# Ecological Health Assessment with the Combination Weight Method for the River Reach after the Retirement and Renovation of Small Hydropower Stations

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Index System

_{5}. Biological characteristics include the fish diversity index and benthic animal diversity index. Morphological structure characteristics include lateral stability, sediment type, and longitudinal connectivity. The index system is shown in Table 1.

^{2}; and $\mathit{BE}$ is the water area before the retirement and renovation of small hydropower stations, m

^{2}.

^{3}/s; $J$ is the gradient of the river; and $B$ is the actual river width, m.

#### 2.2. Index Classification Standard

#### 2.3. Determination of Weight

#### 2.3.1. Analytic Hierarchy Process Determines Subjective Weight

- (1)
- Construct judgment matrixThe ecological health assessment of the river section after the retirement and renovation of the small hydropower station is divided into three levels. The target layer is the ecological health of the river reach (A). The criterion layer includes hydrological characteristics (B
_{1}), water quality characteristics B_{2}, biological characteristics B_{3}, and morphological structure characteristics B_{4}, and the index layer includes the flow reduction degree C_{11}, change degree of water area C_{12}, and other indicators as shown in Table 1. - (2)
- Determine the weight vector and consistency check:$$A\cdot \omega ={\lambda}_{\mathrm{max}}\cdot \omega ,$$
- (a)
- Calculate the numerical product of each row of the judgment matrix A to determine M and calculate the eigenvector $W$:$$Wi=\sqrt[n]{M},$$
- (b)
- Calculate the maximum eigenvalue ${\lambda}_{\mathrm{max}}$ according to the feature vector $W$:$$\lambda \mathrm{max}=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\frac{(A{W}^{T})i}{Wi}},$$
- (c)
- Define the consistency indicators $CI$:$$CI=\frac{\lambda \mathrm{max}-n}{n-1},$$
- (d)
- Introduce the random consistency index $RI$:$$RI=\frac{C{I}_{1}+C{I}_{2}+\dots +C{I}_{n}}{n},$$
- (e)
- Compare $CI$ with the random consistency index $RI$ to obtain the test coefficient $CR$. When $CR$ ≤ 0.1, the consistency is good:$$CR=\frac{CI}{RI}.$$

#### 2.3.2. Entropy Weight Method Determines Objective Weight

- (1)
- Form the original data matrix $\mathrm{X}={\left({x}_{ij}\right)}_{m\times n}$ and normalize ${x}_{ij}$ to determine ${P}_{ij}$:$${P}_{ij}=\frac{{x}_{ij}}{{\displaystyle \sum _{i=1}^{m}{x}_{ij}}}\hspace{1em}\hspace{1em}i=1,2,\dots m;j=1,2,\dots n,$$
- (2)
- Calculate the entropy of index $j$:$${e}_{j}=-{\displaystyle \sum _{i=1}^{m}{p}_{ij}}\cdot \mathrm{ln}{p}_{ij}\hspace{1em}\hspace{1em}i=1,2,\dots m;j=1,2,\dots n,$$
- (3)
- The weight of $j$ indicators is calculated:$${u}_{j}=\frac{1}{{e}_{j}}\hspace{1em}\hspace{1em}j=1,2,\dots n,$$$${\beta}_{j}=\frac{{u}_{j}}{{\displaystyle \sum _{j=1}^{n}{u}_{j}}},$$

#### 2.4. Combination Weighting Method

- (1)
- $U|=\left\{{u}_{1},{u}_{2},\dots {u}_{n}\right\}$ is used to represent a basic set of weight vectors, and linearly combine these n vectors into a possible set of weights:$$U={\displaystyle \sum _{k=1}^{n}{\alpha}_{k}{u}_{k}^{T}}\left({\alpha}_{k}>0\right),$$
- (2)
- The most satisfactory weight vector was found to optimize ${\alpha}_{k}$ to minimize the deviation between $U$ and each ${U}_{k}$:$$\mathrm{min}\parallel {\displaystyle \sum _{j=1}^{n}{\alpha}_{j}}\times {u}_{j}^{T}-{u}_{i}^{T}{\parallel}_{2}\hspace{1em}\hspace{1em}i=1,2,\dots n.$$
- (3)
- According to the differential property of the matrix, the first derivative condition of Equation (16) optimization is:$$\sum _{j=1}^{n}{\alpha}_{j}}\times {u}_{i}\times {u}_{i}^{T}={u}_{i}\times {u}_{i}^{T}\hspace{1em}\hspace{1em}i=1,2,\dots n.$$
- (4)
- Equation (17) was converted into the following set of linear equations:$$\left(\begin{array}{cccc}{u}_{1}\xb7{u}_{1}^{T}& {u}_{1}\xb7{u}_{2}^{T}& \dots & {u}_{1}\xb7{u}_{n}^{T}\\ {u}_{2}\xb7{u}_{1}^{T}& {u}_{2}\xb7{u}_{2}^{T}& \dots & {u}_{2}\xb7{u}_{n}^{T}\\ \vdots & \vdots & \vdots & \vdots \\ {u}_{n}\xb7{u}_{1}^{T}& {u}_{n}\xb7{u}_{2}^{T}& \dots & {u}_{n}\xb7{u}_{n}^{T}\end{array}\right)\left(\begin{array}{c}{\alpha}_{1}\\ {\alpha}_{2}\\ \vdots \\ {\alpha}_{n}\end{array}\right)=\left(\begin{array}{c}{u}_{1}\xb7{u}_{1}^{T}\\ {u}_{2}\xb7{u}_{2}^{T}\\ \vdots \\ {u}_{n}\xb7{u}_{n}^{T}\end{array}\right).$$
- (5)
- The set $\left({\alpha}_{1},{\alpha}_{2}\dots {\alpha}_{n}\right)$ is obtained and normalized as:$${\alpha}_{k}^{\ast}=\frac{{\alpha}_{k}}{{\displaystyle \sum _{k=1}^{n}{\alpha}_{k}}}.$$
- (6)
- Determine the combined weight:$${u}^{\ast}={\displaystyle \sum _{k=1}^{n}{\alpha}_{k}{u}_{k}^{T}}.$$

#### 2.5. Fuzzy Comprehensive Assessment

- (1)
- Determine the evaluation set:$$V=\left\{{V}_{1},{V}_{2},\dots {V}_{i}\right\},$$
- (2)
- Determination of membershipWhen the standard value and weight value of each index are known, the trapezoidal distribution membership function can determine the fuzzy membership of the index layer, criterion layer, and index layer:The membership function of the smaller and better index is:$$r\left(x\right)=\{\begin{array}{l}1,x<a\\ \frac{b-x}{b-a},a\le x\le b\\ 0,x>b\end{array},$$The membership function of the larger and better index is:$$r\left(x\right)=\{\begin{array}{l}0,x<a\\ \frac{x-a}{b-a},a\le x\le b\\ 1,x>b\end{array},$$
- (3)
- According to the characteristics of each index, the membership function of each index is drawn up, and the membership matrix ${R}_{i}$ is established. Then, the comprehensive evaluation model is:$${R}_{i}=\left[\begin{array}{ccccc}{r}_{i11}& {r}_{i12}& {r}_{i13}& {r}_{i14}& {r}_{i15}\\ {r}_{i21}& {r}_{i22}& {r}_{i23}& {r}_{i24}& {r}_{i25}\\ {r}_{i31}& {r}_{i32}& {r}_{i33}& {r}_{i34}& {r}_{i35}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {r}_{im1}& {r}_{im2}& {r}_{im3}& {r}_{im4}& {r}_{im5}\end{array}\right],$$
- (4)
- Multilevel fuzzy comprehensive evaluationThe river health ecological assessment system is composed of three structural levels: the target layer, the index layer, and the criterion layer. It can be divided into an index layer reflecting the criterion layer and criterion layer reflecting the two-level fuzzy comprehensive assessment of the target layer.
- (a)
- The fuzzy evaluation of the indicator layer reflecting the criterion layer is:$$B={\left\{{B}_{1},{B}_{2},{B}_{3},{B}_{4}\right\}}^{T},$$$${B}_{i}={\omega}_{c}\circ {R}_{i}=({\omega}_{ci1},{\omega}_{ci2},\dots ,{\omega}_{cim})\circ \left[\begin{array}{ccccc}{r}_{i11}& {r}_{i12}& {r}_{i13}& {r}_{i14}& {r}_{i15}\\ {r}_{i21}& {r}_{i22}& {r}_{i23}& {r}_{i24}& {r}_{i25}\\ {r}_{i31}& {r}_{i32}& {r}_{i33}& {r}_{i34}& {r}_{i35}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {r}_{im1}& {r}_{im2}& {r}_{im3}& {r}_{im4}& {r}_{im5}\end{array}\right],$$
- (b)
- The recursion to the criterion layer to reflect the fuzzy evaluation of the target layer is:$$A={\omega}_{B}\circ B=({\omega}_{B1},{\omega}_{B2},\dots ,{\omega}_{Bm})\circ \left[\begin{array}{c}{\omega}_{c1}\circ {R}_{1}\\ {\omega}_{c2}\circ {R}_{2}\\ {\omega}_{c3}\circ {R}_{3}\\ {\omega}_{c4}\circ {R}_{4}\\ {\omega}_{c5}\circ {R}_{5}\end{array}\right]=({A}_{1},{A}_{2},{A}_{3},{A}_{4},{A}_{5}),$$

## 3. Study Region

^{2}, a river length of 40 km, and a river slope of 8.1‰. It is a typical mountainous river with a steep slope, a rapid natural flow, and many gravels and boulders at the bottom of the riverbed. Xiyuan, Hongfang, Tufang, Laifang, Shimen, Changqiao, and other diversion-type small hydropower stations are successively distributed from upstream to downstream of the Tufang River. This density of hydropower stations has resulted in a long river cutoff length which does not align with the water usage habits of local residents.

## 4. Results

#### 4.1. Weight Consistency Test

#### 4.2. Assessment Index Weight

#### 4.2.1. Weight Calculation by the AHP

#### 4.2.2. Weight Calculation using the Entropy Method

#### 4.2.3. Weight Calculation Results

#### 4.3. Calculation Results of Fuzzy Evaluation

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Table 1.**Hierarchy of the ecological health assessment system of the river reach after the retirement and renovation of small hydropower stations.

Target Layer | Criterion Layer | Index Layer |
---|---|---|

Ecological health of river reach A | Hydrological characteristics B_{1} | Flow reduction degree C_{11} |

Change degree of water area C_{12} | ||

Water quality characteristic B_{2} | Stability of water temperature structure in upstream and downstream of dam C_{21} | |

DO C_{22} | ||

Total nitrogen (TN) C_{23} | ||

BOD_{5} C_{24} | ||

Biological characteristics B_{3}Morphological structure characteristics B _{4} | Fish diversity index C_{31} | |

Benthic animal diversity index C_{32} | ||

Lateral stability C_{41} | ||

Sediment type C_{42} | ||

Longitudinal connectivity C_{43} |

Index Layer | Very Healthy | Healthy | Subhealthy | Unhealthy | Morbid |
---|---|---|---|---|---|

Quantitative index score | (80–100) | (60–80) | (40–60) | (20–40) | (0–20) |

Qualitative index score *^{1} | 100 | 80 | 60 | 40 | 20 |

Flow reduction degree C1 _{1} | The dam is completely removed, and the water is fully restored to the natural river channel | The dam is not demolished, and the discharge from bottom hole is carried out to restore the water to the river | The dam is not removed, the water-retaining gate is opened, and the water flow is discharged from the gate in the dry season, with diversion in the diversion channel | The ecological renovation is carried out by adding drainage channels and ecological units to maintain the discharge of ecological flow | Diversion power station without ecological flow renovation |

Change degree of water area C1 _{2} | <15 | [15, 30) | [30, 45) | [45, 60) | ≥60 |

Stability of water temperature Structure in upstream and downstream of dam *^{2} C_{21} | The water temperature structure of upstream and down-stream reaches of dam is the same as that of the natural river, stable and without obvious water temperature stratification | The overall water temperature structure of upstream and downstream of dam is less affected by the dam, close to the natural river state | Reservoir section transports water to downstream of dam through surface overflow, and there is difference in water temperature structure between upstream and downstream of dam | The reservoir section transports the bottom water temperature to the downstream through drainage channels and facilities, and the water temperature structure in upstream and downstream of dam is different and unstable | There is no discharge flow in the reservoir area and no water temperature structure in the downstream of dam |

DO [46] C_{22} | ≥6 | [5, 6) | [3, 5) | [2, 3) | [0, 2) |

TN [46] C_{23} | $\le $0.5 | [0.5, 1) | [1, 1.5) | [1.5, 2) | ≥2 |

BOD_{5} [46] C_{24} | $\le $3 | [3, 4) | [4, 6) | [6, 10) | ≥10 |

Fish diversity index [38] C_{31} | ≥2 | [1.5, 2) | [1, 1.5) | [0.5, 1) | <0.5 |

Benthic animal diversity index [39] C _{32} | ≥3 | [2, 3) | [1, 2) | [0, 1) | 0 |

Lateral stability coefficient [37] C _{41} | ≥1.5 | [1.2, 1.5) | [1, 1.2) | [0.8, 1) | <0.8 |

Sediment type [40] C_{42} | Bedrock, cobble and gravel, sand, clay of 4 categories appear | Bedrock, cobble and gravel, sand, clay appear in 3 categories | Bedrock, cobble and gravel, sand, clay appear in 2 categories | Bedrock, cobble and gravel, sand, clay appear in 1 categries | Riverbed is hardened without containing any of the above components |

Longitudinal connectivity C _{43} | The dam is completely removed without retaining the dam foundation. At the same time, the river channel is cleared, which does not affect the normal migration of fish | The dam is re-moved, the dam foundation is retained, and the water is restored to the natural river channel, which has a small impact on fish migration | The dam is not removed, and the bottom hole discharge is carried out to restore the water to the river, which has a partial impact on fish migration | The water-retaining gate is opened, and the water flows from the gate in the dry season, so the fish cannot migrate correctly | The ecological transformation is carried out by adding drainage channels and ecological units to maintain the discharge of ecological flow, and fish cannot migrate normally |

^{1}If the qualitative index meets the description, the corresponding score will be given. *

^{2}For the retired hydropower stations, the upstream and downstream of the dam refers to the upstream and downstream of the original dam site.

**Table 3.**Judgment matrix and weight of water quality and biological characteristics of Tufang River (2018) by AHP.

Water Quality | Stability of Water Temperature Structure in Upstream and Downstream of Dam | DO | TN | BOD_{5} | Weight Value ψ | Parameter Value |
---|---|---|---|---|---|---|

Stability of water temperature structure in upstream and downstream of dam | 1 | 1/5 | 1/3 | 1/3 | 0.078 | ${\lambda}_{\mathrm{max}}$ = 4 CR = 3.33 × 10 ^{−16} |

DO | 5 | 1 | 3 | 3 | 0.522 | |

TN | 3 | 1/3 | 1 | 1 | 0.2 | |

BOD_{5} | 3 | 1/3 | 1 | 1 | 0.2 | |

Biology | Fish diversity index | Benthic animal diversity index | Weight value ψ | Parameter value | ||

Fish diversity index | 1 | 1 | 0.5 | ${\lambda}_{\mathrm{max}}$ = 2 CR = 0 < 0.1 | ||

Benthic animal diversity index | 1 | 1 | 0.5 |

**Table 4.**Weight of water quality and biological characteristics index layer in Tufang River (2018) using the entropy method.

River Reach | Stability of Water Temperature Structure in Upstream and Downstream of Dam | DO | TN | BOD_{5} | ||||

Actual Value | $-\mathrm{Pi}\cdot \mathrm{lnPi}$ | Actual Value | $-\mathrm{Pi}\cdot \mathrm{lnPi}$ | Actual Value | $-\mathrm{Pi}\cdot \mathrm{lnPi}$ | Actual Value | $-\mathrm{Pi}\cdot \mathrm{lnPi}$ | |

Shimen | 100 | 0.24 | 8 | 0.33 | 1.29 | 0.35 | 7.26 | 0.35 |

Changqiao | 40 | 0.36 | 6.9 | 0.36 | 1.38 | 0.34 | 7.4 | 0.35 |

${e}_{j}$ | 0.60 | 0.69 | 0.69 | 0.69 | ||||

${u}_{j}$ | 1.67 | 1.45 | 1.44 | 1.44 | ||||

${w}_{j}$ | 0.28 | 0.24 | 0.24 | 0.24 | ||||

River Reach | Fish Diversity Index | Benthic Animal Diversity Index | ||||||

Actual Value | $-\mathrm{Pi}\cdot \mathrm{lnPi}$ | Actual Value | $-\mathrm{Pi}\cdot \mathrm{lnPi}$ | |||||

Shimen | 1.81 | 0.20 | 1.17 | 0.33 | ||||

Changqiao | 1.15 | 0.37 | 1.02 | 0.36 | ||||

${e}_{j}$ | 0.67 | 0.69 | ||||||

${u}_{j}$ | 1.50 | 1.45 | ||||||

${w}_{j}$ | 0.51 | 0.49 |

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Cai, F.; Hu, Z.; Jiang, B.; Ruan, W.; Cai, S.; Zou, H. Ecological Health Assessment with the Combination Weight Method for the River Reach after the Retirement and Renovation of Small Hydropower Stations. *Water* **2023**, *15*, 355.
https://doi.org/10.3390/w15020355

**AMA Style**

Cai F, Hu Z, Jiang B, Ruan W, Cai S, Zou H. Ecological Health Assessment with the Combination Weight Method for the River Reach after the Retirement and Renovation of Small Hydropower Stations. *Water*. 2023; 15(2):355.
https://doi.org/10.3390/w15020355

**Chicago/Turabian Style**

Cai, Feng, Zhinan Hu, Beihan Jiang, Weifang Ruan, Shujuan Cai, and Huiling Zou. 2023. "Ecological Health Assessment with the Combination Weight Method for the River Reach after the Retirement and Renovation of Small Hydropower Stations" *Water* 15, no. 2: 355.
https://doi.org/10.3390/w15020355