Urban density and transport energy consumption: a fractal model
L. Bourdic* & S. Salat
16th International Conference of Hong Kong Society for Transportation Studies. 17-20 December 2011, Hong Kong.
Cities are dense and complex structures. Fractal theory provides a theoretical framework to analyze complex systems that is worth being used for urban systems analysis. Fractal dimension is one of the useful tools to quantify urban complexity. This paper investigates through the prism of fractal theory the relationships between urban density and private transport energy consumption. A simple model based on a power law function is proposed to provide a physical interpretation to the relationship first coined by Newman and Kenworthy. It notably interprets the exponent of the curve as the fractal dimension of the road network. A robustness analysis is then carried out to discuss the validity of such a fractal model.
Keywords: Transport, Energy, Fractals, Urban Efficiency, Power Laws
The debate around the link between private transport energy consumption (for gasoline first, and then as an aggregated diesel-gasoline figure) and urban density has emerged after Newman and Kenworthy’s seminal work (1989). It has raised since a huge amount of criticisms. Gordon and Richardson (1989) wrote the first violent reply to Newman and Kenworthy’s paper. Based on the U.S. experience, they defended the liberalization of land market, as opposed to any kind of planning based on arguments such as density. The main criticism raised by Gordon and Richardson emerged from the lack of complete multivariate analysis, criticism which has been further developed by Gomez-Ibanez (1991). Further criticisms stressed the fact that socio-economic factors should be the main drivers for mobility behaviors. Urban morphology would then principally be caused by a change in these socio-economic parameters (Kirwan, 1992). Following these critics, the main drivers for gasoline consumption should be econometric parameters such as GDP/inhabitant, household incomes or fuel prices (Allaire, 2007): by increasing car tenure and transport expenses, they are supposed to be responsible for the increase in the average distance traveled, and thus the decrease in density. These authors cast doubt on the causal link established by Newman and Kenworthy between density and gasoline consumption, the better living standards causing both the drop in urban density and the rise in automobile mobility (Allaire, 2007).
This concern about causality though boils down to the chicken-or-egg dilemma. The improvement of living standards is certainly partly responsible for the increased use of cars and the associated energy consumption, fostering at the same time urban sprawl and low density urban tissues. But turning the question round in the way Illich (1973) did reverses this causal link. In the current state, there is no choice for people not to use their car if they live in a non dense city. Lengthening urban distances, the car has excluded all other transportation means, literally following the definition of a radical monopoly.
Another controversy was raised by Brindle (1994), who criticized the statistical theory behind Newman and Kenworthy’s results. He argued that plotting two per capita parameters was a guarantee to find a hyperbolic relationship between the two parameters. The log-log multivariate analysis that will be carried out in this paper will avoid the trap of this spurious correlation. On the statistical standpoint however, Evill (1995) discredited Brindle’s arguments, by showing that the correlation found by Newman and Kenworthy had nothing to do with spurious per capita correlations.
According to Rickwood (2009) yet, despite all the criticisms, there is too little evidence to refute Newman and Kenworthy’s thesis, as it remains one of the most comprehensive international study analyzing the effect of urban form on energy consumption figures. Numerous authors since have confirmed these results. Naess (1993) confirmed them analyzing energy consumption figures for 22 Scandinavian cities, avoiding at the same time the cultural bias Newman and Kenworthy could be criticized for. ECOTEC report (1993) also got to the same result, showing an inverse relationship between modal transfers and density for UK cities. Ewing and Cervero (2001) eventually explored the impact of the built environment on transportation variables and proved that high urban density reduces the average journey distance and thus encourages soft and public transports.
Few if any literature on the topic though proposes a model that could physically explain the link between urban density and private transport energy consumption. Density for sure recovers a wide range of parameters, and the results obtained by Newman and Kenworthy are a radical shortcut, as based on aggregated variables. However, despite the lack of sharpness of this analysis, this relationship is worth being investigated, as it has been fueling the debate around urban form and energy consumption for the last 20 years. Using power laws and fractal theory, authors aim at providing new insights, by proposing a physical interpretation of this curve.
2. Power laws and transport efficiency
“Few if any economists seem to have realized the possibilities that such invariants hold for the future of our science. In particular, nobody seems to have realized that the hunt for, and the interpretation of, invariants of this type might lay the foundations for an entirely novel type of theory.”
Schumpeter (1949), on the subject of power laws and Pareto distributions.
Power laws have a tremendous importance in many natural and man-made phenomena. It is a relation of the type:
With Y and X are variables, α a constant exponent and k a constant. This type of law is also known as a scaling law, as it structures the relationship between size (or scale) of elements in a set and their multiplicity. This type of regularity is omnipresent in both:
• Natural phenomena: river networks (Maaritan et al., 1996), biological networks and size of animals (Brown et al., 2000; West, 1999), trees, lungs, coast lines (Mandelbrot, 1982), etc.
• Socio economic phenomena: city size (Zipf, 1949), income distribution and wealth, quantitative linguistics (Montemurro, 2001), website occurrence in Google (Adamic et al., 2000), citations in scientific literature (Bradford, 1985), etc.
Power laws present a remarkable number of regularities, and allow to describe a wide range of distributions with analogue properties: many small objects and few large objects, many small events, and few large events. The omnipresence of this type of heavy tail distribution has a lot to do with the presence of flows and networks in the background of all these phenomena: blood system, river networks, human networks, financial flows, etc. Salat and Bourdic (2011a) have shown that this type of distribution within networks can be explained on the basis of thermodynamics arguments: for certain types of flows, this type of structure is the most efficient one. Systems and networks thus ‘naturally’ evolve towards this type of distribution.
The omnipresence of this type of structure is directly related to networks efficiency. According to Heitor-Reiss (2006) and Salat & Bourdic (2011b), based on thermodynamics and energy arguments, this type of structure is the most efficient one to distribute a flow (the sap in a tree for instance) into a 2 or 3 dimension space. A two dimension tree-like structure is much more efficient than a grid structure, where all the veins are the same size. This is the intrinsic reason why so many network based structures are fractals: lung, river basins, road networks, internet, electricity networks, etc. The fractal dimension of such a structure is a fundamental parameter, that is called “non-euclidian”. In Euclidian theory (the classic one), a point has a topological dimension of zero, a line 1, a plane 2 and a volume 3. Fractal theory allows this value not to be an integer: fractal structures have a non integer topologic dimension. A simple square grid made of lines has a fractal dimension that can be considered as being one. A two dimension fractal tree on the contrary is somewhere between a line and a plane and its fractal dimension is between one and two.
The better efficiency of such networks is understandable: most of the flow transits via the big veins, where friction is lower, making the energy losses smaller. As one goes along the network, there is fewer flow to be distributed, and the diameter decreases, in order to feed every little part of the surface. This will be referred to as “accessibility” further in this paper. The analogy with urban networks is outstanding: wide highways and boulevards to roughly and quickly irrigate the city, and narrower and narrower streets to irrigate all the places within the city: fractal means accessible. Table 1 displays values of road networks fractal dimensions for UK cities, ranging from 1 to 2.
Table 1: Fractal dimension of road networks in 10 UK cities. Adapted from Baba (2000)
City Fractal dimension of the road network
Norwich 1 .15
Kingston Upon Hull 1 .90
Derby 1 .45
Exeter 1 .83
Leicester 1 .63
Northampton 1 .33
Oxford 1 .68
3. A power law model for private transport energy consumption
The main issue of this paper is to propose a simple model to link private transport energy consumption and urban density. Authors present hereunder a naïve and simple model that could explain the emergence of such a power law linking private transport consumption per capita and urban density. The following reasoning is based on average values. To the first order, average energy consumption for private transport is directly related to the average distance traveled per capita per year. These two parameters are linked by the average efficiency parameter of the car fleet in J/km:
Average distance traveled per year per capita can be split into a product of two terms: the average speed and the average time spent traveling per year:
Following Zahavi’s empirical work (1980), the average travel time budget follows a distribution with a central tendency around one hour per day per capita. The average annual time spent traveling can thus be considered as being 365 hours/(yr.cap). This leads to the following equation, where the average annual consumption for private transport per capita is directly proportional to the average speed of private transports in the city.
This reasoning leads to an outstanding result: urban consumption for transport is proportional to the average speed of private transport. This is not without going against certain common beliefs, defending that faster and more fluid networks have to be fostered for a better efficiency. Based on the 1990 data gathered by Barter (1999) for 43 cities, the correlation between average private transport energy consumption (including both gasoline and diesel) and average traffic speed is high (Adj R²=0.73), which provides a sound argument for this assumption.
The key point is then to link the average speed of private transports in cities with urban density. Intuitively speaking, the denser the city, the more contracted the road network. A high population density means a high building density, letting fewer space for the road network. The correlation between average speed and density is thus expected to be an inverse one, looking for instance like:
But it is possible to refine this equation using fractal theory, which allows describing objects distributions with an invariance of scale. As explained here above, trees for instance are fractals: the size r of the branches is linked to the number n(r) of branches of size r by the following power law, m being the fractal dimension of the structure, A a constant:
The fractal dimension of the road network has a counter effect to density. Density contracts the network, letting less space to roads and paths within the city. But for a given density, and a given area dedicated to the road network, fractal networks are the most efficient ones. It is the result of a constrain optimization. Fractal dimension of the road network is a good proxy for accessibility, in analogy with natural systems such as lungs. In the case of road networks, the closer the fractal dimension to two, the more accessible the city. That is why it is the difference 2-D that matters.
The term 2-D has a significant impact on urban transport phenomena. Baba (2000) corroborates this hypothesis by proving a significant correlation (R²=0.658) between 2-D, density and average journey length for UK cities, using an inverse power law model. The constant average travel time budget hypothesis allows to extend this result to the average speed. This reasoning leads to the following power law, linking average speed, population density and fractal dimension of the network, with A a constant parameter:
It is crucial though to note that diminishing the fractal dimension of the network would decrease the accessibility, and is consequently not desirable. Finally, the average private transport energy consumption can be expressed as a function of urban density and fractal dimension of the network:
is the average private transport energy consumption
d is the population density
D is the fractal dimension of the road network
C0 is a constant, equal to
The statistical test designed to test the model is a log-log regression. Before using an OLS method, it is indeed necessary to transform this relationship into a linear one, by applying the Natural logarithm on both sides of the equation:
Concerning the existence of a spurious correlation (Brindle, 1994) due to the hidden ‘per capita’ term, in this method, both sides of the equation display a per-capita term: energy consumption per capita and area per capita. This transformation insures that there is no spurious correlation (Naess, 1993). The analysis is based on 1990 data from a sample of 43 cities, gathered and updated by Barter (1999). The correlation coefficient is high (Adj R²=0.85). This method also provides the m parameter, that is 0.829 (p<5.10-4) and can be interpreted as the average fractal dimension of the road networks for the 43 cities analyzed: D=1.171.
4. Robustness analysis
It is crucial though to test the robustness of such a model. This will be done by carrying out a multivariate analysis integrating econometric parameters. This concern was at the center of the criticisms that have emerged after Newman and Kenworthy’s paper. This is why it is crucial to test if there is any omission bias due to any hidden econometric explanatory variable. The multivariate analysis will aim at regressing the average energy consumption for private transport versus a set of explanatory variables. These explanatory variables have to be chosen carefully to cover the spectrum of parameters that could explain the consumption associated with private transport. The first explanatory variable is an income proxy. Four other econometric parameters aim at taking into account the differences between countries concerning development and access to automobile: car tenure, road network development, public transport network development and car cost (including both capital and operating costs). The last explanatory used in the analysis is often quoted in the literature as having a significant impact on the demand for private transport: land use mix. If housings, shops, leisure facilities and offices are mixed within the city, demand for private transport is supposed to decrease. If on the contrary offices are concentrated in one zone, whereas housings are concentrated in another, demand for private transport is supposed to increase. The land use mix proxy is meant to take into account this phenomenon. In this analysis, it is based on two density parameters: the inner population density and the inner job density. This ratio is a good proxy for the land use mix, in terms of housing and job diversity: if there are more jobs than housings in the inner city, the ratio is lower than one, jobs are in excess in the inner city, and people have to commute towards the inner city to work.
The multivariate analysis is based on the same sample as previously (Barter, 1999), which is unfortunately incomplete for 14 cities. The core hypothesis of this multivariate analysis is that the average energy consumption for private transport varies as a product of a power of these parameters. This model is analog to the Cobb-Douglas function that is widely used in economics:
With C0, α, β, γ, δ constants, d urban density, D fractal dimension of the road network, A, B, C and D the explanatory variables. The multivariate regression is then based on a log-log transformation:
For the exact same reason as previously, there is no spurious correlation due to the per-capita bias with this method, thanks to the log transformation.
Table 2: Multivariate regression to estimate log(carbon emissions) from several explanatory variables.
Explanatory variable Coefficient
Log(inv. density) 0.150
Log(car per capita) 0.622**
Log(road length per cap) -0.050
Log(GRP per capita) -0.001
Log(land use mix) -0.011
Log(car cost) -0.199
Log(public transport) -0.180.
Significance levels: ***<0.001, **<0.01,*<0.05, .<0.1.
Only two explanatory variables are significantly correlated with private transport energy consumption, car tenure and public transport availability, the significance of the latter remaining low. This analysis shows that there is unfortunately no significant correlation with the variable on which the focus is put in this paper: density. A more accurate analysis is necessary though in order not to draw too early conclusions. The covariance matrix displays information about the colinearity between regressors (explanatory variables): the higher the correlation between regressors, the less robust the analysis. The colinearity analysis leads to consider the results obtained in the previous multivariate analysis with care. The explanatory variables associated to density, car tenure and public transport availability are highly correlated one with the other, R² being higher than 0.64. This correlation introduces a colinearity bias in the multivariate analysis: to improve its robustness, it is better to drop two of the variables. The colinearity between density and public transport development has already been implicitly coined by Newman and Kenworthy (1989), as they postulated a switch towards public transports above a given threshold of urban density. The new multivariate analysis then shows a very significant correlation between density and private transport energy consumption (p<0.001), and a slightly lower correlation with the income proxy (p<0.05).
Table 3: Multivariate regression to estimate log(carbon emissions) from several explanatory variables.
Explanatory variable Coefficient
Log(inv. density) 0.650***
Log(road length per cap) 0.009
Log(GRP per capita) 0.230*
Log(land use mix) 0.114
Log(car cost) -0.433
Significance levels: ***<0.001, **<0.01, *<0.05, .<0.1.
Based on this comprehensive sample of cities, the average energy consumption for private transport is only significantly correlated with urban density and income. A good model for private transport energy consumption in urban areas could then be:
This paper proposes an innovative model to physically explain the relationship between private car energy consumption and urban density that has first been coined by Newman and Kenworthy. It does neither aim at feeding the debate on the validity of this relationship nor at discussing the causality issue. It aims on the contrary at providing a simple model that could account for the link between two aggregated and global variables describing urban areas. It proposes to use insights from recent and innovative scientific fields such as the theory of complex systems to look at old questions through new eyes. The statistical analysis carried out in this paper shows that this model is worth being investigated further. Authors recognize that for this model to be considered as valid, further analysis has to be carried out to verify whether the inverse power law exponent can be interpreted as the fractal dimension of the road network. The highest hurdle though lies in the data availability for such fractal dimensions. Calculating the fractal dimension of the road network in the 43 cities on which this analysis has been based would indeed require a tremendous work of data gathering. Further investigation are being carried out by the Urban Morphology Laboratory on smaller samples to confirm the validity of the model presented in this paper. The focus will notably switch toward city scale analysis, the reasons for this being twofold. The first reason is that the fractal dimension is much easier to calculate for small samples, such as intra-urban zones, which is a way to resolve the data availability issue. The second and principal reason for this is that urban policies are drawn up on the city scale. Comparisons between cities are for sure useful to encourage competition between them. But policy makers should also be able to use urban density as a local policy tool. Density figures averaged on city scale are useful for comparisons, but not as a tool for local policy makers. Urban densities vary a lot when comparing cities throughout the world, from 10 pph in Brisbane to 245 pph in Seoul (Barter, 1999). But it varies in the same range within cities: from 249 pph in Paris historic core to 7 in the suburb (Panerai, 2009). It would thus be tempting to extend and generalize Newman and Kenworthy’s results within cities. This generalization though is not without raising a series of methodological issues. Density is a multi-scale concept. As such, scaling up or down is nothing but trivial. However, if the validity and the robustness of such a model is proven on the city scale, it could reveal to be a useful predictive tool for urban policies, although according to Niels Bohr, prediction is difficult, especially about the future…
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