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Eco-Cruise Control for Passenger Vehicles: Methodology B. Saerensa,1, H. A. Rakhab, M. Diehlc, E. Van den Bulcka a
c
Department of Mechanical Engineering, Katholieke Universiteit Leuven, Celestijnenlaan 300A, 3001 Heverlee, Belgium b Charles E. Via, Jr. Department of Civil and Environmental Engineering, and Center for Sustainable Mobility, Virginia Tech Transportation Institute, 3500 Transportation Research Plaza, Blacksburg, VA 24061, USA Optimization in Engineering Center (ESAT/SCD), Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, 3001 Heverlee, Belgium
Abstract An eco-cruise control system uses road topographical data obtained from a high resolution digital map to control the vehicle velocity to optimize its fuel consumption. The optimal velocity profile is the result of an optimal control problem. This paper compares and assesses different fuel consumption models, cost functions, and solution methods, as they all have an influence on the resulting profile and associated fuel savings of an eco-cruise control system for passenger vehicles. Keywords: optimal control, automotive control, eco-driving, intelligent cruise control 2010 MSC: 34H05, 49N90
1. Introduction Adapting the vehicle velocity on hilly roads can lower the fuel consumption. This approach is used in heavy duty truck driving (Hellström et al., 2010) and is mostly known as predictive cruise control (PCC). PCC is a system that uses the downstream road profile to actively change the vehicle velocity through use of a cruise controller. Evidently, this approach can also be adopted in driving a passenger vehicle (Ahn et al., 2011). In this application it is named eco-cruise control (ECC) in accordance with “eco-driving”. This paper focuses on the underlying calculation methods of an eco-cruise control system. The goal is to find an optimal velocity profile on a hilly road. This is the result of an optimal control problem that minimizes the fuel consumption for a given roadway profile. Three components have a significant 1
Corresponding author. Email address:
[email protected] (B. Saerens).
1
influence on the resulting velocity profile: 1) the fuel consumption model, 2) the cost function that is used in the optimization, and 3) the optimal control solution method. This paper assesses different possibilities for the aforementioned components of the cruise control system.
2. Eco-Cruise Control System Components
Figure 1 shows the scheme of an eco-cruise control system. The driver chooses the reference velocity ref [m/s] and a velocity band in which the vehicle velocity [m/s] can vary (min max ). The eco-
cruise control system also needs topographical information. The road slope [rad] can be obtained by a GPS system. The optimal control unit calculates an optimal velocity profile that can be fed to a
conventional cruise controller as the target velocity cc [m/s]. The optimal controller has three main
components:
vehicle model This contains a model of the longitudinal dynamics of the vehicle, a powertrain model, and a fuel consumption model. cost function This function will be optimized, next to the fuel consumption it can also contain e.g. a penalty for deviating from the reference velocity. solution method The optimal control problem solver can use different solution methods. The choice of these components will have a significant influence on the calculation time, ease of implementation, and potential consumption savings.
2.1. Vehicle Model The vehicle is considered a point mass and its motion is the result of Newton's second law. Basic longitudinal dynamics that include inertia, road friction, aerodynamic drag, and road slope resistance are used. It is considered that there are no gear shifts, and that in case of an automatic transmission the power
converter uses a lock-up. Given a velocity and acceleration ⁄, the needed engine power [W] is: ·
,
(1)
where [kg] is the overall inertia (vehicle mass and rotational inertia of the engine and driveline) and
[~] are lumped coefficients that result from the longitudinal dynamics, they can also include driveline friction. Coefficients and can be a function of the road slope [rad].
2
An appropriate fuel consumption model is essential to minimize the consumption. Because of its simplicity and analytical nature, the VT-CPFM1 model is used (Rakha et al., 2011): f
, if 0,
if 0.
(2)
This model is able to estimate vehicle consumption rates consistent with in-field measurements (
0.9) and can be calibrated using publicly available data. Low-degree polynomial fuel consumption models are commonly used for this type of application (Saerens et al., 2012).
For comparison and validation purposes, a more detailed six-parameter polynomial fuel consumption model is also considered: f " " " "# " # "# ,
with # $⁄$max " [-] the engine load. This model yields a very good fit with experimental fuel
(3)
consumption measurements ( 0.99). However, the calibration of this model requires the gathering of in-field vehicle fuel consumption data.
A combination of dynamics and consumption model specifies the optimal velocity : f &", ss , ' ( % %
opt
0,
(4)
where ss , [W] is the necessary power to maintain a steady state (constant) velocity on a road with
slope . The optimal velocity minimizes the fuel consumption per traveled distance.
2.2. Cost Function
An eco-cruise control system allows a deviation from the reference velocity ref in a specified velocity
band min max in order to minimize the fuel consumption levels. Given a certain distance )e [m]
to travel, this results in the following optimization: min
, ,
, - ), e
-
f ,
(5)
with - [kg/m] the integral cost function. The cost function of equation (5) will ensure that the cruise
controller drives the vehicle around the optimal velocity (bounded by min and max ). This may result in 3
an average velocity that can differ strongly from the reference velocity. For some drivers, this can be considered as unwanted behavior. One could prefer that the cruise controller allows the velocity to fluctuate around the reference velocity, resulting in an average velocity that more or less equals the reference velocity. Different cost functions are used in the literature to tackle the above-mentioned problem. For example Latteman et al. (2004) penalize a deviation from the reference velocity: -
f . ref / .
(6)
f 0 ,
(7)
One can also use a time constraint in order to make the average velocity equal the set velocity: -
where 0 [kg/s] is determined iteratively (Hellström et al., 2010). This would be a computationally
demanding task. An easier solution is to use an indirect time constraint and determine 0 such that the
steady-state velocity on a level road equals the reference velocity: f &", ss , ' 0 ( % %
ref ,
0.
(8)
This can be interpreted as an alteration of the fuel consumption model, such that the new optimal velocity equals the reference velocity on a level road. This new optimal velocity is defined as the steady-state
cruising velocity ss [m/s] and depends on the road slope . On a level road this velocity equals the
reference velocity: ss 0 ref.
2.3. Solution Method The optimal velocity profile is the result of the following optimal control problem: min
subject to:
, ,
, - ) , e
/ / / , ) min max . 4
(9)
Generally speaking, there are three basic approaches to address optimal control problems: 1) dynamic programming, 2) direct methods, and 3) indirect methods (Pontryagin’s maximum principle). This paper compares these different methods for use in ECC. The rest of this section explains these methods in more detail. 2.3.1. Discrete Dynamic Programming Dynamic programming (DP) uses the principle of optimality of subarcs to compute recursively a feedback control (Bellman, 1957). It can easily handle non-convex control problems and inequality constraints. Here, DP on a discretized state grid is used.
To solve the eco-cruise control problem, the total distance )e is divided into : steps of ∆) [m] and the
allowed velocity band is discretized with a grid size of ∆ [m/s]. The basis of DP is a backward or forward iteration. In the case of a backward iteration consider < : / 1, : / 2, …: @) , min&A) , , @) " , ', !
(10)
where @ [kg] is the cost-to-go function that gives the remaining minimum cost to go from a given distance
) and velocity to the end. The transition cost A) , , [kg] gives the cost to go from at ) to at ) " , with a multiple of ∆. Here, this transition is calculated using the integral cost
function - and equation (1) with the trapezoidal rule for integration.
DP on a discretized state grid is used for eco-driving by e.g. Hellström et al. (2010) and Saerens et al. (2010). 2.3.2. Direct Multiple Shooting Direct multiple shooting (DMS) transforms the optimal control problem of equation (9) into a new optimization problem (Bock and Plitt, 1984). The control is discretized on a distance grid: )
