WO2013006893A1

WO2013006893A1 - Automated method and system for analysing transport networks

Info

Publication number: WO2013006893A1
Application number: PCT/AU2012/000805
Authority: WO
Inventors: Majid SARVI; Iradj OUVEYSI
Original assignee: Monash University
Priority date: 2011-07-11
Filing date: 2012-07-05
Publication date: 2013-01-17

Abstract

An automated method for analysing the sustainability of a transport network including a plurality of transport zones interconnected by a plurality of transport nodes and links, includes (a) computing a plurality of traffic matrices, each matrix defining traffic load on the plurality of transport links during a different time period, (b) computing a maximum allowed traffic growth, without a shift in traffic distribution, that the transport network can sustain without the addition of new infrastructure, (c) computing a maximum allowed shift in traffic distribution, without overall network traffic growth, (d) computing maximum feasible growth values for a series of traffic shift values between zero growth and the maximum allowed traffic shift computed in step (c), and (e) generating a network lifetime curve of maximum feasible network growth as a function of traffic shift values from the values in step (d).

Description

AUTOMATED METHOD AND SYSTEM FOR ANALYSING TRANSPORT NETWORKS

Field of the Invention

The present invention relates generally to automated methods and systems for analysing transport networks, and in particular to the analysis of the traffic load that can be borne by such networks.

Background

Transport networks, such as networks providing for the movement of rail, road, marine and aircraft vehicles, are designed, built and maintained by both government bodies and private industry. Such networks are designed with the hope that they will provide sufficient capacity to bear traffic for a number of years. However, transport networks are so complex that it becomes difficult to effectively analyse when network capacity will be reached, and what remedies may be available.

It would therefore be desirable to provide a tool for analysing transport networks which enabled the more efficient and effective analysis of their capacity. It would also be desirable to provide a tool for analysing transport networks which ameliorate or overcome one or more problems or disadvantages of existing transport network analysis tools. Summary of the Invention

According to a first aspect of the present invention, there is provided an automated method for analysing the sustainability of a transport network, the network including a plurality of transport zones interconnected by a plurality of transport nodes and links, the method including the steps of:

(a) computing a plurality of traffic matrices, each matrix defining traffic load on the plurality of transport links during a different time period;

(b) computing a maximum allowed traffic growth, without a shift in traffic distribution, that the transport network can sustain without the addition of new infrastructure, by:

for each traffic matrix, computing the maximum value of a traffic growth multiplier applied to each matrix element at which a network sustainability metric is not exceeded, and

determining a maximum allowed growth factor by selecting the minimum value of the computed maximum growth multipliers;

(c) computing a maximum allowed shift in traffic distribution, without overall network traffic growth by, for a set of feasible traffic shifts scenarios in which traffic volume is increased in one or more zones and correspondingly reduced in one or more other zones, by : for each traffic matrix, computing the maximum value of a traffic shift multiplier applied to each matrix element at which the network sustainability metric is not exceeded, and determining a maximum allowed traffic shift factor by selecting the minimum value of the computed maximum traffic shift multipliers;

(d) computing maximum feasible growth values for a series of traffic shift values between zero growth and the maximum allowed traffic shift computed in step (c); and

(e) generating a network lifetime curve of maximum feasible network growth as a function of traffic shift values from the values in step (d). In one or more embodiments, the method further includes the step of:

setting one or more network variables to optimise travel shift and thereby prolong network lifetime. In this case, the network variables may include road pricing.

In one or more embodiments, the method further includes the step of:

adding links to augment traffic carrying capacity and thereby prolong network lifetime.

According to a second aspect of the present invention, there is provided a computerised system including a display, processing means and a memory device for storing instructions to cause the processor to:

(a) compute a plurality of traffic matrices, each matrix defining traffic load on the plurality of transport links during a different time period;

(b) compute a maximum allowed traffic growth, without a shift in traffic distribution, that the transport network can sustain without the addition of new infrastructure, by:

(c) compute a maximum allowed shift in traffic distribution, without overall network traffic growth by, for a set of feasible traffic shifts scenarios in which traffic volume is increased in one or more zones and correspondingly reduced in one or more other zones, by :

for each traffic matrix, computing the maximum value of a traffic shift multiplier applied to each matrix element at which the network sustainability metric is not exceeded, and determining a maximum allowed traffic shift factor by selecting the minimum value of the computed maximum traffic shift multipliers;

(d) compute maximum feasible growth values for a series of traffic shift values between zero growth and the maximum allowed traffic shift computed in step (c); and (e) generate a network lifetime curve on the display of maximum feasible network growth as a function of traffic shift values from the values in step (d). According to a third aspect of the present invention, there is provided a series of program instructions for use with a computerised system including a display, processing means and a memory device, wherein the instructions cause the processor to:

determining a maximum allowed growth factor by selecting the minimum value of the-> computed maximum growth multipliers;

(d) compute maximum feasible growth values for a series of traffic shift values between zero growth and the maximum allowed traffic shift computed in step (c); and

(e) generate a network lifetime curve on the display of maximum feasible network growth as a function of traffic shift values from the values in step (d).

Brief Description of the Drawings

The invention will now be described in further detail by reference to the accompanying drawings. It is to be understood that the particularity of the drawings does not supersede the generality of the preceding description of the invention.

Figure 1 is a schematic diagram showing a computer-enabled system for method for analysing the sustainability of a transport network;

Figure 2 is a graphical representation of a network lifetime curve generated by the computer-enabled system of Figure 1 ;

Figure 3 is a first exemplary embodiment of a transport network; Figure 4 is a graphical representation of a network lifetime curve generated by the computer-enabled system of Figure 1 for the exemplary embodiment of a transport network shown in Figure 3;

Figure 5 is a second exemplary embodiment of a transport network;

Figures 6 and 7 are graphical representations of network lifetime curves generated by the computer-enabled system of Figure 1 for the exemplary embodiment of a transport network shown in Figure 5; and

Figure 8 is a schematic diagram showing various functional elements of the computer- enabled system of Figure 1.

Detailed Description

Referring firstly to Figure 1 , there is shown an exemplary computer-enabled system for analysing the sustainability of a transport network. The system 10 includes a client user terminal 12 providing a graphic user interface 14, and a database 18 in communication with the server 16 and the client user terminal 12. The client user terminal 12 and server 16 are interconnected by means of a communications network 20.

Traffic network data is maintained remotely from the client user terminal 12 in the database 18 for use by the client user terminal 12 in analysing the sustainability of a transport network. Whilst in this embodiment the traffic network data is maintained remotely in database 18 and processed remotely at the client user terminal 12, it will be appreciated that the traffic network data may also be made accessible to the user terminal 12 in any other convenient form, such as a portable data storage device like a CD-ROM. Alternatively, both traffic network data and processing functions could be performed at the server 16, and the client user terminal 12 would be limited to the generation of analysis requests and the display of relevant results.

Transport networks can be considered to include a number of transport zones interconnected by a number of transport links and nodes. A transport network with m nodes can be represented by a graph G = V_FS^' _r E > where v is the set of nodes (vertices), Z is the set of zones and E is the set of directed links (arcs) in G. A link between two nodes h and q in V can be denoted by (k,q) or simply as o_kq. It can be assumed that each zone s— < , z ε Z covers a subset of G, where V_a c V is the set of nodes covered by∑, E_s = (9_ft£? £

k_F: q £ ¥_s]„ It can also be considered that all zones in Z are mutually disjoint, implying that their node sets (or link sets) are mutually disjoint.

For the sake of simplicity in the description which now follows, the indices of i,j will be used to refer to zones in G (where each zone is represented by its centroid) and to distinguish nodes from zones, the indices k, k will be used to refer to nodes in G. For a given transport network, represented by c, the present travel of traffic in the network can be represented by a finite set S of n traffic matrices:

Since the traffic matrices are defined based on the inter-zone traffic, then their dimensions are equal to ^< X , where m is the number of zones. Each of the matrices in 5 represents the traffic between all OD zone pairs at a certain time of day for different days of the year. For example, for some u and v, a first matrix \T.,(i,,f 1 may be the traffic matrix between 8 AM and 9 AM on a normal working day, whilst a second matrix may be the demand matrix between 8 PM and 9 PM on a normal Sunday. The OD demand matrix [T J-j)] can be generated based on trip-attraction and trip-production vectors.

The transport network topology can be defined as a directed topology by:

• A set of capacities C = | _¾?} where c_h is the capacity of link (¾, q^'j «= K;

• A set of traffic volumes ·*^■<■ = where v_ft¾ is the traffic volume of link e E;

• Traffic congestion indices defined for each link (¾.-. <?) £ E;

· A set of travel time © - where 0_i; is the travel time from zone i to zone /;

• A set of zone trip-production P— f J¾ where P_t is the total trip-production of zone i;

• A set of zone trip-attraction A = {A,} where A_t is the total trip-attraction of zone L Where the link congestion index is defined as the ratio of traffic volume from node h to node q to the capacity of link ζ-_Γ._σ = ^ (noting that can also be an environmental characteristic of a link or zone, such as total emission in terms of C f_z , etc.).

Network Lifetime Analysis with Linear Traffic Growth

In an idealised situation where the network analysed by the system 10 is only subject to linear growth of travel demand, then based on the traffic demand profile defined by S, a "network lifetime curve", defining network sustainability against a shift in traffic patterns and traffic growth, is generated by the system 10 as follows.

Definition 1: For a given network a == V_rz;. K > with link capacities e^, ¥ k.q)€ E, the demand matrix [^. i. ).], VLJ e Z can be said to be sustainable if network G has enough capacity to support this traffic load without excessive traffic congestion. In other words, the offered traffic load is said to be unsustainable if it leads to infeasibility of the transport problem. Recalling the above-discussed measure of link congestion metric, feasibility of transport network implies that the traffic congestion index of each link should not exceed a predefined value.

For any demand matrix [¾^'£, )], the operations carried out by the system 10 rely upon the concept of linear growth factor, denoted by ^ ^€ In particular, the system 10 computes the maximum value of φ for which is sustainable. Since we are interested in traffic growth, the system 10 computes values in the range ψ > 1. For a given topology and for a given demand matrix [7 (;,/)3, we can consider that is the maximal value for such that

is sustainable. Accordingly,

^■ " = rain {^^••.» §»».^'φζ}» (2)

The system 10 uses ψ ' as a measure for the lifetime of a given topology with a given set of link capacities, traffic congestion index and a set of traffic load matrices S. The measure ψ* signifies by how much the current traffic load can grow until new capacity should be added to the current topology. The following description uses the capital letter τ for traffic matrix and small letter τ to denote an element in matrix T. For example, the \[i_tJ] element in the matrix is denoted by r_?(L . Network Lifetime Analysis with UTG and Linear Traffic Growth

So far the definition of ψ ^" assumes that the network growth is uniform. However, the system 10 should preferably take into account also unexpected traffic changes in the traffic distribution. To allow for such unexpected changes in the traffic growth, a parameter associated with UTG, denoted by U,. is introduced.

For the sake of simplicity of the explanation of the embodiments of the present invention described herein, it shall be assumed that individual zones are subject to UTG. This means that the system 10 shall consider scenarios in which only the trip-attraction of an individual zone is subject to UTG (It will be understood however that the present invention can be applied to extended scenarios in which a combination of zones are subject to UTG). In order to incorporate such UTGs in our original framework, a new sets of matrices S U) is considered as follows. Calculating the maximum UTG that a network can sustain

Consider a scenario where the trip-attraction value of zone i, A_t is subject to UTG. Let

Ai = .4/(1 + £/), £/ e 2R⁺ b_e the new trip-attraction value of zone i. Under the assumption of conservation of flow we have:

In order to satisfy this equality the system 10 derives the new A j≠t using the gravity model. This approach is based on the assumption that any trip-attraction value increment of a zone will be balanced by reducing trip-attraction values of other zones. The magnitude of such compensation is estimated by utilizing the gravity model. This implies that the closer competitive zones will have higher reductions.

1

(4)

Where . = ^'∑■,■.—. These values of i, together with A.- maintain the total traffic (the sum of n Sfn

all entries in matrix T~(t, j)) constant, hence guarantee a shift of traffic load without growth. The increment in the trip-attraction of zone £ is bounded by the condition that the trip- attraction values of other zones are non-negative.

Definition 2. For a given network ®— < v„Z_r E > with trip-attraction values A.— {A_*},, V£ s E, any increment in the trip-attraction value of zone i is compensated by the reduction of trip- attraction values of other zones in the network using the gravity model, if Equation 2 is used to derive the modified trip-attraction values of all zones j fc ^.„ I.

Let

A_t = .4.(1 ÷ IF) (5)

For each £ V.,J ^«* l the upper bound on U imposed by non-negativity condition of A_it can be found using Equation 2, as follows:

At ^'≥ 9 (A_* - ΑΛ X -r¹

a. Where «„ , =— . Let -^- = then

A_f (.4 - A_*) X— i4f?¾, - A.

This implies that:

(6)

Using Equations (5) and (6) we have:

Definition 3. For a g/Ven network G =< V„Z„E >,a UTG value denoted by U is feasible for zone i if an increment of UA₍ in the value of A_t does not lead to any negative trip-attraction value in the remaining zones by implementing the gravity model.

Where uj' is the maximum feasible UTG that Zone 1 can sustain considering the limitation imposed by zone /. Recall that, this constraint is due to non-negativity requirements of trip- attraction values. Hence

Where i ^ is the maximum feasible U value when Zone I is subject to UTG (considering all limitations imposed by other zones).

Now we derive A_j~_fr j≠ i based on from Equations (2) and (3) as:

From Equation (6) we can derive i " values as:

Where ^lj! is the reduction factor for the trip-attraction value of zone / (J≠ i) corresponding to U increment for trip-attraction of zone ί . The above formulations provide a closed form formula on the maximum amount of increment of trip-attraction value of Zone 1 and corresponding reduction factors for the trip-attraction values of the remaining zones in the scenario number 1. Considering all the scenarios, the value of U must not exceed any of the bounds LP ·. corresponding to Zone l and if^{' 1*}'; corresponding to Zone 2 and etc. Hence u cannot exceed the bound L?_mfl;f defined by:

U_mi._t - min {

, V∞ .«"«2, ■ 0°⁾

Where & ^" ≤ m is the total number of zones that are considered to be subject to UTG (in this paper it is assumed that any single zone can be subject to UTG and we have λ - m ).

Generating J(tQ

Following the calculation of i/_ma._f , the maximum UTG value that a network can sustain under all scenarios, the next step for the system 10 is to generate the modified traffic matrices S U). The lifetime measure curve should be generated within the feasible range of UTG, which is β≤ W≤ . To perform this task, a significant number of Us in the feasible

UTG region is considered and the associated Ψ W) values are calculated. For a given U≤ U_mi:t and the input demand matrix T_t i,f), the system 10 needs to generate λ traffic matrices fl ' i_rtt , l≤ ϊ≤ λ . The process for developing .f_r' ^Zi' (.t/) based on U value and the demand matrix F_C|¾ for scenario I is as follows. Imposing U would lead to updating the trip-attraction (A) and trip-production (P) values. By implementing the gravity model on the modified A and P values, the corresponding modified demand matrix f^(l_rf) can be generated. This implies that a total of An new matrices should be generated by the system 10 for any given u value. Recall that these matrices construct the new set S lf), now the next task of the system 10 is to measure how much each matrix in S U) can linearly grow before any violation of traffic congestion index experienced by any link in the network.

Note that S(U) can be augmented to include other scenarios, for example when two or more zones simultaneously become more attractive and etc. This augmentation practice is straightforward but not trivial.

Calculating ψ"(β^' } By analogy to the definition of J T¹ (lifetime measure for linear traffic growth only) as our measure for the lifetime of a transport network with given link capacities, traffic congestion index and the set of matrices 5, the function ψ^,ί1 (^'#):■ (lifetime measure for UTG plus linear traffic growth) defines the lifetime of a given transport network with given link capacities, traffic congestion index and, in this case, the set of matrices Notice that in the absence of any UTG ( U = 0), these functions are the same and we have ψ* = ΨΗβ^' . This is a simple observation, recalling that -φ" is defined for the case of linear traffic growth only(without any UTG), but ψ · is a measure defined based on predefined UTG scenarios in conjunction to the linear traffic growth. In the absence of any traffic shift, these two functions are actually identical.

To obtain the lifetime curve Ψ (1Γϊ, the system 10 first calculates

for the case W -€>. For any matrix € 5 (if), 1≤ ί≤ A, the system 10 calculates Ψ₍¾ (ϋ) which is the maximal φ such that Τ^ is feasible. To find Ψ,¾ -/) ^tne system 10 solves a series of user equilibrium assignment problems subject to some feasibility criteria. At each iteration the system 10 tries a value of ψ to determine if the resultant assignment is feasible. Each feasibility problem has two sets of constraints. The first set, the set of traffic requirement constraints, ensures that the traffic demand between any OD zone pair is fully satisfied. The second set (the capacity constraints) ensures that the total flow passing through any individual link s £', does not violate predefined congestion index The above procedure is repeated by the system 10 for all matrices in S(<lf) and then we have

This process yields to the determination of a single point of the lifetime curve (namely W tf). U) )and the system 10 needs to repeat the same methodology for other feasible U values to obtain the whole lifetime curve for display by the graphic user interface 14.

The next section describes the feasibility problem that is outlined above.

Feasibility

Assume that we have only one inter zone demand matrix as f For any OD zone pair |^' ] , the traffic requirement constraint is:

where ¾· = T-( _r}} is the traffic demand from zone i to zone (the fi,/|th element of matrix t '.rJ) ). P*( is the set of acceptable paths (obtained via traffic assumptions and assignment algorithm) from zone i to zone / in the network and .-..·?, is the amount of flow from zone i to zone J through path . The capacity constraint for a given link { k_rq). £ £^' is:

Where c_kq is the capacity of link h_rq} , (_fte is the traffic congestion index of link {fc_r q}, Ω is the set of all OD zone pairs in G and § is an indicator function which is equal to one if path p uses link (i ¾) and is zero otherwise.

The set of all constraints of type (1 1 ) and (12), together with all non negativity constraints {xf:₍ 0), form the feasibility problem (FP). ∑ ^xi j = ^τυ ^{fov al1} i- Z (^Fp)

∑ (∑ ≤ Ch Ou, f r all (/*, q)€ £

·'■^■ > 0 for all /../€ Z and for all p€ Ω

Note that r appears on the right hand side of the first set of constraints in the model.

To find ψ^:* the values are substituted by the system 10 by for some ^'Ψ <= tf¾⁺. The feasibility of (FP) can be checked in polynomial time using the static user equilibrium assignment algorithm. The only task is to find the maximum ψ value such that (FP) remains feasible. This is achieved by the system 10 performing a binary search approach on the assigned ψ values. The static assignment algorithm works in the following manner. Traffic demand (OD trip matrix) is assigned to the network to determine travel time and flows on links. Travel time on a link is, in general, a function of the link flow. Users are free to choose the path which minimizes their travel time. Behaviour of users (i.e. assignment results) can be predicted based on Wardrop principles. Based on these principles at the user equilibrium condition, travel times on used paths are equal and no user can improve his/her travel time by switching paths. Traffic assignment problem in this case is called User Equilibrium (UE) which has been studied extensively in the literature. UE can be solved effectively by the system 10 by the known steepest descend method. Lifetime Curve

The output of the system 10 is a curve which identifies how much the network under study can sustain the travel demand growth. This is analysed by the system 10 in conjunction with the existence of traffic shift in the network. In general, when we shift traffic, it may not be possible to route the new traffic without violating traffic congestion index on the underlying network. In other words, the modified demand matrix after traffic shift may not be sustainable under available network resources. This implies that the traffic assignment might lead to infeasibility in corresponding network flow problem. In such cases, in order to find a feasible solution we will have to consider ψ < l. At present, we are mainly interested in networks with growth potential so it is unlikely that ψ < i will have to be considered. The way the system 10 normalizes traffic shifts, using u and v to keep the total traffic fixed (recall that the sum of all trip-attractions of the network kept fix during a traffic shift), is important and not at all arbitrary. By normalizing the traffic, the shift is independent of the growth (signified by the ψ variable). This way the system 10 covers both shift and growth and are able to have a meaningful single lifetime curve which includes the impacts of different types of traffic shift.

Given a network & =< ¥,2^' _rE^' > , with zonal trip-attraction values A = link capacities c_hq, traffic congestion indices _¾?,. € if, inter-zone travel times © = 6_¾ Yi Z and the set of zones that are subject to UTG, defined as £ 2; the output of lifetime algorithm is a simple curve. A typical lifetime curve has been depicted in Figure 2.

The lifetime curve provides the necessary information for transport network designers to perform a lifetime analysis on the network and to compare different potential future scenarios. A transport network designer might consider a few different topologies or traffic management scenarios for designing purposes. These scenarios/topologies could be considered based on an underlying real life network and some possible network augmentation. The examples of such network augmentation could be constructing the new roads, adding extra lane to some existing roads in one/both direction(s), and changing the lane directions of some roads on different times of the day. Notice that one option could have a higher Ψ'(β) than another, but with the introduction of UTG, when U exceeds a certain value, the situation may be reversed. This provides the designer with insight into the possible effects of unexpected uneven traffic growth on network lifetime.

Pseudo code for the computation of U _{ic :}.

The following pseudo code outlines the details of the procedure that is described above for the system 10 to compute the maximum feasible UTG in the network. Here li_fntMf is the maximum demand shift that the network can sustain. The reduction factor r '" indicates how much the trip-attraction value of each zone , ί= %_r ≠ f. should be decreased when a single zone, say zone i experiences an increment in its trip-attraction value by the amount of U_mex:.

Algorithm 1: Lifetime Curve Analysis of Transport Networks: Calculating U_f 1 : Input: A network ffi -< V,E_f E >, \y\\ - IT*, |£ [ - ift, with zonal trip-attraction values

A = ,,}, inter-zone travel times © = S 2 and the set of zones that are subject to

UTG, defined as C S;

Output: This algorithm finds the maximum allowable UTG value, U._∞nx-

4: Set u_>nsx = «^■

5: while ^ do

6: Let i€ <?

7: Set £ = i

8: u ^:> ■»

9: for / = 1 to m do

10: If ί then

12: If

14:

15: end if

16: end if

17: end if

18: for / = i to m do

19: Set If = U _x

20: if / ¾¾ . then

21 : (f> _

22: end if

23: end for

24 K W^jnct* ^{;> v}mie ^then

26 end if

27 *- *2 \ £

28 end while Pseudo code for the computation of S (<if)

Once the maximum UTG value that the network can sustain has been calculated, the next stage is to implement the potential demand shift in the network. This is achieved by increasing the trip-attraction value of a single zone with a value of U≤ and adjusting the trip-attraction values of the remaining nodes. This would result to a modified travel demand matrix denoted as f . The following pseudo code describes the details of the required procedure to be performed by the system 10.

For a given U, the total number of modified demand matrices is equal to ηλ. This procedure should be repeated by the system 10 for the necessary number of U values in the feasible interval [Ό ,_ηβΛ, . The resultant modified travel demand matrices constitute S ^U .

Algorithm 2: Lifetime Curve Analysis of Transport Networks: Generating S U) 1 : Input: A network ίϊ =^■<: V.. 7,_r κ , Z[ = m, with zonal trip-attraction values

A = {A_f}, zonal trip-production values - (_¾, a set of travel demand matrices

.« = ^'{ί¾υ¾ ^,^ ]_^.._' i. , U)3 TO))}. . r</ViJ€ Z, a selection of

U e [0., U^ex] and the set of zones that are subject to UTG, defined as £ Z;

2: Output: This algorithm generates the modified travel demand matrices S{V) for the selected feasible u.

3: Set 5(10 - 0

4: for t = 1 to n do

5: Set Q = Ζ?_τσ

6: while Q≠ 3 do

7: Let £€ Q

8: Set ί = I

9: A_t *- . (ΐ + if)

10: for = 1 to m do

11 : if /≠ I then

12

13

14 end if

15 end for 6: Use the gravity model to generate modified demand matrix T ^'J

18: Q <r~ q \ l.

19: end while

20: end for

Pseudo code for the computation of ψ ^¾ ( fpj

Once the pairs of feasible shifts and modified travel demand matrices [k^',S (ii)j are generated, the next stage is for the system 10 to solve the assignment problems corresponding to S (U)] for each traffic shift U. The following pseudo code illustrates the details of the procedure for this process. The output of this procedure is the maximum allowable growth for any given traffic shift u.

Algorithm 3: Lifetime Curve Analysis of Transport Networks: Calculating Ψ"[ί*^Γ)

1 : Input: A network G = ¥_rZ,E >, | | = m, \Z\ = :< , with link capacities c_hq, traffic congestion indices V(ft,q) s Π, inter-zone travel times O■ 0_ij{ rallLj! « Z and the pair lU_rS(if}];

2: Output: This procedure calculates Ψ^* &}.

3: ψ -rfi) = tn

4: for t = I to n do

5: for I = 1 to m do

6: f ^«- ?.·£)

7: Set = 1

8: Set ψ^_αχ = W

9: Set r = 1

10: while e > do

11 : Δ = F_maje -

12: e = ^-

14: Generate FP for ψ

15: Check FP feasibility through user equilibrium assignment 16: if FP is Feasible then

18: else 20: end if

21 : end while

23: ¥ TO =

24: end if

25: end for

26: end for

Illustrative Example 1

Figure 3 depicts a small example of a four-zone network 30, to show how to calculate the topology lifetime, namely the Ψ^(^'ΙΠ curve 40 depicted in Figure 4 on the graphic user interface 14 of the system 10. The curve 40 is used by network designers to analyse the sustainability of the transport network 30, notably as a tool for network design or to plan for infrastructure additions or traffic redirection within the network 30.

The four-zone network 30 consists of 5 nodes and 1G links (two lanes per link). All links have the capacity equal to 1600 vehicle per hour and the traffic congestion index can be considered to be _¾ = 1.13 for all links (¾¾·) « E.

A trip-attraction vector, trip-production vector (for the sake of simplicity, A and P are defined to be vectors rather than sets in the example sections) and trip travel matrix are given as:

26 20 26

A = [3200, 3000. 1500, 3600]

0 26 20

Θ

26 0 26

V = [3600. 1500. 3000. 3200]

20 26 0

Using the gravity model and the corresponding equations described previously, the system 10 is configured to produce a trip matrix T — (π and Γ

as follows:

1019 956 478 1147 0.00 3.30 2.54 3.30

425 398 199 478 3.30 0.00 3.30 2.54

T = Γ =

850 79G 98 956 2.54 3.30 0.00 3.30

906 850 425 1019 3.30 2.54 3.30 0.00 In order to better comprehend the present invention, a scenario will be explained whereby Zone 1 is subject to UTG. Other scenarios would be similar to this and very straight forward to be followed by a skilled addressee.

Once Zone l is subject to UTG, the system 10 derives the required u and r values which provide the limits on the magnitude of allowable traffic shift:

Using . a series of computer program instructions to implement Equation (6), the system 10 computes ^ as :

A = mta £ l3100„70Qa^A508O}- 7008

Similarly using a series of computer program instructions to implement Equation (7), the system 10 computes: 'ⁱ \ = raire if ¹" = mln{^¾ = rain { 3.09,1.19,3.71} = 1.19 •^ii:t /si ♦ "l- A^

Using a series of computer program instructions to implement Equation (9), the system 10 computes J*^{'1 "}' values as:

This would result to the following maximum allowable shift in trip-attraction being computed by the system 10 to be:

A - ?α08,1®46,β,2!4ί .6] (13)

The above procedure is then repeated by the system 10 for all scenarios. This results in the following:

ma mlm {[ 1.19 Λ1¾5Λ¾.1.37> - 1.19 In this case, if^^., is the maximum limit that is considered in the process of shifting trip-attraction in any scenario. Since four scenarios have been taken into account in this exemplary embodiment, the number of elements (matrices) in S U is equal to four, denoted as f '·²>?**_Γ audi ϊ ** In order to generate any matrix f the system 10 uses a series of computer program instructions to implement Equation (5) to compute the new trip- attraction value of zone / and accordingly system 10 uses a series of computer program instructions to implement Equation (8) to thereby compute the new trip-attraction values of all zones ≠ i. Using the new trip-attraction values and the gravity model, the modified trip matrix is generated by the system 10. In this example the modified T" matrix is as follows.

1095 9 3 448 1124^'

f - - 56 3S9 187 468

91» 777 373 937

. 97 829 398 999

In this example, the bound on u, is = 1.19. The system 10 performs

1

computations on the points tf U. 1 ft " MI and for each of these points, the system 10 performs derives ^(ti). In the case of

Ψ¾(1 , ¾(* and Ψ, ^) being the maximal values for ψ such that Τ'^, ^■φΐ^'^, ¾p ^w and ψΤ'^^' are feasible, respectively, these four optimization problems are identical to the first one performed by the system 10 to obtain -φ^''. These give rise to the function Ψ 1£), a graphical representation of which is depicted in Figure 4 and displayed on the graphic user interface 14 of the system 10, as follows:

Ψ * U) = rain { Ψ,¾. (iQrVfo (if) , Ψ,¾ (£0-, 0 (14)

Table 1 : The^■ψ ^'" values for the four-zone example = f_ma:f = 1.19). The results of the computations performed by the system 1 are presented in Table 1 above. It can be seen, by comparing ¾¾■({ ) with other Ψ " W). values, that Zone 3 is not of concern in this transport network.

It can also be seen how this measure provides an effective transport network design tool. In the table, Ψ ζΙ!)— Ψ,¾(£Γϊ for all values of U, implying that Zone 4 is a critical zone. Links (1,5) and (5,4 are the bottleneck links in this example. Accordingly, if a network designer was to choose to add capacity, the system 10 enables them to identify that the capacity should be added to these links to improve the lifetime of the transport network 3.

From Figure 4 it can be seen that the maximum feasible linear growth is about ψ" = ¾?.Sl in the case of w = v. This implies that this network would not experience any traffic congestion index violation up to at least 8 years with the assumption of linear traffic growth of S per year in the case of zero UTG. It also indicates that in the case of trip- attraction gain of Zone 4 with the shift amount of U = ©.ST, the network would be congested even without any linear traffic growth. This also shows that with UTG value of U = Q.22S, this network can carry a uniform growth up to 1.33, which can be interpreted as around 6 years of time span, considering a linear traffic growth of per year.

Illustrative Example 2

Figure 5 depicts a large network 50, consisting of 1 Q nodes, 20 zones and 98 links. All links have two lanes. The surrounding links (e.g. Links (23,24) and (33,39)) have the capacity equal to 2^'JW - '—^- and the remaining links all have the capacity equal to leco ^"e^^' . ^' . The traffic congestion index is set to _¾τ = 1.15 for all links (·*&· ¾·) s ε and the volume delay function is given as:

Where η^ησ and .7^ are travel time and free travel time from node k to node q correspondingly. The trip-attraction and trip-production vectors are given in terms of "^'"^Cl-^r as follows:

A = (000, 1000. 1250. 1 100.400.400, 50. 600. 2400, 450. 450, 450. 00. 500, 400, 400, 1 100. 1250, 1200. 1 100) V = [600, 700, 650, 600, 1200, 1200, 1350, 300, 300, 1150, 1350. 1150, 1400, 1400. 1200, 1200, 600, 700, 650, 600)

Assuming that all individual zones have the potential of gaining momentum in terms of trip-attraction values, 20 scenarios are analysed by the system 10. Here at each scenario, an individual zone is subject to UTG. Using the derived equations in earlier section, the maximum UTG shift for this example computed by the system 10 to be equal to ^a_ia_i ⁼ In order to compute values required to display the lifetime curve on the graphic user interface 14, the maximum linear growth values are computed by the system 10 for the points U = , ~ U_ma;t , U_mex , ^■■ ·, _m„ and - if_m,.. . The results of lifetime analysis, as computed by the system 10, for this network are shown in Table 2 below.

Table 2: The ψ* values for the twenty-zone example From this table, it can be seen by comparing ¾¾ς (£0 and Ψ ,₆-. (ί ) with other Ψ" Χί} values, that these three zones do not show any limiting behaviour on network congestion while their trip-attraction values are increased. This simply implies that the network designer does not need to worry if the trip-attraction values of these zones are going to increase in the future. It can also be observed that Ψ/¼ (1 , ¾¾>W.

V ¾¾.(. ) and Ψ_¾¾(ίί) have non-increasing behaviour with respect to U.

The system 10 is configured to determine that Zone 17 is the critical zone for 0≤ U < 1.20 and afterwards Zones 4 and 39 become critical. The network reaches to its capacity limits after t/=lL6S and cannot tolerate any linear growth beyond this point. Concentrating on Zones 17, 39 and 1, the system 10 determined that in the case of Zone 17 which is the critical zone in the early stages of increasing of U, two links, (23,29) and (22,23) exceed the congestion index in sequence. In the case of Zone 39 the congested link is (55,59). The result of this analysis by the system 10 is depicted by the lifetime function curve 60 depicted in Figure 6.

Figure 7 depicts lifetime function curve 70 corresponding to a situation where the capacity of two congested Links (23,? ) and (2?,?a) to SWii ^"^^' has been increased to see how this capacity augmentation is affecting network performance. Curve 70 (Case 1) can be compared in this figure with the lifetime curve 74 derived in the first case. It can be seen that increasing the capacities of these links has significantly elevated the overall performance of the network. In other words, the Ψ ^: ί ) value has increased from IAQ up to i.Sa for u— 0.3L2. This means that the lifetime of the network has been increased up to at least 3 years considering a S% linear growth per year.

In many real life cases due to budget and space conditions, adding a new lane (capacity improvement) is not feasible. In such circumstances, road pricing policies could be used. For example, a flat toll value of 57 can been imposed to Links {15,16), (15,21), (16,2,2) and 21,22). The effect of introducing this road pricing policy on the lifetime curve has been depicted in Figure 7 denoted by curve 72 (Case 2). Comparing to the base case 74, it can be seen that there is an improvement of lifetime from 1.4 up to 1.48 for v - 0.12. This is equivalent to 1.6 years improvement in the network lifetime assuming E<?¾ linear traffic growth per year. The lifetime improvement at Case 2 is consistent for almost entire domain of lifetime curve. It can also be seen that Case 2 outperforms Case 1 and the base case when U≥ 1-08. This indicates that having such a pricing policy in this example is very effective strategy, resulting in a more resilient network for U > 1.08. This means that for higher UTG values, a simple pricing policy could be more effective strategy comparing to capacity expansion in the bottleneck links which is a quite expensive alternative.

This implies that a network design engineer can simply use this technique to perform an intensive sensitivity analysis of the network in order to draw the best strategy for the network expansion.

The system 10 may be implemented using hardware, software or a combination thereof and may be implemented in one or more computer systems or processing systems. In particular, the functionality of the client user terminal 12 and its graphic user interface 14, as well as the server 16 may be provided by one or more computer systems capable of carrying out the above described functionality.

An exemplary computer system 80 is shown in Figure 9. The computer system 80 includes one or more processors, such as processor 82. The processor 82 is connected to a communication infrastructure 84. The computer system 80 may include a display interface 86 that forwards graphics, texts and other data from the communication infrastructure 84 as the graphical input to the display unit 88. The computer system 80 may also include a main memory 90, preferably random access memory, and may also include a secondary memory 92.

The secondary memory 92 may include, for example, a hard disk drive, magnetic tape drive, optical disk drive, etc. The removable storage drive 96 reads from and/or writes to a removable storage unit 98 in a well known manner. The removable storage unit 98 represents a floppy disk, magnetic tape, optical disk, etc.

As will be appreciated, the removable storage unit 98 includes a computer usable storage medium having stored therein computer software in a form of a series of instructions to cause the processor 82 to carry out desired functionality. In alternative embodiments, the secondary memory 92 may include other similar means for allowing computer programs or instructions to be loaded into the computer system 80. Such means may include, for example, a removable storage unit 100 and interface 102.

The computer system 80 may also include a communication interface 104. Communication interface 104 allows software and data to be transferred between the computer system 80 and external devices. Examples of communication interface 104 may include a modem, a network interface, a communication port, a PCMCIA slot and card etc. Software and data transferred via a communication interface 104 are in the form of signals 106 which may be electromagnetic, electronic, optical or other signals capable of being received by the communication interface 104. The signals are provided to communication interface 104 via a communication path 108 such as a wire or cable, fibre optics, phone line, cellular phone link, radio frequency or other communications channels.

Although in the above described embodiments the invention is implemented primarily using computer software, in other embodiments the invention may be implemented primarily in hardware using, for example, hardware components such as ^"an application specific integrated circuit (ASICs). Implementation of a hardware state machine so as to perform the functions described herein will be apparent to persons skilled in the relevant art. In other embodiments, the invention may be implemented using a combination of both hardware and software.

From the foregoing it can be seen that the system 10 enables the performance of a transport network to be analysed as a function of growth and shift in the traffic load. The system 10 determines which networks are likely to last longer if traffic grows in a uniform manner and which networks are likely to last longer under shifts in traffic load towards more attractive zones. The system 10 can be used to analyse the feasibility of transport networks over the time horizon based on linear growth and unexpected traffic shifts. The output of a lifetime curve (either for display as a curve or as a series of values for further computation by the systems 10) demonstrates the maximum linear growth that a network can sustain under a range of unexpected traffic shifts. The interpretation of this curve is not a complex task, and it shows when in the future the network would experience traffic congestion and identifies the elements of the network that are the cause of network failure.

This system 10 provides a powerful tool to be used by network design engineers and traffic network managers. The operations performed by system 10 are based on an scalable mathematical model, easing the way of performing an intensive sensitivity analysis. The features and capabilities of the model enable network managers to implement various potential scenarios such as road pricing and infrastructure growth in order to increase the lifetime of their network. While the invention has been described in conjunction with a limited number of embodiments, it will be appreciated by those skilled in the art that many alternative, modifications and variations in light of the foregoing description are possible. Accordingly, the present invention is intended to embrace all such alternative, modifications and variations as may fall within the spirit and scope of the invention as disclosed.

Claims

1 . An automated method for analysing the sustainability of a transport network, the network including a plurality of transport zones interconnected by a plurality of transport nodes and links, the method including the steps of:

(c) computing a maximum allowed shift in traffic distribution, without overall network traffic growth by, for a set of feasible traffic shifts scenarios in which traffic volume is increased in one or more zones and correspondingly reduced in one or more other zones, by :

(e) generating a network lifetime curve of maximum feasible network growth as a function of traffic shift values from the values in step (d).

2. A method according to claim 1 , and further including the step of:

setting one or more network variables to optimise travel shift and thereby prolong network lifetime.

3. A method according to claim 2, wherein the network variables include road pricing.

4. A method according to any one of claims 1 to 3, and further including the step of:

5. A computerised system including a display, processing means and a memory device for storing instructions to cause the processor to: (a) compute a plurality of traffic matrices, each matrix defining traffic load on the plurality of transport links during a different time period;

6. A series of program instructions for use with a computerised system including a display, processing means and a memory device, wherein the instructions cause the processor to: