Research on Parameter Optimization of the Optimal Schedule Model of Water Resources for the Jiaodong Water Transfer Project Based on the ICCP Model

Abstract

Establishing an optimal operation model of water resources is a crucial mean to promote the social and economic development in the Jiaodong area, where water resources are seriously deficient. Constraints of the optimal operation model mainly include water balance constraint, discharge capacity constraint, and constraint on the full utilization of operating water. For water transfer projects that have been in operation for decades, the parameters of these constraints, such as Discharge Capacity (DC), Water Conveyance Efficiency (WCE), Evapotranspiration (E), and Water Supply Volume (WSV), have changed from their original design values, which in turn affect the results of the operation model. In order to address the uncertainties caused by corresponding parameters, according to the characteristics of each parameter, an Interval-Chance Constrained Programming (ICCP) model for the Jiaodong Water Transfer Project was proposed. Interval Programming (IP) and Chance-Constrained Programming (CCP) were used to optimize the parameters of constraints involving WCE, DC, E, and WSV. Then, the Sobol method was used to analyze the sensitivity of each parameter to the operating objective function. The results reveal that (1) total water shortage ratio decreased by [14.82%, 17.26%], [14.81%, 17.25%], and [14.82%, 17.26%], respectively, under the incoming water condition of 50%, 75%, and 95%, indicating that ICCP model can adequately consider complex uncertainties and effectively alleviate water shortage; (2) WCE and DC are important parameters for optimal operation model of water resources, and therefore, channels should be regularly maintained to ensure that WCE and DC would not reduce; (3) Decision variables in this study are in the form of intervals, which are more reasonable because they provide more decision-making options to managers.

1. Introduction

The shortage of water resources has increasingly become an important bottleneck restricting regional social and economic development. Inter-basin water operating is one of the most effective methods to alleviate the shortage of water resources. One typical example is the Jiaodong Water Transfer Project in China. While the Jiaodong region serves as the political and economic hub of Shandong Province, severe water shortage has posed a threat to the sustainable socioeconomic development of the region in recent decades. The Jiaodong Water Transfer Project plays an important role in alleviating the imbalance between water supply and demand. A lot of relevant studies on optimal operation of water resources have been carried out. Yang, J. L et al. [1] proposed an optimal operation model of water resource for the Jiaodong Water Transfer Project, which could effectively alleviate serious imbalance between water supply and demand in the Jiaodong area. Sun, J. H. [2] studied water resources optimal operation of the Jiaodong Water Transfer Project, and constructed a distributed water operating model and planning index operating model, which could greatly reduce the water shortage rate in water-receiving areas even in the case of emergency.

The parameters of the optimal operation model of water resources were expressed in the form of definite values in the above researches, without considering the uncertainties in the actual operation of the water transfer project. However, in actual water resources management and planning problems, uncertainties may present in terms of multiple formats and exist on multiple levels [3]. As for the current operating operation model of the Jiaodong Water Transfer Project, since it has been completed and operated for more than 30 years, due to sediment, grass in summer, freezing in winter, and other reasons, Discharge Capacity (DC) has changed gradually, which has deviated from its design value. The value of Water Conveyance Efficiency (WCE) fluctuates according to the change of flow volume. The value of Evapotranspiration (E) and Water Supply Volume (WSV) varies each month according to weather or water demand and other reasons. To address such problem, it is significant to consider uncertainties in the establishing model [4]. This way is superior to traditional optimization methods, where parameters are represented by a fixed value [5].

Consequently, a large number of inexact optimization programming methods were developed for coping with uncertainties in water resources management, including Interval Programming (IP) [6,7,8,9,10], Chance-Constrained Programming (CCP) [11,12,13], Stochastic Programming (SP) [14,15], Fuzzy Mathematical Programming (FMP) [16,17], and Fuzzy Chance Constrained Programming (FCCP) [18,19]. IP can solve the uncertainties caused by parameters with a known range. CCP can solve the uncertainties caused by parameters with known distribution. SP can solve the uncertainties caused by parameters as random variables. FMP and FCCP can solve the uncertainties caused by parameters with membership functions that are easy to obtain. The selection of programming methods should consider the characteristics of each parameter. In this research, those parameters include DC, WCE, E, and WSV, where the distribution of E and WSV can be obtained and WCE and DC can describe in the form of intervals. As a result, this study chooses IP to deal with the uncertainties caused by WCE and DC, and CCP to deal with the uncertainties caused by E and WSV.

The main purpose of this article is to consider the uncertainties caused by the deviation between the actual operating situation and design value of the project’s parameters after long-term operation of the Jiaodong Water Transfer Project. In order to meet the actual operating situation, it is imperative to establish an optimal operation model of water resources considering the complex uncertainties of parameters. According to the characteristics of each parameter, Interval-Chance Constrained Programming (ICCP) methods were chosen. Decision variables in this study are in the form of intervals which are more reasonable because they provide more decision-making options to managers. Research can provide a scientific basis for dealing with the uncertainties of other inter-basin water transfer projects.

2. Study Area

The Jiaodong Water Transfer Project is an important part of the backbone water network in Shandong Province. It consists of the Yellow River Diversion Project to Qingdao and the Yellow River Diversion Project in the Jiaodong area. Four main cities are involved in the project (Weifang, Qingdao, Yantai, and Weihai) and the location of the study area can be seen in Figure 1.

The Yellow River Diversion Project to Qingdao is a large-scale, inter-basin, and longdistance water diversion project in Shandong Province. It is a key project during the Seenth Five-Year Plan period. The project draws Yellow River water from the Dayuzhang of Boxing County, Binzhou City, through 4 cites Binzhou, Dongying, Weifang, Qingdao, and 10 counties (or cities, areas), to Qingdao Baisha Waterworks, with a total length of 290 km. The project passes through 36 rivers and has more than 250 km of open channels and more than 450 buildings, including a five-level pumping station and a one-level temporary pumping station, one large reservoir, and one desilting basin.

The Yellow River Diversion Project in the Jiaodong Area is a long-distance, inter-basin, inter-regional, and large-scale operation and allocation project of water resources invested by Shandong Province. The project draws water from the Dayuzhang of Boxing County, Binzhou City, through six cites, i.e., Binzhou, Dongying, Weifang, Qingdao, Yantai, Weihai, and 16 counties (or cities, areas), with a total length of 482 km. The project newly builds 160 km open channels, 150 km pipelines (culverts), a 7-level pumping station, 5 large tunnels, 6 large aqueducts, 19 siphons, and 467 bridges, sluices, and other buildings.

The Jiaodong Water Transfer Project implements a joint operation of the Yangtze River, the Yellow River, and local water sources, making it a very essential allocation project of water resources. By 2016, the Jiaodong Water Transfer Project has diverted 7.91 billion m³ and allocated 6.77 billion m³ of water, effectively alleviating water deficit in the Jiaodong area, in support for the social and economic development of the Jiaodong area, especially for the four cities. Therefore, this study chooses the four cities as the study area.

According to the daily operation demands, 22 main water-diversion outlets on main water transmission lines in the 4 cities are considered. The water supply network of the Jiaodong Water Transfer Project is shown in Figure 2 (adapted from Figure 2 in [20]). The information of each water outlet is shown in

Table 1. Information table of the water-diversion outlet.

City	Water-Diversion Outlet		Design Flow (m3/s)	County	Water Supply Area
	Number	Name
Weifang	1	Sluice of Shuangwangcheng reservoir	8.6	Shouguang	Shuangwangcheng reservoir
	2	Qingshui Lake	9		Shouguang North, Yangkou
	3	Guanzhuanggou front sluice (right)	2		Lianmenghuagong
	4	Xindanhe back sluice (right)	1.5		Longze reservoir
	5	Danhe front sluice (left)	2		Dadiyanhua
	6	Xifengan sluice	5	Hanting	Bailang River
	7	Yinhuangjihuaisluice (right)	20.5	Changyi	xiashan reservoir, Fangjiatun
	8	Huaihe beach sluice (right)	10		Changyi
	9	Jiaolaihe front sluice (right)	3	Gaomi	Gaomi
Qingdao	10	Qiaoxitougou sluice (right)	3	Jimo	Jimo
	11	Jihongtan Reservoir	23	Jihongtan	Jihongtan Reservoir
	12	Shuangyou reservoir	1.3	Pingdu	Shuangyou reservoir
	13	Huibu sluice	1.3		Huibu town
Yantai	14	Xiyang town sluice	1.65	Laizhou	Ningjia reservoir, Xiyang Village
	15	Xinjianwanghe sluice	1.65		JieXi reservoir, Wanghe
	16	Houjia reservoir sluice	1.53	Zhaoyuan	Zhaoyuan Development Zone
	17	Nanshan reservoir sluice	1.65	Longkou	Nanshan reservoir
	18	Nanluan reservoir sluice	1.65		Nanluan reservoir
	19	Wenshitang reservoir sluice	1.53	Penglai	Penglai urban area
	20	Menlou reservoir	5.5	Yantai urban area	Yantai urban area
	21	Mouping	5.5		Mouping urban area
Weihai	22	Meishan reservoir	4.8	Weihai	Meishan reservoir

Note: For the Yantai section, the starting point of 0 + 000 is the Songzhuang sluice; after the Huangshui River Pumping Station, the pile number will not be counted.

.

3. Methodology

3.1. Overall Method

The main purpose of this article is to consider the uncertainty caused by the deviation between the actual operating situation and design value of project’s parameters after long-term operation of the Jiaodong Water Transfer Project. Those parameters include DC, WCE, E, and WSV. Available data to calculate them include partial water allocation data and historical hydrology data of the Jihongtan reservoir. According to the available data, the distribution of E and WSV can be obtained. However, the measurement data of actual water allocation data of some water-diversion outlets in some months are incapable to obtain the distribution; only the fluctuation range of WCE can be obtained. DC fluctuates in a certain range. The upper bounds of DC are the design discharge capacity of channels, which can be achieved when the channels are well maintained. The lower limit should be determined according to the measured flow data. For the above analysis, this study chooses IP to deal with the uncertainties caused by WCE and DC, and CCP to deal with the uncertainties caused by E and WSV. Programming methods were chosen according to the characteristics of each parameter. Each method has a unique contribution in enhancing the accuracy of model results. Moreover, in order to calculate WS (water shortage) of county, this study chooses Principal Component Analysis (PCA) to find out the most influential factor and Analytic Hierarchy Process (AHP) method to calculate the weight of the main impact factor. The rest of the parameters, i.e., DF (design flow) and IGWD (indicator of guest water demand), are constant values. The schematic of the ICCP model is shown in Figure 3.

3.2. IP

IP is an important mathematical programming method to solve the uncertain information caused by parameters with a known range. $A^{\pm }=[a^{-},a^{+}]=\left\{x\right|a^{-}\le x\le a^{+},x\in R\}$ is called an interval number, when $a^{-}=a^{+}$, the interval number degenerates to a real number. The four operations of two interval numbers $A^{\pm }=[a^{-},a^{+}]$ and $B^{\pm }=[b^{-},b^{+}]$ are defined as:

$$\left\{\begin{array}{l}{A}^{\pm }+B^{\pm }=[a^{-}+b^{-},a^{-}+b^{+}]\\ A^{\pm }-B^{\pm }=[a^{-}-b^{+},a^{+}-b^{-}]\\ A\ast B=[\min \{a^{-}{b}^{-},a^{-}{b}^{+},a^{+}{b}^{-},a^{+}{b}^{+}\}, \max \{a^{-}{b}^{-},a^{-}{b}^{+},a^{+}{b}^{-},a^{+}{b}^{+}\}]\\ \frac{A}{B}=[\min \{\frac{a^{-}}{b^{-}},\frac{a^{-}}{b^{+}},\frac{a^{+}}{b^{-}},\frac{a^{+}}{b^{+}}\}, \max \{\frac{a^{-}}{b^{-}},\frac{a^{-}}{b^{+}},\frac{a^{+}}{b^{-}},\frac{a^{+}}{b^{+}}\}]\end{array}\right.$$

(1)

In many cases, it is advantageous to express the data in terms of center matrix $A_c=\frac{1}{2}(a^{-}+a^{+})$ and of the radius matrix $\mathsf{\Delta }=\frac{1}{2}(a^{+}-a^{-})$. Thus, A can be given either as $[a^{-},a^{+}]$ or as $[A_c-\mathsf{\Delta },A_c+\mathsf{\Delta }]$, and consequently, as $A=\left\{A\right||A-A_c|\le \mathsf{\Delta }\}$. Using the solution method mentioned in reference [21]: if $A^{\pm }=[a^{-},a^{+}]$, $B^{\pm }=[b^{-},b^{+}]$, $X^{\pm }=[x^{-},x^{+}]$. A and B are coefficient matrices. X is a decision variable. Then, the system AX ≤ B is strongly solvable if and only if the system $A^{+}{x}^{+}-A^{-}{x}^{-}\le b^{-}$ is feasible. $X=[x^{-},x^{+}]$ is a strong solution of AX ≤ B when X satisfies:

$$A_{c}X-B_c\le -\mathsf{\Delta }\left|x\right|-\delta$$

(2)

We can sum up these results in the form of a simple algorithm (Algorithm 1) [21].

Algorithm 1: IP

If $A^{+}{x}^{+}-A^{-}{x}^{-}\le b^{-}$ has a solution x−, x+ then set x = x1 − x2 and terminate: x is a strong solution of Ax ≤ b; else terminate Ax ≤ b is not strongly solvable; end

3.3. CCP

CCP is an important mathematical programming method to solve uncertain information caused by parameters with known distribution. When the constraints cannot be fully satisfied in some extreme cases with a low probability of occurrence, CCP can cope with such uncertainties by allowing decision variables not to fully satisfy the constraints to a predesigned extent to avoid the over-conservative optimization scheme. Therefore, the CCP method can make traditional rigid constraints flexible, so as to achieve a moderate compromise between the optimization of objective function and the satisfaction of constraints, and provide a systematic trade-off analysis for decision makers. By using the theorem method mentioned in reference [22], CCP can be linearized. If the constraint coefficient a_ij is a definite constant, b_i (i = 1, 2, …, m) is a random parameter of a known probability distribution (mostly normal distribution), and the coefficient c_j of the objective function Z is a constant or a random parameter independent of b, then the CCP model can be expressed as follows:

$$\max E(Z)={\displaystyle \sum _{j=1}^{n}E(c_j)x_j}$$

(3)

such that:

$$P({\displaystyle \sum _{j=1}^n{a}_{ij}{x}_j}\le b_i)\ge {\alpha }_i,i=1,2,\dots ,m, x_j\ge 0, j=1,2,\dots ,n$$

(4)

where α_i ∈ [0, 1] is the minimum probability that the constraint condition i is satisfied, usually close to 1, specifically 1 − α_i.

The above CCP can be transformed into an equivalent deterministic programming. Suppose that the distribution density function f(b_i) and probability distribution function F(b_i) of random parameter b_i are as shown in the Figure 4. The corresponding b_i of probability distribution function and cumulative probability F(b_i) = 1 − α_i is $b_i^{(1-\alpha )}$, and thus:

$$P(b_i\ge b_i^{(1-{\alpha }_i)})={\alpha }_i$$

(5)

The equivalent deterministic constraint of (4) is:

$${\displaystyle \sum _{j=1}^n{a}_{ij}{x}_j}\le b_i^{(1-{\alpha }_i)},i=1,2,\dots ,m$$

(6)

The right end term of the constraint condition is $b_i^{(1-\alpha )}$, which is determined according to the probability distribution function of b_i. For example, b_i conforms to the normal distribution N(μ, σ²), and the random variable z^(1−α_i) corresponding to P = 1 − α_i can be found through the standard normal distribution table, and then:

$$b_i^{(1-{\alpha }_i)}=\sigma z^{(1-{\alpha }_i)}+\mu$$

(7)

If the chance constraint is $P({\displaystyle \sum _{j=1}^n{\alpha }_{ij}{x}_j\ge b_i})\ge {\alpha }_i,i=1,2,\dots ,m$, the equivalent deterministic constraint is:

$${\displaystyle \sum _{j=1}^n{a}_{ij}{x}_j\ge b_i^{(a_i)}},i=1,2,\dots ,m$$

(8)

where the right endpoint $b_i^{({\alpha }_i)}$ is the corresponding variable value of probability distribution function of b_i and cumulative probability and F(b_i) = α_i.

The derivation of this theorem can also be found in [23].

3.4. PCA and AHP

Given that the WS is affected by many factors, this study chooses PCA to find out the most influential factor and AHP method to calculate the weight of the main impact factor. The AHP method was developed by Satty [24], and a detailed explanation of the method can be seen in Hadad and Henny [25]. The principle and calculation steps of PCA and AHP are shown in reference [26] and reference [27].

3.5. Sobol Method

The Sobol method was used to analyze the sensitivity of each parameter to the objective function. The principle and calculation steps of the Sobol method are shown in the reference [28].

4. Optimal Operation of Water Resources

4.1. Objective and Mission

(1)

Operation objective: In order to minimize the total amount of water shortage and alleviate the imbalance between water supply and demand in each city, optimal scheduling of water resources is carried out in the Jiaodong area.

(2)

Operation mission: To transfer water from different sources to each water-diversion outlet according to different water inflow conditions and engineering conditions.

4.2. Establishment of ICCP Model

(1)

Operation period: Due to the actual condition in the Jiaodong area, an annual schedule is selected. The scheduling period is from October of the current year to September of the next year.

(2)

Operation timestep: Taking water conveyance duration from canal head to each water-diversion outlet and operation over years into comprehensively consideration, the operation timestep is determined as a month, which is denoted by t (t = 10, 11, 12, 1, 2, …, 9).

(3)

Decision-making variable: Decision-making variable x_t,i refers to the monthly operation water volume of water-diversion outlet i in month t of the four cities, where water-diversion outlets are denoted by i ($i=1,2,\dots ,22$).

(4)

Objective function:

$$\min z^{\pm }={\displaystyle \sum _{t=10}^9{\displaystyle \sum _{k=1}^{13}{D}_{nt,k}-{\displaystyle \sum _{t=10}^9{\displaystyle \sum _{i=1}^{22}{x}_{t,i}{}^{\pm }}}}}$$

(9)

where $z$ is the total water deficit of each water-diversion outlet during the operation period, $D_{nt,i}$ is water demand of each water-diversion county in month t, and water-diversion counties are denoted by k (k = 1, 2, …, 13).

(5)

Constraints:

(1)

Water balance constraint of channel sections:

$${w_{t+1,i}}^{\pm }=w_{t,i}{}^{\pm }\times a_{t,i}{}^{\pm }-x_{t,j}{}^{\pm }$$

(10)

where $w_{t,i}$, $w_{t,i+1}$, respectively, are the input water volume of channel section i and i + 1 in month t and $a_{t,i}$ is the WCE of the channel section i in month t.

(2)

The DC constraint of channel sections: the operation volume in each canal section cannot be greater than DC:

$$w_{t,i}{}^{\pm }\le W_{t,i}{}^{\pm }$$

(11)

where $W_{t,i}$ is the DC of channel section i in month t.

In order to facilitate comprehensive analysis and solution, combined with the water supply network, the water balance constraint of channel sections (10) and the DC constraint of channel sections (11) are combined and derived:

$$w_{t,i}^{\pm }=\left\{\begin{array}{l}\frac{{\displaystyle \sum _i^{11}[x_i\times {\displaystyle \prod _{i+1}^{11}{a}_i^{\pm }}]}}{{\displaystyle \prod _i^{11}{a}_i^{\pm }}}+{\displaystyle \sum _{n=12}^{22}\frac{x_n}{{\displaystyle \prod _i^8{a}_i^{\pm }}{\displaystyle \prod _{12}^n{a}_n^{\pm }}}},i=1,2,\dots ,8\\ {\displaystyle \sum _i^{11}\frac{x_i}{{\displaystyle \prod _i^{11}{a}_i^{\pm }}}},i=9,10,11\\ {\displaystyle \sum _i^{22}\frac{x_i}{{\displaystyle \prod _i^{22}{a}_i^{\pm }}}},i=12,13,\dots ,22\end{array}\right.\le W_{t,i}^{\pm }$$

(12)

(3)

DF constraint of water-diversion outlet:

$$x_{t,j}{}^{\pm }\le Q_i$$

(13)

where $Q_i$ is the design flow of the water-diversion outlet i.

(4)

MWD constraint of the water-diversion outlet:

$$x_{t,i}{}^{\pm }\ge D_{mt,i}{}^{\pm }$$

(14)

where $D_{mt,i}$ is the MWD of the water-diversion outlet i in month t.

(5)

Constraints of counties WS

The supply volume of each county cannot greater than the WS of each county:

$${\displaystyle \sum _{i=1}^5{x}_{t,i}{}^{\pm }}\le D_{dt,1}{x}_{t,6}{}^{\pm }\le D_{dt,2}{\displaystyle \sum _{i=7}^8{x}_{t,i}{}^{\pm }}\le D_{dt,3}{x}_{t,9}{}^{\pm }\le D_{dt,4}{x}_{t,10}{}^{\pm }\le D_{dt,5}{x}_{t,11}{}^{\pm }\le D_{dt,6}{\displaystyle \sum _{i=12}^{13}{x}_{t,i}{}^{\pm }}\le D_{dt,7}{\displaystyle \sum _{i=14}^{15}{x}_{t,i}{}^{\pm }}\le D_{dt,8}{x}_{t,16}{}^{\pm }\le D_{dt,9}{\displaystyle \sum _{i=17}^{18}{x}_{t,i}{}^{\pm }}\le D_{dt,10}{x}_{t,19}{}^{\pm }\le D_{dt,11}{\displaystyle \sum _{i=20}^{21}{x}_{t,i}{}^{\pm }}\le D_{dt,12}{x}_{t,22}{}^{\pm }\le D_{dt,13}$$

(15)

where D_dt,i is the WS of counties i in month t.

(6)

Constraints of external water transfer volume:

Total amount of diversion water from catchment to each water-diversion outlet cannot greater than the optimal water volume of the four cities:

$$\mathrm{Weifang}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \sum _{t=10}^9{\displaystyle \sum _{i=1}^9{D}_{t,i}{}^{\pm }}\le Q_{d1}}$$

(16)

$$\mathrm{Qingdao}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \sum _{t=10}^9{\displaystyle \sum _{i=10}^{13}{D}_{t,i}{}^{\pm }}\le Q_{d2}}$$

(17)

$$\mathrm{Yantai}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \sum _{t=10}^9{\displaystyle \sum _{i=14}^{21}{D}_{t,i}{}^{\pm }}\le Q_{d3}}$$

(18)

$$\mathrm{Weihai}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \sum _{t=10}^9{D}_{t,22}{}^{\pm }\le Q_{d4}}$$

(19)

where D_t,i is the water distribution volume from canal head to water-diversion outlet i in month t and Q_dj is IGWD (10⁴ m³).

(7)

Constraints on the full utilization of operating water

When the operating water reaches the end of the water transmission line, all the water in the channels will be supplied to the terminal water distribution outlet, so that the water can be fully utilized:

$$V_{Ji,t+1}{}^{\pm }=V_{Ji,t}{}^{\pm }+x_{11,t}{}^{\pm }+p_{Ji,t}-q_{Ji,t}-E_{Ji,t}$$

(20)

$$V_{Jid}\le V_{Ji,t}{}^{\pm }\le V_{Jip}$$

(21)

where V_Ji,t+1 is storage capacity of the Jihongtan reservoir in the t + 1 period, (10⁴ m³), V_Ji,t is storage capacity of the Jihongtan reservoir in the t period, (10⁴ m³), p_Ji,t is the rainfall of the Jihongtan reservoir in the t period, (10⁴ m³), q_Ji,t is the water consumption of the Jihongtan reservoir in the t period, (10⁴ m³), E_Ji,t is the E of the Jihongtan reservoir in the t period, (10⁴ m³), V_Jid is the dead storage capacity of Jihongtan reservoir, (10⁴ m³), and V_Jip is the profiting storage capacity of Jihongtan reservoir, (10⁴ m³).

Since E and WSV are variable with a known distribution, after introducing the theoretical knowledge of CCP, (21) and (22) can be optimized as:

$$V_{Ji,t+1}{}^{\pm }=V_{Ji,t}{}^{\pm }+x_{11,t}{}^{\pm }+p_{Ji,t}-q_{Ji,t(\zeta )}-E_{Ji,t(\zeta )}$$

(22)

$$\mathrm{Pr}\{V_{Jid}\le V_{Ji,t}{}^{\pm }\le V_{Jip}\}\ge \alpha$$

(23)

(8)

Non-negative constraints:

All parameters are non-negative.

4.3. Data Preparation of Model

This study selects 2025 as the planning year and 2017 as the base year. The relevant data sources are shown in

Table 2. Sources of data and information.

Retrieving Data.	Sources of Data
Calibrated flow and design flow of each canal section	Data Collection of Engineering and Economic Indicators of Yellow River Diversion Project in Shandong Province Compilation of Construction and Management Documents of Yellow River Diversion Project in Jiaodong Area,
Design flow of each water-diversion
Balance analysis of supply and demand in 2025	Medium and Long Term Planning for Comprehensive Utilization of Water Resources in Shandong Province
Planning indicator in 2015–2025	Medium and Long Term Planning for Comprehensive Utilization of Water Resources in Shandong Province
Water allocation in the four cities from 2012 to 2017	Shandong province bureau of Jiaodong water diversion
population, GDP, grain production, oil production, vegetable production, fruit production, meat production, milk production, gross industrial product, precipitation, and total water intake of each county, using	Shandong Statistical Yearbook, Weifang Statistical Yearbook, Qingdao Statistical Yearbook, Yantai Statistical Yearbook, Weihai Statistical Yearbook
daily operation data in 24 March 2016~19 February 2019	Shandong province bureau of Jiaodong water diversion
background information detailed in the introduction	Shandong province bureau of Jiaodong water diversion

. Among them, the water demand amount of the four cities in 2025 are from reference [29].

4.3.1. WS

In order to avoid most of the water allocated to the front cities and counties, the WS of the four cities should be calculated and subdivided as the WS of the 13 counties, which can be calculated by the following formula:

$$D_{dk}=D_{dc}\times ({\displaystyle \sum (w\mathrm{imp}\times (im{p}_k/im{p}_c))})$$

(24)

where D_dk is water deficit of the county k, D_dc is water deficit of the city c the county k affiliates to, w_imp is weight value of main impact factors, and imp_k is value of main impact factors of the city c the county k affiliates to.

Using data from Medium and Long Term Planning for Comprehensive Utilization of Water Resources in Shandong Province, the WS in the planning year 2025 of the four cities is shown in

Table 3. WS/10⁹ m³ of the four cities in 2025.

City	p = 50%	p = 75%	p = 95%
Weifang	8.816	10.128	10.788
Qingdao	9.871	9.883	9.889
Yantai	4.388	5.487	8.183
Weihai	2.214	3.789	5.231
Sum	25.289	29.287	34.091

.

Based on the PCA method, SPSS27 software was used to analyze the possible impact factors of WS (population, GDP, grain production, oil production, vegetable production, fruit production, meat production, milk production, gross industrial product, precipitation, and total water intake). The impact factors were selected through consultation with experts from the Jiaodong Water Transfer Bureau and also referred to [30,31] Figure 5 shows the rotated factor loading matrix.

According to the first principal component of the rotated factor loading matrix, population, industrial GDP, precipitation, and total water intake had the greatest impact on WS, so they were selected as the main impact factors of WS in this study. Based on the AHP method, SPSS27 software was used to assign weights to the four main impact factors. The hierarchy and weight of the WS of the country are shown in

Table 4. Hierarchy and weight of the WS of the country.

Target Layer	Criterion Layer	Weight	Scheme Layer	Weight	Weight to Target Layer (wimp)
WS	social system	1/3	population	1	1/3
	economic system	1/3	industrial GDP	1/3	1/9
			total water intake	2/3	2/9
	ecosystem	1/3	precipitation	1	1/3

. The weights of each layer were obtained through consultation with experts from the Jiaodong Water Transfer Bureau.

After inspection, the CR values of the constructed judgment matrix were all less than 0.1, meeting the consistency test, meaning the calculated weights have consistency. The results for the WS per county were calculated by Formula (24) and shown in

Table 5. The WS per county.

City	County	Water Deficit/109 m3
		p = 50%	p = 75%	p = 95%
Weifang	Shouguang	2.963	3.404	3.626
	Hanting	1.619	1.859	1.981
	Changyi	1.806	2.075	2.210
	Gaomi	2.428	2.790	2.972
Qingdao	Jimo	1.661	1.663	1.664
	Jihongtan	6.303	6.311	6.315
	Pingdu	1.907	1.910	1.911
Yantai	Laizhou	0.795	0.994	1.482
	Zhaoyuan	0.645	0.807	1.203
	Longkou	0.766	0.958	1.429
	Penglai	0.564	0.705	1.051
	Yantai urban	1.618	2.023	3.017
Weihai	Mishan	2.214	3.789	5.231

.

4.3.2. MWD

During optimal operation, the Minimum Water Demand (MWD) of each water-diversion outlet, namely, the low limit of water supply, must be guaranteed. In 909 sets of daily operation data of the Jiaodong Water Transfer Project (24 March 2016~19 February 2019), 57 sets of water allocation data whose daily total water allocation amount of each water-diversion outlet is larger than that of the catchment were filtered out. Then, 852 sets of water allocation data were integrated and analyzed. Finally, the MWD of each water-diversion outlet was obtained and shown in Figure 6.

4.3.3. DC

Normally, the DC of the channels cannot exceed the design flow, and the design value of DC can be reached when the channels are well maintained. However, the Jiaodong Water Transfer Project was completed 30 years ago; the value of DC has decreased gradually, and has deviated from the design situation. In this research, the upper bounds of DC are the design discharge capacity of the channels, and the lower bounds are determined according to the minimum value of the measured flow data. The results are shown in Figure 7.

4.3.4. WCE

WCE is the proportion of canal tail flow to canal head flow. Since total flow of external diversion guest water is different in different seasons, the WCE of each channel section of the Jiaodong Water Transfer Project was calculated under different diversion flow (catchment flow) by using the selected 852 sets of water allocation data. Since all canal sections after section 19 were pressurized pipelines, their WCE was set to 1. The WCE of different channel-sections in each month was obtained and shown in Figure 8.

Part of the historical data of E, WSV, and rainfall of the Jihongtan reservoir in 2016–2018 were monitored (see the Abbreviations). In order to deal with the uncertainty caused by this kind of data, the normal distribution of E, WSV, and rainfall of the Jihongtan reservoir was tested. The results are shown in

Table 6. Normal distribution test of parameters.

Parameter	Bilateral Significance Value of k-s Test	Normal Distribution	Mean Value	Variance
Rainfall	0	Not conform	/	/
E/mm	0.2	conform	92.5	31.3
WCE/105 m3	0.2	conform	1765.4	398.6

.

Since rainfall data do not conform to normal distribution, monthly average rainfall data were used to calculate. After linearizing Formulas (22) and (23), Formula (2) can be used to solve the model.

5. Results and Discussion

5.1. Optimal Solutions of Water Operation

ICCP model was programmed by Python language under the incoming water condition of 50%, 75%, and 95%. Figure 9 and Figure 10 show the change of WS and Water Shortage Rate (WSR) after operating with the ICCP model.

The solutions presented as interval numbers indicate that the related decisions are sensitive to the change of input parameters. As we can see from Figure 9, with incoming water conditions ranging from 50% to 95%, the operating results remain unchanged. This can be caused by the fact that despite the WS being different under different incoming water conditions, the planning indicator is constant under different incoming water conditions. Namely, the water supply is certain and always much less than the water demand. Therefore, even after fully operating, the results still cannot fully meet the water demand of water receiving areas under various incoming water conditions. In this study, since the water requirements are decided according to the planning indicator rather than WS, the WS does not affect the model results. However, the calculation method of WS proposed in this research can be applied to other water diversion projects where the WS influences the model results.

Figure 10 presents the change of the WSR of the four cities. After operating, the WSR decreased by [14.82%, 17.26%], [14.81%, 17.25%], and [14.82%, 17.26%], respectively, under the incoming water condition of 50%, 75%, and 95%. During the 13th Five Year Plan period, the external water diversion indicator of the four cities are 4.07 × 10⁸ m³, 3.63 × 10⁸ m³, 2.335 × 10⁸ m³, and 2.02 × 10⁸ m³, respectively. The external water operation volume of the four cities are [3.6, 4.1], [2.8, 3.2], [1.9, 2.3], and [0.8, 0.9] (10⁸ m³), respectively, indicating that the operating efficiency is quite high.

5.2. Discussion

5.2.1. Competitive Analysis

(1)

Competitive analysis of the water operating volume

The results of the water operation volume in this study are compared with those of previous studies [2]. The comparison chart of water dispatching results under different planning years and different incoming water conditions is shown in Figure 11.

As generally known, under the actual operation conditions, due to the influence of various uncertainties, the water operation volume of each water-diversion outlet in each month cannot be a certain value. Figure 11 shows that the water volume amount results calculated by the previous model had the same certainty under all incoming water conditions, while the results calculated by ICCP model are interval values, which are much more in line with the actual operation situation. The uncertainty of various parameters has been fully considered and presented in interval form, making it an effective operating model for the Jiaodong Water Transfer Project.

(2)

Competitive analysis of parameter optimization

As for WCE, Sun, J. H. [2] analyzed the daily operation data of the project, combining the Davis Wilson formula and Cosgakov formula, and obtained an empirical formula to calculate WCE applicable to the project. It was more reasonable compared with the reference [1], where results were obtained directly by calculating the ratio of discharge flow at the canal tail to canal head. However, the WCE obtained by both methods was a constant value. In practice, the WCE fluctuates with the influence of discharge flow, roughness, etc. In this study, the upper and lower bounds of the WCE were calculated and shifted according to measured daily operation data, which was more in line with the actual situation. As for DC, Sun, J. H. [2] calculated its value according to the design condition. However, with the decade-long operation of the Jiaodong Water Transfer Project, the DC of the project has decreased gradually, and has deviated from the design situation. This study used the design flow and measured data to calculate its upper and lower bounds, which was more in line with the actual situation. As for E and WSV, there were three parameters with known history values for the Jihongtan reservoir. CCP was introduced to deal with the uncertainty caused by two parameters satisfying normal distribution: E and WSV. Compared with the method in reference [2], where all parameters were calculated according to the multi-year mean value, this method could better deal with complex uncertainty problems. As for WS, Yang, J. L. et al. [1] only considered the WS of the four cities, while Sun, J. H. [2] divided the WS of the cities into counties according to the proportion of the GDP. This study used the PCA method to determine the main impact factors of WS in this study, then used the AHP method to determine the weight of each impact factor, and finally used Formula (24) to calculate the WS of each county, which was a more reasonable approach.

5.2.2. Sensitivity Analysis

The Sobol method was used to analyze the sensitivity of each parameter to the objective function. DC and WCE are randomly sampled according to their intervals, while E and WSV are sampled according to their distribution functions. The number of samples is set to 300. The results of the sensitivity analysis of the uncertainty parameters of the ICCP model with the objective function are shown in Figure 12.

As shown in Figure 12, WCE and DC have the greatest impact on objective function, as well as WSV and E have a moderate impact. The results indicate that WCE and DC are important parameters for the model results, and their uncertainty should be delicately considered during the establishment of optimal operation model of water resources. The reason why WSV and E have a moderate effect on the objective function may be that WSV and E only affect the water operation volume of the eleventh water-diversion outlet, so it has little effect on the total amount.

5.2.3. Future Implication

Parameters of the water resources optimal operation model constantly contain complex uncertainties, which offers many opportunities for the application of the ICCP model. The ICCP model can not only reveal the uncertainty of WCE, DC, E, and WSV, but also deal with the uncertainty of other parameters with a known range or known distribution. The data used in this study are limited to the Jiaodong Water Transfer Project in China, but it could be expected that the model would also work in different scenarios around the world.

5.2.4. Limitations of the Implications

It is worth mentioning that the constraint value of the external water transfer was based on the planning indicators in the “Medium and Long Term Plan for Comprehensive Utilization of Water Resources in Shandong Province”, which optimized the allocation of external water transfer in the four cities of Jiaodong. In the future, with the improvement of water transfer data, further research on water transfer prediction needs to be carried out to improve the prediction accuracy of external water transfer. In addition, the constraints and parameters of the operating model were determined based on the existing engineering conditions. With the social and economic development of the four cities in Jiaodong, various water resource projects, such as water-saving engineering, water source development and utilization engineering, and water ecological protection engineering, have begun to be constructed. In the future, it is necessary to adjust the constraints and parameters in the operating model according to the conditions of the new construction project, and conduct new research on water resource optimization operating.

6. Conclusions

According to the characteristics of optimal operating model parameters, this study chooses different methods to optimize the parameters. A real case study of optimal operation of water resources has been provided for demonstrating the application of the developed model. Finally, the following conclusions are drawn.

The unique contribution of this article is to consider the uncertainty of the Jiaodong Water Transfer Project. The relevant parameters of the Jiaodong Water Transfer Project have deviated from the design value after long-term operation, resulting in uncertainties of DC, WCE, and WSV of the water transfer project, which in turn affect the results of the water diversion model. In order to better solve the uncertainties of parameters, this paper proposed the ICCP model. Its output is also in the form of interval, which is more in line with the actual situation and can also provide more decision-making options to managers.

The results of the sensitivity analysis show that WCE and DC are important parameters for the output results of the optimal operation model of water resources, indicating channels should be regularly maintained to ensure that WCE and DC would not reduce due to sediment, grass in summer, freezing in winter, and other reasons.