利用傅立葉頻譜分析優化颱風期間水庫運行

Abstract

臺灣地區水庫在颱風期間之操作，涉及一項複雜的多目標最佳化問題：操作人員須同時兼顧枯水期蓄水量之最大化、長期庫容之保育（即減少泥砂淤積）、以及在保護敏感基礎設施之前提下維持防洪功能。本論文建立一套雙重公式化之數學最佳化架構，用以評估石門水庫之歷史操作情形，並量化其可達成之改善幅度。此架構分別於時域（Hybrid 模型，約1,670個決策變數）及頻域（Network 模型，採用傅立葉級數，約168個決策變數）求解相同之最佳化問題。頻域公式化透過維度縮減（變數數量減少為十分之一）、數值條件之改善（正交傅立葉基底），以及解具備之內建平滑性，從根本上簡化了此最佳化問題。本研究並提出一項創新之線性趨勢項，用以打破傅立葉級數之數學週期性限制，使模型得以在保有計算優勢之同時，容許水量之淨累積。 將此架構回溯應用於七場颱風事件（2008–2015年）後，最佳化結果顯示，透過系統性地重新分配七座排水結構之出流，平均每場事件可較歷史操作增加53 Mm³之蓄水量，此為相對於歷史操作之顯著改善。在所有事件中最一致之發現為：於泥砂濃度較高之時期，發電廠取水口（PPI）之使用量在各情境下均可減少63–97%，並將流量改道至專為防洪設計之結構。儘管兩套公式化模型在數學結構上存在根本差異，其於最終蓄水量之預測結果仍能達到±0.15%以內之一致性，驗證了解之穩健性。頻域方法可達成40–60%之運算加速，並使求解器疊代次數減少30–40%，使大量情境模擬與隨機模式預測控制（MPC）在計算上成為可行。泥砂結果顯示，蓄水與排砂之間存在受目標函數權重所支配之內在權衡關係：於淤積型事件中，最佳化之保水策略會保留較歷史操作更多之泥砂；而於歷史沖蝕型事件中，則可降低淨河床沖蝕量。本架構建立了回溯性之績效基準，識別出無需新建基礎設施即可實施之操作性改善措施，並展示頻譜方法如何能大幅簡化水資源管理領域中大規模非線性最佳化問題之求解。

Reservoir operation during typhoon events in Taiwan presents a complex multi-objective optimization problem where operators must simultaneously maximize water storage for the dry season, minimize sediment retention to preserve long-term capacity, and maintain flood control while protecting sensitive infrastructure. This thesis develops a dual-formulation mathematical optimization framework to evaluate historical operations at Shihmen Reservoir and quantify achievable improvements. The framework solves the problem in both the time domain (Hybrid model: ~1,670 variables) and frequency domain (Network model: ~168 variables using Fourier series). The frequency-domain formulation fundamentally simplifies the optimization through dimensional reduction (10× fewer variables), improved numerical conditioning (orthogonal Fourier basis), and inherent solution smoothness. A novel linear drift term breaks the mathematical periodicity of Fourier series, enabling water accumulation while preserving computational advantages. Applied retrospectively to seven typhoon events (2008–2015), the optimization identifies an average storage gain of 53 Mm³ per event – a significant improvement over historical operations – by systematically reallocating releases across seven discharge structures. The most consistent finding across all events is a 63–97% (across all scenarios) reduction in power plant intake (PPI) usage during sediment-heavy periods, redirecting flows to purpose-designed flood-control structures. The two formulations agree to within ±0.15% on final storage despite fundamental mathematical differences, validating solution robustness. The frequency-domain approach achieves 40 – 60% computational speedup and 30 – 40 % fewer solver iterations, making ensemble calculations and stochastic model predictive control tractable. Sediment outcomes reveal an inherent storage-sediment trade-off governed by objective function weights, with optimal water-conservation strategies retaining more sediment than historical operations in depositional events but reducing net bed erosion in historically erosive events. The framework establishes retrospective performance benchmarks, identifies actionable operational improvements requiring no new infrastructure, and demonstrates how spectral methods can dramatically simplify large-scale nonlinear optimization problems in water resources management.