Component sizing of a plug-in hybrid electric powertrain via convex optimization

Abstract

This paper presents a novel convex modeling approach which allows for a simultaneous optimization of battery size and energy management of a plug-in hybrid powertrain by solving a semidefinite convex problem. The studied powertrain belongs to a city bus which is driven along a perfectly known bus line with fixed charging infrastructure. The purpose of the paper is to present the convexifying methodology and validate the necessary approximations by comparing with results obtained by Dynamic programming when using the original nonlinear, non-convex, mixed-integer models. The comparison clearly shows the importance of the gear and engine on/off decisions, and it also shows that the convex optimization and Dynamic Programming point toward similar battery size and operating cost when the same gear and engine on/off heuristics are used. The main conclusion in the paper is that due to the low computation time, the convex modeling approach enables optimization of problems with two or more state variables, e.g. allowing for thermal models of the components; or to include more sizing variables, e.g. sizing of the engine and the electric machine simultaneously.

Real-Time Distributed Economic Model Predictive Control for Complete Vehicle Energy Management

“…As in [35], only the evaporator in the climate control system is modeled, which is assumed to satisfy a set of coupled thermal energy balances given by: (11) where C r and C w are the heat capacities of the refrigerant and the walls of the evaporator, respectively, T r (t) and T w (t) are the temperatures of the refrigerant and walls of the evaporator, respectively, u ccs is the effective cooling power from the compressor, T amb is the ambient temperature, h i and h o are the heat transfer coefficients between the inner and outer walls of the evaporator, respectively and Q l is the heat generated when the inlet air is condensed (latent heat), which is assumed to only depend on the ambient air temperature and humidity Φ amb . Similar to the battery, we can represent the climate control system model in terms of stored energy by defining the thermal energy in the wall and refrigerant relative to the ambient temperature, i.e,…”

“…The proposed solution strategies can be divided into so-called offline and online solution strategies [2,3]. Offline solution strategies have been developed based on, e.g., dynamic programming (DP; see, e.g., [4][5][6]), Pontryagin's minimum principle (PMP; see, e.g., [7][8][9][10]) or convex optimization (see, e.g., [11,12]). The offline solution strategies require all disturbances to be known (e.g., the driving cycle) so that the global optimal solution can be computed and can therefore not be implemented in real time.…”

Convex Optimization for the Energy Management of Hybrid Electric Vehicles Considering Engine Start and Gearshift Costs

“…It is seen as an alternative method for the optimization of the power flows in HEVs with the advantage of being computationally more efficient than for example DP or indirect methods such as PMP are. Convex optimization has already been employed for dimensioning of powertrains [28][29][30], energy efficiency analysis [31] and online control of HEVs [32,33]. However, inherently discrete decision variables, such as the engine on/off decision or the gear decision, are typically determined using heuristic approximations.…”

“…Therefore, in the DP, PMP and SDP approaches, researchers [26,[37][38][39][40][41] have introduced penalties or costs for the changes in the engine or the gear state such that unacceptably frequent starts and shifts can be prevented. To do so in convex optimization-based approaches, heuristics based on signal filtering have been presented so far [29]. However, a corresponding optimal solution for methods relying on convex optimization has yet to be found.…”

“…In order to apply the concepts presented in Section 4 below, the generic, nonlinear vehicle model has to be approximated by a convex model, see [27] for a general introduction to convex optimization, and [29,45] for an introduction to convex modeling and optimization of HEVs.…”