## 3 Model Formulation
### Objectives  
Stop planning should not only satisfy the required travel demand of passengers, but also improve the efficiency of practical operation, where the trade-off between supply and demand is incorporated. In this paper, we study the impact of stop planning on capacity utilization of HSR lines, and mainly focus on two aspects: (a) improving capacity utilization by balancing the stop distributions on segments, and (b) improving the service level for passengers by decreasing passenger additional travel time. (1) Stop distribution imbalance on segments  
As for a segment along the HSR line, more homogeneous trains with approximate travel speeds would improve the capacity utilization of the segment. However, the diverse stop patterns of passing trains result in the difference in stop density in different spatial-temporal zones, leading to the difference in travel times and travel speeds among trains [49]. With a given line capacity, the difference in travel time and travel speed would obviously impact the utilization ratio of line capacity. Thus, to improve the homogeneity among trains, we need to balance the stop distribution of passing trains along a specific segment. The homogeneity of trains along the same segment can be represented by the balance of stop distribution. To reach approximate travel speeds among trains along the same segment, the travel time, especially the stop time (including stop, acceleration, and deceleration time), should be proportional according to the travel distances. We adopt the difference in stop time ratios along the same segment to measure the stop distribution imbalance. Since the applied metric would be independent of the scale of participants in the system, we can adopt the min-max based index to optimize the largest stop time ratio among trains along a specific segment [24]. We decompose the HSR line into \(N_{seg}\) directed segments by high-level stations, especially stations with starting/ending facilities. As for segment \(u\), \(\eta_{l,u}\) represents whether train \(l\in L\) passes this segment: if train \(l\) passes segment \(u\), then \(\eta_{l,u}=1\); otherwise, \(\eta_{l,u}=0\). If \(\eta_{l,u}=1\), then the sequence numbers of the two end stations of train \(l\) at segment \(u\) are, respectively, \(i^{\mathrm{l}}_{l,u},i^{\mathrm{l}}_{l,u}\). The measurement index of stop distribution imbalance along the HSR line is  
\[Z_{1}=\sum_{u=1}^{N_{seg}}\max\biggl{\{}\left(a^{l}_{i^{\mathrm{l}}_{l,u}}-d^{ l}_{i^{\mathrm{l}}_{l,u}}-\sum_{i=i^{\mathrm{l}}_{l,u}}^{i^{\mathrm{l}}_{l,u}-1}t^{ \mathrm{run}}_{l,i}\right)/\left(a^{l}_{i^{\mathrm{l}}_{l,u}}-d^{l}_{i^{ \mathrm{l}}_{l,u}}\right)\bigg{|}\eta_{l,u}=1,l\in L\biggr{\}} \tag{14}\]
2. Additional time ratio in passengers' journeys  
Stop planning should aim to satisfy passengers' travel desire and decrease unnecessary time waste in their journeys. As for time-varying demand, the whole travel time of a passenger includes the boarding deviation time, running time, stop time, and transfer time [1; 55]. In fact, the ideal travel scheme for an O-D pair is direct travel from the origin station to the destination station without any adjustment, intermediate stop, or transfer. Thus, despite the necessary direct travel time, we need to reduce the time consumption of other issues. Let \(T(r,s)\) be the actual total travel time of passengers from station \(r\) to station \(s\) and \(T_{0}(r,s)\) be the total time consumption of the direct travel scheme without any additional issue; then, the additional time ratio is  
\[Z_{2}=\sum_{(r,s)\in RS}[(T(r,s)-T_{0}(r,s))/T(r,s)] \tag{15}\]  
where \(q_{rs}\) is the demand volume of O-D pair \((r,s)\). According to the constraints and objectives, we construct a bi-level programming model to describe the stop planning optimization process. The upper level optimizes the stop plan subject to relevant constraints, while the lower level assigns passenger flow by the schedule-based passenger flow assignment method in [PERSON] et al. [1] and [PERSON] et al. [55] to evaluate the stop plan.