# Short-Term Multi-Objective Optimal Operation of Reservoirs to Maximize the Benefits of Hydropower and Navigation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Navigation Capacity Evaluation

#### 2.1. Hydrodynamic Model

#### 2.2. Analysis of the Flow Velocity

#### 2.2.1. The Dangerous Navigation Area of Channel

#### 2.2.2. The Navigation Capacity Considering the Flow Velocity ($NCv$)

#### 2.3. Analysis of the Variation of Water Level

## 3. Multi-Objective Reservoir Operation Model

#### 3.1. Objective Function

#### 3.1.1. Economic Objective: Maximizing the Daily Total Power Generation

#### 3.1.2. Navigation Objective: Maximizing the Navigation Capacity

#### 3.2. Constraints

## 4. Methodology

- (1)
- When there are no individuals in the external archive set, the non-dominated solutions generated in the iteration are stored directly in the archive set;
- (2)
- If the newly generated individuals are dominated by the individuals in the archive set, the new individuals will be deleted. Conversely, individuals can be deleted from the original archive set and new individuals added to the archive set;
- (3)
- When the number of individuals in the archive set reaches the preset maximum capacity, the individuals with smaller crowding distance are deleted.

## 5. Case Study

#### 5.1. Case Description

^{3}/s, the XJB reservoir can control a basin area of 458,800 square kilometers, accounting for 97% of the Jinsha River basin area. In addition, the total storage capacity of the XJB reservoir is 5.163 billion cubic meters, and the regulating storage volume is 900 million cubic meters. The main parameters of XJB reservoir is shown in Table 1.

^{6}kW, and the annual average power generation is 3.013 × 10

^{10}kWh. Furthermore, it is the only dam in the Jinsha River hydropower base to build a ship lift with a very efficient ship navigation rate. Its scale is equivalent to that of the Three Gorges reservoir, and it takes only 15 min for a ship of one kiloton to cross the dam. Compared with the average dam time of the Three Gorges reservoir of 5 h, the navigation efficiency of XJB reservoir is quite high.

^{3}/s. Moreover, the allowed navigation period of the XJB reservoir is from 08:00 to 18:00 h in a day, and a typical ship with a size of 85.0 m × 10.8 m × 2.0 m (length × width × draft depth) is selected for analysis. For the short-term optimization operation problem in this case study, the length of time in each period is set to one hour.

#### 5.2. Parameter Settings and Simulation Working Conditions

#### 5.2.1. Model Parameter Setting

#### 5.2.2. Simulation Working Conditions

#### 5.3. Results

#### 5.3.1. Numerical Simulation Results

^{3}/s. Besides, in Condition P2 the upstream water level is 370 m in July–September, the highest flow is in August, and the average annual flow is 9970 m

^{3}/s. Finally, Condition P3 and Condition P4 show the minimum and maximum navigable flow in the downstream reaches of the reservoir, respectively.

- (1)
- The numerical simulation results of Condition P1 and Condition P2 are shown in Table 3, Figure 5, and Figure 6. It can be seen that the longitudinal velocity and transverse velocity of the upstream entrance area of approach channel satisfy the navigation requirements. Therefore, it can be inferred that the upstream navigation capacity ($N{C}^{up}$) is less affected by the flow velocity. In this study case, $N{C}^{up}$ is mainly related to the water level variation, and the formula for $N{C}^{up}$ is expressed as follows:$$N{C}^{up}=\frac{1}{C{S}^{up}}\frac{1}{\left(Tn-1\right)}{\displaystyle \sum _{j=1}^{C{S}^{up}}{\displaystyle \sum _{t=2}^{Tn}(1-\left|\frac{{Z}_{t,j}^{down}-{Z}_{t-1,j}^{down}}{\Delta {Z}_{h,\mathrm{lim}\mathrm{t}}}\right|)}}$$
- (2)
- The transverse velocity and longitudinal velocity in Condition P3 and the longitudinal velocity in the Condition P4 are all satisfied by the navigation requirement, as shown in Figure 7 and Table 3. However, the local transverse velocity exceeds the limit value of 0.3 m/s in the Condition P4, which seriously hinders the navigation in the entrance area of the approach channel. As a result, the transverse velocity is the most important factor affecting the downstream navigation, which is considered in this case.

^{3}/s as an example, $NC{v}^{down}$ = 0.403 is obtained through the calculation, as shown in Figure 8.

- (1)
- For downstream navigation of XJB reservoir, when the discharge volume (less than 1700 m
^{3}/s) is relatively small, the downstream navigation capacity considering the flow velocity ($NC{v}^{down}$) of the three working conditions is the same ($NC{v}^{down}$ = 1); - (2)
- When the discharge volume is large (greater than 1700 m
^{3}/s), the $NC{v}^{down}$ is the poorest in Condition 1 (only left-bank turbines work), better in Condition 2 (all turbines work), and the best in Condition 3 (only right-bank turbines work).

#### 5.3.2. Multi-Objective Model Results

- (1)
- There is an obvious inverse relationship between the total power generation ($E$) and the downstream navigation capacity ($N{C}^{down}$) from the results of Figure 10b. The larger the total power generation ($E$), the smaller the downstream navigation capacity ($N{C}^{down}$) in a day. The minimum value of $N{C}^{down}$ is 0.52, and the maximum value is 0.99 with a growth of 90.38%, which varies greatly. At the same time, there is a drop of $E$ from the maximum value at 10,942.07 × 10
^{4}kWh to the minimum value at 10,925.44 × 10^{4}kWh. - (2)
- As shown in the Figure 10c, there is a certain inverse trend between the upstream navigation capacity ($N{C}^{up}$) and the downstream navigation capacity ($N{C}^{down}$). When the upstream navigation capacity increases, the downstream navigation capacity declines. The minimum value of $N{C}^{up}$ in all the schemes is 0.968, and the maximum value is 0.992 with a smaller growth compared to the change in $N{C}^{down}$.
- (3)
- Finally, it can be seen that the relationship between the total power generation ($E$) and the upstream navigation capacity ($N{C}^{up}$) is not obvious shown in the Figure 10d. There is little interaction between these two elements.

#### 5.4. Discussion

^{3}/s, and the average output is maintained at 4213 MW, resulting in a large amount of electricity during this period. In the meantime, the value of $NC{v}^{down}$ per hour is maintained between 0.48 and 0.52, resulting in a large transverse velocity in the entrance area of approach channel, which is not conducive to navigation according to the numerical simulation results in Section 5.3.1. This scheme significantly rises the total power generation, but reducing the downstream navigation capacity in contrast.

^{3}/s, respectively. Consequently, the value of $NC{v}^{down}$ per hour is also the highest with an average value of 1, and it builds beneficial condition for navigations while decreasing the economic benefits of the reservoir.

## 6. Conclusions

- (1)
- The proposed NCEM to evaluate the navigation capability is reasonable and effective, and can comprehensively analyze the influence of flow velocity and water level variation on navigation accurately.
- (2)
- The proposed multi-objective model can obtain a favorable Pareto frontier and explore the relationship between objectives. In the case study of the XJB reservoir, there is an obvious inverse relationship between power generation and the downstream navigation capacity. Also, the relationship between downstream navigation capacity and upstream navigation capability is inverse. However, there is little interaction between power generation and upstream navigation capability.
- (3)
- The results illustrate that the method and model are reasonable and effective, and also indicate that they can provide a series of favorable optimal operation schemes for the reservoir to obtain economic and navigational benefits.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic drawing of the dangerous width of channel and the dangerous navigation region (when the maximum limit value of velocity is 0.3 m/s).

**Figure 3.**Study area: (

**a**,

**b**) The location of the Xiangjiaba (XJB) reservoir; (

**c**) The layout of turbines in the XJB reservoir; (

**d**) The upstream reaches of the reservoir; (

**e**) The downstream reaches of the reservoir.

**Figure 5.**Longitudinal velocity (

**a**) and transverse velocity (

**b**) diagram under Condition P1 of the upstream model.

**Figure 6.**Longitudinal velocity (

**a**) and transverse velocity (

**b**) diagram under Condition P2 of the upstream model.

**Figure 7.**Longitudinal velocity (

**a**) and transverse velocity (

**b**) diagram under Condition P3 of the downstream model.

**Figure 8.**Transverse velocity diagram in the entrance area of approach channel in the upstream model (where $NC{v}^{down}$ = 0.403, $Qx$ = 5700 m

^{3}/s).

**Figure 10.**The Pareto optimal front. (

**a**) The optimal results of three objectives (E, NC

^{down}and NC

^{up}); (

**b**) The optimal results of two objectives (E and NC

^{down}); (

**c**) The optimal results of two objectives (NC

^{down}and NC

^{up}); (

**d**) The optimal results of two objectives (E and NC

^{up}).

**Figure 11.**Operation schemes. (

**a**) The comparison of the objective values of the three schemes; (

**b**) Scheme 1; (

**c**) Scheme 37; (

**d**) Scheme 100.

Characteristics of Water Level (m) | Installed Capacity/MW | Minimum Power/MW | $\mathit{Q}{\mathit{n}}_{\mathbf{max}}\text{}({\mathbf{m}}^{3}/\mathbf{s})$ | $\mathit{Q}{\mathit{n}}_{\mathbf{min}}\text{}({\mathbf{m}}^{3}/\mathbf{s})$ | Flow Velocity (m/s) | $\mathbf{\Delta}{\mathit{Z}}_{\mathit{h},\mathbf{lim}\mathit{t}}\text{}(\mathbf{m}/\mathbf{h})$ | ||
---|---|---|---|---|---|---|---|---|

Dead Water Level | The Normal Height | u_{max}^{1}(m/s) | v_{max}^{2}(m/s) | |||||

370 | 380 | 6400 | 1800 | 12,000 | 1200 | 2.0 | 0.3 | 1.5 |

^{1}u

_{max}is the maximum limit longitudinal velocity;

^{2}v

_{max}is the maximum limit transverse velocity.

Condition | Discharge | Working Turbines | UBL ^{1} | Simulation |
---|---|---|---|---|

1 | Symmetric flow | All turbines | AC | Preliminary simulation and elaborate simulation |

2 | Asymmetric flow | Four left-bank turbines | AB | Asymmetric flow simulation |

3 | Asymmetric flow | Four right-bank turbines | BC | Asymmetric flow simulation |

^{1}UBL is the upstream boundary line of the downstream model.

Model | Condition | UBC ^{1} | DBC ^{2} | u (m/s) ^{4} | v (m/s) ^{4} | UBL |
---|---|---|---|---|---|---|

Q (m^{3}/s) ^{3} | Z (m) ^{3} | |||||

Upstream | P1 | 9880 | 380 | 0.08–0.12 | 0.04–0.05 | —— |

P2 | 9970 | 370 | 0.16–0.28 | 0.02–0.10 | —— | |

Downstream | P3 | 1200 | 265.8 | 0.05–0.35 | 0.025–0.125 | AC |

P4 | 12,000 | 277.25 | 0.50–2.00 | $\ge $0.3 (locally) | AC |

^{1}UBC is the upstream boundary condition;

^{2}DBC is the downstream boundary condition;

^{3}Q is the inflow (I) and Z is the upstream water level (Z

^{up}) in the upstream model; Q is the discharge flow (Qx), and Z is the downstream water level (Z

^{down}) in the downstream model;

^{4}u is the longitudinal velocity and v is the transverse velocity.

No. | UBC | DBC | Dd | Cross-Sectional Water Level | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Qx | ${\mathit{Z}}^{\mathit{d}\mathit{o}\mathit{w}\mathit{n}}$ | No. 10 | No. 11 | No. 12 | No. 13 | No. 14 | No. 15 | No. 16 | No. 17 | No. 18 | ||

1 | 1200 | 265.8 | 0 | 265.80 | 265.80 | 265.80 | 265.80 | 265.80 | 265.80 | 265.80 | 265.80 | 265.80 |

2 | 1700 | 266.7 | 0 | 266.72 | 266.72 | 266.72 | 266.72 | 266.72 | 266.72 | 266.72 | 266.72 | 266.72 |

3 | 2200 | 267.6 | 0 | 267.56 | 267.56 | 267.56 | 267.56 | 267.56 | 267.56 | 267.56 | 267.56 | 267.56 |

4 | 2700 | 268.3 | 0 | 268.29 | 268.29 | 268.29 | 268.29 | 268.29 | 268.29 | 268.29 | 268.29 | 268.29 |

5 | 3200 | 269.0 | 0 | 269.00 | 269.00 | 269.00 | 269.00 | 269.00 | 269.00 | 269.00 | 269.00 | 269.00 |

6 | 3700 | 269.6 | 0 | 269.63 | 269.63 | 269.63 | 269.63 | 269.63 | 269.63 | 269.63 | 269.63 | 269.63 |

7 | 4200 | 270.2 | 25 | 270.24 | 270.24 | 270.24 | 270.24 | 270.24 | 270.24 | 270.24 | 270.24 | 270.24 |

8 | 4700 | 270.8 | 28 | 270.82 | 270.82 | 270.82 | 270.82 | 270.82 | 270.82 | 270.82 | 270.82 | 270.82 |

9 | 5200 | 271.3 | 31 | 271.35 | 271.35 | 271.35 | 271.35 | 271.35 | 271.35 | 271.35 | 271.35 | 271.35 |

10 | 5700 | 271.8 | 34 | 271.84 | 271.84 | 271.84 | 271.84 | 271.84 | 271.84 | 271.84 | 271.84 | 271.84 |

11 | 6200 | 272.3 | 36 | 272.32 | 272.32 | 272.32 | 272.32 | 272.32 | 272.31 | 272.31 | 272.31 | 272.31 |

12 | 6700 | 272.8 | 37 | 272.77 | 272.77 | 272.77 | 272.77 | 272.77 | 272.77 | 272.76 | 272.76 | 272.75 |

13 | 7200 | 273.2 | 38 | 273.21 | 273.21 | 273.21 | 273.21 | 273.21 | 273.21 | 273.20 | 273.20 | 273.19 |

14 | 7700 | 273.6 | 40 | 273.64 | 273.64 | 273.64 | 273.64 | 273.64 | 273.64 | 273.63 | 273.63 | 273.62 |

15 | 8200 | 274.1 | 41 | 274.07 | 274.07 | 274.07 | 274.07 | 274.07 | 274.07 | 274.06 | 274.06 | 274.05 |

16 | 8700 | 274.5 | 42 | 274.49 | 274.49 | 274.49 | 274.49 | 274.49 | 274.49 | 274.48 | 274.48 | 274.47 |

17 | 9200 | 274.9 | 43 | 274.92 | 274.92 | 274.92 | 274.92 | 274.92 | 274.92 | 274.91 | 274.91 | 274.90 |

18 | 9700 | 275.3 | 43.5 | 275.34 | 275.34 | 275.34 | 275.34 | 275.34 | 275.34 | 275.33 | 275.32 | 275.31 |

19 | 10,200 | 275.8 | 43.6 | 275.77 | 275.77 | 275.77 | 275.77 | 275.77 | 275.77 | 275.76 | 275.75 | 275.74 |

20 | 10,700 | 276.2 | 43.7 | 276.19 | 276.19 | 276.19 | 276.19 | 276.19 | 276.19 | 276.18 | 276.17 | 276.16 |

21 | 11,200 | 276.6 | 43.8 | 276.60 | 276.60 | 276.60 | 276.60 | 276.60 | 276.59 | 276.59 | 276.58 | 276.57 |

22 | 11,700 | 277.0 | 43.9 | 277.02 | 277.02 | 277.02 | 277.02 | 277.02 | 277.01 | 277.01 | 277.00 | 276.98 |

23 | 12,000 | 277.3 | 44.5 | 277.27 | 277.27 | 277.27 | 277.27 | 277.27 | 277.26 | 277.26 | 277.25 | 277.23 |

No. | UBC | DBC | Cross-Sectional Water Level | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

I | ${\mathit{Z}}^{\mathit{u}\mathit{p}}$ | No. 1 | No. 2 | No. 3 | No. 4 | No. 5 | No. 6 | No. 7 | No. 8 | No. 9 | |

1 | 5000 | 380.0 | 380.03 | 380.03 | 380.02 | 380.02 | 380.02 | 380.01 | 380.01 | 380.01 | 380.00 |

2 | 5000 | 379.5 | 379.50 | 379.50 | 379.50 | 379.50 | 379.50 | 379.50 | 379.50 | 379.50 | 379.50 |

3 | 5000 | 379.0 | 379.00 | 379.00 | 379.00 | 379.00 | 379.00 | 379.00 | 379.00 | 379.00 | 379.00 |

4 | 5000 | 378.5 | 378.50 | 378.50 | 378.50 | 378.50 | 378.50 | 378.50 | 378.50 | 378.50 | 378.50 |

5 | 5000 | 378.0 | 378.00 | 378.00 | 378.00 | 378.00 | 378.00 | 378.00 | 378.00 | 378.00 | 378.00 |

6 | 5000 | 377.5 | 377.50 | 377.50 | 377.50 | 377.50 | 377.50 | 377.50 | 377.50 | 377.50 | 377.50 |

7 | 5000 | 377.0 | 377.00 | 377.00 | 377.00 | 377.00 | 377.00 | 377.00 | 377.00 | 377.00 | 377.00 |

8 | 5000 | 376.5 | 376.50 | 376.50 | 376.50 | 376.50 | 376.50 | 376.50 | 376.50 | 376.50 | 376.50 |

9 | 5000 | 376.0 | 376.00 | 376.00 | 376.00 | 376.00 | 376.00 | 376.00 | 376.00 | 376.00 | 376.00 |

10 | 5000 | 375.5 | 375.50 | 375.50 | 375.50 | 375.50 | 375.50 | 375.50 | 375.50 | 375.50 | 375.50 |

11 | 5000 | 375.0 | 375.00 | 375.00 | 375.00 | 375.00 | 375.00 | 375.00 | 375.00 | 375.00 | 375.00 |

12 | 5000 | 374.5 | 374.50 | 374.50 | 374.50 | 374.50 | 374.50 | 374.50 | 374.50 | 374.50 | 374.50 |

13 | 5000 | 374.0 | 374.00 | 374.00 | 374.00 | 374.00 | 374.00 | 374.00 | 374.00 | 374.00 | 374.00 |

14 | 5000 | 373.5 | 373.50 | 373.50 | 373.50 | 373.50 | 373.50 | 373.50 | 373.50 | 373.50 | 373.50 |

15 | 5000 | 373.0 | 373.00 | 373.00 | 373.00 | 373.00 | 373.00 | 373.00 | 373.00 | 373.00 | 373.00 |

16 | 5000 | 372.5 | 372.50 | 372.50 | 372.50 | 372.50 | 372.50 | 372.50 | 372.50 | 372.50 | 372.50 |

17 | 5000 | 372.0 | 372.00 | 372.00 | 372.00 | 372.00 | 372.00 | 372.00 | 372.00 | 372.00 | 372.00 |

18 | 5000 | 371.5 | 371.50 | 371.50 | 371.50 | 371.50 | 371.50 | 371.50 | 371.50 | 371.50 | 371.50 |

19 | 5000 | 371.0 | 371.00 | 371.00 | 371.00 | 371.00 | 371.00 | 371.00 | 371.00 | 371.00 | 371.00 |

20 | 5000 | 370.5 | 370.50 | 370.50 | 370.50 | 370.50 | 370.50 | 370.50 | 370.50 | 370.50 | 370.50 |

21 | 5000 | 370.0 | 370.00 | 370.00 | 370.00 | 370.00 | 370.00 | 370.00 | 370.00 | 370.00 | 370.00 |

Scheme | $\mathit{E}$ (10^{4} kWh)
| $\mathit{N}{\mathit{C}}^{\mathit{d}\mathit{o}\mathit{w}\mathit{n}}$ | $\mathit{N}{\mathit{C}}^{\mathit{u}\mathit{p}}$ | Scheme | $\mathit{E}$ (10^{4} kWh)
| $\mathit{N}{\mathit{C}}^{\mathit{d}\mathit{o}\mathit{w}\mathit{n}}$ | $\mathit{N}{\mathit{C}}^{\mathit{u}\mathit{p}}$ |
---|---|---|---|---|---|---|---|

1 | 10,942.07743 | 0.516103 | 0.990381 | 40 | 10,934.87 | 0.775069 | 0.982459 |

2 | 10,942.65972 | 0.522835 | 0.988732 | 41 | 10,938.68 | 0.776607 | 0.981775 |

3 | 10,942.33743 | 0.538384 | 0.988012 | 42 | 10,933.95 | 0.778997 | 0.982459 |

4 | 10,942.11842 | 0.550021 | 0.989657 | 43 | 10,937.91 | 0.783004 | 0.978221 |

5 | 10,941.27857 | 0.553263 | 0.992386 | 44 | 10,934.97 | 0.790506 | 0.982092 |

… | … | … | … | … | … | … | … |

35 | 10,939.44119 | 0.742237 | 0.981028 | 96 | 10,922.86 | 0.979242 | 0.968444 |

36 | 10,939.48026 | 0.745708 | 0.980875 | 97 | 10,925.08 | 0.987635 | 0.96826 |

37 | 10,938.9987 | 0.750805 | 0.979485 | 98 | 10,924.49 | 0.989242 | 0.968253 |

38 | 10,938.77714 | 0.758043 | 0.979526 | 99 | 10,923.75 | 0.989259 | 0.968254 |

39 | 10,938.71673 | 0.763488 | 0.978766 | 100 | 10,925.44 | 0.99352 | 0.968062 |

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## Share and Cite

**MDPI and ACS Style**

Jia, T.; Qin, H.; Yan, D.; Zhang, Z.; Liu, B.; Li, C.; Wang, J.; Zhou, J.
Short-Term Multi-Objective Optimal Operation of Reservoirs to Maximize the Benefits of Hydropower and Navigation. *Water* **2019**, *11*, 1272.
https://doi.org/10.3390/w11061272

**AMA Style**

Jia T, Qin H, Yan D, Zhang Z, Liu B, Li C, Wang J, Zhou J.
Short-Term Multi-Objective Optimal Operation of Reservoirs to Maximize the Benefits of Hydropower and Navigation. *Water*. 2019; 11(6):1272.
https://doi.org/10.3390/w11061272

**Chicago/Turabian Style**

Jia, Tianlong, Hui Qin, Dong Yan, Zhendong Zhang, Bin Liu, Chaoshun Li, Jinwen Wang, and Jianzhong Zhou.
2019. "Short-Term Multi-Objective Optimal Operation of Reservoirs to Maximize the Benefits of Hydropower and Navigation" *Water* 11, no. 6: 1272.
https://doi.org/10.3390/w11061272