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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 3, MAY 2015 1197
Velocity Predictors for Predictive Energy Management
in Hybrid Electric Vehicles
Chao Sun, Xiaosong Hu, Member, IEEE, Scott J. Moura, Member, IEEE, and Fengchun Sun
Abstract— The performance and practicality of predictive
energy management in hybrid electric vehicles (HEVs) are highly
dependent on the forecast of future vehicular velocities, both in
terms of accuracy and computational efficiency. In this brief, we
provide a comprehensive comparative analysis of three velocity
prediction strategies, applied within a model predictive control
framework. The prediction process is performed over each
receding horizon, and the predicted velocities are utilized for
fuel economy optimization of a power-split HEV. We assume
that no telemetry or on-board sensor information is available for
the controller, and the actual future driving profile is completely
unknown. Basic principles of exponentially varying, stochastic
Markov chain, and neural network-based velocity prediction
approaches are described. Their sensitivity to tuning parameters
is analyzed, and the prediction precision, computational cost, and
resultant vehicular fuel economy are compared.
Index Terms—Artificial neural network (NN), comparison,
energy management, hybrid electric vehicle (HEV), model
predictive control (MPC), velocity prediction.
I. INTRODUCTION
SOPHISTICATED energy management strategies have
been developed to provide better fuel economy perfor-
mance in hybrid electric vehicles (HEVs) [1], [2]. This brief
intends to facilitate the performance of predictive energy
management through evaluating different horizon velocity
predicting approaches.
In the literature, dynamic programming (DP) and equiva-
lent consumption minimization strategy (ECMS) are crucial
in resolving the energy management problem for HEVs.
Globally, DP can ensure an optimal result when complete
knowledge of driving conditions is prescribed [3]. However,
the exact future power demand is usually unknown and the
computational burden is prohibitive. The DP solutions are
often realized offline and deployed as benchmarks [2]. The
ECMS is an instantaneous optimization for HEV energy
management [4], [5]. By defining an equivalent fuel cost
for battery energy, ECMS solves the optimal power split at
each time instant rather than over a time horizon. It has
Manuscript received May 9, 2014; accepted September 5, 2014. Date
of publication October 7, 2014; date of current version April 14, 2015.
Manuscript received in final form September 12, 2014. Recommended by
Associate Editor F. Vasca.
C. Sun was with the National Engineering Laboratory for Electric Vehicles,
Beijing Institute of Technology, Beijing 100081, China. He is now with the
Department of Mechanical Engineering, University of California at Berkeley,
Berkeley, CA 94720 USA (e-mail: chaosun.email@gmail.com).
X. Hu and S. J. Moura are with the Department of Civil and Environmental
Engineering, University of California at Berkeley, Berkeley, CA 94720 USA
(e-mail: xiaosong@chalmers.se; smoura@berkeley.edu).
F. Sun is with the National Engineering Laboratory for Electric
Vehicles, Beijing Institute of Technology, Beijing 100081, China (e-mail:
sunfch@bit.edu.cn).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCST.2014.2359176
demonstrated that given an appropriate equivalence factor,
ECMS is comparable with DP [6]. Nevertheless, tuning the
equivalence factor is nontrivial. The ECMS variants, such as
telemetric-ECMS and adaptive-ECMS, are proposed to adjust
the equivalence factor based on information from telemetry
equipment or on-board sensors [7], [8]. Due to the uncertainty
of future driving profiles, endowing the controller with an
appropriate prediction ability can achieve better performance.
The forecast ability in the adaptive-ECMS is different from the
model predictive control (MPC) approach used in this brief.
The MPC has attracted increasing attention in the HEV
energy management research community. Given a finite
moving horizon, an MPC controller can maintain computa-
tional load within a practical range. An MPC controller solves
a short-term energy management problem at each time step,
via nonlinear programming [9], quadratic programming [10],
Pontryagin’s minimum principle [11], or DP [12]. The
performance of MPC strongly depends on the power reference
provided in each prediction horizon. The precision of future
power prediction is instrumental for the overall vehicle fuel
economy. Considering that road grade information is static, we
focus on the horizon velocity prediction. No telemetry devices
or environment detecting sensors are assumed, and the future
driving information is completely unknown.
We investigated three velocity predicting approaches. In [9],
we assumed power demand decreases exponentially over the
prediction horizon. The aim of this simple method was to
provide an intuitive understanding of how velocity predic-
tion affects fuel economy. To systematically investigate this
method, a generalized exponentially varying velocity predictor
is considered in this brief.
Markov-chain models are widely used for vehicular velocity
modeling [13], [14]. An MPC controller with a stochastic
Markov-chain velocity predictor is usually called stochastic
MPC (SMPC) [12], [15]. The 1-stage Markov-chain process
has proven to be effective in generating fixed-route driving
patterns. However, when comprehensive driving tasks are
considered, the accuracy of 1-stage Markov chain may decline.
On the other hand, the predicted power demand relies not
only on the present vehicle states, but also on the historical
values [16]. Typically, the more historical data used, the more
accurate the prediction. For SMPC, multistage Markov-chain
processes can hence be formulated to enhance the velocity
prediction accuracy.
Artificial neural networks (NNs) are a successful method for
time series forecasting [17]. Applications of NNs to predicting
city power load [18], driving handling behaviors [19], and traf-
fic flows [20] have verified its strong capability in predicting
nonlinear dynamic behaviors. In [21], we also employed NN
to predict the road type and traffic congestion to improve the
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1198 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 3, MAY 2015
HEV fuel efficiency. Although tuning the NN parameters may
produce different prediction performances, it is easy to design
an NN-based velocity predictor with reasonable precision.
To the best of our knowledge, NN-based velocity predictor
for predictive energy management of HEVs has not been
investigated.
In this brief, we present a comparative study of three classes
of velocity predictors for MPC-based energy management of
a power-split HEV. This brief adds two original contributions
to the related literature. First, an NN-based velocity predictor
is implemented, for the first time, in MPC-based HEV energy
management. Three different NN structures are examined in
terms of prediction accuracy and consequent fuel economy.
Second, the generalized exponentially varying and Markov-
chain velocity predictors are investigated to fully explore
their potential. The above three classes of velocity predictors
are systematically compared in terms of prediction precision,
computation time, and consequent fuel consumption. Although
the foregoing contributions are made specifically for MPC-
based energy management of a power-split HEV, the proposed
approach extends to other HEV configurations.
The remainder of this brief is organized as follows.
In Section II, the problem formulation is introduced, including
the control-oriented vehicle model and nonlinear MPC opti-
mization strategy. Section III details the three different types of
horizon velocity predictors. Comparison results are illustrated
in Section IV. Finally, the conclusion is drawn in Section V.
II. PROBLEM FORMULATION
A. Control-Oriented Power-Split HEV Model
Power-split HEVs are dominant in the current HEV
market [14]. In a planetary gear set functioned power-split
HEV, the engine and motor/generator 1 (MG1) are connected
to the planet carrier and the sun gear, respectively. A torque
coupler is used to combine the ring gear with MG2 to power
the final drive. The kinematic constraint on the ring, carrier,
and sun gear angular velocities is given mathematically by
ωsS+ωrR=ωc(S+R)(1)
where Sand Rare the radii of the sun gear and the ring gear,
respectively. Angular speeds of the ring, sun, and carrier gears
are denoted as ωr,ωs,andωc, respectively. By neglecting
the inertia of the pinion gears and assuming that all the
powertrain shafts are rigid, inertial dynamics of the powertrain
is derived as
JMG1 dωMG1
dt =TMG1 +FS (2)
Jeng dωeng
dt =Teng −F(S+R)(3)
JMG2 dωMG2
dt =TMG2 −Taxle
gf+FR (4)
where JMG1,Jeng,andJMG2 are inertias of MG1, engine
and MG2, respectively; Teng =Tcis the engine torque,
TMG1 =Tsand TMG2 =Trare MG1 and MG2 torques
(positive in motoring mode), respectively; Frepresents the
internal force on pinion gears; gfis the gear ratio of the
final drive; and Taxle is the torque produced from powertrain
on the drive axle. To reduce the control-oriented model’s
complexity, we disregard the inertial dynamics, and use the
steady-state values of (2)–(4).MG2 torque and vehicle velocity
are given by
ωMG2 =gf
Rwheel V(5)
mdV
dt =Taxle +Tbrake
Rwheel +mgsin(θ) −1
2ρACdV2
−Crmg cos(θ) (6)
where Rwheel is the wheel radius; Vis the vehicle velocity;
mis the vehicle mass; Tbrake is the friction brake torque; gis
the gravitational acceleration; θdenotes the road grade and
is assumed to be zero; (1/2)ρ ACdis the aerodynamic drag
resistance; and Crrepresents the rolling resistance coefficient.
At each time instant, the controller computes an optimal
split between the engine, MG1 and MG2 to minimize fuel
consumption. Fuel flow rate of the engine ( ˙mfuel) and power
transfer efficiencies for MG1 and MG2 (ηMG1 and ηMG2)are
extracted from empirical maps
˙mfuel =ψ1(ωeng,Teng )(7)
ηMG1 =ψ3(ωMG1,TMG1)(8)
ηMG2 =ψ2(ωMG2,TMG2)(9)
where ψ1,ψ2,andψ3are corresponding empirical maps.
Upper and lower boundaries of the battery state of
charge (SoC) need to be specified to maintain the battery
within a safe operating region and extend its life [22]. For
HEV energy management, the SoC is modeled as a single
state. Although more sophisticated battery models have been
developed to describe battery dynamics [23], [24], the internal
resistance model is still the most prevailing model for HEV
supervisory control due to its simplicity, described as
˙
SoC =−
Ibatt
Qmax (10)
Pbatt =Voc Ibatt −I2
batt Rbatt (11)
where Ibatt and Qmax are battery current and maximum
capacity, respectively; Pbatt and Rbatt are battery power and
internal resistance; and Voc represents the open circuit voltage.
Positive Pbatt denotes discharge. The battery in a power-split
HEV is connected to a bidirectional converter to supply power
or recuperate energy from the electrical machines. Terminal
battery power is described by
Pbatt =PMG1/(ηMG1ηinv)kMG1 +PMG2/(ηMG2ηinv)kMG2 (12)
where PMG1 and PMG2 are MG1 and MG2 shaft powers,
respectively (positive in motoring mode); ηinv is the inverter
efficiency
ki=1,if Pi>0
−1,if Pi≤0for i={MG1,MG2}. (13)
A complete description of the battery SoC dynamics can be
obtained from (9) to (13). Equations (1)–(13) summarize the
control model used for MPC. Throughout this brief, MPC is
applied to a detailed plant model furnished by the QSS-toolbox
developed at ETH Zürich [25].
SUN et al.: VELOCITY PREDICTORS FOR PREDICTIVE ENERGY MANAGEMENT IN HEVs 1199
B. Nonlinear MPC
A hierarchical MPC [26] is employed for fuel consumption
minimization, formulated as a constrained nonlinear optimiza-
tion problem and solved by DP at each time step. The engine
torque and speed are selected as control variables. Denoting x
as the state variable, uas the control variable, das the system
disturbance, and yas the output, the proposed control-oriented
powertrain model can be represented as
˙x=f(x,u,d), y=g(x,u,d)
with x=SoC, u=[Teng,ω
eng]T,d=Vpredict ,y=
[˙mfuel,Pbatt,TMG1,ω
MG1,TMG2]T,whereVpredict is the future
velocity sequence provided by a velocity predictor in each
control horizon.
Considering a 1 s time step (t=1), at time step k,the
cost function Jkis formulated as
Jk=(k+Hp)t
kt
([˙mfuel(u(t))]2+λO(t)) dt (14)
where Hpis the prediction horizon length, which is herein
equal to the control horizon length [9]; O(t)is the engine
ON/OFF switching time; and λis the penalty for engine
ON/OFF switching. In addition, the following physical con-
straints must be enforced:
SoCmin ≤SoC ≤SoCmax
Tmin
eng ≤Teng ≤Tmax
eng
ωmin
eng ≤ωeng ≤ωmax
eng
Tmin
MG1 ≤TMG1 ≤Tmax
MG1
ωmin
MG1 ≤ωMG1 ≤ωmax
MG1
Tmin
MG2 ≤TMG2 ≤Tmax
MG2
ωmin
MG2 ≤ωMG2 ≤ωmax
MG2
Imin
batt ≤Ibatt ≤Imax
batt
Pmin
batt ≤Pbatt ≤Pmax
batt .(15)
For HEV structures, the final battery SoC is requested to be
the same as the initial SoC
SoC(T)=SoC(0). (16)
In this brief, the driving information of an assigned trip task
is completely unknown, which complicates the task of satisfy-
ing (16) over the global time horizon via MPC. This problem
can be addressed by approximating a cost-to-go function, or by
regulating the terminal SoC reference in each control horizon.
The second approach is used in this brief. Terminal SoC
references in all control horizons are specified to be SoC(0)
to guarantee the battery SoC converges to the initial value at
the end of the trip. This approach is conservative in utilizing
battery energy, but effective in enforcing constraint (16)
SoC(k+Hp)=SoC(0)(17)
where SoC(k+Hp)is the terminal SoC reference in control
horizon k.
The MPC controller is applied in the supervisory level of
the control architecture. The optimization task is solved using
DP when predicted vehicle velocities are provided in each
receding horizon. For compactness of notation, denote Vk=
V(kt). The control procedure is described as follows [27].
1) A horizon velocity predictor is used to estimate the
control horizon driving profile, based on current velocity
request Vkand historical velocities. Assume fpis the
velocity prediction function
Vpredict =fp(...,Vk−3,Vk−2,Vk−1,Vk)
=Vk+1,Vk+2,...,Vk+Hp−1.
2) Given Vpredict , calculate the optimal control policy
minimizing the objective function (14).
3) Apply the first element of the control policy. Feedback
states, and repeat the control procedure.
In this brief, we assume the target vehicle is equipped with-
out any radar, global positioning system or similar devices.
The road grade is zero and future driving profiles are com-
pletely unknown.
III. HORIZON VELOCITY PREDICTORS
Three horizon velocity predictors are investigated in
this brief: 1) generalized exponentially varying predictor;
2) Markov-chain-based predictor; and 3) artificial NN-based
predictor. The NN-based velocity predictor is implemented for
the first time in the HEV energy management problem, and is
compared with the other two approaches.
A. Exponentially Varying Velocity Predictors
The relationship between predicted future velocities and
total fuel consumption is complex. To obtain an intuitive
understanding of this relationship, exponentially varying future
velocities were considered in [28].
In each receding horizon, the exponentially varying horizon
velocities are formulated as
Vk+n=Vk×(1+ε)n,for n=1,2,...,Hp(18)
where Vkis the initial velocity at time step kand εis the
exponential coefficient. Different εvalues are considered to
examine the sensitivity of fuel economy to the predicted future
velocities.
B. Markov-Chain Velocity Predictors
Markov chain is an important methodology used in model-
ing driving velocities or power demands in HEVs [12]–[15].
In this brief, the Markov-chain states and emissions are defined
on discrete-valued domains given by vehicle velocity V(0
to 36 m/s) and vehicle acceleration αin (−1.6to1.6m/s
2),
respectively.
Note that the driving behavior studied in this brief is com-
prehensive in an average sense. Thus, the Markov emission
probability distribution is computed from a comprehensive
dataset. Eight different driving cycles are included in this
dataset, considering both highway and urban driving scenarios.
Six of the sample cycles are standard driving cycles and the
other two are real collected driving data. Suppose the vehicle
velocity and acceleration are discretized into pand qintervals
and indexed by iand j, respectively. Velocity at time step kis
1200 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 3, MAY 2015
Fig. 1. One-stage Markov emission probability matrix (60×60).
Vk+n−1, and next step acceleration is αk+n,wherenindexes
time in the receding horizon. The Markov-chain process is
defined by an emission probability matrix T∈Rp×qwith
[T]ij =Pr[αk+n=αj|Vk+n−1=Vi](19)
for i∈{1,..., p},j∈{1,...,q}and n∈{1,...,Hp}[29].
Suppose p=60 and q=60, the probability matrix extracted
from the sample dataset is shown in Fig. 1.
A random Markov-chain model can update its emission
probability matrix online by utilizing historical driving
data [29]. To reduce complexity and ensure fairness in the
comparison, we do not consider such adaptive Markov-chain
models in this brief. The complexity of the Markov-chain
model will also increase if more conditions, such as tempera-
ture, position, and road grade, are considered.
The precision of Markov-chain velocity predictors relies
heavily on the emission probability matrix. To increase the
prediction accuracy, 1-stage Markov-chain process can be
extended to multistage [30], using the following probabilities:
Pr[αk+n=αj|Vk+n−1=Vi1
Vk+n−2=Vi2,...,Vk+n−s=Vis](20)
where sis the Markov-chain stage number. When the stages
increase, more historical speeds are needed for computing the
probability matrix, and the number of Markov states increases
exponentially. Thus, the resolution of the emission probability
matrix can be improved, leading to higher velocity prediction
precision. In the meantime, the sample velocity dataset for
generating multistage Markov-chain processes needs to cover
all possible input states. This requirement is inevitable and
may cause both the size of the emission probability matrix
and the online computation to grow substantially.
C. NN Velocity Predictors
The NNs can be trained to learn a highly nonlinear
input/output relationship by adjusting weights to minimize the
error between the actual and predicted output patterns of a
training set [31]. Three different types of network structures
are examined in this brief: 1) back propagation (BP-NN);
2) layer recurrent (LR-NN); and 3) radial basis function
NN (RBF-NN).
A three-layer BP-NN has a hierarchical feed forward
network structure, which is also the basis of other network
architectures. The input layer is used to receive and distribute
the input pattern, followed by a hidden layer that depicts the
nonlinearities of the input/output relationship. Output layer
yields the desired output patterns. For a BP-NN, the activation
function is hyperbolic tangent sigmoid function. The basic
formula of BP-algorithm is
a1=tan sig(n)=en−e−n
en+e−nn=Wa0+b(21)
where a1and a0are the neural outputs of the current layer and
prior layer, respectively, nis the accumulator output, Wis the
weight, and bis the bias. The literature shows that a three-layer
BP-NN with a sigmoid function as the activation function can
approximate any nonlinear systems with arbitrary precision.
The main disadvantage of BP-NN is that it has a slow learning
convergence and is at risk of getting trapped into local minima.
With a self-connected hidden layer, the LR-NN has an
internal state, which allows the network to exhibit temporal
dynamic behavior [32]. However, this may increase the train-
ing convergence time. The basic formula of the recurrent layer
in a LR-NN is
a1(t)=tan sig(n(t)) =en−e−n
en+e−n
n(t)=Wa0(t)+Wa1(t−t)+b(22)
where Wis the feedback weight; a1(t−t)is the delayed
output at time (t−t).
An RBF-NN is a widely used feed forward network for time
series forecasting. In the standard approach to RBF-NN imple-
mentation, an RBF needs to be predefined at first. Then, the
number of hidden layer neurons is determined. An RBF-NN
usually has better convergence speed and performance
compared with BP-NN. In our simulation, the Gaussian
function is used as the RBF in the hidden layer to activate
the neurons, formulated as
a1=exp −n−c2
2b2n=Wa0+b(23)
where cis the neural net center and bis the spread width.
Both cand bcan be fit using a gradient descent method.
For velocity prediction purpose, the inputs of NNs are
historical velocity sequences, and the outputs are predicted
horizon velocity sequences. Each input–output pattern is com-
posed of a moving window of fixed length, which can be
expressed as
[Vk+1,Vk+2,...,Vk+Hp]= fNN(Vk−Hh+1,...,Vk)(24)
where Hhis the dimension of the input velocity sequence and
fNN represents the nonlinear map function of an NN-based
predictor. In an MPC framework, the prediction horizon
length is Hp, so that the NN velocity predicting process is
Hp-step ahead.
The size of the NN depends on the number of input nodes
and the number of hidden nodes. Here, we only focus on eval-
uating the speed prediction performance of NNs with different
numbers of hidden layer nodes. The same driving dataset,
which is used for Markov emission probability computation,
is used for networking training. About 85% of the data is
employed as training sample to establish the network and the
rest 15% is used for performance validation.
SUN et al.: VELOCITY PREDICTORS FOR PREDICTIVE ENERGY MANAGEMENT IN HEVs 1201
Fig. 2. Sample (training) and testing (validation) dataset. Real HW means data real life highway driving trip. Real UB means real life urban driving trip.
IV. SIMULATION RESULTS AND COMPARISON
In this section, we provide a comprehensive comparative
analysis of the three velocity forecasting methods. Our dis-
course begins with an explanation of the training and val-
idation data. Then, a definition of performance metrics are
defined.
The eight driving cycle samples used for training include
four highway types and four urban types. The same data
is used both for probability computation in Markov-chain
approach and network training in NN approach. A different
set of eight driving cycles are used for testing and comparing
performance of the velocity predictors, which are standard
driving cycles HWFET, WVUINTER, UDDS, WVUSUB and
real collected driving data Real HW2, Real HW3, Real UB2,
and Real UB3. The real collected cycles in sample and testing
data are from [33] and [34]. All sample and test cycles are
concatenated for ease of presentation in Fig. 2.
The vehicle parameters and efficiency maps are obtained
from the Toyota Prius HEV model in Autonomie [35]. Sim-
ulation was performed on a personal computer with an Intel
Corel i7-3630QM CPU at 2.4GHz. The prediction and con-
trol horizon length Hpis specified to be 10, and the step
computation is completed within 0.75 s. The initial SoC and
final SoC in all the simulations are set as 0.60. Upper and
lower SoC bounds are 0.8 and 0.3, respectively. Simulations
for eight testing trips are conducted. Detailed results for the
UDDS testing cycle are provided in this section, along with
the performance distributions across all eight validation cycles.
The following four metrics are used to assess the tested
velocity predictors.
1) Average root mean squared error (RMSE, in
meter/second) of the predicted velocities in all of
the control horizons.
2) Online computation time Tcal (in microseconds) of the
velocity prediction process at each time step.
3) Violating frequency eof the velocity and acceleration
constraints from the predicted velocity sequences.
4) Compensated fuel consumption (in grams), which is the
most important metric to evaluate vehicle fuel economy.
Due to the property of prediction, MPC cannot guarantee an
exact final SoC constraint at the end of a global time horizon.
TAB LE I
SIMULATION RESULTS OF EXPONENTIALLY VARYING
VELOCITY PREDICTOR FOR UDDS
For fair comparison in the following analysis, the produced
fuel consumption results are compensated by converting
the final battery SoC deviation into equivalent engine fuel.
Simulation results for the UDDS cycle are illustrated in
Sections IV-A–IV-D. The same simulation procedure is
conducted for the other seven testing cycles. A comprehensive
fuel consumption performance comparison is demonstrated
in Section IV-E based on all simulation results.
A. Results of Exponentially Varying Predictor for UDDS
As shown in Table I, improved fuel efficiency can be
achieved when the predicted velocity decreases slightly, as
opposed to an increase or a large decrease. Terminal SoC of
MPC simulation becomes larger as εgrows. This is because
when the predicted velocity changes aggressively, the engine
tends to provide more power. Thus, more engine power will
be absorbed by the battery through the generator. The average
RMSE reaches its minimum when ε=−0.01, yielding the
best fuel economy.
B. Results of Markov-Chain Predictor for UDDS
Different types of driving cycles are included in the Markov
probability generation dataset: 1) four highway types and
2) four urban types. In this case, the driving behavior we
studied is comprehensive and blended, which is the main
reason that the 1-stage Markov-chain model has the worst
performance in characterizing future horizon velocities, as
1202 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 3, MAY 2015
TAB LE I I
SIMULATION RESULTS OF MARKOV-CHAIN PREDICTOR FOR UDDS
TABLE III
SIMULATION RESULTS OF NN-BASED PREDICTOR FOR UDDS
shown in Table II. The 1-stage model is, however, the most
computationally efficient.
For multistage Markov-chain models, one requires an often
prohibitively large set of driving data to identify the transition
matrix. This phenomenon exemplifies the poor applicability of
multistage Markov-chain velocity predictors. To address this
problem in realistic implementation, a sufficiently rich sample
database is required. In this brief, the emission probabilities
for unavailable high-stage Markov states are replaced by the
ones derived from lower-stage states to overcome the lack of
a large dataset. The maximum Markov stage in this simulation
is five, since when the stage number is larger than five, the
extracted probability matrix changes negligibly.
Simulation results of Markov-chain velocity predictor for
UDDS testing are shown in Table II, from which we can see
that more stages results in smaller RMSE. Hence, increased
fuel efficiency can be achieved via multistage Markov-chain
velocity predictors. However, in real-world implementation,
it is usually difficult to guarantee that all possible Markov
states be contained in the probability generation dataset. In
addition, the magnitude of the probability matrix for a mul-
tistage Markov-chain model scales exponentially. Therefore,
Markov-chain models with greater than three stages are
rarely used. Note that adaptive or self-learning improving
approaches are not considered here for fair comparison of the
three velocity predictors.
C. Results of NN-Based Predictor for UDDS
For NN-based predictors, the dimension of the input, Hh,is
specified to be 10. Although more neural nodes result in
higher training precision, excessive complexity can lead to
over-fitting. Table III shows the comparison results between
different types of NNs with different numbers of nodes.
Note that the error caused by velocity/acceleration constraint
violations (e) of the predicted horizon speed is included in the
average RMSE computation.
TAB LE I V
VELOCITY PREDICTOR COMPARISON FOR UDDS
The RBF-NN structure achieves better fuel economy than
the LR-NN or BP-NN velocity predictors, despite a similar
RMSE, since there are substantially fewer constraint
violations (e). Besides, RBF network structure needs much
less training time and tuning effort.
D. Velocity Predictor Assessment
The DP, deterministic MPC (DMPC) and ECMS are bench-
marks. In DMPC, the controller has full knowledge of the
horizon velocity profile at each time step. In ECMS, the
equivalence factor is tuned from the sampling cycles and
implemented in the testing trips as a constant (−357.5). To
fully exploit the potential of ECMS, we also tuned the optimal
equivalence factor for each testing trip through trial-and-error.
This well-tuned ECMS is noted as optimal ECMS (OECMS)
and demonstrated for comparison.
The 1-stage and 5-stage Markov-chain velocity predictors
are selected to compare with the best velocity predictors from
the exponentially varying and NN types, denoted as 1-stage
MC, 5-stage MC, −0.01 EV, and RBF-100, respectively. These
four simulations together with benchmark results are shown
in Table IV.
1) Prediction Precision: As can be seen from Table IV,
the average RMSE of the 1-stage MC velocity predictor is
larger than that of the −0.01 EV velocity predictor. This
means that the 1-stage MC process behaves poorlyin modeling
comprehensive and blended driving cycles. The prediction
precision from 5-stage MC predictor is improved by 16%
compared with 1-stage MC, proving that employing more
states can increase the prediction precision. Consequently, the
average RMSE of RBF-100 is the minimum one as opposed
to the other approaches, although the constraints are violated
slightly by 0.18%. This indicates that the RBF-100 predictor
is preferable in modeling comprehensive driving behaviors.
The nature of each prediction method can be seen visually
in Fig. 3. This figure elucidates how a 1-stage MC is too
short-sited to capture velocity profiles longer than a few time
steps. One can also observe how the 5-stage MC roughly pre-
dicts constant acceleration. In contrast, the RBF-100 captures
microtrip-like behaviors better than the other three.
2) Fuel Consumption: From Table IV, we can see that the
RBF-100 predictor is the most energy-saving predictor, which
is reasonable as it achieves the minimum prediction precision.
Fig. 4 shows the SoC trajectories for DP, DMPC, and the
selected MPC controllers given in Table IV. The difference
SUN et al.: VELOCITY PREDICTORS FOR PREDICTIVE ENERGY MANAGEMENT IN HEVs 1203
Fig. 3. Prediction result of the four predictors in Table IV for UDDS. Predicted velocity [meter/second] is shown across the prediction horizon at each time
step.
Fig. 4. SoC trajectories of MPC approaches compared with DP result for
UDDS testing.
between the DP and MPC-based trajectories is due to DP being
a global optimization; whereas the MPC controller yields a
locally optimal solution for each control horizon. With the
same terminal SoC constraint in each receding horizon, the
SoC trajectories of RBF-100 MPC and DMPC are overlaid.
The SoC trajectory for the 5-stage MC approach differs
noticeably from the DMPC benchmark, due to poor velocity
prediction.
3) Computation Time: From Table IV, it is obvious that
the Markov-chain method is computationally heavier than the
NN-based and exponentially varying methods. This is because
the process of generating a stochastic Markov emission needs
additional calculations to form the probability distributions.
4) Constraint Violation: The exponentially-varying and
Markov-chain approaches do not violate the velocity and
acceleration constraints. In the RBF-100 case, the constraints
are violated 24 times (0.18%) in the UDDS cycle. The
constraint violation pertaining to the RBF-100 predictor is thus
negligible.
Fig. 5. Normalized fuel consumption and final SoC comparison. SE is simple
ECMS, OE is optimal ECMS, and DM is DMPC.
E. Summarized Simulation Results
Similar outcomes are achieved for the remaining seven tests.
Fig. 5 shows the average values and standard deviations
of normalized fuel consumption and final SoC. The DP
and OECMS produce the optimal/near-optimal results if the
driving profiles are known a prior. The NN-based predictors
generally maintain the average fuel optimality over 92%,
which is considerably better than the other two predictors and
fairly close to DMPC, both in terms of average and standard
deviation. Compared with simple ECMS, MPC with NN-based
velocity predictors saves more than 7% fuel, and guarantees
the final SoC constrained.
Furthermore, the 4% gap between DP and DMPC fuel
economy is because DP is a global approach, whereas
DMPC conservatively optimizes the fuel consumption locally.
Compared with DMPC, the extra fuel consumption caused
from velocity predicting errors varies from 2% to 21%
based on different methods. Note these results are evaluated
across both emission certification cycles and real-world drive
1204 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 3, MAY 2015
cycle data. Consequently, we conclude the NNs provide a
promising blend of prediction capability and computational
efficiency for MPC energy management in HEVs.
V. CONCLUSION
This brief presents a comparative study of three classes
of velocity predictors for MPC-based energy management
in a power-split HEV: 1) generalized exponentially varying;
2) Markov chains; and 3) artificial NNs. The NN-based
horizon velocity predictor is proposed, and the sensitivity of
its prediction precision on different NN structures is examined
to elucidate a successful template. Generalized exponentially
varying and Markov-chain velocity predictors are systemat-
ically described and compared with the NN-based predictor
in terms of the prediction precision, computational cost, and
fuel economy. Results demonstrate that NN-based velocity
predictors provide the best overall performance across a range
of certification and real-world drive cycles. This brief provides
better understanding for predictive HEV energy management.
ACKNOWLEDGMENT
The authors would like to thank Prof. J. K. Hedrick at
the University of California at Berkeley, Berkeley, CA, USA,
for substantial help and enlightening discussions on model
predictive control theory and implementation.
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