Minimising Fuel Consumption of a Series Hybrid Electric Railway Vehicle Using Model Predictive Control

Master of Science Thesis in Electrical Engineering Department of Electrical Engineering, Linköping University, 2017

Niklas Sundholm Master of Science Thesis in Electrical Engineering

Minimising Fuel Consumption of a Series Hybrid Electric Railway Vehicle Using Model Predictive Control

Niklas Sundholm

LiTH-ISY-EX--17/5095--SE

Supervisor: Måns Klingspor isy, Linköping University Keiichiro Kondo Department of Electrical and Electronic Engineering, Chiba University Examiner: Martin Enqvist isy, Linköping University

Automatic Control Department of Electrical Engineering Linköping University SE-581 83 Linköping, Sweden

With the increasing demands on making railway systems more environmentally friendly, diesel railcars have been replaced by hybrid electric railway vehicles. A hybrid system holds a number of advantages as it has the possibility of recuperating energy and allows the internal combustion engine (ice) to be run at optimal eﬃciency. However, to fully utilise the advantages of a hybrid system the hybrid electric vehicle (hev) is highly dependent on the used energy management strategy (ems). In this thesis, the possibility of minimising the fuel consumption of the series hybrid electric railway vehicle, Ki-Ha E200, has been studied. This has been done by replacing the currently used ems, based on heuristics, with a model predictive controller (mpc). The heuristic ems and the mpc have been evaluated by comparing the performance results from three diﬀerent test cases. The performance of the implemented mpc seems promising as it yields more optimal operation of the ice and improved control of the battery state of charge (soc).

iii

Acknowledgments

Firstly, I want to express my utmost gratitude to Shigetomo Shiraishi, Toshiba Corporation Railway Systems Division, for making this thesis work possible and Keiichiro Kondo, Chiba University, for accepting me to his laboratory. Further, I want to thank Martin Enqvist and Måns Klingspor for their valuable help and discussions through out this thesis work.

Tokyo, June 2017 Niklas Sundholm

Contents

Notation ix

1 Introduction 1 1.1 Background ...... 1 1.2 Purpose ...... 2 1.3 Objective ...... 2 1.4 Methodology ...... 3 1.5 Limitations ...... 3 1.6 Thesis Outline ...... 3

2 Series Hybrid Electric Railway Vehicle Modelling 5 2.1 Hybrid Electric Vehicle Powertrain ...... 5 2.2 Train Motion ...... 7 2.2.1 Train Resistances ...... 10 2.3 Internal Combustion Engine ...... 11 2.4 Battery ...... 12 2.5 Model Equation ...... 15

3 Controller Design 17 3.1 MPC Basics ...... 17 3.2 Hybrid MPC ...... 19 3.2.1 Piecewise Aﬃne Model ...... 19 3.2.2 Mixed-Integer Programming ...... 21 3.2.3 Objective Function ...... 22 3.2.4 Constraints ...... 23 3.2.5 Softened Constraints ...... 25 3.2.6 Move-Blocking ...... 26

4 Results 29 4.1 Heuristic Controller ...... 29 4.2 Test Cases ...... 31 4.3 Simulator ...... 32 4.4 Simulation Results ...... 32

vii viii Contents

4.5 Discussion ...... 41

5 Conclusions and Future Work 45 5.1 Conclusions ...... 45 5.2 Future Work ...... 46

Bibliography 47 Notation

Abbreviations Abbreviation Interpretation hev Hybrid Electric Vehicle ems Energy Management Strategy ice Internal Combustion Engine ac Alternating Current dc Direct Current soc State of Charge mpc Model Predictive Control pwa Piecewise Aﬃne mip Mixed-Integer Programming qp Quadratic Programming miqp Mixed-Integer Quadratic Programming

1 Introduction

This master’s thesis presents the results of the work that was conducted at Toshiba Corporation Railway Systems Division in cooperation with Chiba University, Ja- pan.

1.1 Background

With the rise of environmental issues, due to global warming, emphasis has been put on making systems more environmentally friendly. In the case of the railway industry, different measures have been taken to reduce the total energy consumption and the environmental burden of trains. Among the measures that have been taken are weight saving, improved component efficiencies and utilisation of regenerative braking. Within the railway transportation system there are many non-electrified sections, mainly in rural areas, where diesel trains are operated. One of the major drawbacks of diesel trains, apart from their exhaust emissions, is their inability to recuperate energy through regenerative braking. In 2003, East Japan Railway Company addressed the environmental burden of diesel railway vehicles by developing the world’s first railcar, Ki-Ha E200, utilis- ing a hybrid system. A hybrid electric vehicle (hev) combines the conventional internal combustion engine (ice) with an electric motor that will, depending on the hybrid configuration, fully or partially act as the main propulsion system. The presence of an electrochemical storage device is common in most modern hybrid systems. The railcar Ki-Ha E200 uses a diesel engine as ice and a lithium-ion battery as storage device [6]. Ki-Ha E200 started operational services in 2007 [13]. A hev holds a number of advantages compared to a diesel railcar. A hybrid system

1 2 1 Introduction has the possibility of recuperating energy through regenerative braking, down- sizing of the engine, running the diesel engine at its optimal eﬃciency, reducing toxic substances in the exhaust emission and reducing noise levels when running inside stations.

In order to fully utilise the advantages of a hybrid system, a supervisory energy management strategy (ems) is needed to determine the optimal combination of the available power sources. A great amount of research has been done in optimal control of hevs, with the majority targeted towards automobiles [16]. As the computational power of modern computers has increased it has allowed the use of computationally heavy control methods such as model predictive control (mpc) [18], [17]. The principles used for automobiles can similarly be applied to locomotives as well. In [12] the optimal operation of railway vehicles, minimising total energy consumption, is analysed by comparing the use of dynamic programming, the gradient method and sequential quadratic programming. It is stated that none of the aforementioned methods are applicable for real-time control. However, mpc can be seen as an approximation of dynamic programming and could be realisable in railway applications if a control cycle of 1 second or less is achieved.

1.2 Purpose

The series hev Ki-Ha E200 currently uses a heuristic ems, based on the hevs speed and the state of charge (soc) of the storage battery, to determine the operation of the ice. The performance of the heuristic ems is highly dependent on the running pattern of the train and can therefore not guarantee optimal control. The purpose of this master’s thesis is to investigate the possibility of minimising the fuel consumption of the Ki-Ha E200 by replacing the current ems with a hybrid mpc instead.

1.3 Objective

The objective of this master’s thesis can be divided into a number of sub-objectiv- es. Firstly, a basic model of the series hev Ki-Ha E200 should be developed. The model will be used to simulate the target hev and in the design of the hybrid mpc. The existing heuristic ems should be implemented and its performance should be used as a benchmark for the hybrid mpc. The performance of the two ems should be compared and analysed. 1.4 Methodology 3

1.4 Methodology

A general study of the target railway system was conducted to determine the scope of the thesis work and to understand the basic principles. Secondly, a more specific literature study of emss used in the control of hevs was carried out, where the focus was specifically set on earlier applications of mpc. To implement an mpc, with satisfactory performance, a precise model is required. Since no data was available for system identification, theoretical studies of the train components were carried out. To simplify the modelling of the hev certain components have been assumed to be static, while emphasising the modelling of the train motion, diesel engine and battery dynamics. Since previous modelling of the Ki-Ha E200 varied from the derived train model in this thesis work, the existing heuristic ems was implemented and used as a performance benchmark. Due to the non-linearities in the derived train model, the model was approximated by a number of piecewise affine (pwa) models instead that could be used as an internal model in the hybrid mpc. The performance of the hybrid mpc and heuristic emss was evaluated by applying them to predetermined test cases and comparing the acquired results.

1.5 Limitations

The thesis is limited to the modelling of a series hybrid electric railway vehicle. The modelling is based on the train speciﬁcations and performance limitations of the railway vehicle Ki-Ha E200. Furthermore, only longitudinal motion of the train has been considered in this thesis, thus a straight track has been assumed in all of the test cases. All implementation and simulation is done using Mat- lab and Simulink. Due to performance restrictions in the platform used to run simulations, the mpc prediction horizon is limited to a maximum of 40 time steps.

1.6 Thesis Outline

Chapter 2 introduces the target hev Ki-Ha E200 and gives a basic overview of its components. More detailed modelling of the train motion, ice and battery dynamics is given. The results of the modelling is summarised in a model equation used to simulate the train dynamics and to develop the hybrid mpc. Continuing, Chapter 3 explains the basic theory of mpc. The basic mpc theory is expanded to describe hybrid mpc and miqp optimisation problems. After cov- ering the hybrid mpc prerequisite, the design and implementation of the hybrid mpc constraints and objective function is described. 4 1 Introduction

In Chapter 4, the simulator implementation is presented. A short description of the heuristic controller is given followed by a presentation of the test cases that are used to analyse the performance of the emss. The simulation results are then presented, followed by a discussion and analysis of the obtained results. Finally, Chapter 5 concludes the thesis and a few ideas of possible future work are mentioned. 2 Series Hybrid Electric Railway Vehicle Modelling

This chapter aims at deriving a realistic model of a series hev. The train parameters are based on the Ki-Ha E200 speciﬁcations given in [13] and [14]. The purpose of the model is to describe the train dynamics, which can be used to simulate the train and implement an mpc. Modelling a complete railway vehicle is a complex task, thus various simpliﬁcations have to be done. The modelling of the train motion and battery dynamics is mainly based on the theory presented in [9].

2.1 Hybrid Electric Vehicle Powertrain

An hev combines an ice with an electric propulsion system. The combination of these two propulsion systems allows for a more efficient system at the cost of a more complex structure and control. Depending on the objective of the system, different hev configurations are used. hevs are classified into series, parallel or series-parallel hybrid systems depending on how the powertrain’s components are configured. The hev configuration determines how the power from the ice and/or electric motor is delivered to the wheels of the vehicle. The train Ki-Ha E200 is a series hev, which is the simplest kind of hybrid configuration. The usage of this configuration holds a number of advantages. Firstly, energy can be recovered through regenerative braking and stored in the battery. This is done by using the traction motor as a generator to convert the train’s kinetic energy into electrical energy during deceleration of the train. Secondly, the diesel engine can be operated at its optimal fuel efficiency, with minimum exhaust fumes, since the engine operation does not directly depend on the power

5 6 2 Series Hybrid Electric Railway Vehicle Modelling requirements of the train. This also allows for the ice to be completely shut off, to avoid idling, when the train is standing still. Furthermore, the train can be powered solely using the battery to reduce noise levels and emissions when approaching and stopping at train stations. A schematic figure of the component configuration of the series hev Ki-Ha E200 is depicted in Figure 2.1. A description of the notation used in Figure 2.1 and a brief overview of the components is given below. The internal combustion engine (ICE) converts chemical energy, stored in the fuel, to mechanical energy. The ice receives the input signal Pice∗ , which is a reference for the desired ice power, and outputs the power Pice. Since the knowledge about the diesel engine is limited, the ice dynamics have been approximated by a simple expression presented in Section 2.3. The mechanical energy from the ice is transformed into electrical energy by the electric generator (EG). The generator is a three-phase induction generator that outputs alternating current (ac). The dynamics of the electric generator will not be modelled and will be considered static, described by the efficiency constant ηeg . The direct current (dc) link (DC) connects the ac circuits in the hev. It is assumed that there are no losses in the dc link. To connect the electric generator to the dc link the ac output has to be converted to dc first. The conversion is done by using a converter (CONV), which is a pulse width modulation recti- fier. The converter is considered static and is described by the efficiency constant ηconv. A battery (B) and auxiliary power supply unit (AUX) is connected to the dc link. These are both dc components that use the same voltage as the dc link, hence no conversion is required. The battery transforms electrical energy to chemical energy and vice versa. A more detailed description and model of the battery can be found in Section 2.4. The auxiliary power supply unit supplies power to the air-conditioning and interior lights of the hev. The supplied power is assumed to be the constant power Paux at all times. The electric motor (EM), or traction motor, is a three-phase induction motor. The electric motor is controlled by an inverter (INV), which transforms dc to ac.

Figure 2.1: Component conﬁguration of the series hev Ki-Ha E200. The arrows indicate the power ﬂow directions. 2.2 Train Motion 7

The inverter is bi-directional meaning that it can pass power in both directions, making it suitable for both regeneration and motor control. This component is considered static in this thesis and is described by the efficiency constant ηinv. The electric motor can be used either to propel the train or as a generator in the case of regenerative braking. When the train is propelled the electric motor converts electrical energy into mechanical energy and when regenerative braking is used the electric motor converts the train’s kinetic energy into electrical energy. In this thesis, the electric motor is considered static with the same efficiency, ηem, for both operations. The gear-box (GB) provides an angular velocity and torque conversion between the electric motor output and the wheels of the hev. This component is considered static and is described by the efficiency constant ηgb. The power output at the wheels of the hev is denoted Pout. Based on the brief description of the hev’s component’s functionality and their circuit connection, an analysis of the power flow at the dc link yields

P (t) + P (t) P (t) P = 0, (2.1) conv b − inv − aux where Pconv(t) is given by

Pconv(t) = ηconv ηeg Pice(t). (2.2)

As mentioned above, Pinv(t) depends on if the electric motor is operating as a traction motor or as a generator. The eﬃciency is assumed to be the same for both operations, thus the inverter power is calculated according to

 P (t)  out , powering P (t) =  ηinv ηemηgb (2.3) inv  ηinv ηemηgbPout(t). regenerating

More detailed descriptions and modelling of the vehicle motion, the ice and the battery dynamics are described in the sections below.

2.2 Train Motion

This section aims at developing a simple model that describes the main dynamics of the train motion. By limiting the train to only move in one direction, the curvature of the track can be neglected and only the longitudinal dynamics need to be studied. The longitudinal dynamics of a railway vehicle, running at a certain speed v along a track with inclination α, can be expressed by Newton’s second law of motion as dv(t) m = F (t) F (t) F (t) F (t), (2.4) dt t − mb − tr − g 8 2 Series Hybrid Electric Railway Vehicle Modelling

Figure 2.2: Forces acting on a railcar running at speed v along a track inclined by the angle α.

where Ft is the tractive force, Fmb the mechanical break force, Ftr the train resistance and Fg the gravitational force. The equivalent mass of the hev’s rotating parts, such as the wheels and ice, is assumed to be included in m in (2.4). The mass that has been added due to the inertia of the train’s rotating parts, is assumed to be 10% of the train weight. Figure 2.1 depicts the forces acting on a railway vehicle when running along an inclined track. The amount of energy that the train consumes per unit time is referred to as the output power of the railway vehicle and is calculated as

Pout(t) = Ft(t)v(t), (2.5) where the tractive force Ft(t) is the force exerted on the railway vehicle’s driving wheels. Apart from the train resistance, which will cause the train to decelerate, the tractive and gravitational force determines the acceleration or deceleration of the train. When a train is designed, suitable components have to be chosen to meet the requirements that have been set on the performance of the train. The chosen components will determine the characteristics of the traction system which is described by the tractive and brake force curves.

The tractive force Ft(t) is constrained to a value between Ft,min and Ft,max. The value of the tractive force determines if the train is powering, braking or coasting. A positive value corresponds to propelling (powering) the train, a negative value corresponds to braking and a value of zero corresponds to coasting. Depending on the speed of the train, diﬀerent factors limit the max- and minimum tractive force. The tractive and brake curves can be divided into the constant torque region, the constant power region and the characteristics region. In the constant torque region the tractive force is limited due to the maximum possible current being used. In the constant power region the tractive force is limited due to the maximum possible vehicle output and in the vehicle characteristics region the tractive force is limited due to the characteristics of the train’s components. The regenerative brake force is limited in the higher speed regions because of the characteristics of the traction motor. A mechanical brake force Fmb is available to achieve constant deceleration over the entire speed range. 2.2 Train Motion 9

40 F t,max F t,min F 30 mb

10 [kN]

-10

-20

-30 0 10 20 30 40 50 60 70 80 90 100 v [km/h]

Figure 2.3: hev maximum and minimum traction force (blue), regenerative brake force (red) and mechanical brake force (black) curves.

The tractive and regenerative brake force curves used in this thesis are based on the Ki-Ha E200 specifications in [14]. Unfortunately, the curves need to be modi- fied as they contain errors. The power should be constant in the constant power region, which it is not. The tractive and brake force curves have been recalculated using the same constant torque, constant power and characteristics regions as in the Ki-Ha E200 specifications. The curves have been calculated assuming a maximum acceleration of amax = 2.3 km/h/s and a maximum deceleration of amin = 2.0 km/h/s. The maximum passenger capacity of the Ki-Ha E200 is 117 passengers.− The average weight of a passenger has been assumed to be 60 kg, resulting in a maximum passenger weight mp = 7020 kg. A common practice in the design of a traction system, in the railway industry, is to add an additional force of 29.43 N (3 kgf) per ton of the maximum weight mmax = m + mp to the maximum tractive force in the constant torque region. This is done to guarantee that the train’s internal starting resistances are overcome. The maximum tractive force curve Ft,max can be calculated according to

  + 0 02943 0 20 mmaxamax . mmax, v  v ≤ ≤ Ft,max = (mmaxamax + 0.02943mmax)· , 20 < v 70 (2.6)  20 ≤ (m a + 0.02943m )· v · v , 70 < v 100 max max max 20 70 ≤ and Ft,min as   mmaxamin, 0 v 42 Ft,min =  ≤ ≤ (2.7) m a · v . 42 < v 100 max min 42 ≤

The resulting tractive and brake curves are shown in Figure 2.3. 10 2 Series Hybrid Electric Railway Vehicle Modelling

2.2.1 Train Resistances

An important aspect in the modelling of a railway vehicle is the resistances that aﬀect the train. Knowing which resistances that are present during the train operation improves the understanding of their eﬀects and ability to avoid them. The train resistances can be divided into external and internal resistances. Internal resistances are directly associated with the train, whilst external resistances refer to the terrain conditions where the train is operating. Further, internal resistances can be separated into starting resistances and running resistances.

External Resistances

External resistances are caused by the nature in which the train is operating. A few examples of external resistances are the track gradient, track curvature, wind and air conditions, such as humidity. In this thesis, only longitudinal dynamics are considered and a track gradient of α = 0 is assumed. This assumption allows us to neglect the term Fg in (2.4) as the gravitational force, induced by the gravity acting on the train, is given by

Fg (t) = mgsin(α). (2.8) The eﬀect of wind and air conditions, along with any other external resistance, has been neglected.

Internal Resistances

When a train is standing still the couplers between carriages are usually in a slack position. Before a train starts moving all couplers need to be tightened. Doing so is part of the starting resistance of a train. The starting resistance is only present when a train is accelerating from stand still and rapidly decreases as the train speed increases. As mentioned earlier, a rule of thumb is to use a force of 29.43 N per ton of the train’s maximum weight to overcome the starting resistance. In the railway industry it is generally agreed that the internal running resistance can be described by a quadratic function as 2 Frr (t) = r2v (t) + r1v(t) + r0, (2.9) where the constants r2, r1 and r0 vary depending on the train. The constants are estimated through measurements of the train’s acceleration and deceleration while the train is running.

The constant r2 accounts for the aerodynamic resistance of quiet air and depends on the frontal area of the train. The aerodynamic resistance is mainly noticeable at high speeds.

The constant r1 describes the resistance that is directly proportional to the train speed. As a train is running along a track, the bogie and carriage are subject to 2.3 Internal Combustion Engine 11 lateral oscillations. This is known as bogie hunting and increases with the speed of the vehicle. The oscillations causes the bogie to move side to side across the rail, retarding the train due to the sliding friction.

Lastly, r0 describes the resistances that do not depend on the speed of the vehicle. The value of r0 is determined by the rolling resistance, track resistance and bearing resistance.

If a ﬂat track is assumed and external resistances are neglected, Ftr (t) in (2.4) can be replaced with the quadratic approximation of the running resistance (2.9). The train motion is then expressed by

dv(t) m = F (t) F (t) (r v2(t) + r v(t) + r ). (2.10) dt t − mb − 2 1 0

The values of r2, r1 and r0, speciﬁc for Ki-Ha E200, can be found in [14].

2.3 Internal Combustion Engine

The Ki-Ha E200 uses a diesel engine as its ice. The ice transforms the chemical energy stored in the fuel into mechanical power, which in turn is transformed into electrical power by an electrical generator. Due to the hybrid configuration, the ice does not directly propel the train. This is one of the major benefits of a hybrid system as it allows the engine to be run at its optimal efficiency. Due to the lack of information about the ice that is used in Ki-Ha E200, approx- imations of the engine dynamics have been made. The ice input is restricted to one of four predetermined values for the power output reference signal Pice∗ . The ice power output is given by Pice. The available power output reference signals are denoted Pice,of∗ f , Pice,low∗ , Pice,opt∗ and Pice,max∗ . The output power reference values are determined based on the ice output power and fuel efficiency specification shown in Figure 2.4. A summary of the ice output power reference values is given in Table 2.1.

Table 2.1: Summary of the ice output power reference values.

ice Engine Speed [rpm] ice Output Power [kW] ice Fuel Eﬃciency [g/kWh]

Pice,of∗ f 0 0 0 Pice,low∗ 1200 200 220.25 Pice,opt∗ 1500 280 201.89 Pice,max∗ 2000 330 216.13

Since no ice parameters are known, modelling the actual dynamics of the diesel engine is unachievable. The dynamic of the ice power output is therefore approx- 12 2 Series Hybrid Electric Railway Vehicle Modelling

350 240

300 230

250

220 200 [kW] ice [g/kWh]

P 150 210

100

200 50

0 190 1000 1200 1400 1600 1800 2000 2200 1000 1200 1400 1600 1800 2000 2200 [rpm] [rpm] (a) ice output power. (b) ice fuel eﬃciency.

Figure 2.4: ice output power and fuel efficiency as a function of engine speed. The circled data points indicate the power reference values, from left to right, Pice,low∗ , Pice,opt∗ and Pice,max∗ . imated by a first order differential equation P (t) P (t) ˙ ice∗ ice Pice(t) = − , (2.11) Tice where Tice is the ice time constant.

2.4 Battery

The Ki-Ha E200 uses a rechargeable lithium-ion battery as storage battery due to its high power and energy density. The battery can either be used as an energy supply, used to propel the hev, or it can be used for energy storage. The storage battery can be charged by the ice or by recuperating energy from regenerative braking. The battery model described in this section is based on the Ki-Ha E200 storage battery speciﬁcation given in [13]. It is stated that the storage battery consists of two groups of batteries, connected in parallel, where each group is composed of a 4 series, 2 parallel connection of battery modules. Each battery module holds 48 battery cells connected in series. The battery cell has a rated voltage of 3.6 V and a nominal capacity of 5.5 Ah. However, the resistance of the battery cell is not stated. Therefore the battery cell resistance has been chosen as 0.7 mΩ based on the "LIM25H" battery speciﬁcations given in [15]. In this thesis, the battery temperature will not be taken into account and will be assumed to be constant at 25◦C. The physical battery model can be approximated by an equivalent circuit, shown in Figure 2.5. The battery is represented by an ideal voltage source Ub,oc, in series with an internal resistance Rb,i. 2.4 Battery 13

Figure 2.5: Battery equivalent circuit.

Knowing the battery configuration and specifications stated above one can calculate that the open circuit voltage is Ub,oc = 693 V, the internal resistance Rb,i = 33.6 mΩ and the nominal battery capacity Q0 = 22 Ah. Further, the nominal energy capacity of the battery is approximately 15.2 kWh. The maximum and minimum battery power is 300 kW and -300 kW, respectively. By applying Ohm’s law and Kirchhoff’s voltage law to the equivalent circuit, the following relations are obtained

Pb(t) = Ub(t)Ib(t), (2.12a) U (t) = U R I (t). (2.12b) b b,oc − b,i b An important aspect to consider when using a battery as a storage device is the state of charge (soc) of the battery. The soc is the ratio

Q(t) q(t) = , (2.13) Q0 between the current electric charge Q, that can be delivered by the battery, and the nominal battery capacity Q0. In order to reduce losses in the conversion from stored to delivered energy, it is desired to keep the soc within certain limits. Furthermore, the longevity of the battery will increase, which is important both from an economic and sustainability viewpoint.

Both the open circuit voltage Ub,oc, which is the voltage between the terminals when there is no load applied, and the internal resistance Rb,i varies with the battery soc. According to [9] these relationships can be approximated by the linear expressions

Ub,oc(t) = κ2q(t) + κ1, (2.14a) Rb,i(t) = κ4q(t) + κ3. (2.14b)

Due to the coefficients κ2 and κ4 being unavailable for the studied storage battery these values have been assumed to be zero if the soc is kept within a certain range. This leaves Ub,oc and Rb,i as constants. The internal resistance of the 14 2 Series Hybrid Electric Railway Vehicle Modelling battery is not only affected by the battery soc, but also if the battery is being charged or discharged. This effect has been neglected to simplify the modelling of the battery. There are no possible ways of directly measuring the soc of the battery. However, one possible method of estimating it is known as Coulomb Counting. This method estimates the soc based on accurate direct measurements of the charge\discharge current Ib(t) as Q˙ (t) = I (t). (2.15) − b In the case of charging, a small fraction of the charging current is not transformed into charge. For simplicity this effect has been neglected in this thesis. The change in soc can be expressed by inserting (2.15) into the time derivative of (2.13) Q˙ (t) I (t) q˙(t) = = b . (2.16) Q0 −3600Q0

Note that Q0 is measured in the unit Ah and Ib(t) in A, thus Q0 needs to be multiplied by 3600 in order to express the soc time derivative in per second instead of per hour.

It is favourable to express the change in soc in terms of battery power Pb instead of the current Ib. Solving (2.12b) for Ib and inserting it in (2.16) gives

U U (t) q˙(t) = b,oc − b . (2.17) − 3600Q0Rb,i

Combining (2.12a) and (2.12b) results in the quadratic expression

U 2(t) U U (t) + P (t)R = 0. (2.18) b − b,oc b b b,i

Solving the preceding equation gives the following expression for the battery terminal voltage s 2 U U U (t) = b,oc b,oc P (t)R , (2.19) b 2 ± 4 − b b,i which, inserted into (2.17) yields

q 2 Ub,oc Ub,oc 4Pb(t)Rb,i q˙(t) = ± − , (2.20) − 7200Q0Rb,i where Pb can be expressed, using the relations (2.1) and (2.2), as

P (t) = P (t) η η P (t) + P . (2.21) b inv − conv eq ice aux 2.5 Model Equation 15

The battery soc can thus be calculated by only knowing the hev power ﬂow. If Pb(t) = 0 is inserted in (2.20) the change in the battery soc should be equal to zero. Thus, the correct solution is given by

q 2 Ub,oc Ub,oc 4Pb(t)Rb,i q˙(t) = − − . (2.22) − 7200Q0Rb,i

2.5 Model Equation

The non-linear state-space model of the hev, where the state derivatives depend on the current states and inputs, can be written as

x˙(t) = f(x(t), u(t)), (2.23a) y(t) = h(x(t), u(t)). (2.23b)

The state vector x(t) and input vector u(t) are deﬁned as

  s(t)     F (t)  v(t)   t  x(t) ,   , u(t) , F (t) . (2.24) P (t)  mb   ice  P (t)  q(t)  ice∗

Equation (2.10), (2.11), (2.22) and (2.21) can be written on the same form as (2.23a) resulting in

 v(t)      s˙(t)  F (t) F (t) r v(t)2+r v(t)+r     t mb 2 1 0   v˙(t)   m m m     − P ∗ (t−) P (t)  x˙(t) =   =  ice − ice  . (2.25) P˙ (t)  T   ice   q ice     2  q˙(t)  Ub,oc U 4(Pinv (t) ηconv ηeqPice(t)+Paux)Rb,i   − b,oc− −  − 7200Q0Rb,i

Note that Pinv(t) is determined by whether the train is powering or braking according to (2.3). This equation is valid in the special case of zero track curvature and track gradient.

The values of the parameters used in the hev model are found in Table 2.2. 16 2 Series Hybrid Electric Railway Vehicle Modelling

Table 2.2: Values of the parameters used in the hev model.

Notation Value Description m 43560 kg Equivalent mass of the hev. 2 2 r2 3.56 Ns /m Running resistance polynomial coeffi- cient. r1 20.3 Ns/m Running resistance polynomial coeffi- cient. r0 554.9 N Running resistance polynomial coeffi- cient. ηeg 0.90 Electric generator efficiency. ηconv 0.95 Converter efficiency. ηinv 0.95 Inverter efficiency. ηem 0.90 Electric motor efficiency. ηgb 0.98 Gear-box efficiency. Paux 30 kW Auxiliary power supply. Tice 1 s ice time constant. Pice,of∗ f 0 kW Off ice output power reference. Pice,low∗ 200 kW Low ice output power reference.. Pice,opt∗ 280 kW Optimum ice output power reference. Pice,max∗ 330 kW Maximum ice output power reference. Ub,oc 692 V Battery open circuit voltage. Rb,i 33.6 mΩ Battery internal resistance. Q0 22 Ah Battery nominal capacity. 3 Controller Design

The hev model, derived in Chapter 2, is used in the mpc that has been developed. Firstly, this chapter gives a short introduction and explains the fundamental principles of mpc. The mpc basics will serve as a prerequisite to hybrid mpc. A hybrid mpc can be seen as a framework that covers the use of both continuous and discrete variables. This allows one to model discrete valued input signals and system logic. One of the possible uses of the hybrid mpc framework is to approximate the non-linear train model by a number pwa models instead. Since both continuous and discrete variables are used, the hybrid mpc optimisation problem results in what is known as a miqp problem. This is a far more challenging optimisation problem compared to a standard qp problem. The theory presented in this chapter is mainly based on [5], [11] and [3].

3.1 MPC Basics

In the past, mpc has mainly been applied to systems with large time constants due the computational eﬀort that is needed. With the rapid development of the computational capabilities of computers, mpc has gained signiﬁcant popularity even for systems with small time constants. Another contributing factor why mpc is widely accepted in the industry is because the concept of mpc is simple and easy to understand. One of the major advantages of mpc, compared to other control methods, is that it can explicitly consider input, state and safety limit constraints. This will not only keep the inputs and states within the given constraints, assuming an accurate model, but it will allow the target system to safely operate near its physical and safety limits. Another advantage of mpc is that both SISO and MIMO systems are naturally handled the same way, which can ease the

17 18 3 Controller Design control of large, more complex, systems. When working with mpc it is favourable to work with linear discrete-time dynamic models that describe the evolution of the plant as a function of the current system state x and the manipulated input u according to

x(kTs + Ts) = Fx(kTs) + Gu(kTs) (3.1a) y(kTs) = Cx(kTs) (3.1b) z(kTs) = Mx(kTs) (3.1c) where Ts is the sampling interval, y a vector of measured outputs and z a vector of states to be controlled. If the model accurately describes the plant we wish to control, the mpc can utilise the prediction power that the model holds. The idea of mpc is to formulate the control problem as an optimisation problem, and to solve this on-line, to retrieve the optimal input sequence over a speciﬁed number of time steps. The number of time steps is called the prediction horizon and is denoted N. The optimal input sequence, also called the optimiser, for a given prediction horizon N is the one that minimises an objective function V as , U ∗(x) min V (x, U) = [u0∗ (x), u1∗ (x), ..., uN∗ 1(x)]. (3.2) U − The objective function, or cost function, is a function of the current system state x and the input sequence U. The system states need to be measured or estimated by the use of an observer. Knowing the current system state allows one to calculate the optimiser. The ﬁrst element of the optimiser u0∗ (x) is applied to the plant, resulting in a complex feedback controller. Due to disturbances and inac- curacies in the plant model the optimiser U ∗(x) has to be recalculated at every time step. The objective function needs to be designed based on the purpose of the controller and can generally be expressed as

N 1 X− V (x(k),U) = q(x(k),U) + p(x(k + N)), (3.3) k=0 where q(x(k),U(k)) denotes the stage cost and p(x(N)) denotes the terminal cost. The terminal cost is designed to guarantee stability. In mpc the objective function is commonly expressed as a quadratic function subject to a set of linear constraints. The reason for this is that it can easily be rewritten as a qp problem, which is a well known optimisation problem. Concluding the details above, the basic mpc algorithm can be summarised as: 1. Measure the system states x(k) or estimate them by the help of an observer.

2. Calculate the optimal input sequence U ∗(x) that minimises a cost function V (x, U) by solving an optimisation problem on-line.

3. Implement the ﬁrst element of the optimal input sequence u0∗ (x). 3.2 Hybrid MPC 19

4. Update time k = k + 1. 5. Repeat procedure from Step 1.

3.2 Hybrid MPC

In many applications it is common that both linear and discrete dynamics are present. These systems are referred to as hybrid systems. The theory presented in the previous section can be expanded to hybrid systems, resulting in a hybrid mpc. A popular case of discrete-time hybrid systems is the pwa system. Instead of working with a general non-linear internal model in the mpc, a pwa system will switch between several aﬃne models depending on the location of the state vector. To model the discrete dynamics, binary variables are introduced. When binary variables are introduced, the optimisation problem turns into a mixed integer programming (mip) problem. This type of optimisation problem has a complexity that grows exponentially with the number of introduced binary variables. The hybrid mpc has been implemented in Matlab with the help of the free optimisation modelling toolbox Yalmip [10].

3.2.1 Piecewise Afﬁne Model

As mentioned above, an alternative to working with a non-linear internal model in the mpc is to approximate it with several affine models instead. The logic of switching between the affine models, depending on the location of the state space vector, can be realised by introducing binary variables in the model. The affine models are obtained by dividing the state and input space into multiple regions. Each region is associated with a certain affine model. Following the theory presented in [3] the pwa, model can be expressed as

x(t + 1) = Ai(t)x(t) + Bi(t)u(t) (3.4a) y(t) = Ci(t)x(t) + Di(t)u(t) (3.4b) H i(t)x(t) + Ji(t)u(t) K i(t) (3.4c) ≤ The matrices Ai(t), Bi(t), Ci(t), Di(t), H i(t), Ji(t) and K i(t) are constants. The variable i(t) denotes the current mode of the system and determines which affine model that is active at a certain time. In this thesis the non-linear model (2.25) has been approximated by five affine models. These models are obtained by partitioning the input and state space into five regions and linearising the non-linear model about a suitable point in each region. The points of linearisation are chosen as 20 3 Controller Design

 0   0   0   0   0             10/3.6   30/3.6   50/3.6   70/3.6   90/3.6  x =   , x =   , x =   , x =   , x =   . 0,1   0,2   0,3   0,4   0,5   280000 280000 280000 280000 280000  0   0   0   0   0  (3.5) The linearisation point for the distance s0 and battery soc q0 are both set to 0 as they do not aﬀect the linearised model. The maximum vehicle speed is 100 km/h, thus the total speed range has been divided into ﬁve equally large regions. The linearisation point v0 has been chosen as the middle point of each respective region. The linearisation point Pe,0 is set to 280 kW because it is the ice’s optimal point of operation and therefore also the desired operating point.

The aﬃne models were acquired by ﬁrst calculating the stationary points using the chosen points (3.5). Inserting x0 in (3.6) allows one to calculate u0. Knowing x0 and u0, the matrices A, B, C and D can be calculated. None of the chosen points (3.5) can actually be used to calculate a stationary point due to s˙ = v. Therefore, s˙ = v has been neglected and the standard procedure of calculating stationary points has been used to calculate x0 and u0. Following the theory regarding linearisation in [7], a non-linear system described by (2.23) can be linearised about a stationary point x0, u0, y0 such that

f (x0, u0) = 0, y0 = h(x0, u0). (3.6)

The linear model is obtained by observing small deviations from the stationary point x = x0 + ∆x, u = u0 + ∆u and calculating the ﬁrst order Taylor expansion about this point. The Jacobians of f are given by

 0 1 0 0  0 2r2v(t)+r1 0 0  − m   0 0 1 0 fx =  − Tice  ,  Ft(t) ηconv ηeq   0 q q 0  2 2  − Q U 4(Ft(t)v(t) ηconv ηeqP (t)+Paux)R Q U 4(Ft(t)v(t) ηconv ηeqP (t)+Paux)R 0 b,oc− − ice b,i 0 b,oc− − ice b,i  0 0 0   1 1 0   m − m   0 0 1  fu =  Tice  ,  v(t)   q 0 0   2  − Q U 4(Ft(t)v(t) ηconv ηeqP (t)+Paux)R 0 b,oc− − ice b,i hx = I, hu = 0. (3.7) With the notation A = fx(x0, u0), B = fu(x0, u0), C = hx(x0, u0) and D = hu(x0, u0) the linearised model can be written as

x˙(t) = ∆x˙(t) = A∆x(t) + B∆u(t), (3.8) ∆y(t) = C∆x(t) + D∆u(t). 3.2 Hybrid MPC 21

The linearised model is a continuous-time system. Sampling the continuous-time system (3.8) with a sampling time of Ts and approximating the time derivative, using Euler forward, yields the discrete time model

x(kTs + Ts) x(kTs) − = A∆x(kTs) + B∆u(kTs), Ts (3.9) ∆y(t) = C∆x(t) + D∆u(t), which can be rewritten as x(kT + T ) = (I + T A)∆x(kT ) + T B∆u(kT ), s s s s s s (3.10) ∆y(t) = C∆x(t) + D∆u(t).

The discrete time models are used in the hybrid mpc implementation.

3.2.2 Mixed-Integer Programming

To model the logic of switching between multiple affine models, binary variables are introduced. An optimisation problem containing both continuous and discrete variables is known as a mip problem. The discrete variables are included in the mip model through integrality constraints. mip problems can be more finely categorised. In the case of a quadratic objective function and linear constraints the optimisation problem is known as a miqp problem. When solving a miqp problem a number of different solvers are available e.g. Gurobi, Mosek and Cplex. In this thesis, Gurobi has been used to solve the miqp problem [8]. A miqp problem can be formulated as

1 min xT Hx + f T x (3.11a) x 2 subject to Aeqx = beq (3.11b) Ax b (3.11c) ≤ x Z i = 1, ..., n. (3.11d) i ∈ Note that for the special case of n = 0 we end up with a qp problem. The first step of solving a miqp problem is to enlarge the feasible set. This is done by relaxing the constraints, e.g. ignoring all integrality constraints. The relaxation of the original miqp can then be used to calculate a lower bound to the optimal solution. If the result happens to fulfil all the integrality constraints the optimal solution has been found. This is however unlikely to occur, thus the relaxed problem needs to be refined or tightened. This is typically done using a branch and bound method where each node contains a qp problem. The number of nodes in the branch and bound tree grows exponentially with the number of binary variables 2n. Solving miqp problems can therefore become very computational heavy as the worst case would result in having to solve 2n qp problems. This is of course not reasonable in the case of real time applications. Therefore the miqp 22 3 Controller Design solvers use various methods of reducing the computational complexity. Further details about branch and bound and how miqp problems can be solved can be found in [1] and [2].

3.2.3 Objective Function

As mentioned in Section 3.1 the idea of mpc is to formulate the control problem as an optimisation problem and calculate the optimiser that minimises the objective function, subject to a set of constraints. The states we want to control and the reference tracking signals are deﬁned by z(k) and r(k), respectively. The design of the objective function is determined by how we want the hev to behave. The objective function used in the hybrid mpc can be written as

N 1 X− 2 V (x(k),U) = z(k + j) r(k + j) + Q2F (k + j) + Q3F (k + j) k − kQ1 t mb j=0

+ Q4ηice(Pice∗ (k + j)). (3.12)

In the operation of trains it is of utter importance that time tables are kept and speed limits along the track are followed. Therefore the objective function in- cludes reference tracking of a given speed profile. The reference tracking is included as a quadratic term since we want to avoid running faster or slower than the desired speed. One of the major advantages of series hybrid systems is the possibility of running the ice at its optimal efficiency. The ice efficiency is denoted ηice in (3.12) and depends on Pice. Running the ice at the optimal efficiency will not only minimise engine loss, but also the total fuel consumption over time. The ice efficiency is included in the objective function to minimise the engine loss and total fuel consumption.

The use of the mechanical break Fmb will cause energy loss in terms of heat dissipated in the breaks. The mechanical brake force Fmb is included in the objective function to make sure that it is only used when the required deceleration can not be achieved solely using regenerative braking. After designing the objective function one has to tune the length of the prediction horizon N, as well as the weights Q1, Q2, Q3 and Q4. The prediction horizon should typically be long enough to cover the settling time of the slower dynamics of the target system. However, due to limitations in the computational power of the used simulation platform the prediction horizon has been limited to N 40. The objective function weights are tuned by the user and will determine≤ how the mpc will prioritise the minimisation of the various terms in the objective function. 3.2 Hybrid MPC 23

3.2.4 Constraints

In Section 3.1 it is mentioned that one of the major advantages of mpc is that constraints can be explicitly included when setting up the control problem. In a hybrid mpc a miqp problem has to be solved at every time step. When setting up an miqp problem all states and inputs are required to be explicitly bounded in order for Yalmip to be able to perform Big-M reformulations. To understand the meaning of Big-M reformulations the interested reader is referred to [10]. The constraints used in the hybrid mpc can be stated explicitly as

min V (x(k),U) (3.13a) U subject to x(k + 1) = Ai x(k) + Bi u(k) (3.13b) i 1, 2, 3, 4, 5 (3.13c) ∈ { } s s s (3.13d) min ≤ ≤ max v v v (3.13e) min ≤ ≤ max P P P (3.13f) ice,min ≤ ice ≤ ice,max q (s) q q (3.13g) min ≤ ≤ max F (v) F F (v) (3.13h) t,min ≤ t ≤ t,max F (v) F F (v) (3.13i) mb,min ≤ mb ≤ mb,max P (v) P P (v) (3.13j) b,min ≤ b ≤ b,max P ∗ P ∗ ,P ∗ ,P ∗ ,P ∗ (3.13k) ice ∈ { ice,of f ice,low ice,opt ice,max} As stated in (3.11), the miqp problem only allows linear inequality, equality and integrality constraints. Therefore, non-linear constraints need to be linearised. The discrete valued variables and the logic in the switching of affine models need to be modelled as well. In this thesis, five affine models are used to approximate the non-linear hev model. The constraint (3.13b) calculates the predicted future state using one of these linearised discrete-time models. The active model is given by the value of i, which is determined by the vehicle speed. To implement the logic of switching between the pwa models, five binary decision variables are introduced, denoted di,j , j = 1, ..., 5. To ensure that only one model is active, a constraint is set on the decision variables

i = 1 · di,1 + 2 · di,2 + 3 · di,3 + 4 · di,4 + 5 · di,5, (3.14a) X5 di,j = 1. (3.14b) j=1

1 1 To give an example, di,1 = 1 implies that A and B are active and used to calculate x(k + 1). 24 3 Controller Design

In a similar way the constraint (3.13k) can be realised by introducing the decision variable dice,j , j = 1, ..., 4 and assigning Pice∗ as

Pice∗ = Pice,of∗ f dice,1 + Pice,low∗ dice,2 + Pice,opt∗ dice,3 + Pice,max∗ dice,4, (3.15a) X4 dice,j = 1. (3.15b) j=1

Constraint (3.13h) and (3.13i) depend on the vehicle speed and vary non-linearly. The tractive, regenerative brake and mechanical brake force curves, shown in Figure 2.3, are linearised for ﬁve equally large vehicle speed regions, see Fig- ure 3.1.

The battery power constraint (3.13j) has to be linearised due to the inclusion of the non-linear term Pout(t) = Ft(t)v(t), in (2.21). Similarly to the previous constraint linearisation, the vehicle output is approximated for ﬁve equally large vehicle speed regions as shown in Figure 3.2. The linearisation of the battery power constraint is done by replacing the non-linear vehicle output term through a Taylor expansion

F (t)v(t) = F v + F (v(t) v ) + v (F (t) F ), (3.16) t t,0 0 t,0 − 0 0 t − t,0 and linearising about one point in each region. The points of linearisation are chosen as F = 0 for all ﬁve points and v = 10 , 30 , 50 , 70 , 90 . t,0 0 { 3.6 3.6 3.6 3.6 3.6 }

40 40 F F t,max t,max F F t,min t,min F F 30 mb 30 mb

20 20

10 10 [kN] [kN]

0 0

-10 -10

-20 -20

-30 -30 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 v [km/h] v [km/h]

(a) Ft and Fmb curves. (b) Linearised Ft and Fmb curves.

Figure 3.1: Linearisation of the tractive, regenerative brake and mechanical brake force curves. 3.2 Hybrid MPC 25

800 800

600 600 1000 1000

400 400 500 500

200 200

[kW] 0 [kW] 0 out out

P 0 P 0

-500 -500 -200 -200

40 -400 40 -400

20 100 20 100 80 80 -600 -600 0 60 0 60 40 40 -20 -20 F [kN] 20 F [kN] 20 t 0 v [km/h] t 0 v [km/h]

(a) Pout. (b) Linearised Pout.

Figure 3.2: Linearisation of Pout used to approximate the non-linear constraint on the battery power Pb, with linear constraints.

The vehicle speed regions used in the linearised constraints above are chosen to be the same as the regions used for determining the active pwa model. Thus, the speed of the vehicle will not only determine the active pwa model, but also which linearised constraints that are active. An important concept in the energy management in hevs is that the sum of the vehicle’s kinetic energy and the energy of the storage battery should be kept constant. This concept implies that if the initial battery soc is q0, the soc should return to its initial value when the train is approaching its destination. Due to the length of the mpc prediction horizon being limited, this becomes a challenging task. In this thesis an attempt to solve the problem has been done by introducing a lower bound constraint on the soc of the battery (3.13g), that depends on the distance travelled by the vehicle according to

s(k) qmin + (q0 qmin) q(k) qmax, (3.17) stot − ≤ ≤ where stot denotes the total distance between the origin and the destination.

3.2.5 Softened Constraints

One of the issues that can arise in an mpc controller is that the optimisation problem becomes infeasible due to a constraint violation. This can occur when the mpc is operating close to a constraint and a large disturbance affects the system. Another example is if the internal model differs from the plant model. Infeasibility is of course an unacceptable behaviour by the mpc when operating on-line. There are a number of different measures that can be taken if infeasibility occurs. In this thesis the control signals will use the same values as in 26 3 Controller Design the previous time step if infeasibility occurs. More advanced measures could be taken, such as increasing the feasible set by relaxing less important constraints and recalculating the optimisation problem. One strategy to avoid infeasibility is to soften some of the hard constraints [11]. This means that instead of observing hard constraints as boundaries, that can never be crossed, we allow a small violation if necessary. However, not all constraints are possible to relax. Input constraints are often hard constraints that cannot be violated. State constraints are many times user-defined and can therefore be relaxed. Relaxing constraints can be done by introducing slack variables. If we consider a standard qp problem, the slack variable can be added to the objective function and the linear constraints as

1 T T min x Hx + f x + ρ 1 (3.18a) x, 2 k k Ax b + (3.18b) ≤ 0 (3.18c) ≥ The slack variable has the same dimension as b and is non-negative. By setting the weight ρ we can decide to what degree the constraint should be relaxed, where ρ = 0 corresponds to a completely unconstrained problem and ρ to the original problem with a hard constraint. → ∞ In this thesis a slack variable was introduced to soften the battery soc constraint. The hard constraints are not set because of physical limitations of the battery, but rather to keep the soc within a desired range to promote the battery eﬃciency and longevity.

3.2.6 Move-Blocking

One of the challenges of solving miqp problems within the sampling time of the mpc is the computational complexity, which grows exponentially with the number of binary optimisation variables used. Thus, using a long prediction horizon will have a significant impact on the computation time. Different strategies can be used to reduce the computational effort. One possibility is to introduce move-blocking. In this thesis the simple move-blocking scheme input blocking has been implemented. This strategy decreases the degrees of freedom in the optimisation problem by fixing certain input signals to be constant over several time- steps [4]. This is done by restating the original optimisation problem to optimise ˆ T T T T T T U = [uˆ0 uˆM 1] instead of U = [u0 uN 1] , where M < N. By introduc- ··· − N M ··· − ing the blocking matrix T R × the relation between U and Uˆ can be written as ∈

T T T T T T U = [u0 uN 1] = (T Im)[uˆ0 uˆM 1] , (3.19) ··· − ⊗ ··· − 3.2 Hybrid MPC 27

where is the Kronecker product, m is the number of input signals and Im is an identity⊗ matrix with m rows. As an example we can assume a prediction horizon N = 4, input blocking u0 = u1 and u2 = u3 and the optimisation variables Uˆ = T T T [uˆ0 uˆ1 ] . This gives the blocking matrix

1 0   1 0 T =   . (3.20)   0 1 0 1

Depending on the choice of M a blocking matrix T can be deﬁned. The blocking matrix can then be used to reformulate the original optimisation problem and reduce the degrees of freedom. In this thesis, input blocking has only been used for the input signal Pice∗ because it is a discrete variable.

4 Results

The performance of the hybrid mpc, described in Chapter 3, is evaluated using software simulations. Before describing the simulator implementation, a brief presentation of the heuristic ems which is currently being used in Ki-Ha E200, and the design of three diﬀerent test cases are presented. Finally, this chapter presents the simulation results followed by a discussion of the obtained results.

4.1 Heuristic Controller

The heuristic ems implementation is based on the principle that the sum of the kinetic energy of the railway vehicle and the energy stored in the storage battery should be kept constant. If this principle is strictly followed it implies that the initial and ﬁnal value of the soc of the storage battery are equal. The output power reference Pice∗ is determined by whether the train is powering, coasting or braking, and by the vehicle speed and the battery soc. Figure 4.1 shows how the vehicle speed and battery soc space is divided into ﬁve regions.

29 30 4 Results

100

80 D

70 A 60

50 B SoC [%] 40 E 30 C 20

0 0 10 20 30 40 50 60 70 80 90 100 v [km/h]

Figure 4.1: The battery soc and vehicle speed space is divided into ﬁve regions. The regions are denoted A, B, C, D and E. The active mode is used in the heuristic ems to determine the operation of the ice.

It can be seen that an upper and lower limit has been set for the battery soc. These limits are set to ensure that the battery is operating at a high efficiency and to ensure the longevity of the battery. Within the upper and lower limit there are five different regions denoted mode A, B, C, D and E. Knowing the mode, i.e. the vehicle speed and soc of the battery, and train operation, Pice∗ is determined according to Table 4.1.

Table 4.1: The ice output power reference Pice∗ is determined by the mode and whether the hev is powering, coasting or braking. E.g. if the hev is in mode E and braking, Pice∗ should be set to Pice,low∗ .

Mode Powering Coasting Braking A OFF OFF OFF B OPT OFF OFF C MAX MAX LOW D OFF OFF OFF E MAX MAX LOW

The ideal case is to operate the train in mode A and B as one wants to avoid running the ice at low or maximum. The heuristic ems suﬀers from rapid switching between modes when operating near region borders. To avoid this a hysteresis loop is implemented. An example of how the rapid switching between mode A and E can be eliminated using a 4.2 Test Cases 31

Figure 4.2: Hysteresis loop used to avoid rapid switching between modes. The ﬁgure shows an example of how this is avoided at the boundary between mode A and E. The upper and lower soc limits that triggers the change of mode are set to 0.55 and 0.45, respectively. hysteresis loop is shown in Figure 4.2. The soc limits that trigger a change in mode need to be tuned to obtain acceptable performance.

4.2 Test Cases

In order to evaluate and compare the performance of the heuristic ems and the hybrid mpc, three test cases with different speed reference patterns have been designed, see Figure 4.3. The test cases are not based on actual running data. The designed test cases include sections of powering, coasting and braking similar to real train operation. Constant acceleration and deceleration has been used for powering and braking. All test cases assume zero track curvature and track gradient. Test case 1 and 2 are designed to examine how the length of the track affects the performance of the controllers. Test case 3 is designed to investigate the effect of speed limitations along the track. The total travel time and total distance travelled of the three test cases are sum-

80 80 80

70 70 70

60 60 60

50 50 50

40 40 40 [km/h] [km/h] [km/h] ref ref ref v v v 30 30 30

20 20 20

10 10 10

0 0 0 0 50 100 150 200 250 0 100 200 300 400 500 600 700 0 100 200 300 400 500 Time [s] Time [s] Time [s] (a) Test case 1 (b) Test case 2 (c) Test case 3

Figure 4.3: Three test cases, with diﬀerent speed reference patterns, used to evaluate and compare the performance of the hybrid mpc and the heuristic ems. 32 4 Results marised in Table 4.2. These values are used to determine how much the total travelled distance and total time of the hybrid mpc result deviates from that of the heuristic ems.

Table 4.2: Total travel time and distance travelled for the three test cases used in the controller evaluation.

Test Case t [s] s [m] 1 254 4230 2 733 13715 3 555 8286

4.3 Simulator

The simulator is implemented using Matlab and Simulink. The top level view of the simulator in Simulink is shown in Figure 4.4. The Controller block can be set to use either the heuristic ems or the hybrid mpc. The chosen controller will calculate the control signal u(k) based on the input r(k), ..., r(k + N) and the state x(k). The Reference block outputs the future reference values for the vehicle speed. If the heuristic ems is the active controller the ﬁrst value of the sequence r(k), ..., r(k + N) will be used as the actual vehicle speed. The HEV block contains the non-linear model (2.25) and takes the control signals u(k) and calculates the system states x(k).

4.4 Simulation Results

In this section the simulation results for the heuristic ems and the hybrid mpc are presented for three diﬀerent test cases. Each test case has been simulated using three diﬀerent values for the initial soc of the battery. The objective function weights, slack variable weight, prediction horizon and input blocking settings used in the hybrid mpc are summarised in Table 4.3.

Figure 4.4: Top level view of the simulator in Simulink. 4.4 Simulation Results 33

Table 4.3: Values of the hybrid mpc tuning parameters used in the simulation.

Tuning Parameter Value

Q1 100000 Q2 0 Q3 0.001 Q4 500 ρ 80000 N 40 M 5

The sample time of the hybrid mpc is set to 1 second. However, the control signal Pice∗ is only updated every five seconds. This is assumed to be a requirement of the controller in order to avoid the ice from rapidly switching on and off, similar to the effect of the hysteresis loop used in the heuristic ems. To give an overview, and a basis for the analysis of the performance of the two studied controllers, simulation data has been collected and is summarised in Table 4.4. In order to allow a qualitative comparison of the acquired results, ∆q has to be converted into a fuel equivalent mf ,∆q. This is done by calculating how much fuel is required, if the engine is operating at its optimum point, to obtain the amount of energy that yields ∆q. The fuel equivalent can be calculated as

R t Pb,MP C(t) Pb,H (t)dt t0 − 201.89 mf ,∆q = · . (4.1) 1000 · 3600 · ηconv ηeg 1000

The fuel equivalent is used to calculate the percentage reduction or increase in fuel consumption.

To further support the analysis of the acquired results, the collected data from three different test scenarios are shown in Figures 4.5-4.10. The presented test case scenarios are Test case 1 with an initial soc of 30%, Test case 2 with an initial soc of 50% and Test case 3 with an initial soc of 70%. For each scenario the results from both the heuristic ems and hybrid mpc are given. 34 4 Results [m] , 0 s s q ∆ ∆ and [s] t t ∆ ∆ [%] controller. In the table, Reduction mpc and fuel consumption, between , respectively. The fuel equivalent [kg] f soc q m ∆ ∆ f , m and q the total amount of fuel consumed when the ∆ erence in f [kg] ff m f m ∆ and soc [%] q ∆ travelling for a longer time or a longer distance, when using the [kg] hev f mpc m erence in arrival time and the total distance travelled is given by [%] ff q for a specific test case, is denoted has increased (positive) or decreased (negative) the total fuel consumption equivalent, mpc [kg] the final value of the battery . The di f mpc q m , ems soc Heuristic Hybrid [%] q and the hybrid 5070 50.92 66.93 1.557 1.398 55.4550 73.1170 40.01 1.727 48.31 1.649 3.907 4.53 3.429 58.4750 6.18 76.9070 32.89 4.554 0.171 52.89 4.554 1.937 0.251 18.46 1.937 51.29 28.59 0.186 71.02 0.647 2.669 0.254 1.125 2.669 18.40 18.13 0.721 -0.87 1.103 0.732 -0.19 0.732 0 0.708 -1.60 0 0.706 0.48 -7 0 -7 0 0.92 -4 0.99 3 0 0 -8 -9 [%] ems 0 , compared to the test case specification given in Table 4.2. q Summarised data from the simulation results of the heuristic and the hybrid mpc makes it possible to calculate the reduction in the total fuel consumption equivalent. The reduction column states has reached its destination and come to a complete stop. The di q ∆ 1 302 42.34 2.001 37.91 303 41.28 1.806 4.728 -4.43 38.91 30 36.48 -0.195 4.554 2.856 -2.36 32.27 0.148 -0.174 2.669 -4.21 0.052 -2.32 -0.187 0 0.136 -2.59 -7 0 -1.70 -4 0 -8 f , m how many percent the hybrid hev the heuristic compared to the heuristic hybrid Table 4.4: gives the initial battery respectively. A positive value corresponds to the Test Case 4.4 Simulation Results 35

Speed Position 80 4500 simulated speed position

4000 70

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20 1000

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Tractive Force Mechanical Break Force 40 12 tractive force mechanical break force

30 10

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[kN] 6 [kN] t mb F 0 F

4 -10

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SoC Battery Power 50 0.44 battery power SoC 0.42 0

0.4

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-150 0.34 [kW] q [-] b P 0.32 -200

0.3 -250 0.28

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(e) Pb(t) (f) q(t)

Fuel Consumption Engine Power 2.5 350 fuel mass engine power

300 2

250

1.5 200 [kg] [kN] f * ice m P 150 1

100

0.5

0 0 0 50 100 150 200 250 0 50 100 150 200 250 t [s] t [s]

(g) Pice∗ (t) (h) mice,f (t) Figure 4.5: Simulation results using the heuristic soc-based energy management controller, for Test case 1, with initial soc q0 = 0.3. 36 4 Results

Speed Position 80 4500 reference speed position simulated speed 4000 70

3500 60

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Tractive Force Mechanical Break Force 40 12 tractive force mechanical break force

30 10

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[kN] 6 [kN] t mb F 0 F

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SoC Battery Power 300 0.4 battery power SoC 0.38

200 0.36

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0.22

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(e) Pb(t) (f) q(t)

Fuel Consumption Engine Power 2 300 fuel mass engine power 1.8

250 1.6

1.4 200 1.2 [kg]

[kN] 1

150 f * ice m P 0.8

100 0.6

0.4 50

0.2

0 0 0 50 100 150 200 250 0 50 100 150 200 250 t [s] t [s]

(g) Pice∗ (t) (h) mice,f (t) Figure 4.6: Simulation results using a hybrid mpc controller, for Test case 1, with initial soc q0 = 0.3. 4.4 Simulation Results 37

Speed Position 80 14000 simulated speed position 70 12000

10000

8000 40 s [m] v [km/h] 6000 30

4000 20

2000 10

0 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 t [s] t [s] (a) s(t) (b) v(t)

Tractive Force Mechanical Break Force 40 12 tractive force mechanical break force

30 10

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[kN] 6 [kN] t mb F 0 F

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SoC Battery Power 300 0.5 battery power SoC

200 0.45

100 0.4

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(e) Pb(t) (f) q(t)

Fuel Consumption Engine Power 4 350 fuel mass engine power

3.5 300

3 250

2.5

200 [kg]

[kN] 2 f * ice m P 150 1.5

100 1

50 0.5

0 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 t [s] t [s]

(g) Pice∗ (t) (h) mice,f (t) Figure 4.7: Simulation results using the heuristic soc-based energy management controller, for Test case 2, with initial soc q0 = 0.5. 38 4 Results

Speed Position 80 14000 reference speed position simulated speed 70 12000

10000

8000 40 s [m] v [km/h] 6000 30

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Tractive Force Mechanical Break Force 40 10 tractive force mechanical break force 9 30 8

20 7

6 10

[kN] 5 [kN] t mb F 0 F 4

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0.55 200

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(e) Pb(t) (f) q(t)

Fuel Consumption Engine Power 5 300 fuel mass engine power 4.5

250 4

3.5 200 3 [kg]

[kN] 2.5

150 f * ice m P 2

100 1.5

1 50

0.5

0 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 t [s] t [s]

(g) Pice∗ (t) (h) mice,f (t) Figure 4.8: Simulation results using a hybrid mpc controller, for Test case 2, with initial soc q0 = 0.5. 4.4 Simulation Results 39

Speed Position 80 9000 simulated speed position

8000 70

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Tractive Force Mechanical Break Force 40 12 tractive force mechanical break force

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200 0.65

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(e) Pb(t) (f) q(t)

Fuel Consumption Engine Power 2 300 fuel mass engine power 1.8

250 1.6

1.4 200 1.2 [kg]

[kN] 1

150 f * ice m P 0.8

100 0.6

0.4 50

0.2

0 0 0 100 200 300 400 500 0 100 200 300 400 500 t [s] t [s]

(g) Pice∗ (t) (h) mice,f (t) Figure 4.9: Simulation results using the heuristic soc-based energy management controller, for Test case 3, with initial soc q0 = 0.7. 40 4 Results

Speed Position 80 9000 reference speed position simulated speed 8000 70

7000 60

6000 50

5000 40 s [m] 4000 v [km/h] 30 3000

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0 0 0 100 200 300 400 500 0 100 200 300 400 500 t [s] t [s] (a) s(t) (b) v(t)

Tractive Force Mechanical Break Force 40 12 tractive force mechanical break force

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[kN] 6 [kN] t mb F 0 F

4 -10

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-30 0 0 100 200 300 400 500 0 100 200 300 400 500 t [s] t [s]

SoC Battery Power 300 0.75 battery power SoC 0.7

200 0.65

100 0.6

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0 [kW] q [-] b

P 0.5

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Fuel Consumption Engine Power 3 300 fuel mass engine power

250 2.5

200 2 [kg]

[kN] 1.5

150 f * ice m P

100 1

50 0.5

0 0 0 100 200 300 400 500 0 100 200 300 400 500 t [s] t [s]

(g) Pice∗ (t) (h) mice,f (t) Figure 4.10: Simulation results using a hybrid mpc controller, for Test case 3, with initial soc q0 = 0.7. 4.5 Discussion 41

4.5 Discussion

The results that are presented in the previous section will mainly be analysed from three different viewpoints. The overall goal is to reduce the total fuel consumption of the Ki-Ha E200. However, in the case of an hev the analysis can not simply focus on the total fuel consumption, but needs to include the operation of the ice and the use of the storage battery. As mentioned in Section 2.1 one of the benefits of a hybrid system is that the ice can be operated at its optimal efficiency as it does not directly propel the vehicle. Another advantage is that the train can recuperate energy through regenerative braking, which can be stored in the storage battery. Therefore, before analysing the quantitative results, presented in Table 4.4, the operation of the ice and soc control will be analysed through observations of Figures 4.5-4.10. To understand the effect that the operation of the ice has on the fuel consumption, a comparison of the efficiency values given in Table 2.1 can be done as 220.25 201.89 − 0.0834, 220.25 ≈ (4.2) 216.13 201.89 − 0.0659. 216.13 ≈ From the calculations in (4.2) we obtain that running the ice at optimal efficiency, compared to running it at low or maximum efficiency, will result in a fuel reduction of 8.34% and 6.59%, respectively. The ice operation is easily analysed by observing Figures 4.5-4.10. It can be seen that the hybrid mpc runs the ice at its optimal efficiency or turned off for all test cases. Unlike the hybrid mpc, the heuristic ems also runs the ice at low or maximum at some points. It is therefore reasonable to expect that the fuel reduction will, at best, only be a few percent. One of the drawbacks of the heuristic ems is its inability to control the final value of the soc of the storage battery. The deviation from the initial value varies greatly for the different test cases. The case of the final soc being lower than the initial value is particularly troublesome. To reach the initial value of the soc, when arriving at a station, the ice would have to run to charge the battery. This would increase noise levels and exhaust emissions inside train stations, which is undesirable. The hybrid mpc partially solves this problem by ensuring that the soc is returned to a value equal to, or higher than, its initial value. This is done by adjusting the lower boundary value of the soc constraint such that it varies with the position of the hev. This is not ideal but can be seen as an acceptable behaviour with regards to the short prediction horizon. One could make the soc return to its initial value, using a short prediction horizon, but that could possibly require the mechanical brake to be used instead of regenerative braking. The energy would then be lost as heat is dissipated in the brakes instead of recuperated and stored in the battery. To understand the quantitative results, a few words need to be said about the design of the test cases and the control objective. The test cases are made up of 42 4 Results sections of powering, coasting and braking. The powering and braking sections use maximum acceleration and deceleration, respectively. This is unfavourable in case of the hybrid mpc as the control signal constraints have been linearised. The use of the linearised constraints will increase the feasible set in the optimisation problem, making it possible for the hybrid mpc to output control signals that violate the physical limitations of the hev. To avoid this issue, the control signals from the hybrid mpc are saturated. This could result in the use of non-optimal control signals, which will result in worse performance.

With regards to how the control objective has been set up to follow a certain speed profile, rather than reaching a certain position within a given time, it is mainly the design of the speed pattern that will determine the efficiency of how the train is operated, rather than the hybrid mpc. Depending on how the hybrid mpc follows the speed profile the performance will be slightly better or worse than that of the heuristic ems.

In order for the comparison of the quantitative results to be fair it is important that the run-time and travelled distance for a speciﬁc test case do not vary too much for the two controllers. From Table 4.4 it can be seen that ∆t is equal to zero for all test cases, meaning that the train stops at the same time for both controllers. The deviation in the train’s position is denoted ∆s and is at most 9 meters, which is small enough to be considered acceptable. To obtain a quantitative value that can be used to compare the fuel consumption of the two controllers, ∆q has been converted into a fuel equivalent mf ,∆q. The fuel equivalent is added to the fuel consumption of the controller with the lower ﬁnal soc. Doing so makes it possible to calculate how much the total fuel consumption has increased or decreased.

It is apparent that the hybrid mpc can achieve a reduction in fuel consumption for both Test case 1 and 2 for an initial soc value of 30% and 50%. In these cases the heuristic controller is running the ice at low and maximum, thus the hybrid mpc is able to reduce the total fuel consumption as it runs the ice at optimum power at all times.

If the initial soc is set to 70% the heuristic ems is running the ice at optimum power close to or at all times. The hybrid mpc performs worse than the heuristic ems in these cases. There are a few possible reasons for this result. As mentioned earlier, the design of the test cases will make the hybrid mpc operate near the constraint limits, resulting in a non-optimal output. Doing so repeatedly leads to a less eﬃcient use of energy when propelling the train, resulting in a higher fuel consumption. The way the hybrid mpc follows the given speed proﬁle could also result in poorer performance.

For Test case 3 the hybrid mpc only performs better in the case of an initial soc value of 30%. If the initial soc value is set to 50% or 70%, the heuristic ems operates the ice at optimum power throughout the whole simulation apart from a short instance at the end where it is run at low, see Figure 4.9. If we instead study the hybrid mpc for this case, see Figure 4.10, it shows that the hev struggles to 4.5 Discussion 43 follow the speed reference. This implies that the tuning is sub-optimal in this case and will contribute to a higher fuel consumption.

5 Conclusions and Future Work

This chapter gives a short conclusion of the thesis and presents a few possible ideas for future work.

5.1 Conclusions

The purpose of this thesis was to investigate the possibility of minimising the fuel consumption of the series hev Ki-Ha E200. The train currently uses a heuristic ems based on whether the train is powering, coasting or braking, and on the soc of the storage battery and the vehicle speed. In this thesis, a hybrid mpc was designed and implemented to study the possibility of achieving a reduced fuel consumption. To begin with, a model of the series hev was developed to express the system dynamics. The model was used as an internal model in the hybrid mpc and to calculate the evolution of the system states. The heuristic ems was implemented as a benchmark for the performance of the developed hybrid mpc. The two controllers have been tested for three different test cases, where each test case has been run with three different initial values for the storage battery’s soc. In all test cases, we have assumed a straight and flat track. The overall performance of the hybrid mpc seems promising. The hybrid mpc runs the engine at its optimal operating point for all test cases and manages to return the battery’s soc to a value larger than or equal to its initial value. In terms of total fuel consumption the hybrid mpc performs better than the heuristic ems in most cases. However, in the case of a high initial value of soc of the battery, the hybrid mpc performs slightly worse in some of the test cases.

45 46 5 Conclusions and Future Work

5.2 Future Work

As this thesis only deals with a very simplified model of the actual system and the developed controller has only been tested and analysed in simulation, a lot of interesting work remains to be done. Applying the controller to the actual system would of course be very interesting and valuable for the analysis of the controller performance, but more work is required before reaching that point. Due to hardware limitations in the simulation platform, the mpc prediction horizon has been restricted to a maximum of 40 time steps. Running simulations on a platform with more computation power, or possibly rewriting the hybrid mpc implementation to reduce the computational effort needed, would allow one to use a longer prediction horizon. Evaluating the effect of a longer prediction horizon is an interesting aspect to evaluate as it could further improve the performance of the hybrid mpc. The hybrid mpc has only been tested for the special case of a straight and flat track. Therefore, a suitable first step for future work would be to include the track gradient and curvature. This result is of great interest as the performance of the heuristic ems greatly decreases in environments with varying elevation, which increases the possibilty of improvement. Bibliography

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