Abstract
For cascade hydropower system (CHS), multi-objective long-term comprehensive operation (MOLTCO) is important for balancing the conflicting objectives of hydropower production and ecological sustainability. This paper proposes an Improved Multi-objective RIME algorithm (IMORIME) to address the MOLTCO problem, aiming to simultaneously maximize power generation and minimize ecological flow deviations. The presented IMORIME introduces Good Point Set method for uniform population initialization and incorporates a hybrid constraint handling strategy combining boundary clipping and penalty functions to effectively ensure solution feasibility. A crowding-weighted roulette selection strategy is adopted to determine guided solutions, adaptively balancing convergence and diversity. Moreover, IMORIME introduces an improved hard rime puncture mechanism inspired by Differential Evolution to mitigate stagnation at local optima and to maintain Pareto front diversity; a Whale Optimization Algorithm spiral enhancement strategy to reinforce local exploitation and accelerate convergence; and a Quasi-Opposition-Based Learning strategy to broaden the search space and preserve global diversity. The efficacy of the proposed algorithm is verified through benchmark tests and practical applications in the Three Gorges and Gezhouba cascades. Quantitative comparisons against five classical algorithms reveal that IMORIME consistently achieves the top average rank (1.0) in both Hypervolume and objective function values across all inflow scenarios. Specifically, in the highly constrained low-flow scenario, IMORIME reduces the ecological flow deviation by 10.5% and improves the Hypervolume metric by 3.1% compared to the second-best algorithm. Furthermore, IMORIME consistently maximizes total hydropower generation while maintaining the most well-distributed Pareto front (yielding the optimal average Spacing rank of 1.3). In the practical cascade system application, IMORIME demonstrates superior convergence stability and robustness, effectively balancing generation benefits and ecological protection in MOLTCO for CHS with coupled generation–ecology objectives.
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The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
Abbreviations
- f 1 :
-
The objectives of maximizing total hydropower generation
- f 2 :
-
The objectives of minimizing the squared relative ecological flow deviation
- t :
-
The indices of the scheduling period
- Δt :
-
The length of each scheduling period
- m :
-
The indices of the reservoir
- T :
-
The total numbers of scheduling periods
- M :
-
The total numbers of reservoirs
- P m,t :
-
The hydroelectric output of reservoir m at period t
- C m :
-
The power output coefficient of reservoir m
- \({Q}_{m,t}^{P}\) :
-
The power discharge rate of reservoir m at period t
- \({Q}_{t}^{eco}\) :
-
The ecological flow requirements at period t
- H m,t :
-
The hydraulic head of reservoir m at period t
- V m,t :
-
The initial storage volumes of reservoir m at period t
- V m,+1t :
-
The final storage volumes of reservoir m at period t
- \({Q}_{m,t}^{\text{I}}\) :
-
The inflow rates of reservoir m at period t
- \({Q}_{m,t}^{\text{O}}\) :
-
The total outflow of reservoir m at period t
- \({Q}_{m,t}^{S}\) :
-
The spillage flow of reservoir m at period t
- \({Q}_{m,t}^{O,min}\) :
-
The lower bounds of total outflow of reservoir m at period t
- \({Q}_{m,t}^{\text{O},\text{max}}\) :
-
The upper bounds of total outflow of reservoir m at period t
- vm(⋅):
-
The storage–capacity relationship of reservoir m
- Z m,t :
-
The water level of reservoir m at period t
- \({Z}_{m,t}^{down}\) :
-
The average tailwater level of reservoir m at period t corresponding to \({Q}_{m,t}^{\text{O}}\)
- Zm(⋅):
-
The function between the tail water level and outflow of reservoir m
- \({\text{V}}_{m}^{-1}(\cdot )\) :
-
The inverse function of the storage–capacity curve of reservoir m
- \({Z}_{m}^{\text{begin}}\) :
-
The initial water levels of reservoir m
- \({Z}_{m}^{\text{end}}\) :
-
The final water levels of reservoir m
- \({Z}_{m,t}^{min}\) :
-
The minimum allowable water levels of reservoir m at period t
- \({Z}_{m,t}^{\text{max}}\) :
-
The maximum allowable water levels of reservoir m at period t
- \({P}_{m,t}^{min}\) :
-
The lower limits of hydroelectric output of reservoir m at period t
- \({P}_{m,t}^{max}\) :
-
The upper limits of hydroelectric output of reservoir m at period t
- q(⋅):
-
The functional relationship between water level and maximum discharge capacity
- R :
-
The rime population
- UB :
-
The lower bounds of the search space
- LB :
-
The upper bounds of the search space
- N :
-
The number of rime agents in the population
- D :
-
The number of decision variables of the optimization problem
- R ij :
-
The j th decision variable of the rime agent j
- rand, r 2 , r 6 :
-
The random number in (0,1)
- R best, j :
-
The position of the optimal individual in j th dimension
- h :
-
The adhesion degree of rime agents, a random value within (0,1)
- Factor :
-
The composite control coefficient that controls the rime agent movement
- θ :
-
The iteration-dependent angle controlling the movement direction
- β :
-
The environmental factor
- r 1 ,r 3 :
-
The random number within the range (-1,1)
- E :
-
The attachment coefficient
- w :
-
The number of segments of the step function
- K :
-
The maximum iteration number
- k :
-
The current iteration number, where k=1,2,…,K
- [⋅]:
-
The rounding operation
- Fnorm (Ri):
-
The normalized fitness value of the rime agent i
- GD :
-
The unit cube in D dimensional Euclidean space
- {⋅}:
-
The fractional part of a number
- u j :
-
The j th component of the generator point in the D dimensional unit cube GD
- PN(i):
-
The Good Point Set
- F 1 :
-
The first nondominated front
- CD λ :
-
The raw crowding distance of a solution λ∈F1
- CD max :
-
The maximum finite crowding distance within the front
- P λ :
-
The normalized selection probability of the rime agent i
- Ω:
-
The ordered guidance set
- \({R}_{i}^{gui}\) :
-
The specific guided solution assigned to the rime agent i for the current iteration
- r 4 , r 5 :
-
The mutually exclusive indices randomly selected from the population of size N
- D′ :
-
The element-wise distance between the current individual and the guided solution
- b :
-
The form of the logarithmic spiral
- l :
-
The random number uniformly distributed in [-1,1]
- R qopp :
-
The quasi-opposite population
- A :
-
The maximum size of the external archive
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Acknowledgements
This work was financially supported by the Natural Science Foundation of Henan Province (242300420309). Appreciation is extended to the Pymoo group for offering access to the computational codes applied in this research.
Funding
This work was financially supported by the Natural Science Foundation of Henan Province (242300420309).
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Aolin Gao: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Resources; Software; Visualization; Writing - original draft; Writing - review & editing. Hu Hu: Supervision; Project administration; Funding acquisition. He Li: Data curation; Validation; Writing - review & editing. Lyuwen Su: Data curation; Writing - review & editing Zhe Yang: Investigation; Resources; Writing - review & editing. Kaixu Geng: Data curation; Writing - review & editing. Cihang Shan: Investigation; Validation.
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Gao, A., Hu, H., Li, H. et al. Enhancing long-term comprehensive operation of cascade hydropower system using an improved multi-objective RIME optimization. Sci Rep (2026). https://doi.org/10.1038/s41598-026-45836-z
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DOI: https://doi.org/10.1038/s41598-026-45836-z
