Page 170 - 360.revista de Alta Velocidad - Nº 5

Castillo, Enrique. Grande, Zacarías.

KEYNOTE

1. Introduction and motivation

Though when you think of high-speed lines you immediately tend to associate them with double
track lines, not all high-speed lines must necessarily be built in double-track. In fact, the single
track multiplies its capacity almost proportionally to the speed of the trains that circulate through
it. Therefore, an increase of speed immediately produces an increase in the efficiency of the single
track line. This means that single track lines today have important advantages with respect to past
single track lines because train speeds have been increased substantially.
When high speed lines are used in a country for the first time, they are usually oriented to link two
large populations (case 1). However, other lines may involve small and at most one large population
(case 2). In this paper we point out that the above two cases 1 and 2 are completely different and
require different solutions, because: (a) the number of users is necessarily much lower in the second
than in the first case leading to very different train frequencies, which are determined mainly by
the small size city, and (b) we must question ourselves whether or not the expensive double track is
necessary in Case 2, or a new alternative should be contemplated. In addition, since decisions with
optimal criteria are required, computer programs are unavoidable to design, develop preliminary
projects and evaluate the proposed solutions.
It appears that the first known application of the mathematical programming methodology applied
to railway optimization problems was due to (Amit & Goldfarb, 1971) Optimization and simulation
were already the most commonly used methods, even before the eighties, when computers had
no power enough to deal with these complex problems (see (Assad, 1980) and (Haghani, 1987)).
However, nowadays problems related to railway network management and similar cannot be
conceived of without computers (see (Petersen, et al., 1986), (Hellström, 1998), (Yang & Hayashi,
2002) or (Ouyang, et al., 2009)). An exhaustive review of existing optimization methods was done
by (Cordeau, et al., 1998).

One of the main reasons for these railway optimization problems to be complicated is that a huge
amount of continuous and binary variables and constraints are involved, leading to mixed integer
(MIP) linear and non-linear related programming problems of high complexity (see, for example,
(Kraay & Harker, 1995) (Carey & Lockwood, 1995) , (Higgins, et al., 1996), (DÁriano & Pranzo, 2004),
(DÁriano, et al., 2007)), which require a lot of memory and CPU resources (see (Burdett & Kozan,
2010)). As indicated in (Castillo, et al., 2011), (Castillo, et al., 2015), a reduction in binary variables
has an important effect on the CPU time required.

A possible alternative to single track and double track lines is the alternate double-single track
(ADST) line, in which single track segments are combined with double track segments. As will be
verified, the capacity of a single track line is clearly overcome by an ADST line with a very low
construction and maintenance costs increase. In this paper, first, a general formula for dimensioning
the lengths of single track and double track section is derived. Then, these formulas are applied
to several cases that will serve us to illustrate the possibilities that the ADST lines provide from a
practical point of view.

The design of a railway line must always include the predicted demand. If the demand is very high a
double track is the right solution. Contrary, if the demand is very low, the right solution is the single
track. However, in the most common cases of intermediate demands the optimal solution involves
segments in single and segments in double track. Since in this intermediate case we must decide
which segments must be in single and which in double tracks, the demand plays a relevant role in
the optimal solution because the timetable must be optimized in order to reduce travel time. This
implies that the costs of single and double tracks must be included in the objective function of our
optimization problem. In fact, we are in front of a bi-level problem in which the first level decides
about single or double tracks and the second level optimizes the timetables to reduce travel times
and reduce the occupancy of the line.

168 360.revista de alta velocidad