# A Comprehensive Model for Assessing Synergistic Revenue–Cost for the Joint Operation of a Complex Multistakeholder Reservoir System

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

_{m}reservoirs on the m-th tributary, and the sub-reservoir i is denoted by ${R}_{i}^{m}$ $\left(i=1,2,\cdots ,{N}_{m}\right)$.

_{1}) in the downstream cascade and the lateral inflow of downstream reservoir F

_{j}are influenced by the streamflow regulation of the upstream reservoir system. On the one hand, when the upstream reservoir system conducts joint operations for complementing the downstream reservoirs, synergistic revenue will be generated downstream. On the other hand, deviation from the individual optimal operation strategy of upstream reservoirs owing to joint operation will reduce revenue, corresponding to operation cost. Moreover, revenue and cost could be totally different under various combinational scenarios of the upstream stakeholders. Therefore, this study focused on evaluating the synergistic revenue and operation cost corresponding to joint scenarios of individual and combinational stakeholders. An equivalent aggregated reservoir model based on the topology of the river system is proposed, which aggregates the upstream compensating reservoirs into a group of M aggregated reservoirs.

#### 2.1. Construction and Simplification of Multiobjective Joint Operation Indexes

^{3}/s); and $O{e}_{sum,t}^{\left(\varnothing \right)}$ is the sum of the average outflow of all aggregated reservoirs in the individual operation scenario in period t (m

^{3}/s).

#### 2.2. Generalization Method for Equivalent Aggregated Reservoirs

^{3}/s), and $I{l}_{i,t}$ is the lateral inflow of sub-reservoir i in period t (m

^{3}/s).

^{3}), and ${V}_{i,t}^{\left(\mathrm{K}\right)}$ is the storage of sub-reservoir i in scenario $\mathrm{K}$ and period t (m

^{3}).

^{3}/s), and $f\left(\cdot \right)$ is the joint reservoir operation model or function for calculating the specific outflow of each aggregated reservoir under a specified level of complementarity (the corresponding model is established in Section 2.3).

^{3}/s), and ${\Phi}_{j}$ is the set of aggregated reservoirs with direct hydraulic connection to compensated reservoir j.

#### 2.3. Equivalent Aggregated Multiobjective Joint Optimal Operation Model for a Complex Reservoir System

#### 2.3.1. Objective Functions

#### 2.3.2. Constraints

^{3}/s).

^{3}), and $\underset{\xaf}{{V}_{j,t}}$ and $\overline{{V}_{j,t}}$ are the lower and upper storage limits, respectively, of compensated reservoir j at the beginning of period t (m

^{3}).

^{3}), and $Ve{E}_{m}$ and $V{E}_{j}$ are the end storage of aggregated reservoir m and compensated reservoir j, respectively (m

^{3}).

^{3}/s).

#### 2.3.3. Solving Method

#### 2.4. Synergistic Revenue and Cost Assessment

## 3. Case Study

_{1}–F

_{6}). The 21 upstream reservoirs are converted to aggregated reservoirs on each tributary, namely A, B, C, D, and E, which belong to five different stakeholders. Aggregated reservoirs A and B are located upstream of the entire compensated reservoir system, while aggregated reservoirs C, D, and E are located downstream tributaries. Table 1 and Table 2 list the various constraints such as storage and flow of each aggregated reservoir and compensated reservoir.

#### 3.1. Analysis of Model Precision of Aggregated Reservoir Operation and Computational Effort

#### 3.1.1. Model Precision

#### 3.1.2. Computational Effort

#### 3.2. Multiobjective Synergistic Revenue Assessment and Cost Analysis of a Reservoir System

^{8}kWh.

^{8}kWh, while the maximum synergistic revenue under combinational scenarios with two–five aggregated reservoirs is 90.93 × 10

^{8}, 98.80 × 10

^{8}, 102.70 × 10

^{8}, and 105.04 × 10

^{8}kWh, respectively. In comparison with the maximum synergistic revenue in energy production contributed by a single aggregated reservoir, the synergistic revenue increases by 13.08%, 22.87%, 27.72%, and 30.63%, and the incremental increase from each additional aggregated reservoir is reduced from 10.52 × 10

^{8}to 2.34 × 10

^{8}kWh.

#### 3.3. Multiobjective Joint Operation Strategy for a Reservoir System

_{3}generates the highest synergistic revenue in energy production, accounting for 34.73% and 28.95% of the total under the two scenarios.

_{5}and F

_{6}, which are most affected by the streamflow regulation of the aggregated reservoirs, are taken as examples. Three scenarios—status quo, scenario {B}, and combinational scenario {A,B,C,D,E}—are selected as typical scenarios. For the three scenarios, the total power output and total synergistic revenue processes in energy production of the compensated reservoirs system are plotted in Figure 8. The inflow process and the average head process of reservoir F

_{5}are presented in Figure 9 and Figure 10, while the power release processes of reservoir F

_{6}are shown in Figure 11.

_{5}is increased by 2.51 and 2.86 m (Figure 10), and the power release of F

_{6}is increased by 7.91% and 19.98%, respectively (Figure 11). Figure 9 also shows that current inflow in the flood season (periods 15–31) is reduced by 2.67% and 7.20% under joint operation scenario {B} and scenario {A,B,C,D,E}, respectively, resulting in a corresponding reduction in spillage by 4.40 × 10

^{9}and 1.17 × 10

^{10}m

^{3}, accounting for 6.70% and 17.76% of the total spillage, respectively. Consequently, the synergistic revenue in energy production of the compensated reservoirs system is improved by 80.41 × 10

^{8}kWh (2.48%) and 105.04 × 10

^{8}kWh (3.24%).

^{9}and 1.59 × 10

^{10}m

^{3}. Moreover, the postponed refilling of the aggregated reservoirs during the late flood season homogenizes the outflow process temporally, thereby increasing the dry season inflow and the water head of the compensated reservoirs and also providing additional water for their refilling operations.

#### 3.4. Identification of Key River Systems and Aggregated Reservoirs for Achieving the Highest Synergistic Revenues

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Haddad, O.B.; Moravej, M.; Loáiciga, H.A. Application of the Water Cycle Algorithm to the Optimal Operation of Reservoir Systems. J. Irrig. Drain. Eng.
**2015**, 141, 4014064. [Google Scholar] [CrossRef] - Zhou, Y.; Guo, S.; Xu, C.; Liu, P.; Qin, H. Deriving joint optimal refill rules for cascade reservoirs with multi-objective evaluation. J. Hydrol.
**2015**, 524, 166–181. [Google Scholar] [CrossRef] - Yeh, W.W. Reservoir Management and Operations Models: A State-of-the-Art Review. Water Resour. Res.
**1985**, 21, 1797–1818. [Google Scholar] [CrossRef] - Chang, X.; Liu, X.; Zhou, W. Hydropower in China at present and its further development. Energy
**2010**, 35, 4400–4406. [Google Scholar] [CrossRef] - Labadie, J.W. Optimal Operation of Multireservoir Systems: State-of-the-Art Review. J. Water Resour. Plan. Manag.
**2004**, 130, 93–111. [Google Scholar] [CrossRef] - Li, X.; Liu, P.; Ming, B.; Huang, K.; Xu, W.; Wen, Y. Joint Optimization of Forward Contract and Operating Rules for Cascade Hydropower Reservoirs. J. Water Resour. Plan. Manag.
**2022**, 148, 4021099. [Google Scholar] [CrossRef] - Dau, Q.V.; Adeloye, A.J. Influence of Reservoir Joint Operation on Performance of the Pong–Bhakra Multipurpose, Multireservoir System in Northern India. J. Water Resour. Plan. Manag.
**2021**, 147, 4021076. [Google Scholar] [CrossRef] - Chen, J. Long-Term Joint Operation of Cascade Reservoirs Using Enhanced Progressive Optimality Algorithm and Dynamic Programming Hybrid Approach. Water Resour. Manag.
**2021**, 35, 2265–2279. [Google Scholar] [CrossRef] - Lu, S.; Shang, Y.; Li, W.; Peng, Y.; Wu, X. Economic benefit analysis of joint operation of cascaded reservoirs. J. Clean Prod.
**2018**, 179, 731–737. [Google Scholar] [CrossRef] - Hu, H.; Tian, G.; Dai, Z. Multi-dimensional interest game between reservoir and city stakeholders in the Yellow River Basin: A case study of the lower reaches. Appl. Water Sci.
**2023**, 13, 1–17. [Google Scholar] [CrossRef] - Madani, K.; Hooshyar, M. A game theory–reinforcement learning (GT–RL) method to develop optimal operation policies for multi-operator reservoir systems. J. Hydrol.
**2014**, 519, 732–742. [Google Scholar] [CrossRef] - Xie, J.; Zhang, L.; Chen, X.; Zhan, Y.; Zhou, L. Incremental Benefit Allocation for Joint Operation of Multi-Stakeholder Wind-PV-Hydro Complementary Generation System With Cascade Hydro-Power: An Aumann-Shapley Value Method. IEEE Access
**2020**, 8, 68668–68681. [Google Scholar] [CrossRef] - Wei, N.; He, S.; Lu, K.; Xie, J.; Peng, Y. Multi-Stakeholder Coordinated Operation of Reservoir Considering Irrigation and Ecology. Water
**2022**, 14, 1970. [Google Scholar] [CrossRef] - Cheng, C.; Yan, L.; Mirchi, A.; Madani, K. China’s Booming Hydropower: Systems Modeling Challenges and Opportunities. J. Water Resour. Plan. Manag.
**2017**, 143, 2516002. [Google Scholar] [CrossRef] - Zhang, X.; Liu, P.; Feng, M.; Xu, C.; Cheng, L.; Gong, Y. A new joint optimization method for design and operation of multi-reservoir system considering the conditional value-at-risk. J. Hydrol.
**2022**, 610, 127946. [Google Scholar] [CrossRef] - Ma, C.; Xu, R.; He, W.; Xia, J. Determining the limiting water level of early flood season by combining multiobjective optimization scheduling and copula joint distribution function: A case study of three gorges reservoir. Sci. Total Environ.
**2020**, 737, 139789. [Google Scholar] [CrossRef] [PubMed] - Wang, L.; Zheng, H.; Chen, Y.; Long, Y.; Chen, J.; Li, R.; Hu, X.; Ouyang, Z. Synergistic management of forest and reservoir infrastructure improves multistakeholders’ benefits across the forest-water-energy-food nexus. J. Clean Prod.
**2023**, 422, 138575. [Google Scholar] [CrossRef] - Wheeler, K.G.; Hall, J.W.; Abdo, G.M.; Dadson, S.J.; Kasprzyk, J.R.; Smith, R.; Zagona, E.A. Exploring Cooperative Transboundary River Management Strategies for the Eastern Nile Basin. Water Resour. Res.
**2018**, 54, 9224–9254. [Google Scholar] [CrossRef] - Wu, L.; Bai, T.; Huang, Q. Tradeoff analysis between economic and ecological benefits of the inter basin water transfer project under changing environment and its operation rules. J. Clean Prod.
**2020**, 248, 119294. [Google Scholar] [CrossRef] - Ni, X.; Dong, Z.; Xie, W.; Jia, W.; Duan, C.; Yao, H. Research on the Multi-Objective Cooperative Competition Mechanism of Jinsha River Downstream Cascade Reservoirs during the Flood Season Based on Optimized NSGA-III. Water
**2019**, 11, 849. [Google Scholar] [CrossRef] - Chang, J.; Meng, X.; Wang, Z.; Wang, X.; Huang, Q. Optimized cascade reservoir operation considering ice flood control and power generation. J. Hydrol.
**2014**, 519, 1042–1051. [Google Scholar] [CrossRef] - Feng, Z.; Niu, W.; Cheng, C. China’s large-scale hydropower system: Operation characteristics, modeling challenge and dimensionality reduction possibilities. Renew. Energy
**2019**, 136, 805–818. [Google Scholar] [CrossRef] - Zhang, Z.; Zhang, S.; Wang, Y.; Jiang, Y.; Wang, H. Use of parallel deterministic dynamic programming and hierarchical adaptive genetic algorithm for reservoir operation optimization. Comput. Ind. Eng.
**2013**, 65, 310–321. [Google Scholar] [CrossRef] - Huang, L.; Li, X.; Fang, H.; Yin, D.; Si, Y.; Wei, J.; Liu, J.; Hu, X.; Zhang, L. Balancing social, economic and ecological benefits of reservoir operation during the flood season: A case study of the Three Gorges Project, China. J. Hydrol.
**2019**, 572, 422–434. [Google Scholar] [CrossRef] - Dai, L.; Zhang, P.; Wang, Y.; Jiang, D.; Dai, H.; Mao, J.; Wang, M. Multi-objective optimization of cascade reservoirs using NSGA-II: A case study of the Three Gorges-Gezhouba cascade reservoirs in the middle Yangtze River, China. Hum. Ecol. Risk Assess. Int. J.
**2017**, 23, 814–835. [Google Scholar] [CrossRef] - Giuliani, M.; Herman, J.D.; Castelletti, A.; Reed, P. Many-objective reservoir policy identification and refinement to reduce policy inertia and myopia in water management. Water Resour. Res.
**2014**, 50, 3355–3377. [Google Scholar] [CrossRef] - Xu, B.; Sun, Y.; Huang, X.; Zhong, P.; Zhu, F.; Zhang, J.; Wang, X.; Wang, G.; Ma, Y.; Lu, Q.; et al. Scenario-Based Multiobjective Robust Optimization and Decision-Making Framework for Optimal Operation of a Cascade Hydropower System Under Multiple Uncertainties. Water Resour. Res.
**2022**, 58, e2021WR030965. [Google Scholar] [CrossRef] - Jia, B.; Zhong, P.; Wan, X.; Xu, B.; Chen, J. Decomposition–coordination model of reservoir group and flood storage basin for real-time flood control operation. Hydrol. Res.
**2015**, 46, 11–25. [Google Scholar] [CrossRef] - Li, J.; Zhong, P.; Yang, M.; Zhu, F.; Chen, J.; Xu, B.; Liu, W. Dynamic and Intelligent Modeling Methods for Joint Operation of a Flood Control System. J. Water Resour. Plan. Manag.
**2019**, 145, 10. [Google Scholar] [CrossRef] - Li, J.; Zhong, P.; Yang, M.; Zhu, F.; Chen, J.; Liu, W.; Xu, S. Intelligent identification of effective reservoirs based on the random forest classification model. J. Hydrol.
**2020**, 591, 125324. [Google Scholar] [CrossRef] - Ma, Y.; Zhong, P.; Xu, B.; Zhu, F.; Lu, Q.; Wang, H. Spark-based parallel dynamic programming and particle swarm optimization via cloud computing for a large-scale reservoir system. J. Hydrol.
**2021**, 598, 126444. [Google Scholar] [CrossRef] - Yao, H.; Dong, Z.; Li, D.; Ni, X.; Chen, T.; Chen, M.; Jia, W.; Huang, X. Long-term optimal reservoir operation with tuning on large-scale multi-objective optimization: Case study of cascade reservoirs in the Upper Yellow River Basin. J. Hydrol. Reg. Stud.
**2022**, 40, 101000. [Google Scholar] [CrossRef] - Archibald, T.W.; Mckinnon, K.I.M.; Thomas, L.C. An aggregate stochastic dynamic programming model of multireservoir systems. Water Resour. Res.
**1997**, 33, 333–340. [Google Scholar] [CrossRef] - Zhou, Y.; Guo, S.; Liu, P.; Xu, C. Joint operation and dynamic control of flood limiting water levels for mixed cascade reservoir systems. J. Hydrol.
**2014**, 519, 248–257. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of the topology of a complex reservoir system and details of reservoir classification.

**Figure 2.**Flow chart of equivalent aggregated multiobjective joint operation and synergistic revenue–cost assessment model for a complex multistakeholder reservoir system.

**Figure 3.**Schematic of an equivalent aggregated reservoir model: aggregation of (

**a**) cascade reservoirs, (

**b**) parallel reservoirs, and (

**c**) mixed reservoirs.

**Figure 5.**Schematic of the topological relationship between the compensated reservoirs group and the aggregated reservoirs in the upper reaches of a basin.

**Figure 6.**Simulated outflow process of the reservoirs in comparison with the actual outflow for aggregated reservoirs: (

**a**) A, (

**b**) B, (

**c**) C, (

**d**) D, (

**e**) E, and (

**f**) the entire system.

**Figure 7.**Synergistic revenue–cost relationships under different combinational scenarios: (

**a**) singular aggregated reservoir, (

**b**) two aggregated reservoirs, (

**c**) three aggregated reservoirs, and (

**d**) four and five aggregated reservoirs.

**Figure 8.**Total power output and total synergistic revenue processes of the compensated reservoirs system under each of three scenarios.

**Figure 10.**Average head process of typical compensated reservoir F

_{5}under each of three scenarios.

**Figure 11.**Power release processes of typical compensated reservoir F

_{6}under each of three scenarios.

**Figure 14.**Spatial streamflow composition and storage capacity coefficient of the aggregated reservoirs.

Aggregated Reservoir | Sub-Reservoir Number | Storage Capacity (10^{6} m^{3}) | Storage (10^{6} m^{3}) | Outflow (m^{3}/s) | ||
---|---|---|---|---|---|---|

Upper Limit | Lower Limit | Upper Limit | Lower Limit | |||

A | 6 | 1801 | 6507 | 4706 | 7000 | 439 |

B | 4 | 8281 | 13,560 | 5279 | 7800 | 401 |

C | 2 | 4668 | 6062 | 1394 | 7500 | 0 |

D | 3 | 2401 | 5568 | 3167 | 7500 | 0 |

E | 6 | 8997 | 15,982 | 6985 | 5500 | 280 |

Compensated Reservoir | Storage Capacity (10^{6} m^{3}) | Storage (10^{6} m^{3}) | Outflow (m^{3}/s) | Power Output (10^{4} kW) | |||
---|---|---|---|---|---|---|---|

Upper Limit | Lower Limit | Upper Limit | Lower Limit | Upper Limit | Lower Limit | ||

F_{1} | 3010 | 5857 | 2847 | 20,000 | 900 | 1020 | 0 |

F_{2} | 10,409 | 18,981 | 8571 | 30,000 | 1100 | 1600 | 0 |

F_{3} | 6461 | 11,574 | 5112 | 32,000 | 1300 | 1386 | 0 |

F_{4} | 903 | 4977 | 4074 | 32,000 | 1300 | 640 | 0 |

F_{5} | 22,044 | 39,227 | 17,183 | 55,000 | 6000 | 2250 | 0 |

F_{6} | - | 706 | 706 | 55,000 | 0 | 272 | 0 |

**Table 3.**Comparison of computational parameters of the equivalent aggregated multiobjective joint optimization model and the traditional model.

Model | Maximum Reservoir Number | Time (min) | Variables Number | Constraints Number |
---|---|---|---|---|

The equivalent aggregation model | 11 | 5.9 (−96.67%) | 3408 (−67.18%) | 4001 (−39.75%) |

The traditional model without aggregation | 27 | 176.83 | 10,384 | 6641 |

Maximum Outflow Change Degree $\left(\Delta {C}^{\left(\varnothing \right)}\right)$ | Energy Production (10^{8} kWh)$\left({\beta}^{\left(\varnothing \right)}\right)$ | Reliability | ||
---|---|---|---|---|

Water Supply $\left({\gamma}^{\left(\varnothing \right)}\right)$ | Ecological Protection $\left({\upsilon}^{\left(\varnothing \right)}\right)$ | Shipping $\left({\mu}^{\left(\varnothing \right)}\right)$ | ||

0 | 3240.16 | 90.20% | 92.10% | 90.50% |

**Table 5.**Average synergistic revenue of each compensated reservoir under the two compensating aggregated reservoir scenarios (×10

^{8}kWh).

Scenarios | $\sum _{\mathit{j}=1}^{6}{\mathit{F}}_{\mathit{j}}$ | F_{1} | F_{2} | F_{3} | F_{4} | F_{5} | F_{6} |
---|---|---|---|---|---|---|---|

{B} | 80.41 | 5.94 | 11.73 | 27.93 | 20.06 | 9.82 | 4.92 |

{A,B,C,D,E} | 105.04 | 8.43 | 14.26 | 30.41 | 19.52 | 21.64 | 10.78 |

**Table 6.**Key combinational scenarios set under different maximum outflow change degrees and numbers of aggregated reservoirs (×10

^{8}kWh).

Compensating Reservoirs Number | Low Change Degree (Degree = 49%) | Medium Change Degree (Degree = 77%) | High Change Degree (Degree = 95%) | |||
---|---|---|---|---|---|---|

Combinational Scenario | Synergistic Revenues | Combinational Scenario | Synergistic Revenues | Combinational Scenario | Synergistic Revenues | |

1 | {B} | 47.28 (89.07%) | {B} | 73.42 (87.33%) | {B} | 80.09 (80.19%) |

2 | {A,B} | 52.36 (98.64%) | {A,B} | 81.56 (97.01%) | {A,B} | 88.43 (88.54%) |

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**MDPI and ACS Style**

Quan, Y.; Xu, Y.; Mo, R.; Huang, X.; Ji, S.; Wang, H.; Li, Z.; Xu, B.
A Comprehensive Model for Assessing Synergistic Revenue–Cost for the Joint Operation of a Complex Multistakeholder Reservoir System. *Water* **2023**, *15*, 3896.
https://doi.org/10.3390/w15223896

**AMA Style**

Quan Y, Xu Y, Mo R, Huang X, Ji S, Wang H, Li Z, Xu B.
A Comprehensive Model for Assessing Synergistic Revenue–Cost for the Joint Operation of a Complex Multistakeholder Reservoir System. *Water*. 2023; 15(22):3896.
https://doi.org/10.3390/w15223896

**Chicago/Turabian Style**

Quan, Yufei, Yang Xu, Ran Mo, Xin Huang, Saijin Ji, Huili Wang, Zirui Li, and Bin Xu.
2023. "A Comprehensive Model for Assessing Synergistic Revenue–Cost for the Joint Operation of a Complex Multistakeholder Reservoir System" *Water* 15, no. 22: 3896.
https://doi.org/10.3390/w15223896