Energy management for series-production plug-in-hybrid electric vehicles based on predictive DP-PMP

Appendix A Convergence analysis

This section provides a convergence analysis of the optimization strategy of subsection 4.3. For this we consider the update law shown in Equation 37:

To visualize the central aspects of the following discussion, Figure 11 shows an exemplary depiction of the resulting terminal state-of-energy values resulting for a set of predefined, constant co-states. In this context, two cases are considered, the control of the engine on/off decision in combination with the power split by combining PMP with DP (left part of Figure 11) and a sole control of the power split based on PMP (right part of Figure 11).

Considering the PMP-DP-Solution in the left part of Figure 11, one may notice that there exist terminal state of energy values which may not be reached for a given constant co-state. This effect can be explained by the fact, that for certain engine operation points a transition of the electrical driving decision occurs at a DP decision boundary present for a co-state s∗ when

with Cequ as the equivalent cost of fuel and electrical energy presented in Equation 29. Given the present transition of the electrical driving decision, this will lead to different terminal SOE-values for the co-states on both sides next to s∗ (as depicted in the left part of Figure 11). By applying Equation 27 and Equation 28, Cequ can be written as

with

and as

Due to the convex modelling of Pf (compare Equation 27) and the physical requirement given during pure electrical driving it applies that γ1<0, γ2>0, PS,off>0, A<0 and B>0. Consequently Cequ(x2,k,u2,k=1) is concave and Cequ(x2,k,u2,k=0) is linearly increasing. Hence there exists an intersection point at a boundary value s∗ where a transition in the electrical driving decision occurs. Consequently for increasing co-states, less electrical energy will be applied what results in increased terminal SOE-values (compare Figure 11 and Figure 12).

Figure 12

Exemplary drawing of the resulting equivalent costs.

Accordingly the convergence of the update law can be shown as follows: We first consider the case that si<s∗, the actual value of si is located on the left side of the DP decision boundary as depicted in Figure 11. Accordingly, si results in a terminal SOE value which is smaller than the desired terminal SOE target. To compensate the resulting deviation from the terminal SOE target, the recalculated co-state scomp has to be greater than the actual value of si (scomp>si) (compare the right part of Figure 11 where SOE(s) is monotonically increasing in s). This implies, that scomp−si>0. Consequently, given a value of κ>0, the applied update law increases the value of the updated co-state si+1. Accordingly one can state, that in theory, there exists a value of κ∗>0 (which is unknown in practice), which directly results in the co-state s∗. Further, values of κ∈]0,κ∗[ will result in an updated co-state si+1∈]si,s∗[ and values of κ>κ∗ will result in a co-state si+1>s∗. Consequently we need to distinguish two cases in the update steps: If si<s∗, then we increase s and approach s∗ from the left. If si>s∗ then we decrease s and approach s∗ from the right. Hence one can summarize, that for a sufficiently small choice of the value of κ, the value of s converges to an arbitrary small neighbourhood of the value s∗. (Note: If the target SOE does not result in a DP decision boundary, the terminal SOE value is reachable and s converges to the optimal value.)

In practice these aspects result in a trade-off between computationally efficiency (due to the number of iterations required through the choice of κ) and accuracy in approaching the value of s∗ which needs to be considered during the application process of the algorithm.