A spatially disaggregated model for the technology selection and design of a transit line

Abstract

Our research question is the usefulness of a high level of spatial granularity for the travel demand when planning a transit line. We formulate a new optimization model for the technology selection and design of a transit line where the spatial attributes of the travel demand can be finely set. The solution method relies on approximated formulae, and we establish relationships with a classic result for the optimal stop spacing. We also present a refinement of the in-vehicle passenger crowding for an existing transit design model where demand spatial attributes are set synthetically. We call “spatially disaggregate” and “spatially aggregate” the former and the latter model, respectively. These two models are compared by numerical experiments on a scenario for three semi-rapid transit technologies where two variants consider opposite demand profiles in terms of spatial distribution. We conclude that the spatially aggregated model is sufficient when the main goal is technology selection, whereas the spatially disaggregate model is better for design and benchmarking purposes.

Appendix A — Formulae shared by the spatially disaggregated and aggregated models

Here we report formulae that are common to the spatially disaggregated model presented in this paper and in the spatially aggregated model of Moccia et al. (2018).

The average speed excluding user service at stops, $S_{\text{run}}$, is

$$\begin{aligned} S_{\text{run}} = \frac{1}{\frac{1}{S_{max}} + \frac{t_u}{60}}&\,\,\,\,\,S_{max}[\text {km/h}], t_{u}[\text {min/km}]. \end{aligned}$$

(38)

Let ${\bar{a}}$ and ${\bar{b}}$ be the average acceleration and deceleration rates of a TU. The incremental time loss caused by the acceleration and deceleration phases is denoted by $T_{a}$, and is equal to

$$\begin{aligned} T_{a} = \dfrac{S_{\text{run}}}{25920} \left( \dfrac{ 1 }{{\bar{a}}} + \dfrac{ 1 }{{\bar{b}}} \right)&\,\,\,\,\,S_{\text{run}}[\text {km/h}], {\bar{a}}, {\bar{b}} [\text {m/s}^{2}], \end{aligned}$$

(39)

(see, e.g., Vuchic and Newell 1968).

The lost time for acceleration, deceleration, and door opening and closing, $T_{l}$, is

$$\begin{aligned} T_{l} = T_{a} + \frac{t_d}{3600}&\,\,\,\,\,T_{a}[\text {h}], t_{d}[\text {s}]. \end{aligned}$$

(40)

The average waiting time $t_w$ of a user is

$$\begin{aligned} t_{w}(f) = {\left\{ \begin{array}{ll} \dfrac{w}{60} + \mu \dfrac{\epsilon }{f} &{} \text {if} \,\,\,\, f < f_{l}\\ \dfrac{\epsilon }{f} &{}\text {if} \,\,\,\, f_{l}\le f \le f_{m},\\ \dfrac{\epsilon }{f_{m}} &{} \text {if} \,\,\,\, f > f_{m} \, \,\,\,\, f[\text {TU/h}], w[\text {min}] \end{array}\right. }. \end{aligned}$$

(41)

The lower bound for the frequency, $f_{\text{min}}$, is

$$\begin{aligned} f_{\text{min}} = \max \left\{ f_{pol}, \dfrac{\alpha q \tau }{\nu k n}\right\}&\,\,\,\,\,f_{pol}[\text {TU/h}],q[\text {pax/h}], k[\text {pax/veh}], n[\text {veh/TU}].&\end{aligned}$$

(42)

The threshold value of the stop spacing, $d_{\text{min}}$, depends on acceleration and deceleration rates as follows

$$\begin{aligned} d_{min} = \dfrac{S_{max}^{2}}{25920} \left( \dfrac{ 1 }{{\bar{a}}} + \dfrac{ 1 }{{\bar{b}}} \right)&\,\,\,\,\,S_{max}[\text {km/h}], {\bar{a}}, {\bar{b}} [\text {m/s}^{2}], \end{aligned}$$

(43)

(see, e.g., Vuchic and Newell 1968).

The capital amortization per hour of service is computed as

$$\begin{aligned} \frac{ P(1 - \varXi ) \iota }{H (1 - (1+\iota )^{-y})}, \end{aligned}$$

(44)

where P is the purchase price, H is the number of service hours in a year, $\iota$ is the discount rate, $\varXi$ is the fraction of the residual value, and y is the one-stage technical life. The one-stage technical life is lower than a typical service lifetime because it expresses the equivalent years including the cost of a mid-life rebuild at the prevalent discount rate.