Economic MPC for online least costly energy management of hybrid electric vehicles

Abstract

In this work, the problem of online energy management of hybrid electric vehicles is addressed. A least costly objective function accounting for battery energy consumption and aging, and for the auxiliary power unit fuel consumption and noise emissions is considered. In this scenario, all the cost terms are expressed as monetary variables. This allows to assess the economic effectiveness of the proposed hybrid powertrain solution. Therefore, the online energy management policy is computed relying on the economic model predictive control framework. Some dissipativity properties for steady-state and periodic operation of the system under investigation are proved. Therefore, some results for close to optimum convergence of the economic model predictive control are provided. An electric bus case-study is illustrated in detail to show the performance of the proposed online management strategy.

Introduction

In the last years, the quest for vehicular emissions reduction, fuel economy improvement, and energy efficiency have led automotive industries to devote time, effort, and money for the development of new powertrain solutions for the next generation of vehicles. In this scenario, Electric Vehicles (EVs) have become effective means for urban mobility thanks to the high efficiency, the absence of local emissions (Helmers & Marx, 2012), and the low price of the electrical energy. One of the obstacles limiting the spread of this technology is the low energy density of batteries if compared to carbon fuels (Warner, 2015), which leads to reduced all electric ranges. Thus, a common pathway to resolve this issue is the introduction of an auxiliary power unit called Range EXtender (REX). This device is composed of a diesel Internal Combustion Engine (ICE) coupled with an Electric Generator (EG). The internal combustion engine delivers mechanical power, which is converted into electrical power by the electric machine and then used for the vehicle motion. In this scenario, the architecture composed of the REX and the EV is a series hybrid electric powertrain (also known as Extended Range Electric Vehicle (EREV)) where two movers are available: the REX and the Li-ion battery.

It is widely recognized that, to evolve the standard ICE technology, hybrid electric vehicles come to hand. As a matter of fact, the introduction of multiple power sources allows for the improvement of the fuel economy and for the reduction of pollutants emissions (Sciarretta & Guzzella, 2007). However, to benefit from hybrid architectures it is of paramount importance to introduce effective Energy Management Strategies (EMSs) aiming at optimizing the power split between the available movers. In accordance with Onori, Serrao, and Rizzoni (2016), the EMS development may follow two principal routes: the rule-based one or the model-based optimization one. Rule-based strategies have been widely used for real-time implementation. In this scenario, heuristics are employed to choose the best control action at each time instant. The computational effort is reduced to its minimum, however, no optimality of the solution is guaranteed (see, e.g., Banvait et al., 2009, Jalil et al., 1997, Lin et al., 2001). In model-based optimization methods the EMS problem is recast as an optimal control problem. Therefore, the optimal energy management policy is retrieved minimizing a suitable objective function. To this end, different optimization methodologies can be adopted. Solutions based on dynamic programming are proposed by Pérez, Bossio, Moitre, and García (2006) and Sundstrom and Guzzella (2009). Moreover, in Elbert et al., 2014, Murgovski, Johannesson, Hellgren, Egardt, and Sjöberg, 2011, and Murgovski, Johannesson, Sjöberg, and Egardt (2012) solutions to the EMS problem based on convex programming tools are discussed. These policies are computed in a non-causal fashion and the optimization over the entire driving cycle is carried out offline. Eventually, real-time approximations of the non-causal optimal policies are obtained relying, e.g., on the equivalent consumption minimization strategy, a Pontryagin’s minimum principle-based heuristic (Musardo et al., 2005, Pisu and Rizzoni, 2007). The general drawback of model-based optimization approaches is the necessity to design a proper cost function with appropriate weights to balance the different cost terms.

In this work, the problem of online EMS development for hybrid electric vehicles is addressed. According to Pozzato et al., 2018, Pozzato et al., 2019, Pozzato et al., 2020, a least costly objective function is introduced. This formulation allows to express all the cost terms as monetary variables. Therefore, the solution of the EMS optimal control problem returns the overall vehicle operating cost, which is useful to assess the economic effectiveness of the proposed mobility solution. In this scenario, a control-oriented model for the EREV is first described. The REX provides power continuously and its usage is function of some idling heuristics. Then, the REX is characterized in terms of noise emissions, allowing for high penalization of unpleasant operating conditions. Thus, the EMS problem is formalized as a mixed-integer convex program with the battery modeled as a convex difference inclusion. The EMS problem is then solved online using an Economic Model Predictive Control (EMPC) framework. EMPC is a particular case of model predictive control where general, possibly economically motivated, performance criteria are considered (Faulwasser, Grüne, Müller, et al., 2018). As usual in Model Predictive Control (MPC), the control input at each time step is computed by solving a finite-horizon optimal control problem and then applying the first component of the computed optimal predicted input sequence. In the context of economic MPC, dissipativity properties have turned out to play a crucial role.¹ Namely, dissipativity allows to characterize the optimal operating behavior of a system (with respect to the given performance criterion) and to analyze closed-loop convergence, compare Faulwasser et al., 2018, Müller et al., 2015. Available results in the literature establish dissipativity properties for certain types of difference equations (see, e.g., Berberich et al., 2020, Damm et al., 2014).

Against this background, the paper is not only focused on providing an EMPC solution for the EMS problem of EREVs, but also to derive dissipativity properties that allow to conclude closed-loop performance and convergence results of EMPC schemes for more general system classes. Therefore, the main innovative contributions of the paper are summarized as follows:

• On the one hand, considering generic dynamical systems described by convex difference inclusions (which include also the battery energy model (23) of the EMS problem), dissipativity is proven for both cases of optimal steady-state and periodic operation. Proving this property allows to retrieve important information on the performance of a generic EMPC for such systems;

• On the other hand, the implications of the aforementioned dissipativity results for the EMS problem at hand are assessed. Moreover, for the first time the online least costly solution is computed rewriting the EMS problem in the EMPC framework. This allows to obtain an online energy management strategy with convergence guarantees given by dissipativity.

The remainder of the paper is organized as follows. First, in Section 2 a backward powertrain model is introduced. Therefore, Section 3 describes the energy management strategy problem for the EREV. Then, theoretical results on dissipativity for steady-state and periodic operation of systems modeled by convex difference inclusions are discussed in Section 4. In Section 5, the validity of such dissipativity conditions and of their implications is shown, in a simulation environment, for the particular case of an extended range electric bus. Finally, in Section 6 some final remarks are carried out.