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<p>Realizing the comprehensive benefits of cascade reservoirs requires the relationships analysis between multi-objectives. The study developed an integrated framework to assess these interactions quantitatively and qualitatively. First, a multi-objective optimal operation model was established and solved to generate Pareto optimal solution sets. Next, the cloud model was used to classify these objective values into different grades. This step effectively handled the fuzziness inherent in setting grade boundaries, preventing evaluation biases and ensuring the robustness of the results. Finally, set pair analysis techniques, specifically the exponential set pair potential <i>shi</i>(<i>u</i>) and the five-element subtraction set pair potential <i>S</i><sub><i>f</i></sub>(<i>u</i>), were applied to these graded results to evaluate the multi-objective relationships. The proposed procedure was applied to the cascade reservoirs in the lower Yalong River. Results revealed a strong synergy between total hydropower production <i>f</i><sub>1</sub> and output stability index <i>f</i><sub>2</sub>. This pair was mathematically characterized as an identity potential, with <i>shi</i>(<i>u</i>) greater than 1 and a high average <i>S</i><sub><i>f</i></sub>(<i>u</i>) of 0.92. Conversely, relationships involving the ecological benefit index <i>f</i><sub>3</sub> exhibited clear conflicts. The pairs <i>f</i><sub>1</sub>-<i>f</i><sub>3</sub> and <i>f</i><sub>2</sub>-<i>f</i><sub>3</sub> were identified as partial anti-potentials, indicated by <i>shi</i>(<i>u</i>) values below 1 and average <i>S</i><sub><i>f</i></sub>(<i>u</i>) values of -0.25 and − 0.30. These metrics demonstrated a synergy between power generation and output stability, both of which conflict with natural ecological rhythms. The proposed method offers an alternative for analyzing the multi-objective feedback relationships in cascade reservoirs.</p>

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Multi-Objective Relationship Analysis of Cascade Reservoirs Based on Set Pair Analysis and Cloud Model

  • 期刊论文

  • 发表日期：

    2026年4月20日

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  • Jian Wang,
  • Zhangling Xiao,
  • Yude Zhu,
  • Zhongmin Liang,
  • Yan Xing,
  • Xinyue Liu,
  • Zongmo Yang

摘要

Realizing the comprehensive benefits of cascade reservoirs requires the relationships analysis between multi-objectives. The study developed an integrated framework to assess these interactions quantitatively and qualitatively. First, a multi-objective optimal operation model was established and solved to generate Pareto optimal solution sets. Next, the cloud model was used to classify these objective values into different grades. This step effectively handled the fuzziness inherent in setting grade boundaries, preventing evaluation biases and ensuring the robustness of the results. Finally, set pair analysis techniques, specifically the exponential set pair potential shi(u) and the five-element subtraction set pair potential Sf(u), were applied to these graded results to evaluate the multi-objective relationships. The proposed procedure was applied to the cascade reservoirs in the lower Yalong River. Results revealed a strong synergy between total hydropower production f1 and output stability index f2. This pair was mathematically characterized as an identity potential, with shi(u) greater than 1 and a high average Sf(u) of 0.92. Conversely, relationships involving the ecological benefit index f3 exhibited clear conflicts. The pairs f1-f3 and f2-f3 were identified as partial anti-potentials, indicated by shi(u) values below 1 and average Sf(u) values of -0.25 and − 0.30. These metrics demonstrated a synergy between power generation and output stability, both of which conflict with natural ecological rhythms. The proposed method offers an alternative for analyzing the multi-objective feedback relationships in cascade reservoirs.