigation water supply system and relevant operation data corresponding to the study area.
• Step (3): Calculate and hypothesize the statistics of the uncertainty factors, including the mean, coefficient of variance, coefficient of skewness, coefficient of kurtosis, and the suitable probability distribution.
• Step (4): Carry out the multivariate Monte Carlo simulation (MMCS) method with the above statistical properties of the uncertainty factors to simulate a significant number of uncertainty factors.
• Step (5): Emulate the water supplies at the irrigation canals through the RIBASIM model for the study area with the simulated uncertainty factors.
• Step (6): Calculate the exceedance probabilities (i.e., reliability) of irrigation water supplies at the irrigation canals by carrying out the AFOSM method, using a significant number of simulated irrigation water supplies and corresponding uncertainty factors.
• Step (7): Derive the logistic regression equation regarding the exceedance probabilities corresponding to the water supply estimates calculated, considering various water demands and uncertainty factors selected in Step (6).
• Step (8): Evaluate the reliability of irrigation water supply at the canals, attributed to the variation in the specific uncertainty factors selected in Step (7) through the exceedance probability calculation equations established in Step (7).

3. Study Area and Data

3.1. Introduction to the Study Area

To demonstrate the applicability of the proposed reliability analysis model for water supply on the water allocation in a multi-canal irrigation zone (named RA_IWS_Canal), this study selects the complicated irrigation channel zones (Zhudon canal zone) in Northern Taiwan (see Figure 1) as the study area.

The study area of the Zhudong irrigation zone is located within the Touqian River watershed in Northern Taiwan, and its total irrigation area is approximately 800 ha, in which rice is the main crop during the growing season, i.e., the middle of February and the end of November (the fifth to thirty-third 10-day periods). The Zhudong irrigation zone comprises 15 irrigation canals, the irrigation extent of which can be seen in Figure 2a; two reservoirs (Baoshan and Baoshan II) as well as a water treatment plant (Yuandon) (see Figure 1). Note that the irrigation management of the 15 canals is carried out by supplying the irrigation water from upstream to downstream in accordance with the corresponding water demands.

In the Zhudong irrigation zone, the irrigation water mainly comes from the Shanping and Yulon Rivers through the Zhudong Weir. Figure 3 presents the observed 10-day runoff in the Shanping and Youluo Rivers. At the Zhudong Weir, the surface runoff can be delivered into two regions, the Zhudong irrigation zone and the Baoshan II Reservoir, with the maximum intake runoffs of 20 m³/s and 2.4 m³/s, respectively. In addition, the Yuandon treatment plant, located between the second and third canals, on average, carries 1126 tons/year (about 0.386 m³/s) of water (See Figure 4). Moreover, the Baoshan and Baoshan II Reservoirs mainly obtain water between the seventh and eighth canals and the Shanping Weir, respectively, according to their operation rules, with the annual initial water levels in the reservoirs referred to in Figure 5.

Supplying irrigation water is generally performed based on the corresponding water demand assigned in accordance with the irrigation extent and growth season. However, there is a pattern of historical 10-day irrigation water demands in the Zhudong irrigation zone (see Figure 6); accordingly, to estimate the irrigation water demands at 15 canals, this study defines the weights of the irrigation water demands of the canals by calculating the ratio of the irrigation area corresponding to the area of the Zhudong irrigation zone; the canal-based planned irrigation water demand is then obtained via the irrigation water demand for the Zhudong irrigation zone of corresponding weight (see Figure 2b).

3.2. Configuration of Water Allocation Model (RIBASIM)

According to the irrigation canal network of the Zhudong irrigation zone introduced above, the corresponding water allocation (RIBASIM) model could be configured as shown in Figure 6. Within the RIBASIM model configured for the Zhudong irrigation zone, the 10-day surface runoff from the Shanping Weir is regularly allocated based on the intake ranking of the canals and hydraulic structures from upstream to downstream. Moreover, in this study, determining the irrigation water allocation using the RIBAISM model, the inflow from the rainfall and outside hydrological structures and specific outflow, i.e., hydrological loss (e.g., the evapotranspiration and infiltration) are not considered; the unique operations of the irrigation water supply during the drought periods are also excluded in the RIBAIM. In summary, the RIBAIM model adopted in the development of the proposed RA_IWS_Canal model focuses on simulating the irrigation water supplies at the canals, corresponding to various conditions of the inflow from the rivers and water intakes at the other canals and hydraulic structures within the multi-canal irrigation zone.

4. Model Development

In this section, we present the results from the process of developing the proposed RA_IWS_Canal model. In addition, in order to demonstrate the applicability of the proposed RA_IWS_Canal model, we quantify and evaluate, considering the variations in the relevant uncertainty factors, the reliability of irrigation water supply under consideration in the Zhudong canal basin.

4.1. Simulation of Uncertainty Factors Concerned

Within the framework of developing the proposed RA_IWS_Canal model, the generation of the uncertainty factors considered is carried out using the statistics of Wu’s model [21]. As for the hydrological factors, the inflow in the Shanping and Youluo River and the water intake of the hydraulic structures, including the Baoshan and Baoshan II Reservoirs and the Yuandon water treatment plant were used. Except for the Yuandon water treatment plant, which carries water from the Zhudong canal basin, the remaining water intake of the Baoshan and Baoshan II Reservoirs should result from the reservoir operation rules that regulate the annual initial water levels; thus, the water intake of the Yuandon water treatment plant could be directly simulated based on the associated statistical properties, whereas the simulations of the water intake of the Baoshan and Baoshan II Reservoirs could be achieved by coupling generated annual initial water levels with the corresponding operation rules (see Figure 5a). Figure 7 shows the 1000 simulations of the river runoff, annual initial water levels in the reservoirs, and the water treatment plant’s water intake.

Furthermore, according to Figure 8, since the planned irrigation water demands for the rice within the Zhudong irrigation zone vary noticeably with the space (irrigation canal) and time (10-day), the simulations should be achieved using the MMCS method with the first two statistical moments and the correlation coefficients of the various 10-day periods. These can be multiplied by the weights of the estimated planned irrigation water demands (see Figure 2b) to achieve 1000 simulations of the canal-based planned irrigation water demands.

Figure 9 illustrates the 1st, 250th, 500th, 750th, and 1000th simulation cases of the planned irrigation water demands at the 2nd, 4th, 8th, and 14th irrigation canals.

Nevertheless, the irrigation factors, i.e., the planned irrigation water demand and the maximum water intake of the Shanping Weir and canal-based gates, should be artificially determined without the observations; thus, the resulting statistical properties can be assigned according to historical operating data as listed in

Table 2. Summary of hypothetical statistics of the irrigation uncertainty factor.

Irrigation Factor	Statistical Properties
	Mean	Standard Deviation
Maximum water intake at the Shanping Weir (m3/s)	1.5	1.0
Maximum water intake of canal-based gate (m3/s)	0.2–0.4	0.07–0.98

, and the corresponding 1000 simulations can be then accomplished via the MMCS method, with normal distribution, as shown in Figure 10.

4.2. Simulation of Canal-Based Irrigation Water Supply

After generating the 1000 uncertainty factors, the estimations of the 10-day inflow at the center of the Zhudong irrigation zone through the Shanping Weir carried by the Shanping and Youluo Rivers, based on the associated simulations of the maximum water intake (see Figure 10), the resulting water supplies of 15 irrigation canals from the RIBASIM model can be then accomplished. Figure 11 and Figure 12 illustrate the simulations of the inflow at the Shanping Weir and resulting simulated water supplies at the 2nd, 4th, 8th, and 14th canals for the 1st, 250th, 500th, 750th, and 1000th simulation cases, respectively. It was shown that when the high inflow of over 1.5 m³/s at the Shanping Weir was provided, the irrigation canals carried irrigation water. For example, in the 500th simulation case, with the inflow of 0.5 m³/s, only the first canal can obtain the irrigation water of about 0.025 m³/s; most of the inflow may be carried by the Baoshan Reservoir and Yuandon water treatment plant. Accordingly, the simulations of the corresponding water intakes of the Baoshan and Baoshan II Reservoirs are then obtained (see Figure 13) in response to the impact on the irrigation water allocation in the Zhudong irrigation zone. As a result, the above reservoirs’ water intake, instead of the annual initial water levels in the reservoirs, are defined as the hydrological factor.

4.3. Development of the Proposed RA_IWS_Canal Model

As mentioned earlier, when developing the proposed RA_IWS_Canal model, the exceedance probabilities of the canal-based irrigation water supplies (i.e., reliability) were quantified via the AFOSM using 1000 irrigation water allocation simulation cases in advance. These were utilized in the establishment of the exceedance probability calculation equation. The detailed model development is addressed below:

4.3.1. Derivation of the Relationship of Water Supply with Uncertainty Factors

Within the framework of developing the proposed model introduced in Section 4.2, it was required that a functional relationship between the canal-based irrigation water supply and the uncertainty factors was derived in advance. As mentioned in Section 4.1, the uncertainty factors involve the hydrological factors, i.e., the runoff at the Shanping Weir and water intakes of the Baoshan Reservoir and Yuandon water treatment plant. In contrast, the planned irrigation water demands and the maximum water intakes of the water gate at all canals are regarded as the irrigation factors. Since the irrigation water is allocated from upstream to downstream, the irrigation water carried at a specific canal might be influenced by its upstream branches; thus, the total water supplies obtained by the previous canals should be defined as the uncertainty factors in this study.

Consequently, this relationship of the water supply for various 10-day periods at the specific irrigation canal could be derived from the above hydrological and irrigation factors as follows:

$$\begin{gathered} W{S}_{IC}^{N_{10-day}}=\alpha +{\beta }_{WD}\left(Q_{WD}^{}\right)+{\beta }_{SW}\left(Q_{SW}^{N_{10-day}}\right)+{\beta }_{BR}\left(Q_{BR}^{N_{10-day}}\right)+ \\ {\beta }_{BRII}\left(Q_{BRII}^{N_{10-day}}\right)+{\beta }_{YW}\left(Q_{YW}^{N_{10-day}}\right)+{\beta }_{Qmax,g}\left(Q_{max,g}^{}\right)+ \\ {{\displaystyle \sum }}_{i=1}^{IC-1}{\beta }_{WS}^i\left(W{S}_i^{N_{10-day}}\right) \end{gathered}$$

(12)

where IC and N_10-day indicate the irrigation branch and 10-day period, respectively; $\alpha$ is the intercept; ${\beta }_{WD},{\beta }_{SW},{\beta }_{BR},{\beta }_{BRII},{\beta }_{YW},{\beta }_{Qmax,g},and{\beta }_{WS}^i$ are the regression coefficients of the uncertainty factors; $W{S}_{IC}^{N_{10-day}}$ serves as the water supply at the ICth canal in the Nth 10-day period; $Q_{WD}^{}$ represents the planned irrigation water demand; $Q_{SW}^{N_{10-day}}$ represents the inflow at the Shanping Weir in the Nth 10-day period; $Q_{BR}^{N_{10-day}}$ and $Q_{BRII}^{N_{10-day}}$ are the water intake to the Baoshan and Baoshan II Reservoirs, respectively, in the Nth 10-day period; $Q_{YW}^{N_{10-day}}$ stands for the water intake to the Yuandon water treatment plant; $Q_{max,g}^{}$ accounts for the maximum water intake of the water gate; and $W{S}_i^{N_{10-day}}$ serves as the water supply at the ith canal in the Nth 10-day periods.

Table 3. Summary of the regression coefficients of the uncertainty factors at the 2nd, 4th, 8th, and 14th irrigation canals in the 24th 10-day period.

Indicator of Irrigation Canal (IC)	Regression Coefficients
	$\mathit{\alpha }$	${\mathit{\beta }}_{\mathit{W}\mathit{D}}$	${\mathit{\beta }}_{\mathit{B}\mathit{R}}$	${\mathit{\beta }}_{\mathit{B}\mathit{R}\mathit{I}\mathit{I}}$	${\mathit{\beta }}_{\mathit{S}\mathit{W}}$	${\mathit{\beta }}_{\mathit{Y}\mathit{W}}$	${\mathit{\beta }}_{\mathit{Q}\mathit{m}\mathit{a}\mathit{x},\mathit{g}}$	${\mathit{\beta }}_{\mathit{W}\mathit{S}}^{\mathit{i}}, \left(\mathit{i}=\mathit{I}\mathit{C}-1\right)$
IC = 2	−0.029	0.131	0.000074	0.00042	−0.0012	0.0524	−0.00036	0.01
IC = 4	−0.039	0.26686	−0.01517	−0.00046	0.016317	0.03947	0.000437	−0.0141, −0.0076, 0.01736,
IC = 8	−0.042	−0.106	−0.08933	0.00006	0.0959	−0.0655	0.00561	−0.096, −0.109, −0.063, 0.057, 0.04, 0.153, 0.104, 0.026
IC = 14	−0.03	−0.138	−0.1145	0.0016	0.12986	−0.14173	0.0338	−0.266, −0.159, −0.143, −0.254, −0.155, −0.161, −0.207, −0.147, −0.098, 0.0467, 0.045, 0.063, −0.029, 0.0466

illustrates the results from the determination of the regression coefficients of the uncertainty factors considered in the above relationship of water supply at the study spots, the 2nd, 4th, 8th, and 14th canals in the 24th 10-day period in which the number of regression coefficients of the water supply at the upstream canals are 1, 3, 7, and 14.

4.3.2. Reliability Quantification of Irrigation Water Supply

By using the advance first-order second-moment (AFOSM) method, with the relationship of water supply to the uncertainty factors established via Equation (10), the probability of the irrigation water exceeding the various water demands (i.e., reliability) can be quantified at the canals in 33rd 10-day periods within the Zhudong irrigation zone, as shown in Figure 14.

In reference to Figure 14, the exceedance probability declines with the irrigation water demand, and usually approaches zero in cases where the water demand is over 0.3 m³/s. This shows the exceedance probabilities of the irrigation water supplies of 0–0.6 m³/s at the 2nd canal from the 5th to 33rd 10-day periods, indicating that the exceedance probability curves have different varying trends with the 10-day periods. Specifically, the exceedance probabilities at the 3rd and 8th canals sharply decline with the irrigation water supply, with a higher variation compared to the remaining canals, due to the water intake of the Baoshan Reservoir and the Yuando water treatment plant.

Moreover, the 13th and 14th irrigation canals have relatively large irrigation areas (about 30–50 ha) located at the downstream nodes in the Zhudong irrigation zone. Accordingly, their carried water supplies might be impacted due to uncertainty in the upstream canals’ and hydraulic structures’ water intakes; therefore, their water supplies may exhibit great variation, leading to a high likelihood of a shortage risk. Thus, the corresponding irrigation water supply reliabilities, in terms of the exceedance probability (on average 0.15), gradually decrease with water supplies from 0 m³/s to 1.25 m³/s, especially in the drought season.

To summarize the above results from the reliability analysis via the AFOSM, the hydrological and irrigation factors considered in this study are shown to significantly impact the canal-based irrigation water supply. Therefore, 1000 simulations of irrigation water allocations and the resulting reliabilities could be employed to model development in response to the variations in the water intake of the 15 irrigation canals and hydraulic structures (Baoshan Reservoir and Yuandon water treatment plant) in the Zhudong irrigation zone.

4.3.3. Derivation of the Exceedance Probability Calculation Equation

Although the reliability of the canal-based 10-day irrigation water supply can be accomplished using the AFOSM, it may expend too much computation time to identify the failure points when calculating the exceedance probabilities [29]. In this study, to efficiently quantify the reliability of water supply at the canals in various 10-day periods, considering the uncertainty factors, we adopt logistic regression analysis to derive a functional relationship regarding the probability of canal-based irrigation water supply ($W_{IS}$) exceeding the water demand ($w_{ID}$) $Pr\left(W_{IS}>w_{ID}\right)$ with the uncertainty factors selected (called an exceedance probability calculation equation) (see Equation (9)).

According to the results from reliability quantification in Section 4.1 and Section 4.2, the hydrological factors, the inflow at the Shanping Weir $Q_{SW}^{N_{10-day}}$, the water intakes of the Baoshan Reservoir $Q_{BR}^{N_{10-day}}$ and Yuando water treatment plant $Q_{YW}^{N_{10-day}}$, and the irrigation factors, the maximum water intake of the canal-based water gate at the canal-based water gates $Q_{max,g}^{}$ make a greater difference to the irrigation water allocation in the Zhudong irrigation zone; thus, the above four uncertainty factors are defined as the independent variables in the logistic regression equations of the irrigation water supply reliabilities. In particular, based on the model framework as mentioned in Section 2.4, the probabilities of a canal’s 10-day water supply exceeding the specific water demands are calculated via the AFOSM, with the numerous combinations of the above four uncertainty factors and the 10 irrigation water demands (from 0.0001 m³/s to 0.15 m³/s) in the growth season (i.e., the fifth and thirty-third 10-day periods); subsequently, the corresponding logistic regression equation can be established as follows:

$$\begin{gathered} Ln\left(\frac{Pr\left(W_{IS}>w_{ID}\right)}{1-Pr\left(W_{IS}>w_{ID}\right)}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} ={\alpha }_0+{\alpha }_1\left(N_{10-day}\right)+{\alpha }_2\left(w_{ID}\right)+{\alpha }_3\left(Q_{max,g}\right)+{\alpha }_4\left(Q_{SW}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} + {\alpha }_5\left(Q_{BR}\right)+{\alpha }_6\left(Q_{YW}\right) \end{gathered}$$

(13)

where $Pr\left(W_{IS}>w_{ID}\right)$ denotes the probability of the water supply ($W_{IS}$) exceeding the specific water demand ($w_{ID}$) at the target canal; ${\alpha }_0,{\alpha }_1,\dots ,{\alpha }_6$ are the regression coefficients; $N_{10-day}$ denotes the desired 10-day period; $Q_{max,g}$ represents the maximum intake of the target water gate; $Q_{SW}$ serves as the inflow at the Shanping Weir; $Q_{BR}$ stands for the water intake of the Baoshan Reservoir; and $Q_{YW}$ is the water intake of the Yuandon water treatment plant.

Note that, because the first canal is the first intake canal, its water supply is only impacted by the inflow at the Shanping Weir, the planned irrigation water demand, and the gate-based maximum delivered water. Thus, in the case of the magnitudes of the above factors being provided, the corresponding irrigation water supply reliability can be identified by comparing the difference among the above factors. Therefore, its exceedance probability calculation equation is excluded in the proposed RA_IWS_Canal model. Furthermore, since the Yuandon water treatment plant is located between the second and third canals, its water intake ($Q_{YW}$) variation poorly affects the irrigation water supply of the second canal; accordingly, the coefficient of $Q_{YW}$ is excluded in the Equation (11) for the second canal. Similarly, the water intake of the Baoshan Reservoir, located between the seventh and eighth canals, hardly impacts the irrigation water supplies at the second–seventh canals; thus, the coefficients of $Q_{BR}$ in Equation (11) for the second–seventh canals can be assigned as zero.

Table 4. Summary of the uncertainty factors used in the exceedance probability calculation equations for the irrigation canals. Note: √ indicates that the uncertainty factor is accepted in the exceedance probability calculation equation.

No. of Canal	Uncertainty Factor Used
	${\mathit{w}}_{\mathit{I}\mathit{D}}$	${\mathit{Q}}_{\mathit{m}\mathit{a}\mathit{x},\mathit{g}\mathit{a}\mathit{t}\mathit{e}}$	${\mathit{Q}}_{\mathit{S}\mathit{W}}$	${\mathit{Q}}_{\mathit{B}\mathit{R}}$	${\mathit{Q}}_{\mathit{Y}\mathit{W}}$
2nd	√	√	√
3rd–7th	√	√	√		√
8th–14th	√	√	√	√	√

and

Table 5. Regression coefficients of the uncertainty factors in the exceedance probability calculation equations at the 2nd, 4th, 8th, and 14th canals.

No. of the 10-Day Period	Constant	Uncertainty Factor
		${\mathit{N}}_{10-\mathit{d}\mathit{a}\mathit{y}}$	${\mathit{w}}_{\mathit{I}\mathit{D}}$	${\mathit{Q}}_{\mathit{m}\mathit{a}\mathit{x},\mathit{g}\mathit{a}\mathit{t}\mathit{e}}$	${\mathit{Q}}_{\mathit{S}\mathit{W}}$	${\mathit{Q}}_{\mathit{B}\mathit{R}}$	${\mathit{Q}}_{\mathit{Y}\mathit{W}}$
	${\mathit{\alpha }}_0$	${\mathit{\alpha }}_1$	${\mathit{\alpha }}_2$	${\mathit{\alpha }}_3$	${\mathit{\alpha }}_4$	${\mathit{\alpha }}_5$	${\mathit{\alpha }}_6$
2nd	2.292	0.027	−24.349	−88.175	1.165	0	0
4th	−6.678	0.011	−367.805	211.523	4.397	0	6.776
8th	−0.05	−0.010	−63.142	34.096	1.177	−1.472	0.481
14th	−1.585	−0.008	−10.470	8.665	0.827	−0.482	1.786

summarize the uncertainty factors used in the exceedance probability calculation equations and illustrate the associated regression coefficients regarding the above uncertainty factors at the 2nd, 4th, 8th, and 14th canals.

Furthermore, by observing Equation (14), the exceedance probability calculation equation can also be improved to design the planned irrigation water demand under a specific exceedance probability (i.e., reliability) with the remaining factors provided as:

$$\begin{gathered} w_{ID}=\frac{1}{{\alpha }_4}\left\{Ln\left(\frac{Pr\left(W_{IS}>w_{ID}\right)}{1-Pr\left(W_{IS}>w_{ID}\right)}\right)-\left[{\alpha }_0+{\alpha }_1\left(N_{10-day}\right)+{\alpha }_3\left(Q_{max,gate}\right)+ \\ {\alpha }_4\left(Q_{SW}\right)+{\alpha }_5\left(Q_{BR}\right)+{\alpha }_6\left(Q_{YW}\right)\right]\right\} \end{gathered}$$

(14)

Instead, using Equation (14), the resulting planned irrigation water demand for the $N_{10-day}$ period at the specific canal can be decided with the desired reliability under conditions of the hydrological and irrigation factors provided.

Therefore, using the exceedance probability calculation equations adopted in the proposed RA_IWS_Canal model, the reliability analysis for the irrigation water supply in the 10-day periods can be accomplished in response to the resulting variation from the uncertainty factors, excluding those used in Equation (11). In addition to the reliability quantification, the canal-based planned irrigation water demands can be determined using the exceedance probability calculation of Equation (12) with a reliable planned irrigation water.

4.4. Model Application

To illustrate the application of the proposed RA_IWS_Canal model in the reliability assessment of the irrigation water supply in the multi-canal irrigation zone, the exceedance probabilities of the irrigation water supply are calculated and evaluated due to various conditions of the uncertainty factors regarding five study cases (see

Table 6. Conditions of uncertainty factors used in the model application.

Study Cases	No. of the 10-Day Period	Inflow at the Shanping Weir (m3/s)	Water Intake of the Baoshan Reservoir (m3/s)	Water Intake of the Yuandon Treatment Plant (m3/s)	Maximum Water Intake of Canal Gates (m3/s)	Irrigation Water Demand (m3/s)
I	18	0.5–2.4	1.1	0.3	0.09	0.035
II		2.4	0.0–1.8	0.3	0.09
III		2.4	1.1	0.0–0.4	0.09
VI		2.4	1.1	0.3	0.05–0.1
V		2.4	1.1	0.3	0.09	0.1–1.0

), and the corresponding results from the reliability quantification are shown Figure 14.

4.4.1. Variation in Inflow at the Shanping Weir

According to results from the study case (see Figure 15a), as the inflow at the Shanping Weir changes from 0.5 m³/s to 2.3 m³/s, the water supply exceedance probabilities at the canals sharply increase with the inflow and nearly exceed 0.8, excluding the 14th canal (about 0.4). Specifically, at the 13th canal, the resulting irrigation water supplies are at low risk with a high likelihood (about 0.6); otherwise, when providing more than 1.6 m³/s of irrigation water from the Shanping Weir, the water supply reliabilities at the 13th canal show a significant increase of 0.57. The above results reveal that the inflow at the Shanping Weir makes an apparent contribution to the irrigation water supply at all canals in the Zhudong canal basin, reducing the insufficient supply risk by approximately 0.2, especially at the canals downstream from the intake-related hydraulic structures (about 0.4).

4.4.2. Variation in Water Intake from the Hydraulic Structures

The results from Case II and III, shown in Figure 15b, demonstrate that the exceedance probability of the irrigation water supply at the canals in the Zhudong canal basin gradually declines by 0.04 with the water intake of the Baoshan Reservoir increasing from 0.0 m³/s to 1.8 m³/s. Specifically, the 8th and 14th canals have a more noticeable decrease (about 0.23) in the exceedance probabilities of the irrigation water supply than the remaining canals. This indicates that more water intake of the Baoshan Reservoir possibly induces a higher water shortage risk at its downstream canals. In particular, the water intake of the Baoshan Reservoir has a high likelihood of adversely impacting the irrigation water supply reliability at the canals.

Similar to the reliability assessment for the irrigation water supply due to the variation in the water intake of the Baoshan Reservoir, the effect of the variation in the Yuandon water treatment plant on the irrigation water supply can be evaluated by changing its water intake from 0 m³/s to 0.4 m³/s. However, Figure 15b reveals that, unlike the results from the Baoshan Reservoir, the exceedance probabilities of the irrigation water supply with varied water intake of the Yuandon water treatment plant are approximately constant, except at the 14th canal. This implies that the irrigation water supply at most of the canals in the Zhudong canal basin might be barely influenced by the variation in the water intake of the Yuandon water treatment plant. However, the water supply exceedance probabilities at the 14th canal generally increase from 0.48 to 0.67, indicating that enhancing the reliability of supplying irrigation water at the 14th canal could been achieved with more intake of the Yuandon water treatment plant. Accordingly, despite the water supply reliability of the 14th canal being considerably less than the remaining canals, attributed to its place at the bottom of the water allocation sequence in the Zhudong canal basin, it is highly likely that it can carry enough water from the Zhudong canal to satisfy the desired water intake of the Yuandon water treatment plant. Furthermore, the remaining canals could obtain the irrigation water with low insufficient risk as a result of the lesser impact of water intake of the Yuandon water treatment plant.

4.4.3. Variation in Maximum Water Intake of the Canal-Based Gate

Regarding Figure 15c, the reliability of the canal-based irrigation water supply has a significantly positive correlation with the maximum water intakes of the canal-based gates with an increasing exceedance probability. This is because the irrigation water supplies obtained at the canals are determined based on the remaining irrigation water in the central canal channel with the maximum intake of the gates and the planned water demand. In general, the irrigation water supply carried by the canal should be the lowest amount among the remaining inflow in the main channel, the gated-based maximum water intake, and the planned water demand. For example, the maximum water intakes of the gates at the 8th–14th downstream canals of the Baoshan Reservoir make a significant impact on the exceedance probability of irrigation water supply (from 0.1 to 0.8). However, although raising the maximum water intake of the gates can efficiently enhance the reliability of the irrigation water supply, its performance should be restrained to the associated planned water demands.

4.4.4. Variation in Planned Irrigation Water Demand

As mentioned in the above sections, the water supply is mainly constrained by the planned water demand. Thus, the effect of the variation in the planned irrigation demand should be evaluated according to the impact on water supply reliability. From Figure 15c, it can be seen that the second and third canals can carry an irrigation water supply of 0.06 m³/s with a reliability of over 0.9. Nevertheless, regarding the downstream canals from the Yuandon water treatment plant, their probability of exceeding the water supply fall from approximately 0.9 to 0.7 as a result of the water demand of less than 0.05 m³/s. Especially at the 14th canal at the end of the Zhudong canal basin, the probability of the irrigation water supply exceeding the low water demand of fewer than 0.08 m³/s reaches 0.6. It summarizes that the effect of variation in the planned irrigation water demand on the reliability of irrigation water supply is possibly attributed to the conditions of the inflow and outflow known in a crop growth area.

In addition, the proposed RA_IWS_Canal model can be employed to formulate the planned irrigation water demands under acceptable reliability via Equation (12). Figure 16 represents the planned water demands at 15 irrigation canals as a result of the specific hydrological and irrigation factors (see Table 6) under a reliability of 0.7, in which the planned water demand at the first canal is assigned as the gated-based maximum water intake (i.e., 0.09 m³/s), without the corresponding exceedance probability calculation equation. This can be seen in Figure 16a. Despite a high reliability of 0.7, the original estimated water demands at the first, second, third, and seventh canals exceed the maximum delivered water of 0.09 m³/s to the total irrigation water demand added into the water intakes of the Baoshan Reservoir (1.1 m³/s) and Yuando water treatment plant (0.3 m³/s), greater than the inflow at the Shanping Weir of 2.4 m³/s. Thus, when the irrigation water demands at the above canals are assigned as the maximum delivered water of 0.09 m³/s, as shown in Figure 16b, the water demands at the remaining canals can then be estimated. The corresponding total water carried, including the canal-based water demands and water intakes of the Baoshan Reservoir and Yuandon water treatment plant, is approximately 2.4 m³/s. The proposed RA_IWS_canal model could be applied to formulate the planned irrigation water demands by the probabilistic decision-making algorithm with the desired reliability.

4.4.5. Summary

According to the above results, using the proposed RA_IWS_Canal model, the reliability analysis of the irrigation water supply at the canals can be efficiently implemented with varying uncertainty factors. In addition, the uncertainty factor, the inflow at the Shanping Weir, and the maximum water intakes of the canal-based gates are proportional to the irrigation water supply reliability. Specifically, the inflow at the Shanping Weir is demonstrated to markedly facilitate, with high reliability (about 0.8), the irrigation water supply at all canals in the Zhudong irrigation zone).

The water intake of the Baoshan Reservoir possibly induces an insufficient risk of the irrigation water supply (approximately 0.23), especially for its downstream canals. In contrast with the Baoshan Reservoir, the Yuandon water treatment plant could be supplied water of about 0.4 m³/s from the main channel of the Zhudong irrigation zone, with a low likelihood (0.19) of influencing the canal-based irrigation water supply. Additionally, despite the high planned water demands at the canals being expected to obtain more irrigation water, they possibly lead to the high likelihood of low water, especially at the downstream canals (about 0.6); thus, regulating the appropriate and reasonable planned irrigation water demands for the canals plays an essential role in the allocation and management of the irrigation water system.

5. Conclusions

This study aims to propose a probabilistic reliability analysis model (named the RA_IWS_Canal model) for the irrigation water supply in the multi-canal irrigation zone, attributed to uncertainties in the hydrological and irrigation factors considered. The hydrological factors include the surface runoff in the rivers and the water intake of the hydraulic structures. The planned irrigation water demands, as well as the maximum water intakes of the canal-based gates, are regarded as the irrigation factors. In detail, the proposed RA_IWS_Canal model comprises two equations: the irrigation water supply estimation and exceedance probability equations. Of the above equations, the irrigation water supply estimation equation could be established using a significant number of irrigation water allocation simulations via the multivariate Monte Carlo simulation approach. It is utilized in the quantification of the water supply reliability (i.e., the exceedance probabilities) using the uncertainty method (advance first-order second-moment, AFOSM). Eventually, the exceedance probability calculation equation can be derived using logistic regression analysis with uncertainty factors considered, meaning that the irrigation water supply’s resulting reliability from the uncertainty factors’ variations can be quantified using the above equation in the case of the uncertainty factors provided.

To demonstrate the application of the RA_IWS_Canal model on the reliability of the irrigation water supply, 15 irrigation canals in the Zhudong canal basin in Northern Taiwan were selected as the study area. The uncertainty factors considered include the 10-day surface runoff in the Shanping and Youluo Rivers, the water intake of the Baoshan Reservoir and Yuandon water treatment plant, and the maximum water intakes of the gates and planned irrigation water demand at the 15 irrigation canals. Using the multivariate Monte Carlo simulation approach and the RIBASIM water allocation simulation model, 1000 simulations of irrigation water allocation, with the uncertainty factors, can be achieved for the development of the proposed model. The results from the model development and application indicate that the inflow at the Shanping Weir and the maximum water intakes of the canal-based gates are proportional to the irrigation water supply reliability. Specifically, the inflow at the Shanping Weir is demonstrated to markedly facilitate the irrigation water supply at all canals in the Zhudong irrigation zone, with high reliability, by 0.8. Additionally, the water intake of the hydraulic structures at Baoshan Reservoir exhibits a more significant impact on the reliability of the irrigation water supply at its downstream canals (about 0.77) than the Yuandon water treatment (approximately 0.19), revealing that the Baoshan Reservoir probably carries water to reduce the greater water supply reliabilities at the canals, in comparison to the Yuandon treatment plant. As a result, the proposed RA_IWS_Canal model is demonstrated to evaluate the effect of the variations, not only in the canal-based uncertainty factors (i.e., the planned irrigation water supply and maximum water intake of the canal-based gates), but also in the regional factors, including the water supplies and water intakes of the other canals and hydraulic structures, on the canal-based irrigation water supply reliability.

The resulting water allocation simulation RIBASIM model is configured without considering the water intake from other water resources (e.g., groundwater) and hydrological loss (e.g., infiltration and evapotranspiration) and delivery loss, which make a significant contribution to irrigation water supply [16,18,36,37], especially in the drought season. Thus, the above hydrological variables would be taken in account in the model development. Furthermore, although higher planned irrigation water demands are advantageous to growing crops, they may trigger a shortage risk. Therefore, formulating the appropriate planned water demands is necessary to accomplish irrigation water allocation with high reliability. Thus, if further work could be carried out coupling the probabilistic decision-making algorithm integrated with the proposed RA_IWS_canal model, the planned irrigation water demands at various canals within a multi-canal irrigation zone could be determined. In addition to the reliability quantification, since this study reproduces a significant number of irrigation water supplies at all canals under the generated uncertainty factors for the model development, the resulting irrigation water allocation simulations would be adopted for training an artificial intelligence (AI) model (e.g., machine learning and artificial neural network) for efficiently allocating the irrigation water with high reliability and accuracy within an irrigation system, instead of a complicated network–structure water allocation simulation modeling [38,39,40].