鐵路運輸系統的能源消耗佔了營運成本的相當部分，因此採取節省能源之列車運行策略，將能降低鐵路運輸的營運成本。影響列車行駛耗能的重要因素為路線條件如坡度、曲率半徑等，以及列車行駛速率。因此，列車運行所消耗的能源最小化為規劃列車運轉曲線的重要目標。
　　傳統上列車運轉曲線之規畫多以系統模擬為之。此種方式較難以系統化的方法解得最佳曲線。因此本研究嚐試利用數學規劃方法，求解列車之最佳運轉曲線。本研究以列車的牛頓運動方程式為基礎，建立列車機械特性、允許最大速率與加速度、行駛模式、速率、距離、與耗能間之數學關係。所考慮的列車運行方式有加速、等速、惰性運轉、減速及再生式電力煞車五種運行模式。各種運行模式各有其不同的行駛特性，因此所需之數學式亦不盡相同。本研究以數學方法分析在不同的運行模式下，行駛距離、時間、速率、及耗能之間的關係。之後利用這些關係式建構非線性混合整數規畫數學模式。該模式假設列車行駛路段可分割為多數等長度之區間，每一區間中不變化運行模式。模式之主要決策變數為列車在各區間之分界點之行駛速率、各區間之運行模式、以及各區間之耗能。其目標在追求總耗能量極小。因此在所擬之假設下，該數學模式可描述列車之運轉狀況，模式每一個可行解為一組滿足所有規定與假設之運轉曲線，而其最佳解則為所有可行運轉曲線中耗能量最低者。
　　上述模式之許多限制式以及目標函數為高度非線性函數，且決策變數中含有大量整數變數。因此難以利用解析方法求解。為此本研究採用基因演算法求解並獲得良好成果。求解時將每一區間之運行模式及加速度編碼成染色體，以使基因演算法中之每一個解對應一組運轉曲線。為了提高搜尋效率，求解時並鬆弛部份限制式以擴大可行解區間。除了各解之耗能量之外，各個解對這些限制式的違反量則計入適存值函數中，以在求解過程中逐步淘汰不可行及不良解。演算法依列車行駛之特性分為內外兩層，外層以運行模式基因為進行交配與突變的主要對象，其目的在搜尋優良之運行模式組合。內層以加速度基因為主要變異對象，求解對應於外層運行模式之最佳加速度組合。外層的每個個體，均進入內層演算以求得最佳之加速度組合。經由內外層反覆運算，逐世代演化，達演算停止條件後結束。模式驗證方面則以C語言撰寫程式，求解本模式之問題。經測試例驗證模式之正確性，並由求解結果探討運行模式之選擇對列車運行的影響。

　　Energy cost makes a significant part of the operation cost of a railroad system. Therefore, an energy-efficient operating strategy can potentially be benificial to railroad systems. Major factors that affect the energy consumption of a train are track parameters such as grade and slope, as well as the speed and operating mode of the train. Therefore, minimizing the amount of energy consumed becomes an important goal for the task of speed profile planning.
　　Traditionally, speed profiles for trains are generated by simulation. It is difficult to optimize the plan with this approach. In this research we attempt to optimize the speed profile with mathematical programming methods. Based on fundamental Newtonian equations, we first establish the mathematical relationships between allowed maximum speed and acceleration, operating mode, train speed, driving distance, and amount of energy consumed. The five operating modes considered are accelerating, uniform motion, coasting, braking and regenerative braking. The train behaves differently under each of the modes and different formulae are developed for each mode. After formulating the relationships between these factors, we develop a non-linear mixed integer programming optimization model based on these relationships. The optimization model divides the entire trip into segments of equal length, and assumes that only one operating mode is used in each segment. Major decision variables used in the model are the traveling speed of the train at the dividing points between segments, the mode used in each segment, and the energy consumed in each segment. The optimization goal is to minimize the total energy consumption. Under the assumptions made, the model is capable to discribe the train’s operation. Every feasible solution of the model correspond to a speed profile of the train that satisfies all regulations and assumptions, and the optimum solution is the feasible speed profile that uses the least amount of energy.
　　The objective function as well as many constraints of the resulting model are highly non-linear, and many of the decision variables are binary integers. Thus it is extremely hard to be solved analytically. Therefore, we developed a Genetic Algorithm for the model and yielded good results. In the heuristic the operating modes and acceleration of the segments are encoded to form chromosomes so that each solution corresponds to a speed profile. Some constraints are relaxed to enlarge the feasible region in order to achieve better efficiency. The energy consumption as well as violation of the relaxed constraints are weighted and added into the fitness function, and infeasible or inferior solutions can be gradually phased out in the evolution process. The heuristic is consisted of two tiers. The outer tier processes genes corresponding to operating modes and searchs for good combinations of modes. The inner tier solves for good allocations of accelerations with respect to a given mode combination. When solving, each solution is processed in the outer tier first, then enters the inner tier. The heuristic terminates when a stopping criteria is reached. We implimented the heuristic with the C language. Testing results confirmed the correctness of the model and heuristic, and the effects of mode allocation to energy consumption is discussed.

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