Optimization of Fuel Consumption in Hybrid Electric Vehicles

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Optimization of Fuel Consumption in Hybrid Electric Vehicles

F R E D R I K B Å B E R G E L L I O T D A H L

Master of Science Thesis Stockholm, Sweden 2013

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Optimization of Fuel Consumption in Hybrid Electric Vehicles

F R E D R I K B Å B E R G E L L I O T D A H L

Master’s Thesis in Optimization and Systems Theory (30 ECTS credits) Master Programme in Mathematics (120 credits) Royal Institute of Technology year 2013 Supervisor at KTH was Xiaoming Hu Supervisor at Sophia University Tokyo was Tielong Shen Examiner was Xiaoming Hu

TRITA-MAT-E 2013:39 ISRN-KTH/MAT/E--13/39--SE

Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

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Abstract

There are various technologies used for reducing fuel consumption of automobiles. Hybrid electric vehicles is one approach that has been used, which can reduce fuel consumption by 10-30% compared to conventional vehicles.

In this master thesis the minimization of fuel consumption of a power-split type HEV along a given route is considered, where the vehicle speed has been assumed to be known a priori. This minimization was made by first de- riving a model of the HEV powertrain, followed by creating a Dynamical programming based program for finding the optimal distribution of torques.

The performance was evaluated through the commercial software GT-Suite. The resulting control from the Dy- namic program could follow the reference speed in many situations. However the battery state-of-charge calculated in the Dynamic program did not update properly, resulting in a depleted battery in some cases.

The model derived could follow the dynamics of the vehicle, but there are some parts which could be improved.

One of them is the dynamical model of the rotational speed for the engine, ωe.

The Dynamic program works for finding the controller, and can be modified to work with improved state-equations.

Keywords: Fuel optimization, Hybrid electric vehicle, Dy-

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Sammanfattning

Det finns olika sätt att minska bränsleförbrukningen hos bilar, men ett sätt som använts är el-hybrider. Dessa kan minska bränsleförbrukningen med 10-30% jämfört med kon- ventionella bilar.

I det här examensarbetet undersöks optimering av bräns- leförbrukning för en el-hybrid, där hastigheten antas vara känd i förväg. Optimeringen skedde genom att först härle- da en modell för drivlinan, och därefter skapades ett Dy- namisk programerings baserat program för att hitta den optimal kombinationen av moment.

Bränsleförbrukning och prestanda jämfördes genom pro- gramvaran GT-Suite. Dynamiska programmeringen gav lo- vande resultat som följde referenshastigheten i många fall.

Däremot uppdaterades inte laddningen för batteriet lika bra, vilket ledde till att batteriet i vissa fall blev urladdat.

Modellen som härleddes visade i många fall liknande respons som GT-Suite, men viss förbättring kan ske. En utav dessa förbättringar är rotationsekvationen för bräns- lemotorn, ωe.

Den Dynamiska programmeringen som skapades fun- gerade, och kan modifieras för förbättrade tillståndsekva-

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Acknowledgements

We would like to express our gratitude to Professor Tie- long Shen at Sophia University, Professor Xiaoming Hu at Royal Institute of Technology and Doctor Jiangyan Zhang at Sophia University for all the guidance, help and for giv- ing us the opportunity to work on this project. We would also like to thank all the people we have had the pleasure to meet while in Japan for making our stay so much more memorable.

Tokyo, July 2013 Elliot Dahl

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List of Figures

3.1 Schematic figure of the planetary gear. . . 5

4.1 Torques working on the planetary gear set. . . 10

4.2 Power balance between the ring gear and the power acting externally on the vehicle. . . 11

4.3 Model of the battery. . . 13

4.4 A polynomial approximation of the voltage as a function of SOC . . . . 15

4.5 A polynomial approximation of the resistance as a function of SOC when the battery is discharging. . . 16

4.6 A polynomial approximation of the resistance as a function of SOC when the battery is charging. . . 17

4.7 Approximation of BSF C as a function of Te and Ne . . . 18

4.8 ωe from derived and empirical model, with GT-Suite output, for 160 s. . 19

4.9 ωe from derived and empirical model, with GT-Suite output, for 2600 s. 20 4.10 ωm from derived and empirical model, with GT-Suite output, for 160 s. 21 4.11 ωm from derived and empirical model, with GT-Suite output, for 2600 s. 21 4.12 Comparison of SOC between GT-Suite and the derived model. . . 22

6.1 Speed and driver satisfaction results for Sunday, driving away from home. 32 6.2 SOC and fuel economy results for Sunday, driving away from home. . . 33

6.3 Engine power used during Sunday, driving away from home. . . 33

6.4 Motor power used during Sunday, driving away from home. . . 34

6.5 Generator power used during Sunday, driving away from home. . . 34

6.6 Fuel used during Sunday, driving away from home. . . 35

6.7 Fuel used during Sunday, driving away from home. Using higher SOC grid. . . 36

6.8 Speed and driver satisfaction results for Sunday, driving back home. . . 37

6.9 SOC and fuel economy results for Sunday, driving back home. . . 38

6.10 Engine power used during Sunday, driving back home. . . 38

6.11 Motor power used during Sunday, driving back home. . . 39

6.12 Generator power used during Sunday, driving back home. . . 39

6.13 Fuel used during Sunday, driving back home. . . 40

6.14 Speed and driver satisfaction results for Tuesday, driving to work. . . . 41

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6.16 Engine power used during Tuesday, driving to work. . . 42

6.17 Motor power used during Tuesday, driving to work. . . 43

6.18 Generator power used during Tuesday, driving to work. . . 43

6.19 Fuel used during Tuesday, driving to work. . . 44

6.20 Speed and driver satisfaction results for Tuesday, driving back home. . . 45

6.21 SOC and fuel economy results for Tuesday, driving back home. . . 46

6.22 Engine power used during Tuesday, driving back home. . . 46

6.23 Motor power used during Tuesday, driving back home. . . 47

6.24 Generator power used during Tuesday, driving back home. . . 47

6.25 Fuel used during Tuesday, driving back home. . . 48

6.26 Calculating the power given from Pm= Tmω_m. . . 51

6.27 Calculating the power given from Pg= Tgωg. . . 51

6.28 SOC as a function of ωe and ωm. . . 53

B.1 Comparison of ηeP_e+ ηbP_b with (^PF_R+ M ˙va) va for 1000 s. . . 60

B.2 Comparison of ηePe+ ηbPb with (^PFR+ M ˙va) va for 200 s. . . 61

D.1 Estimation of throttle as a function of BMEP and Ne. . . 67

D.2 Comparison of GT-Suite throttle and throttle from the estimated function. 67 E.1 Speed and reference speed from the DP program for the simplified prob- lem, 100 time steps. . . 72

E.2 Control and reference acceleration from the DP program for the simpli- fied problem, 100 time steps. . . 72

E.3 Speed and reference speed from the DP program for the simplified prob- lem, 20 time steps. . . 73

E.5 Speed and reference speed from the DP program for the simplified prob- lem, 20 time steps. . . 74

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List of Tables

4.4 Comparison of coefficients from the derived model and the GT-Suite

empirically derived model. . . 21

6.1 Results for different days. . . 49

6.2 Estimation of required SOC grid. . . 50

6.3 Amount of fuel needed to charge the battery to the same SOC level as the reference solution. . . 52

A.1 Physical Parameters and Nomenclature . . . 57

C.1 Values of coefficients for ˙ωe, ˙ωm and Tg. . . 63

D.1 Coefficients for throttle. . . 65

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Chapter 1

Introduction

In order to avoid global warming and to resolve energy resources issues it is necessary to improve the fuel consumption of automobiles. This has led to various different technologies for internal combustions engines and hybrid electric vehicles (HEV).

Many of these technologies incorporate very advanced algorithms in order to resolve the problem. Unfortunately, although Europe and the US have already strength- ened their cooperation with the academic world in order to implement and find new and better control algorithms [1], Japan lags behind [2]. This is a major drawback, considering that Japan is one of the major car exporters in the world with house- hold names such as Toyota, Mazda, Mitsubishi, Honda and Subaru just to mention a few. Therefore it is encouraged that the Japanese industry and Japanese Univer- sities start a joint research for a sustainable future. This is the reason why JSAE (Society of Automotive Engineers of Japan), SICE (The Society of Instrument and Control Engineers) and Japanese automobile industry created ”The Technical Com- mittee on Vehicle Control and Modeling” which came about in the year 2009. The committee posts so called benchmark problems, which are good research themes in the academics as well as having useful applications in the industry. The benchmark problem investigated in this report will be the ”Fuel Consumption Optimization of Commuter Vehicle Using Hybrid Powertrain”.

What is so special with a hybrid powertrain is that it may combine driving force from two power sources, for instance an internal combustion engine and an electrical motor, and can use the energy accumulated by the internal combustion engine freely as either a driving force or as to charge a battery. The internal combustion engine can therefore work under optimal conditions where it can achieve high thermal efficiency on the hybrid powertrain, but not without advanced algorithms. These algorithms are what will be researched in this report such that the fuel consumption of a hybrid powertrain is minimized.

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Chapter 2

Background Material

An interest of the hybrid vehicles has grown considerably in the last 15-20 years, although they have existed since before the 1900 [3]. One of the reasons is that different control systems can be used to improve fuel consumption within ranges of 10 − 30% compared with conventional vehicles. The main objective of the energy- management strategy for HEV are all the same, to minimize fuel consumption along some given route. In their article [3], Sciaretta and Guzella brings forth the different methods used for fuel consumption optimization and classifies some research groups active in HEV energy control with their respective method used.

The energy-management approaches are typically divided into two groups, they are heuristic strategies and optimal strategies [4].

The heuristic approaches are usually implemented in real time and are rule based. An example is the Fuzzy logic approach which is used by Schouten et al. [5]

and Baumann et al. [6].

The second approach, optimal strategies, uses tools from the optimal control theory and appropriate dynamical models to find the optimal controllers. There are two different kinds of methods [7],[8], global optimal methods that solves the entire problem as a whole, and local optimization methods which divides the global problem into smaller local problems. Under the global methods we find for example dynamic programming (DP) and Pontryagins minimum principle (PMP). Under the local methods, which may use some information about future or past driving conditions although not all of it, we find for example stochastic dynamic programming (SDP), model predictive control (MPC) and equivalent energy consumption minimization (ECMS).

In order to apply theories from optimal control one first need to find some kind of mathematical model describing the dynamics of the HEV. Such attempts have thoroughly been done by Syed et al. [9], Liu et al. [10] and Mansour et al. [11]

among others.

After arriving at a successful model, the optimal control strategy chosen depends strongly on what kind of prior information one has. If for instance all knowledge about the future route is known, then DP is a good alternative. This approach is

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CHAPTER 2. BACKGROUND MATERIAL often used as an benchmark [1],[4]. For example, Liu et al. uses DP to compare the performance of an SDP and an ECMS approach in [12]. Although there are existing articles that try to simplify the model enough in order to be able to use DP in real time implementation [13].

There are also approaches that use DP to extract specific rules [7], thus a mixture between an optimal strategy and a heuristic method.

The other global method mentioned above is PMP. The problem though with PMP is that the solution might not be the global optimal solution, however it is faster than DP, which is one of the reasons for Kim et al. to use this method to try to find a real- time PMP algorithm in their article [14]. Also here we find an intermediate approach which this time bases their control rules on PMP [15].

However, although the global optimal solutions are the most attractive, they do demand full knowledge of the given route, which might not always be available.

Local methods are therefore well suited for the challenge of finding satisfying energy management controllers with less prior knowledge.

One of the problem faced without knowledge of the routes speed and acceleration is that of the demanded power. This problem is directed in Lin et al. [16], where the power demand is modeled as random Markov process. The optimal control strategy is then obtained using SDP.

Future driving conditions can also be predicted with the help of Intelligent Trans- portation Systems, this has for example been done in [17], where they use this information in order to establish a prediction based real-time controller structure using MPC. This approach is showed to be comparable to ECMS when the velocity and load prediction is noise free.

Borhan et. al [18] considers two approaches to MPC, and compares them to a controller available in the commercial software PSAT. The first is linear time- varying MPC, which results in an improvement of fuel consumption compared to the PSAT controller, except for one case where the result is similar. For a second approach nonlinear MPC is used, where an improvement of fuel consumption is achieved compared to both LTV-MPC and the PSAT controller.

The last local method mentioned above, ECMS, was first introduced by Paganelli in the year 1999. The strategy is based on the idea that the stored electric energy only functions as a energy buffer and that in the end the energy always comes from the fuel, even the energy from the battery. This is because the the energy coming from the battery always has to be replenished later by the fuel from the engine, either directly from the engine or indirectly trough regenerative braking. So in both charge and discharge phase, a virtual fuel consumption may be added to the actual fuel consumption to obtain the instantaneous equivalent fuel consumption which is to be minimized at each moment [1]. The ECMS approach has proven very effective, for example in Sciaretta et al. paper [19], they show how the ECMS almost achieves the same fuel consumption results as that coming from the benchmark DP approach. This might be the reason for Sivertsson to develop an adaptive control strategy based on a map-ECMS approach for the PHEV benchmark problem that was organized by IFP Energies nouvelles in 2012 in [20].

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Chapter 3

Problem Description and Background Information

There are at least two kinds of HEVs: Charge-sustaining and plug-in HEV (PHEV).

The former aims to keep the state of charge of the battery at a given level, while the latter does not have this requirement. In the benchmark paper [2] it is not specified which kind of HEV is used. However the problem formulation is equivalent in the two different cases with the exception of the end state for the battery, and therefore doesn’t need to be included in the model derivation.

3.1 Hybrid electrical vehicle

The HEV in the considered benchmark problem is equipped with a split type hybrid powertrain. Two electrical motors, EM1 and EM2, plus an engine which are linked together by a planetary gear set, see Fig. 3.1.

Ring gear Carrier gear

Planet gear Sun gear

Figure 3.1. Schematic figure of the planetary gear.

EM1 is linked to the sun gear, the engine is linked to the carrier gear and EM2 is linked to the ring gear in the planetary gear set. Both EM1 and EM2 are able to drive the vehicle by using energy from a battery and they can both generate electric power to charge this same battery. EM1 can also be used to start the engine.

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CHAPTER 3. PROBLEM DESCRIPTION AND BACKGROUND INFORMATION However in this study we will only consider the following 5 modes to be possible, and use them in order to construct an energy management strategy.

1. The engine drives the vehicle.

2. EM2 drives the vehicle.

3. Both the engine and EM2 drives the vehicle.

4. The engine generates electrical power using EM1 and charges the battery.

5. EM2 generates electric power when the vehicle is decelerated.

3.1.1 Traffic conditions

For series production HEV, the control algorithm managing the hybrid powertrain is usually designed so that it will achieve moderate fuel consumption for driving situations of all users [2]. This means that it is not optimized for the best fuel consumption for a specific user and driving condition, so an optimization taking regards to the specific driving condition can potentially lower the fuel consumption in comparison to the general case, although more knowledge about the specific route is needed. For a general user, the following can describe common situations,

• Going to the office.

• Going back home.

• Driving on weekends.

When driving to the office on weekdays, there may be traffic jams due to a huge number of commuter vehicles, which results in lower and more varying speeds.

While when going home, the traffic might be more disperse, resulting in less traffic jams and higher speeds. The third situation, during weekends, might show a very different driving pattern from that of the weekdays, usually depending on the agenda of the day, see [2] for samples of the three situations.

3.1.2 Problem formulation

The problem considered is to minimize the fuel consumption of an HEV that rec- onciles the drivers demanded vehicle velocity, vd, by designing a control algorithm for the powertrain. The control algorithm should decide on the power distribution between the combustion engine and the electric motors in order to let the vehicle have a desired speed. At the same time, other constraints should be considered, such that keeping the battery charge (State of charge, SOC) within a given level.

Also, it is stated in [2] that it is generally known that the fuel economy of a vehicle is improved if the acceleration performance is restricted. However, there must be a trade-off between the drivers demand to accelerate fast and the benefit of restricted

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3.1. HYBRID ELECTRICAL VEHICLE

acceleration. Therefore, one of the constraints is also to keep a so called driver satisfaction parameter, Sd, at 90% or higher. This parameter is defined as

S_d(k) = Sd(k − 1) + ∆Sd(k), (3.1) where

∆Sd(k) =



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





0, |v_d(k) − va(k)| ≤ δV s

0.1Pn, δ_{V s} < |v_d(k) − va(k)| ≤ δV L

Pn, δV L < |vd(k) − va(k)| .

We have δV s = 7.5 km/h, δV L= 15 km/h, Pn= −1 and Sd(0) = 100. vd(k) denotes the demanded reference speed and va(k) denotes the actual speed at timestep k.

The number of steps of each route varies.

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Chapter 4

Mathematical Model of the HEV

In order to derive an optimal control algorithm for the HEV, system equations are needed in state space form describing the vehicle. A mathematical model based on some simplified physics is therefore derived below. Notation of variables and values of parameters are given in Table A.1 in Appendix A.

4.1 Model derivation

4.1.1 Vehicle dynamics

The HEV uses a planetary gear which enables the vehicle to use different control modes, as described in the previous chapter. EM1 is connected to the sun gear, the engine is connected to the carrier gear and EM2 is connected to the ring gear.

Fig. 4.1 is an illustration on how the different parts are linked together and how they affect each other. The torques direction is positive to the left as indicated by ω+. The dotted box indicates the planetary gear set. It is assumed that the planetary gears share a collective loss which does not depend on which part the power is coming from, i.e if it is from the engine, EM1 or EM2 in order to simplify the derivation. Since the primary function of EM1 is as a generator, and of EM2 as a motor, they will be called generator and motor, abbreviated g and m.

By applying the rigid body equation for a fixed axis [21], the following equations may be derived for the planetary gears seen in Fig. 4.1,

˙ωsI_s= F · Rs− T_s, (4.1)

˙ωcI_c= Tc− F(Rr+ Rs), (4.2)

˙ωrI_r= F · Rr− T_r, (4.3) where Ts, Tcand Tr are the torques on the sun gear, carrier gear and the ring gear.

Their respective inertia is denoted Is, Ic and Ir. Rr and Rs represent the ring

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CHAPTER 4. MATHEMATICAL MODEL OF THE HEV

Figure 4.1. Torques working on the planetary gear set.

gear and the sun gears radius from the center, F denotes the internal force working between the gears.

The equations for the generator, engine and motor may be derived in similar fashion,

˙ωgI_g = Tg+ Ts, (4.4)

˙ωeIe= Te− T_c, (4.5)

˙ωmI_m= Tm+ Tr− T_d, (4.6) where Td is shown in Fig 4.2. The definition of Tg is such that when Tg and ωg has opposite signs, i.e. Pg = Tgωg<0, the generator will charge the battery. This will be more clear when the model of the battery is developed.

Now Eq. (4.4) and (4.1) together with ˙ωs= ˙ωg gives

˙ωg(Ig+ Is) = F · Rs+ Tg. (4.7) Eq. (4.5) and (4.2) together with ˙ωc= ˙ωe gives

˙ωe(Ie+ Ic) = Te− F ·(Rr+ Rs). (4.8) And Eq. (4.6) and (4.3) together with ˙ωr= ˙ωm gives

˙ωm(Ir+ Im) = F · Rr+ Tm− T_d. (4.9) From Fig 4.2 we may derive the following relationships, which will eventually lead to an expression for ˙ωm. Note that for the inertia of the vehicle, we consider the

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4.1. MODEL DERIVATION

Figure 4.2. Power balance between the ring gear and the power acting externally on the vehicle.

vehicle as a point mass, so that Iv = MR²_tire. For the driveshaft, differential-gear, 4 axles and finally the vehicle we get

˙ωdI_d= ηmG_fT_d− T_dg, (4.10)

˙ωdgI_dg= Tdg− T_a, (4.11)

˙ωa4Ia= ηdgTa− T_v, (4.12)

˙ωvIv = Tv− T_f. (4.13)

η_m represents the energy losses that are present in the planetary-gear. Gf is the differential gear ratio, i.e how fast the rotational speed of the driveshaft is compared to the rotational speed of the ring gear. ηdgrepresents the transmission efficiency of the differential gear. Eq. (4.10), (4.11), (4.12) and (4.13) with ˙ωv = ˙ωa= ˙ωdg= ˙ωd, eliminating Tdg, Ta and Tv, gives

˙ωd(Iv+ 4Ia+ ηdgI_dg+ ηdgI_d) = ηdgηmG_fT_d− T_f. (4.14) Eq. (4.9) and (4.14) with ωd= ωm/G_f gives

˙ωm(I_v+ 4Ia+ ηdgI_dg+ ηdgI_d

Gf + ηdgη_mG_f(Ir+ Im)) = ηdgη_mG_f(F · Rr+ Tm) − Tf. (4.15) In Eq. (4.15) Tf is an expression involving the friction brake and the opposing forces [22]. The expression for Tf is

T_f = Tf b+ Rtire(µrM gcos(θ) +1

2ρAC_dv²+ Mg sin(θ)). (4.16)

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CHAPTER 4. MATHEMATICAL MODEL OF THE HEV The first term on the right hand side of Eq. (4.16) is Tf bwhich is the torque caused from friction braking. The next one µrM gcos(θ) is caused by the friction forces between the wheels and the road. ¹₂ρAC_dv² is the friction forces caused by the air resistance and last Mg sin(θ) is the forces caused by gravity.

The planetary gear functions as a speed-summing unit [22], with the angular velocities related as

˙ωmR_r+ ˙ωgR_s= ˙ωe(Rr+ Rs). (4.17) There are now four unknown variables, ˙ωg, ˙ωe, ˙ωm and F and four equations, (4.7), (4.8), (4.15) and (4.17). With these equations we may solve for ˙ωe and ˙ωm, which will constitute two out of three state variables that will be considered. The last one will be the state of charge (SOC). The results are,

˙ωe = (Iv⁰R²_s+ Ig⁰R_r²η)Te+ (Rr+ Rs)Ig⁰R_rηT_m I_g⁰I_v⁰(Rr+ Rs)²+ Ie⁰I_v⁰R²_s+ Ie⁰I_g⁰R²_rη +(Rr+ Rs)Iv⁰R_sT_g− I_g⁰R_r(Rr+ Rs)Tf

I_g⁰I_v⁰(Rr+ Rs)²+ Ie⁰I_v⁰R²_s+ Ie⁰I_g⁰R²_rη (4.18) and

˙ωm= (Rr+ Rs)RrI_g⁰ηT_e+ (Ig⁰η(Rr+ Rs)²+ IcR²_sη)Tm

I_g⁰I_v⁰(Rr+ Rs)²+ Ie⁰I_v⁰R²_s+ Ie⁰I_g⁰R²_rη +−I_e⁰RrRsηTg−(Ie⁰R²_s+ Ig⁰(Rr+ Rs)²)Tf

I_g⁰I_v⁰(Rr+ Rs)²+ Ie⁰I_v⁰R²_s+ Ie⁰I_g⁰R²_rη , (4.19) where

I_g⁰ = Ig+ Is, (4.20)

I_e⁰ = Ie+ Ic, (4.21)

I_v⁰ = I_v+ 4Ia+ ηdgI_dg+ ηdgI_d

Gf + ηdgη_mG_f(Ir+ Im) (4.22) and

η= ηdgηmGf. (4.23)

4.1.2 Battery

The third state is the state of charge, or SOC, given in % where 100% represents a fully charged battery and 0% a depleted battery. The SOC will be represented by a number between 0 and 1. The dynamics is given by

SOC˙ = −I_b

Q_b, (4.24)

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where Ib is the battery current and Qb is the maximum capacity of the battery. For modeling the battery an internal resistance model is used, as shown in Fig. 4.3, where U is the battery voltage, Uoc the open circuit voltage of the battery and Rb

denotes the internal resistance of the battery. The model is simplified in that it does not describe all the dynamics in detail. This is an approach that has been used earlier, for instance in [10] and [11].

Figure 4.3. Model of the battery. U is the battery voltage, Uocthe open circuit voltage of the battery, Ib the batery current and Rb the internal resistance of the battery.

From basic electric circuit analysis, we have Ohm’s law U = R · I,

P = U · I. (4.25)

From this it follows that the power output from the battery can be written as

P_b= UIb. (4.26)

Using Kirchoff’s voltage law to find U = Uoc− U_b, where Ub = RbI_b, Eq. (4.26) becomes

P_b = UocI_b− I_b²R_b. (4.27) By solving Eq. (4.27) for Ib, we get

I_b = Uoc±^pU_oc² −4PbR_b 2Rb

. (4.28)

From this the state equation for SOC, which is the final state equation needed, is given by

SOC˙ = − Uoc−^pU_oc² −4PbR_b 2RbQ_b

!

, (4.29)

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CHAPTER 4. MATHEMATICAL MODEL OF THE HEV with the minus sign in Eq. (4.28) since with Pb = 0 the SOC should be constant.

It can be noted that when Pb <0 the battery is charging, since ˙SOC >0 and when P_b > 0 the battery is discharging. Furthermore, since the battery power flow is either from or to the motor or the generator we have the following relationship,

Pb = ηmePm+ ηgePg, (4.30) where

ηme=

(ηme,disscharge, P_m >0 ηme,charge, Pm <0,

while ηge only takes on one value since it is considered to only be able to charge the battery.

Voltage

Since the voltage Uoc varies with the SOC, a mathematical model is needed. GT- Suite¹, which will be considered the reference model, includes a map of the battery voltage at different levels of the SOC. From the map it can be seen that the voltage takes values between 202 and 237.29 V .

By performing a polynomial fit to this data, the expression obtained was U_oc(SOC) = au0+au1SOC+au2SOC²+au3SOC³+au4SOC⁴+au5SOC⁵. (4.31) The values of the coefficients are

au0= 202.0361 au1= 95.3121 au2= −284.6488 a_u3= 604.1528 au4= −696.0737 au5= 316.5064.

A plot of Uoc(SOC) compared to the data from the map is seen in Fig. 4.4.

1GT-Suite HEV model is provided by Yuji Yasui, Honda R&D Co.

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0 0.2 0.4 0.6 0.8 1

200 205 210 215 220 225 230 235 240

SOC [−]

U oc [ V ]

Map from GT−Suite Polynomial approximation

Figure 4.4. A fifth degree polynomial approximation of the voltage as a func- tion of SOC.

Battery Resistance

The battery resistance Rb also needs to be modelled. In a similar way as for the voltage, a map from GT-Suite is used. From this it can be seen that the resistance depends on both the SOC and if the battery is charging or discharging. The approximation is given as

R_b(SOC) =











a_d0+ ad1SOC+ ad2SOC²+ ad3SOC³+ · · ·

a_d4SOC⁴+ ad5SOC⁵+ ad6SOC⁶, discharging a_c0+ ac1SOC+ ac2SOC²+ ac3SOC³+ · · ·

ac4SOC⁴+ ac5SOC⁵+ ac6SOC⁶, charging

(4.32)

with coefficients

a_d0= 0.7022 ad1= 0.1398 ad2= −17.1832 a_d3= 76.0844 a_d4= −138.7320 ad5= 115.9729 ad6= −36.5850

ac0= 0.7012 ac1= 0.1422 ac2= −15.1353 ac3= 64.0950 ac4= −112.5153 ac5= 91.2406 ac6= −28.1699.

See Fig. 4.5 and 4.6 for an illustration of Eq. (4.32). From the map it is also found that the resistance is between 0.357 and 0.7 Ω.

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0 0.2 0.4 0.6 0.8 1

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

SOC [ − ] R b,discharge [ Ω ]

Figure 4.5. A polynomial approximation of the resistance as a function of SOCwhen the battery is discharging.

Electric efficiency of the motor and generator

In the problem description [2] the electric efficiency of the generator and motor, ηge

and ηme, are not given. Therefore they are estimated by using data from GT-Suite.

From the equation for Pb (4.30),

Pb = ηmePm+ ηgePg

and the state equation for the SOC (4.29),

SOC˙ = − U_oc−^pU_oc² −4PbR_b 2RbQb

! ,

we find

SOC˙ = −





U_oc−^qU_oc² −4Rb(ηmeP_m+ ηgeP_g) 2RbQb



. (4.33)

By obtaining the state of charge, SOC from GT-Suite in specific intervals were the motor and generator power is either only positive or negative, we can estimate Uoc

from Eq. (4.31), Rb from Eq. (4.32) and approximate SOC˙ from Euler forward.

With these values together with the power of the motor, Pm and the generator, Pg, 16

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0 0.2 0.4 0.6 0.8 1

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

SOC [ − ] R b,charge [ Ω ]

Figure 4.6. A polynomial approximation of the resistance as a function of SOCwhen the battery is charging.

also received from GT-Suite we can find an approximate value of the efficiencies through an iterative least squares estimation. After this had been done we plugged these values into our model and compared it to the GT-Suite response, we found that by manually adjusting these values we could come closer to the GT-Suite re- sponse. The estimates obtained were found to be ηge= 0.8425,

η_me =

(ηme,disscharge = 1.258, Pm>0 η_me,charge= 0.818, P_m<0.

4.1.3 Fuel Consumption

For modeling the fuel consumption mf, the equation

BSF C= ˙mf

Pe

, (4.34)

relating the fuel flow with the power and the BSFC (Brake Specific Fuel Consumption)[23], will be used. The BSFC is measured from the engine, and is provided as a map in GT-Suite. By approximating the map as a function of the engine torque and the

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engine rotational speed expressed in [rpm], the following function can be found BSF C(Te, Ne) = b0+ b1Ne+ b2Te+ b3N_e²+ b4NeTe+ b5T_e²+ b6N_e³

+b7N_e²T_e+ b8N_eT_e²+ b9T_e³ (4.35) with coefficients

b0= 6.9933 · 10² b1 = −6.5760 · 10⁻³ b2= −1.8639 · 10¹ b₃= 3.1841 · 10⁻⁶ b₄ = −9.7399 · 10⁻⁵ b₅= 2.27 · 10⁻¹ b₆= −2.6412 · 10⁻¹⁰ b₇ = 2.1534 · 10⁻⁸ b₈= −8.4366 · 10⁻⁷ b9= −8.5085 · 10⁻⁴.

The unit of BSFC is _{kW ·h}^g . Since P = T ω the fuel consumption per second can be calculated as

˙mf = TeωeBSF C(Te, Ne) 3.6 · 10⁶ ,

g s

. (4.36)

A plot of BSF C as a function of Te and Ne can be seen in Fig. 4.7. The BSFC takes values between 201 and 720 _{kW ·h}^g according to the GT-Suite map.

Figure 4.7. Approximation of BSF C as a function of Te and Ne, together with map data from GT-Suite

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4.2. MODEL VERIFICATION

4.2 Model verification

4.2.1 Verification of ω_e and ω_m

In order to verify the derived model, Eq. (4.18) and (4.19) was implemented in Simulink and compared with the GT-Suite response. It showed considerable dynamical similarities with the GT-Suite output. However, Jiangyan Zhang, at Sophia University 2013, had also found an empirically based model which showed even greater similarities with the GT-Suite response. In Fig. 4.8 and 4.9 the two mod- els for ωe are compared with the GT-Suite response, and in Fig. 4.10 and 4.11 a comparison is made for ωm. For ωe and ωm we also found that the two derived dynamical equations would follow better with some constraints. For ωe we set if ωe(k − 1) ≤ 0 and ˙ωe(k) < 0 or ωe(k) < 0 then ωe(k) = 0. If ωe(k − 1) ≥ ωe,max

and ˙ωe(k) > 0 or ωe(k) > ωe,maxthen ωe(k) = ωe,max. We found similar constraints for ωm, if ωm(k − 1) ≤ 0 and ˙ωm(k) < 0 or ωm(k) < 0 then ωm(k) = 0. These constraints are set so that ωe and ωm does not go outside the feasible range.

0 20 40 60 80 100 120 140 160

0 500 1000 1500 2000 2500 3000

Time [s]

Angular velocity [rad/s]

Derived Empirical GT−Suite

Figure 4.8. Comparison of ωebetween the derived model, the empirical model, and the GT-Suite output, for a time period of 160 s.

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0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500 3000

Time [s]

Figure 4.9. Comparison of ωebetween the derived model, the empirical model, and the GT-Suite output, for a time period of 2600 s.

20

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4.2. MODEL VERIFICATION

0 20 40 60 80 100 120 140 160

0 200 400 600 800 1000 1200 1400 1600 1800

Time [s]

Figure 4.10. Comparison of ωm between the derived model, the empirical model, and the GT-Suite output, for a time period of 160 s.

0 500 1000 1500 2000 2500

0 500 1000 1500 2000

Time [s]

Figure 4.11. Comparison of ωm between the derived model, the empirical model, and the GT-Suite output, for a time period of 2600 s.

The coefficients from Eq. (4.18) and (4.19), with values from data given in Table A.1 in Appendix A, is presented together with the coefficients from the empirically derived model in Table 4.4.

Table 4.4: Comparison of coefficients from the derived model and the GT-Suite empirically derived model.

Parameters Te Tm Tg Tf

ωe, Derived 2.0086 0.0567 7.0838 -0.0148 ωe, Empirical 2.6841 0.0876 10.2663 -0.0153 ω_m, Derived 0.0567 0.0785 -0.0947 -0.0300 ωm, Empirical 0.0616 0.1249 -0.1030 -0.0293

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4.2.2 Verification of SOC

The mathematical model of the SOC was also compared with the SOC from the GT-Suite when the same power was applied. In Fig. 4.12 the SOC model with the derived efficiencies is compared to the actual SOC output from GT-Suite.

Since the SOC becomes complex when Uoc² −4PbR_b is negative, we set a restriction so that the square root term in Eq. (4.29) is excluded whenever it is does not take on real numbers.

0 500 1000 1500 2000 2500

0.55 0.56 0.57 0.58 0.59 0.6

Time [s]

SOC [−]

Estimated GT−Suite

Figure 4.12. Comparison of SOC between GT-Suite and the derived model.

4.3 State dynamical equations used

Although the derived dynamical models for ωe and ωm, Eq. (4.18) and (4.19) shows great dynamical similarities with the GT-Suite response, we decided to continue our research with the more accurate empirically derived model in order to eventually approach a lower fuel consumption. The dynamical models that are to be used are,

˙ωe = 2.6841Te+ 0.0876Tm+ 10.2663Tg−0.0153Tf (4.37)

˙ωm= 0.0616Te+ 0.1249Tm−0.1030Tg−0.0293Tf (4.38) and the unchanged

SOC˙ = −(Uoc−^pU_oc² −4PbRb

2RbQ_b ). (4.39)

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Chapter 5

Control and Optimization

We will now derive and state the optimization problem.

5.1 General form of the optimization problem

The general mathematical form of an optimal control problem usually contains similar features [24]. They are seen in Eq.(5.1)-(5.5),

minimize

u(·) J(t0, x_o, u(·)) = Φ(x(tf)) +^Z ^t^f

t0

f₀(t, x(t), u(t))dt (5.1) subject to

˙x(t) = f(t, x(t), u(t)) (5.2)

x(0) = x0 (5.3)

x(t) ∈ X(t) (5.4)

u(t) ∈ U(t), (5.5)

where x(t) is the state vector and u(·) an admissible control vector on [t0, tf].

5.2 Dynamic Programming

5.2.1 General concepts

Dynamic programming is a method that solves an optimal control problem by di- viding the problem into several sub-problems which are to be solved and eventually added into a final solution [25]. The method is based on Bellman’s principle of optimality, which can be stated as follows,

Let u^∗ : [t0, t_f] → R^m be an optimal control for minu(·)J(t0, x₀, u(·)) that generates the optimal trajectory x^∗ : [t0, tf] → Rⁿ. Then, for any t⁰ ∈ (t0, t_f], the restriction of the optimal control to [t⁰, t_f], u^∗|_[t0,t_f], is

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CHAPTER 5. CONTROL AND OPTIMIZATION optimal for minu(·)J(t⁰, x^∗(t⁰), u(·)) and the corresponding optimal tra- jectory is x^∗|_[t⁰_,t_f_] [24].

This means that the optimal path from any of the intermediate steps to the end corresponds to the optimal solution at this point to the end.

5.2.2 Discrete Dynamic Programming

If the continuous optimal control problem in Eq.(5.1)-(5.5) instead is discretised, we find the multistage decision problem

minimize_u

k J(0, x0) = Φ(xN) +

N −1

X

k=0

f₀(k, xk, u_k) (5.6) subject to

x_k+1 = f(k, xk, u_k) (5.7)

x₀ given (5.8)

x_k∈ X_k (5.9)

uk ∈ U(k, xk). (5.10)

The principle of optimality can in the discrete state be formulated as, If {u^∗_k}^{N −1}_k=0 is an optimal control for the problem (5.6)-(5.10), then {u^∗_k}^{N −1}_k=n is optimal for the subproblem obtained by considering an opti- mization on the problem (5.6)-(5.10) but with initial condition (n, x^∗(n)) i.e., we restart the optimization from somewhere along the optimal path [24].

With

J^∗(n, xn) = min Φ(xN) +

N −1

X

k=n

f₀(k, xk, u_k) (5.11) and the principle of optimality we can find the following theorem.

Theorem 1 Suppose there exist a finite solution to the backwards dynamic pro- gramming recursion

J(N, xN) =

(Φ(xN), xN ∈ X_N

∞, xN ∈ X/ _N

J(n, xn) = min

u∈U (n,xn){f₀(n, xn, u) + J(n + 1, f(n, xn, u))}, n = N − 1, N − 2, . . . , 0 where the optimization over U(n, xn) is restricted to those control variables for which f(n, xn, u) ∈ Xn+1. Then there exists an optimal solution to problem (5.6)-(5.10) and

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Optimization of Fuel Consumption in Hybrid Electric Vehicles

Optimization of Fuel Consumption in Hybrid Electric Vehicles

Optimization of Fuel Consumption in Hybrid Electric Vehicles

Abstract

Sammanfattning

Acknowledgements

Contents

List of Figures

List of Tables

Chapter 1

Introduction

Chapter 2

Background Material

Chapter 3

Problem Description and Background Information

3.1 Hybrid electrical vehicle

Chapter 4

Mathematical Model of the HEV

4.1 Model derivation

4.2 Model verification

4.3 State dynamical equations used

Chapter 5

Control and Optimization

5.1 General form of the optimization problem

5.2 Dynamic Programming