Abstract
One of the main challenges during the development of operating strategies for modern diesel engines is the reduction of the CO2 emissions, while complying with ever more stringent limits for the pollutant emissions. The inherent trade-off between the emissions of CO2 and pollutants renders a simultaneous reduction difficult. Therefore, an optimal operating strategy is sought that yields minimal CO2 emissions, while holding the cumulative pollutant emissions at the allowed level. Such an operating strategy can be obtained offline by solving a constrained optimal control problem. However, the final-value constraint on the cumulated pollutant emissions prevents this approach from being adopted for causal control. This paper proposes a framework for causal optimal control of diesel engines. The optimization problem can be solved online when the constrained minimization of the CO2 emissions is reformulated as an unconstrained minimization of the CO2 emissions and the weighted pollutant emissions (i.e., equivalent emissions). However, the weighting factors are not known a priori. A method for the online calculation of these weighting factors is proposed. It is based on the Hamilton–Jacobi–Bellman (HJB) equation and a physically motivated approximation of the optimal cost-to-go. A case study shows that the causal control strategy defined by the online calculation of the equivalence factor and the minimization of the equivalent emissions is only slightly inferior to the non-causal offline optimization, while being applicable to online control.
Highlights
Today, almost 17% of the carbon dioxide (CO2 ) emissions are caused by road transportation [1].By 2035, the global number of road transport vehicles is expected to have almost doubled compared to 2009 [2]
The optimal solution is a repetition of the optimal solution of the single section. The reason for this result is the fact that the only state variable represents the cumulated nitrogen oxide (NOx) emissions, while the rest of the model does not depend on that state variable
The plot shows the full range of attainable NOx emissions, the upper limit corresponding to λNOx = 0 and the lower limit corresponding to λNOx = ∞
Summary
Almost 17% of the carbon dioxide (CO2 ) emissions are caused by road transportation [1]. Because the cumulated emissions at the end of the cycle are constrained, it is always necessary to consider the full driving cycle in the optimization This problem can be solved by removing the final state constraint and including the weighted pollutant emissions in the performance criterion. In [12], the weighted NOx emissions are included in an optimization of the operating costs of a diesel engine. Weighting factor can be obtained by solving the non-causal optimal control problem In both cases, the control system is sensitive to disturbances and model errors. A simple, yet effective method for the online adaptation of the weighting factors is derived Using these weighting factors, the equivalent emissions can be minimized online to realize a causal optimal operating strategy.
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