IN
DEGREE PROJECT MATHEMATICS, SECOND CYCLE, 30 CREDITS
,
STOCKHOLM SWEDEN 2017
Optimal Energy Management for
Parallel Hybrid Electric Vehicles
using Dynamic Programming
Optimal Energy Management for
Parallel Hybrid Electric Vehicles
using Dynamic Programming
NIKITA TAGNER
Degree Projects Systems Engineering (30 ECTS credits) Degree Programme in Aerospace Engineering (120 credits) KTH Royal Institute of Technology year 2017
TRITA-MAT-E 2017:45 ISRN-KTH/MAT/E--17/45--SE
Royal Institute of Technology School of Engineering Sciences
Abstract
In this thesis, two optimal control problems for the control of hybrid electric vehicles are formulated. The first is general formulation where both velocity and state of charge can vary. The second is a formulation where the velocity is prespecified and therefore only the state of charge can vary. The first for-mulation takes significantly more time to solve with dynamic programming than the second formulation.
For the most hilly drive cycle that was evaluated, 4.45% fuel savings were obtained by using the general formulation over the formulation with prespec-ified velocity. For the least hilly cycle, this number dropped to 1.75%. When the lowest admissible velocity was lowered from 75 to 70 km/h, fuel savings of 0.52% were obtained. From 80 to 70 km/h, the number increased to 1.92%.
Sammanfattning
I denna avhandling formuleras tv˚a optimala styrningsproblem f¨or reglering av hybridelektriska fordon. I den f¨orsta, mer generella, formuleringen kan b˚ade hastighet och batteriladdning variera. I den andra formuleringen ¨ar hastigheten specifierad i f¨orv¨ag and d¨armed kan endast batteriladdningen variera fritt. Den f¨orsta formuleringen tar betydligt l¨angre tid att l¨osa med
dynamisk programmering ¨an den andra formuleringen.
Av dem utv¨arderade k¨orcyklerna gav den som var mest kuperad br¨anslebesparningar p˚a 4.45% om den l¨ostes med den generella formuleringen ist¨allet f¨or den d¨ar
hastigheten ¨ar specifierad i f¨orv¨ag.
N¨ar den l¨agsta till˚atna hastigheten s¨anktes fr˚an 75 till 70 km/h sparades 0.52% br¨ansle. D¨aremot, om den l¨agsta till˚atna hastigheten s¨anktes fr˚an 80 till 70 km/h ¨okade besparingen till 1.92%.
Acknowledgements
First and foremost, I would like to express my sincerest gratitude to Niklas Pettersson, my supervisor at Scania. He came up with the idea for this thesis, remembered that I asked him about potential projects during my summer job at his department and has been extraordinarily helpful and patient with me throughout these past months. Without him, this work would not have been possible.
My supervisor at KTH, Johan Karlsson, can not go unmentioned. He painstak-ingly read through several drafts of my report and helped me find an excel-lent, if not optimal, way of presenting the finished optimization problems. If it weren’t for him, the structure of the report would be completely different and certainly not for the better.
I would also like to thank Henrik Sv¨ard at Scania, who gave valuable mathe-matical insight and helped me every week even though he wasn’t technically my supervisor. Thanks for caring and cutting your lunch short on Thurs-days.
In addition, I would like to thank Javier and Ahmed for many great laughs and afternoon walks.
I cant let my family go unmentioned through this. Thank you mom, dad, my little sister Viktoria, my babushka and my best little friend Sonya for being there when I was feeling down and supporting me throughout not only this part of my education but also through primary school and high school. Without you, I wouldn’t have made it. I love you.
Contents
1 Introduction 1
1.1 Objective . . . 1
1.2 Outline . . . 2
2 Background 4 2.1 Impact of the Transport Sector . . . 4
2.2 Hybrid Electric Vehicles . . . 5
2.3 Energy Management Problem . . . 7
3 Mathematical Preliminaries 10 3.1 Optimal Control Problem . . . 10
3.2 Dynamic Programming . . . 11
3.2.1 Numerical Aspects . . . 12
4 Mathematical Model 15 4.1 Reformulation from time to position . . . 15
4.2 Vehicle Dynamics . . . 16 4.3 Battery Dynamics . . . 19 4.4 Fuel Consumption . . . 21 5 Problem Formulation 22 5.1 General Formulation . . . 22 5.2 Cost Function . . . 24 5.2.1 Choice of weight β . . . 26
5.3 Problem with fixed velocity . . . 27
6.2 Changing slope . . . 31
6.3 Changing velocity constraints . . . 37
6.4 Changing state of charge constraints . . . 40
7 Discussion 44 7.1 Performance of the solutions . . . 44
7.2 Optimal behaviour . . . 45
7.3 Changing Slopes . . . 46
7.4 Changing velocity constraint . . . 47
7.5 Changing State of Charge constraint . . . 48
7.6 Future work . . . 48
Chapter 1
Introduction
Due to concerns about climate change, poor air quality and depleting oil supplies getting vehicles powered by alternative fuels to market is highly im-portant.
There are many alternatives to using solely internal combustion engines both on the market and in development. Some examples are Fuel cell vehicles, battery electric vehicles, hybrid electric vehicles and vehicles running on al-ternative fuels such as bio diesel.
In this report, only Hybrid Electric Vehicles will be covered. These are
vehicles powered by both an internal combustion engine using some chemical fuel, like diesel, and by an electrical machine which uses electrical energy stored in a battery. Hybrid electric vehicles have been shown to reduce fuel consumption by between 5 and 30 % depending on the vehicle and driving conditions [22].
1.1
Objective
The first problem is to minimize the fuel consumption over a drive cycle while making sure that the charge in the battery exhibits charge sustain-ability. Both the velocity and the battery charge are allowed to vary within a set of admissible bounds. The slope of the road is known for the entire cycle. The second problem is also to minimize the fuel consumption over a drive cycle while making sure that the charge in the battery exhibits charge sus-tainability. However, the velocity is assumed to be known and prespecified. The battery charge is allowed to vary within a set of admissible bounds. The slope of the road is known for the entire cycle.
The two solutions will be compared in order to evaluate how much fuel can be saved by implementing the problem with free velocity, which should take more time to solve, rather than the problem with prespecified velocity. This will be done by solving the problems for the three different drive cycles with varying slopes.
In addition, the set of admissible velocities and charges in the battery will be varied in order to evaluate how the two different energy management strategies will perform under different conditions.
1.2
Outline
First, in Chapter 2, a more comprehensive background will be given with dis-cussions on impact of the transportation sector on the environment, hybrid electric vehicles in general and the energy management problem in particular. Then, in Chapter 3, some mathematical prerequisites will be presented and discussed.
In Chapter 4, mathematical models of the vehicle, battery and fuel con-sumption will be formulated.
In Chapter 5, the two mathematical problems to be solved will be formu-lated as so called Optimal Control Problems.
man-agement strategies will be presented.
In Chapter 7, the results and the methodology will be discussed.
Chapter 2
Background
In this chapter, a thorough background will be given to the impact of freight transport on society, what hybrid vehicles are and how they are controlled. In section 2.1, the effects of the transportation sector on the environment will be discussed. Then, in section 2.2, the concept of Hybrid Electric Vehi-cles will be defined and explained. Next, in section 2.3 the energy manage-ment problem will be introduced and various approaches to solving it will be discussed.
2.1
Impact of the Transport Sector
Transportation, the movement of goods and people, is a cornerstone of any economy. A portion of this movement is accomplished by using road trans-port. This portion can be significant. In the European Union, for instance, roughly 75% of all freight transport is performed using road vehicles [19]. However, like for all industries, there are challenges (or opportunities, if you will) on the horizon for the transportation sector.
released into the atmosphere has been found to be a major contributor to the warming of our planet. Heavy duty vehicles are responsible for about 25% of the total carbon dioxide emissions from road transport and some 6% in total in the European Union [2]. Part of the solution to this challenge is the hybrid electric vehicle which can run on both an internal combustion engine and on electricity by utilizing a battery and electrical machines for propulsion. This can reduce both fuel consumption and emissions [22].
2.2
Hybrid Electric Vehicles
Conventional vehicles are typically propelled by a high capacity energy source, often a chemical fuel, and converted to mechanical energy using an internal combustion engine. For heavy duty vehicles, it is quite common to use diesel as the energy source.
A hybrid vehicle is propelled by a combination of a high capacity energy source and some lower capacity rechargeable energy storage system. This system acts mainly as a storage buffer and can assist the internal combus-tion engine in propelling the vehicle or can recover kinetic energy during braking and store it for later use. The rechargable energy storage system can, for instance, be a rotating flywheel, supercapacitor or some pneumatic device [25]. In this thesis, however, the energy storage system is an electro-chemical battery. The electrical energy stored in the battery is converted to mechanical energy using one (or more) electrical machines. In this case, the vehicle is referred to as a hybrid electric vehicle [25].
Figure 2.1: Layout of a parallel hybrid electric vehicle
internal combustion engine and the electrical machine are coupled together and their combined energy is transmitted through the rest of the drive line to the wheels. This allows the vehicle to be propelled either by only one of the two fuel sources, or by both combined [29]. The full arrow in Figure 2.1 de-notes the chemical energy path whereas the dashed line the electrical energy path i.e how the chemical and electrical energies respectively are transmitted to the wheels. Note that the dashed arrow points in two direction. This indicates the ability of a hybrid electric vehicle to use regenerative braking - by applying a braking torque with the electrical machine the battery can be recharged with electrical energy that can be re-converted to kinetic at a later time.
2.3
Energy Management Problem
For conventional vehicles, all the power required to move the vehicle is pro-duced from a single energy source. In hybrid electric vehicles, however, the necessary power can come from both a chemical energy storage and an elec-trical energy storage. This introduces an extra control task - how much of the total energy required at every given time instant should be taken from each fuel source.
Clearly, it is desirable to keep fuel consumption down. Furthermore, unless the vehicle is a plug-in hybrid which can be recharged from the electricity grid, there must be charge sustainability over any given drive cycle mean-ing that the energy in the battery at the beginnmean-ing of a drive cycle must be roughly equal to the energy at the end of the drive cycle. In addition, minimization of exhaust emissions as done in [5] [28] and tear on components like the battery as in [14] [18] can also be desirable to minimize but these approaches will not be studied here.
The methods used to solve this problem can be split into two categories - rule based and model-based optimization methods [17].
Rule based methods are, as the name suggests, implemented using rules and based on heuristics, intuition or optimal solutions obtained from the more precise model-based methods [17]. Their main advantage is the low compu-tational burden which allows them to easily be implemented on embedded controllers. Furthermore, they were often the first methods to be tried when control of hybrid electric vehicles became a research topic [22].
In [12], deterministic rules are set using engineering intuition and the nec-essary parameters are fine tuned using experiments in order to reduce pol-lutant emissions, keeping the battery charge, reducing noise and decreasing fuel consumption. This does, however, require parameter tuning which may only work as desired for the case for which the parameters were tuned. Fur-thermore, significant time can be spent on tuning [22].
Model-based optimization methods, on the other hand, output an optimal control by minimizing a cost function subject to constraints and model dy-namics. This will give an optimal solution for the stated problem but requires knowledge of future driving conditions, for instance slope on the road ahead. Once a cost function and relevant constraints and dynamics have been for-mulated into what is called an optimal control problem, it can be solved using various approaches [17].
One common approach to use for solving the arisen optimal control prob-lem is Pontryagins Minimum Principle. Like all model-based approaches it requires information about future driving conditions. It is a set of neces-sary, but not sufficient, conditions for optimality [20]. It has been found to produce results very close to that of dynamic programming and also, by assuming that the so called co-state is constant which is often valid in the correct charge interval for the battery, implementable in real time without knowledge of the future road. However, it requires significant simplification of the problem [13].
Another important method is the Equivalent Consumption Minimization Strategy (ECMS) which relates electrical energy in the battery to chemi-cal energy in the fuel tank. This method does not require future information about the drive cycle and can be used in real time due to very low computa-tional burden [11]. It is however quite sensitive to the set relation between electrical and chemical energy and might need to be changed depending on the drive cycle [17].
In recent years, however, dynamic programming has been implemented in real time on embedded controllers for conventional vehicles [8] [9] [24] where the necessary information about future slope has been obtained using GPS data and the algorithm implemented in a receding horizon fashion with de-creased fuel consumption when compared to the previously used controllers as a result. Using GPS-data to obtain information about future driving con-ditions is also called look-ahead control.
Chapter 3
Mathematical Preliminaries
In this chapter, the background knowledge necessary to understand the math-ematical methods used in this thesis will be presented.
In Section 3.1, a general optimal control problem will be presented.
Thereafter, in Section 3.2, the mathematical technique that will be used in this thesis to solve optimal control problems, Dynamic Programming, will be presented and discussed in detail.
3.1
Optimal Control Problem
A general optimal control problem in a continuous spatial coordinate s is minimize u Z sf si f0(s, x(s), u(s))ds + φ(x(sf)) subject to dx ds = f (s, x(s), u(s)) x(si) given x ∈ X(s) u ∈ U(s) (3.1)
where si is the initial position, sf is the final position, x is the state vector,
u is the control vector, f0 is the cost function, φ is the final cost function, f
as a function of the position [23].
By discretizing the trajectory s in (3.1) into N stages which results in
sf−si
N = h sized steps and approximating
dx
ds using Euler Forward, one
ob-tains minimize N −1 X k=0 hf0(k, xk, uk) + φ(xN) subject to xk+1 = xk+ hf (k, xk, uk) x0 given xk ∈ Xk k = 0 . . . N − 1 uk∈ Uk k = 0 . . . N − 1 (3.2)
where k is an integer which denotes the k : th stage which corresponds to position s = ksf
N on the discretized trajectory and h is the distance between
stages h = sf
N. Likewise, xk denotes the state at stage k and uk the control
at stage k [23].
Later in this thesis, two optimal control problems for the energy manage-ment problem of hybrid electric vehicles will be defined on the form (3.1). It is then straightforward to obtain the discretized version (3.2).
3.2
Dynamic Programming
Developed by Richard Bellman in [1], Dynamic Programming is a mathemat-ical optimization method. It can be used to solve problems where sequential decision making occurs - meaning that, with each action, the stage and state evolves and a reward is acquired. It can be used to solve problems on the form (3.2) since it is a sequential optimization problem; being in state xk
and applying a control uk evolves the state to xk+1 = xk+ hf (xk, uk) and a
reward hf0(xk, uk) is obtained. This then continues for state xk+1.
as J∗(n, xn) = min {uk}N −1k=n φ(xN)+ N −1 X k=n f0(k, xk, uk) subject to xk+1 = xk+ hf (k, xk, uk) xk∈ Xk uk ∈ Uk (3.3) where the states xk and controls uk are subject to the same constraints and
dynamics as in (3.2). The optimal solution to (3.2) is then clearly J∗(0, x0).
Dynamic programming is based on the ”Principle of Optimality” which can be expressed using (3.3) and is shown in Theorem 1.
Theorem 1. Assume that the control sequence {u∗k}N −1
k=0 is optimal for J ∗(0, x
0)
and leads to the optimal trajectory {x∗k}N
k=0 where x∗0 = x0. Then, the control
sequence {u∗k}N −1
k=n is optimal to J
∗(n, x∗
n), i.e if we restart the optimization
of the problem (3.2) along the optimal trajectory.
Clearly, J∗(N, xN) = φ(xN) since no control can be applied at stage
k = N . Then, by Theorem 1, J∗(k, xk) = min
uk∈Uk
hf0(k, xk, uk) + J∗(k + 1, f (k, xk, uk)) (3.4)
for k = N − 1, N − 2, . . . , 0. Essentially, (3.4) says that if the optimal control sequence and state trajectory from k + 1 is known, it is possible to calculate the optimal trajectory from k by minimizing over the stage cost hf0(k, xk, uk) and the cost-to-go from stage k + 1 at the state where the
control uk leads. Doing this recursively will, eventually, lead to J (0, x0)
and the optimal solution for (3.2) will be found. This is what is called the backwards dynamic programming recursion and if it has a solution then that solution is optimal to (3.2) [23].
3.2.1
Numerical Aspects
The numerical implementation of backwards dynamic programming recursion (3.4) requires discrete and finite sets of the state variables and controls over which the cost-to-go functions (3.3) will be calculated. By sampling the state vector x into Q samples, the set
¯
is obtained. Likewise, by sampling the control vector u into Z samples, the set
¯
U = {¯u1, ¯u2, . . . , ¯uZ} (3.6)
is obtained. Note that the state and control variables in their respective vec-tors do not necessarily have to be sampled into the same number of discrete values as done in (3.5) and (3.6).
Using the sets defined in (3.5) and (3.6), an numerically implementable pseudo-code can be written and is displayed in Algorithm 1. The algorithm Algorithm 1 Dynamic Programming
1: for all x ∈ ¯X do 2: JN∗(N, x) ← φ(x)
3: end for
4: for k = N − 1 to 0 do
5: for all x ∈ ¯X do
6: J∗(k, x) ← minu∈ ¯U{hf0(x, u) + J (k + 1, x + hf (k, x, u))}
7: u∗k(x) ← arg minu∈ ¯U{hf0(x, u) + J (k + 1, x + hf (k, x, u))}
8: end for
9: end for
can be seen as counting backwards from the final stage N where it begins by initializing JN(x). Then, at stage k and for every x in the discrete and finite
set ¯X, the function hf0(k, x, u) + J∗(k + 1, x + hf (k, x, u)) is minimized over
all u in the discrete set ¯U where J (k + 1, x) has already been calculated in the previous iteration and denotes the optimal cost to go at stage k + 1 for the state x + hf (k, x, u).
As can be seen in Algorithm 1, the cost-to-go function J∗(k, x) is only defined for x ∈ ¯X. However, there is in general no guarantee that x + hf (x, u) ∈ ¯X. This is schematically shown in Figure 3.1, where a control ¯uj ∈ ¯U is applied
at state ¯xi and leads to a state ¯xi+ hf (k, ¯xi, ¯uj) 6∈ ¯X. In order to deal with
this, linear interpolation between the values J∗(k + 1, ¯xi+ hf (k, ¯xi, ¯uj)) and
J∗(k + 1, ¯xi+1+ hf (k, ¯xi, ¯uj)), which are known from the previous itertation,
Chapter 4
Mathematical Model
In this chapter, a mathematical model for the vehicle will be presented. First, in Section 4.1, a procedure to reformulate a functions derivative with respect to time to a derivative with respect to position on the drive cycle using the chain rule will be presented.
Then, in Section 4.2 the longitudinal dynamics of the vehicle will be formu-lated as a derivative of the vehicles kinetic energy with respect to position. Afterwards, in Section 4.3 the dynamics of the electrical charge in the battery will be modeled as a derivative of the charge with respect to position. Finally, in Section 4.4 a model for the fuel consumption will be presented.
4.1
Reformulation from time to position
In this thesis, the distance traveled along a drive cycle will be considered as the ”time”-variable. This is due to the fact that the slope of the road, θ, depends on the position of the vehicle rather than time, θ = θ(s) where s denotes position on the drive cycle. It is therefore convenient to rewrite all derivatives with respect to time into derivatives with respect to position. For any smooth function f , the relationship
df dt = df ds ds dt = df dsv (4.1)
4.2
Vehicle Dynamics
The longitudinal dynamics of the vehicle are given by Newtons second law as
mdv
dt = Ft− Fb− Fouter (4.2)
where m is the mass of the vehicle, Ft is the traction force from the wheels,
Fb is the force from the wheel brakes and Fouter is the sum of all outer forces
acting on the vehicle. By using (4.1) on (4.2),
mvdv
ds = Ft− Fb− Fouter (4.3)
is obtained. However, it has been shown in [9] that using velocity as a state in the control of heavy vehicles with dynamic programming may lead to oscillatory solutions. Instead, kinetic energy
e = 1
2mv
2 (4.4)
will be used to represent the state. By recognizing, through (4.4), that
mvdv ds = 1 2m dv2 ds = de ds the formulation (4.3) can be rewritten using (4.4) as
de
ds = Ft− Fb− Fouter (4.5)
which denotes the derivative of the kinetic energy with respect to position and is the same approach as the one in [9].
Assuming that the angular velocity of the internal combustion engines and the electrical machines output shafts are equal, the traction force Ft can be
expressed as
Ft= j
Tice+ Tem
rwheel
(4.6)
where Tice denotes the torque provided by the internal combustion engine,
Figure 4.1: Schematic figure of the longitudinal dynamics of a truck (Original image from [24])
wheels and j the gear ratio between the wheels and both the internal com-bustion engine and the electrical machine and is assumed to be constant. In addition, Fb can also be expressed as a torque,
Fb =
Tb
rwheel
(4.7) where Tb is the torque applied to the wheels. Furthermore, according to [21],
Fouter can be decomposed as
Fouter(s, v) = Froll+ Faero(v) + Fgrade(s) (4.8)
where Froll, Faero and Fgrade are defined in Table 4.1 where croll denotes the
roll coefficient, Cd the drag coefficient, ρair the density of the surrounding
air, and A the frontal area of the vehicle. In Figure 4.1, all forces acting on the vehicle are displayed.
In conclusion, by combining (4.5), (4.6) and (4.7) while recognizing through (4.4) that velocity is a function of the kinetic energy
v(e) = r
Table 4.1: Definition of outer forces acting on the vehicle
Force Description Expression
Froll Roll resistance crollmg
Faero Air drag 12CDρairAv2
Fgrade Part of gravity due to slope mg sin θ(s)
an expression for the derivative of kinetic energy with respect to position is de ds = fe(s, e, Tem, Tice, Tb) where fe(s, e, Tem, Tice, Tb) = jTice+ Tem rwheel − Tb rwheel − Fouter(s, e) (4.9) and Fouter is given by (4.8). This expression for the derivative will be used
in the following optimal control problem formulations. Moreover, Tice, Tem and Tb are constrained by
Tice,min≤ Tice ≤ Tice,max, (4.10)
Tem,min≤ Tem≤ Tem,max, (4.11)
and
0 ≤ Tb < ∞ (4.12)
respectively. Note that Tem,min can be negative which results in the battery
being charged. In addition, Tice,min can be negative as well which can occur
if the fuel injected into the internal combustion engine does not generate a sufficient amount of torque to overcome the friction forces of the shaft. Finally, the kinetic energy is constrained to
emin ≤ e ≤ emax (4.13)
Figure 4.2: Equivalent circuit model of a battery
4.3
Battery Dynamics
The state of charge, q, is a measure of the current electrical charge in a battery and is defined as
q = Q
Q0
(4.14) where Q is the electrical charge and Q0 is the nominal electrical charge of a
battery. By taking the derivative of (4.14) with respect to time, an expression for the time derivative of the state of charge is obtained as
dq
dt = −
i Q0
(4.15) where i is the current which is denoted as positive when it discharges the battery and negative when it charges. Proceeding by using (4.1) on (4.15),
dq ds = − 1 v i Q0 (4.16) is obtained.
In Figure 4.2, an equivalent circuit of a battery is shown where U0 is the
open-circuit voltage of the battery and R0 is the internal resistance.
Ac-cording to [21], an expression for the current can be obtained by using an equivalent circuit model as
where ω denotes the angular velocity of the electrical machine and Pem,loss
the losses its energy losses. Furthermore, R0 is assumed to be constant.
This is as approach often used for modelling the battery in hybrid electric vehicles [4] [9] and even in studies that take battery ageing into account [15]. The open circuit voltage can be approximated using the polynomial
U0(q) = p1q4+ p2q3+ p3q2+ p4q + p5 (4.18)
and the losses in the electrical machine can be approximated using,
Pem,loss(ω, Tem) = c2Tem2 + c1|Tem| + c3|Tem|ω + c4ω + c5 (4.19)
where ω denotes the angular velocity of the electrical machine. By combining (4.15) with (4.17), the expression
d dsq = fq(e, q, Tem) is obtained where fq(e, q, Tem) = 1 2R0Q0 1 v(e) U0(q)− − q
U0(q)2− 4R0(Temω(e) + Ploss,em(ω(e), Tem))
(4.20)
and U0 is given by (4.18), Pem,loss by (4.19), the velocity v is given by the
kinetic energy
v(e) = r
2e m
and the angular velocity of the electrical machine is given by
ω(e) = j
rwheel
r 2e m
due to the definition of kinetic energy in (4.4). This expression for the deriva-tive will be used in the optimal control problem formulations.
Lastly, q is limited by
qmin ≤ q ≤ qmax (4.21)
4.4
Fuel Consumption
In general, the mass flow of fuel into an internal combustion engine is mod-elled using fuel maps on the form
dmf
dt = fmap(Tice, ω)
where mf denotes the mass of fuel and fmap is often slightly non-linear. This
non-linearity can lead to behaviors which are difficult to interpret which is why the non-linear map is linearized as
dmf
dt = k1Ticeω + k0ω (4.22)
where ω is assumed to be equal to the angular velocity of the electrical machine. By using (4.1) on (4.22),
dmf
ds =
1
v (k1Ticeω + k0ω) (4.23)
is obtained. The energy flow out of the internal combustion engine into the coupling is assumed to be related to the mass flow into the engine by
dEf
ds = Qlhvηf dmf
ds (4.24)
where Qlhv is the lower heating value of the fuel and ηf is the efficiency of the
engine, an approach previously used in [9]. By combining (4.23) with (4.24), dEf
ds = ff uel(e, Tice)
is obtained where Ef denotes the amount of energy being generated by the
internal combustion engine and ff uel(e, Tice) = Qlhvηf
1
v(e)(k1Ticeω(e) + k0ω(e)) . (4.25) where the velocity v is given by the kinetic energy
v(e) = r
2e m
and the angular velocity of the electrical machine is given by
ω(e) = j
rwheel
r 2e m
Chapter 5
Problem Formulation
In this chapter, two optimal control problems concerning the energy man-agement of hybrid electric vehicles will be formulated on the form (3.1) and discussed.
In Section 5.1, a general optimal control problem for the energy manage-ment of hybrid electric vehicles will be formulated. This problem will have kinetic energy and state of charge in the battery as state variables and torque from the internal combustion engine, the electrical machine and the wheel brakes as control inputs.
In Section 5.2, the choice of cost function for the general optimal control problem will be motivated and discussed.
In Section 5.3, the velocity of the general optimal control problem will be assumed to always be given which results in a problem with only state of charge as state variable and only torque from the internal combustion engine as control input.
5.1
General Formulation
Consider the energy management problem for a hybrid electric vehicle where it is desirable to minimize fuel consumption, keep the trip time T , which denotes the total time that the vehicle drives from the start of the drive cycle to the end, below some value Tmax while keeping the state of charge q
inputs u = [Tice, Tem, Tb]T, the problem can be written as minimize Tice,Tem,Tb ηeU0(q(sf))Q0(q(si) − q(sf)) + e(si) − e(sf) ηd + Z sf s=si
ff uel(e(s), Tice) + β
1 v(e(s))ds subject to de ds = fe(s, e, Tem, Tice, Tb) dq ds = fq(e, q, Tem) e(0), q(0) given e ∈ [emin, emax] q ∈ [qmin, qmax]
Tice ∈ [Tice,min, Tice,max]
Tem ∈ [Tem,min, Tem,max]
(P1) where fe is given by (4.2), fq by (4.20), ff uel by (4.25) and
v(s) = r
2e(s)
m .
Furthermore, ηe is the energy efficiency of the electrical energy path (from
battery to coupling) and
ηe =
(
ηe,charge if q(sf) ≤ q(si)
ηe,discharge if q(sf) > q(si)
(5.1)
is the energy efficiency from the wheels to coupling and ηd,charge > 1 whereas
ηd,discharge< 1 [21]. Finally, β is a weight on the total trip time
T = Z sf
s=si
1
v(e(s))ds (5.2)
and will be discussed more extensively in Section 5.2.
and a downhill is so steep that the electrical machine braking torque cant keep the constraint by itself. Thus, the only admissible action is to use the wheel brakes. This observation serves to reduce the computation time of the algorithm significantly - for every solution it is known how to implement Tb
which means that it does not need to be discretized and solved for in the Dynamic Programming algorithm.
5.2
Cost Function
To motivate the cost function, an energy analysis of the vehicle will be con-ducted. In a hybrid electric vehicle, chemical energy is converted to rotational energy which is transmitted through the driveline and to the wheels which results in an increase of the vehicles kinetic energy. By braking with the elec-trical machine, some of this kinetic energy can be stored as electric energy in the battery and be used later by converting energy back to kinetic energy through the electrical machine. There are losses of energy associated with transmitting energy through the driveline, losses in converting it from one form to another with both the internal combustion engine and the electrical machine and losses in the battery. This is illustrated schematically in Figure 5.1 where ηf denotes the efficiency of converting chemical energy in the fuel
to rotational energy in the coupling, ηe the efficiency of converting electrical
energy to rotational energy in the coupling or vice-cersa and ηd is the
effi-ciency of converting the rotational energy in the coupling to kinetic energy of the vehicle. These energies can only be compared fairly at the same point in the drive line which in this thesis is chosen to be the coupling, shown as the summation node in Figure 5.1. The costs in the optimal control problem (P1) are derived from this energy interpretation.
In the problem (P1), the cost function consists of four parts. The first one,
ηemU0(q(sf))Q0(q(si) − q(sf)), (5.3)
is part of the final cost and denotes how much energy can be transferred from the battery to the coupling at the end of the drive cycle. It makes sure that charge sustainability is maintained over the cycle. The second one,
e(si) − e(sf)
ηd
Figure 5.1: Energy flow in the hybrid electric vehicle
represents the rotational energy that can be transferred to the wheels from the coupling at the end of the drive cycle. If this term is omitted, the optimal control would be to spend as little chemical energy as possible at the end of the drive cycle, resulting in the kinetic energy going to its lower bound at the end of the cycle, e(sf) = emin. The third part,
Z sf
si
ff uel(e(v(s)), Tice)ds (5.5)
denotes the total amount of chemical energy in the fuel converted to rota-tional energy at the coupling for the entire drive cycle and corresponds to the consumed fuel mass. The fourth one,
β Z sf
si
1
v(e)ds, (5.6)
5.2.1
Choice of weight β
The weight on the trip time, β, will be chosen by following the approach in [26] with slight modifications to make the procedure work for hybrid electric vehicles as well as linearized fuel models. Assume that the slope is constant, Tem = Tb = 0 and that there exists one applicable control ˆTice that keeps
the constant velocity ˆv. A constant velocity corresponds to constant kinetic energy, ˆe, which results in deds = 0 and using (4.5), one obtains
ˆ Tice=
rwheel
j Fouter(e(ˆv)) (5.7)
where Fouter no longer depends on s since the slope is constant. Using this
control, the cost function of (P1) becomes J (ˆv) = ηeU0(q(sf))Q0(q(si) − q(sf)) + Z sf si ff uel(ˆe, ˆTice) + β 1 v(ˆe)ds. (5.8) By utilizing (4.23), (5.7) and recognizing that
ω = j v
rwheel
, the cost function (5.8) becomes
J (ˆv) =ηeU0(q(sf))Q0(q(si) − q(sf)) + + Z sf si Qlhvηf k1Fouter(e(ˆv)) + k0 j rwheel + β ˆ vds. (5.9)
The minimum of 5.9 can be found at the stationary point, dJ dˆv(ˆv) = Z sf si Qlhvηfk1 dFaero(e(ˆv)) dˆv − β ˆ v2ds = 0
Solving for β, the expression
β = Qlhvηfk1
dFaero(e(ˆv)
dˆv vˆ
2 (5.10)
is obtained where Faerois defined in Table 4.1. Choosing this β for the general
5.3
Problem with fixed velocity
Now, consider the general formulation (P1) where the velocity, and therefore the kinetic energy, is known and prespecified throughout the entire drive cycle, i.e the kinetic energy is a given function of the position e = e(s). Then, the cost term denoting the net kinetic energy at the coupling available at the end of the drive cycle, (5.4), is a constant which is known and can be taken out of the minimization. Likewise, the integeral (5.6) which denotes the total trip time T multiplied by a weight β is also known and can be taken out. In addition, deds is also always known and therefore the kinetic energy e can be removed from the state vector. Furthermore, (4.2) can be rewritten as Tice+ Tem= rwheel j de ds + Fouter(s, e) − Tb rwheel (5.11) where the right hand side is known for all s and e since it is obvious that Tb is never optimal unless absolutely necessary as discussed in Section 5.1.
Therefore, for either any Tice or Tem, the other is given by the torque split
(5.11) which necessitates only one control variable for the optimization. In conclusion, if the kinetic energy of the vehicle is known for the entire drive cycle, e = e(s), using state vector x = q and control vector u = Tice,
the general formulation (P1) can be rewritten as minimize
Tice
ηeU0(q(sf))Q0(q(si) − q(sf)) +
Z sf
s=0
ff uel(e(s), Tice)ds
subject to dq ds = fq(e, q, Tem) where Tem = rwheel j de ds + Fouter(s, e) − Tb rwheel − Tice q(0) given q ∈ [qmin, qmax]
Tice ∈ [Tice,min, Tice,max]
Tem ∈ [Tem,min, Tem,max]
(P2)
Chapter 6
Results
In this chapter, the results for both the general formulation with free velocity, (P1), and the formulation with prespecified velocity, (P2), will be shown. In Section 6.1, the drive cycles to be used, the state and control constraints and sampling intervals will be presented.
In Section 6.2, the general formulation (P1) and the formulation with pre-specified velocity (P2) will be solved for three different drive cycles with varying slope. The trip time, whether final state of charge difference and the final velocity will be presented. Then, the velocity, state of charge, torque from the internal combustion engine and electrical machine will be displayed for each cycle and for both (P1) and (P2). Finally, a fuel consumption com-parison will be shown.
In Section 6.3, the lowest admissible velocity will be increased and (P2) will be solved. The trip time, final state of charge difference and the final veloc-ity will be presented. Then, the velocveloc-ity, state of charge, torque from the internal combustion engine and electrical machine will be displayed for (P1). Finally, a fuel consumption comparison will be shown.
6.1
Drive cycle and sampling data
The 235 km long drive between S¨odert¨alje and Norrk¨oping then back will be used. The slope is known for the entire road i.e θ(s) is a given function. Furthermore, the slope will be multiplied by three scaling factors, thus gen-erating three drive cycles. The histograms for each cycle are shown in Figure 6.1 where Figure 6.1a shows the histogram where the slope θ has been mul-tiplied by 1.15, Figure 6.1b the histogram for the drive cycle where the slope θ has been multiplied by 1 (the original cycle) and Figure 6.1c the histogram for the drive cycle where the slope θ has been multiplied by 0.85.
In Table 6.1, the minimum and maximum values of the state variables [e, q], the control variables [Tice, Tem, Tb] and the sampling intervals that will be
used for dynamic programming are presented. These values will be used for solving both (P1) and (P2) except for the cases when the velocity limits and the state of charge limits will be changed.
In addition, the road trajectory is sampled using h = 200 m.
Table 6.1: Sampling information for discretizing (P1) and (P2)
Tice[N m] Tem[N m] v[km/h] q[−]
Max value 2600 1050 85 1
Min value -150 -1050 70 0
(a) Cycle 1 - Scaling factor 1.15 (b) Cycle 2 - Scaling factor 1
(c) Cycle 3 - Scaling factor 0.85
6.2
Changing slope
By solving (P1) for the all drive cycles displayed in Figure 6.1 and for the data in Table 6.1 the trip time, state of charge difference ∆q = q(sf) − q(si)
and the final velocity are obtained. They are displayed in Table 6.2. Like-wise, by solving (P2) where the velocity is prespecified as being 80 km/h for the entire drive cycle, the corresponding results are displayed in Table 6.3.
Furthermore, Figure 6.2 displays state and control variable trajectories Table 6.2: Trip time, difference in state of charge and the final velocity obtained by solving (P1) for three different drive cycles
Cycle T [h] ∆q[%] v(sf)[km/h]
1 2.97 0.17 84.93
2 2.96 0.18 84.96
3 2.96 0.17 84.93
Table 6.3: Trip time, difference in state of charge and the final velocity obtained by solving (P2) for three different drive cycles
Cycle T [h] ∆q[%] v(sf)[km/h]
1 2.96 0.29 80
2 2.96 0.29 80
3 2.96 0.29 80
for a 25 km stretch of road for the general formulation (P1). This stretch of road has been chosen because it contains both small and large hills. In Figure 6.2a, the altitudes of the three drive cycles are displayed. In Figure 6.2b, the velocity profiles are displayed. In Figure 6.2c, the states of charge are displayed. In Figure 6.2d, the torques from the internal combustion en-gine are displayed. In Figure 6.2e, the torques from the electrical machine are displayed.
Figure 6.3d, the torques from the internal combustion engine are displayed. In Figure 6.3e, the torques from the electrical machine are displayed.
(a) Altitude
(b) Velocity
(c) State of Charge
(d) Torque from internal combustion engine
(e) Torque from electrical machine
(a) Altitude
(b) Velocity
(c) State of Charge
(d) Torque from internal combustion engine
(e) Torque from electrical machine
6.3
Changing velocity constraints
By solving (P1) for the drive cycle which has the histogram displayed in Fig-ure 6.1b and the data in Table 6.1 (except for the lower limit on the velocity which will be set to 70, 75 and 80 km/h) the trip time, the state of charge difference ∆q = q(sf) − q(si) and the final velocity are obtained. They are
displayed in Table 6.4.
Furthermore, Figure 6.6 displays state and control variable trajectories for Table 6.4: Mass of fuel consumed, difference in state of charge and total trip time for (P1) for different minimal velocities
Minimal velocity [km/h] T [h] ∆q[%] v(sf) [km/h]
70 2.96 0.18 85
80 2.95 0.19 85
85 2.92 0.14 85
the same 25 km stretch of road as in Section 6.2 for the general formulation (P1). In Figure 6.6a, the altitude drive cycles is displayed. In Figure 6.6b, the velocity profiles are displayed. In Figure 6.6c, the states of charge are displayed. In Figure 6.6d, the torques from the internal combustion engine are displayed. In Figure 6.6e, the torques from the electrical machine are displayed.
(a) Altitude
(b) Velocity
(c) State of Charge
(d) Torque from internal combustion engine
(e) Torque from electrical machine
6.4
Changing state of charge constraints
By solving (P1) and (P2) for the drive cycle which has the histogram dis-played in Figure 6.1b and the data in Table 6.1 (except for the limit on the state of charge which will be set to 0 and 1 for one case and 0.85 and 0.15 for another) the trip time, the state of charge difference ∆q = q(sf) − q(si) and
the final velocity are obtained. They are displayed in Table 6.5 where
”Reg-ular SOC Window” refers to the case when qmin = 0 and qmax = 1 whereas
”Smaller SOC Window” refers to the case when qmin = 0.15 and qmax = 0.85
Furthermore, Figure 6.8 displays state and control variable trajectories for the entire drive cycle for the general formulation (P1). The entire cycle has been chosen because the trajectories are almost identical except for a small section near the middle of the cycle. In Figure 6.8a, the altitudes of the three drive cycles are displayed. In Figure 6.8b, the velocity profiles are dis-played. In Figure 6.8c, the states of charge are disdis-played. In Figure 6.8d, the torques from the internal combustion engine are displayed. In Figure 6.8e, the torques from the electrical machine are displayed.
Also for the entire cycle, Figure 6.9 displays the state and control variables for the formulation with prespecified velocity (P2). In Figure 6.9a, the al-titudes of the three drive cycles are displayed. In Figure 6.9b, the velocity profiles are displayed. In Figure 6.9c, the states of charge are displayed. In Figure 6.9d, the torques from the internal combustion engine are displayed. In Figure 6.9e, the torques from the electrical machine are displayed.
Finally, in Figure 6.10, the mass of fuel saved by using (P1) instead of (P2) for varying state of charge limits are obtained.
Table 6.5: Trip time, difference in state of charge between the beginning and the end of a drive cycle and the final velocity for (P1)
T [h] ∆q[%] v(sf)[km/h]
Regular SOC Window 2.95 0.19 85
(a) Altitude
(b) Velocity
(c) State of Charge
(d) Torque from internal combustion engine
(e) Torque from electrical machine
(a) Altitude
(b) Velocity
(c) State of Charge
(d) Torque from internal combustion engine
(e) Torque from electrical machine
Table 6.6: Trip time, difference in state of charge between the beginning and the end of a drive cycle and the final velocity for (P2)
T [h] ∆q[%] v(sf)[km/h]
Regular SOC Window 2.96 0.27 80
Smaller SOC Window 2.96 0.27 80
Chapter 7
Discussion
In this chapter, the results will be discussed.
In section 7.1, the performance of the solutions will be discussed with em-phasis on the charge sustainability, trip time and final velocity.
In section 7.2, the state variable trajectories, control inputs of the general formulation (P1) and the formulation with prespecified velocity (P2) will be examined in order to gain understanding about the optimal behaviour of a hybrid electric vehicle.
In section 7.3, the impact of varying slope on the solutions of (P1) and (P2) and their respective fuel consumptions will be assessed.
In section 7.4, the impact of having a higher minimal velocity on the solution of (P1) will be discussed.
In section 7.5, the impact of a having tighter window of admissible state of charges on the solution of (P1) and (P2) will be analysed. In section 7.6, possible future work will be suggested.
7.1
Performance of the solutions
explanation might be that there is a downhill at the end of the cycle, as seen in Figure 6.9a. However, this effect was observed for other roads as well, with no downhill at the end. This is likely due to the cost term
e(si) − e(sf)
ηd
which is negative if the final velocity, or kinetic energy, is higher than the inital. It seems that the formulation (P1) finds it optimal to spend chemical energy in the internal combustion engine to increase the kinetic energy right at the end of the cycle. One way around this problem could be to have a cost term that penalizes any deviation from the initial value, for instance
(e(si) − e(sf))2
ηd
but this could also produce undesirable results since if e(s) > e(si) near the
end of the cycle and there is a downhill, a braking torque might be applied to achieve e(sf) = e(si) even though it is intuitevly better to let the vehicle
accelerate. This is clearly not desirable. Instead, a tuning parameter to the cost function could be added as
γe(si) − e(sf) ηd
where γ should be tuned appropriately so that desirable solutions at the end of the cycle are obtained.
7.2
Optimal behaviour
The state and control trajectories for (P1) and (P2) are shown in Figure 6.2 and 6.3 respectively.
from position 90 km to about 105 km since the hills are fairly small there and the velocity is never at risk of accelerating above the desired limits. However, past 105 km the hills become larger and it is no longer possible for the vehicle to keep its velocity below the maximal value during downhills. Therefore, negative torque from the electrical machine has to be applied thus increasing the state of charge. In addition, the torque from the electrical machine is applied during uphills to help the internal combustion engine.
When the velocity is prespecified as being 80 km/h, the electrical machine has to brake during downhills in order to maintain the velocity. Therefore, the formulation with prespecified velocity (P2) uses the electrical machine significantly more than (P1) which can be seen by comparing Figure 6.2e with Figure 6.3e. This also implies that the electrical machine is used to generate positive torque at higher values than when the velocity is not prespecified be-cause otherwise the battery would be overcharged at the middle of the cycle. However, the energy losses in an electrical machine scale quadratically with the torque, as seen in (4.19). Therefore, it is wisest to use it as conservatively as possible.
For both solutions, the torque from the internal combustion engine is roughly the same. The major difference between the two strategies is therefore which energy buffer is used most - the kinetic energy of the vehicle or electrical energy in the battery.
7.3
Changing Slopes
Note that the reason fuel masses are compared rather than the cost func-tion is that the net charge in the battery over a cycle is very small and trip times are roughly equal, as seen in Tables 6.2, 6.3, 6.4, 6.5 and 6.6. In ad-dition, the fact that the velocity of the vehicle goes to 85 km/h at the end of each cycle when solving (P1) is not much of a problem for comparing fuel consumptions since it only happens right at the end of a 236 km long drive and therefore does not have a big impact on the mass of fuel consumed.
7.4
Changing velocity constraint
As seen in Figure 6.6, which displays the state variable and control input trajectories when solving (P1) using the original drive cycle with histogram 6.1b for different minimal values of the velocity, the major difference between the solutions when the minimal allowed velocity is higher is that the vehi-cle cant deaccelerate as much during uphills which limits how much it can accelerate in the upcoming downhill. In order not to violate the velocity con-straints braking torque needs to be applied with the electrical machine. This is especially evident at 106 km for the case when the minimal velocity is 80 km/h - a braking torque is applied for that case but not for the case when the minimal velocity is 70 km/h since it can accelerate more during the downhill. Judging by Figure 6.7, there is an increase in fuel consumption when the minimal admissible velocity is higher. For cycle 2 (with scaling factor 1), 0.52% fuel can be saved by allowing the velocity to go down to 70km/h in-stead of 75km/h and 1.92% can be saved by allowing the velocity to go down to 70 km/h instead of to 80 km/h.
7.5
Changing State of Charge constraint
As seen in Figure 6.8 and 6.9, which display the state variable and control input trajectories for an entire cycle when solving (P1) and (P2) using the drive cycle with histogram displayed in Figure 6.1b. For both the general formulation (P1) and the formulation with prespecified velocity (P2), there is very little difference in the control trajectory. The only major difference is at the middle of the drive cycle. There, when the state of charge limits are tightened, the electrical machine generates a lot of torque so that it can apply a braking torque in the next downhill and not violate the state of charge constraints. This, albeit necessary for this case, is in general undesir-able because the losses in an electrical machine are quite large and increase quadratically with larger torque, evident from the loss polynomial (4.19). In Figure 6.10, it is seen that limiting the state of charge window for the general formulation (P1) and the formulation with prespecified velocity (P1) does not have a big impact on the fuel consumption. This is due to the drive cycle in particular since for both state of charge limits, the electrical machine generates rather high torques at the middle of the cycle in order to discharge the battery and allow for regenerative braking in the upcoming cycle.
7.6
Future work
For starters, the next logical step would be to evaluate how much fuel savings can be achieved using the formulation with prespecified velocity and includ-ing either engine on/off, gear shiftinclud-ing or both. Then, complemented by this report, a more more or less clear picture could emerge of for what driving conditions the formulation with prespecified velocity (P2) is preferable to use over (P1).
ex-plored by optimizing separately over gear shifting and engine on/off.
Chapter 8
Conclusion
Implementing dynamic programming on line as a cruise controller has been done for conventional vehicles. To implement online, the computational ef-fort must be small. However, the number of calculations necessary for solving a problem with dynamic programming increases exponentially with the num-ber of states. Therefore, it is imperative to choose as few states as possible while not loosing too much optimality.
In this study, two optimal control problems for the energy management of hybrid electric vehicles were formulated. The first formulation, the general formulation, has both kinetic energy, which is equivalent to velocity, and state of charge in the battery as states. The second formulation has only state of charge as state whereas the velocity is prespecified. The general for-mulation requires significantly more time to solve than the forfor-mulation with prespecified velocity.
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