Complexity Issues in Fuel Economy Optimization of Advanced Automotive EnginesJanuary 14, 2006
Figure 1. (a) Brake specific fuel consumption (BSFC) in kg/kWh at a fixed speed and torque operating point versus intake and exhaust cam timing; (b) BSFC versus intake cam timing and spark advance.
Recent steady increases in sales of automotive vehicles in the U.S., where 16 to 17 million vehicles are now sold annually, provide an opportunity to magnify improvements in fuel economy and emissions achieved with new powertrain technologies.
Driven in their quest for improved fuel economy by the federal Corporate Average Fuel Economy standard, as well as by the desire to benefit customers directly, automakers have taken two routes. The first is development of new propulsion systems, such as hybrid electric, clean diesel, or, more recently, homogeneous charge compression ignition engines. These new technologies improve fuel economy by about 30%, but the cost is higher than for the conventional ones and can result in different vehicle drivability.
The second route is incremental improvements in fuel economy via the addition of auxiliary devices to conventional engines. Features of these advanced engines include exhaust gas recirculation valves, intake and/or exhaust variable cam timing (VCT), displacement on demand, and/or cam profile-switching systems. Each of these devices increases complexity, which makes the control system's task of realizing expected benefits more difficult.
This article addresses engine modeling (via steady-state mapping) and optimization aspects of the control system design. In a conventional gasoline engine, speed and torque are constrained by driver demand, and the air–fuel ratio is regulated to stoichiometry (equal to 14.6 for gasoline) to minimize emissions with three-way catalysts. The only optimization parameter, therefore, is spark timing.
For the advanced engine considered in this article, two independently controlled optimization parameters are added: intake and exhaust VCT. Figure 1(a) depicts fuel consumption at one speed/torque operating point as intake (IVO) and exhaust (EVC) cam timing vary. The circle denotes a typical choice for nominal cam timing, the combination that provides a tradeoff between requirements at different operating conditions. Fuel economy can be improved by more than 5% by varying cam timing and using the appropriate spark setting. The surface shown in Figure 1(b) is obtained by fixing the exhaust cam timing at 30 degrees, and plotting fuel consumption versus intake cam timing and spark advance. The bold curves in Figures 1(a) and 1(b) correspond to optimal spark timing (called MBT) and represent the same set of points.
The response surfaces in Figure 1 are generated via a "spark sweep," a set of measurements at 8 to 10 spark timings around MBT, at each intake/exhaust cam timing grid node, with the engine speed and torque held constant. Obtaining the engine model by this "full-factorial" approach is expensive and time consuming, as the process described above needs to be repeated for many additional speed–torque combinations. Because a single measurement can take several minutes, the curse of dimensionality strikes much earlier for modeling than for computation. Faster computers do not help--accurate measurement of fuel consumption can be obtained only when the transients have died down, and this is governed by time constants of the thermal response of the engine.
The standard approach to mapping complexity is to use design-of-experiments (DOE) techniques. In general, these techniques provide acceptable performance with a fraction of the measurements needed for the full-factorial method. The smaller number of carefully selected mapping points are fit with response surfaces, which are then used to find optimal parameter settings. In a DOE mapping experiment simulated on our engine data set, about two thirds of the fuel economy benefit is achieved with only 16% of the measurements needed for the full-factorial approach. Nevertheless, recovering the last one third of the benefit within the DOE framework has proved challenging. In particular, it is well known that DOE methods provide lower accuracy at the edges of the operating space. As it happens, most of the best BSFC points in our engine are found at extreme values of cam timings, as illustrated in Figure 1.
An alternative approach is to try to find the optimum directly, by a search in the parameter space. We have experimentally tested several gradient search methods, including simultaneously perturbed stochastic approximation, persistently exciting finite differences, and modified Box and Wilson search. In each case, an approximate gradient direction was found from several successive "rough" measurements requiring only several seconds each. The parameters were then updated in the direction of the negative (approximate) gradient. The idea behind this approach is that in a steep region of the BSFC surface, large gradients dominate measurement inaccuracies, and in a flat BSFC region, thermal equilibria change little between adjacent points so that thermal transients have no noticeable effect. Even if a wrong direction is occasionally picked, subsequent steps will correct the course and ensure that BSFC will decrease on average. Figure 2 shows a typical optimization run with the modified Box and Wilson gradient search method at the same speed and torque point shown in Figure 1. The initial point is at the nominal cam timing (the circle in Figure 1(a)), and the minimum coincides with the lowest BSFC point in Figure 1(a).
Figure 2. Optimization of BSFC by the modified Box and Wilson gradient search method.
As shown in Figure 2, finding optimal parameter combinations takes about 15 minutes, which is the time required to obtain about half of a spark sweep. This method provides the optimal parameter combination in a small fraction of the time a conventional mapping method would take. But this is not the only information the control system utilizes: To operate an engine during transients, more information is needed because of the different speeds of the actuator response, which is in general slower than the change in operating conditions. In particular, the MBT spark is needed at every cam timing combination in which the actuators can find themselves during transients. Obviously, a gradient search method does not provide this information.
What gradient search methods can provide in a short time is the scatter plot of optimal points in the optimization space, each at a different operating condition. For our example, it turns out that practically all of the points fall on two lines in the intake–exhaust cam timing plane, most of them on the line defined by EVC = 40 degrees. Thus, we restrict the mapping to the two lines, and allow optimal points to be selected only on those lines. Simulated DOE mapping showed about 84% of the steady-state fuel economy benefit with only about 13% of the measurements of the full-factorial approach. To interpolate between the lines as required for transient accuracy, we selected a kernel-based method, a version of "inverse distance" interpolation. Experimental testing and analysis have not shown any perceptible loss in fuel economy from transient inaccuracy.
The key point to be made is that a better accuracy–efficiency tradeoff can be achieved when the mapping and optimization steps are interlaced. Conventional mapping methods treat the engine as a black box and start optimization only after the mapping step has been completed. The caveat is that these ideas have been tested only on one engine hardware configuration, and an important question remains: To what extent is the approach scalable and portable to other mapping, modeling, and optimization problems?
Mrdjan Jankovic is a researcher in the Scientific Research Laboratories at the Ford Motor Company.