# A Provincial Initial Water Rights Incentive Allocation Model with Total Pollutant Discharge Control

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## Abstract

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## 1. Introduction

- (1)
- (2)
- Using a single-objective decision making model, Gao Zhu et al. [15] carried out research on the initial allocation of pollutant discharge based on water function regionalization. Wan Shan et al. [16] established an initial pollutant discharge allocation model based on the simulation study of both economic optimality and fairness of allocation.
- (3)
- With a multi-objective decision making method, Huang Xianfeng et al. [17] took into consideration a two level allocation model of river emission rights. The economical optimization and water quality optimization were regarded as the objective functions. The concentration and total quantity of pollutant were the constraints and the principle of equality and continuity of the production program of the dischargers were taken into account. In addition, Huang Binbin et al. [18] carried out research on an optimal allocation model of river pollutant discharge based on pollution limits.
- (4)
- Using a hybrid allocation model, Liu Gang et al. [19] constructed a cooperation allocating initial discharge permits system for industrial source points in the Taihu Lake basin which was regulated by the policy-oriented government, led by economy-oriented government subjects and included the participation of many stakeholders. In addition, Liu Nianlei et al. [20] studied the pollutant discharge with an entropy weighing method and improved the proportional distribution method.

- Firstly, the most stringent water resources management set the benchmark of the “three red lines”, which means a “red line” for total water use, a “red line” for water use efficiency, and a “red line” for controlling pollution. The initial water rights allocation models focused on the first two “red lines”, but were less concerned with the third “red line” which illustrates the pollution limits of the water functional zones that should be enhanced. The third “red line” can be considered by the allocation of initial pollutant discharge, but current models of provincial initial pollutant discharge rights allocation are insufficient, and existing research in this area also lacks a comprehensive consideration of utilization benefits including economy benefit, society benefit, and ecological environmental benefit.
- Secondly, the incentive mechanism of rewarding excellence and punishing inferiority is crucial. In the field of water resources management, it is important to punish the over standard pollutant discharge during environmental protection of water resources. However, in reality, it is also necessary to implement positive incentives to the less than standard pollutant discharge. How to embed water pollution problems into the initial water right allocation and establish the incentive function is rarely considered and really needs further discussion.

- Firstly, in contrast to the existing models of provincial initial pollutant discharge rights allocation, we focus on the comprehensive consideration of utilization benefits including economic benefits, societal benefits, and ecological environmental benefits. Hence, the first contribution of this paper is that we set up three objective functions separately including the maximum comprehensive economic benefits of basin, the optimal fairness and coordination of provincial pollutant discharge rights, and the minimum losses to the ecological environment of the basin. Furthermore, based on the three objective functions and the constraints of the total pollutant discharge, we put forward a multiple-objective provincial initial pollutant discharge rights allocation model with a self-adaptive chaotic optimization algorithm.
- Secondly, none of the aforementioned papers consider the incentive mechanism for rewarding excellence and punishing inferiority in the water environmental protection; however, awarding the less than standard pollutant discharge is as important as punishing the over-standard pollutant discharge. Hence, the second contribution of this paper is that we establish an incentive function by embedding the amount of provincial initial pollutant discharge and we follow the mechanism of rewarding excellence and punishing inferiority.
- Thirdly, by using this incentive function, we attempt to embed pollutant discharge allocation into the process of initial water quantity rights allocation, so as to internalize externalities of over-standard or under-standard pollutant discharge into the initial water rights allocation. Hence, the third contribution of this paper is that we set up a provincial initial water rights allocation model with the above incentive function considering both water quantity and water quality.

## 2. Research Framework

## 3. Multi-Objective Provincial Initial Pollutant Discharge Rights Allocation Model

#### 3.1. Model Construction

#### 3.1.1. The Objective Functions of the Model

#### 3.1.2. The Constraint Condition of Model

#### 3.2. Model Calculation

#### 3.2.1. Non-Dimension of Objective Functions

#### 3.2.2. The Transformation of the Multi-Objective Planning Model

#### 3.2.3. Solving the Model

## 4. Provincial Initial Water Rights Incentive Allocation Model

#### 4.1. Basic Design Idea of the Incentive Mechanism

- Set the real amount of pollutant discharge of province ${S}_{k}$ to ${{x}_{{S}_{k}}}^{R}$. When the real amount of pollutant discharge ${{x}_{{S}_{k}}}^{R}$ exceeds the initial pollutant discharge rights ${x}_{{S}_{k}}$, the negative incentive should be implemented to reduce its initial water quantity rights.
- When the real provincial amount of pollutant discharge ${{x}_{{S}_{k}}}^{R}$ is lower than the initial pollutant discharge rights ${x}_{{S}_{k}}$, the positive incentive should be implemented to increase its initial water quantity rights.

#### 4.2. Model of the Incentive Allocation

## 5. Empirical Study

#### 5.1. Data Source

#### 5.2. Calculation of Provincial Initial Pollutant Discharge Rights Alloction

#### 5.3. Provincial Initial Water Rights Incentive Allocation Calculation

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Province | Year | Population (Ten Thousand Persons) | Per Capita GDP (Yuan) | Pollutant Discharge (One Hundred Million Cubic Meters) |
---|---|---|---|---|

Jiangsu | 2011 | 2939 | 8.21 | 29.0 |

2012 | 2960 | 8.97 | 29.1 | |

2013 | 2985 | 9.84 | 29.2 | |

2014 | 2996 | 10.68 | 28.3 | |

Zhejiang | 2011 | 1763 | 6.54 | 12.1 |

2012 | 1775 | 6.97 | 12.1 | |

2013 | 1791 | 7.53 | 12.6 | |

2014 | 1797 | 7.98 | 12.9 | |

Shanghai | 2011 | 1175 | 12.01 | 22.6 |

2012 | 1184 | 12.53 | 23.1 | |

2013 | 1194 | 13.44 | 22.9 | |

2014 | 1198 | 14.47 | 22.9 |

**Table 2.**Solutions of provincial initial pollutant discharge rights allocation (Taihu Basin, planning year 2030).

Provincial Initial Pollutant Discharge Rights Allocation | Calculation Results |
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Objective function F_{1} | Revenue function of Jiangsu’s pollutant discharge rights ^{1}: ${F}_{11}=GD{P}_{1}\left({x}_{1}\right)=1410.8{x}_{1}-32200$ |

Revenue function of Zhejiang’s pollutant discharge rights: ${F}_{12}=GD{P}_{2}\left({x}_{2}\right)=1200.7{x}_{2}-10586$ | |

Revenue function of Shanghai’s pollutant discharge rights: ${F}_{13}=GD{P}_{3}\left({x}_{3}\right)=4248.8{x}_{3}-80597$ | |

${F}_{1}=\mathrm{max}{f}_{1}(X)=\mathrm{min}[-{f}_{1}(X)]=\mathrm{min}[(32200-1410.8{x}_{1})+(10586-1200.7{x}_{2})+(80597-4248.8{x}_{3})]$ Solving the Equation (7), ${f}_{1}({X}^{*})=16359$; Using Equation (10), ${F}_{1}^{\prime}=\mathrm{min}(0.086{x}_{1}+0.073{x}_{2}+0.26{x}_{3}-8.542)$ | |

Objective function F_{2} | Parameters selected ^{2}: relative importance of province population, area, and GDP index: ${\beta}_{\text{}1}=1/3$, ${\beta}_{\text{}2}=1/3$, ${\beta}_{\text{}3}=1/3$ |

Using Equation (2) to calculate: ${F}_{21}={({x}_{1}/{x}_{2}-{\beta}_{1}\times 1.667-{\beta}_{1}\times 1.604-{\beta}_{1}\times 2.500)}^{2}$; ${F}_{22}={({x}_{1}/{x}_{3}-{\beta}_{1}\times 2.500-{\beta}_{1}\times 3.746-{\beta}_{1}\times 0.625)}^{2}$; ${F}_{23}={({x}_{2}/{x}_{3}-{\beta}_{1}\times 1.500-{\beta}_{1}\times 2.335-{\beta}_{1}\times 0.250)}^{2}$; ${F}_{2}={F}_{21}+{F}_{22}+{F}_{23}=\mathrm{min}[{f}_{2}(X)]=\mathrm{min}[{({x}_{1}/{x}_{2}-1.92)}^{2}+{({x}_{1}/{x}_{3}-2.29)}^{2}+{({x}_{2}/{x}_{3}-1.36)}^{2}]$; Solving the Equation (8), ${f}_{1}({X}^{*})=0.0172$; Using Equation (11), ${F}_{2}^{\prime}=\mathrm{min}[{({x}_{1}/{x}_{2}-1.92)}^{2}+{({x}_{1}/{x}_{3}-2.29)}^{2}+{({x}_{2}/{x}_{3}-1.36)}^{2}-0.0172]/0.0172$ | |

Objective function F_{3} | Parameters selected: environment damage coefficient ^{3}: Jiangsu ${\eta}_{1}$ = 0.6; Zhejiang ${\eta}_{2}$ = 0.5; Shanghai ${\eta}_{3}$ = 0.5 |

Environmental damage function of Jiangsu province ^{4}: ${P}_{1}\left({x}_{{S}_{1}}\right)=3075.6{x}_{1}-47695$; Environmental damage function of Zhejiang province: ${P}_{2}\left({x}_{{S}_{2}}\right)=2890.2{x}_{2}-9226.2$; Environmental damage function of Zhejiang province: ${P}_{3}\left({x}_{{S}_{3}}\right)=1062.9{x}_{3}-13426$; ${F}_{3}=\mathrm{min}[{f}_{3}(X)]=\mathrm{min}[0.6\times (3075.6{x}_{1}-47695)+0.5\times (2890.2{x}_{2}-9226.2)+0.5\times (1062.9{x}_{3}-13426)]$; Solving the Equation (7), ${f}_{3}\left({X}^{*}\right)=12292$; Using Equation (11), ${F}_{3}^{\prime}=\mathrm{min}(0.150{x}_{1}+0.118{x}_{2}+0.043{x}_{3}-4.25)$ | |

Constraint condition | ${x}_{1}+{x}_{2}+{x}_{3}\le 54.46;\text{}{x}_{1}\ge 0$; ${x}_{2}\ge 0$; ${x}_{3}\ge 0$ |

Algorism parameter selecting | Initial population size B = 200, Chaotic iteration M = 100, Initial temperature T_{0} = 100 |

Initial pollutant discharge of province (a hundred million cubic meters) | Using the Equation (14) to obtain the results as follows: Jiangsu ${x}_{{S}_{1}}=27.55$; Zhejiang ${x}_{{S}_{2}}=14.93$; Shanghai ${x}_{{S}_{3}}=11.98$ |

^{1}Revenue function of pollutant discharge can be obtained mainly by fitting analysis between the GDP of the province and the total pollutant discharge of the province;

^{2}Considering the population, area, and GDP as equally important to the initial water rights allocation, we assume ${\beta}_{\text{}1}=1/3$, ${\beta}_{\text{}2}=1/3$, ${\beta}_{\text{}3}=1/3$;

^{3}The environmental damage coefficients are obtained by the combination of actual provincial situation and expert opinion approach;

^{4}Environmental damage function can be obtained by fitting analysis between the pollution footprint and the pollutant discharge of the province according to the evaluation model of the water environmental ecological damage [27].

**Table 3.**The normalized allocation ratios of initial water rights incentive allocation (Taihu Basin, planning year 2030).

Variables for Initial Water Rights Incentive Allocation | Calculation Results |
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Parameters Selected | According to the Ratio between Real Pollutant Discharge and Initial Pollutant Discharge and the Expert Opinion Approach, We Select the Adjustment Coefficient $\partial =3$ |

Adjusted value of incentive function for provinces | Using Equation (18) to calculate: Jiangsu Province: $(1-27.55/28.97{)}^{2}=0.002$; Zhejiang Province: $(1-11.97/14.93{)}^{2}=0.039$; Shanghai City: $(1-11.98/22.66{)}^{2}=0.222$ |

Pre-configuration allocation ratio of the initial water quantity rights of province | According to [28], the pre-configuration allocation ratio of the initial water quantity rights of province can be expressed as follows: Jiangsu Province ${\omega}_{{S}_{1}}=35.63\%$; Zhejiang Province ${\omega}_{{S}_{2}}=31.94\%$; Shanghai City ${\omega}_{{S}_{3}}=32.45\%$ |

Adjusted allocation ratio of initial water rights of province | Using Equation (19) to calculate: Jiangsu Province ${{\omega}_{{S}_{1}}}^{\prime}=35.53\%$; Zhejiang Province ${{\omega}_{{S}_{2}}}^{\prime}=33.20\%$; Shanghai City ${{\omega}_{{S}_{3}}}^{\prime}=25.24\%$ |

The normalized allocation ratio of initial water rights of province | Using Equation (20) to calculate: Jiangsu Province ${{\omega}_{{S}_{1}}}^{\u2033}=37.81\%$; Zhejiang Province ${{\omega}_{{S}_{2}}}^{\u2033}=35.33\%$; Shanghai City ${{\omega}_{{S}_{3}}}^{\u2033}=26.86\%$ |

**Table 4.**The solutions and comparison for different allocation methods of provincial initial water rights in the Taihu Basin (under water frequencies of 50%, 75%, and 90%, planning year 2030).

Different Scenarios | Jiangsu Province | Zhejiang Province | Shanghai City | ||||||
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Different Water Frequency | Different Water Frequency | Different Water Frequency | |||||||

50% | 75% | 90% | 50% | 75% | 90% | 50% | 75% | 90% | |

Our incentive allocation plans (a hundred million cubic meters) | 124.47 | 137.36 | 148.44 | 116.31 | 128.35 | 138.71 | 88.42 | 97.58 | 105.45 |

“Water use efficiency” allocation model (a hundred million cubic meters) | 117.25 | 129.39 | 139.83 | 105.13 | 116.02 | 125.38 | 106.82 | 117.89 | 127.39 |

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**MDPI and ACS Style**

Ge, M.; Wu, F.-P.; You, M.
A Provincial Initial Water Rights Incentive Allocation Model with Total Pollutant Discharge Control. *Water* **2016**, *8*, 525.
https://doi.org/10.3390/w8110525

**AMA Style**

Ge M, Wu F-P, You M.
A Provincial Initial Water Rights Incentive Allocation Model with Total Pollutant Discharge Control. *Water*. 2016; 8(11):525.
https://doi.org/10.3390/w8110525

**Chicago/Turabian Style**

Ge, Min, Feng-Ping Wu, and Min You.
2016. "A Provincial Initial Water Rights Incentive Allocation Model with Total Pollutant Discharge Control" *Water* 8, no. 11: 525.
https://doi.org/10.3390/w8110525