4.1. State equation
The state of the transportation system under consideration is represented by the capacity of it. The capacity x is therefore describing the state of the system. Suppose the infrastructure can be treated as a production-inventory system. For a description see (Sethi and Thompson, 2000).
The system can be improved by new production with a production rate p: 1x&(t)= p(t). This production rate depends on time. The system also looses quality at a constant rateδ in time: 2x&(t)=−δ⋅x(t). The total deterioration depends linearly on the existing capacity. The deterioration rateδis constant in time. The autonomous deterioration can be, at least partially, counteracted by conducting maintenance at rate m(t):
) ( ) ( )
3x&(t =m t ⋅x t . Combining these effects together, we get a first form of the state
equation:
(
( ))
( )) ( )
(t p t m t x t
x& = − δ − ⋅ . (4.1)
As the capacity is measured in vehkm/h, the dimensions of p are vehkm/(yr⋅h), and those ofδand m are yr-1. The initial state of the system is given by x(0) = x0> 0.
Other constraints on the system are:
p(t) ≥ 0 for all t > 0; δ > 0; 0 ≤ m(t) ≤ δ for all t > 0. This upper bound on m is necessary, since the maintenance (i.e. repair of damage) cannot lead to completely new capacity.
The total construction effort u(t) is defined as the sum of the production and maintenance activities: u(t)=p(t)+m(t)⋅x(t). Substituting in equation (4.1) gives a relation with a single control parameter u:
) ( ) ( )
(t x t u t
x& =−δ⋅ + . (4.2)
The construction effort is limited by the maximum construction effort:
0≤u(t)≤umax.
4.2. Production and maintenance
It is difficult to give a precise definition of both production and maintenance. Both are construction works, but often the construction work for new production is combined with necessary maintenance. A uniform classification of maintenance does not exist, but some distinction in road maintenance and improvement work is possible (Paterson, 1987):
Routine maintenance m Localized repairs (typically less than 150m in continuous length) of pavement and shoulder defects, and regular maintenance of road drainage, side slopes, verges and furniture.
Resurfacing m Full-width resurfacing or treatment of the existing pavement or roadway (inclusive of minor shape correction, surface patching or restoration of skid resistance) to maintain surface characteristics and structural integrity for continued serviceability.
Rehabilitation m or p Full-width, full-length surfacing with selective strengthening and shape correction of existing pavement or roadway (inclusive of repair of minor drainage structures) to restore the structural length and integrity required for continued serviceability.
Improvement p Geometric improvements related to width, curvature or gradient of roadway, pavement, shoulders, or structures, to enhance traffic capacity, speed or safety; and inclusive of associated “rehabilitation” or “resurfacing” of the pavement.
Reconstruction p Full-width, full-length reconstruction of roadway pavement and shoulders mostly on existing alignment, including rehabilitation of all drainage structures generally to improved roadway, pavement and geometric standards.
New construction p Full-width, full-length construction of a road on a new alignment, upgrading of a gravel or earth road to paved standards, and provision of additional lanes or carriageways to existing roads.
Choosing the rehabilitation to be part of m implies that m(t)=δin normal maintenance conditions. The Dutch administrative maintenance practice is to make a distinction between continuous preventive maintenance m and discreet rehabilitation maintenance would be included in p. In this case m(t) <δ. The former has theoretical, mathematical advantages, while the latter has the benefit of the possibility to research decision to let additional capacity construction coincide with necessary rehabilitation.
In the case that it is possible to use one expression for all the construction works, thus the usage of the construction effort u(t), the choice in classification between p(t) and m(t) becomes arbitrary.
4.2. Deterioration
The following formulas are taken from a publication of the World Bank (Paterson, 1987). The deterioration of the road surface can be measured in the International Roughness Index (IRI). A specific definition of IRI can be found in the World Bank publication. Roughness itself can be defined as “the deviations of a surface from a true planar surface with characteristic dimensions that affect vehicle dynamics, ride quality, dynamics loads and drainage.” An empirical formula to predict the roughness R is:
[
R S L t]
e tt
R( )= 0 +725⋅(1+ )−5.0⋅ 4( ) ⋅ 0.0153 . (4.3) The roughness R(t) in m/km IRI, at age t in year since construction depends on two major parameters (R0is typically between 1 and 3 for new roads):
Li(t) is the cumulative traffic loading at time t, in million ESA (assuming that the load damage increases with power i). Mostly it is predicted that i=4. ESA is the number of equivalent 80 kN single axle load.
S is the so-called modified structural number of pavement strength. It can be calculated using the formula:
∑
+ ⋅ − ⋅ −=
i i
ih B B
a
S 0.04 3.51 ln( ) 0.85 ln2( ) 1.43 ai: material and layer strength coefficients;
hi: layer thickness in mm (Σh≤700 mm);
B: in situ California Bearing Ratio of subgrade in %.
Table 5 Emperical values for aiand CBR to be used in formula above.
Pavement layer Strength coefficient ai
Surface course
Asphalt concrete 0.30-0.45
Base course
Granular materials 0.0-0.14
Cemented materials 0.075+0.039⋅UCS-0.00088⋅UCS2
Subbase and subgrade layers
Granular materials 0.01+0.065⋅ln(B)
Cemented materials UCS>0.7 Mpa 0.14
UCS: unconfined compressive strength in MPa after 14 days.
Typical values for S are between 2 and 6. Let, for argument sake, S=2.4; R0=1.5.
Formula (4.3) would then lead to (with t in years):
R(t)≈[1.5+1.60⋅L4(t)]⋅e0.0153t.
The cumulative traffic loading L4 can be computed as: =
∑
⋅ a
n a
n
N a
L 80 with Na the
number of passing axle loads a. For n the figure commonly used is n=4. The commonly used dimension is ESA.
Also, a relation needs to be established between v and R. (Paterson, 1987) gives a graphical representation of such a relation. The velocity is a slowly decreasing function of R, see figure 6.
0 50 100 150
0 2 4 6 8 10 12 14 16 18 20
Roughness R (IRI m/km)
Velocityv(km/h)
Figure 6 The relation between R and v. The function shown in gray is given by:
v(R)=166.5/(1+0.162⋅R(t)).
The assumption that the capacity of a road decreases proportionally to the velocity allows an estimation of the autonomous deterioration of the capacity possible. So:
x(t)/x0=v(t)/v(0). It follows that:
) (
1 )
6 . 1 ( 162 . 0 1
5 . ) 166
(
0 0153 . 0 4
0 L t e v R
t R
x t ⋅
⋅
⋅ +
= + x0.
If the transport demand y is constant in time, than L4(t) is constant in time. Now, using the state equation (4.2) and setting p=0 and m=0, it is possible to determine the autonomous deterioration rateδ, since
t x t t x
x t
x d
(t) d ) ( ) 1
( )
( =−δ⋅ ⇒δ =− ⋅
& . In figure 7
this is numerically determined for R0=1.5 m/km and L4= 1 ESA. It shows that R is not constant in time, but – for traffic densities common in the Netherlands – mostly ranges between 0 and 0.3.
The maintenance standards in the Netherlands state that roads with IRI<2.6 do not require maintenance. Roads where 2.6≥IRI <3.5 need maintenance planning, since roads with IRI≥3.5 require immediate maintenance. In the Netherlands, 0.2% of the state roads have IRI≥3.5, while 98.7% have IRI < 2.6 (MinV&W, 1999).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0 5 10 15 20 25 30
Time t (yr) Deteriorationrateδ (yr-1) Relativecapacityx/x0
0 20 40 60 80 100 120 140 160
Averagevelocityv(km/h)
Relative capacity Deterioration rate Average velocity
Figure 7 The deterioration rate δ declines as a function of time. Regular maintenance in the Netherlands is conducted every 6 to 8 years (Alberts, 2002).
0.0 0.10.2 0.3 0.4 0.5 0.6 0.70.8 0.91.0 1.11.2 1.31.4 1.51.6 1.71.8 1.92.0 0
1 2 3 4 5
0 0.05
0.1 0.15
0.2 0.25
0.3
Deteriorationrateδ(yr-1 )
Axle loads L4(106ESA) Initial
roug hn
ess R
0(
m/k m
Figure 8 The deterioration rate after 2 years for several cumulative axle loads L4(horizontal axis) and values of roughness R0(vertical axis).
As maintenance is conducted every 6 to 8 years, then it follows, if maintenance m is defined such that m(t)≈ δ, that 0.13 yr-1<δ <0.17 yr-1 for normal Dutch road conditions. The conclusion is that on average:δ=0.15 yr-1.
4.3. Objective function
The objective function J representing the time discounted life cycle energy use of the system is in its core the summation of the energy use of the different life stages:
[ ]
∫
∞− ⋅ + +
=
0
) , ( ) , ( )
(p H m x F x y dt E
e
J ρt . (4.4)
The three utility functions E, H and F represent the energy use of – respectively – the production phase of the infrastructure E, the maintenance and operation phase of the infrastructure H, and the operation phase of the vehicles F. These functions are all positive: E(p)≥0 for all p≥0; H(m,x)≥0 for all m≥0, x≥0; F(x,y)≥0 for all x≥0, y≥0.
E(p) is a function representing the energy use for the construction of new capacity.
Every capacity increasing measure on a road has a specific influence on the increase in capacity, with a specific energy requirement. All capacity increasing actions exist of one or more distinctive engineering measures. For most measures i, the relationship will be of a linear type: Ei(p)=αi⋅p. The parameter αi is assumed to be constant, but might decrease slowly in time as technology improves. For some measures, the relation between the energy use E and the production rate p might be less than linear in p, as initial installation costs may be high (lighting, electronic traffic regulation). For asphalt construction, is it presumably more than linear in p. The reason for the latter is that a third lane on a highway has less effect than a second one, and a fourth less then a third, etcetera, while the energy needs are largely determined by the amount of asphalt that is equal for every lane. Similar relationships will also exist in maintenance requirements:
Hi(m,x)=ßi⋅m⋅x. The parameter ßi has a similar function as parameter αi in the production energy function. There does not exist a clear correlation between αi and ßi. Some measures with lowα will have a high ß. (One should think about measures which require large continuous electricity supply in the operation phase).
So it is assumed that most relations can be characterized by:
E(p(t))=αi⋅p(t),αi> 0;
H(m(t),x(t))=ßi⋅m(t)⋅x(t), ßi>0.
A thorough analysis for the whole life cycle of infrastructure is carried out by Bos (Bos, 1998). This study results in energy for total production of a standard freeway of 64⋅106MJ/km. This accounts for αroad= 8.0⋅103MJ⋅h⋅veh-1⋅km-1. The materials are accountable for most of the energy requirements, both for the construction and the maintenance phase. If one assumes that at one point in time all the materials will have to be replaced (*), the assumption α ≈ ß is valid. However, the current data suggests that maintenance requirements are substantially lower than construction requirements. The current data looks at the material requirements during the functional lifetime of the road.
That is, until the road subbase (sand bed and lower asphalt layers) and the concrete artworks (bridges) need replacement. Therefore, only asphalt renewal in the top layers is included. Taking data from current Dutch studies (Bos, 1998; Alberts, 2002), one can estimate ß at 150-500 MJ⋅h⋅veh-1⋅km-1. For the reason (*) mentioned above, this value of ß is an underestimation. Therefore, with the assumptionα=ß and only taking traditional road construction into consideration, part of the objective function can be expressed in terms of the total construction effort: E(p)+H(m,x)=α⋅p(t)+ß⋅m(t)⋅x(t)=α⋅u(t).
4.4. Main utility function
4.4.1. Estimating transport demand
The energy function that relates to the vehicular fuel consumption F(x(t),y(t)), however, is determined by a more complicated relationship. Suppose an exogenous function y(t) exists that forecasts the transport demand. See figure 9 for an example. The flux of trafficΦat one point is, on average, given by: Φ(t)= y(t)/l. With the case of figure 9, in which the length ℓ=54.3 km, the average throughput for 2010 is given by:
veh/h 5065 veh/day
10 120 3
3 . 54
10 6 . A12 6 2010 A12 2010
A12Φ = y l≈ ⋅ 6 = ⋅ = . Note that this formula
does not include rush-hour peak traffic. In this paper, the transport demand is considered constant in time: y(t)=y. For the answer to the question to which extent the transport infrastructure should grow, is it best to use the value of y(t)=ymax in the equations.
0 0.5 1 1.5 2 2.5 3
1985 1990 1995 2000 2005 2010 2015 2020
Year Transportperformancey(109 veh·km/yr)
Figure 9 The transport performance on the Dutch highway A12, The Hague-Utrecht, including an extrapolation until 2020. The baseline of t in years is t=0 for the year 1986. The extrapolation curve is given byA12y(t) = A12ymax/(1+e-0.136⋅t+0.258), with ymax=2.48·109vehkm/year. The curve is fitted using numerical least squares methods, but it should be noted that the form of the outcome of the figure is subjective to the chosen fit curve.
4.4.2. Determining flux for two traffic states
The flux Φ(t) represents the actual number of vehicles that are passing one point in a certain amount of time. The flux cannot exceed the point capacity, or:
Φ(t)≤x(t)/ℓ.
The actual fluxΦ(t) on a road section determines the velocity v(Φ). For this purpose an experimental function is created to establish a relationship between the velocity and the flux. It should be noted that the traffic flow on a road can exists in two different regimes: the 'normal' free flow state, and the 'congested' forced flow state. The system can almost instantaneously jump from one state to the other. An article by Wahle et al.
shows an example of such an occurrence (Wahle et al., 1999).
The maximum possible flux approaches the theoretical capacity of a road at the optimum velocity vˆ : max(Φ(v))=x/lorΦ(vˆ)=x/l. Αccording to literature, the optimum velocity lies commonly between the 50 km/h and 75 km/h, depending on the architecture of the road (Kreuzberger and Vleugel, 1992).
For the forced flow system, it is assumed that all the vehicles are queued. The vehicles in this case have to maintain a safe distance between them to avoid collision. This safe distance is∆safe=c1+c2⋅v+c3.v2. The flux is given by:
v v v
c v c c
w
Φ(v) v 2 for ˆ
3 2 1
+ <
+
= ⋅ . (4.5a)
The variable w is the width of the road in the number of lanes. For low speeds, the velocity and the average distance between vehicles determine the flux. c1 is the minimum distance between vehicles, set at 7.5⋅10-3 km/veh (Wahle et al., 1999); c2 is the reaction time of an individual driver, thus c2⋅v is the approximate safe distance to avoid collisions. c3 is a higher order term, since the braking distance increases slightly more than linear with velocity. c3 is determined by stating that dΦmax dv=0at the turning point of vˆ =60 km/h; this implies that:
/km h 10 08 .
ˆ 2 6 2
km/h) 60 (
km 10 5 . 2 7 1
3 2
3 −
⋅ = ⋅
=
=c v −
c . c2 is set by the constraint on the highest
possible flux of a single lane ofΦ
( )
vˆ =2000veh/h. It results in an average reaction time of 0.25⋅10-3 h = 0.9 seconds, if w=1 and y/ℓ=2000 veh/h. See figure 10 for a graphical representation.Figure 10 The solid line represents the highest possible flux at given velocity. The dashed line indicates the density that results from given combinations of flux and velocity.
The second part of the graph is part of a Gauss-curve. The idea is that the more vehicles are driving on a road, the likelier it is that they will interact resulting in reduction of velocity. So, it is chosen that this should be a probabilistic curve of form:
v v
Φ(v)=c4⋅e(v−c5)2/c6 for ≥ ˆ. (4.5b) The parameters of this curve are c4=Φmax=x/ℓ=2000 veh/h (for a single lane road);
c5= vˆ =60 km/h and c6 that performs a similar function as the deviation in the standard distribution curve. c6=3929 by demanding that Φmax(120)=800 for a single lane road, or
lx
max v f
Φ = ~⋅
~)
( with ~f = 52 in general. These figures are valid for Dutch highways with a maximum speed allowed of 120 km/h.
4.4.3. Determing velocity
For the free flow system y/ℓ<Φ(vˆ)=2000 veh/h, the velocity is – following equation (4.5b) – given by: 5 ln( ) 6
4 c
c
v= + Φc ⋅ . For high transport demands y/ℓ≥2000 veh/h, it is assumed that all cars want to move at the optimum velocity of vˆ =60 km/h. Now the amount of vehicles per kilometer, or the density d (veh/km) is determining the velocity.
A certain flux Φ implies, at 60 km/h, a necessary average density of cars of d=Φ/60 (veh/km). Thus, higher throughputs imply higher densities. However, judging by figure 10, a certain density correlates to a specific velocity. As one can see, at a flux of more than the maximum of 2000 vehicles per hour, the velocity will drop below the optimum velocity. Therefore, the road will accommodate even less than 2000 vehicles.
The remaining vehicles will either have to change route, or will be put on a ‘waiting list’. In this paper, it is assumed that they will be accommodated elsewhere on the
0 500 1000 1500 2000 2500
0 20 40 60 80 100 120 140
velocity v (km/h) fluxΦ(veh/h)
0 20 40 60 80 100 120 140
densityd(veh/km)
flux (veh/h) density (veh/km)
network with the same relative energy efficiency. Let d*=d/(w⋅vˆ ) denote the theoretical density. ForΦ≥2000 veh/h, the velocity is given by:
2 .
Since (d*)-1> c1, or the cars cannot move closer together than the minimum distance, it goes that c1−(d*)−1<0,therefore c22−4c3(c1−(d*)−1) >c2, thus:
It is now possible to rewrite some constants following the explanations of those constants as mentioned in section 4.4.2. Equation (4.6) shows the final formula to determine the velocity.
( )
The parameters used in (4.6) with their default values in this paper:
c, c>0 Effective length of vehicle at rest 7.5⋅10-3 km⋅veh-1 vˆ , 0< vˆ < v~ Optimum velocity (velocity with highest capacity) 60 km⋅h-1 v~, v~ > vˆ Arbitrary velocity: ~ 1
<1 Arbitrary fraction: ~ 1
~)
( = ⋅ ⋅ −
Φ v f x l 0.4
ℓ>0 Length of road (network) km
w≥1 Width of road (network) (# lanes)
4.3.2. Relation between energy consumption and velocity
The velocity v can be used to calculate the average fuel consumption of the traffic. The curve in figure 11 is a result of the equations of section 4.3.1, and shows the dependence of velocity on flux.
0 20 40 60 80 100 120 140 160 180 200
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
FluxΦ(veh/h)
Velocityv(km/h)
Figure 11 This curve shows the relation between fluxΦand velocity v. The discontinuity in the first derivative atΦ=2000 veh/h is where the system jumps from the free flow state to the forced flow (jam) state. In reality, there exists a range of fluxes in which there is a certain probability that the system will jump between the states. Velocities higher than legally permitted should be excluded of course.
As mentioned earlier, the actual flux at low speeds differs from the theoretical flux, since at low speeds not all throughputs can be accommodated. As said, an approximation is to assume that those throughputs beyond capacity will be accommodated at some other place without interfering with any other transport or traffic system, but with the same efficiency as the throughput that is accommodated. Thus, the energy use of the traffic F(x(t),y(t)) would be given by the relative energy use g(v) and the traffic y(t):
F(x(t),y(t))=g(v)⋅y(t).
The formula of g(v) can be approximated in interpolating some empirical data of the actual energy consumption in congested traffic in the Netherlands (Veurman et al., 2002). The empirical data was divided into classes. Between the classes, the interpolation is made. See figure 12 for the resulting curve.
The curve in figure 12 is presented by g(v)=-0.730⋅(1-9.29⋅e-0.0101v-1.86⋅10-6⋅v3). The third order is related to the aerodynamic resistance of vehicles at high speeds. In the low regions, a fit with an exponential curve is made. The total energy use of all the vehicles is thus:
F(x,y)=y(t)⋅{-0.730⋅(1-9.29⋅e-0.0101v(x,y)-1.86⋅10-6⋅v(x,y)3)}. (4.7)
0 1 2 3 4 5 6 7
0 20 40 60 80 100 120 140
velocity v (km/h)
relativeenergyuseg(MJ/vehkm)
Figure 12 An interpolation is made of the empirical class data as presented by the solid dots. The open dots indicate the range of the classes. At the high end, velocities above 90 km/h, a third order power becomes dominant. The fit is made using the least squares method.
In the next graph, figure 13, the total energy consumption of traffic is illustrated.
0 20 40 60 80 100 120 140
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Density d (veh/h)
Velocityv(km/h)
0 10 20 30 40 50 60 70
Thousands EnergyuseF(MJ/km×h)
Velocity Energy use
Figure 13 The final energy curve is presented by the dashed line; the energy use is shown in MJ per km⋅h. The velocity curve is the solid line. Note that the velocity is limited to 120 km/h.
4.3.3. Discount rateρ
The objective function has a discount factor of e-ρt. This model chooses the discount rate such that ρ-1 the mean expected lifetime of a road is. Since it is not clear how the infrastructure will function after the end of the roads existence, there is reason to include a discount factor. It is a factor that is associated with uncertainty in time.
Discount rates in an economic context describe "the inter-temporal preference structure of the economic agents." (Haurie, 2001). In this case, the discount rateρassociated with uncertainty about the functionality of the infrastructure during the usage period, induces the discounting process. In the case of a discount factor of e-ρt, one could consider ρa killing rate, since ρ dt is the elementary probability that "death occurs" in the elementary time interval [t,t+dt], given that the construction survived up to time t. A discount rate of 1.25% (t in years) corresponds to a random life duration with expected value 1/0.0125=80 years.
Only for later consideration, this paper suggests another type of discount factor. An alternative discount factor could be formulated on ground of allowed greenhouse gas emissions. It would be a factor that is associated with uncertainty in sustainable energy consumption. Considering that the transportation sector is fully dependent on fossil fuels throughout the present century, and considering that the emissions of greenhouse gases have to be reduced significantly in the same period, one could formulate a discount factor, which is based on the highest emission rate allowed. That would be of a form ∆=(1+et*−t)−1, with t* the time at which half of the emissions should be reduced.
This would be a rising discount rate.
A proposed discount factor of ∆=e−ρtcan be based on the speed of transition to clean energy in the transportation sector. The discount rateρwould then typically be defined as the inverse of the time at which half of the transportation energy is clean energy. The combination of both factors: t t
t
e r e
−
−
+
= ⋅
∆ *
1
ρ
, r being the reduction rate to be achieved at time t*, would include both a reduction path for greenhouse gas emissions and a
, r being the reduction rate to be achieved at time t*, would include both a reduction path for greenhouse gas emissions and a