# An Optimal Allocation Model for Large Complex Water Resources System Considering Water supply and Ecological Needs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Construction for Water Resources Allocation

#### 2.1. Water Supply Systems of Reservoirs

#### 2.2. Objective Function

_{1}and F

_{2}, respectively, represent the water supply and riverine ecological objectives. The subscript j and k represent water users and sub-areas, respectively. Subscript y and m in Equation (2) represent the year and month numbers, respectively. TWS and TWD are the total water supply and water demand, respectively, and TWDR is the total water deficit ratio. Q and QN are the actual reservoir release and natural flow, respectively. $\overline{Q}$ is the average annual natural flow. The parameter AAPFD is used to reflect the deviation between ecological reservoir release and the natural flow [20,41]. Gehrke [42] argued that annual proportional flow deviation (APFD) was related to the diversity of fish species, but the amended method, AAPFD, is proved to be better to be applied in this study using Equation (2). Actually, AAPFD can quantify the degree of deviation of the natural flow, and it is sensitive to modification of natural flow. Ladson [41] also found that the condition of the river ecosystem is negatively correlated with the value of AAPFD. The smaller the value of AAPFD, the smaller the alteration of the natural flow regime. Greater values of AAPFD signify more serious cases of hydrological alteration. When AAPFD is greater than five, the river ecosystem will be seriously damaged. It should be stressed that the objectives F

_{1}and F

_{2}were normalized and transformed into non-dimensioned variables ranging from 0 to 1.

#### 2.3. Constraints

_{min}and V

_{max}are the dead storage and useful storage of the reservoir, respectively. V

_{t}is the current storage of the reservoir.

_{t}is the basic ecological demand of the downstream reservoir, ${\xi}_{t}$ is the minimum proportion of the ecological demand in the time period t.

_{nj}is the j

^{th}sector’s total water supply for the n

^{th}water supply project.

_{kt}is the total water coming into the sub-area(s). WR is the water recession of the k

^{th}sub-area. ${W}_{kt}^{0}$ and ${W}_{kt}^{1}$ are the initial and terminal total available water quantity.

^{th}and k

^{th}sub-areas. Ω is the sum of the direct upper reaches of the subsystem k.

## 3. System Analysis Technique for Finding the Optimal Solution

#### 3.1. Decomposition-Coordination (DC) Method

#### 3.1.1. System Decomposition

_{kt}and μ

_{kt}are the Lagrange multipliers, which are also called coordinative variables. Equation (12) is the expression of the Lagrange function. Meanwhile, Equation (12) can be separated as follows to make the function additive:

_{1,kt}is the water supply objective and ${F}_{1}=\frac{1}{T}{\displaystyle \sum _{k=2}^{K}{\displaystyle \sum _{t=1}^{T}{F}_{1,kt}}}$. F

_{2,t}is the ecological objective and ${F}_{2}=\frac{1}{T}{\displaystyle \sum _{t=1}^{T}{F}_{2,t}}$. The water balance equation is satisfied in this model; thus, the last two items of Equation (13) are equal to 0. Then, Equation (13) is defined as being the total system model Lagrange function, and can be regarded as a summary of all subsystems. In absolute terms, the equation is also the sum of the expression of the objectives for each subsystem. In view of this, the problem of a large complex problem can be decomposed into several sub-problems, expressed as follows:

#### 3.1.2. System Coordination

_{kt}and μ

_{kt}to obtain the overall optimal solution. The decomposition and coordination processes are closely related. According to the dual theory, if we consider the Lagrange multiplier as variables, Equation (12) can be translated into a dual function, the maximum value of which can be determined, formulated as follows:

^{th}subsystem is regarded as the inflow of the k

^{th}subsystem, as well as the new correlation estimate value and constraints. At this level, WR

_{kt}and I

_{kt}are feedback variables, and they are optimized in each subsystem and used to respond to the coordination level. The following condition should also be satisfied if the Lagrange function achieves its maximum.

#### 3.2. Subsystem Solving

_{0,t}and W

_{0,t}.

_{t}(V

_{t}) and ${f}_{t}^{*}\left({W}_{t}\right)$ are the minimum values of an objective function of subsystem 1 and subsystem k from the beginning of the t

^{th}time step to the T

^{th}time step, which can indirectly reflect the ecological and water deficit ratio. f

_{t+}

_{1}(V

_{t+}

_{1}) and ${f}_{t+1}^{*}\left({W}_{t+1}\right)$ are the minimum values of an objective function of subsystem 1 and subsystem k for the rest period (from (t+1)

^{th}time step to T

^{th}time step). g

_{t}(V

_{t},Q

_{t}) is the minimum ecological water deficit ratio of stage t with the decision variables Q

_{t}at the state V

_{t}. ${g}_{t}^{*}\left({W}_{t},TW{S}_{t}\right)$ is the minimum water deficit ratio of stage t with the decision variable TWS

_{t}at state W

_{t}.

#### 3.3. Whole Solving Procedures

_{1t}) and the total water supply of each sub-area (M

_{kt}). Reservoir ecological release can be calculated using the Tennant method, and total water supply can be calculated based on the water balance equation. In this step, the initial allocation scheme is generated.

^{th}subsystem. The optimization process is within the given width of the corridor by normal DP using the inverse sequence method to obtain the new allocation scheme and narrow the corridor. If the error-adjacent iteration of the allocation is less than ε, then go to the next step; otherwise, repeat this step, and set m = m + 1.

## 4. Case Study

#### 4.1. A Brief Description of the Study Area

^{8}m

^{3}, will be constructed in the short-term year to supply water for these towns. The topological structure of the reservoir and the towns is demonstrated in Figure 4b. The Xihe reservoir will supply water for six towns, either by riverbed or channel. The water system in Lingui Town is different from that in the other towns, which are manually interconnected. For the towns upstream of the reservoir and Lingui Town, water is supplied by channel from the reservoir.

#### 4.2. Data and Methodology

#### 4.3. Results and Discussion

#### 4.3.1. Analysis of the Optimal Allocation Result

_{2}is provided in Table 3. We find that the value of F

_{2}in 2030 is higher than that in 2020 under the same water conditions, apart from particularly dry years, which only suggests the total amount of reservoir release is decreasing, and the trend in years with different levels stays the same (Figure 6 and Table 2 and Table 3). This reduction occurred due to the sharp increase in economic index and water demand in the study area (Table 1), and the water release of the reservoir can be reduced as appropriate to supply more water for maintaining the economic development on the condition that the riverine environment is protected. This means that the optimization result for reservoir release can be flexibly adjusted based on both socio-economic water demands and internal water conditions, and the system model construction is applicable for reservoir operation and management, which is beneficial to sustainable development.

#### 4.3.2. Trend and Sensitivity Analysis

_{1}and F

_{2}. F

_{1}= 0 means that water demand can be fully satisfied for each sub-area in each time period, and F

_{2}= 0 implies there is no water alteration of natural flow. Figure 8a,b show the effects of the change of average reservoir release on the value of Lagrange function (L), the water supply objective (F

_{1}) and the ecological objective (F

_{2}) in both scenarios, respectively. There obviously exists a balance point between these two variables, because F

_{1}and F

_{2}have an opposite trend. It can also be observed that the value of the x-coordinate corresponds to the minimum value of the Lagrange function and is exactly the critical point where the value of F

_{1}begins to increase. The critical point and the minimum Lagrange value are on the optimal line that is vertical to the x-axis. This also reveals that the point is exactly the biggest value of the average actual reservoir release, and also the smallest value of F

_{2}under the condition of minimum water deficit (F

_{1}) when L achieved the optimal value. This phenomenon implies that it is just the balance point, and that the optimal result of the water allocation scheme is when L reaches its minimum.

_{1}was much more drastic than F

_{2}when actual reservoir release was greater than the optimal line, meaning that the water supply objective was much more sensitive to the whole system model than the ecological objective. In addition, the high sensitivity of the water supply element is also manifested by the effects on guaranteed water supply for different water users (see Figure 8c,d). Water supply guarantee rate, especially that for industrial and agricultural users, declines dramatically as F

_{1}increases slightly because of their secondary priority. Moreover, the different sensitivities were also reflected in the changing rate of the Lagrange function. When the reservoir release exceeds the optimal line, it means the extra release to protect the riverine environment is at the expense of a reduction in the water supply, and the more sensitive water supply element also made L increase dramatically. Conversely, reservoir release that is less than the optimal line has no effect on the water supply because it only influences the ecological objective, and the changing trend of L is much smoother. Different water demand levels could also influence the value of L and F

_{1}, as well as the average reservoir release. The total water demand is increasing as the economy develops, so the layout of water allocation will also be adjusted. Decision-makers should satisfy the socio-economic water demand due to its higher sensitivity, and ensuring water supply only slightly decreases reservoir release and decreases the guarantee rate of ecological water demand.

#### 4.3.3. Comparisons with Traditional Allocation Schemes

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

TWD | Total water demand |

TWS | Total water supply |

TWDR | Total water deficit ratio |

Q | Actual reservoir release |

QN | Natural streamflow |

V | Actual reservoir storage |

W | Water quantity of sub-area |

WS | Water supply only from reservoir |

WEVP | Surface evaporation of reservoir |

QB | Basic ecological flow |

WR | Water recession |

WIF | Intermediate flow |

## Appendix A

Parameters | Corresponding Meanings |
---|---|

I_{akt} | Individual input of subsystem k at t^{th} time step |

O_{akt} | Individual output of subsystem k at t^{th} time step |

I_{bkt} | Intermidiate input of subsystem k at t^{th} time step |

O_{bkt} | Intermidiate output of subsystem k at t^{th} time step |

M_{kt} | Decision variable of subsystem k at t^{th} time step |

Subsystem | Parameter for System Modeling | The Corresponding Parameter for Modeling of Water Resources | The Corresponding Meaning of Parameters |
---|---|---|---|

Subsystem 1 | I_{a}_{1t} | QN | The natural flow income of reservoir |

O_{a}_{1t} | AAPFD | The extent to which actual release to natural flow, the manifestation of the objective function expressed by F_{2} | |

O_{b}_{1t} | $\sum _{j=1}^{J}}{\displaystyle \sum _{k=2}^{K}}{WS}_{jkt$ | Water supply from the reservoir to each sub-area | |

M_{1t} | Q | The actual water release of reservoir | |

Subsystem k | I_{bkt}^{(1)} | $\sum _{j=1}^{J}}{WS}_{jkt$ | Water supply from the reservoir to k^{th} sub-area |

I_{bkt}^{(2)} | WI_{kt} | Water income from upstream sub-area(s), equal to the summary of the water recession (WR) of upstream sub-area(s) | |

I_{akt} | WIF_{kt} | Intermediate water flow between the (k − 1)^{th} and k^{th} sub-area | |

O_{akt} | TWDR_{k} | The total water deficit ratio for k^{th} subsystem, the manifestation of the objective function expressed by F_{1} | |

O_{bkt} | WR_{kt} | Water recession of k^{th} sub-area | |

M_{kt} | $\sum _{j=1}^{J}}{TWS}_{jkt$ | The total water supply for k^{th} sub-area, including local water, upstream recession and water supply from the reservoir |

_{bkt}) denotes different variables in water resources field. For accuracy, the superscript number (1) and (2) is defined to explain the difference. (1) is the water supply from the reservoir to subarea(s) and (2) is water income from upstream sub-area(s). (ii) For convenience, the serial number of each subsystem included the reservoir, and the sub-areas would not be numbered separately. When k only refers to the sub-area in some formulas or equations, its subscript begins with 2.

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**Figure 1.**Reservoir topology structure. (

**a**) Mixed-connected; (

**b**) parallel; (

**c**) based on (

**b**), the small region divided into several areas; (

**d**) the topology structure among sub-areas and reservoir.

**Figure 4.**Location and water resources condition of the study area. (

**a**) Actual river system and sub-area (

**b**) the topology structure of rivers and sub-areas.

Level Year | Population/Million | GDP/Million RMB | Domestic Water Demand/10^{8} m^{3} | Industrial Water Demand/10^{8} m^{3} |
---|---|---|---|---|

2013 | 0.31 | 2.26 | 0.29 | 0.87 |

2020 | 0.51 | 5.33 | 0.59 | 1.67 |

2030 | 1.09 | 10.35 | 1.26 | 2.10 |

Scenarios | Normal Year | Moderate Dry Year | Particular Dry Year | |||
---|---|---|---|---|---|---|

Total Release | F_{2} | Total Release | F_{2} | Total Release | F_{2} | |

2020 | 13.08 | 0.1848 | 10.80 | 0.2384 | 6.59 | 0.2803 |

2030 | 11.85 | 0.2214 | 10.21 | 0.2768 | 6.59 | 0.2803 |

Scenario | Optimal Reservoir Release for Each Month (10^{8} m^{3}) | Total | F_{2} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |||

2020 | 0.26 | 0.39 | 0.71 | 1.55 | 2.46 | 3.08 | 2.17 | 1.22 | 0.58 | 0.39 | 0.37 | 0.29 | 13.47 | 0.2948 |

2030 | 0.23 | 0.35 | 0.61 | 1.41 | 2.36 | 2.89 | 2.04 | 1.13 | 0.54 | 0.33 | 0.32 | 0.24 | 12.46 | 0.3266 |

Level Year | Parameters | |||||
---|---|---|---|---|---|---|

L | F_{1} | F_{2} | ||||

Traditional | Optimal | Traditional | Optimal | Traditional | Optimal | |

2020 | 33.91 | 16.72 | 0.0299 | 0.0004 | 0.5094 | 0.2948 |

2030 | 36.67 | 19.45 | 0.0445 | 0.0043 | 0.5094 | 0.3266 |

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## Share and Cite

**MDPI and ACS Style**

Tan, Y.; Dong, Z.; Xiong, C.; Zhong, Z.; Hou, L.
An Optimal Allocation Model for Large Complex Water Resources System Considering Water supply and Ecological Needs. *Water* **2019**, *11*, 843.
https://doi.org/10.3390/w11040843

**AMA Style**

Tan Y, Dong Z, Xiong C, Zhong Z, Hou L.
An Optimal Allocation Model for Large Complex Water Resources System Considering Water supply and Ecological Needs. *Water*. 2019; 11(4):843.
https://doi.org/10.3390/w11040843

**Chicago/Turabian Style**

Tan, Yaogeng, Zengchuan Dong, Chuansheng Xiong, Zhiyu Zhong, and Lina Hou.
2019. "An Optimal Allocation Model for Large Complex Water Resources System Considering Water supply and Ecological Needs" *Water* 11, no. 4: 843.
https://doi.org/10.3390/w11040843