## 1 Introduction
proposed a DP algorithm to generate a driving route by sending the driver's destination information to a server in the cloud, collecting traffic and geographic information of the route, and using it to determine the optimal speed trajectory [11]. In 2004, [PERSON] et al. applied DP to generate an optimal train running trajectory that minimizes the total energy consumption under a fixed starting point and destination, regulated running time, limited electric power, and electric braking in a train driven by a variable-voltage-variable-frequency controlled induction motor/generator [12]. In 2009, [PERSON] et al. applied the DP algorithm to determine the optimal acceleration and deceleration in a wireless-mode railway vehicle [13]. In [14], a DP algorithm was used to address a problem that satisfies the constraints of time, distance, and speed in three dimensions. The authors in [15] used an approach wherein weights were applied to the cost function in the two-dimensional space of distance and speed to reduce the burden of calculation time due to multi-dimensionality. The driving mode for energy optimization was determined by classifying the electric-train operating conditions into four modes (acceleration, cruising, coasting, and braking) and optimizing the sequence of these modes. In [15; 16; 17; 18; 19; 20], to solve the problem of energy optimization in cases when the travel distance and the target travel time are fixed, a weight is applied to the cost function and is repeatedly changed until the target travel time is satisfied. To reduce the computational time, [PERSON] first applied the weighting factor to reduce the three-dimensional DP to a two-dimensional DP [19]. In the first study [19] that optimized train operation by applying weights to DP, the topic of obtaining weight coefficients is not mentioned at all. In [15], a solution that satisfies the target condition was obtained after the weights were changed at least nine times. Here, numerical iterative approaches, such as the bisection method or secant method, are used to find the optimal weights. Based on the four basic operating modes (acceleration, cruising, coasting, and braking), [PERSON] et al. determined the energy-optimization trajectory by optimizing the sequence of the operating modes and their corresponding switching points [16]. [PERSON] et al. used an optimization method selected in consideration of the traffic constraints, unlike previous research results, for eco-driving in car operation [17]. [PERSON] et al. described an optimization approach and presented a method for determining suitable optimization parameters to consider drivers' preferences [18]. Among the three optimization algorithms (ACO, DP, and GA), DP yielded the best results in energy optimization, but more time was required in terms of calculation time [6]. In this study, the DP method was selected as an optimization method considering the adaptability to the disturbance and the speed limit within the operating route. Due to the nature of train operation, if the distance between stations is fixed, it is operated according to the timetable, and, thus, strict adherence to the arrival time is essential. Therefore, in several existing studies, to find a speed profile that optimizes energy while satisfying given constraints, such as distance, arrival time, speed limit, gradient, and curve, a weight is applied to the cost function and is repeatedly changed to find a velocity profile that satisfies the target travel time. Here, the time-consuming process of finding the optimal solution that reflects this change is repeated whenever the weights are changed. This paper presents a method to shorten the iterative process of changing the weights and finding the optimal solution for electric trains. To this end, we, herein, propose an approach to obtain an optimal solution, by composing weights in an array multiplexed by a desired interval and number and processing them in batches using the DP algorithm.