## 2 Multi-Objective Operation Model
### Objective Function and Constraints  
Speed profile optimization is a multi-objective optimization problem, requiring the formulation of a comprehensive model to analyze and determine optimal operational profiles. This study considers four key performance indicators: energy consumption, punctuality, comfort, and safety. Each indicator is formally defined as follows:  
The punctuality indicator \(K_{t}\) is quantified as the absolute deviation between the actual running time \(T\) and the planned running time \(T_{0}\), which is expressed as follows:  
\[K_{t}=T-T_{0}, \tag{4}\]  
\[T=\sum_{i=1}^{n}\Delta t, \tag{5}\]  
where \(T\) represents the total elapsed time (s), \(T_{0}\) denotes the scheduled running time (s), \(\Delta t\) denotes the time step (s), and \(n\) indicates the total number of time steps.  
Given the lack of a regenerative braking capability in railway catenary maintenance vehicles, this study employs an energy consumption calculation method based on the work performed by the traction force. The energy consumption indicator \(K_{e}\) is defined as follows:  
\[K_{e}=\frac{1}{\eta_{e}}\int_{0}^{T}F(t)\cdot v(t)dt, \tag{6}\]  
where \(F(t)\) represents the traction force, \(v(t)\) denotes the operating speed, and \(\eta_{e}\) denotes the efficiency of the traction system.  
The comfort indicator \(K_{c}\) is quantitatively defined using the rate of change of acceleration [19] and is expressed as follows:  
\[K_{c}=\frac{1}{n}\sum_{i=1}^{n}\frac{|\Delta a_{i}|}{\Delta t}, \tag{7}\]  
where \(\Delta a_{i}\) represents the change in acceleration within the \(i\)-th time step \(\Delta t\).  
The safety indicator \(K_{v}\) is defined according to railway operational requirements as follows [6]:  
\[K_{v}=\max\{\exp\biggl{(}\frac{v(t)-v_{\text{lim}}(s(t))}{v_{\text{lim}}(s(t)) }\biggr{)}\}, \tag{8}\]  
where \(v_{\text{lim}}(s(t))\) and \(v(t)\) represent the speed limit and the vehicle's speed at time \(t\), respectively. The safety indicator \(K_{v}\) exhibits an inverse relationship with the proximity between the vehicle's speed \(v(t)\) and the speed limit curve \(v_{\text{lim}}(s(t))\). Specifically, \(K_{v}\) is a dimensionless indicator; a smaller value of \(K_{v}\) indicates a greater distance between the operating speed \(v(t)\) and the speed limit curve \(v_{\text{lim}}(s(t))\), reflecting a higher level of operational safety.  
Therefore, the multi-objective optimization function for the running curves of section operations, along with the function's associated constraints, can be formulated as follows:  
\[F=\min\{K_{t},K_{e},K_{c},K_{v}\}, \tag{9}\]  
\[s.t.\left\{\begin{array}{l}v=ds/dt\\ F(v)-W(s,v)-B(v)=m\cdot dv/dt\\ v(S_{SP})=v(S_{TP})=0\\ v(t)\leq v_{\text{lim}}(s(t))\\ |S-S_{TP}|\leq\Delta S\\ T\leq 1.3T_{0}\end{array}\right., \tag{10}\]where \(m\) represents the mass of the operational vehicle and \(F(v)\), \(W(s,v)\), and \(B(v)\) denote the traction, resistance, and braking forces, respectively. The system dynamics are governed by these forces.  
The multi-objective optimization function integrates four key performance indicators: energy consumption, punctuality, comfort, and safety. In addition, the optimization framework is governed by several operational constraints to ensure practical applicability, safety, and precision. First, the vehicle's speed must be zero at both the starting point \(S_{SP}\) and stopping point \(S_{TP}\). Second, the operational speed \(v(t)\) must not exceed the designated speed limit \(v_{\lim}(s(t))\) at any point during the journey. Additionally, the stopping position error \(\Delta S\), defined as the distance between the actual stopping point \(S\) and the target stopping point \(S_{TP}\), must not exceed 0.5 m. Finally, the total running time for each section \(T\) must not exceed 1.3 times the planned time \(T_{0}\). These constraints ensure that the optimization framework aligns with real-world operational requirements while maintaining safety and precision.