Modelling of Transport · 2014-04-08 · Session 4. Modelling of Transport 147 MESOSCOPIC APPROACH TO MODELLING A TRAFFIC SYSTEM Yuri Tolujew1,2, Mihail Savrasov2 1 Otto …

Session 4.

Modelling of Transport

Session 4. Modelling of Transport

147

MESOSCOPIC APPROACH TO MODELLING A TRAFFIC SYSTEM

Yuri Tolujew1,2, Mihail Savrasov2

1 Otto von Guericke University Magdeburg

Universitätsplatz, 2, Magdeburg, 39106, Germany Tel.: +49391 4090310. E-mail: [email protected]

2 Transport and Telecommunication Institute Lomonosova 1, Riga, LV-1019, Latvia

Tel.: +371 29654003. E-mail: [email protected]

A lot of new mathematical models for traffic systems have been developed in the past. Two approaches are widely used, namely

microscopic and macroscopic models. Both approaches have several deficits. The mesoscopic approach presented here eliminates the deficits inherent in both the microscopic and the macroscopic approach. The paper shows that the mesoscopic approach is suitable for reproducing process sequences in flow systems and describes the use of the mesoscopic approach to modelling and analysing a crossroad, which can be presented as a flow system. Furthermore, the paper presents the conceptual mesoscopic model of the crossroad and its implementation with MS Excel. The modelling task is to estimate the dynamics of all 12 queues and the crossroad capacity utilization. The developed mesoscopic model can easily be extended and adapted. For example, a set of connected crossroads with different parameters can be modelled.

Keywords: mesoscopic modelling, crossroad modelling, crossroad capacity estimation 1. Introduction

In the past a lot of new mathematical models for traffic systems have been developed. Almost all of them can be categorized as macroscopic or microscopic models. Macroscopic models [1] use differential equations and describe the behaviour of traffic flows. In such models long periods of time (days, hours) can be observed. Microscopic models use standard simulation models based among other things on discrete events [2] and cellular automats [3]. Models of this class are used to model short periods of time with a very high level of detail. The mesoscopic approach is known in the field of the traffic simulation for a long time. In traffic simulation, the term mesoscopic is often applied to refer to a combination of macroscopic and microscopic simulation [4]. In [5, 6] a new class of mesoscopic models has been described. The purpose of this model class is to take advantage of the two traditional approaches to modelling flow systems by avoiding their disadvantages like the time and labour consuming creation and implementation of microscopic models.

The basic principles of mesoscopic modelling can be described with “algorithmic management and analytical calculation” and “discrete time and continuous quantities”. The second phrase shows that the philosophy of mesoscopic modelling has similarities with macroscopic modelling with differential equations. However, this analogy can only be observed in the presentation of numerical results. Results are presented as process graphs with the time step Δt. In mesoscopic models relationships between variables are often implemented as complex algorithms and not as concrete formulas. This is characteristic to microscopic simulation.

A mesoscopic model uses mathematical formulas to calculate the results as continuous quantities in every step ∆t of the discrete modelling time. In contrast to the microscopic approach, the mesoscopic approach monitors quantities of objects that belong to a logical group instead of individual flow objects (e.g. customers in Queuing Systems). In contrast to macroscopic models, in mesoscopic models any number of groups of objects can exist at the same time and interactions between them can be implemented.

In [6] it has been shown that “multi-channel funnels” can be effectively used as the main structural component of mesoscopic models. The process of product accumulation and processing can be modelled with this component. Flow processing is done through different strategies of resource usage.

Advantages of the mesoscopic approach are its high flexibility in preparing input data for the simulation, its universal and easy structure of the internal model data, that no restrictions for modelling complex control algorithms exist, its high performance for computing the model code, its clear presentation of simulation results and that the use of commercial simulation software is unnecessary.

The goal of this work is the experimental estimation of the potential of using the mesoscopic approach for the transport flow modelling of a city transport network.

2. Modelling and Source Data Description

The modelling object is a symmetric crossroad with traffic light control as in Figure 1. Transport flows are created in sources 1 to 4. They can pass the crossroad by using the following directions: right (r), straight (s) and left (l). Two opposite sources have their green phase at the same time. During this time the two remaining sources have their red phase. That means that they do not interfere with each other. The modelling task is to estimate the dynamics of all 12 queues and the crossroad capacity utilization. Transport flows are given and traffic lights phases will be determined.

The International Conference “Modelling of Business, Industrial and Transport Systems – 2008”

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Sour

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Fig. 1. Conceptual model of the crossroad

The mesoscopic approach uses the queue length of vehicles waiting at a crossroad for determining the

quantity of vehicles (q). This concept is also used to describe the number of created vehicles and vehicles passing a crossroad. If the number of vehicles in the incoming flow is known, the queue length of vehicles can be easily estimated using empirical data. The flows in the model will be described in meter/minutes (m/min).

Random values are used for the description of all 12 incoming flows. Table 1 shows the numerical parameters of stationary incoming flows, which are used in the example presented here. The number of vehicles for each traffic light cycle is generated by given distribution laws. In this example the cycle duration is 50 seconds (20s +5s+20s+5s). Table 1. Parameters of the incoming vehicles flows

Incoming flow Distribution law Mean (m/min) Left border (m/min) Right border (m/min) Crossroad passing

r1 uniform 20 15 25 0,6 s1 uniform 65 40 90 0,7 l1 uniform 10 5 15 0,6 r2 uniform 20 15 25 0,6 s2 uniform 65 40 90 0,8 l2 uniform 10 5 15 0,6 r3 uniform 15 10 20 0,6 s3 uniform 55 40 70 1,2 l3 uniform 10 5 15 0,6 r4 uniform 20 15 25 0,6 s4 uniform 65 40 90 0,8 l4 uniform 10 5 15 0,6

An empirical function (see Figure 2) is used to

create more realistic process of crossroad passing. This function is estimated through direct observing of real crossroad passing processes. This function can be used for all directions. The function is used in the model as direct and as inverse function. Figure 2 shows that during the first 25.5 seconds of a traffic light cycle a vehicle flow with the length q1=122m can pass the crossroad and in the first 32.5 seconds a flow of q2=185m.

Fig. 2. Dynamic of vehicles flow during crossroad passing

Exemplary chart

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3. Principles of a Mesoscopic Model for a Crossroad

The principal structure of the crossroad model is presented in Figure 3. The model consists of four autonomic components, because direct relationships do not exist between the traffic flows of the four directions. Each component has three parallel channels. The channels generate, delay and release traffic flows. The delay and the release are realized by a multi-channel funnel described in [6]. The content of each channel of the funnel is numerically equal to the length of the queue. The control component of the model (Flow Control) defines the quantity of vehicles which can pass the crossroad in each traffic light cycle for the different directions.

So1

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Fu1 Fu2 Fu3 Fu4Legend:sourcefunnelmaterial f lowinformation flowcontrol datatype of productr1

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Fu1 Fu2 Fu3 Fu4Legend:sourcefunnelmaterial f lowinformation flowcontrol datatype of productr1

SoFu

Legend:sourcefunnelmaterial f lowinformation flowcontrol datatype of productr1

SoFu

Fig. 3. Principal structure of the mesoscopic model

The traffic light cycle length is used as the discrete time step Δt (in this example the cycle length is 50 seconds). In each step Δt, the data shown in Table 2 are calculated and updated. The presented Table 2 is an example of a standard realization of an internal data structure in a mesoscopic model.

Table 2. Kernel of the crossroad mesoscopic model

The calculation of the data for flows of type r and s is trivial. The value of the “phase capacity” is determined with the exemplary chart (Figure 2). If the accumulated length of the waiting flow and the incoming vehicles is smaller than the “phase capacity” the “duration of pass flow” can be calculated with the exemplary chart. Line 1 in Figure 4 shows the process of vehicles entering during a traffic light cycle. Curve 2 presents the process of all vehicles passing the crossroad before the green phase ends. Then tleft begins the interval for the left turn.

The data calculation for direction l (left turn) is done after the data calculation for straight on flow s. If the condition tleft=0 applies during a cycle (Curve 3 in Figure 4), it is assumed that the defined minimal flow of vehicles can complete the left turn. Such case is presented for flow l2 in Table 2. It is assumed that a vehicle flow of 8 meters can pass the crossroad during yellow light. For tleft>0 the “phase capacity” (Curve 4 in Figure 4)


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can be calculated using the exemplary chart for direction l. Then the value of passed vehicles can be calculated taking into account driving during yellow light. Table 2 shows that the flow s2 has passed the crossroad in 17.31s which results in tleft=2.69s for flow l2. Initial values for the queues of all 12 directions can be defined before the model execution begins. The number of cycles for modelling has to be provided. Model execution can be done in two modes: full execution and step-by-step execution, which allows monitoring data changes in every traffic light cycle.

tyellow1 tyellow2tred tgreen tleft

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1qmax qend

tyellow1 tyellow2tred tgreen tleft

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1qmax qend

Fig. 4. Schema of the process for one traffic light cycle

4. Data Output and Interpretation of Simulation Results

Necessary data are copied from Table 2 to the process trace file during mesoscopic model execution. Diagrams of incoming flows (column “arrival per cycle”) and outgoing flows (column “passed through volume”) can be presented in differential and integral (cumulative) forms for all components of the model. The trace file also contains the contents of the funnel (column “remaining on current cycle”).

Table 3 presents the outcomes of modelling 20 cycles. The frames “input (sum)” and “output (sum)” show the total length values of funnel enter and funnel exit. The data in the frame “queue (maximum)” present the maximal lengths of the queues during model execution. The queue lengths are changing during a traffic light cycle. The column “remaining on current cycle” in Table 2 does not present the maximal values of the queues but just the minimum values because the values are written in the trace file at the end of every green phase. Table 3. Output data of the mesoscopic model of a crossroad

The method used for estimating the maximal queue length for each traffic light qmax can be explained using Figure 4. It is assumed that the number of incoming vehicles grows linearly (straight line 1). It is also assumed that the queue length does not grow anymore when the green phase starts. This assumption can be verified through real process surveys. If the queue end reacts to the beginning queue moving during a green phase start the value of qmax approaches to qbegin+qinput (Figure 4). Figure 5 presents the value of qmax for 20 cycles of traffic light. The incoming queues are aggregated for each direction. Table 3 presents the maximum value of a queue in the column “total”.

The source data for modelling (Table 1) are chosen in such a way, that flow s1 is slower (coefficient 0.7) than the other straight on flows. Figure 5 shows that the queue for source 1 has a growing tendency. The queue dynamics for source 2 and 4 is similar since the parameters of all the flows are the same. The queue for source 3 is the shortest queue since flow s3 has the smallest flow intensity (on average 55 m/min) and the biggest passing speed (coefficient value is 1.2).


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Fig. 5. Dynamics of queues of incoming flows 5. Conclusions

The presented mesoscopic model of a crossroad has the following characteristics:

1. The main principles of mesoscopic modelling of flow systems are being realized during the development of the model. The model does not present individual flow objects, but only defines sets of objects (groups of vehicles coming to the crossroad during one traffic cycle).

2. All parameters of the model can be directly estimated. Any empirical data can be used to model the flow dynamics. Incoming flows are modelled as random values of the length of the flow with any distribution law. The duration of a traffic light phase is defined as a parameter.

3. The model allows studying a stationary and non-stationary mode of crossroad processes. 4. First results of modelling can be shown as a detailed trace file of processes for every structural

component and type of flow. Any characteristics of the crossroad processes can be calculated on the basis of the trace file. Graphs of process evaluation can also be constructed.

The developed mesoscopic model can easily be extended and adapted. For example, a set of connected

crossroads with different parameters can be modelled. Furthermore, any traffic light control algorithms can be realized and tested with the model. References 1. Kühne, R. D., Rödiger, M. B. Macroscopic simulation model for freeway traffic with jams and stop-start

waves. In: Winter Simulation Conference. 1991, pp. 762-770. 2. Yatskiv, I., Yurshevich, E., Savrasov, M. Investigation of Riga Transport Node Capacity on the Basis of

Microscopic Simulation. In: 21st European Conference on Modelling and Simulation (ECMS 2007). Prague, Czech Republic, 2007, pp. 584-589.

3. Esser, J., Schreckenberg, M. Microscopic simulation of urban traffic based on cellular automata, Int. J. Mod. Phys. C 8, Vol.5, 1997, pp. 1025-1036.

4. Burghout, W., Koutsopoulos, H., and Andreasson, I. Hybrid mesoscopic-microscopic traffic simulation, Transportation Research Record: 1934. 2005, pp. 218-225.

5. Tolujew, J., Alcalá, F. A Mesoscopic Approach to Modelling and Simulation of Pedestrian Traffic Flows. In: 18th European Simulation Multi-conference / G. Horton (Ed.). Ghent: SCS International, 2004, pp. 123-128.

6. Savrasov, M., Toluyew, Y. Application of Mesoscopic Modelling for Queuing Systems Research. In: 7th International Conference, Reliability and Statistics in Transportation and Communication / I. V. Kabashkin, I. V. Yatskiv (Eds.). Riga: Transport and Telecommunication Institute, 2007, pp. 94-99.


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ANALYSIS OF MOBILITY AND RELIABILITY MEASURES ON THE PARTICULAR TRANSPORT NODE AT THE STAGE

OF ITS RECONSTRUCTION

Irina Yatskiv1, Elena Yurshevich2, Irina Pticina3

Transport and Telecommunication Institute Lomonosova Str.1, Riga, Latvia

E-mail: 1 [email protected], 2 [email protected], 3 [email protected]

The goal of this article is to analyse the complex transport node in the capital of Latvia from the traveller’s point of view. The

attempt of applying the system of measures of reliability and mobility for the analysis of the situation in two transport corridors crossing this problematic transport node in Pardaugava has been done. This system of measures was used usually for macroscopic transport models. The offered measures - total travel time, weighted total travel time, travel time index, travel delay, planning time index and buffer index – have been calculated on the basis of the results obtained in micro simulation model experiments.

Keywords: transport node, micro simulation, corridor, mobility, reliability, measures, travel time

1. Introduction

One of the main problems arising during the transport planning is the inaccuracy of the forecasts of

transport situation. It resulted in the inefficient decisions during planning and reconstruction of the transport network. If we consider the experience of the leading European countries, Germany, for example, this problem is being solved by various methods and the major of them is the developed system of information displaying the utilized capacity of streets and the system of transport inquiries [1]. In Munich, for example, the traffic count is performed in 1700 points twice per year. Unfortunately, in Latvia and Riga, in particular, the system of such inquiries and measurements on the national and regional level is absent. As a consequence, there is a great probability of erroneous forecasts about the progress of the situation related to the conformity of the transport infrastructure to the travel demands of population.

One of the ways of possible solution of this problem is using the simulation model of the area under reconstruction for analysing and forecasting of the progress of situation in the model constructed on basis of the existing data.

For today, aids and opportunities of simulation modelling at micro [2] and macro levels [3] are mostly developed and demanded. However, the task is often formed in such way that it is difficult to solve it by both methods: for microscopic modelling it is too complicated, but the macroscopic approach does not allow providing the required level of adequacy of the model construction. The mesoscopic approach existing in practice is not so developed and has no own methods and aids.

The object of the mesoscopic modelling might be the complex transport node consisting of several signal controlled and non-signal controlled intersections, alternative corridors between the same nodes of transport network etc. For this purpose the simulation model might be constructed at the micro level. However, the analysis characteristics reflecting the functioning of the node will mainly refer to separate intersections (LOS), directions (route time and queue length) etc. While, it is desirable to obtain mainly some integral characteristics of the node.

In the given paper, the attempt of applying the mobility and reliability measures system [4] usually used for the analysis of the transport system (TS) at the macro level of the analysis of the complex transport node but not to the system, in general, has been made. Commonly, the given system of measures is based on the real traffic measurements. However, in case of their absence the results obtained on the basis of technologies of these data evaluation by the simulation modelling might be used. 2. Task Setting

The simulation model for the considered transport node was described in [5]. It was constructed by means

of special transport system simulation package VISSIM of PTV Company. This complex transport node of the Riga transport network is located between two bridges connecting the right and left banks of the river in Pardauguva and includes several crossroads:

1. Uzvaras avenue – Slokas street 2. Uzvaras avenue – Ranka dambis street


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3. Ranka dambis street – Trijadibas street 4. Slokas street – A.Grina avenue 5. Balasta dambis street – Daugavgrivas street (1 part) 6. Daugavgrivas street – Ranka dambis street 7. A.Grina avenue – Ranka dambis street (1 part) 8. A.Grina avenue – Ranka dambis street (2 part) 9. Balasta dambis street – Daugavgrivas street (2 part)

The model has been developed and validated using available statistical data for 2005 [ ]; then the

forecasts for 2007 on the basis of the extrapolation method have been made. A full description of this approach was presented in the article [5].

The structure of the transport flow in this node Car:Lorry:Bus is the next 0.8:0.05:0.15 and also 6 routes of trams crossing the territory of the considered node were implemented in the model.

In this paper two transport corridors crossing this transport node are considered. The transport corridor “Kalnciema–Ranka dambis street–Uzvaras avenue” (KRU) connects the Kipsala district of Riga with the Akmenu bridge in the city centre direction and vice versa. The transport corridor “Agenskalns–Slokas street–Uzvaras avenue” (ASU) connects the Agenskalns district of Riga with the Akmenu bridge in the city centre direction and vice versa. These two corridors provide travelling from the bedroom communities to the business centre of the city in the morning and homecoming in the evening. In this paper the inhabitants’ morning pick-hour (8:00–9:00) travelling by the public and private transport modes has been considered. The scheme of the both transport corridors is presented in Fig.1. The main transport corridors characteristics are presented in Table 1.

Fig. 1. Scheme of the two transport corridors traffic direction in the morning pick-hour

Table 1. Characteristics of the Considered Transport Corridors

№ Name Length Permitted Speed (km/h)

1 Kalnciema - Ranka dambis street - Uzvaras avenue 2.36 km 50

2 Agenskalns - Slokas street - Uzvaras avenue 2.38 km 50

The total lengths of both transport corridors are approximately similar. The permitted traffic speed is 50

km/h, but the free flow speed is lower because of the signal controllers, which are located in both considered transport corridors. The KRU corridor has the traffic lights in the Ranka dambis street and Uzvaras avenue intersection, but the ASU – in Slokas street and Uzvaras avenue. Also there are signal controllers for the transport following from the Akmenu bridge in the Uzvaras avenue direction and special signal controllers for trams. The signal controller’s cycles are presented in Table 2.


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Table 2. Time Cycles of Signals’ Controllers

The signal controllers group location

Cycle time, s

Red/ amber, s

Amber, s Red end, s

Green end, s

3 3 44 81

0 0 84 47

Slokas – Uzvaras avenue Ranka dambis street – Uzvaras avenue 84

0 3 44 78

Akmenu bridge – Uzvaras avenue 84 3 3 4 41

0 0 84 47 Uzvaras avenue (for the trams) 84 0 0 44 81

2. Transport Corridors Mobility and Reliability Measures

The most known and widely practically used approach to the design of the TS service quality measures system has been developed in the Texas Transportation Institute [4]. The offered system of measures includes reliability and mobility measures.

Mobility is the ability to reach a destination in time and cost that are satisfactory [6]. Travel mobility impacts are typically measured as a change in travel time, speed or delay.

Reliability is the level of consistency in transportation service (e.g., hour to hour and day to day) [6]. Measures of travel time reliability are important for evaluating of ramp management improvements and the analysis of many other operational improvements. These measures are intended to capture the impact of reducing travel time variability and to make travel times more predictable.

The following measures have been offered by the Texas Transportation Institute [7]: Mobility measures:

• Total Travel Time (TTT), which takes time for a vehicle to travel a given distance. • Travel Time Index (TTI), which compares peak period travel and free flow travel while accounting for

both recurring and incident conditions. • Travel Delay (TD) is the extra amount of time spent travelling because of congested conditions.

Reliability measures: • Buffer Index (BI), which calculates the extra percentage of travel time a traveller should allow when

making a trip in order to be on time in 95 % cases. • Planning Time Index (PTI) – the ratio of travel time or the worst day of the month compared to the time

required to make the same trip at free-flow speeds. • Volume-to-Capacity Ratio (VCR) – measures the relative levels of volume and capacity for section of

roadway. It estimates the relationship between the physical infrastructure (supply) and traffic volume (demand).

The first five measures were considered in this investigation. Let us discuss them in details. Total Travel Time (TTT) can be calculated as the sum of Travel Time (TTi) on all segments:

∑=

=n

iiTTTTT

1, (1)

where TTi – travel time on i segment (i = 1, …, n),

or can be measured in vehicle-hours or person-hours for estimating area-wide impacts and calculated as the weighed sum:

∑=

×=n

iiiiw VVTTTTT )( , (2)

where VVi – vehicle volume on i segment as the weighed factor . Travel Time Index (TTI) – the ratio of the travel time during the peak period to the time required to

make the same trip at free-flow speeds. The estimated effects reflect the conditions of the physical infrastructure and operations programs. The corridor values can be computed for hourly conditions and weighted by the number of travellers to estimate the peak-period or daily index values. The index values can be related to general public as an indicator of the length of extra time spent in the transportation system during a trip:


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ff

act

TRTR

TTI = , (3)

where TRact – actual travel rate and calculated as:

actact V

TR 1= , (4)

where Vact – actual travel speed. TRff – free-flow travel rate and calculated as:

ffff V

TR 1= , (5)

where Vff – free-flow travel speed. The free-flow speed (Vff) can be calculated by various ways, for example it may be measured with ITS

data or estimated from the speed limit. It is also possible to assume a free-flow travel speed (e.g., 50 km/h) for a given facility type.

In this work free-flow speed has been estimated by means of simulation experiments. Travel Delay estimates the extra time spent travelling as a result of congestion and the most basic form

of calculation is the difference between the actual (TTTact) and free-flow period (TTTff) travel times. Travel delay on i segment iTD (i = 1,…,n) is calculated as

iffiacti TTTTTTTD −= . (6)

To obtain a system-level calculation, several corridors (freeway and arterial) can be summed to get the total delay in hours. In our case we deal with the data of simulation model and the delay was calculated as the sum of delays in all segments of the modelled corridor:

∑=

=n

iiTDTD

1. (7)

The method of the 90th or 95th percentile travel times (TTI0.95) is the simplest method of measuring the travel time reliability and it estimates the harmful effect of on specific routes during the heaviest traffic days.

Buffer Index is calculated as the ratio between the difference of the 95th percentile travel time and the average travel time (TTT ) divided by the average travel time:

%10095.0 ×⎥⎦

⎤⎢⎣

⎡ −=

TTTTTTTTTBI . (8)

The extra time (buffer) is needed to ensure on-time arrival for most trips. It estimates the additional time that a traveler needs to budget during peak-period travel to be assured of arriving on time with the 95% confidence.

The buffer index can be calculated using the weighed Total Travel Time on the basis of (2). The calculations basically consist of calculating the average and the 95th percentile travel time for each

section of roadway for each combination of days and time periods. The Buffer Index values of each road section can be calculated and then combined to calculate the Buffer Index for a corridor or area.

Planning Time Index (PTI) is statistically defined as the 95th percentile of Travel Time Index (TTI0.95) and also represents the extra time most travelers add to a free-flow TTT when planning trips:

ffTRTRTTIPTI 95.0

95.0 == , (9)

where TR0.95 – 95th Percentile Travel Rate. In other terms, the Planning Time Index marks the upper limit for the nearly worst (95% of the time)

travel conditions.


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3. Experiment Description The mobility and reliability of transport corridors in the morning peak hours from 8:00 till 9:00 AM were

considered. As the outcome of the model there have been collected the information about the average travel time, delay and vehicle volume on the all segments of transport corridors. The data of one experiment were collected during one hour. The length of the transition behaviour of the model was chosen as 15 minutes with the aim of the system saturation achievement. The total number of experiments was 50. The TTT and the weighted TTT for both transport corridors have been calculated taking into account these data.

Also the free flow time was measured and estimated on the basis of the free flow speed in the model to calculate in future the characteristics of mobility and reliability of transport corridors. As a result the free flow speed was estimated taking into account the allowable speed in the city and the delay of signal controllers. The characteristics of free flow traffic are presented in Table 3.

Table 3. Free Flow Traffic Characteristics

KRU ASU

Average travel time, min 3.3 3.2 Average speed, km/h 41.81 31.95 Travel Rate, min/km 1.44 1.88

On the basis of the measured average actual and free flow travel time and the traffic volume, the Total

Travel Time (TTT) and Weighted Total Travel Time (TTTw) were calculated by the formulae (1) and (2). Also the average actual speeds for both transport corridors were calculated. Taking into account the formulae (3-9) and estimated characteristics of traffic there were calculated the following measures: the travel time index (TTI), the travel delay (TD), the planning time index (PTI) and the buffer index (BI). 4. Analysis of Results

The estimated characteristics of the transport corridors mobility are presented in Table 4. Firstly when considering the mobility measures, the fact has attracted the attention that the characteristics of mobility for transport corridor Agenskalns-Slokas street-Uzvaras avenue (ASU) are much less than for transport corridor Kalnciema-Ranka dambis street-Uzvaras avenue (KRU). For example, the actual speed of the ASU (13.74 km/h) corridor is approximately half as great as the actual speed of KRU (29.29 km/h). In 95% cases the actual speed of the KRU corridor will be not more than 36,71 km/h, but in case with the ASU corridor – not more than 16.33 km/h only. In both cases the actual speed is less than the free flow speed, but the worst situation occurs in the ASU transport corridor. The values of TTT and TTTw confirm these conclusions. The mean value of TTT for the KRU corridor is 5.06 minutes, but in case with the ASU corridor – 20.64 minutes. Taking into account the fact that the total lengths of both transport corridors are approximately the same, it is necessary four times more for travelling through the ASU corridor than through the KRU. It is easy to explain this fact on the basis of infrastructure analysis: the ASU transport corridor has the one-lane way and two signal controllers, but the KRU – the two-lane way and only one signal controller. Also, the transport traffic from Ranka Dambis Street additionally disturbs the flow of the ASU transport corridors, because it is the first flow into the main transport traffic in Uzvaras Avenue (direction to the Akmenu Bridge).

Table 4. Transport corridors mobility measures

Measures Unit Tr. Corr. min max mean median 95%

percentile Std.Dev. Variat. Coeff.

KRU 19.78 36.83 29.29 28.73 36.71 5.07 0.17 Actual Speed (km/h)

ASU 11.90 18.51 13.74 13.61 16.33 1.44 0.10 KRU 3.65 10.92 5.06 4.63 7.42 1.41 0.28 TTT (minutes) ASU 9.82 36.39 20.64 18.80 33.17 6.25 0.30 KRU 1552.70 3542.35 2202.35 2126.09 2990.76 429.52 0.20

TTTw (Vehicle-minutes) ASU 2844.12 4045.12 3662.88 3685.08 4005.35 255.76 0.07

KRU 1.26 5.00 2.05 1.78 3.65 0.81 0.39 TTI ASU 5.01 16.04 6.91 6.49 10.28 1.92 0.28 KRU 0.80 8.08 2.19 1.77 4.58 1.41 0.65

TD (minutes) ASU 7.38 32.20 16.96 14.99 28.81 5.94 0.35


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The weighted TTT has been calculated and it takes into account the traffic volume on each segment of transport corridors. As a result, we can see that the variation of traffic mobility of the ASU (0.07) corridor is much more less than for the KSU corridor (0.20). It is easy to explain, because the simulation experiments have demonstrated that the level of congestion of the ASU traffic is much higher than in the KRU case. This fact is illustrated by histograms (Fig.2) and line plots (Fig.3).

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14

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18

20

No

of o

bs

0%

4%

8%

12%

16%

20%

24%

28%

32%

36%

40%

b)

Fig. 2. Histograms of the total travel time(a) and weighted total travel time(b) for the KRU and the ASU transport corridor traffic

TTT of KRU corridor TTT of ASU corridor

Case 1Case 5

Case 9Case 13

Case 17Case 21

Case 25Case 29

Case 33Case 37

Case 41Case 45

Case 490

5

10

15

20

25

30

35

40

a)

Weighted TTT of KRU corridor Weighted TTT of ASU corridor

Case 1Case 5

Case 9Case 13

Case 17Case 21

Case 25Case 29

Case 33Case 37

Case 41Case 45

Case 491400

1600

1800

2000

2200

2400

2600

2800

3000

3200

3400

3600

3800

4000

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b)

Fig.3. The total travel time (a) and weighted total travel time(b) for the KRU and the ASU transport corridor in 50 experiments

Considering the TTI it is possible to establish the fact that in KRU case the actual travel rate is greater

than the free-flow rate by 26%, but in case of the ASU traffic – it is necessary 5 times more to reach the end of the transport corridor in the morning peak hours than in the case of the free-flow traffic. The average time spent in the ASU transport corridor is 16 minutes (but it can achieve the value of 30 minutes also).

All these measures of mobility demonstrate the fact that both corridors have the problem with traffic in the morning peak hours. But the worst situation is in the ASU corridor.

Also, the transport corridors reliability measures were calculated (see Table 7) on the basis of simulated characteristics. The value of the PTI illustrates that for the KRU transport corridor in peak hours a traveller should take into account that it is needed 3.67 times more than in case of free flow travelling to get the destination point in time in 95% cases. Considering the value of the PTI for the ASU we should point out that a traveller needs much more additional time to achieve the destination point in time.

Table 7. Transport corridors reliability measures

Transport Corridor Planning Travel Index (PTI) Buffer Index (BI),

% Weighted Buffer Index (BIw),

%

KRU 3.67 47% 36% ASU 10.29 61% 9%


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The buffer index for the KRU corridor demonstrates that the traveller planning his/her travel time in peak hours and taking into account the non stability situation with congestion (the high variation coefficient) has to provide himself/herself the additional time. A traveller needs approximately 50% more time of the average travel time in peak hours if he/she wants to be sure that he/she will be in time in the destination place with probability of 0.95. In the case of the second corridor (ASU) the buffer index shows that the additional time is needed - 61% more than average travel time. But, considering the last column in Table 7 we can see that the calculated weighted buffer index (taking into account the value of traffic volume) is much less than the non weighted one for both corridors. It can be explained by the fact that the variability of the ASU corridor traffic flow is much lower than of the KRU corridor. In reality, the ASU corridor has the jam state along the whole corridor length. The variability of the weighted total travel time is about 7% (see Table 6 and Fig.2b, 3b). So, in the case of planning the travel time it is necessary to take into account the weighted coefficient, for the purpose of getting more precise information about the properties of the transport corridor traffic flows. 5. Conclusions

The evaluation of the current and future situations in reconstructing transport node is the important task for the transport planner and analyst. The reliability and mobility of the urban transport system and its fragments (nodes) are more interesting properties from the user’s point of view. Calculation and analysis of these measures are used usually for the transport system as a whole, but can be considered also for fragments – transport corridors comparing. This approach can be used for comparing various scenarios of the transport node reconstructing. References 1. Design Characteristics of National Travel Surveys. Cost 355 Materials. Madrid. May, 2007. 2. Denos C.Gazis Traffic Theory, Kluwer Academic Publishers: Boston/Dordrecht/London, 2002, pp.259. 3. Juan de Dios Ortuzar, Luis G.Willumsen. Transport modelling. 3-rd edition. Wiley: NY, 2005, 499 p. 4. Development of congestion performance measures. Using its information. Final report. Sarah b. Medley.

Michael j Demetsky, Virginia transportation research council. Charlottesville, Virginia. January 2003 5. Yatskiv I., Yurshevich E., Savrasov M. Investigation of Riga transport node capacity on the basis of

microscopic simulation. 21 st European Conference on Modelling and Simulation (ECMS-2007), Prague, Czech Republic. Prague, 2007, pp.578-586.

6. The Keys to Estimating Mobility in Urban Areas. Applying Definitions and Measures. That Everyone Understands. A White Paper Prepared for the Urban Transportation Performance Measure Study by Texas Transportation Institute. The Texas A&M University System. Second Edition. May 2005

7. Odot operations performance measures. Final report. Bill Eisele, Tim Lomax. Texas transportation institute June 2004


159

EQUILIBRIUM WAITING TIME AND QUEUE LENGTH ON AN INTERSECTION: GROUP ARRIVAL OF CARS

Ilya Gertsbakh

Ben Gurion University

E-mail: [email protected]

We consider an intersection with Poisson car arrival and fixed service time per car. It is assumed that cars arrive in “packs”, in the middle of the green-red cycle. Using standard Markov’s chain techniques, we find

out the equilibrium queue length and waiting time. We suggest to compute the travel time index (TTI) for a given route by assuming that the car joins the queues at the intersections along its route when they are in the equilibrium state. Our approach has some features of macroscopic and mesoscopic modelling of traffic congestion.

Keywords: waiting time, queue length, equilibrium, TTI 1. System’s Description

We consider an intersection which works in a periodic regime. Its cycle for one particular flow, e.g. for the cars turning left, has green and red phase during ,g rT T seconds, respectively. m cars can pass through the

intersection during the green phase, so that mTg /=δ is the service time of one car. The intersection receives a Poisson flow of cars, with rate .λ The number of cars arriving at the intersection during one cycle is a Poisson random variable with parameter ,TλΛ = where T is the cycle duration, rg TTT += . We assume that the input flows going into different directions (left, right, straight), are served independently and there is no interaction between these flows.

Our principal assumption which considerably simplifies the solution is that all cars, which are supposed to arrive during one cycle and going in the same direction, e.g. left, arrive simultaneously, in one particular instant. This instant is defined as the time when the green light changes to red. In other words, the intersection receives a “pack” of X cars each T seconds, where X is a Poisson random variable with parameter Λ . Denote by nY the queue length right after the n -th “pack” arrival. Assuming that the system is in equilibrium, i.e.

,Λ>m the sequence { ,...}2,1, =nYn is a stationary Markov’s chain whose limiting distribution ],...,,[ 10 ππ can be found in a standard way by using the transition matrix of the chain.

Suppose the state of the chain is 0, i.e. .0=nY The obviously XYn =+1 , the number of cars in the

“pack”, and !/)exp()( kkXP kΛΛ−== If ,,...,2,1, maaYn == then all a cars will be served during the nearest green period and again XYn =+1 . If ,, maaYn >= then m cars will pass through the intersection and ma − will be left for at least one cycle. So,

1 max(0, )n nY X Y m+ = + − . (1)

Denote by B= ][ ijp the transition matrix for the chain. Obviously, ,!/)exp( jp jij ΛΛ−= for

,,...2,1,0 mi = and .,1,)!/()exp(, ijiijp ijjim ≥≥−−= −

+ ΛΛ Finally, 0, =+ jimp for j<i. Let us put aside for a while the computational aspects of finding the stationary distribution for the

Markov’s chain and suppose that we already know the values of ].0,[ ≥iiπ 2. Queue Length and Waiting Time in Equilibrium

Suppose that all cars arriving at the intersection and waiting in a line are “white” and consider a virtual yellow car, which arrives immediately after the pack of white cars arrive, at the instant when the green light changes for red. We assume that the yellow car sees the queue in its equilibrium state, i.e. it sees i white cars in front of him with probability iπ . What is his waiting time, since its arrival until it leaves the intersection? Obviously, the yellow car always waits one red phase, then it waits a number of green-red cycles equal to the integer part of the ratio )/(,/ miIntmi . In addition, its waiting time includes also the time during which the last portion of Res(i/m) cars leaves the intersection. It gives the following average virtual “yellow” waiting time [Gross&Harris, Chapter 2]:


160

W= δδπ +⋅+⋅+∑=

})/(Re)/({1

misTmiIntTM

iir , (2)

where }.0:max{ >= kkM π W includes also the service time δ of the yellow car itself. The cars wait for service in the line in which they have arrived to the intersection. We assume that each

car occupies the space of l meters, which includes the car length and the space between cars. Usually, it is assumed that the cars are of 5 meters long and the space between them in the waiting line is 1.0 meter. So, we assume that l=6 meters. With probability iπ , the yellow car sees in front of him i white cars, and joins the queue whose length is i times l . Therefore the average virtual queue length for the yellow car is

.1∑=

⋅=M

iiilL π (3)

3. Estimation of the Travel Time and the Travel Time Index (TTI)

Suppose that our yellow car travels in urban area from some point A to point B. For a fixed route, the distance between A and B is D(A,B). The traffic-free travel time in the urban area equals Tf = D(A,B)/V, where V is the accepted average car speed in the urban area. (For example, V= 36 km/hr, or 10 meters/sec).

Now suppose that along the chosen route from A to B, there are Q intersections, intersection q, q=1,…,Q, receives a Poisson input flow with intensity )(qλ and has red-green cycle of length T(q). An important assumption is that the input flows to each of these intersections are independent. This assumption is realistic if the input flows are formed by many flows from different sources, which join the route AB from side streets.

Let us assume that the queues at all Q intersections are in stationary regime and the yellow car meets anyone of them in its equilibrium state, i.e. waits the corresponding virtual waiting time in each one of them by joining the corresponding waiting line. Suppose that the distance between the s-th and (s+1) –st intersection is L(s,s+1). L(A,1) and L(Q,B) are the distances from the point A to the first intersection and from the Q-th intersection to point B, respectively.

The yellow car has a “free drive“ leg between q-th and (q+1)-st intersection of length L(q,q+1)- L(q+1), where L(q+1) is the queue length at the intersection (q+1) computed by the above formula (3). The time needed for driving this leg is T(q)=( L(q,q+1)- L(q+1))/V(q),

where the speed V(q) may depend on the free-drive distance. In addition to the free driving time, there will be a waiting time W(q+1) at the (q+1)-st intersection according to (2). Denoting A as intersection with number q=0 and B as intersection with number Q+1, the total travel time from A to B, T(A,B) equals

))1()((),(0

++=∑=

qWqTBATQ

q

. (4)

Finally, the TTI (travel time index) will be (see [Downs, Chapter1])

TfBATBATTI ),(),( = . (5)

4. Example

An intersection has 40 sec of red phase and 20 sec of green phase. m= 4 cars can be served during the green phase, i.e. =δ 5 sec. The intersection receives a Poisson flow of incoming cars with 2=Λ cars per minute, i.e. 2 cars on the average during one cycle.

The following are the probabilities p(k) of receiving a “pack” of k cars: Table 1. p(0) p(1) p(2) p(3) p(4) p(5) p(6) p(7) 0.135 0.271 0.271 0.180 0.090 0.036 0.012 0.006


161

Formally, the transition matrix B is infinite. Finding the stationary distribution analytically is rather involved, see [Gross & Harris]. We prefer to use a numerical solution, which can be obtained as follows. Consider the matrix B and cut out its “North-Western” part B(M) of size MxM. Usually, it is enough to take M equal 3 times the index of the last nonzero term in the above Table 1, i.e. for our example, M= 21. Modify the last column of B(M) so that the last term equals to the sum of all cut-off probabilities. This guarantees that B(M) remains a stochastic matrix. Find out the stationary distribution by standard techniques. The author found it convenient to read the limiting distribution from the matrix Q=(B(M))^(N), where N is about 30-40. In our example, the stationary probabilities are as follows:

,126.00 =π 259.01 =π , ,267.02 =π ,185.03 =π 098.04 =π , 042.05 =π , 016.06 =π , 006.07 =π , 001.08 =π .

By (2), W=58 sec. By (3), L= 15 meters. The average queue is rather small – about 2-3 cars.

Now suppose that the route AB consists of 5 segments of length 400 meters each, at the end of a segment there is an intersection. Each one of them works exactly as the above described intersection. The yellow car has an average speed 8.5 m/sec on the free parts of the route. So, the T(A,B)= (400-15)/8.5)*5 + 5*58=516 sec. Tf=5*400/10=200 sec. Finally, we have TTI(A,B)=516/200 =2.58. This figure is rather high, mainly due to the long red periods on the intersections. 5. Is Our Approach Mesoscopic or Macroscopic?

Our approach, probably, gives conservative estimates of the travel time since it assumes that the yellow car arrives at the intersection at the most inconvenient moment, when the green light turns to red. On the other hand, it ignores the interaction between cars in different (parallel) lines and the speed reduction resulting from this interaction. Hopefully, the above two phenomena will compensate each other.

Our approach is certainly not a microscopic one. It is also more “macro” than the mesoscopic one proposed in [Wilco Burghout, Chapter 3] Our model is not macro, since we do take into account the probabilistic nature of the car arrival mechanism and its interaction with the service at the intersection. It would be correct, in our opinion, to position our model as an intermediate between the macro- and the meso-type schemes.

Our model has a potential for estimating the travel time in an urban area and thus may be useful for calculating the TTI, which is the main congestion measure of the transportation system in a whole. A principal issue remains establishing, by means of simulation and/or direct observations, how accurate is the TTI estimation based on the proposed approach. References 1. Gross, D. and C. Harris. Fundamentals of Queueing Theory. Wiley, 1974. 2. Downs, A. Still Stuck in Traffic. Brookings Institution Press, 2004. 3. Burghout W. Hybrid Microscopic-Mesoscopic Traffic Simulation: Doctoral Dissertation. Stockholm,

Sweden: Royal Institute of Technology, 2004.


162

A MODEL OF CHOOSING PASSENGERS’ AIR FLIGHT

Catherine Zhukovskaya

Riga Technical University Institute of Transport Vehicle Technology

Lomonosova 1, LV-1019, Riga, Latvia E-mail: [email protected]

The article deals with a creation of the passenger’s behaviour model at a flight choice. From the variety of those different factors, which have the greatest influence on the mentioned choice, are identified. Taking them into account the original flight choice models by a passenger for one Origin-Destination those n pair of cities is developed. The simulation approach has been used for practical calculations. Simulation is carried out with MathCAD software.

Keywords: choice model, choice of flight, simulation approach 1. Introduction

Choice models describe the general enough situation when it is necessary to make only one decision from the several available. In the choice models subject is the person who is making a decision, and alternatives are his possible decisions.

There are a lot of different mathematical theories describing the various choice models, for example, their great number concerned with the transport processes is possible to find at M. Ben-Akiva and S. Lerman [1].

In the Soviet Union the first choice models were developed and applied within the limits of creation the Automatic Control Systems (ACS) of Civil Aviation in the seventieth years of the last century. The first here are the two-alternative models of a choice by a passenger of the mode of transport (aviation or railway), offered by J. Paramonov [7]. The next group of such models are three-alternative choice models in which the third alternative designates “the refusal of a trip”. The different examples concerning with the mentioned choice problem we can see at the work of A. Andronov [6].Modern choice models are building on a multi-alternative basis and have received a wide distribution at the description of the passenger behaviour at a flight choice. The similar model example creation can be found at J. Paramonov [8, 9, 10].

The creation of corresponding model includes two problems: firstly, it is necessary to create the model, secondly, it is necessary to estimate its parameters. But, it is essential to note, that creation of the analytical expression for multi-alternative choice models is complex enough. And for practical calculations the simulation approach is often used [4].

When choosing an air flight the passenger may take into account the number of different factors [2, 8]. For example: the cost of the air-ticket, the length of flight, the comfort of schedule (here we mean the time of the aircraft departure), how far the airport is situated from the city centre, the comfort on board, the information about potential technical problems with aircrafts of this or that airline, a potential possibility of the terrorist act in this or that airport, etc.

In this research the multifarious factors defining this choice are carefully studied. From the variety of the considered factors are identified those which have the greatest influence. These factors are listed below and the brief characteristic to them is given.

At the initial stage of simulation the different approaches and concepts of the choice models creation have been considered. Firstly, the elementary model has been created: the model of a flight choice by passenger for a day. In this model the choice process is considered according to the stream of passengers arrived this day. This model is the basic for construction of all subsequent models [5].

The following model simulating the choice process takes into account several days (for the certain time horizon H). It is the logic continuation of the previous model. But at the corresponding computer program creation we have collided with necessity of storage and recalculation of huge quantity of data by which the passengers of each day from the period of consideration H are characterized. However, the output data analysis has allowed to find out the basic regularities inherent in passengers of various days from the time period H, and to generalize their characteristics. We have come to conclusion about inexpediency of the passengers’ initial characteristics storage during all period of simulation. Therefore, for the subsequent models the initial concept of model has been completely reprocessed, and having left the initial problem statement without changes, the new approach for its decision is elaborated.

Within the limits of this approach the original model considering the choice process not from the just arrived passengers, but from already existent queue of passengers who are situated in a mode of the flight expectation during some time interval t <= h (in days) has been developed, where t = 1, 2, ..., h.


163

The developed model belongs to a class of multi-alternative choice models [1]. As the stochastic model corresponding to it is complex enough for the description and simulation, for practical calculations the simulation approach has been used [4]. Simulation is carried out with MathCAD software. The detailed description of model and corresponding algorithm is given below. 2. Initial Preconditions of the Model

Let us consider the following problem. There is m number of flights for a day for one Origin-Destination pair of cities. Each j-th flight, where j = 1, 2, ..., m, will be described by the following parameters:

• the number of seats ( ) ( ) ( ) ( )( )mj rrrr 112

11

11 ...,,...,,,=r ;

• the cost of air-ticket ( ) ( ) ( ) ( )( )mj rrrr 112

11

12 ...,,...,,,=r ;

• the time of departure ( ) ( ) ( ) ( )( )mj rrrr 112

11

13 ...,,...,,,=r . There is a stream of passengers wishing to depart with one of these m flights. It is necessary to note, that

in the considered choice model the passengers’ characteristics vary depending on the number of days from the passenger’s lookup horizon h (number of days when the flight for a passenger is possible), i.e. the passengers’ characteristics from the different days are not identical. The fullest extent of these characteristics belongs to the current day passengers.

It is accepted that the number of passengers for a day is a random variable N having a normal distribution with parameters Nμ and Nσ . The values { }kN for different days (from the consideration horizon H) are independent and identically distributed random variables.

It is assumed that each current day passenger is described by four parameters having the stochastic character: • 1X is the desirable time of departure; • +Δ is the positive passenger-flight deviation (or the value of maximal allowed positive deviation the

actual time of flight from the desirable by passenger time of departure, where the positive deviation is understood as the moment of departure after the specified time);

• −Δ is the negative passenger-flight deviation (or the value of maximal allowed negative deviation the actual time of flight from the desirable by passenger time of departure, where the negative deviation is understood as the moment of departure before the specified time);

• h is the passenger’s lookup horizon (the maximal number of days during which the passenger is ready to wait a flight).

Each of the listed parameters is a random variable and has its own distribution. The random variable 1X defines the desirable time of departure within a day. It is described by the

corresponding density of distribution. It takes whole values from 1 up to 24 with certain probabilities which correspond to the change of intensity of demands within a day.

In our research it is accepted, that distribution of a random variable 1X corresponds to uniform distribution in interval ( )23,7∈t .

Obviously, that for each passenger the values +Δ and −Δ have the random meanings, but in the considered model it is accepted that const=Δ=Δ −+ , and, moreover, these values are identical for all the passengers. The value of h has been taken identical for all passengers as well (h = const). It is supposed, that at the presence of the real corresponding data the fitting of the simulation model will be carried out by changing of values +Δ , −Δ and h . 3. Model of a Flight Choice by a Passenger for a Day

At the description of the process of a flight choice by a passenger for a day it is supposed that passenger buys the air-ticket in a day of flight. However, if there are no air-tickets suitable for any flight this day, i.e. in the day of the expected departure the passenger receives the refusal, then he transfers the air-ticket purchase the next day. If the next day he is refused also, the air-ticket purchase will be transferred the day after the next too, and so on. Thus, the flight can be postponed for some days, but no more than h days, after that the passenger finally cancels the trip. So the passenger can get the air-ticket during the certain time horizon h.

Additional remarks: 1. The time horizon h can accept only positive values. 2. Generally the time horizon h for each passenger is a random variable. For simplification of

calculations it is accepted that this variable is identical for all passengers. So, there is a stream of passengers wishing to depart by certain flight, but due to various reasons, they are

compelled either to transfer a flight on other day, or some other days, but no more than h days, or to refuse a trip.


164

Thus, during the horizon of consideration H the queue of expecting passengers ( )ht QQQQ ...,,...,,, 10=Q , where tQ is a number of the expecting t days passengers (t = 0, 1, …, h), gradually grows.

The original algorithm for decision of the task in view, as it has been already noted earlier, has been developed. Their novelty and originality consists in consideration of the simulation process not from the side of newly arrived passenger, but from the side of already existent queue of expecting passengers.

It is accepted, that the characteristics of the passengers from the every t-th day from the passenger’s lookup horizon h are homogeneous values, but the passengers characteristics from different days differ among themselves. For example, the passengers of different days of expectation (t ≠ 0) are characterized by different values of the flight refusals probabilities: the value t increase leads to the increase of the flight refusal probability, and reaches its maximum last day of expectation (h).

In this case, the process of choice of the flight j (j = 1, 2, ..., m) by the current day passenger i (i = 1, 2, ..., N) can be described in the following way.

From all flights m only those moments of departure ( )jr3 are selected which are situated within the

corresponding time interval ( ) ( ) ( ) ( ) ( )( )iiiij XXr +− Δ+Δ−∈ 113 , . If there are some flights available then we select

one with the minimal cost of the air-ticket ( )jr2 :

( ) ( ) ( ) ( ) ( ) ( ) ( )( ){ }iiiijj

j

i XXrrj +− Δ+Δ−∈= 1132 ,:min , (1)

i.e. we take into account two characteristics of passenger at once: the desirable time of departure and the cost of a flight. If there is no any suitable flight, the passenger passes in queue of the current day expecting passengers.

For each passenger from t-th day of expectation, where 1...,,2,1 −= ht , the defining characteristic at the flight choice is the minimal cost of the air-ticket:

( ){ }j

jrj 2min= , (2)

but if there is no any suitable flight, the passenger passes in queue of the t-th day expecting passengers. All the passengers of h-th day of expectation receive refusal. The queue of expecting passengers for this

day is cleared. The detailed description of the process of the flight choice by a passenger for a day is given below.

3.1. Process description at a flight choice by a passenger for a day

The simulation process can be divided into two stages: the preliminary and the basic ones.

The preliminary stage of the simulation process: 1. In the beginning of a day all passengers whose expecting time has exceeded h days receive refusal. 2. Next we rewrite the expecting passengers queue, i.e. increase the expecting time (in days) for not

departed passengers: tt QQ =+1 , where t is the day of consideration, t = 0, 1, ..., h-1.

The basic stage of the simulation process: 1. Firstly, we try to send those passengers who expected more than one day. Moreover, we try to send

them in reverse order: at first hQ passengers with “the age h”, then 1−hQ passengers with “the age h-1”, etc. The defining characteristic of a flight for passengers who are expecting more than one day is the flight cost 2r , i.e. the passenger chooses the flight with the minimal cost ( )jr2 , here j is a number of the chosen flight, where j = 1, 2, ..., m (2).

2. Further, we generate N passengers who have arrived the current day ( 0=t ), and put them into the current day queue NQ =0 . The number of the current day passengers N have the normal distribution with parameters Nμ and Nσ .

3. Next, we generate the desirable time of departure for the current day passengers ( )iX1 , where

i = 1, 2, ..., N. The desirable time of departure have the uniform distribution in interval ( ) ( )23,71 ∈iX . 4. After that we try to send new coming passengers 0Q taking into account already two factors. Firstly,

we consider the desirable by passenger time of departure, i.e. from all m flights we take out those which departure time ( )jr3 get in a corresponding time interval ( ) ( ) ( ) ( ) ( )( )iiiij XXr +− Δ+Δ−∈ 113 , , here j is a number of the chosen flight (j = 1, 2, ..., m), i is a number of the passengers, where


165

(i = 1, 2, ..., N). Secondly, we take into account the cost of a flight ( )jr2 , i.e. if there are some suitable flights we get out one of them with the minimal cost (1).

5. All current day passengers who were refused we put into the current day queue. 6. As output values we receive four ones: the new vector of expecting passenger number at the end of

current day Q , the number of satisfied claims of current day passengers 0QN − , the general number of all refusals for a day Ref, and new vector of free seat number 1r .

The formal description of the algorithm for a day is the following. 3.2. Algorithm of simulation for a day

Input: the number of flights m, the characteristics of flights 1r , 2r , 3r , the parameters of the current day passenger number distribution Nμ and Nσ , the meanings of the random variables +Δ , −Δ , h, and the vector of the number of expecting passengers Q .

Output: the new vector of expecting passenger number at the end of current day Q , the number of satisfied claims of the passengers who came in current day 0QN − , number of refusals for a day Ref, and a new vector of the free seats number 1r .

Intermediate variables: the number of the current day passengers N; the vector of the current number of free seats ( )jr1 , where j is the number of current flight; J is the number of the chosen flight (J = 1, 2, …, m-1).

Algorithm

Begin 1. Input of the initial data: m, 1r , 2r , 3r , Nμ , Nσ , −Δ , +Δ , h, and ( )hQQQ ...,,, 10=Q . 2. Initialisation of the counter of the current day refusal number Ref ( hQRef =: )

3. Identification of the variable of the available seats numbers ( ) ( )jj rR 11 := , where j = 0, 1, …, m-1 4. Increment of the expecting time of not departed passengers ( tt QQ =+ :1 , where t is the day of consideration, t = 0, 1, …, h-1) 5. Begin a cycle by the consideration horizon (by the variable t in the back order 1...,,1, −= hht )

5.1. Begin a cycle by passengers who are expecting tQ days (by the variable i, where 1...,,1,0 −= tQi ) Initialisation of the variable J (J = -1, i.e. none flight is chosen)

5.1.1. Begin a cycle by flights (by the variable j, where j = 0, 1, …, m-1)* If there are available seats on the j-th flight ( ( ) 01 >jR ), then

- Choosing of the flight jJ =:

- Decrement of the number of the available seats on chosen flight ( )JR1 ( ( ) ( ) 1: 11 −= JJ RR )

- Decrement of the expecting passenger queue tQ ( 1: −= tt QQ )

End the cycle by flights

End the cycle by passengers 6. Generations of the current day passenger number N and the passenger characteristic 1X . 7. Placing of the new coming passengers into the current day passenger queue ( NQ =:0 ) 8. Begin a cycle by the current day passengers 0Q (by the variable i, where 1...,,1,0 0 −= Qi )

Initialisation of the variable ( )iJ ( ( ) 1−=iJ , i.e. none flight is chosen) 8.1. Begin a cycle by flights (by the variable j , where 1...,,1,0 −= mj )*

- ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ){ }iiiijjj

j

i XXrRrJ +− Δ+Δ−∈>= 11312 ,,0:min:

- Decrement of the number of the available seats on chosen flight )(1

JR ( 1: )(1

)(1 −= JJ RR )

- Decrement of the current day passenger queue 0Q ( 1: 00 −= QQ )

End the cycle by flights End the cycle by current day passengers 9. Calculation and output variables tQ , where ht ...,,1,0= , 0QN − , Ref, and 1r End


166

Comment to the algorithm: The cycles by flights (*) are organized in the way that they are looked through in the sorted order by 2r (the cost of air-tickets order) 4. The Model of Passenger Distribution by Flights for Several Days 4.1. The description of the simulation process

For a basis of the model of passenger distribution by flights for several days the previous model of a flight choice by a passenger for a day has been taken. We left all preconditions of above mentioned model, but made three essential additions in a new one:

1. The passengers of the different days of the expectation queue are characterized by the different values of the flight refusal probabilities ( )hppp ...,,, 21=p , and an increase of the value h leads to the values { }ip increase, where i = 1, 2, ..., h. This probability reaches a maximum in the last day of expectation (h).

2. Thus, not all passengers “with the age h” refuse from trip; there is very small part of passengers of this day, who, with the probability hp−1 , choose this or that flight. At a choice of the flight the defining characteristic for them is the cost of flight (2).

3. The general value of the refusals of a flight consists of the sums of refusals each day from the horizon h.

Thus, the changes have taken place only in the first preliminary stage of simulated process which now is described as follows:

1. In the beginning of the day of consideration with the probability hp all passengers whose expecting time has exceeded h days are refused.

2. Remaining passengers with “the age h” we distribute on flights. Here the defining characteristic at a flight choice is its cost 2r , i.e. the passenger chooses the flight with the minimal cost ( )jr2 , here j is a number of the chosen flight, where j = 1, 2, ..., m (2).

3. For the passenger from the t-th expectation day we introduce the value of refusals Ref. According to the assumption of the model, in the t-th expectation day the probability of rejection will reach the value tp from the passenger numbers of expectation queue for t-th day, where t = 1, 2, …, h-1.

4. Further, as before, we rewrite the queue of expected passengers, i.e. we increase an expecting time (in day) for not departed passengers: tt QQ =+1 , where t is considered day, t = 1, 2, …, h-1.

The description of the simulating process basic stage has not suffered absolutely any changes. It is necessary to notice especially that the given model is the previous model generalization and at the following vector of the flight refusal probabilities ( )1...,,0,0=p completely describes the previous model. Changing the meanings of the flight refusal probabilities vector p we can simulate the huge value of different possible situations.

As output values on the end period of consideration H we obtain: the percentage of the commercial loading of flights, the average number of refusals for a day, the average number of passengers which have chosen the flight in the desirable for them time in a day, the percentage of the passengers which didn’t fly in the desirable day, the average time of expecting (in days) for passengers.

The percentage of the commercial loading of j-th flights jCom (in %), where j = 1, 2, …, m, can be defined in the following form:

( )

( ) %1001

1 ⋅= j

j

j rRCom , (3)

where ( )jr1 is the number of seats on the j-th flight, ( )jR1 is the number occupied seats on the j-th flight. 5. Numerical Example

In the considered numerical example the vector of the flight refusal probabilities p have the following

meanings ( )T1000=p . At these meanings of probabilities we can simulate the following conditions: all passengers of the t = h day of expectation receive the refusal, passengers of other days from the queue of expectation 0 < t < h are distributed on flights according to the minimal cost of the air-ticket, the current day


167

passengers t = 0 are distributed on flights according to two conditions at once: the desirable time of departure and the minimal cost of the air-ticket.

There are 5=m flights between one Origin-Destination pair of cities, the vectors of flight characteristics are sorted in the cost of air-tickets order. From this point of view the initial data of flight characteristics have the following meanings: the vector of seats number ( )T1008010080801 =r , the vector of the air-tickets

cost ( )T15012510075502 =r , and the vector of the plane departure time

( )T1917912213 =r . The passenger characteristics take the following meanings: desirable departure

time corresponds to the uniform distribution in interval ( )23,7∈t , the values of the positive and negative passenger-flight deviation (further, absolute value of passenger-flight deviation) are identical for each passenger ( +− Δ=Δ=D ), the passenger’s lookup horizon h = 3 is identical to all passengers, the initial queue of the

expecting passengers is empty ( )T0000=Q , the consideration period is 100=H days, and the number of passengers N from each current day (t = 0) from the consideration period H have the normal distribution with parameters 300=Nμ and 20=Nσ .

The main task of this example is to investigate the dependence on the percentage of the commercial loading of j-th flights jCom (3), where j = 1, 2, …, m, from the value of absolute passenger-flight deviation D.

As we can see from the meanings of the vector ( )T1008010080801 =r , the total sum of seats on

flights for each current day ( ) 4601

1 =∑=

m

j

jr considerably exceeds the average number of the current day

passengers Nμ . It allows us to investigate the commercial loading of flights only for the current day passengers. The obtained results of the commercial loading of j-th flights jCom (in %), where j = 1, 2, …, m, dependence on the value of absolute passenger-flight deviation D are presented in the table 1. Table 1. The percentages of the comercial loading for m flights dependence on the value D

D (hour) 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

1Com (%) 100 100 100 100 99 99 100 100 100 100 100

2Com (%) 49 50 55 65 82 92 97 99 100 100 100

3Com (%) 18 31 42 47 52 49 43 41 40 45 44

4Com (%) 24 37 55 68 83 92 99 99 100 100 99

5Com (%) 19 29 40 36 23 18 15 20 22 27 26

Other purpose of this example is the investigation of the current day passengers service quality K dependence on the value of absolute passenger-flight deviation D, where K is the number of the current day passengers who were departed during desirable time of day. Results of simulation are shown on figure 1.

Fig. 1. Value K dependence on value D


168

Analysis of the received results We would like to remind, that capacity of planes had only two meanings: 80 and 100 seats. The following

regularity has been noted: 1. With the increase of absolute passenger-flight deviation value D the meaning of the quality of service

K increases and converges to the value 300=Nμ (see fig. 1). 2. The increase of absolute passenger-flight deviation value D till the meaning D = 1 leads to the growth

of all flights commercial loading, but the further increase of value D (D > 1) has shown the steady growth the commercial loading of flights only with the number of seats 80.

3. The commercial loading of flights with number of seats is equal 100 is considerably smaller and did not converge to a level of 100 %, but stabilized at levels of 40 % for the 3-rd flight and 22 % for the 5-th.

Therefore, it is possible to conclude, that results of simulation are quite adequate to the simulated process. Thus, two tendencies are detectible:

1. The less the air-ticket cost, the higher the percent of the flights commercial loading. 2. The flights filling process takes place proportionally enough, therefore, the flights with greater

number of seats is characterized by not enough capacity (the less percent of the flights commercial loading for the 3-rd and the 5-th flights).

Possible recommendations: the low percentage of the commercial loading of the 3-rd flight is connected with the inconvenient moment of the flight departure for the passenger (9 PM) and with high enough cost of the air-ticket, therefore, it is necessary either to reduce the cost of the air-ticket, or to change the plane departure time, or to offer the plane with the smaller number of seats. Note, that each variant can be simulated separately for the optimum decision acceptance. 6. Conclusions

The passenger’s behaviour model at a flight choice for one Origin-Destination pair of cities is created out in this research. This model taking into account three characteristics of flight: the number of seats, the cost of air-ticket and the time of departure; and four characteristics of passenger as well: the desirable time of departure, the positive and negative passenger-flight deviations, and the passenger’s lookup horizon. The originality of the model consists in the choice process consideration from the side of the expecting passengers queue. For practical calculation the simulation approach has been used. Numerical example confirm the adequacy of the considered model. Acknowledgements

The author is obliged to the supervisor Professor Alexander Andronov for his help at writing this paper. The paper is prepared by support of the European Science Foundation.

References 1. Ben-Akiva, M., Lerman, S. Discrete Choice Analysis: Theory and Applications to Travel Demand, 6th ed.

Cambridge, MA: The MIT Press, 1994. 416 p. 2. Benítez, R. Factores determinantes de la demanda de transporte aéreo y modelos de previsión, Boletín

económico de ICE, Información Comercial Española, No 2652, 2000, págs 41-48. 3. Gentle, J. Elements of Computational Statistics. New York, Berlin: Heidelberg Springer, 2002. 420 p. 4. Ross, S.M. Applied Probability Models with Optimisation Applications. New York: Dover Publications Inc,

1992. 198 p. 5. Zhukovskaya, C. Model of the Potential Passenger Behaviour when Choosing an Air Flight. In: Scientific

Papers of the RTU, Machinery and Transport, 6th series, Intelligent Systems of Transport, issue 19. Riga: RTU Publishers, 2007. (In printing).

6. Andronov, A., Hizhnyack, A., Shvarztkiy, I. Forecasting of the Passengers’ Air Transportations. M.: Transport, 1983.183 p. (In Russian)


169

MODEL-BASED RESULTS OF INVERSE PROBLEM SOLUTION FOR RADAR MONITORING OF ROADWAY COVERAGE

Alexander Krainyukov, Valery Kutev

Transport and Telecommunication Institute

Lomonosova1, Riga, LV-1019, Latvia Tel.:+371 7100634. Fax: +371 7100660. E-mail : [email protected], [email protected]

This work deals with the development of a model-based approach to the inverse problem of subsurface radar sounding solution in

frequency domain. We propose to use an Ultra Wide Band (UWB) pulse radar combined with a line transmitter and receiver electromagnetic antennas.

Forward modelling is based on using of three-component mixing formula for estimation of soil electromagnetic properties in frequency domain, as well as on linear system response functions for the radar antenna system, and on the equations for wave propagation in a horizontally multi-layered medium representing the subsurface.

Model inversion, realized by numerically, using results of forward modelling simulation in limited range of the soil electromagnetic properties changing. For realization of model inversion data was choose genetic algorithm of calculations.

Keywords: radar, soil dielectric properties, radar monitoring, forward modelling, inverse modelling

1. Introduction

Radar monitoring of roadway coverage is an electromagnetic (EM) method for quality estimation of road covers, which is the most efficient for natural soil roads. System of radar monitoring executes the two kinds of processing: primary signal processing, which gives continuous subsurface profile of probing subject, and secondary signal processing, which gives estimation of a roadway physical properties.

For formalization of inverse problem vector of parameters { }1 2 nP p ,p ,...,p= is entered, which components pi determines a great number of legitimate values of parameters for model vector mod limP P⊆ .

Results of the reflected signal spectrum calculations compare with these for model signal spectrum, which is calculated for all search range of data on electrical properties of probing object. The end of the calculations will be by meeting the condition [1]:

( ) ( )( )2L

refl i mod i modi 1

A S ,P S ,Pω ω α=

= − <∑ , (1)

where ( )r efl iS ,Pω – a module of the reflected signal spectral density for estimated vector P ;

( )mod i modS ,Pω – a module of the model signal spectral density for model vector; α – mean of threshold,

which is limited by a permissible absolute error of the parameter estimations. Modern techniques utilize personal computer to perform comparison in frequency domain, but it is

necessary to have an efficient models for solution of radar probing forward problems. 2. Modelling Electromagnetic Properties of Natural Soils

Moisture content and bulk density largely characterize physical and mechanical soil status and behaviour. A non-destructive determination of these soil properties is essential.

The electromagnetic properties of a two-phase mixture of soil and water can be analysed by using many empirical or theoretical formulas. Among them, Maxwell-Garnett mixing formula

( )( )

31

2w s

eff sw s w s

WW

ε εε ε

ε ε ε ε⎡ ⎤−

= +⎢ ⎥+ − −⎢ ⎥⎣ ⎦

(2)

and Odelevsky mixing formula

11

3

e ff ww

s w

WW

ε εε

ε ε

⎡ ⎤⎢ ⎥−⎢ ⎥= +⎢ ⎥+⎢ ⎥−⎣ ⎦

(3)


170

are well-known in literature [2,3]. If soil volumetric water content of soil W%, as well as dielectric permittivity

of mixture components Sε•

(soil) and wε•

(water) are known, then equivalent dielectric permittivity effε•

of two-phase mixtures soil-pore water may be estimated in according to (2) and (3).

Unfortunately, calculation data are not a correct in wide frequency range therefore more adequate for practice is three-component mixing formula. This formula is developed to calculate the equivalent dielectric permittivity of the three-phase mixture air-soil-bulk pore fluid in the following form:

1

1

2Re Im

/( 2 )

in

ii

i

effeff eff n

iii

P

jP

ε ε

ε εε ε ε ε

ε ε

• •

• •=

• • • •

• •

=

−

+= − = +

+

∑

∑, (4)

where i – number of the mixture component; Pi=Vi/V – relative volumetric content of i-th component;

1

n

iii

Pε ε• •

=

=∑ – average mean of mixture complex dielectric permittivity.

If W is volumetric water content of soil with P soil porosity, then one can see that 1 ( )P P W= − and

1 1ε = (air); 2 (1 )P P= − and ' "2 2 2jε ε ε= − (non-clay dry minerals); 3P W= and 3 wε ε= (pore water). For

calculations of real 2'ε and imaginary "

2ε parts of the dry mineral complex dielectric permittivity 2ε it is possible to use Krotikov empirical formulas [3]:

( )22 1 0.5 mε γ′ = + ⋅ and 3

2 2 10mε ε γ′′ ′= ⋅ ⋅ , (5)

where mγ – density of soil mineral foundation. The complex dielectric permittivity of the pore water wε can be described by the Debye relaxation

function as follows:

( ) ( ) ( )0

0j ,

1w

wwj

ε ε σε ε

ωτ ωε− ∞

= ∞ + −+

(6)

where ( )0

0 lim wω

ε ε→

= and ( ) lim wω

ε ε→∞

∞ = – are the pre-relaxation and post-relaxation real permittivities of the

pore water molecules; wτ – relaxation time constant; 910

36ε

π

−

Ο = – electromagnetic constant of free space;

ω – angular frequency; wσ – water DC electrical conductivity.

According to [2], ( )ε ∞ is approximately 5.5, and this value is almost unaffected by the temperature and

pore fluid salinity. In contrast, ( )0ε and τ w are affected by the temperature KT . Temperature dependencies

of ( )0ε and wτ may be presented in form of empirical expressions such as:

( ) 4 2 6 30 87.74 0.4 9.4 10 1.4 10ε − −≈ − Τ + ⋅ Τ − ⋅ Τ , (7)

( )10 12 14 21 1.1 10 3.8 10 6.9 102wτ π

− − −≈ ⋅ − ⋅ Τ + ⋅ Τ . (8)

DC electrical conductivity of pore water is affected by pore fluid salinity. The salinity S is defined as the total mass of solid salts in grams dissolved in one kilogram of solution and expressed as parts per thousand (ppt). For 424 ,,, MgSOMgClNaSONaCl solutions with salt concentration 3/10 dmgC ≤ may be used the following empirical expression [3]:


171

{}

0 2 5 2

5 5 5 6

( , ) (25 , ) 1 1.962*10 8.08*10

3.02*10 3.922*10 (1.721*10 6.584*10 ,

w wT N С N

N N

σ σ − −

− − − −

= − Δ + Δ

⎡ ⎤− Δ + Δ + − Δ⎣ ⎦ (9)

where saltMCN /= – normality of soil water; saltM – salt equivalent

2 4 2 4SO( 58.45 , 71 , 47.45 , 60 );NaCl Na MgCl MgSOM g M g M g M g= = = =

0 0(25 , )C N T CΔ = − and 0 2 3(25 , ) (10.394 2.38 0.683 0.135 )w С N N N N Nσ = − + − .

Using the three-component mixing formula (3), the frequency dependence of theoretical equivalent dielectric permittivity of a soil-water mixture at different volumetric water content W[%] and pore fluid salt concentration 3[ / ]С g dm is calculated. Some results of such calculations are plotted in Fig.1. For comparison, calculation results for two-phase mixtures in accordance Maxwell-Garnett (2) and Odelevsky (3) formulas are presented on Fig.1 too.

10 9 1010 1011 10 12 3

4

5

6

7

8

real

E

Frequency, Hz

10 9 1010 1011 10 12 0

0.5

1

1.5

imag

e E

Frequency, Hz

Fig. 1. Frequency dependencies of real ( eff

•

εRe ) and imaginary ( eff

•

εIm ) parts of equivalent dielectric permittivity of sandy soil ( %5;/67,1 3 == Wcmgmγ ), calculated for three-phase mixture model (3)

and two-phase mixture formulas: Maxwell-Garnett (1) and Odelevsky (2) 3. Forward Modelling of Subsurface Radar Probing

The possibility to estimate accurately the subsurface electric properties from subsurface radar signals using inverse modelling is obstructed by the appropriateness of the forward model describing the subsurface radar system.

A typical subsurface radar system has three main components: transmitter and receiver that is directly connected to antennas, and display with a timing control unit [3]. The transmitting antenna radiates a short high-frequency EM pulse into the inspected medium, where it is refracted, diffracted and reflected primarily as it encounters changes in dielectric permittivity and electric conductivity. Waves that are scattered back toward the medium's surface induce a signal in the receiving antenna, and are recorded as digitised signals for display and further analysis.

According to that, a channel of the signal forming for subsurface radar probing in frequency domain may be presented as it is shown on Fig.2 [4]. There ( )ANTK ω is complex transfer function of the antenna system,

( )reflR ω is reflection coefficient of inspected object, and ( )FWK ω is complex transfer function of antennas direct coupled EM field.


172

Fig. 2. Channel the signal forming for subsurface radar probing in frequency domain As result, complex transfer function of radar subsurface probing model may be written in the following

form:

( ) ( ) / ( ) [ ( ) ( ) ( )]RAD l pr FW ANT reflK U U K K Rω ω ω ω ω ω= = + . (10)

It depends on a conditions of the subsurface radar probing, as well as on geometry of antennas location. Now let us consider a forward modelling of the subsurface radar probing for line radar antennas with

geometry as it is shown on Fig. 3.

Fig. 3. Geometry of subsurface radar antennas location

If a height of antennas location H over upper medium boundary is sufficiently small ( 0H ≈ ), then complex transfer function of the antenna system may be presented as [4]:

2 20 2 2

22 ( ) (2 )0

2 2 2 22 2 0 2 0

60 ( ) ( ) cos( )(2 ) [ ( )] ( )sin ( ) cos

eff L jk d hANT

L in

L RK j e

l d h R Zε ωπ ε ω ω θ

ωω ε ω θ ε ω θ

− += − ⋅+ + −

, (11)

where 1j = − – image unity; l – half length of liner antennas Ant.1 and Ant.2; d – distance between Ant.1 and Ant.2 antennas; ck /0 ω= – wave number for free space; −= fπω 2 angular frequency of

monochromatic wave with liner frequency f ; )2/( 20 hdarctg=θ – incident angle; 2h and 2ε are

thickness and complex permittivity of inspected layer; ( )inZ ω – input impedance of transmitting antenna; LR –

load resistor of receiving antenna; ( )effL ω – effective antennas length

1 cos2( )sin

Leff

L L

k lLk k l

ω −= ⋅ , (12)

for complex wave number [2]

)(ωFWK

)(ωANTK )(ωreflR ∑ )(ωprU )(ωlU

R – ρ ρ – R0 R0

r'0

η

ρ ϕ

d

ξ

Аnt. 2 θ Аnt.1

Y

Z

X

R

H

r0 n0


173

12

31 0 1 0 0 0

0 20 00

(2 ) (2 ) 2 (2 )2 11 2 2 4 3 45(2 )lnL

K k nH I k nH k nH k nHk k j jH k nH k nHk nHa

π⎧ ⎫

⎡ ⎤⎪ ⎪⎛ ⎞= + − + − + +⎢ ⎥⎜ ⎟⎨ ⎬

⎢ ⎥⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭

… ,

which depends on antennas diameter a, antennas high H over upper boundary of inspected medium, and complex refraction coefficient 2 ( )n ε ω= . There (.)1K and (.)1I are modified Bessel functions of the first order.

Following [4], complex transfer function of antennas direct coupled field we write as

0 2 0

2( )

222

60 ( )( ) ( )

( ( ) 1) [ ( )]eff L jk d jk d

FWL in

L RK j e e

ld R Zε ωπ ω

ω ε ωε ω ω

− −⎡ ⎤= ⋅ −⎣ ⎦− +

(13)

and the model reflection coefficient we write as

23 0 2 3 0'

2 3 0 23 0 2 3 0

( ) cos ( ) ( )sin( ) ( , )

( ) cos ( ) ( )sinreflR R

ε ω θ ε ω ε ω θω ω θ

ε ω θ ε ω ε ω θ−

− −= =

+ −. (14)

Equations (11), (13), (14) may be used for further modelling of radar subsurface probing of soils in frequency domain. If we want to see signals in time domain, then we must perform transformation

1( ) ( ) ( )2

j tl RAD probeu t K S de ωω ω ω

π

∞−

−∞

= ∫ , (15)

where ( )probS ω – the spectrum of probe signal, which used for excitation of the transmitting antenna. Signals on output of receiving antenna, which are received by help of (11),(13),(14), and (15) equations,

are shown on Fig.4, for example.

Fig. 4. Model signals on output of receiving antenna for H=0,01m, l=0,25m, 2h =1,5m

4. Numerical Results of Model Inversion

Model inversion, we realized by numerically in the limited range of the soil electromagnetic properties changing according to equation (1). For realization of model inversion data is chosen genetic algorithm of calculations [5] with the following characteristics:


174

• Maximal number of generations ........................................50 • Coding alphabet of an estimated parameters ......................binary • Number of bit per parameter ..............................................6 • Probability of crossing over ...............................................0,9 • Probability of mutation. ......................................................0,05 • Admitted number of population into generation

for the aim functional means improving .............................500.

We have investigated inverse problem using the signal of transmitter given as:

0( ) sin 2tmu t tfU e α π−=

for 1=U m V, 72 10α = ⋅ 1−s and 0

100f = MHz. The spectrum of reflected signal is computed as

2 31( , ) ( ) ( ) ( , )exp( ) ,

2 ANTrefl prob K R P j t dPS Sω ω ω ω ω ωπ

∞ →

−−∞

→= ∫

where )(ωS prob – the spectrum of probe pulse signal )(tu prob , given by

1( ) ( )exp( )2prob probS u t j t dω ω ωπ

∞

−∞

= ∫ ;

−)(ωANTK complex transfer function of the radar antenna system, given by (11); ),(32 PR→

− ω – complex

reflection coefficient (14 ) of the model for true vector of parameters P→

. Product of ( ) ( )ANT probK Sω ω , module of which is shown on Fig. 5, determinate a frequency range

min max( ... )F F for calculations of the task (1) aim functional A , as well as a signal energy

max

min

0 2 ( ) ( )F

ANT probF

E K S dfω ω= ∫ ,

which in principle may be used for normalization of the computation process parameters. As it is seen on Fig. 5, frequency range from 20MHz up to 500 MHz may be used for efficient estimation components of vector parameters P .

Fig. 5. Frequency dependence of )()( ωω probANT SK module


175

The probing object is modelled by two-layer media with a changing of the primary parameters as it is given in Table 1.

Table 1. The primary parameters changing for model of probing object

Range of the parameter changing

N Primary parameters of model soils Medium 2 Medium 3

1. Thickness of layer )0,5...5,1(,2 =mh ∞→3h

2. Bulk density )0,2..35,1(, 32 =−gcmγ )5,2..8,1(, 3

3 =−gcmγ

3. Porosity of soils )35...12(,%2 =P )20...15(,%3 =P

4. Volumetric moisture content )5...3(,%2 =W )12...5(,%3 =W

5. Salt concentration of pore water )0,3....0,1(, 3 =−gdmC

6. Type of salt 424 ,,, MgSOMgClNaSONaCl

For estimation of model secondary parameters we choose a module of effective complex dielectric

permittivity of medium 2, averaged in used frequency range min max( ... )F F , as well as a thickness of this medium. Model inversion are performed for different relative means of threshold ,%αδ and maximal frequency

maxF in order to investigate influence of factors on accuracy of the parameter estimations and an efficiency of calculation process. The search range of parameter vector has been limited by the ±50% changing of parameters.

The main results of such estimations are presented in Table 2.

Table 2. The main results of accuracy estimation of model secondary parameters

Maximal frequency of analysis F max , MHz (Relative means of used signal energy En)

100 (88,0%)

200 (98,4%)

300 (99,6%)

500 (99,9%)

N

Relative mean of threshold

,%αδ

,%2ε

δ ,%2hδ ,%

2εδ ,%

2hδ ,%2ε

δ ,%2hδ ,%

2εδ ,%

2hδ

1. 0,005 7,7 4,29 4,19 2,86 3,96 2,39 4,08 2,0

2. 0,025 6,73 6,20 4,11 2,61 4,32 2,22 3,72 1,96

3. 0,05 7,37 7,49 5,69 3,33 4,67 2,31 4,36 2,85

4. 0,5 7,33 7,14 6,4 6,26 6,56 6,5 6,88 5,86

5. 5,0 7,23 6,63 6,89 6,72 7,05 6,63 6,39 5,92

As it is seen from the presented data the relative errors of the secondary parameter estimation not exceed

10% both for effective dielectric permittivity and thickness of medium 2. Increasing of relative means of threshold more than ,%αδ >0,05 gives essential growing of errors for all used means of maximal frequency of analysis F max .

As it is known from technical literature [5], one from three conditions is necessary for completing of the data processing by genetic algorithm. These are as follows:

• appropriate quality of solution is received (for radar subsurface inspection of road covers it means that parameters of object are estimated with an enough accuracy);

• local extremity is found but algorithm can’t come out from this state; • maximal number of population into generation is formed but appropriate quality of solution is not

received.

We investigated these items for our process of the model inversion. For these purposes in all performed estimations of the secondary model parameters we determinate the means of aim functional A , which corresponds to the completion of a computation process. Then the data are averaged and used for analyses.

In Table 3 are presented averaged means of aim functional A , which corresponds to the completion of a computation process for different relative means of threshold ,%αδ and maximal frequency F max .


176

Table 3. Averaged means of aim functional A , which corresponds to the completion of a computation process

Maximal frequency of analysis F max , MHz (Relative means of used signal energy En) N

Relative mean of threshold

,%αδ 100 (88,0%)

200 (98,4%)

300 (99,6%)

500 (99,9%)

1. 0,005 0,056 0,026 0.031 0.031

2. 0,025 0.017 0.028 0.031 0.031

3. 0,05 0.030 0.039 0.039 0.039

4. 0,5 0,062 0,182 0,206 0,191

5. 5,0 0.065 0.189 0,244 0,211

5. Conclusions The main results of these model-based experiments are the following:

• The value of threshold α influences on errors of estimation of the model parameters, as well as on the number of population into generation, which must be formed;

• Mean of thresholdα must be set in consideration of energy of spectral constituents in the reflected signal, which are used for decision of inverse problem. Such choice of value of threshold allows to determine electrical properties of the roadway coverage with the set error, using the eventual number of generation, and, consequently, substantially reduce a duration of the inverse problem decision

• In our experiments influences of chromosome length, terms of mating and mutations on the errors of the model parameter determination are investigated. Finding the dependencies, this by finding may be useful to optimise the structure of genetic algorithm for realization of the secondary signal processing in the radar monitoring system.

• In order to decrease error of estimation of the model parameters for sounded object it is necessary to increase the chromosome length (number of bits per parameter).

References

1. Krainyukov, A., Kutev, V. Intelligent system for radar monitoring of transport communications. In: 7th

International Conference “Reliability and Statistics in Transportation and Communication (RelStat’07), 25–27 October 2007, Riga, Latvia”. Riga: Transport and Telecommunication Institute, 2007, pp. 312-320.

2. King, R.W.P., Smith, G.S. Antennas in Material Fundaments, Theory, and Applications. The MIT Press, 1981.

3. Finkelstein, M.I., Kutev, V.A., Zolotarev, V.P. Application radar-tracking under superficial sounding in engineering geology. M.: Nedra, 1986. (In Russian)

4. Krainykov, A.V., Kutev, V.A. Modelling of a path of formation of the reflected signals in problems under superficial radar-locations of layered terrestrial covers. In: Scientific papers of the Riga Aviation University. Riga: RAU, 1995, pp.17-23. (In Russian)

5. Gladkov, L.A., Kureichick, V.V., Kureichick, V.M. Genetic algorithms / Edited by V.M. Kureichick, 2-nd issue. М.: Fizmatlit, 2006. (In Russian)


177

TIME-DEPENDENT PROBLEM FOR DETERMINATION OF EXHAUST CONCENTRATION IN URBAN TRANSPORT SYSTEM

Stanislav Grishin1, Janis Rimshans2,3, Eugene Kopytov1

Sharif Guseynov1,2, Oleg Schiptsov1

1Transport and Telecommunication Institute

Lomonosova 1, Riga, LV-1019, Latvia 2Institute of Mathematical Sciences and Information Technology

Liela 14, Liepaja, LV-3401, Latvia 3University of Latvia

Rainis Blvd. 29, Riga, LV-1459, Latvia

The paper proposes the 3-D mathematical model for analytic determination of exhaust concentration dynamics in the city regions under a priori information on airflow velocity. In this model is provided that the turbulent and molecular diffusion coefficient changes depending on the vertical remoteness above the ground surface. The numerical example of problem solving is presented. The created model can be used for solutions of operative problems on urban traffic organization, of long-term planning of urban agglomeration development and new highway building.

Keywords: auto transport, urban ecology, time-dependent mathematical model, exhaust concentration, airflow velocity 1. Introduction

Environmental pollution, noise and vibration exploration of the urban traffic becomes more serious with

every year. Evaluation problem of transport negative impact on urban ecology is the subject of many scientific researches. The evaluation of air pollution by automobile transport takes a special place in the analysis of town pollution level, since hazardous pollutants emission into the atmosphere is resulted by burning of fuel. According to the data [1] the environmental pollution from automobile transport is 60,6% of all the atmosphere pollution. The chemical constitution of emissions depends on the fuel type and quality, production technology, the type of fuelling in the car engine, and its technical stage in future emission filtration. Among the harmful substances that are being emitted into the atmosphere, carbon dioxide, nitrogen oxides, hydrocarbons, aldehydes, sulphides and lead have to be pointed out in the first place. In this list carbon dioxide (СО2) has the main influence on the atmosphere. It exerts maximal influence on the glasshouse effect specifically. Besides, nitrogen oxides, hydrocarbons, aldehydes, sulphides, lead and other substances have negative impact on urban people health. The proof of that is the increase in quantity of urban people's oncological diseases, heart-vascular system and lungs diseases. As an example, cancer incidences rate in Switzerland is 9 times higher for people who live close to highways with the intensive traffic rather than for people who live in suburbs located 400 meters away from the highway.

Annual air pollution management costs are 100 million pounds sterling in Great Britain, 700 billion yen in Japan, and 1,5 billions dollars in the USA. There is an opinion in the USA that pure air could reduce the expenses on medicine by more than 2 billions dollars per year [1].

Issues of urban environment protection against transport pollution require development and environment protection actions plan implementation. The scheme of urban environment protection inter-relation problems caused by transport is overviewed by authors in the work [2].

The main subjects of authors’ research are the air pollution status evaluation and dynamics forecasting in urban traffic circumstances; actions development for clearing and prevention of pollution in a town area, and foremost, living areas. The present article analyses the mathematical model for the evaluation of such factors as traffic and transport structure, urban agglomeration planning and meteorological conditions influence on air pollution in urban quarters. The introduced model can be used for long-term planning of urban agglomeration, new urban quarters designing, and new highway building. This model can also be useful in operative questions solutions on a traffic organization matter within a living area.

2. The General Definition of the Problem

In general case the following definition of the problem takes place. Let the set

1 2( ) { ( ), ( ),..., ( )}nS t s t s t s t= be defining air condition in the given point of tree-dimensional space at the time t , where ( ), 1,2,...,js t j n= is determining the concentration of j-th substance.

It is assumed that at some moment of time et the condition of the air can be specified from the functional equation


178

* * *1 1( ) ( ), 1,2,..., ; ( ,..., ; ); ( ,..., ; )e i k e k eS t F S t i k Y t t t Z t t t⎡ ⎤= =⎣ ⎦ , (1)

where F is some functional, 1 2( ), ( ),..., ( )kS t S t S t is the set specifying the condition of the air at the analysed

point at the moments of time 1 2, ,..., kt t t , that are anteceding to the moment et , i.e.; ( )*1,..., ;k eY t t t is the set of

uncontrolled parameters, that are describing specification of external environment, influencing to the concentration of the substances in the air, meteorology in the first place (a temperature of the air, direction and speed of the wind) et al, ( )*

1,..., ;k eZ t t t is the set of partly or fully control external environment parameters, regulation of which influences on the concentration of the substances in the air, for example, a traffic intensity, in total and per a car type.

Note that depending on the problem observed, some indexes of an external environment can be uncontrolled parameters of set ( )*

1,..., ;k eY t t t and controlled parameters of set ( )*1,..., ;k eZ t t t . For

example, such interesting indexes as town quarters planning, a profile and specification of highways, specification of green space are related to the uncontrolled parameters of the set ( )*

1,..., ;k eY t t t , if problem of

operational control of transport flow is considered, and to the controlled parameters of the set ( )*1,..., ;k eZ t t t ,

if the problem of urban development long-term planning and execution of ecology actions, for example, a green space implantation is considered. The tasks for operational control of transport flow, urban development long-term planning and building of new highways in general case are stated as follows: with the stated set

( )*1,..., ;k eY t t t and ( )*

1,..., ;k eZ t t t is required to determinate a concentration of hazardous substances

( )eS t in the examination area of town air space, and with the excess of j-th substance concentration allowed

level maxjs is required to change the controlled parameters ( )*

1,..., ;k eZ t t t in the way to have eligible set

elements *( )eS t , i.e. concentrations *( ) ( )j e es t S t∈ would be within tolerance limits: max 1,2,...j e js (t ) s , j = ,n≤ .

Note, that solving of problem (1) should be executed for different variants of external environment conditions ( )*

1,..., ;k eY t t t , besides, a corresponding managerial decision (the set of control impacts) will be

defined for each variant. The set of these solutions allows formulating the urban transport system management strategy and finding the proposals of its development.

For determination of the functional F in equation (1) different mathematical methods can be used, primarily, the methods that are based on mathematical description of physical processes, and statistical methods (for example, the methods of multi-regression), that use the statistics which has been gathered in the process of ecological monitoring of the city. The main advantage of physical methods is the possibility to use them for each new situation, while the statistical methods are applied only after a large number of observations has been gathered. The physical models allow receiving more exact solutions, but in opposite to statistical models they are more complicated and require more sophisticated mathematical tools for their solution.

In this research the authors are building exactly the physical model, namely, under the functional F in the functional equation (1) a differential operator is being formed, which is describing the process of turbulent diffusion when mass transfer in the space is dictated by the turbulent motion of environment. In the present work the solution of the building model is analytically in closed formula. Besides, this research carries out numerical implementation of the obtained analytical solution with certain simplifying assumptions. 3. Mathematical Model

In present work the 3-D mathematical model for analytic determination of exhaust concentration

dynamics in the city regions under a priori information on air flow velocity is proposed. In this model is provided that the turbulent and molecular diffusion coefficient changes depending on the vertical remoteness above the ground surface, i.e. the condition of vertical layering of town air is designed-in the proposed mathematical model owing to long-term harmful substances accumulation therein produced by vehicles.

The mathematical statement of the problem formulated above is following: it is required to find the concentration ( ){ }

1 2 3, , ,nC x x x t of n -th ( )1,n N= harmful substance at any spatial point ( )1 2 3, ,x x x in the

bounded and closed domain [ ] [ ] [ ]1 2 30, 0, 0,l l l× × at any moment of time [ ]0,t T∈ from the equation


179

{ } ( ) ( )( ) { } ( )( ) ( ) { } ( )

( ) ( )1 2 3

,, , , , , 0,

, , : 0 1,3 ;

nn n

i i

C x tdiv D x t grad C x t x t grad C x t t

tx x x x x l i

ϑ ϑ∂

= ⋅ − ⋅ ≥∂

= < < =

(2)

from the initial condition { } ( ) { } ( ) ( ) ( )0 1 2 30

, , , , : 0 1,3 ;n ni it

C x t C x x x x x x l i== = ≤ ≤ = (3)

from the boundary condition (for each fixed 0, 1j M= − ) (see [3-5])

{ }{ } ( ) { } { } ( ) { } { }( ) ( )

,

,

,1, ,2, , , ,

,, , , 1,3 , 0,

i i j

i i j

nn n n n

i j i j i j i i j i i jx ai x a

C x tC x t C x x t a x b i t

xγ γ

==

∂⋅ − ⋅ = ≤ ≤ = ≥

∂ (4)

{ }{ } ( ) { } { } ( ) { } { }( ) ( )

,

,

,3, ,4, 3, , ,

,, , , 1,3 , 0,

i i j

i i j

nn n n n

i j i j i j i i j i i jx bi x b

C x tC x t C x x t a x b i t

xγ γ +

==

∂⋅ + ⋅ = ≤ ≤ = ≥

∂ (5)

from the matching condition

{ } ( ) { }{ } ( ) { } ( )

3 33 30 0, , , 1, 1, 0 1,2 ,

j j

n ni ix l x l

C x t C x t j M x l i= − = +

= = − ≤ ≤ = (6)

( )( ){ } ( )

{ }

( )( ){ } ( )

{ }

( )3 33 3

3 30 0

, ,, , , 1, 1, 0 1,2 .

j j

n n

i i

x l x l

C x t C x tD x t D x t j M x l i

x xϑ ϑ

= − = +

∂ ∂⋅ = ⋅ = − ≤ ≤ =

∂ ∂ (7)

In the unsteady-state initial boundary value (2)-(7) problem the coefficient of turbulent and molecular diffusion is designated as ( )( ),D x tϑ , which is assumed as piecewise constant function in this work:

( )( )

{ } { }

{ } { }

{ } { }

0 11 3 3 3

1 22 3 3 3

min

13 3 3

0 ,

,0 ,.................................................

,

def

M MM

D const if l x l

D const if l x ld D x t

D const if l x l

ϑ

−

⎧ = = ≤ ≤⎪

= ≤ ≤⎪< ≤ ≡ ⎨

⎪⎪ = ≤ =⎩

for [ ] ( )0, , 1,2 ,i ix l i∀ ∈ =

where M is the quantity of layered domains on the vertical axis 3OX , i.e. these layered domains are parallel to

the plane 1 2X OX ; the vector–function ( ),x tϑ signifies an airflow velocity, which is purposed experimentally known and piecewise constant function:

( )

{ } { } { }

{ } { } { }

{ } { } { }

0 11 3 3 3

1 22 3 3 3

13 3 3

0 ,

,,.........................................................

,

av

avdef

av M MM

const if l x l

const if l x lx t

const if l x l

ϑ

ϑϑ

ϑ −

⎧ = = ≤ ≤⎪

= ≤ ≤⎪≡ ⎨⎪⎪ = ≤ =⎩

for [ ] ( )0, , 1, 2i ix l i∀ ∈ = ;

and finally, the points ,i ja and ,i jb mean the boundary points of layers, namely, the points ,i ja and ,i jb are definition as

{ }

{ }, 3

03

0 1,2,

3; 0,

3; 0,

defj

i j

if i

a l if i j

l if i j

=⎧⎪

≡ = ≠⎨⎪

= =⎩

{ }, 13

1, 2; ,

3; .

def ii j j

l if i jb

l if i j+

= ∀⎧⎪≡ ⎨= ∀⎪⎩

In the problem (2)-(7) we assume that the following constants are known initial constants: N ∈ , M ∈ , 1T +∈ , ( )1 1,jD j M+∈ = , { } ( )1 1,3; 0,j

il i j M+∈ = = , { } ( )1,avj j Mϑ = ,

{ } ( )1, , 1, 2; 1, 4; 0, 1n

i k j i k j Mγ +∈ = = = − , and for 1,n N∀ =


180

{ }

{ }, ,

, ,

1 0; 1; 1,3,0 0; 1; 2,4,

0 0; , .

ni k j

ni k j

if j i kif j i k

if j i k

γ

γ

⎧ = = =⎪⎪= = = =⎨⎪ > ≠ ∀⎪⎩

Besides, in the problem (2)-(7) we suppose that the initial functions { } ( ) ( )0 1,nC x n N∀ = ,

{ } ( ) ( ), , 1,6; 0, 1ni jC t i j M= = −i are given functions, and for 1,n N∀ = it is

{ } ( ){ } ( ) { } { },

,, 0 3,6 ;

,0 .

nn i j

i jC t if j i

C totherwise

⎧ = ∧ =⎪= ⎨⎪⎩

ii

Finally, besides the conditions (1)-(7) in the mathematical statement of the considered problem we require the satisfaction of the corresponding consistency conditions, i.e. the initial function { } ( )0

nC x and the boundary

functions { } ( ), ,ni jC ti satisfy the consistency conditions at the corresponding conjugate points for each 1,n N=

harmful substance. Using standard approach (see [6]) we can prove that the formulated problem (2)-(7) has an unique

solution. In present paper we will suppose that the quantity of vertical layers, which are between the ground surface and the upper bound 3l , equals to four, i.e. 4M = , and the bounds of each layer are given empirically.

Now we start to solve the formulated problem (2)-(7). For the purpose we will introduce the following new function

{ } ( )

{ } ( ) { } { }

{ } ( ) { } { }

{ } ( ) { } { }

( )

0 11 3 3 3

1 22 3 3 3

13 3 3

, 0 ,

, ,, 0 1, 2 ,

..........................................

, ,

n

ndefn

i i

n M MM

w x t if l x l

w x t if l x lw x t x l i

w x t if l x l−

⎧ = ≤ ≤⎪

≤ ≤⎪≡ ∀ ≤ ≤ =⎨⎪⎪ ≤ =⎩

where

{ } ( ){ }( ) { }

{ } ( ) ( ){ } { } { } { }( )

23

1

3

4 2

1 03 3 3 3 3 3

, , , 0, 0 1, 2 ,

1, ; 0, .

av avj j

ij j i

t xdefD Dn n

j i i

j j M

w x t e C x t t x l i

l x l j M l l l

ϑ ϑ

=

⋅⋅ − ⋅

⋅ ⋅

−

∑≡ ⋅ ≥ ≤ ≤ =

≤ ≤ = = =

(8)

Taking into account the notation (8) in the equation (2) and in the initial condition (3), we have the following new problem: it is necessary to find the functions { } ( ) ( ), 1, 2n

jw x t j = from the equation

{ } ( ) { } ( ) ( )

{ } { } { } { }( )

23

21

1 03 3 3 3 3 3

, ,, 0, 0 1,2 ,

1, ; 0, ,

n nj j

j i ii i

j j M

w x t w x tD t x l i

t x

l x l j M l l l

=

−

∂ ∂= ⋅ > < < =

∂ ∂

< < = = =

∑ (9)

from the initial condition { } ( ) ( ) ( ) { } { } { } { }( )1 0

0 3 3 3 3 3 30, , 0 1,2 , 1, ; 0, ,n j j M

j i itw x t w x x l i l x l j M l l l−

== < < = < < = = = (10)

where

( ){ }

{ } ( )3

120 0

avj

ij i

xdefD nw x e C x

ϑ

=

− ⋅⋅ ∑

≡ ⋅ ,

and from the corresponding boundary and matching conditions. Thus, with the help of the non-degenerate transformation (8) the original problem (2)-(7) with

inhomogeneous equation (2) is reduced to the problem (9)-(10) with homogeneous equation (9). The obtained problem is simpler than the original problem (1)-(7). Now we will find the solution of the received problem (9)-(10) in the form (see [6])


181

{ } ( ) ( ) ( ) ( ) ( ) ( )1 2 3 1 2 3, , , , 0 1,3 .ni iw x x x t W x W x W x T t x l i= ⋅ ⋅ ⋅ ≤ ≤ = (11)

Substituting (11) into (9)-(10) and having some not difficult transformations we have the solution of the reduced problem (9)-(10):

{ } ( ) { } { } ( ){ }( ) { } { } 222

1 2 3 , , , , 1 2 31 1 1

, , , , , ,nnn

j ki tn n n

i j k i j ki j k

w x x x t A W x x x eλ λ λ

⎛ ⎞⎛ ⎞∞ ∞ ∞ − ⋅ ⋅ ⋅⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= = =

= ⋅ ⋅∑∑∑ (12)

where

{ }( ) { } ( )

{ } ( ){ }

( ) { } ( )

{ } ( ){ }

( )

31 2 1 2

3 31 2 1 2

1 2 0 1 2 3 , , 1 2 3 3 1 3,0 1 2 , , 1 2 3 20 0 0 0 0 0

, ,2 2

1 2 , , 1 2 3 3 1 2 , , 1 2 3 30 0 0 0 0 0

1 6,0 1 2 , ,

, , , , , , , ,

, , , ,

, ,

ll l l lTn n

i j k i j kn

i j k l ll l l ln n

i j k i j k

i j

dx dx w x x x W x x x dx dt dx w x x t W x x x dxA

dx dx W x x x dx dx dx W x x x dx

dt dx w x x t W

⋅ ⋅= − +

⋅+

∫ ∫ ∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫ ∫

{ } ( )

{ } ( ){ }

1 2

31 2

1 2 3 20 0 0

2

1 2 , , 1 2 3 30 0 0

, ,,

, ,

l lTn

k

ll ln

i j k

x x x dx

dx dx W x x x dx

∫ ∫ ∫

∫ ∫ ∫

{ } { } { }( ), , , ,

nnni j k i j kλ λ λ ∈ are roots of transcendental equation, respectively,

{ }( ) { }

{ }

{ }

{ }

{ }

{ }{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }{ }

{ }

{ }

{ }

1,2,1 1,4,1 1,2,2 1,4,2 1,2,3 1,4,3

1,1,1 1,3,1 1,1,2 1,3,2 1,1,3 1,3,31

1,2,1 1,4,1 1,2,2 1,4,2 1,2,3 1,4,3

1,1,1 1,3,1 1,1,2 1,3,2 1,1,3 1,3,3

n n n n n n

n n n n n nn n

n n n n n nn n n

n n n n n n

tg l

γ γ γ γ γ γγ γ γ γ γ γ

λ λγ γ γ γ γ γ

λ λ λγ γ γ γ γ γ

⎧+ + +⎪

⋅ = ⋅ + + +⎨− ⋅ − ⋅ − ⋅

{ }

{ } { }

1

12

1

1

,

average

averagen

D

D

ϑ

ϑλ

⎫⎪⎪⎪⎬

⎛ ⎞⎪ ⎪− ⎜ ⎟⎪ ⎪

⎩ ⎝ ⎠ ⎭

{ } { }

{ }

{ }

{ }

{ }

{ } { }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ } { }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ } { }

{ }

{ }

2,2,0 2,4,0 2,2,1 2,4,1 2,2,2 2,4,2

2,1,0 2,3,0 2,1,1 2,3,1 2,1,2 2,3,22

2,2,0 2,4,0 2,2,1 2,4,1 2,2,2 2,2,2

2,1,0 2,3,0 2,1,1 2,3,1 2,1,2 2,1,

n n n n n n

n n n n n nn n

n n n n n nn n n

n n n n n

tg l




+ + +⎛ ⎞⋅ = ⋅ + +⎜ ⎟⎝ ⎠

− ⋅ − ⋅ − ⋅ { }

{ }

{ }

{ }

{ }

{ } { }

{ }

{ }

{ }

2,2,3 2,4,3

2,1,3 2,3,3

2,2,3 2,4,3

2 2,1,3 2,3,3

,

n n

n n

n nn

n n n

γ γγ γ

γ γλ

γ γ

⎧ ⎫+⎪ ⎪

⎪ ⎪+⎨ ⎬⎪ ⎪− ⋅⎪ ⎪⎩ ⎭

{ } { }

{ }

{ }

{ }

{ }

{ } { }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ } { }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ } { }

{ }

{ }

3,2,0 3,4,0 3,2,1 3,4,1 3,2,2 3,4,2

3,1,0 3,3,0 3,1,1 3,3,1 3,1,2 3,3,22

3,2,0 3,4,0 3,2,1 3,4,1 3,2,2 3,4,2

3,1,0 3,3,0 3,1,1 3,3,1 3,1,2 3,3,

n n n n n n

n n n n n nn n

n n n n n nn n n

n n n n n

tg l




+ + +⎛ ⎞

⋅ = ⋅ + +⎜ ⎟⎝ ⎠ − ⋅ − ⋅ − ⋅ { }

{ }

{ }

{ }

{ }

{ } { }

{ }

{ }

{ }

3,2,3 3,2,3

3,1,3 3,1,3

3,2,3 3,2,3

2 3,1,3 3,1,3

,

n n

n n

n nn

n n n

γ γγ γ

γ γλ

γ γ

⎧ ⎫+⎪ ⎪

⎪ ⎪+⎨ ⎬⎪ ⎪− ⋅⎪ ⎪⎩ ⎭

For each layer the function { } ( ), , 1 2 3, ,ni j kW x x x from the formula (12) can be defined as following:

– the first layer { }11 1 2 2 3 30 , 0 , 0x l x l x l≤ ≤ ≤ ≤ ≤ ≤ is circumterraneous layer, and in this lowermost layer the

function { } ( ), , 1 2 3, ,ni j kW x x x has the form

{ } ( ){ } { }( ){ } { } { }( )

{ }( ) { }

{ }{ }( ) { } { } { }

{ } { }

{ }, 1

, , 3 2,2,0 1 11, , 1 2 3 1 1 2 2, 1 1

, , 3 1 1 2,1,0

cos 2, , cos sin cos sin ;

cos 2 2

n n avav n nn ni j kni i i ii j k n nn n

i j k i i

x DW x x x x x x x

l D D

θ γ ϑϑβ β β βθ β γ β

⎡ ⎤⎡ ⎤ ⋅ −⎛ ⎞ ⎛ ⎞⎢ ⎥⎢ ⎥= − −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥ ⋅⎣ ⎦ ⎢ ⎥⎣ ⎦

– in the second layer { } { }1 21 1 2 2 3 3 30 , 0 ,x l x l l x l≤ ≤ ≤ ≤ ≤ ≤ the function { } ( ), , 1 2 3, ,n

i j kW x x x has the form


182

{ } ( ){ } { } { }( )( ){ } { } { } { }( )( )

{ }( ){ } { }

{ } { }{ }( )

{ } { }

{ } { }

{ }

{ }

, 2 2, , 3 3 2 1,2,1 2 2 2,2,1 2

, , 1 2 3 1 1 2, 2 2 1, , 3 3 2 1,1,1 2 2,1,1

2

cos 2 2, , cos sin sin

cos 2 2

cos ;

n av n av n ni j k n nni i ii j k n nn n n

i j k i i

n

i

l x D DW x x x x x x

l l D D

x

θ ϑ γ ϑ γβ β β

θ γ β γ β

β

⎡⎡ ⎤− − − ⎛ ⎞⎢⎢ ⎥= + +⎜ ⎟⎢⎢ ⎥ ⎝ ⎠− ⎣ ⎦ ⎢⎣⎤⎛ ⎞+ ⎜ ⎟⎥⎝ ⎠⎦

– in the third layer { } { }2 31 1 2 2 3 3 30 , 0 ,x l x l l x l≤ ≤ ≤ ≤ ≤ ≤ the function { } ( ), , 1 2 3, ,n


{ } ( ){ } { } { }( )( ){ } { } { } { }( )( )

{ } { } { }

{ } { }{ }( )

{ } { }

{ } { }

{ }

{ }

, 3 3, , 3 3 3 1,2,2 3 3 2,2,2 3

, , 1 2 3 2 1 2, 3 3 2, , 3 3 3 1,1,2 3 2,1,2

2


cos 2 2

cos ;

n av n av nn ni j k nni i ii j k n nn n n

i j k i i

n

i


l l D D

x


θ γ β γ β

β

⎡⎡ ⎤− − − ⋅⎛ ⎞ ⎛ ⎞⎢⎢ ⎥= + +⎜ ⎟ ⎜ ⎟⎢⎢ ⎥⎝ ⎠ ⎝ ⎠− ⎣ ⎦ ⎢⎣⎤⎛ ⎞+ ⎜ ⎟⎥⎝ ⎠⎦

– the last layer { }31 1 2 2 3 3 30 , 0 ,x l x l l x l≤ ≤ ≤ ≤ ≤ ≤ is the most removed from a ground surface layer in

considering bounded and closed parallelepiped [ ] [ ] [ ]1 2 30, 0, 0,l l l× × . In this uppermost layer the function { } ( ), , 1 2 3, ,n


{ } ( ){ } { } ( )( ){ } { } { }( )( )

{ }( ){ } { }

{ } { }{ }( )

{ } { }

{ } { }

{ }

{ }

, 4, , 3 3 4 1,2,3 4 4 2,2,3 4

, , 1 2 3 1 1 2, 4 3, , 3 3 4 1,1,3 4 2,1,3

2


cos 2 2

cos .

n av n av n nn ni j kni i ii j k n nn n n

i j k i i

n

i


l l D D

x


θ γ β γ β

β

⎡⎡ ⎤− − − ⎛ ⎞⎢⎢ ⎥= + +⎜ ⎟⎢⎢ ⎥ ⎝ ⎠− ⎣ ⎦ ⎢⎣⎤⎛ ⎞+ ⎜ ⎟⎥⎝ ⎠⎦

Here

{ } { }{ } { } { }

( )2 2 2 2

,, , 2 2

1 2

1, 1,4 ,

nnndef

i j kn mi j k

m

i j m MD l l

λ λ λ π πθ ⋅ ⋅ ⋅≡ − − = =

{ }niβ and

{ }n

iβ are positive roots of the transcendental equation, respectively,

2 2 2 2

1 22 2 2 21 1 1 1 2 1 2 1

1 1 1 1 ,D ctg D ctgD l D l D l D l

π π π π⋅ − ⋅ − = ⋅ − ⋅ −

2 2 2 2

1 22 2 2 21 2 1 2 2 2 2 2

1 1 1 1 .D ctg D ctgD l D l D l D l

π π π π⋅ − ⋅ − = ⋅ − ⋅ −

Thus, the formula (12) is the solution of the reduced problem. Then the solution of the original problem (1)-(7) can be found by the non-degenerate inverse transformation:

{ } ( ){ } { }( )

{ } ( ) ( )2

3

1

3

2 41 2 3 1 2 3, , , , , , , 0, 0 1,3 .

averageaveragejj

ij ji

x tD Dn n

i iC x x x t e w x x x t t x l i

ϑϑ

=

⋅⋅ − ⋅

⋅ ⋅∑= ⋅ ≥ ≤ ≤ = (13)

The formula (13) lets to determine the required concentration { } ( )1 2 3, , ,nC x x x t of the n -th harmful

substance at any moment of time [ ]0,t T∈ in any spatial point ( )1 2 3, ,x x x for each of N harmful substances.

4. Example of Results

Let’s consider the example of calculation of concentration change of harmful substance on a different height above the road area at the post-initiation fixed time moments 1, 2, 6 and 12 hours, respectively. We consider the road section with the width of 21 m and the length of 165 m. It is supposed that there are multi-story


183

buildings in both sides of considered road, at that the average height of buildings is 20 m. Besides, we assume that the number of cars driving through the road section per 12 hours is known and equals to 11000 units, i.e. the traffic flow rate in the considered road is accepted as equi-distributed, namely, it equals to 917 cars per hour approximately. Other initial data are following: depending on the altitude wind velocity on layers changes from 4 m/s till 1 m/s; the coefficient of turbulent diffusion depends on altitude and it changes from 0.13 (highest) to 0.16 (lowest); average concentration of the investigated harmful substance (as investigated material is taken CO2 particularly) is assumed as 179 g/km; exhaust speed near cars is 60-100 m/s.

Computations will be performed for an "imaginary vertical column", the foundation of which is exactly in the middle of the considered road and it is determined for the point ( )1 210.5 m; 82.5 mx x= = . Numerical implementation of the considered mathematical model has been realized by the packaged MathCAD. The results of calculations for the different moments of time, passing after the beginning of turbulent diffusion process, are presented in Figure 1.

a) b)

c) d)

Fig. 1. Change of the concentration of harmful matter on a different height above a road area

Figure 1a shows a change of concentration ( )1 2 210.5, 82.5, , 1C x x x t= = = depending on the variable

3x , i.e. the constructed curve reflects a change of harmful substance concentration depending on a height 3x in 1 hour after the beginning of process of supervision of harmful substance turbulent diffusion in the fixed point of the road area ( )1 210.5 m; 82.5 mx x= = .

Changes of concentration depending on a height in the same point at the moments 2, 6 and 12 hours after the beginning of turbulent diffusion process are presented in Figure 1b, 1c and 1d, respectively. Note that in Figure 1a scale of ordinates is compressed 210 times less, and in the other figures scales of ordinates are taken 10 times less, i.e. there is the graph of function ( ) ( )

1 2

23 1 2 3 10.5; 82.5; 1

10 , , ,x x t

C x C x x x t−

= = == ⋅ in Figure 1a, and

there are the graphs of functions ( )1 2

11 2 3 10.5; 82.5; 2

10 , , ,x x t

C x x x t−

= = =⋅ , ( )

1 2

11 2 3 10.5; 82.5; 6

10 , , ,x x t

C x x x t−

= = =⋅ and

( )1 2

11 2 3 10.5; 82.5; 12

10 , , ,x x t

C x x x t−

= = =⋅ in Figures 1b, 1c and 1d, respectively.

Let's note that this example considered by authors has an illustrative character, because the solving of the offered mathematical model with respect to the considered example has been executed under some simplifying assumptions. For wide application of the offered mathematical model in practical questions is necessary to develop more complex program using the high-level language.


184

5. Discussion

As can be seen from the mathematical statement (2)-(7) of the investigated problem, there is an assumption to the effect that a priori information on airflow velocity ϑ and its direction, at that this vector velocity is considered the known piecewise constant function of variable 3x only and, consequently, this is

independent of other two spatial variables 1 2,x x and temporary variable t , i.e. ( )3xϑ ϑ= . In addition, in the

present work it is assumed that the coefficient of turbulent and molecular diffusion ( )D ϑ is also piecewise

constant function, which changes depending on a vertical remoteness from the ground surface. Obviously, these assumptions simplify the equation (2) and the matching conditions (6)-(7). In general, both the vector velocity ϑ and the coefficient of turbulent and molecular diffusion ( )D ϑ are not of necessity to be piecewise constant

functions. Moreover, these functions cannot be considered a priori known. In the near future, the authors of present work intend to investigate a more general case preliminarily relinquishing above-mentioned strict requirement.

In the process of this investigation the program system must be develop allowing computations of atmospheric air pollution in dwelling zones of city in view of manifold initial data. As evident from the example of results, these computations are intricate and laborious problem. 6. Conclusions • The present work proposes the 3-D mathematical model for analytic determination of exhaust concentration

dynamics in the city under a priori information on airflow velocity. In this model is provided that the turbulent and molecular diffusion coefficient changes depending on the vertical remoteness above the ground surface.

• With the help of the non-degenerate transformation an original problem, where in equation has summands answering to the turbulent phenomena, is reduced to the equivalent problem, where already in equation has no terms in an explicit form answering to the turbulence effect.

• In the research it has been proved that the reduced problem has unique solution. Consequently, with the help of the non-degenerate transformation the formulated original mathematical model has unique solution also. This solution is determined analytically in the closed form.

• The results of these researches can be useful at the decision of operative questions on organization of urban traffic within the limits of dwellings boroughs. The offered model can be used for the perspective planning of urban agglomeration and building new motorways.

References 1. Troickaja, N.A. Integrated transport system. Moscow: Akademija, 2003. 240 p. (In Russian) 2. Grishin, S., Kopytov, E., Shchiptsov, O. Research of Transport System Influence on Ecology of the City. In:

Proceedings of the 7th International Conference “RELIABILITY and STATISTICS in TRANSPORTATION and COMMUNICATION” (RelStat'07), October 24-27, 2007, Riga, Latvia. Riga: Transport and Telecommunication Institute, 2007, pp. 2-9.

3. Davidson, P.A. Turbulence: An Introduction for Scientist and Engineers. Oxford: Oxford University Press, 2006. 657 p.

4. Occendon, J., Howison, S., Lacey, A., Movchan, A. Applied Partial Differential Equations. Oxford: Oxford University Press, 2006. 449 p.

5. Abramovich, G.N. Applied Gas Dynamics, Part I. Moscow: Nauka Press, 1991. 601 p. (In Russian) 6. Tikhonov, A.N., Samarsky, А.А. The mathematical physics equation. Moscow: Moscow State University

Press, 2004. 798 p. (In Russian)

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