Global Sensitivity of Roughness-Induced Fuel Consumption to Road Surface Parameters and Car Dynamic Characteristics

Abstract

Pavement roughness is one of the key contributors to rolling resistance and thus vehicle fuel consumption. Roughness-induced fuel consumption is the result of energy dissipation in the suspension system of vehicles and therefore depends on both road surface characteristics and vehicle dynamic properties. In this paper, the sensitivity of roughness-induced excess fuel consumption to all involving factors, i.e., road roughness metrics, vehicle dynamic properties, and speed is investigated, and the dominant factors affecting fuel consumption are identified. This is achieved by using the Sobol’s method—a robust analysis of variance (ANOVA)-based technique for global sensitivity analysis. To this end, Monte-Carlo (MC) simulation is performed by generating realizations of all input parameters according to their probability distributions and estimating the energy consumption via a mechanistic roughness model. The results of the simulation are then used to obtain global Sobol sensitivity indices. Finally, the comparison between the Sobol sensitivity indices and the previously employed indices based on Spearman Rank Correlation Coefficient (SRCC) is illustrated. It is found that roughness metrics, i.e. the International Roughness Index (IRI) and the waviness number, account for 88–93% of the total variations in energy dissipation and are the most influential factors affecting the excess fuel consumption. It is also observed that among vehicle dynamic properties, the stiffness of tire is the most important parameter accounting for 2–7% of the total variance of the excess energy consumption.

With fast increase in the need for travel and mobility, the greenhouse gas (GHG) emission due to the transportation sector has increased significantly since 1990 ( 1 ). In the United States, for instance, the transportation sector generates the largest share of GHG emissions, accounting for more than 28% of the total emission in 2016 ( 1 ). Among different types of transportation, road transportation has been identified as the most significant contributor to the environmental impact ( 2 ). United States Environmental Protection Agency (EPA) has reported that the number of vehicle miles traveled (VMT) by passenger cars and trucks has increased by 45% since 1990 ( 1 ). In general, the total GHG emission from road transportation originates from the following phases: material production, construction, maintenance, and use phases of the pavement ( 3 – 7 ). Life Cycle Assessment (LCA) of pavements suggests that for heavily congested traffic roads, the impact of use phase manifests itself the most and surpasses the emission from the other phases ( 8 , 9 ).

Besides the fuel needed to propel the vehicle and overcome the air drag, extra energy must be added to the system to compensate rolling resistance forces and to maintain a constant speed ( 10 ). Deflection-, texture-, and roughness-induced pavement–vehicle interactions (PVI) are the most prominent mechanisms of rolling resistance that lead to excess fuel consumption ( 2 , 11 , 12 ). Deflection-induced PVI accounts for the dissipated energy within the viscoelastic material of the pavement ( 11 , 12 ). Texture-induced PVI is responsible for the dissipated energy in the viscoelastic tire due to the interaction between the texture of the road and tire treads ( 13 , 14 ). Finally, roughness-induced PVI takes into account the dissipated energy in vehicles’ suspension system due to road unevenness ( 2 , 15 ).

This paper focuses on roughness-induced PVI for which empirical and mechanistic-based models have been developed. The inherent uncertainty of the inputs to the roughness-induced PVI model raises the question of how the roughness-induced fuel consumption is influenced by the uncertainty of different input factors. To answer such question, global sensitivity analysis is used, which is a powerful tool to quantify the uncertainty of the model output and properly allocate that to different sources of uncertainty in inputs ( 16 ). This technique can be used to identify the parameters with the most and the least significant impact on the output. It thus allows the main focus to be on the individual or groups of input variables with higher impact on the output.

Recently, Giustozzi et al. performed a sensitivity analysis of deflection-induced PVI to account for the impact of resolution and accuracy of input data and investigated how sensitive the deflection-induced PVI model is to the use of average data versus recorded inputs ( 17 ). In another study, Louhghalam et al. used normalized Spearman Rank Correlation Coefficient (SRCC) as a measure of sensitivity to quantify the effect of roughness variables as well as speed on roughness-induced PVI model ( 2 , 15 ). In the research described in this paper, sensitivity analysis is performed on the mechanistic model for roughness-induced PVI. The impact of both road characteristics and vehicle dynamic properties is investigated, unlike in the previous studies ( 2 ). This is achieved by employing a special technique for global sensitivity analysis, Sobol’s method, to investigate the global sensitivity of roughness-induced excess fuel consumption (EFC) to different individual as wells as groups of input factors ( 18 ).

Roughness-Induced PVI

In the following, the fundamental definitions of roughness indices and the equations for the roughness-induced PVI model are briefly reviewed. For the complete derivations of the roughness-induced model see Reference ( 15 ).

International Roughness Index (IRI)

The IRI is defined as accumulated suspension motion of the Golden Car, a standard quarter car model, traveling at a constant speed of $V_{0} = 80$ km/h ( 19 , 20 ):

I R I = \frac{1000}{V_{0} L} \int_{0}^{L} | {\dot{z}}_{G C} | d x

(1)

In the above equation $L$ is the total distance traveled and $\overset{\cdot}{z} = {\overset{\cdot}{u}}_{u} - {\overset{\cdot}{u}}_{s}$ is the relative motion between the unsprung mass $m_{u}$ and sprung mass $m_{s}$ (see Figure 1) with super-dot denoting the time derivative. The subscript $GC$ denotes the variables corresponding to the Golden Car, which refers to a 2-degree-of-freedom quarter car in Figure 1. The two masses are connected with a Kelvin-Voight suspension system with damping coefficient $C_{s}$ , and spring stiffness $k_{s}$ and the tire is represented by spring with stiffness $k_{t}$ .

Figure 1.

Two-degree-of-freedom quarter car model and the dynamic properties of the Golden Car.

Roughness Power Spectral Density (PSD)

The PSD of roughness is the Fourier transformation of autocorrelation function of the road profile and describes the distribution of different wavelengths within the profile. For most roads, roughness PSD can be expressed in the form of a power function as: $S_{ξ} (Ω) = c Ω^{- w}$ with $c$ the unevenness index, $w$ the waviness number, and $Ω$ the angular wavenumber ( 21 ). Roughness PSD is thus a property of solely the road surface and its definition does not require any reference vehicle.

Mechanistic Model for Roughness-Induced PVI

For a vehicle traveling on an uneven road with speed $V$ , the dissipated energy in the suspension must be provided to the system via extra engine power that leads to excess fuel consumption. Thus the quantity of interest, i.e., the dissipated energy in the suspension system per distance traveled ( $δ ϵ$ ), reads:

δ ϵ = \frac{C_{s}}{V} {\overset{\cdot}{z}}^{2}

(2)

where $C_{s}$ and $\overset{\cdot}{z}$ are respectively the damping coeffiecient and the relative motion of the suspension. The dynamic response of the quarter car when traveling on an uneven surface with profile elevation $ξ (x = Vt)$ can be obtained via two different approaches: (i) time-domain method also used in PROVAL ( 20 ); and (ii) frequency-domain method ( 15 ). The frequency-domain analysis is a faster approach but only calculates the steady-state (particular) solution, while the time-domain method accounts for both the steady-state and the transient (free vibration) solutions. For damped systems, the transient solution dissipates exponentially and it only influences the initial time intervals of the dynamic response, thus can be disregarded. Figure 2(b) shows the relative displacement of the two masses using both methods when the Golden Car travels on road profile $ξ (x)$ illustrated in Figure 2(a) with the speed of 80 km/h. As observed, the responses via the two methods perfectly match except at the initial time intervals due to the effect of transient response. The impact of suspension system on the ride comfort is also illustrated in Figure 2(c) and ( d ), where the unsprung mass experiences large velocity and acceleration amplitudes compared with the low-amplitude vibrations of sprung mass felt by passengers.

Figure 2.

Dynamic response of the Golden Car: (a) road profile, (b) relative displacement of the masses, (c) absolute velocity, and (d) absolute acceleration.

The dynamic response can be directly related to the input in the frequency domain via the frequency response function (FRF) as: $\hat{z} (ω) = H_{z} (ω) \hat{ξ} (ω)$ , $where (\hat{.})$ represents the Fourier transform operator, $ω = V Ω$ is the angular frequency, and $H_{z} (ω)$ is the FRF for the relative displacement of the two masses and is given by:

H_{z} (ω) = \frac{ω^{2} k_{t} m_{s}}{(- m_{u} ω^{2} + i C_{s} ω + k_{t} + k_{s}) (- m_{s} ω^{2} + i C_{s} ω + k_{s}) - {(i C_{s} ω + k_{s})}^{2}}

(3)

(2, 15)

It should be noted that the FRF solely depends on vehicle dynamic properties and is not related to the road surface characteristics.

Using the fundamentals of random vibrations, the PSD of relative velocity of the two masses is related to the PSD of the input, i.e., roughness PSD: $S_{\overset{\cdot}{z} (Ω)} = Ω^{2} {| H_{z} (Ω) |}^{2} S_{ξ} (Ω)$ . The mean square of suspension motion is then obtained by integrating its PSD $E [{\dot{z}}^{2}] = \int_{0}^{\infty} S_{\dot{z} (Ω)} d Ω = \int_{0}^{\infty} Ω^{2} {| H_{z} (Ω) |}^{2} S_{ξ} (Ω) d Ω$ ( 22 ), where $E []$ is the expectation operator. Finally, substituting the power function for road roughness PSD the expected value of dissipated energy per distance travelled reads:

E [δ ϵ] = \frac{C_{s}}{V} E [{\dot{z}}^{2}] = C_{s} V^{w - 2} c \int_{0}^{\infty} ω^{2 - w} {| H_{z} (ω) |}^{2} d ω

(4)

(2, 15)

The above relation is sufficient to estimate the average dissipated energy and fuel consumption for a vehicle traveling on a specific road. In the above fundamental relation, the properties of the vehicle are manifested in the FRF, $H_{z} (ω),$ and the road parameters are embodied in the two PSD parameters, unevenness index $c$ , and waviness number $w$ . However, for practical applications, e.g. network analysis, it useful to employ the widely used roughness index IRI and rewrite the expected dissipation in terms of the expected value of IRI ( 23 ):

E [δ ϵ] = \frac{π}{2} C_{s} V_{0} {(\frac{V}{V_{0}})}^{w - 2} \frac{\int_{0}^{\infty} ω^{2 - w} {| H_{z} (ω) |}^{2} d ω}{\int_{0}^{\infty} ω^{2 - w} {| H_{z - GC} (ω) |}^{2} d ω} E {[IRI]}^{2}

(5)

In the above equation $V$ is vehicle speed and $V_{0}$ = 80 Km/h is the reference speed in calulation of IRI. This relation accounts for both car dynamic properties and roughness characteristics and is used in the next sections to perform Monte-Carlo (MC) simulations and sensitivity analysis.

Sensitivity Analysis

The relative impact of different input parameters to the model output is investigated via sensitivity analysis. In a broad sense, sensitivity analysis is categorized into local and global approaches. Local approach calculates the sensitivity of the model output to input parameters at nominal points, especially when the variables are relatively certain. In such deterministic approaches, the sensitivity indices can be estimated by numerically evaluating the partial derivatives of model output with respect to input variables at a specific point ( 16 , 24 ). Unlike local approaches to sensitivity analysis, global approaches consider a practical range and a probability distribution for each input variable. These methods are particularly more effective when the input parameters (and subsequently the model output) are random. Different global approaches have been used recently for sensitivity analysis; among them Sobol’s method has gained a lot of attention because of its capability to estimate normalized sensitivity indices for individual and groups of input parameters. The Sobol’s method, which is an analysis of variance (ANOVA) technique, is significantly more powerful than correlation-based methods such as SRCC. One of the main drawbacks of SRCC as a means of calculating global sensitivity indices is that it measures correlation between variables assuming monotonic relation between input variables and the model output that can be violated in non-linear models such as PVI models. Furthermore, SRCC can only measure first-order indices and thus fails to calculate the sensitivity of the output to groups of variables. For the sake of comparison with previous studies, the results of sensitivity analysis using SRCC are also presented.

Sobol’s Method: A Brief Review

Let scalar value function $g (\vec{x})$ be the nonlinear mathematical model, e.g., the roughness PVI model in Equation 5, where $\vec{x} = (x_{1}, x_{2}, \dots, x_{n})$ is the set of input variables. Sobol assumes joint uniform probability distribution function for the uncorrelated input variables in the form of a unit hypercube of dimension $n$ represented herein by $ℓ^{n}$ . Sobol decomposition of the output function $g (\vec{x})$ is then written as an ANOVA-representation ( 18 ):

g (\vec{x}) = g_{0} + \sum_{i} g_{i} (x_{i}) + \sum_{i < j} g_{i, j} (x_{i}, x_{j}) + \dots + \sum_{i < j} g_{1, 2, \dots, n} (x_{1}, x_{2}, \dots, x_{n})

(6)

with the orthogonality condition: $\int_{ℓ^{n}} g_{i_{1}, i_{2}, \dots, i_s} (x_{i_{1}}, x_{i_{2}}, \dots, x_{i_{s}}) g_{j_{1}, j_{2}, \dots, j_{s}} (x_{j_{1}}, x_{j_{2}}, \dots, x_{j_{t}}) dx = 0$ if at least one of the subscripts on the two functions in the integral argument are not equal. While the functional relationship is developed for uniform random inputs, Arwade et al. extended the method to Gaussian random variables by proposing a new orthogonality condition and showed that the Sobol’s method can be used for uncorrelated random inputs with any distributions ( 25 ). Using the above-mentioned representation, the zeroth-, first-, and second-order Sobol functions can be written as:

\begin{matrix} g_{0} = \int_{ℓ^{n}} g (\vec{x}) d x_{1} d x_{2} \dots d x_{n} \\ g_{i} (x_{i}) = \int_{ℓ^{n - 1}} g (\vec{x}) d x_{1} d x_{2} \dots d x_{i - 1}, d x_{i + 1}, \dots d x_{n} - g_{0} \\ g_{i, j} (x_{i}, x_{j}) = \int_{ℓ^{n - 2}} g (\vec{x}) d x_{1} d x_{2} \dots d x_{i - 1}, d x_{i + 1}, \dots, \\ d x_{j - 1}, d x_{j + 1}, \dots d x_{n} - g_{i} - g_{j} - g_{0} \end{matrix}

(7)

Using the orthogonality condition, the total variance of the model $D = Var (g) = \int_{Ω} g^{2} (\vec{x}) d x_{1} d x_{2} \dots d x_{n} - g_{0}^{2}$ can be decomposed as:

D = \sum_{i} D_{i} + \sum_{i < j} D_{i, j} + \dots + D_{1, 2, \dots, n}

(8)

where $D_{i_{1} i_{2} \dots i_{s}}$ is the $s$ -order variance of group of variables $(x_{i_{1}}, x_{i_{2}}, \dots, x_{i_{s}})$ given by:

D_{i_{1}, i_{2}, \dots, i_{s}} = \int_{l^{s}}^{g} i_{1}, i_{2}, \dots, {i_{s}}^{2} (x_{i_{1}}, x_{i_{2}}, \dots, x_{i_{s}}) d x_{i_{1}} d x_{i_{2}} \dots d x_{i_{s}} - g_{0}^{2}

(9)

Finally, the $s$ -order global sensitivity Sobol index is defined as the normalized variance: $S_{i_{1} i_{2} \dots i_{s}} = D_{i_{1} i_{2} \dots i_{s}} / D$ .

The Sobol indices vary between zero and one, and for a model with $n$ input parameters the sum of $2^{n} - 1$ Sobol indices equals to unity. Small values of Sobol indices indicate that the model is almost insensitive to an individual or a group of input parameters. Thus, a dimension reduction can be performed from $n$ to $n - m$ by fixing the $m$ unessential set of variables at their mean value, if the sum of Sobol indices corresponding to the remaining $n - m$ variables is close to one.

Monte-Carlo Simulations

The MC simulation is performed by generating realizations of input parameters $\vec{x}$ and calculating the roughness-induced dissipation using the PVI model in Equation 5, i.e., $g (\vec{x}) = E [δ ϵ]$ with $\vec{x} = (IRI, w, V, k_{t}, m_{u}, k_{s}, m_{s})$ . This requires that probability density function (PDF) of each input parameter in $\vec{x}$ be specified. The PDF of IRI and waviness number for asphalt concrete (AC) and Portland concrete cement (PCC) are obtained from the research described in References ( 2 ) and ( 26 ) and summarized in Table 1. The PDF for vehicle speeds, illustrated in Figure 3, is obtained from Weigh in Motion (WIM) data provided by the Virginia Department of Transportation. In this research, two classes of vehicles are studied, namely medium cars and trucks. For each class of vehicle, the dynamic properties are gathered from data available in literature ( 23 – 32 ). The data is then used to obtain a PDF for each input parameter and the results are summarized in Table 2.

Table 1.

Probability Distributions Function for Road Roughness Characteristics (Adapted from the Research Described in References [ 2 ] and [ 26 ])

Roughness metrics	Distribution	Mean		Standard deviation
Roughness metrics	Distribution	PCC	AC	PCC	AC
IRI	Lognormal	1.617	1.320	0.492	0.541
$w$	Truncated Normal	2.479	2.475	0.314	0.417

Figure 3.

Empirical PDF for vehicle speed.

Table 2.

Probability Distribution for Dynamic Properties of Different Classes of Vehicles

Dynamic properties (unit)	Distribution	Mean		Standard deviation
Dynamic properties (unit)	Distribution	Sedan	Truck	Sedan	Truck
$m_{u}$ ( $kg$ )	Lognormal	42.5	353.6	10.1	61.6
$k_{t}$ ( $KN / m)$	Lognormal	164.9	1300.9	57.6	500.6
$m_{s}$ ( $kg$ )	Lognormal	340.1	2600.1	40.8	282.6
$k_{s}$ ( $KN / m$ )	Lognormal	23.8	442.2	8.7	183.7

For each realization in the MC simulation, all random input variables must be generated according to their cumulative distribution function (CDF). To generate a random input sample $x_{i}$ with CDF of $F_{X_{i}} (x_{i})$ the inverse transformation technique is used, i.e. a random variable with uniform distribution $U ~ u [0, 1]$ is generated and the inverse of CDF is evaluated at $U$ . It can be shown that $x_{i} = F_{X_{i}}^{- 1} (U)$ is a random variable with CDF $F_{X_{i}} (x_{i})$ . Once all samples are generated and the corresponding dissipations are evaluated, the variances and sensitivity indices are determined via Equation 9.

To minimize the error in the estimation of variances, MC simulation must be performed with sufficiently large number of realizations $N$ . This is especially important for parameters with relatively low variance. In this research, the MC simulation is performed with different number of realizations to ensure the convergence of the Sobol indices. The residual and total variances of the output are respectively obtained from the following sums over $N$ realizations:

\begin{matrix} g_{0} = \frac{1}{N} \sum_{j = 1}^{N} g ({\vec{x}}_{j}) \\ D = \frac{1}{N} \sum_{j = 1}^{N} g^{2} ({\vec{x}}_{j}) - g_{0}^{2} \end{matrix}

(10)

To evaluate the variance corresponding to the subset $y = (x_{i_{1}}, \dots x_{i_{s}})$ of input variables it is necessary to generate $2 N$ realizations of input variables, i.e. ${\vec{x}}_{j} = (x_{j 1}, \dots . x_{jn})$ and $\vec{x}'_{j} = (x'_{j 1}, \dots . x'_{jn})$ . Let $z = (x_{i_{s + 1}}, \dots x_{i_{n}})$ be the complementary set of $y$ (a set that includes all input variables that are not included in $y$ ). The corresponding variance then reads:

D_{y} = \frac{1}{N} \sum_{j = 1}^{N} g ({\vec{x}}_{j}) g ({\vec{y}}_{j}, {\vec{z}}_{j}^{'}) - g_{0}^{2}

(11)

(18)

The partial variances obtained from Equation 11 are normalized by the total variance to determine the global sensitivity indices (GSI). Figure 4 shows the GSI for the individual input variables calculated using different number of realizations with convergence observed at $N = 10^{6}$ . It is found that the sum of GSI for individual variables ( $\sum_{i = 1}^{7} S_{i}$ ) is less than 0.9 which indicates that higher-order terms that account for the impact of group of variables must be considered in the analysis.

Figure 4.

First-order global sensitivity indices.

For variables with very small sensitivity indices, e.g., $V$ , $k_{s}$ , $m_{s}$ , and $m_{u}$ , we observe negative GSI values in Figure 4 when the number of realizations is less than $10^{4}$ . The reason is that for small sensitivity indices, the sum in Equation 11 can be less than $g_{0}$ due to the lack of accuracy in the MC simulation. This can be resolved if the number of realizations is significantly increased, which is computationally expensive. Alternatively, the modified Monte-Carlo (MMC) method can be used in a more computationally efficient manner ( 24 ). In MMC method, sample generation and calculation of residual and total variance remains the same as the classical MC approach, while a direct higher-order expression is used to estimate $D_{y}$ ( 33 ):

D_{y} \approx \frac{1}{N} \sum_{j = 1}^{N} g ({\vec{x}}_{j}) [g ({\vec{y}}_{j}, {\vec{z}}_{j}^{'}) - g ({\vec{x}}_{j}^{'})]

(12)

Using the MMC approach, small sensitivity indices can be estimated with fewer realizations. As an illustration, Figure 5(a) and ( b ) compares the performance of classical MC with that of the MMC simulations for inputs with respectively small ( $V$ and $m_{u}$ ) and large ( $IRI$ and $w$ ) first-order GSIs. It is observed that, while for large GSIs both methods converge with almost the same number of realizations, for small sensitivity indices classical MC requires almost two orders of magnitude more realizations compared with the MMC method. It should be noted that MMC requires evaluating two second-order expressions, i.e., $g ({\vec{x}}_{j}) g ({\vec{y}}_{j}, {\vec{z}}_{j}^{'})$ and $g ({\vec{x}}_{j}) g ({\vec{x}}_{j}^{'})$ , for estimation of variance compared with one second-order function evaluation in MC. It is thus more computationally expensive for the same number of realizations.

Figure 5.

Comparison of MMC and MC for (a) large sensitivity indices and (b) small sensitivity indices.

Results of the sensitivity analysis with $N = 10^{7}$ realizations are summarized in Figure 6(a). Large values of $S_{IRI}$ , $S_{w}$ and $S_{IRI, w}$ in Figure 6 suggest that EFC is mostly affected by the road roughness parameters. In fact, the accumulated impact of $w$ , IRI, and the combined effect of $w$ and IRI accounts for 88% to 93% of total EFC variations. Among vehicle properties, the first-order GSI of tire stiffness $k_{t}$ is the largest index and ranges between 3% and 7%. Influence of the remaining inputs denoted by the index $S_{r}$ is less than 5% as observed in Figure 6.

Figure 6.

Global sensitivity indices using (a) original high-dimensional model and (b) reduced model.

It is found that for all classes of vehicles and road properties $S_{IRI} + S_{w} + S_{IRI, w} + S_{k_{t}} \geq 0.95$ , thus it is useful to divide the input parameters into two sets: the parameters with large GSI, $\vec{y} = (IRI, w, k_{t})$ , and small GSI, $\vec{z} = (V, m_{s}, k_{s}, m_{u})$ . Using the definition of Sobol decomposition and noting that the two sets are complementary, the total sensitivity—the accumulated sensitivity of individual and group variables—of the remaining variable $S_{r} = S_{z}^{tot} = 1 - S_{y}$ is less than 5%. Hence, it is possible to reduce the dimension and complexity of the model by fixing input variables in $\vec{z}$ at their mean values. Similar to the original model, the reduced model requires $N = 10^{6}$ realizations to converge to GSI with desired accuracy, and reducing the number of random variables significantly reduces the simulation time. The sensitivity indices obtained via the reduced method, illustrated in Figure 6(b) and Table 3, show a close agreement between the two models.

Table 3.

Introduced Errors on the Total and First-Order Variances

Pavement type	Variance	Original model		Reduced model		Error (%)
Pavement type	Variance	Sedan	Truck	Sedan	Truck	Sedan	Truck
AC	$D_{IRI}$	0.04	3.02	0.04	2.99	4.91	0.95
	$D_{w}$	0.01	1.12	0.01	1.11	5.38	0.47
	$D_{k_{t}}$	0.002	0.17	0.002	0.17	11.76	0.65
	$D$	0.07	5.50	0.07	5.42	4.78	1.44
PCC	$D_{IRI}$	0.03	2.64	0.03	2.64	7.10	0.07
	$D_{w}$	0.01	1.14	0.01	1.13	9.09	0.98
	$D_{k_{t}}$	0.003	0.35	0.003	0.35	8.57	0.63
	$D$	0.06	4.83	0.05	4.79	7.18	0.68

Comparison with SRCC

SRCC shows the correlation between rank of dissipated energy $g (\vec{x}) = E [δ ϵ]$ and each input factor $x_{i}$ in $\vec{x} = (IRI, w, V, k_{t}, m_{u}, k_{s})$ . Let $d_{i}$ be the difference between the ranks of $x_{i}$ and $E [δ ϵ]$ . The SRCC can be obtained from $ρ_{i} = 1 - 6 \sum_{i = 1}^{N} d_{i}^{2} / N (N^{2} - 1)$ where $N$ is the number of realizations in MC simulation ( 34 ). The corresponding sensitivity indices are calculated by normalizing the SRCC as: ${\bar{ρ}}_{i} = ρ_{i} / \sum_{j = 1}^{6} ρ_{j}$ and compared with the first-order Sobol indices for individual input variables in Table 4.

Table 4.

Comparison of Sensitivity Indices of Individual Input Variables using Sobol’s Method and SRCC

Pavement type	Input variable	$S_{i}$		${\bar{ρ}}_{i}$
Pavement type	Input variable	Sedan	Truck	Sedan	Truck
AC	IRI	0.577	0.549	0.492	0.466
	w	0.194	0.203	0.311	0.336
	$k_{t}$	0.025	0.031	0.142	0.154
	$V$	< 0.001	< 0.001	0.023	0.022
	$m_{u}$	< 0.001	< 0.001	0.015	0.007
	$k_{s}$	< 0.001	< 0.001	0.010	0.006
	$m_{s}$	< 0.001	< 0.001	0.008	0.009
PCC	IRI	0.591	0.547	0.463	0.438
	w	0.212	0.235	0.293	0.317
	$k_{t}$	0.059	0.072	0.176	0.188
	$V$	< 0.001	< 0.001	0.027	0.027
	$m_{u}$	< 0.001	< 0.001	0.018	0.008
	$k_{s}$	< 0.001	< 0.001	0.013	0.009
	$m_{s}$	< 0.001	< 0.001	0.009	0.012

Once again, the results indicate that the excess dissipated energy is mostly influenced by IRI, $w$ as well as the tire stiffness $k_{t}$ . The accumulated impact of the remaining input parameters is less than 7%. It should be noted that, unlike the Sobol indices, SRCC fails to calculate the sensitivity of the dissipation to groups of input variables, thus comparison of higher-order sensitivity indices corresponding to the combined impact of variables is not possible.

Conclusion

Sensitivity analysis is performed to investigate the sensitivity of the roughness-induced energy consumption to different input factors, namely road roughness parameters, vehicle speed and dynamic characteristics. The following results deserve attention:

The results of the global sensitivity analysis using the ANOVA-based Sobol’s method indicates that energy consumption is very sensitive to the road roughness parameters IRI and $w$ . These parameters are responsible for 88–93% of total variation in energy consumptions, with IRI and $w$ respectively accounting for 55–59% and 19–23% of the total EFC variation, in addition to their combined effect that associates with 10–17% of the total sensitivity.

Among different vehicle dynamic properties, stiffness of the tire is shown to be the variable with the highest impact on the energy consumption. Tire stiffness is responsible for respectively 2–3% and 5–7% of the total sensitivity of the excess fuel consumption in AC and PCC pavements. It is observed that energy consumption is slightly sensitive to the remaining vehicle properties such as sprung and unsprung masses and suspension stiffness. The accumulated sensitivity of these variables is less than 3%. The small sensitivity indices are obtained using MMC approach, which requires two orders of magnitude fewer realizations than for the MC simulations.

Fuel consumption in cars is more sensitive to the combined impact of IRI and $w$ , resulting in higher sensitivity for road parameters compared with vehicle properties. For trucks, on the other hand, the impact of tire stiffness on fuel consumption is more significant.

The comparison of the results of ANOVA-based Sobol’s method with that of correlation-based SRCC indicates that while the two approaches are very different in nature they predict the same trend for the sensitivity of the input variables, with SRCC overestimating the sensitivity of insignificant input variables.

The results of sensitivity analysis suggest that road surface characteristics have a significant impact on the roughness-induced dissipation and, while vehicle dynamic properties and travel speed can affect roughness-induced PVI, their impact is not significant.

Footnotes

Acknowledgements

The authors acknowledge the technology development award from the University of Massachusetts president’s office. The Weigh in Motion (WIM) data used for speed distribution is provided by Virginia Department of Transportation and Center for Transportation Innovation and Research.

Author Contributions

The authors confirm contributions to the paper as follows—study conception and design: A. Louhghalam, M. Tootkaboni; analysis and interpretation of results: M. Botshekan, A. Louhghalam, M. Tootkaboni; draft manuscript preparation: M. Botshekan, A. Louhghalam. All authors reviewed the results and approved the final version of the manuscript.

The Standing Committee on Pavement Surface Properties and Vehicle Interaction (AFD90) peer-reviewed this paper (19-00382).

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