MODELING OF PUBLIC PASSENGER TRANSPORT SYSTEMS

 

ABSTRACT

This paper addresses the complex problem of mathematical modeling and simulation of the structure of passenger public transport networks in urban agglomerations. The general design requirements and the objective function of optimizing the public transport networks are formulated. As a result, the modeling and simulation cycle, the mathematical model and the software necessary for the synthesis of the optimal public transport networks, were developed and adapted structurally and functionally to existing transport demand. Trip distribution and modal split was performed by probabilistic methods.

The mathematical model was applied for the synthesis of optimal solutions for the reorganization of the Chisinau public transport network. 

АННОТАЦИЯ

В статье рассмотрена комплексная задача математического моделирования сетей пассажирского общественного транспорта крупных городов. Сформулированы общие требования к проектированию и целевые функции оптимизации сетей общественного транспорта. Разработаны и адаптированы к спросу на перевозки основные этапы моделирования, математическая модель и программное обеспечение, необходимые для проектирования оптимальной структуры сетей общественного транспорта. В процессе моделирования широко используются вероятностные методы.

Математическая модель использована для поиска оптимальных решений по улучшению сети пассажирского общественного транспорта города Кишинева.

 

Keywords: transport, network, model, simulation, zoning, chart, flow, optimization, software, distribution, split.

Ключевые слова:  транспорт, сеть, модель, моделирование, зонирование, диаграмма, поток, оптимизация, программное обеспечение, распределение.

 

1. Introduction

Accelerated urbanization, combined with technological innovations and digitization of most areas, generates significant changes in the economic environment, influencing the quality of life of people. The comfort of the urban environment is strongly determined by mobility, as the basic support of economic growth. Improving the quality of life in modern cities is directly linked to solving environmental problems and improving modern urban mobility systems for people and goods.

The main objective of the network designer is to improve public access to a safe and sustainable transport system by improving the scheme and operational program of the urban public transport system in conditions of technological and economic efficiency, safety and environmental impact reduction.

The judicious location of residential neighborhoods and centers of economic and social attraction in large cities is the pillar of sustainable development of the town and public transport system, ensuring the optimal level of mobility, harmonization of urban architecture and rational use of land.

The integration of the public transport system in the urban environment continues to be a challenge of great complexity and relevance because the transport solutions make their mark on the planning of urban spaces. According to current studies, the economic effect of modernizing public transport systems exceeds more than five times the investments made for this purpose.

 The prospects of evolution of public transport systems depend on the ability of the authorities to respond promptly to public expectations. Transport comfort and safety remain the most important quality criteria.

Traditional planning of public transport systems requires focusing efforts on increasing the capacity of transport infrastructure.

While modern sustainable transport systems aim to increase integration with the urban environment, intensify coordination of different types of public transport and develop green transport systems.

The complex subject of the design of public transport systems in large cities is a constant scientific concern due to the importance of this component for ensuring the quality of transport services [1], [2], [3].

Within the elaboration of the general urban plans and complex transport schemes of the big cities, the compartment dedicated to the public transport system is approached as a correlated one. However, lately the interest of urban administrations in the impact of urban modernization on land use and the public passenger transport system has increased [5], [6], [7].

Modeling of urban trip flows and public transport systems are widely applied to better understand the specifics of the city and for the synthesis of the most appropriate transport optimization solutions. The simulation on models ensures the identification of priorities for the development of public transport systems.

Both in practical and scientific terms, the improvement of the modeling of public passenger transport systems is of major interest.

2. General requirements for modeling public passenger transport systems

Remarkable successes have been achieved in the field of public transport system modeling in terms of algorithms and adaptation of the mathematical apparatus to the specifics of the field. Nevertheless, the problems of evaluating the legitimacy of the emergence and evolution of real transport demand in space and time have not been definitively solved [1], [2], [10].

The statement of the problem of modeling public transport systems is described as follows: for a city with known territorial, demographic, economic, social and traffic characteristics, it is necessary to determine the demand for public transport and to model the structure of the public passenger transport network that fully and safely covers the real mobility of the public, applying the pre-established efficiency criterion of that system [10], [11].

The foundation of urban transportation system modeling was developed in the United States in the 1950s [1]. Most modern models of public passenger transport systems are based on heuristics of choosing the optimal option by comparing possible alternatives for structuring the transport system based on technical and economic optimization criteria [8], [9].

The demand for public passenger transport represents the category n of persons, the places i of generation and j of destination, the type k and the route r of public transport, used by the public for travel, which is a function of the specific parameters of the city [9].

All models of public transport systems apply the classic procedure of zoning the urban territory (figure 1).

The main characteristics of zones, necessary for the modeling of the public transport system, are the following: total population, age and social structure of the population, active population rate, number of jobs, places and work regime of the active population, places of attraction for passive population, distribution of trips by existing modes of transport (mode choice), distribution of population by income.

The general conditions for the models of public passenger transport system are as follows [1], [10]:

▪ to fully satisfy the demand for public passenger transport, to ensure the distribution of the trips by directions so that the trips are operated on the shortest roads, with a minimum number of transfers, taking into account the intensity of road traffic and the transport capacities of the road infrastructure;

 

Figure 1. Transport zoning of the urban territory. Case study: Chisinau city

 

▪ to contribute to uniform distribution of passenger flow along the length of each route of the public transport system, by route sectors and by types of public transport;

▪ to ensure the full use of the dynamic and technological potential of public transport vehicles by rationalization of operational regimes and eliminating road sectors with major traffic problems;

▪ to design the configuration of the public transport system based on the economic and/or technological criterion pre-established by the city administration, in conditions of maximum collinearity of the network and high route frequency;

▪ ensure high functional flexibility and minimize the costs of adapting the system to changes in public transport demand, as well as adequate coordination of the operational program with interurban transport.

Most models for the design of traditional public passenger transport systems use as optimization criterion the minimum value of the total public travel time T_T for the calculated network of routes. The value of total public travel time is determined by the relation [6], [7]:

                          (1)

where: r = 1, 2,…, u are the routes that form the analyzed public transport system;

s = 1, 2,…, f – stop stations on the route r;

Q_sr - number of waiting passengers at the stop s of route r, passengers;

t_ar - average of passenger waiting time on route r, in minutes;

Q_ij - the number of passengers traveling from zone i to zone j, passengers;

t_dij - travel time of passengers from zone i to zone j, in minutes;

t_tij - transfer time in network nodes between zones i și j.

Another possible comprehensive criterion for the efficiency of the public transport system, used in the modeling process, is the criterion of minimizing the operational costs for companies providing public passenger transport [8]. This criterion is in clear contradiction with the criterion mentioned previously and can be used only to compare various network alternatives.

In the last period of time, an attempt is made to apply a general optimization criterion, which on one hand should take into account the minimization of the total time expenses of passengers, and on the other hand to contribute to the reduction of expenses for transport operators. This type of criterion is a reasonable compromise in the sense that it ensures the technological and economic balance between all participants involved in the public passenger transport process [8].

The density of the passenger flow along the routes forming the synthetic transport system, as a new comprehensive criterion for the efficiency of the public passenger transport system, is determined by the relation [9]:

, passenger/hour ∙km                         (2)

for the following modeling conditions:

, km                                         (3)

, passengers                                      (4)

where: D_ij is the average density of passenger flow at line ij, in passenger /hour ∙km;

Q_ij - the flow of passengers on the public transport line ij, passengers /hour;

L_ij - length of public transport line ij, km;

L_min - minimum length of the public transport route, km;

L_max - maximum length of the public transport route, km;

Q_min - minimum flow of passengers on the route, passengers/hour.

This criterion describes the distribution of the intensity of passenger public transport demand along the chosen route.

The paradigm of this type of models presumes the combination in a public transport route of the successive sectors of the urban transport network with major values of the intensity of passenger transport demand.

Maximizing the density of passenger flow can be achieved by varying the length of the designed routes (L_ij = var). The optimal density of passenger flow determines a reasonable value for the length of the synthesized public transport route.

The limit values L_min, L_max are determined depending on the dimensional characteristics of the urban agglomeration and its sectors.

In order to improve the quality of services on lines with different values of passenger flows and to correlate the real flow with the available transport capacities, the model should determine a set of maximum passenger flow densities along the route, adapted according to capacities to the type of vehicles available.

The modeling of public passenger transport systems is based on the application of the classic four-step calculation procedure, as follows [1], [10]:

1) trip generation - for each urban area, determine the total number of trips, arrivals and departures based on mobility indicators and socio-economic data related to zones;

2) trip distribution - passengers’ trips are distributed in pairs of zones (i, j) based on a computational algorithm, creating the origin-destination matrix of the city;

3) modal distribution - journeys are distributed according to the mode of transport used on the basis of modal distribution curves, obtaining the origin-destination matrix for each mode of transport;

4) route assignment (traffic distribution) - estimates the travel volume by mode of transport considered for each direction of the network.

3. Modeling of public passenger transport systems

Modeling and simulation on validated models is the most appropriate solution for cases when direct experimentation on the real object is impossible.

 This situation is also valid for public transport systems [1].

The mathematical model of a system is its simplified image, which is sufficiently accurate, applicable to the multilateral study of the modeled object. Modeling accentuates potential critical processes and ensures the comparison of a wide range of decision-making alternatives.

The representativeness of the model can be improved through an iterative process of increasing sequentially the complexity of the simplest initial model, based on the best argued hypotheses.

Simulation is the activity of practical application of a real system model and carrying out of numerical experiments to establish the dependence between input and output data. Thus, the functioning of a real system is imitated by generating artificial situations and observing them in time and space, anticipating the evolution of the modeled structure.

The structure of the modeling process of the public passenger transport system is presented in table 1. The procedure involves seven stages of modeling, preceded by a series of activities to prepare the initial database. A clearly formulated final result is planned for each modeling step.

The initial data applied in the model developed by the authors [12] include:

- coordinates of the nodes of the urban transport network (graph peaks), m;

- the graph of the road transport network L_ij, with the components l_ij, km;

- transport demand matrix Q_ij, with sectorial components q_ij, pas.;

Table 1.

The structure of the modeling process of the public passenger transport system

Initial data		Modeling stage		The result of modeling
1. Urban road network	→	A. Elaboration of the graph of the public passenger transport system	→	Graph of the public passenger transport system
2. The urban road traffic     scheme
3. Charts of public     passenger transport
4. Past transport demand	→	B. Choosing the method          of studying passenger      flows	→	Study methodology of transport demand
		C. Study of passenger flows		Current demand for public transport
		D. Distribution of demand in the public transport network		Passenger flow charts
5. Zoning the city	→	E. Calibration of the public transport system model	→	The model of the public transport system
6. Characteristics of the areas		F. Model validation
7. Analysis of the strategy of system development      public transport	→	G. Simulation on the model in order to synthesize the optimal structure of the public transport system	→	Optimal configuration of the public passenger transport system

 

- boarding capacity of public passenger transport units, q, seats;

- average transfer times for each transfer node, t_tx , min.;

- headway, I_max, min.;

- the coefficient of non-uniformity for passenger’s arrival at the station, k_s;

- the time interval for which the calculation is performed, T_c, min.;

- the coefficient of non-uniformity, k_n.

The initial phase of the simulation of the public transport system is based on the model [12]. With the help of classical algorithms, the problem of determining the minimum length roads L_ij^min is solved, the solution being exposed in matrix form (figure 2, figure 3).

Subsequently, combinatorial analysis summarizes the initial list of routes, which includes all possible combinations, except for the short routes that connect the neighboring nodes. Routes that do not meet pre-established requirements are deleted from the initial list [1], [10].

The routes that do not fall under the headway limit established for the modeled transport system, are excluded from the list of short routes that directly connect the neighboring nodes of the city graph.

 

Figure 2. Software’s interface based on the model [12]

 

As a result of simulation experiment, the initial configuration of the public passenger transport system for the selected zoning of the city is summarized, for which the value of the applied efficiency criterion of the system is calculated.

To validate the model [12], presented above, simulations were performed for the examples presented in some monographs in the field [10], [11], which confirmed in general terms its practical applicability. In order to improve this model, a long-term comparison with the results of calculations based on high-performance transport planning software (TransCAD, VISUM) is required.

4. Modeling the „origin-destination” matrix

The compliance between the elaborated configuration of the public transport system and the optimal one reflects the accuracy of the modeling for the distribution of transport demand on the arcs of the graph of the public transport network.

The closer the performed distribution is to the actual one, the more appropriate the transport system will be modeled.

 

Figure 3. Results of a model simulation experiment [12]

 

The „origin-destination” matrix is a square one with the same number of rows and columns, equal to the zones number of the city. The elements of the matrix represent the number of trips between all pairs of zones of the city in both directions, respectively.

Model [12] applies as the key method the gravitational model for calculating the passenger correspondence between two zones i and j of a city [10]:

, passengers                               (2)

where: Q_i , Q_j are the populations of zone i and j, respectively, persons,

R_ij - resistance function, representing a value for the difficulty of movement from zone i to zone j;

α, β, γ - empirical calibration coefficients of the model, calculated according to the measurements of passenger flows.

Usually α = β = 1 is adopted, and the value of γ is calculated from the flow measurement data [10].

To compare the results of modeling the „origin-destination” matrix by the deterministic and, respectively, probabilistic method, in the model [12] the entropic model is also applied, which determines the correspondence of passengers between zones i and j with the formula:

, passengers                                  (3)

where: P_ij is the probability that the trip from zone i will end in zone j, taking into account the complexity of the trip and the hypothesis that the trips from zone i are distributed to zones j depending on the share of jobs (studies, etc.) in the zone j from all destination zones.

                                          (4)

where: k_E represents the equilibrium coefficient, iteratively evaluated.

The model also applies the opportunity method for calculating the probability of travel from zone i to zone j.

The hypothesis that the probability that a destination becomes the purpose of a move is constant allows the ordering of the areas from the nearest to the farthest, the journey to the nearest area having the probability P, to the second P·(1-P), until to n·(1-P)^n-1.

Thus, in order to distribute Q_i passengers, the distances from the initial zone and to all other zones will be ordered increasingly.

For the zone i given in the case of m possible destinations and the number of destinations between i and j equal to n the passenger flow shall be:

, passengers                        (5)

It should be noted that in the case of applying the opportunity method for assessing passenger flows, it is not the distance between zones that matters, but the rank of the zone in the increasing by value set.

5. Demand modal split for public transport

In the case of modeling a public passenger transport system that integrates the activity of several types of public transport, the procedure for splitting the transport demand is applied, which determines the proportion in which the transport demand belongs to a certain mode of public transport.

Usually, modern models integrate all possible types of trips: on foot, by bicycle, by car, by bus, trolleybus, tram, subway or a combination of several modes of transport.

Currently, experts in the field highlight the following factors for the choice of the preferred type of transport by individuals [1], [10]:

▪ the level of comfort and safety, as well as the operational characteristics of the modes of transport;

▪ the financial potential and economic status of the passenger;

▪ particularities of the trip process.

In the empirical models the main criterion of modal split of the transport demand is the duration of the trip, the promptness of the public transport being appreciated as very important by the absolute majority of the passengers [3], [9]. 

Empirical models are frequently replaced by probabilistic models of modal split of transport demand [1], [11].

The public passenger transport system, consisting of z modes of public transport, can be modeled with a multinomial logit model, the probability of choosing a certain mode of transport is determined by the following relation:

                                             (7)

where : C_tij   represents the cost of trip with mode t of transport,

μ - coefficient of the model.

6. Aspects of the application of the modeling of the public passenger transport system

The development of the model [12] was conditioned by the lack of cheap, simple and efficient tools for modeling public passenger transport systems at local level. There is no financial potential for the acquisition of modern urban passenger transport planning software, and the abilities and motivation for the application of these instruments by the central and local public administration are still lacking.

Lately, due to the accelerated increase in the level of motorization, the road traffic situation during peak hours in Chisinau has worsened considerably. As a result, the activity of the public passenger transport system is increasingly affected by this negative phenomenon.

The improvement and application of the analyzed model were done systematically for scientific and practical reasons.

In recent years, following the simulation on the model based on the methodological diversity set out above, some relevant suggestions can be made:

▪ none of the projects of the public transport system of Chisinau, developed in the last 15 years by foreign institutions in European projects with the application of the most modern software, can be implemented for various reasons, including due to chronic budget deficit and the need of cardinal change of the existing system;

▪ currently the real demand for public passenger transport in Chisinau and the 2025-2030 forecasts do not argue the implementation of the routes of high-capacity articulated vehicles with heavy traffic, suggested following the simulation on the models applied in European projects;

▪ the development of the municipal passenger transport system is more intuitive and chaotic, with the arbitrary, politically motivated inclusion of new routes in the existing system, without analyzing the efficiency and impact of route opening on the existing transport system;

▪ public transport planning imposes to do transport analysis not only in the city of Chisinau, but also for the entire urban agglomeration, based on the major importance of the capital.

7. Final conclusions

Modeling public passenger transport systems is an effective tool for improving the configuration and operation of this vital element for the urban community. Modeling offers reliable solutions for restructuring public transport systems for better adaptation to the real structure of transport demand.

The structure of the modeling process of the public passenger transport system reflects the consecutive order of generally accepted logic regarding the analysis and synthesis of rational public transport networks.

The proposed model uses the most appropriate probabilistic methods of network distribution and modal split of transport demand.

It offers the opportunity for free practical application by specialized subdivisions of the local public administration.

The main problem to apply the model is to accurately assess the real demand for transport, an activity that requires significant financial and human resources.

 

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