# Ecological Adaptive Cruise Control for City Buses based on Hybrid Model
Predictive Control using PnG and Traffic Light Information

Sai Krishna Chada1, Jitin Mathew Thomas1, Daniel Görges2, Achim Ebert3, Roman
Teutsch4 1Electromobility Research Group, University of Kaiserslautern,
Germany
2 German Research Center for Artificial Intelligence (DFKI), Kaiserslautern,
Germany
3Computer Graphics and HCI Group, University of Kaiserslautern, Germany
4Institute for Mechanical and Automotive Engineering, University of
Kaiserslautern, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>&<EMAIL_ADDRESS>
###### Abstract

This paper proposes an ecological adaptive cruise control (EACC) concept with
the primary goal to minimize the fuel consumption in a city bus with an
internal combustion engine (ICE). A hybrid model predictive control (HMPC) is
implemented in this work to control both continuous and discrete-time
variables. Moreover, a multi-objective optimization problem for EACC is
formulated in time-domain as a mixed-integer quadratically constrained
quadratic programming (MIQCQP) problem. The proposed HMPC-EACC performs robust
vehicle-following while tracking a leading vehicle and plans fuel-efficient
acceleration and deceleration maneuvers for the host vehicle. Additionally, it
uses the signal phase and timing (SPaT) information to compute a green wave
reference speed for the host vehicle to cross the signalized intersections at
a green phase. Moreover, the proposed controller performs pulse and glide
(PnG) to optimally control the engine ON and OFF states and save additional
fuel. Furthermore, the performance of the proposed strategy is evaluated on a
real-world driving profile and compared against a baseline controller from the
literature. Finally, the influence of different prediction horizons on the
fuel savings and computation times are studied. The results reveal significant
reduction in fuel consumption with HMPC-EACC and demonstrate that the proposed
controller is real-time capable.

###### Keywords:

pulse and glide, vehicle-following, green-wave, traffic lights, signal phase
and timing, fuel consumption, mixed integer quadratic programming

##  I Introduction

Local public transportation using city buses is regarded as an effective
solution to mobilize people collectively in urban environments and reduce the
need for private transportation, which is a major cause for traffic congestion
and environment pollution. According to [1], 98.3% of the medium and heavy
commercial vehicles in the European Union are diesel-powered, thus positioning
the internal combustion engines (ICEs) as the dominant energy source in this
segment. Although modern commercial vehicles are equipped with highly
efficient ICEs which adhere to stringent emission norms, there is still a need
to reduce the fuel consumption in the vehicle fleet, to lower the air
pollution in the urban areas and to reduce the overall fleet operational costs
as well. To address this concern, previous studies have investigated on varied
aspects ranging from alternative fuels [2], electrified powertrain
technologies [3] and fuel-efficient routing approaches [4]. Over the past
decade, researchers have been actively investigating on advanced driver
assistance systems (ADAS) that can promote energy savings in commercial
vehicles and in turn help to reduce their CO2 emissions. Previous literature
on energy-oriented ADAS in commercial vehicles focused on predictive/look-
ahead cruise control [5] and eco-driving assistance systems [6] which can
achieve up to 12% reductions in fuel consumption. Recent studies point towards
more advanced adaptive cruise control (ACC) concepts, in which a multi-
objective optimal control problem (OCP) is solved with the aim to minimize the
host vehicle’s energy/fuel consumption, in addition to improving safety and
driving comfort. One such concept is ecological adaptive cruise control
(EACC), in which the host vehicle plans an optimal speed trajectory while
tracking a preceding vehicle in a car-following scenario [7], [8]. Algorithms
such as dynamic programming (DP)[5] and pontryagin’s minimum principle (PMP)
[9] have shown to yield a global optimum solution for the OCP, however were
found to be computationally intensive and not suitable for the online
implementation. To address this limitation, model predictive control (MPC) is
being widely used to facilitate online-control and effectively deal with state
and input constraints.

Fuel consumption in a conventional city bus is primarily influenced by the
acceleration and deceleration maneuvers, resulting due to the lead vehicle
behavior. To determine the interactions between a lead vehicle and host
vehicle, previous works have explored car-following in passenger vehicles for
highway driving and simplified urban scenarios. The influence due to other
important factors in the urban environments such as traffic light (TL)
signals, bus stops and stop signs have often been neglected in the literature.
An effort was made in our previous work [10] to propose a novel EACC concept
to minimize the energy consumption in an electric vehicle (EV) by taking the
advantage of signal phase and timing (SPaT) information from traffic signals.
Recent advancements in the intelligent transportation systems (ITS) and
vehicle-to-infrastructure (V2I) technology have made it possible to
communicate between urban vehicles, signalized intersections and road
infrastructure in real-time. One promising driving strategy to reduce the fuel
consumption in a conventional vehicle is Pulse and Glide (PnG), in which the
vehicle is accelerated to a higher speed than the desired speed in the pulse
phase and then allowed to coast until a lower speed by turning the engine OFF
in the glide phase. Such a strategy has been proven to be efficient in
achieving fuel savings up to 20% [11, 12]. To control the engine ON/OFF state,
a binary variable must be incorporated into the optimization problem, thus
making the hybrid model predictive control (HMPC) approach an ideal choice for
the longitudinal control in conventional vehicles with ICE. To the best
knowledge of the authors, EACC for city buses using HMPC is not yet addressed
in the literature.

In this work, the following contributions are made: (i) An ecological ACC
strategy for a city bus with ICE based on HMPC is proposed, which is capable
of reducing the host vehicle’s fuel consumption, in addition to improving
vehicle safety and driving comfort. For this purpose, a unique problem
formulation is designed in time-domain to propel the host vehicle fuel-
efficiently in the vehicle-following and freeway driving scenarios. (ii) The
additional constraints due to TL signals, bus stops and regulatory speed
limits present in an urban scenario are incorporated into the problem
formulation. Moreover, the proposed HMPC-EACC uses the future information of
the preceding vehicle velocities, route elevation and SPaT information to
derive fuel-efficient velocity trajectories for the host vehicle. (iii) The
optimization problem is formulated as a mixed-integer quadratically
constrained quadratic programming (MIQCQP) problem and serves as an extension
to the existing state-of-the-art EACC concepts. The performance of the
proposed controller is evaluated against a problem formulation from the
literature. (iv) Furthermore, a detailed comparison on the energy savings for
different prediction horizons and the computation capability is presented in
this work.

The remainder of this paper is structured as follows. In Section II, the
control objectives are introduced and a method for green-wave velocity
calculation is presented. Later in Section III, the longitudinal dynamics of a
city bus and the ICE scaling approach are elaborated. In Section IV, different
problem formulations for EACC are compared and their respective properties are
discussed in Section V. Furthermore, a detailed investigation on the
computaton time for the proposed strategy is presented in Section VI. Finally,
conclusions and hints on future work are given in Section VII.

##  II Control Objectives

###  II-A Control objectives

The primary objective of HMPC-EACC is to minimize the fuel consumption in a
conventional city bus in the following scenarios: (i) While tracking a leading
vehicle in a vehicle-following scenario by maintaining a relative distance not
too large (to prevent cut-in scenarios) and not too small (to ensure safety).
(ii) In a freeway scenario by tracking a reference green wave optimal speed
(GWOS) devised for the host vehicle to pass through the signalized
intersections at a green phase, thereby avoiding frequent stops at the TL
signals. (iii) While approaching the bus stops and unavoidable red TL signals,
the host vehicle must plan to stop fuel-efficiently. Furthermore, in such a
scenario, the host vehicle must stop as close as possible to the stop line at
TL signals or bus stops. (iv) In the aforementioned scenarios, the host
vehicle must apply the PnG strategy and optimally turn the engine OFF when
necessary. Besides these, other objectives such as vehicle safety and driving
comfort must be ensured.

###  II-B Green-wave velocity calculation

It is assumed in this work that the SPaT and bus stop information are
communicated in real-time to the host vehicle. An examplary information is
shown in Fig. 1 using a space-time diagram. It can be noticed that the city
bus driving route consists of two signalized intersections
$\text{TS}_{\\{1,2\\}}$ and a bus stop $\text{BS}_{\\{3\\}}$. The approach
used to calculate a green-wave velocity intersection is inspired from [13] and
can be determined by,

$\displaystyle[v_{\text{ref,min}},v_{\text{ref,max}}]=\left[\frac{d_{\text{TS},i\
}}{r_{ij\ }},\frac{d_{\text{TS},i\ }}{g_{ij\
}}\right]\cap[v_{\text{min}},v_{\text{max}}]\ $ (1)

where, $d_{\text{TS},i}$ represents the distance to the $i^{th}$ TL signal and
$r_{ij}$ represents the start time of the $j^{th}$ red phase of the $i^{th}$
TL signal. Similarly, $g_{ij}$ represents the start time of the $j^{th}$ green
phase of the $i^{th}$ TL signal. Furthermore, $v_{\text{min}}$ and
$v_{\text{max}}$ are the minimum and maximum road speed limits respectively.
It is to be noted that GWOS intersection using (1) is calculated for the
upcoming TL signals alone and bus stops are exempted because, it is considered
in this work that the host vehicle makes a compulsory halt at all the bus
stops. From the possible GWOS intersection range obtained using (1),
$v_{\text{ref,min}}$ is chosen as the new reference velocity, because
travelling at lower velocities leads to better fuel-efficiency as compared to
higher velocities. Therefore, the host vehicle controller tracks
$v_{\text{ref,min}}$ to cross the TL signals at a green phase.

Figure 1: Space-time diagram showing an exemplary urban driving scenario

If a green-wave intersection range is not found in (1), it means that stopping
at the next TL signal is unavoidable. In such a case, the host vehicle is
advised to travel at a driver’s set speed until it reaches the upcoming TL
signal. Similarly, if the host vehicle is approaching a bus stop, the phase is
assumed to be red, so as to force the host vehicle to stop at bus stops for a
designated time period called as bus dwell time (DT), usually for boarding or
alighting the passengers, which is considered as
$s_{\text{t}}=10\,\mathrm{s}$. Although the bus DT is not always constant in
real scenarios, $s_{\text{t}}$ can easily be updated based on real
measurements. Upon completion of this scenario, the GWOS is recalculated for
the next TL signal.

##  III System Modeling

###  III-A Vehicle dynamics

The longitudinal dynamics of the host vehicle is given by

$\displaystyle
a_{\text{h}}=\frac{dv_{\text{h}}}{dt}=\frac{1}{m_{\text{eq}}}(F_{\text{t}}-F_{\text{b}}-F_{\text{res}})$
(2)

where $a_{\text{h}}$ and $v_{\text{h}}$ are the acceleration and velocity of
the host vehicle, $m_{\text{eq}}$ is the equivalent mass which is the sum of
vehicle weight, driver and cargo weight and rotational equivalent masses,
$F_{\text{t}}$ is the traction force and $F_{\text{b}}$ is the braking force.
The resistance force is
$F_{\text{res}}=F_{\text{a}}+F_{\text{r}}+F_{\text{g}}$, where $F_{\text{a}}$
is the aerodynamic resistance, $F_{\text{r}}$ is the rolling resistance and
$F_{\text{g}}$ is the gradient resistance. The aerodynamic resistance force
$F_{\text{a}}$ is calculated from the drag coefficient $c_{\text{w}}$, frontal
cross-sectional area of the vehicle $A_{\text{f}}$, vehicle speed
$v_{\text{h}}$ and density of the air $\rho$, as given in (3). The presence of
the term $v^{2}$ makes the equation non-linear. $F_{\text{a}}$ can be
linearized as

$\displaystyle F_{\text{a}}=\frac{1}{2}\rho
A_{\text{f}}c_{\text{w}}v_{\text{h}}^{2}\approx\frac{1}{2}\rho
A_{\text{f}}c_{\text{w}}(p_{1}v_{\text{h}}+p_{\text{2}}),$ (3)

where the coefficients $p_{\text{1}}=27.711$ and $p_{\text{2}}=-168.459$ are
identified through line approximation. Rolling resistance $F_{\text{r}}$ and
gradient resistance $F_{\text{g}}$ are expressed as

$\displaystyle\ \ \ F_{\text{r}}=c_{\text{r}}m_{\text{v}}g\cos\theta$ (4a)
$\displaystyle\ \ \ \ F_{\text{g}}=m_{\text{v}}g\sin\theta$ (4b)

where $c_{\text{r}}$ is the rolling resistance coefficient, $m_{\text{v}}$ is
the vehicle weight, $g$ is the gravitational acceleration and $\theta$ is the
road elevation angle. The minimum safety distance $d_{\text{s}}$ is defined
using the constant headway $h_{\text{s}}$ and the minimum safety distance at
standstill $d_{\text{min}}$. To prevent frequent cut-in scenarios, the desired
inter-vehicle distance $d_{\text{c}}$ is defined using the constant headway
$h_{\text{c}}$ and maximum allowable distance $d_{\text{max}}$.

$\displaystyle d_{\text{s}}=d_{\text{min}}+h_{\text{s}}v_{\text{h},k}$ (5a)
$\displaystyle d_{\text{c}}=d_{\text{max}}+h_{\text{c}}v_{\text{h},k}$ (5b)

A reference engine map OM668 of a passenger vehicle is chosen from the ADVISOR
library and is scaled to represent the city bus engine speed-torque
characteristics using a simplified approach. Firstly, a torque scaling factor
$f_{\text{T,scale}}$ is determined by dividing the maximum torque of the
scaled engine $T_{\text{e,max,new}}$ with the maximum torque of the reference
engine $T_{\text{e,max,ref}}$ as given in (6a). By multiplying
$f_{\text{T,scale}}$ with the torque values of the reference engine
$T_{\text{e,ref}}$, the torque values for the scaled engine $T_{\text{e,new}}$
can be obtained. Similarly, a scaling factor for angular velocity
$f_{\text{w,scale}}$ is derived by dividing the maximum angular velocity of
the reference engine map $w_{\text{e,max,ref}}$ with the maximum angular
velocity of the scaled engine $w_{\text{e,max,new}}$ as given in (6b).
Moreover, the new scaled angular velocities $w_{\text{e,new}}$ can be
calculated by dividing the reference angular velocities $w_{\text{e,ref}}$
with the scaling factor $f_{\text{w,scale}}$.

$\displaystyle
f_{\text{T,scale}}=\frac{T_{\text{e,max,new}}}{T_{\text{e,max,ref}}}\
\Longrightarrow\ T_{\text{e,new}}=f_{\text{T,scale}}T_{\text{e,ref}}$ (6a)
$\displaystyle
f_{\text{w,scale}}=\frac{w_{\text{e,max,ref}}}{w_{\text{e,max,new}}}\
\Longrightarrow\ w_{\text{e,new}}=\frac{w_{\text{e,ref}}}{f_{\text{w,scale}}}$
(6b)

Furthermore, the new engine efficiency $\eta_{\text{e,new}}$ can be determined
with

$\eta_{\text{e,new}}=\frac{T_{\text{e,new}}w_{\text{e,new}}}{\dot{m}_{\text{e,ref}}(w,T)\
H_{\text{u}}}$ (7)

where $H_{\text{u}}$ is the lower heating value of the diesel fuel,
$T_{\text{e,new}}$ and $w_{\text{e,new}}$ are the scaled torque and angular
velocity values obtained from (6a) and (6b) respectively. Here, the fuel flow
rate map $\dot{m}_{\text{e,ref}}(w,T)$ of the reference engine is used.
Furthermore, the engine power map is calculated by

$P_{\text{e}}=\frac{F_{\text{t}}v_{\text{h}}}{\eta_{\text{e,new}}\eta_{\text{t}}}$
(8)

where $\eta_{\text{t}}$ is the transmission efficiency, $F_{\text{t}}$ and
$v_{\text{h}}$ are the traction force and velocity respectively. The city bus
is equipped with a 4-speed automatic transmission. The operating region for
the gears is determined in this work using a rule-based approach with the
primary objective to operate the engine at its best efficiency. These gears
divide the power consumption map into four regions as shown in Fig. 2.

###  III-B Approximation of ICE power consumption map

The ICE power consumption map is approximated seperately for each gear using
(9) as shown in Fig. 2. Here, the power consumption is a function of the
velocity $v_{\text{h}}$ and traction force $F_{\text{t}}$.

$f_{\text{app}}(v_{\text{h}},F_{\text{t}})=x_{00}+x_{10}v_{\text{h}}+x_{01}F_{\text{t}}+x_{20}v_{\text{h}}^{2}+x_{02}F_{\text{t}}^{2}+x_{11}v_{\text{h}}F_{\text{t}}$

(9)

Figure 2: Power consumption map approximation

The coefficients $x_{00}$, $x_{10}$, $x_{01}$, $x_{20}$, $x_{02}$, $x_{11}$
are obtained by solving a convex optimization problem given in (10a). The
objective is to minimize the approximation error between the polynomial
function $f_{\text{app}}$ and the original power consumption map (8) across
the operating range $i=1,..,R$ and $j=1,..,S$ for the traction force and
velocity respectively. Moreover, (10b) ensures that $f_{\text{app}}$ is
positive semidefinite. The optimization problem is solved using CVX semi-
definite programming toolbox.

$\displaystyle\min_{\textit{{x}}_{\text{00}},\textit{{x}}_{\text{10}},\textit{{x}}_{\text{01}},\textit{{x}}_{\text{20}},\textit{{x}}_{\text{02}},\textit{{x}}_{\text{11}}}\sum_{i=1}^{R}\sum_{j=1}^{S}\|Ax-b\|^{2}$
(10a) $\displaystyle\ \ \ \ \ \ \text{s.t.}\ \ \begin{bmatrix}x_{02}&x_{11}\\\
x_{11}&x_{20}\end{bmatrix}\geqslant 0$ (10b) $\displaystyle\text{where},\
A=\left[1\ F_{\text{t},i}\ v_{\text{h},j}\ F_{\text{t},i}^{2}\
v_{\text{h},j}^{2}\ F_{\text{t},i}v_{\text{h},j}\right]$ $\displaystyle\ \ \ \
\ \ \ \ \ \ b=\frac{F_{\text{t},i}\
v_{\text{h},j}}{\eta_{\text{e,new}}\eta_{\text{t}}}$ $\displaystyle\ \
x=[x_{00},x_{01},x_{11},x_{02},x_{20},x_{11}]^{T}$

Approximating the non-linear power consumption map using (9) has a deviation
in the accuracy at lower traction values. For instance when $F_{\text{t}}$ is
zero, the approximated power $f_{\text{app}}$ must ideally be zero. Thus, in a
PnG glide phase while engine is OFF, the resulting energy consumption will be
also zero. However, $f_{\text{app}}$ calculated using (9) at $F_{\text{t}}=0$
is found to have a value greater than zero, thus motivates the engine to
remain ON at all times, which is not desirable. To tackle this issue, an idea
suggested in [14] is used in this work which will be explained in the next
section.

##  IV Problem Formulations

###  IV-A HMPC-EACC for a vehicle-following scenario

Figure 3: Typical vehicle-following schematic

$\displaystyle\min_{\textit{P}_{\text{ICE},k},\textit{F}_{\text{t},k},\textit{F}_{\text{b},k},\textit{n}_{k},\varepsilon_{1,k},\varepsilon_{2,k},\varepsilon_{3,k}}\sum_{k=0}^{N-1}P_{\text{ICE},k}\Delta
T+\zeta_{1}F_{\text{b},k}^{2}+\zeta_{2}\varepsilon_{\text{1},k}^{2}+$ (11a)
$\displaystyle\leavevmode\resizebox{399.2946pt}{}{$\text{s.t.}\ \
v_{\text{h},k+1}=v_{\text{h},k}+\frac{\Delta
T}{m_{\text{eq}}}(F_{\text{t},k}-F_{\text{b},k}-c_{\text{r}}m_{\text{v}}g\cos\theta_{k}$}-$
(11b) (11c) $\displaystyle\leavevmode\resizebox{211.3883pt}{}{$\ \ \ \ \ \ \ \
\ n_{k}-n_{k-1}=\ \varepsilon_{\text{1},k}$};\ \ \ n\in\mathbb{B}:=\\{0,1\\}$
(11d) (11e) (11f) (11g) (11h) (11i) (11j) $\displaystyle\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ d_{\text{rel},k}\geq d_{\text{min}}+h_{\text{s}}v_{\text{h},k}$ (11k)
$\displaystyle\ \ \ \ \ \ \ \ \ \ d_{\text{rel},k}\leq
d_{\text{max}}+h_{\text{c}}v_{\text{h},k}+\varepsilon_{\text{3},k}$ (11l)

The HMPC-EACC optimization problem for a typical vehicle-following scenario is
reformulated in time-domain based on the idea from [14] and is given in (11).
Here, $N$ is the prediction horizon length. The first term in the cost
function represents the energy consumption of the host vehicle which is the
product of the power consumption $P_{\text{ICE}}$ and the sample time $\Delta
T$. The variable $P_{\text{ICE}}$ is constrained by (11f) and (11g). In order
to compensate the error in power consumption approximation as discussed in
Section III-B, an additional term $P_{\text{max}}(1-n_{k})$ is subtracted from
$f_{\text{app}}$ in (11f), where $P_{\text{max}}$ is a constant term, which is
the maximum available power at each gear and is computed offline from the
original map and $n_{k}$ is given by

$n_{k}=\begin{cases}0&:\text{Engine OFF}\\\ 1&:\text{Engine ON}\end{cases}$
(12)

If the engine is OFF, both (11f) and (11g) force $P_{\text{ICE}}$ to zero.
Expanding $f_{\text{app}}$ term in (11f) by substituting (9), makes it a
quadratically constrained problem. Using the second term in (11), an excessive
braking force $F_{\text{b}}$ is penalized. In the third term, a slack variable
$\varepsilon_{1}$ is introduced to avoid frequent switching between the engine
ON and OFF states. A linear constraint describing $\varepsilon_{1}$ is given
in (11d), where $n_{k}$ is the current and $n_{k-1}$ is the previous engine
state. Moreover, the variable $n$ takes a binary value either 0 or 1 as
described in (12). Furthermore, in the fourth term the variation in the
traction force at successive time steps is penalized using a slack variable
$\varepsilon_{2}$ if it exceeds a constant value $\Delta F_{\text{t,max}}$,
with the aim to minimize the jerks and improve the driving comfort, as given
in (11j). To motivate the host vehicle to stay within the desired region
$d_{\text{c}}$, another slack variable $\varepsilon_{3}$ is penalized in the
last term. The host vehicle velocity and relative distance to the leading
vehicle in discretized form is given in (11) and (11c), where $v_{\text{p}}$
is the velocity of the preceding vehicle. Moreover, $\zeta_{1}$, $\zeta_{2}$,
$\zeta_{3}$, $\zeta_{4}$ are the corresponding weighting factors for the
aforementioned terms in the cost function (11). The limits for the traction
force $F_{\text{t}}$ are set using (11e), in which $F_{\text{t,max}}$ is the
maximum traction force calculated offline for each gear. Moreover, $n_{k}$ is
multiplied to $F_{\text{t,max}}$ so as to force $F_{\text{t}}$ to zero when
the engine is OFF. The physical limitations of the vehicle with respect to
velocity and braking force are addressed in (11h) and (11i). In order to
maintain a safe distance to the preceding vehicle, a hard constraint is
introduced in (11k) based on (5a). To motivate the host vehicle to stay within
the desired region a soft constraint is formulated using (11l) based on (5b).
Overall, the presence of a quadratic cost function (11), a binary variable
(11d), linear (11e), (11g), (11h), (11i), (11j), (11k), (11l) and quadratic
inequality constraints (11f) make the optimization problem as MIQCQP, which
can be solved efficiently using Gurobi solver.

###  IV-B HMPC-EACC for a typical urban scenario

Figure 4: Typical urban scenario schematic

The previous optimization problem (11) is extended in this section to
accomodate the additional constraints due to TL signals and bus stops during
the longitudinal control. The new cost function is presented in (13), which
has two new terms. The penultimate term contains the slack variable
$\varepsilon_{4}$ to motivate the host vehicle to stop as close as possible to
the stop line $d_{\text{m}}$ while stopping at a TL or bus stop as defined in
(13d). Here, $d_{\text{ITS}}$ is the relative distance to the upcoming stop
that could be either a TL signal or a bus stop, where
$\text{ITS}=\\{\text{BS},\text{TS}\\}$.

$\displaystyle\ \ \ \ \ \min_{u_{k}}\sum_{k=0}^{N-1}P_{\text{ICE},k}\Delta
T+\zeta_{1}F_{\text{b},k}^{2}+\zeta_{2}\varepsilon_{\text{1},k}^{2}+\zeta_{3}\varepsilon_{\text{2},k}^{2}+$
(13a) s.t.

$x_{k+1}=Ax_{k}+Bu_{k}+w_{i}$

(13b) (13c) (13d)

where

The last term in (13) enforces a penalty for the deviation of the host vehicle
velocity $v_{\text{h}}$ from the GWOS reference velocity $v_{\text{ref}}$
calculated from (1). The inequality constraint (13) is enforced so that the
host vehicle does not cross TL signals when the phase is red. The other
constraints remain the same as in (11). The discrete state space
representation of system dynamics in time-domain is given by (13b). Here, the
state variables are
$x_{k}=[v_{\text{h},k},d_{\text{rel},k},d_{\text{ITS},k}]^{\intercal}$ and the
optimization variables are
$u_{k}=[P_{\text{ICE},k},F_{\text{t},k},F_{\text{b},k},n_{k},\varepsilon_{\text{1},k},\varepsilon_{\text{2},k},\varepsilon_{\text{3},k},\varepsilon_{\text{4},k}]^{\intercal}$.
It is important to note that, $\zeta_{5}$ is initially zero and is activated
when approaching a red traffic light. Moreover, in a vehicle-following
scenario, $\zeta_{6}$ is deactivated and in a freeway scenario, $\zeta_{4}$ is
deactivated by choosing them as zero.

###  IV-C Baseline controller for a typical urban scenario

To validate the performance of the HMPC-EACC controller proposed in the
previous section, a problem formulation for the baseline controller is chosen
from [8]. Although this problem formulation did not consider the influence of
the TL signals on the host vehicle during car-following, it has been
improvised in this section to have a fair comparison with HMPC-EACC. The new
problem formulation is given in (15).

$\displaystyle\min_{\textit{F}_{\text{t},k},\textit{F}_{\text{b},k},\varepsilon_{1,k},\varepsilon_{2,k},\varepsilon_{3,k}}\sum_{k=0}^{N-1}f_{\text{app}}(v_{\text{h},k},F_{\text{t},k})+\zeta_{1}F_{\text{b},k}^{2}+\zeta_{2}\varepsilon_{\text{2},k}^{2}+$
(15a) s.t.

$\ \ \ \
(\text{\ref{eq:Online_Computation_13}}),(\text{\ref{eq:Online_Computation_14}}),(\text{\ref{eq:Online_Computation_16}}),(\text{\ref{eq:Online_Computation_7}}),(\text{\ref{eq:Online_Computation_8}}),(\text{\ref{eq:Online_Computation_9}}),(\text{\ref{eq:Online_Computation_10}}),(\text{\ref{eq:Online_Computation_11}})$

(15b)

In comparison to the HMPC-EACC problem formulation (13), one major difference
in (15) is that the first term represents the approximated power consumption
obtained from (9), meaning that the PnG strategy cannot be realized in the
baseline controller. Therefore, as there is no necessity for the slack
variable $\varepsilon_{1}$, it is eliminated in (15). Another major difference
is that the baseline controller does not have the possibility to track a
reference velocity $v_{\text{ref}}$ for green-wave and performs vehicle-
following throughout the trip. Therefore, the reference tracking term is
eliminated in (15). The additional constraints due to TL signals and bus stops
are added to (15) as defined in the previous problem formulation.

##  V Results

###  V-A Vehicle-following scenario

To demonstrate the performance of the HMPC-EACC problem formulation discussed
in Section IV-A, it is tested on a city bus velocity profile gathered from
real-world tests. In this simulation study, this velocity profile is
considered as the actual speed profile of a leading vehicle and is tracked by
the proposed controller. Two deterministic control approaches are chosen for
this investigation, namely frozen-time model predictive control (FTMPC) and
prescient model predictive control (PMPC). FTMPC has the knowledge of only the
current velocity information of the preceding vehicle and assumes the velocity
to be constant throughout the prediction horizon, whereas PMPC assumes that
the future velocity information is perfectly known within the prediction
horizon. The results for a typical vehicle-following scenario are illustrated
in Fig. 5 and it is evident that the host vehicle equipped with PMPC and FTMPC
have smoother velocity profiles $v_{\text{h,PMPC}}$ and $v_{\text{h,FTMPC}}$
respectively, as compared to the preceding vehicle $v_{\text{p}}$ while
performing vehicle-following. Moreover, the host vehicle strictly adheres to
the regulatory road speed limits $v_{\text{max}}$. The host vehicle is seen to
robustly maintain the inter-vehicle distance $d_{\text{rel}}$ and resides
within the desired region $d_{\text{c}}$ without crossing the safe distance
$d_{\text{s}}$ to the preceding vehicle. Furthermore, the host vehicle applies
the PnG strategy at several time intervals, resulting in switching the engine
OFF and ensures that the traction force is zero, for example at 104 s, 375 s,
1760 s and 2500 s.

Figure 5: Results of HMPC-EACC for a vehicle-following scenario TABLE I: Simulation results | Prediction
---
Horizon
Controller |  | Distance
---
[km]
| Fuel
---
savings
N=8 | Preceding vehicle | 2.582 | -
HMPC-EACC (FTMPC) | 2.578 | +7.11%
HMPC-EACC (PMPC) | 2.578 | +12.11%

To demonstrate the fuel consumption for the proposed controllers, a
longitudinal vehicle model for the city bus has been developed using QSS
toolbox. The results for the aforementioned scenario are summarized in Table
I. It can be noticed that the HMPC-EACC host car equipped with PMPC and FTMPC
can achieve up to 12.11% and 7.11% reductions in fuel consumption as compared
to the preceding vehicle. The reason is that the HMPC-EACC uses the future
information of the route elevation and the preceding vehicle future velocities
to derive optimal control inputs for the host vehicle. Moreover, gliding by
switching the engine OFF using the PnG strategy is another important reason
for obtaining better energy savings. The reason for higher energy savings with
PMPC as compared to FTMPC is primarily due to the availability of perfect
knowledge of the preceding vehicle’s future velocities. It can be noticed from
the enlarged view in Fig. 5, that the host vehicle velocity with FTMPC has
bigger fluctuations while tracking the preceding vehicle as compared to the
PMPC and corresponding traction force curves show higher peaks for FTMPC as
compared to PMPC.

###  V-B Evaluation of HMPC-EACC and baseline controller

Figure 6: Comparison of the HMPC-EACC and baseline controller TABLE II: Simulation results | Prediction
---
Horizon
Controller |  | Distance
---
[km]
| Trip time
---
[s]
| Fuel
---
savings
$N$=8 | Baseline | 2.48 | 2739 | -
HMPC-EACC | 2.48 | 2737 | +10.21%
$N$=15 | Baseline | 2.48 | 2740 | -
HMPC-EACC | 2.48 | 2735 | +12%

In this section, the performance of the proposed HMPC-EACC and baseline
controller from Section IV-B and IV-C respectively, is evaluated in a typical
urban scenario in the presence of a preceding vehicle, signalized
intersections and bus stops. Moreover, it is assumed that both the controllers
have the perfect knowledge of the preceding vehicle future velocities and
location of the upcoming bus stops. However, the major difference is that the
host vehicle with HMPC-EACC receives the SPaT information of the upcoming TL
signals in real-time, whereas the baseline controller receives only the
current phase information of the TL signals. Both the controllers are tested
in a simulative urban driving route of approximately 2.5 km stretch with
signalized intersections $\text{TS}_{\\{1,3,5\\}}$ and bus stops
$\text{BS}_{\\{2,4,6\\}}$ as illustrated in the space-time diagram in Fig. 6a.
In this simulative study, the HMPC-EACC initially begins by tracking the same
preceding vehicle velocity profile introduced in Section V-A. At the first
signalized intersection $\text{TS}_{1}$ around 460 s, the host vehicle
decelerates to stop behind the preceding vehicle because of a red TL phase.
Later at 1000 s, the host vehicle is seen to make the compulsory stop at the
bus stop $\text{BS}_{2}$ for a constant dwell time $s_{\text{t}}$, therefore
the relative distance $d_{\text{rel}}$ between the host vehicle and preceding
vehicle becomes larger. As soon as $d_{\text{rel}}>$100 m (outside sensor
range) the HMPC-EACC stops tracking the preceding vehicle and follows a GWOS
to reach the upcoming TL signals at a green phase. Moreover, two scenarios
encountered at the signalized intersections $\text{TS}_{\\{3,5\\}}$ are
enlarged and depicted in Fig. 6a. A more detailed analysis of these scenarios
is presented in Fig. 6(i) and Fig. 6(ii) respectively. At 1322 s and 1986 s,
the baseline vehicle decelerates to stop at these TL signals, due to the fact
that the baseline controller lacks the SPaT information of the immediate TL
signals. On the other hand, the host vehicle equipped with HMPC-EACC has the
perfect knowledge of SPaT and crosses the TL signals at a green phase.
Moreover, as compared to the host vehicle traction force $F_{\text{t,HMPC-
EACC}}$, it can be noticed that the traction force values for the baseline
vehicle $F_{\text{t,baseline}}$ are found to be very large, as a result of the
strong deceleration and acceleration maneuvers incurred while stopping at the
TL signals. Furthermore, the fuel mass flow rate of both the host and baseline
vehicles for the entire simulation length is plotted in Fig. 6b. It is evident
that the baseline vehicle has consumed more fuel in comparison with the HMPC-
EACC, because of the aforementioned reasons. The overall test results are
summarized in Table II. The host vehicle equipped with the proposed HMPC-EACC
has outperformed the baseline controller by achieving fuel savings up to 10.2%
and 12% for a prediction horizon length of $N$= 8 and $N$= 15 respectively in
a realistic urban scenario.

##  VI Computation Time

The execution time for the proposed HMPC-EACC strategy at different prediction
horizons is summarized in Table III. The evaluation is performed using Matlab®
R2020a profiler on a Windows 10 PC equipped with an Intel® Core™ i7-7500U CPU
processor with 2.70 GHz clock frequency and 12 GB RAM. For a predition horizon
of $N$= 8 , $N$= 15 and $N$= 25 , the mean execution time for the proposed
controller is found to be 23 ms, 71 ms and 224 ms respectively. The sample
time is chosen as $\Delta T$=200 ms in this work and therefore the results
indicate that the prediction horizon length must be less than $N$= 25 steps (5
s) to realize online-implementation.

TABLE III: Computation time for different prediction horizons Prediction horizon |  $N$=8 |  $N$=15 |  $N$=25
---|---|---|---
Computation time | 23 ms | 71 ms | 224 ms

##  VII CONCLUSIONS

In this work, with the goal to minimize the fuel consumption in a city bus
with ICE, an ecological ACC concept based on HMPC is proposed. Firstly, a
problem formulation for HMPC-EACC is designed in time-domain for a typical
vehicle-following scenario and is evaluated on a real-world city bus profile.
The results revealed that the host vehicle can achieve fuel savings up to
12.1% while tracking a leading vehicle by implementing the PnG strategy.
Furthermore, a new problem formulation for HMPC-EACC is proposed that can
handle additional constraints due to TL signals and bus stops in an urban
scenario, besides tracking a preceding vehicle. The proposed controller uses
the SPaT information to devise a GWOS for the host vehicle, to avoid
unnecessary stops at the TL signals, which has shown additional fuel saving
benefits up to 12% when tested against a baseline controller. Furthermore, a
deeper analysis on the computation time for different prediction horizons is
made. The results showcased that the proposed controller has the capability
for online-implementation. In future work, the proposed HMPC-EACC concept will
be evaluated on more driving datasets, in the presence of both deterministic
and actuated traffic light signals. Furthermore, an extension of EACC with a
possibility for lateral control, which is a limitation in the current work
will be investigated in the future work.

## Acknowledgment

This work is funded by the German Ministry for Education and Research (BMBF)
and partly supported by the Center for Commercial Vehicle Technology (ZNT) at
the University of Kaiserslautern, Germany.

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