Slope Optimization (or “Sloop”): Customized Optimization for Road Longitudinal Profile Eco-Design
- the optimization problem is fully described;
- the choice of the algorithm is justified;
- the algorithm is described and specially tailored for the problem.
- the evaluation model with respect to the parameters of the longitudinal profiles;
- the optimization with respect to the main construction parameters and traffic assumptions.
2. Degrees of Freedom: Parameters of the Longitudinal Profile Model
2.1. Road Longitudinal Profile
2.2.1. Linear Constraints
2.2.2. Non-Linear Constrains
- they are nonlinear;
- they are equality constraints, hence they must be taken into account at each iteration of the optimization process.
3. Objective Function
- the assessment of the construction phase;
- the assessment of the operation phase.
3.2. Volume Computation
3.3. Assessment of Ghg Emissions from the Volume Computation
4.1. Choice of Algorithm
- Optimization type The problem is a finite dimensional nonlinear optimization with various linear and nonlinear inequalities and one nonlinear equality.
- Global/Local optimization The optimized profile cannot be too dissimilar from the initial profile for economic and mechanical reasons: earth movements are costly, embankments too large are not stable. The optimization is rather local.
- Initial solution An initial designed profile is available; it is considered feasible to the extent that it satisfies the constraints.
- Smoothness We analyze this mathematical property for the objective function and for the constraints.
- The objective function is relatively smooth as the sum of two smooth functions, i.e.:
- The construction assessment is based on a volume computation. This function has continuous derivatives of all orders ().
- The operational assessment is the sum of the fuel consumption of various vehicles rolling virtually on the road. This sum numerically regularizes the integrand. However, the integrand is not as smooth as the construction cost. For example, a slight change in slope can lead to a drop in engine efficiency that locally alters the consumption or can lead to mechanical braking, which changes overall fuel consumption on the road. This function is at least .
- The nonlinear constraints stem from geometric considerations; they are .
- Computational burden The construction assessment and nonlinear constraints do not require a large amount of computational operations since they rely on simple geometric analysis. On the other hand, the operational assessment requires the simulation of vehicle dynamics, which is a computation time-consuming task.
- Nearly quadratic equality According to Section 2.2.2, the equality constraint on altitude is nearly quadratic. Given that such a constraint must be verified at each iteration, an efficient algorithm needs to take advantage of this property.
- Generic SQP algorithms do not take the quadratic equality constraints into account. We have developed a dedicated solution to take advantage of this feature.
- SQP imposes that the cost function and constraints can be continuously differentiated twice (), whereas the operational assessment is at least . At each iteration, we verify that the quadratic approximation of the Lagrangian is validated by comparing this approximation with the actual value of the objective function.
- SQP requires a computation of the Hessian of the Lagrangian and then of the Hessian of the objective function. The numerical computation of this Hessian is not straightforward because:
- the number of inputs may be high;
- the computation of the operational assessment is time-consuming.
4.2. Description of the Algorithm
- by the descent direction solving the Karush Kuhn Tucker (KKT) conditions  of the quadratic optimization with linear equality constraints. The quadratic program is as follows:
This step requires to compute the gradient and hessian of the objective function and constitutes the main computational cost of the algorithm. The gradient is computed numerically by centered finite difference; this requires simulations of traffic. We have applied the BFGS approximation to compute the hessian according to the equations given in  so as to avoid to carrying out traffic simulations in order to compute the Hessian by finite difference.
- the linear approximation of the non linear equality constraint on the altitude;
- the linear approximation of the active inequalities, i.e., the inequalities, numbered l, which are saturated and hence become equalities, .
- by reducing the length of the increment with being accommodated:
- activation of one non active linear inequality;
- activation of one non active linearized nonlinear inequality;
- an increment can respect linearized nonlinear inequalities while not respecting the nonlinear inequalities themselves. In this case, the increment’s length is reduced until the inequalities are validated, with this reduction factor being due to the linear approximation of non linear constrains;
- along the same line, a reduced increment can output a solution that fails to respect the linear approximation of the non linear equality on altitude. In this case, a reduced factor is computed, as will be described in the next subsection;
- if the reduced increment yields a solution which increases the objective, the increment is again reduced until the objective decreases with this reduction factor being due to the quadratic approximation of the cost;
4.3. Adaptation of Sqp to the Nonlinear Equality on Altitude
5.1. Case Study
Construction Phase Parameters
5.2. Simulation Parameters of the Operations Phase
- number of vehicles/day in each direction over the studied part of the road;
- mass of each type of vehicle (including load and passengers);
- product of the drag coefficient and the front surface area, denoted ;
- tire rolling resistance coefficients, () represents the constant part () the part proportional to vehicle speed.
5.3. Optimized Profiles
- 6 abscissas defining the intersection points between two consecutive straight lines; the displacements of abscissas are limited to +/− 100 m in order to obtain a profile not to far from the initial.
- 6 slopes, one per section; Slopes can vary from −5.3% to 5.3%. These bounds are set in order to ensure stability of the embankment.
- 7 radii for the circle linking the sections, since the two radii of the two extremities are also degrees of freedom; Radii vary from +/− 2000 m around the initial radii in order to keep a profile relatively smooth.
- The tolerance on the altitude constrain is +/− 0.1 m. This step with a maximum height of 10 cm is easily flattened during construction.
- : activation of one inactive linear inequality;
- : activation of one inactive linearized non linear inequality;
- : linear approximation of the non linear equality constraints;
- : compatibility of the quadratic approximation of the altitude constraint ( is the solution of Equation (24));
- : quadratic approximation of the altitude constraint;
- : quadratic approximation of the objective function.
5.4. Sensitivity Analysis
5.4.1. With Optimization Variables
5.4.2. On Construction Parameters
5.5. Evolution of Vehicle Fleet Scenario
- Since vehicle consumption accounts for roughly 90% of GHG emissions, we anticipated finding an optimized profile with milder slopes than the initial profile because, in this case, consumption would be lower. However, the optimized profile proved to have steeper slopes. This finding can be explained by the fact that the ramps remain sufficiently mildly inclined so as to not significantly modify engine efficiency and not trigger braking. If consumption remains rather stable, the algorithm will minimize construction-related emissions, which are largely due to the lime treatment of embankments as underlined by the sensitivity analysis. For this reason, the optimized profile minimizes the embankment while holding vehicle consumption stable. Yet an optimization that only takes construction into account would output a profile with a steep ramp, as is the case for Iteration 5 in Table 4 in order to avoid the embankment. In this case, the traffic-related emissions increase significantly.
- As expected, according to Table 4, the nearly quadratic constraint on the altitude often limits the descent during several iterations: lies below 1 for two thirds of the iterations and in half of the iterations, is the appropriate reduction factor of the increment to validate the altitude constraint ( and ), a finding that underscores the benefit of taking this property into account by the algorithm under the real-world scenario.
- This algorithm explores various feasible profiles with distinct features, thus providing stakeholders with different possibilities. They can refine one of the proposed profiles if deemed relevant. For example, given that the optimized solution includes two successive slopes very close to one another, stakeholders can propose a simpler profile to replace these two successive ramps by just one ramp. In its practical implementation, our algorithm takes this special case into account by automatically equalizing successive slopes when they are nearly equal. This type of ad hoc procedure is not robust (i.e., a discontinuity is artificially created) and difficult to tune (e.g., how to numerically define what constitutes “nearly equal” successive slopes?).
- This last point suggests a limitation of the algorithm, which depends on the number of slopes defined in the initial profile by road designers. An alternative would be to directly optimize the profile in an infinite dimensional space through the use of variational methods. The optimized solution would then be approximated by lines and circles in order to transform this optimized, but not necessarily feasible, solution into a longitudinal road profile that can indeed be built. Research along these lines is currently underway; this approach does not guarantee a feasible solution.
- The degrees of freedom are different in nature and are expressed in different units: angles, abscissa, radii. The implementation of the optimization algorithm does not directly use them: a scaling is first performed.
- The optimization with just a single criterion oversimplifies the problem. One perspective would call for a multi-criteria optimization in order to propose to stakeholders a set of efficient solutions (a Pareto front). It should be noted that stakeholders already have a feasible profile at each iteration in using our algorithm.
- Our vehicle consumption models are more sensitive to slope than the usual models used in environmental assessment. However, they do not take into account the specific deformation of the tire on uphill slopes as described in the article . This phenomenon depends on the tire pressure which is not taken into account in our model. This is a point of improvement in our modeling.
- Our case study does not include civil engineering structures such as bridges or tunnels. Our methodology is based on Project Energy Assessment Method (PEAM) and PEAM also models civil engineering structures as mentioned in , so there are no theoretical limits to include civil engineering structures in our methodology. Implementation can be complicated because the construction cost function may not be continuous. We plan to implement them according to users’ requests.
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|Tasks||GWP Label||Value (kgeq )|
|Per km of transport||1.235|
|Vehicle Class||Traffic |
(Per Day/Per Direction)
|Distribution||Actual Average Speed |
|Van (<3.5 t)||130||4%||103 (64)|
|Truck and articulated vehicle||802||23%||88 (61)|
|Articulated vehicle (40 t)||198||40||5.2||4.824||39.5|
|Articulated vehicle (25 t)||198||25||5.2||4.824||39.5|
|Truck (19 t)||406||19||3.27||4.824||39.5|
|Small class diesel Car||540||1.12||0.68||11.84||5.25|
|Medium class diesel Car||540||1.23||0.97||11.84||5.25|
|Large class diesel Car||540||1.51||0.82||11.84||5.25|
|Iter|| CO2 veh |
| CO2 cons |
| cost |
|2||23.4||−605||−581||0||1.1 × 10||1||1||1|
|3||−44.1||38.8||−5.34||0||1.6 × 10||0.53||1||0.06|
|4||2.06||−85.3||−83.2||0||1.7 × 10||1||1||1|
|5||851||−2838||−1987||0||7.2 × 10||1||1||0.25|
|6||96.3||−321||−225||−0.01||1.7 × 10||1||1||1|
|7||−579||528||−50.9||0.01||2.5 × 10||1||1||0.50|
|8||−209||37.2||−172||0.01||1.1 × 10||1||1||0.12|
|9||−119||−23.1||−142||0.08||2.5 × 10||0.18||0.90||1|
|10||189||−445||−256||0.10||3.2 × 10||0.40||1||1|
|12||3.02||−7.67||−4.65||0.10||1.2 × 10||0.04||0.73||1|
|13||−0.41||−0.04||−0.45||0.10||0.03||2.2 × 10||0.90||1|
|14||−0.53||−0.34||−0.87||0.10||0.01||3.7 × 10||1||1|
|15||−0.28||−0.51||−0.79||0.10||0.01||2.8 × 10||1||1|
|16||−0.25||−0.53||−0.77||0.10||0.01||2.9 × 10||1||1|
|Iter|| CO2 veh |
| CO2 cons |
| cost |
Increased by 5%
|Small class electric Car||1.45||0.68||11.84||5.25|
|Medium class electric Car||1.68||0.97||11.84||5.25|
|Large class electric Car||1.98||0.82||11.84||5.25|
|Class||First Year||Vehicles Par Day|
|Scenario 1 |
In 25 Years
|Scenario 2 |
In 25 Years
|Small class diesel Car||540||0||270|
|Medium class diesel Car||540||0||270|
|Large class diesel Car||540||0||270|
|Small class electric Car||0||403||202|
|Medium class electric Car||0||403||202|
|Large class electric Car||0||403||202|
|Scenario||Profil||CO2 Veh |
|CO2 Cons |
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Vandanjon, P.-O.; Vinot, E. Slope Optimization (or “Sloop”): Customized Optimization for Road Longitudinal Profile Eco-Design. Energies 2020, 13, 6575. https://doi.org/10.3390/en13246575
Vandanjon P-O, Vinot E. Slope Optimization (or “Sloop”): Customized Optimization for Road Longitudinal Profile Eco-Design. Energies. 2020; 13(24):6575. https://doi.org/10.3390/en13246575Chicago/Turabian Style
Vandanjon, Pierre-Olivier, and Emmanuel Vinot. 2020. "Slope Optimization (or “Sloop”): Customized Optimization for Road Longitudinal Profile Eco-Design" Energies 13, no. 24: 6575. https://doi.org/10.3390/en13246575