Energy Management Strategy Design for Plug-in Hybrid Electric
Vehicles with Continuation/GMRES Algorithm*
Jiangyan Zhang
†,‡
and Tielong Shen
‡
Abstract— This paper introduces applications of a numerical
method for realizing real-time optimization for energy manage-
ment of plug-in hybrid electric vehicles (PHEVs). Without pre-
vious information of driving route, on-line power-split decision
of driver demand power is formulated as nonlinear receding
horizon control (RHC) problem. Fuel economy optimization for
both of power-split and parallel PHEVs is investigated. The
Continuation/GMRES (generalized minimum residual) algo-
rithm is applied to solve the proposed nonlinear RHC problems
for real-timeness. Testing results of the proposed strategies are
demonstrated by using GT-Suite HEV simulators.
I. INTRODUCTION
Development of effective energy management strategies
for hybrid electric vehicles (HEVs) is essential for further
improving the inherent advantages in terms of fuel economy,
emission, etc. A number of strategies have been proposed
to deal with this issue in hybrid electric vehicle (HEV)
control [1]. In addition to rule-based strategies, significant
efforts are paid to develop optimal control strategies to
achieve the goals of minimizing the fuel consumption and/or
emissions. The optimal energy management of HEV aims to
minimizing the fuel consumption with all constraints in terms
of the physics of the powertrain structure and the vehicle
driveability. Generally, the problem formulation is nonlinear
with extensive constraints. These cause challenges in solving
the faced optimal control problems, especially, when real-
time online implementation is anticipant by the automotive
engineering [2].
Using analytical approaches, including the dynamic pro-
gramming algorithm and the Pontryagin’s minimum prin-
ciple, can provide global optimal solutions which however
cannot be directly applied to adaptive the real driving con-
ditions [1]. Model predictive control (MPC) algorithm can
be applied on-line by predicting the future driving pattern
according to real-time responses from the driver and the route
information. However, analytical nonlinear model predictive
control (NMPC) approaches are usually suitable for special
cases [3]. Hence a general NMPC approach deserves to be
investigated to deal with the optimal control problems for
the HEV control.
On the other hand, PHEV has received increased inter-
esting due to the distinctive advantages in the all electric
*The first author of this work is supported by the National Natural Science
Foundation of China (No. 61304128)
†
College of Electromechanical and Information Engineer-
ing, Dalian Nationalities University, Dalian 116600, China
zhangjy@sophia.ac.jp
‡
Department of Engineering and Applied Sciences, Sophia University,
Tokyo 102-8854, Japan
tetu-sin@sophia.ac.jp
range and the charge depleting operation and the consequent
fuel economy [4]. Moreover, charge sustaining operation is
required when the battery state of charge (SoC) achieves the
low limiting value during a driving cycle. The distinctive
characteristic in terms of operating modes switching makes
the power management more complex in PHEVs than in
HEVs [4].
This paper gives a review based on the work [5], [6].
NMPC strategies are proposed to deal with energy manage-
ment issues for power-split and parallel PHEVs, respectively.
The formulated optimal control problems are solved by
employing the Continuation/GMRES algorithm [7]. By using
two HEV simulators built in the GT-Suite Software, valida-
tion results of the proposed control strategies are presented.
II. OVERVIEW OF RECEDING HORIZON
CONTROL
In this section, a brief review of RHC technique
is shown, especially introducing more details of the
Continuation/GMRES-based nonlinear RHC approach.
Consider the dynamical system
˙x(t)= f (x(t),u(t), p(t)) (1)
with state vector x ∈ R
n
, control input vector u ∈ R
m
and
vector of other external input p ∈ R
p
. The RHC design is to
find the control u by minimizing a defined cost functional J
over a fixed moving time horizon T (> 0), that is
u(t)=argminJ(x(t),u(t), p(t)) (2)
with
J(x(t),u(t), p(t)) =
ϕ
(x(t + T ), p(t + T ))+
t+T
t
L(x(t
),u(t
), p(t
))dt
(3)
where
ϕ
denotes a terminal cost and L denotes the per-
formance index. Moreover, the constraints in terms of the
states x and control inputs u (i.e. x(t) ∈ X and u(t) ∈ U)
are handled. Then, the RHC scheme solving the above
constrained optimization problem using the predicted future
cost by using system dynamics (1), that is why it is also
known as MPC.
The design result of RHC is a generated control sequence
over the predictive horizon T at each time step. For linear
systems with quadratic cost functional, the standard linear
quadratic (LQ) approach provides a general framework for
solving the optimal control problems [8]. However for non-
linear systems, there are proposed analytical methods which
are applicable for particular systems only. For example,
2015 European Control Conference (ECC)
July 15-17, 2015. Linz, Austria
978-3-9524269-4-4 © EUCA 2969