Railway Power Supply System Models for Static Calculations in a Modular Design Implementation

Railway power supply system models for static calculations in a modular design implementation

Usability illustrated by case-studies of northern Malmbanan

RONNY SKOGBERG

Master’s Degree Project Stockholm, Sweden 2013

XR-EE-ES 2013:006 Railway power supply system models for static calculations in a modular design implementation Usability illustrated by case-studies of northern Malmbanan

RONNY SKOGBERG

Master of Science Thesis Royal Institute of Technology School of Electrical Engineering Electric Power Systems Stockholm, Sweden, 2013

Supervisors: Lars Abrahamsson, KTH Mario Lagos, Transrail AB Examiner: Lennart Söder

XR-EE-ES 2013:006 Abstract Several previous theses and reports have shown that voltage variations, and other types of supply changes, can inﬂuence the performance and movements of trains. As part of a modular software package for railway focused calculations, the need to take into account for the electrical behavior of the system was needed, to be used for both planning and operational uses.

In this thesis, different static models are presented and used for train related power flow calculations. A previous model used for converter stations is also extended to handle different configurations of multiple converters.

A special interest in the train type IORE, which is used for iron ore transports along Malmbanan, and the power systems inﬂuence to its performance, as available modules, for mechanical calculations, in the software uses the same train type.

A part of this project was to examine changes in the power systems performance if the control of the train converters were changed, both during motoring and regenerative braking.

A proposed node model, for the static parts of a railway power system, has been used to simplify the building of the power system model and implementation of the simulation environment.

From the results it can be concluded that under normal conditions, for the used train schedule, the voltage variation should not restrict the trains traction performance. It can also be seen from the results that a more optimized power factor control with a higher regenerative brake power or generation of reactive power could be used to limit the need for investments in infrastructure or to increase the traﬃc for a given system layout.

i Sammanfattning I ett ﬂertal tidigare undersökningar och rapporter har konstaterats att spän- ningsvariationer, och andra förändringar, hos strömförsörjningen till tåg kan påverka dess prestanda och dess färd längs rälsen. Som en del av ett modu- lärt programpaket för tågrelaterade beräkningar uppstod därför ett behov av elkraftsberäkningar, både för planering och operativ drift.

I denna rapport sammanställs och används ett antal olika statiska modeller för tågrelaterade effektflödesberäkningar. Modellen för omformarstationer har även utökats för att hantera konfigurationer då olika typer av omformare används.

Ett särskilt intresse för tågtypen IORE, som används för malmtransporter längs Malmbanan, och dess påverkan av en förändrad strömförsörjning, har funnits då olika typer av mekaniska beräkningar för denna tågtyp utförs i andra beﬁntliga moduler.

En del av projektet bestod i att undersöka förändringar i elförsörjningen, på grund av en ändrad styrning av tågens omformare, både vid återmatning och motordrift.

En föreslagen nodmodell för den statiska delen av elnätet har använts för att förenkla elsystemsmodellen och uppbyggnaden av simuleringsmiljön.

Av resultaten från simuleringarna kan man anta att under normala förhållanden, och med det använda körschemat, bör ej spänningen vara en begränsande faktor för tågens drift. Övriga simuleringar visar också att en mer optimerad effektfaktor för högre återmatad bromseffekt eller för generering av reaktiv effekt kan användas för att slippa investeringar i infrastrukturen, eller för att utöka trafikmängden för ett givet system.

ii Contents

1 Introduction 1 1.1 Background ...... 1 1.2 Aim ...... 2 1.3 Limitations ...... 3 1.4 Structure of thesis ...... 3 1.5 Previous related work ...... 4

2 Power ﬂow analysis 5 2.1 Introduction ...... 5 2.2 Power ﬂow ...... 6 2.3 Losses ...... 7 2.4 Admittance matrix ...... 8

3 Railway power systems 10 3.1 Overview ...... 10 3.2 Power supply ...... 11 3.3 Rail current return systems ...... 12 3.3.1 Booster transformer ...... 13 3.3.2 Auto transformer ...... 13 3.4 High-voltage transmission system ...... 15 3.4.1 Transformers ...... 16 3.5 Converter stations ...... 18 3.5.1 Rotary converters ...... 18 3.5.2 Static converters ...... 21 3.6 Trains ...... 22 3.6.1 Asynchronous trains ...... 23 3.6.2 Thyristor based trains ...... 25 3.6.3 Regulation for motoring ...... 27 3.6.4 Regulation for regenerative braking ...... 28

4 Computer model implementation and calculations 29 4.1 Program layout ...... 29 4.2 A modular standard node ...... 30 4.3 Converters ...... 31 4.3.1 Converter losses ...... 31 4.3.2 Parallel converters ...... 33

iii 4.4 Trains ...... 34 4.4.1 IORE locomotives ...... 34 4.4.2 Thyristor based locomotives ...... 38 4.5 Mathematical model ...... 39 4.5.1 Solver ...... 39 4.5.2 Equations and constraints ...... 40 4.5.3 Optimization ...... 45 4.6 Java-GAMS interaction ...... 47

5 Case-study and simulation 48 5.1 The iron ore line ...... 48 5.2 Railway line model ...... 49 5.3 Train models ...... 50 5.4 Simulations ...... 52 5.4.1 Normal operation ...... 53 5.4.2 Converter station outages ...... 55 5.4.3 Eﬀects of alternative power factor control ...... 59

6 Conclusions and future work 63 6.1 Conclusions ...... 63 6.2 Future work ...... 64

A Numerical data used 65 A.1 Per-unit system ...... 65 A.2 Converters and grid-connection ...... 66 A.3 Catenaries ...... 66 A.4 High-Voltage transmission lines ...... 67 A.5 Trains ...... 69 A.6 Electrical layout ...... 70

Bibliography 72

iv 1| Introduction

1.1 Background

Electric power systems are spread throughout the society and they are an important part of every day life. A special area within electric power systems is electric railways, which is used to power trains for transportation of both freight and people.

Some of the diﬀerences compared to an ordinary power system is the large voltage variation allowed and that the loads are not stationary, which makes the power system layout change over time. The railway power supply system also uses a single-phase design to transfer power to the trains compared to a more common three-phase transmission line used for other power systems.

One important element of an electric railway system is the availability of electric power as needed for the intended traﬃc situation. The infrastructure itself is limited as to the amount of installed capacity and strength of the grid, and over sizing the system could be costly. The tractive power available for the trains could also be limited because of the voltage level of the power system, either by design or from a standard [1]. Both a too low or a too high voltage level could impair the power transfer to, or from, the trains.

Modern trains with a power electronic based traction system, where the power factor can be controlled in software, could be used to improve the power supply systems performance. One solution to keep the catenary voltage high, and with less losses, is if the locomotives were given the possibility to compensate the reactive power in the network by operating with a leading power factor [2]. As this is not always allowed for certain railway systems [3], other methods needs to be considered for better system performance such as changes in the timetable or limiting the trains power demand.

As part of a bigger software suite, TRAINS, developed by the company Transrail Sweden AB, an implementation of the power ﬂow calculations of an AC railway power supply system, named TRAINS AC Supply, is needed to give information about the power system. Electrical models for a railway power system is not available in the existing software, and is to be addressed in this thesis to be able to combine the electrical and mechanical calculations of

1 a railway system. During the infrastructure planning, time table construction and train operation, the information from the calculations could be used to alter the design or suggest operational changes.

The train operations software, TRAINS Performance, keeps track of all trains and the status of the power system after obtaining the results from TRAINS AC Supply. The mechanical aspects of train movements is already implemented in other modules of TRAINS, and can be used to evaluate travel time and power demand. TRAINS AC Supply can be used to calculate if the intended tractive power is feasible or if there is any system violations that needs to be addressed, and for a given system situation give a recommended power demand for the diﬀerent trains.

One of Transrail’s other products CATO, Computer Aided Train Operation, is used nowadays to send information about optimum speed and tractive power to the driver. This speed recommendation is calculated with the intent to minimize the number of stops and speed limits due to nearby trains on the same track. The speed calculated for each train assumes that the maximum tractive power is always available. Together with information about the power system, tractive power limits occurring in the system could also be taken account for.

Information about an unfeasible tractive power could be useful during both planning of a train schedule and during realtime operation. For a degraded state of the power supply system, as in the case with a power outage, the maximum tractive power recommended by CATO for each train could be changed to try to mitigate the eﬀects on the time table.

1.2 Aim

The aim of this thesis is to make it possible to integrate power flow calculations into the TRAINS software suite, considering the consequences of voltage levels in the power system for an AC railway. An additional part of this thesis is to evaluate if it is possible to increase the system performance by optimization of the reactive power. The TRAINS software is implemented in Java [4] and the power flow models are implemented in GAMS [5] using a suitable solver for power flow calculations at fundamental frequency.

For all parts of the simulated power system the following is going to be calculated, where applicable:

• Active and reactive power ﬂow in substations and locomotives.

• Voltage levels and angles at all nodes.

• Currents and losses in all links of the network.

• Converter losses.

2 • Consequences of a limited currents, and voltage levels higher or lower than normal.

• Consequences of limited train tractive power and limited regenerative braking due to the state of the power system.

1.3 Limitations

The programming part of this thesis is limited to calculations of relevant model parameters needed for the calculations of the electrical power system. The connection to any database or evaluation of the calculated data is handed over to the TRAINS Performance application and not part of the program, but formatting of data is implemented as needed.

The power system bounds of the simulated network include the railway power system between Rombak and Kiruna, and the bound towards the supply is between the 50 Hz power lines and the ﬁrst transformer that connects to a converter station.

Transients and faults in the power system is not included in the models as the system is assumed to be stable for the duration of the calculation of the voltages and currents.

Models and simulations included are focused on the available infrastructure and locomotives on the northern part of the Swedish iron ore railway, Malmbanan, between Kiruna and Riksgränsen. As most of the transports on this part of the railway are traﬃcked by MTAB’s, Malmtraﬁk i Kiruna AB, IORE locomotives [6], train modeling is focused on this type of locomotive.

1.4 Structure of thesis

This thesis main focus is divided into four parts:

• Chapter 2 - Power ﬂow analysis A short introduction to basic power ﬂow calculations for non-electrical engineers.

• Chapter 3 - Railway power system Description and models of the diﬀerent electrical parts of an AC railway power supply system used in this thesis.

• Chapter 4 - Model implementation Description of program structure and implementation of models from previous chapters.

• Chapter 5 - Simulation Simulations and case study of power system behavior at Malmbanan for diﬀerent power system and electrical locomotive control strategies.

3 1.5 Previous related work

Plenty of previous work about calculation of power flow in power systems exist, and therefore only work specific to railway power systems is listed below. Papers and theses about load flow problems and power flow calculations are in abundance, more specific for this thesis is therefore papers focused on railway power systems.

In Biedermann’s master thesis [7], simulation results of the eﬀect of a low catenary voltage for common trains in Sweden is analyzed. The conclusion from this thesis is that using a mean value of the catenary voltage is a poor indicator to estimate the performance of the railway system.

In Olofsson’s licentiate and doctoral theses [8,9], models of rotary converters and locomotives with thyristor based converters are derived, which in [10] have been further developed for better computational properties.

Some parts of the Rc-train and other models used in [10] are reﬁned by Boullanger in [11] as to give more realistic results compared to other professional simulation software.

4 2| Power ﬂow analysis

The aim of this chapter is to give engineers with limited knowledge from studies in the ﬁeld of electric engineering an understanding of the fundamental equations and variables used in the following chapters. The equations derived in this chapter are generic, and not only speciﬁc for railway power systems.

2.1 Introduction

Analysis of the state of an electrical power system, in steady-state, is often done using the node-voltage model, where the solution of a linear system of complex equations gives all the currents in the system as a function of the voltages [12]. If all node currents and voltages are known, the power ﬂow for the whole system can be calculated.

In power system analysis, the injection of currents and the power ﬂow from or to loads are often modeled as constant powers [13], however, there are also other types of models that can be used depending on the problem to be solved. Some of the other type of load models used can be based on a voltage source or a current source in combination with an impedance, impedance models with the use of lumped components or constant current models [14]. For loads where the power demand is based on a mechanical force, for example with trains, a simple relationship exist between the demanded force, F , and the needed power, P [15]:

P = F v (2.1)

To be able to solve the node voltage equations when loads are deﬁned as powers, and thus non-linear, an iterative solution is necessary.

For easier writing of the equations for the power system network, all impedances are converted into admittances, and all voltage sources are changed into current sources, a Norton equivalent.

To perform calculations on a power system, an equivalent line model is used to simplify calculations. The electrical properties of a transmission line can be described with two

5 equations, series impedance, zseries, in equation (2.2), and shunt admittance, yshunt, in equation (2.3) [12].

zseries = r + jωl = r + jx (2.2)

yshunt = g + jωc = g + jb (2.3)

To diﬀerentiate an electrical current from the imaginary unit, the letter j is used instead to represent the imaginary unit.

The variables r, l, x, g, c and b in equation (2.2) and (2.3) describes the electrical properties of a transmission line, where the reactance x and the susceptance b includes the value of the angular frequency, ω, that is used in the calculations. The series impedance is used here to describe the properties along a conductor, whereas the shunt admittance is used to describe the properties between the electrical conductors.

The parameters used are r, the resistance along a line, and l, the inductance in the line. The conductance g describes the leakage current through the insulation, or air, and c is the capacitance between the conductors.

The relationship between an impedance and its admittance is

1 y = (2.4) z and can be converted to one or the other as needed to give simpler equations.

The same power ﬂow equations can also be used for symmetric three-phase power systems if they are ﬁrst converted to a single-phase representation.

To be able to easily compare the inﬂuence of diﬀerent network components and simplify numerical calculations, a transformation to "per-unit" is performed for every component in the power system. A common MVA base is selected for the whole power system, and a voltage base is selected for one point in the system [12]. All other base values can then be calculated, see appendix A.1.

2.2 Power ﬂow

To be able to solve the power flow through a transmission line, the equations for the complex power, S, is needed. If Kirchhoff’s current law is used, which gives the currents in the system as a function of the voltages, admittances is preferred to describe the system [12]. For transmission lines with a length less than 250 km, the nominal π model in figure 2.1 can be used for a sufficient accuracy [12]. The total shunt admittance of the analyzed line is in

6 this model divided into two equal parts, and placed as lumped elements at each end of the transmission line.

k Ikm ykm m

Uk yk0 ym0 Um

Figure 2.1. Nominal-π model of a single-phase transmission line.

The current through a transmission line, from node k to node m, using the nominal-π model in ﬁgure 2.1 can be expressed as:

Ikm = ykm(Uk − Um) (2.5)

The complex power from node k to node m is with (2.5) given as:

∗ ∗ ∗ ∗ Skm = UkIkm = Ukykm(Uk − Um) (2.6)

If rewriting (2.6) in rectangular form, with Skm = Pkm + jQkm, separating active and reactive power as the real and imaginary part results in:

2 Pkm = gkmUk − UkUm(gkm cos θkm + bkm sin θkm) (2.7) 2 Qkm = − bkmUk + UkUm(bkm cos θkm − gkm sin θkm) (2.8) where the diﬀerence in voltage angle between node k and node m, θkm, is deﬁned as:

θkm = θk − θm (2.9)

2.3 Losses

The losses in a transmission line can be calculated as the sum of the complex power inserted at each end of the line, Sloss,km = Skm +Smk, and the total system losses is given if summing all powers inserted into all nodes. Separating the active power results in

2 2 Ploss,km = Pkm + Pmk = gkm(Uk + Um) − 2gkmUkUm cos θkm. (2.10)

7 The total system losses can be calculated as:

X Ploss = Pkm. (2.11) k,m

For a lossless line, where

gkm = gk0 = 0 (2.12) the active power into a line is equal the active power received at the other end. For an overhead line, often the shunt conductance, gk0, is neglected as the leakage current is small [12].

2.4 Admittance matrix

To describe a power system in a systematic way, and be able to use that description of the system to calculate the power ﬂow, an admittance matrix is constructed. There are several methods to produce an admittance matrix, and one method is shown in this chapter.

An admittance network of an electrical system can be formulated as:

I = YU (2.13) where Y , the bus admittance matrix, describes all the admittances and connections in the system in complex form, U is a vector of all node voltages, and I is a vector of the injected currents into the nodes. This system of linear equations can be uniquely solved if either the voltage or the current for every node in the system is known.

The total current into a node k, can with the help of the equation for the current through a transmission line, equation (2.5), be written as:

n n X X Ik = Ikm = ykm(Uk − Um). (2.14) m=0 m=0

If the index 0 is chosen as reference, and normally ground, which gives that U0 = 0, equation (2.14) can be separated into:

n n X X Ik = Uk ykm − ykmUm , (2.15) m=0 m=1 where the number of nodes in the system, excluding the reference node, is n.

8 Expanding the sums for an arbitrary node k for all values of m results in:

Ik = Uk(yk0 + yk1 + yk2 + ykk + ··· + ykn) − U1yk1 − U2yk2 − Ukykk − · · · − Unykn (2.16) and eliminating Ukykk gives

Ik = Uk(yk0 + yk1 + yk2 + ··· + ykn) − U1yk1 − U2yk2 − · · · − Unykn. (2.17)

From equation (2.17) it can be seen that the values in the bus admittance matrix Y can be expressed in two parts. The diagonal elements is the sum of all admittances connected to node k, (2.18), and the oﬀ-diagonal elements is the negative value of the admittance between node k and node m, (2.19) [12].

n X Ykk = ykm k 6= m (2.18) m=0

Ykm = −ykm (2.19)

9 3| Railway power systems

3.1 Overview

In 1879, at the Berlin Exhibition, the first glimpse of a new era of electric traction was shown, and just two years later the first electric tramway was built in the same city. All the performance benefits over other types of traction techniques was not known at this time, but as the technology and science moved forward, the advantages of an electrified railway, especially with AC systems, became obvious [15].

Because of limitations in the electrical machines available, and the need for AC to transform the voltage, a low frequency was used in the early days of development. Some of the countries which began the electriﬁcation at the time of the ﬁrst world war decided to use

15 kV 162⁄3 Hz, and several of them still do, while others changed to both a higher voltage and frequency [15].

A railway power system is often by design much weaker, or under-dimensioned, compared to other power systems [10, 15]. The voltage level is more sensitive to the power flow, and the transmission losses can also be higher [10]. The allowed voltage variations on different parts of a railway power supply system is larger compared to other power systems, up to double the percentage of voltage variation under normal conditions [16, 17]. To keep the voltage as stiff as the national grid would increase the cost drastically. Trains are however more capable to handle the voltage variation compared to more general types of power system loads.

Several diﬀerent types of power systems for railways exists today:

• DC

• AC with same frequency as the national grid

• AC with diﬀerent frequency than the national grid

For the AC systems it is also possible to operate either synchronous or asynchronous to the national grid.

10 For Europe, the most common systems used for main railways are 3 kV DC, 15 kV 162⁄3 Hz AC and 25 kV 50 Hz AC. But not only the power system is different in different countries, also the signaling system can vary between them, forcing trains traveling over national borders to be able to handle multiple power systems as well as different signaling systems [15].

The national railway power grid in Sweden is for most parts operating at a nominal voltage of

15 kV, 162⁄3 Hz and is synchronous with the national grid. There is also no power generation directly connected to the railway power system as compared to some other countries [18].

The main parts of a railway power system in this thesis can be grouped into four categories: power supply lines, converter stations, high-voltage transmission system and trains. The power supply lines for a railway power system is in the text further divided into two parts, transmission towards the trains and the current return system. The models used in each part are explained in the following sections.

3.2 Power supply

Two types of power supply system to trains are commonly used, an overhead contact line or a third rail. The third rail solution is most common for subways/metros [15], and therefore only the overhead contact line will be studied in this thesis.

An overhead contact line of a traction system is often called catenary and is usually made of a copper alloy. The name comes from the shape of the line that actually holds the contact line below.

As compared to other power systems, the factor R/X is in the vicinity of one. This implies that the resistance can not be neglected during calculations, as often done for other types of high voltage grids. A train is usually more capable of handling a large voltage drop, in contrast to other types of loads in a power supply system. The larger resistance of the catenary is therefore not a traffic operational problem for most of the time. Even though the trains in most situation can operate at a lower voltage level than normal, the maximum power could under those situation be limited [19]. The effects of the voltage drop on the catenary is one of the effects that will be studied in chapter 5.

There are several types of catenary systems, either just a single contact line, where the ground acts as the return circuit, or in different combinations of return conductors and line feeders. The additional line feeders is of importance to lower the impedance of the catenary system and give a lower voltage drop, especially for lines with a high power demand. Four factors that affects the power demand for trains are: the intensity of the traffic, the use of heavy trains, the gradient of the track and when the speed of the trains are high. For parts of the railway where there is more than one track, the catenary of the separate tracks could be interconnected at regular intervals to even further lessen the impedance [15].

11 The impedance model used in this thesis for a transmission line is the π-model with lumped impedances, varying linearly with the line length dx, see ﬁgure 3.1. The capacitance of the line is divided in two and placed at both ends of the line. The validity for this model, compared to using a distributed impedance model, is good as long as the line length is much shorter than the wavelength. The model is assumed accurate if the wavelength is twenty times longer than the length between two nodes of a contact line [12].

I r dx jx dx k km · · m

jb jb Uk k0 dx m0 dx Um 2 · 2 ·

Figure 3.1. Nominal-π transmission line model with impedances and shunt admittances represented as lumped parts.

Using this model for a railway power supply system with either a 162⁄3 Hz or 50 Hz frequency, would give an accurate enough result upto a line length of about 900 km or 300 km, respectively.

3.3 Rail current return systems

A simple system for powering the trains in a railway power system is with an overhead contact line as the supply conductor, and using the rail as the return conductor. As the total area between the track and the ground is large, the impedance as seen from the train is relative small. It is therefore possible that stray currents would ﬂow through the ground, instead of just through the track [15]. This could lead to undesirable currents and voltages in parts of the surrounding, as the rail, fences or nearby cables. The stray currents could also be a source of induced disturbances into nearby communication lines, especially when the ground resistance is high [10].

To limit the amount of stray current ﬂowing in the ground, a separate return conductor could be connected to the track at regular intervals to help the current to ﬂow in a more controlled way, especially for countries with a low impedance soil [7]. Another way to force the current through the track is to use a transformer based current return system.

The two main types of transformer based current return systems used to control the return current in AC-systems are booster-transformer, BT, and auto-transformer, AT. Both are variants of systems for reducing the current ﬂow through the ground and instead make it

12 preferable for the current to ﬂow through the track and/or a separate return conductor. For a DC-system, one method to force the current to ﬂow through the rail is complete isolation of the rail from the ground [15].

3.3.1 Booster transformer

The booster transformer is connected to the catenary and rail as in ﬁgure 3.2, and consist of a transformer with equal number of turns on each winding, forcing the current through the catenary and the return circuit to be equal. The return circuit can consist of either a simple connection to the rail, or separate conductors along the track [7]. The main purpose of the BT-system is to minimize the leakage current through the ground.

The use of transformers in the catenary gives an increased impedance compared to a system without, and thereby lessen the power transferability. The needed gap in the catenary, where the connection of the transformer is made, could also give problems with arcing as the train passes by [15]. The impedance of a BT-system, as seen from the train, changes smoothly between the transformers, however, when passing a transformer a distinct change in the impedance is caused by the transformer winding.

A typical distance between booster-transformers is 3-5 km, with the longer distance for catenaries with one or more separate return conductors [7].

Booster Transformer Return circuit

Catenary

Track Figure 3.2. Overview of a BT-system with currents in red for a train with a single feeding station.

3.3.2 Auto transformer

To mitigate the influences of the impedance of booster-transformers on the railway power system, a system of auto-transformers, AT, could be used instead of, or together with booster-transformers. The connection of an AT-system, as pictured in figure 3.3, gives a secondary conductor, also called negative, with a 180◦ phase difference compared to the catenary. As a result the voltage level of the system is virtually doubled giving much less losses for the same power drawn from the system [15].

13 There are many different connection configurations for an AT-system, the one pictured in figure 3.3 shows the configuration used in Sweden [20]. In that configuration, the converter station is connected directly to the catenary and track, while for other configurations the converter could be connected between the negative feeder and catenary. If the line current out from the closest converter station of an AT-system exceeds 600 A, the first auto-transformer is normally doubled [21].

Compared to a BT-system, the auto transformers only makes it more preferable for the return current to ﬂow in the rail or the secondary conductor. Even if using ideal transformers in the calculations, the AT-system would not keep the return current from ﬂowing in the ground, as compared to a BT-system [15].

A simple model used for impedance calculations on an AT-system is described and exempliﬁed in [22], where the impedance for the catenary is divided in two parts. An initial impedance, placed at each end of an AT-system, and a length dependent π-model impedance. In ﬁgure 3.4 the impedance approximation is plotted against a more detailed calculation of the catenary impedance for a double- and single-fed system, as seen from the train [22].

One problem when using an initial impedance approximation, is the discontinuous impedance that is seen from the train when passing a converter station with diﬀerent types of catenaries on either side. A solution for this is to move the initial impedance to follow the train instead, still, this only mitigates the problem with diﬀerent initial impedances. For short distances between AT-transformers, the initial impedance could be neglected [23].

Negative feeder Auto Transformer

Catenary

Track Figure 3.3. Overview of a single feed AT-system, with one of several possible types of connections between a converter station and catenary.

A more detailed model of an AT-system is developed in [14] where every transformer consist of a 3-by-3 admittance matrix, however, this adds nine more elements to the admittance matrix for every AT-transformer in the system, as compared to the model used in [22].

For an auto-transformer system in Sweden, a typical distance between the transformers are 10-15 km [7]. If the distance to the ﬁrst transformer from the converter station is more than 2.8 km, a booster-transformer is used in between [10].

14 Single−fed catenary impedance Double−fed catenary impedance 6 6

] ] Phase ° ° 5 Phase 5

4 4

] − Phase/10 [ 3 ] − Phase/10 [ 3 Ω Ω 2 2 Magnitude Magnitude 1 1 Magnitude [ Magnitude [ 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Distance[km] Distance[km] (a) Single-fed AT-system. (b) Double-fed AT-system.

Figure 3.4. Dashed line represents the magnitude and phase of the impedance used as approximation for single and double-fed catenaries. In the ﬁgures a 100 km line with auto- transformers at every 10 km is used as an example. The contact line approximated is a 100 2AT FÖ, and the approximated initial impedance Zo has been chosen to four times the more detailed calculated initial impedance, and with a phase angle of 43◦ [22]. The resulting approximated impedance was Zo = 0.215 + j0.343 Ω and Z = 0.034 + j0.032 Ω/km

3.4 High-voltage transmission system

To strengthen the weak railway power system, a high-voltage transmission system is used in parts of Sweden and other countries. The higher voltage used for most of these power lines gives less losses compared to a normal catenary for the same power ﬂow, and thereby helps to keep the voltage level of the catenary higher. It is also possible to increase the distance between the converter stations if a high-voltage transmission system is used.

At least three voltage levels at the high-voltage transmission lines are in use in the Swedish railway power grid, 132 kV, 30 kV and 15 kV. There are also diﬀerent high-voltage cables used for some distances. The transformers used for the most used voltage level, 132 kV, is in almost all cases arranged as two 25 MVA transformers at a converter station, and one 16 MVA transformer along the catenary. The high-voltage line itself is arranged as a two-phase system, 2x66 kV, with the transformer midpoint directly earthed [3]. The second 25 MVA transformer is used as a backup in case of a failure in the ﬁrst one [22].

The distance between the ends of a high-voltage overhead line could be considerable longer than in the case for a catenary as the voltage is often much higher. From a Swedish point of view, the longest high-voltage transmission line for railway power systems is 70 km between connection points [22] and within the limits of the nominal-π model [12].

In ﬁgure 3.5 the principle of the transformer sizes in a high-voltage transmission system is shown.

15 50 Hz 162⁄3 Hz 25 MVA Frequency

16 MVA

Catenary 25 MVA

16 MVA High-voltage trasmission line 25 MVA

Grid Generator Nominal 15 kV 132 kV voltage voltage voltage

Figure 3.5. High-voltage transmission system arrangement with multiple frequency converters, FC.

3.4.1 Transformers

One of the important parts of a high-voltage transmission system is the transformers connecting the high-voltage transmission line to the catenary. The used model of the transformer inﬂuences the calculations of the power ﬂow in the system, and the total losses.

A detailed model of a transformers equivalent circuit is shown in ﬁgure 3.6, and it consist of an ideal transformer, T1, the winding resistances, R1 and R2, core loss and magnetization impedance, Rm and Xm, and ﬁnally the leakage reactances, X1 and X2 [12].

Z1 = R1 + jX1 I1 I2 Z2 = R2 + jX2

U1 Rm jXm E1 E2 U2

N1 : N2

Figure 3.6. Detailed equivalent circuit model of a transformer.

The voltage and current equations for an ideal transformer is

E N I 1 = 1 = 2 (3.1) E2 N2 I1

16 and is dependent on the turn-ratio N1:N2, where N1 and N2 represents the number of turns on the primary and secondary coil.

0 Transforming an impedance from the secondary side, Z2, to the primary side, Z2, using (3.1) gives the relationship: 2 0 N1 Z2 = · Z2 (3.2) N2

The first step in the simplification of the more advanced model in figure 3.6 is to transfer all impedances to one side of the model using equation (3.2), and in this case the primary side is chosen. If the impedances are converted to their per-unit values, the ideal transformer T1 can be removed as the voltage on either side is the same in p.u. The magnetization current through Rm and Xm is small compared to the current through a fully utilized transformer, and the effect of moving the two impedances to the left side of R1 and X1, in the schematics in figure 3.6, is assumed to be negligible. Combining the impedances on the primary side with the transferred impedances from the secondary side into a single short-circuit impedance as 0 Zk1 = Z1 + Z2 , (3.3) where Zk1 also can be expressed in its real and imaginary parts as

Zk1 = Rk1 + jXk1. (3.4)

Power transformers are commonly designed to give a very low magnetizing current [12], and therefore it is assumed that the inﬂuence of the magnetization reactance, Xm, can be neglected, which gives an approximation of a transformer that can be seen in ﬁgure 3.7.

Even though the magnetizing current is low, the core loss resistance Rm has been kept as segments of railway power systems could have periods of low utilization and the sum of all transformer losses are not insigniﬁcant. This model still contains most of the electrical properties of a transformer, including losses, as needed to give a reasonable level of accuracy for the loss calculations.

Rk1 jXk1

U1 Rm U2

Figure 3.7. Simple equivalent circuit model of a transformer.

17 3.5 Converter stations

As the railway power system delivers its power with only a single phase and the public grid often uses three phases, and sometimes with different frequency, there is a need for power conversion in between the systems. Depending on the frequency used in the railway power system, different connections to the supply grid is possible, either directly through a transformer, or with the use of a converter if the frequency is different in the two power systems.

For a system, as in Sweden, where the frequency is not equal to the public grid, rotary converters and static converters are used. The grouping of the converter types is based on the physical energy conversion, rotary converters uses a mechanical force to transfer power whereas static converters is based on electrical conversion using semiconductors. Rotary converters also gives an electrical separation of the the power systems whereas static converters do not. The electrical separation limits the amount of electrical disturbances that propagate through the converter. Compared to a railway power system that uses only transformers to connect to the utility grid, both static and rotary converters give a symmetrical loading of the supplying grid.

The output voltage from the converter station is variable in the Swedish railway system as another means of controlling of the power ﬂow. The voltage level is commonly set at 16.5 kV as the no-load value. The output voltage control use an amplitude compounding, or voltage drop, as a function of the reactive power to change the converter station voltage level. The no-load voltage level for a converter is in Sweden speciﬁed as to be settable between 15.1 kV and 17.25 kV, and with a compounding factor of between 0 % and -15 % [24].

Over voltage or under voltage in the catenaries could inﬂuence the operation of trains, as their capability to consume, or produce, a speciﬁc tractive power could be limited [7].

3.5.1 Rotary converters

A rotary converter consist of a motor and a generator connected together with a common axis, both constructed as synchronous machines with a salient pole design. As the number of pole pairs in the generator is one third of the number of pole pairs in the motor, for example either 6 and 2 or 3 and 1, the electrical frequency from the generator is also one third of the input frequency of the motor [15].

The rotary converters used in Sweden are all mobile, and is constructed on top of a railway carriage. This way it is possible to transport the converters to a diﬀerent location for service and repair [7].

18 d Iq E q

Id φ U 0 U sin(δ) jXqIq RI I U

jXdId

Figure 3.8. Phasor diagram for a salient-pole generator [12].

Given the phasor diagram for a salient pole generator, in figure 3.8, a static model of the phase difference δ between the terminal voltage, U, and the internal EMF, E, as a function of active and reactive power can be formulated [9]. The two equations (3.5) and (3.6), which can be derived from inspection of figure 3.8, in combination with the used definition of active and reactive power (3.7) and (3.8) is used to give a combined equation.

U · sin(δ) = Xq · Iq (3.5)

Iq = I · cos(δ + φ) (3.6)

P = U · I · cos(φ) (3.7) Q = U · I · sin(φ) (3.8)

The combined equation derives into Xq · P δ = arctan 2 (3.9) U + Xq · Q to give the resulting phase angle for a speciﬁed power, where Xq represent the quadrature-axis synchronous reactance.

For a single rotary converter unit, as shown in ﬁgure 3.9, U m and U g represent the converter m g voltages, Q50 and QG represent the reactive power ﬂow and Xq and Xq represent the quadrature-axis synchronous reactance, at the motor side and generator side of the converter g unit respectively. To simplify the equations, Xq is the combined reactance of both the generator and the converter output transformer.

This model neglects any eﬀects of the losses in the motor and generator, and therefore the active power through a converter unit, as shown in ﬁgure 3.9, is PG on both side of the

19 unit. Neglecting of the losses results in simpler equations and a separate loss calculation depending on the number of active converters in a converter station is used in chapter 4.3.

P jQ P + jQ G − 50 G G M G U m U g

m g Xq Xq

Figure 3.9. A rotary converter unit.

The total phase diﬀerence between the motor and generator voltages for a single rotary converter, ψ, is

m ! g 1 X · PG X · PG ψ = − arctan q − arctan q , (3.10) m2 m g2 g 3 U + Xq · Q50 U + Xq · QG and where the phase diﬀerence contribution by the motor is only one third as discussed in [9].

The voltage phase angle at the 50 Hz side of a converter station, θ0 in ﬁgure 3.10, is dependent of the loading of the converters [9], the short circuit MVA of the grid, the reactance of the used grid transformer, and can be expressed as

1 X · P θ0 = θ50 − arctan 50 50 (3.11) m2 3 U + X50 · Q50 where X50 represent the reactance of the transformer connected to the motor side of a converter combined with the short circuit reactance of the public grid, θ50 is the no-load voltage phase angle, P50 is the total active power into the converter station, and Q50 is the reactive power delivered to the public grid. If an agreement with the grid owner exist, reactive power can be sent to the grid to keep the voltage level at a desired level [24].

Public grid M G Catenary 50 Hz 162/3 Hz

θ50 θ0 θ M G

X50 . . .

M G

Figure 3.10. Connections in a rotary converter station with multiple units, and the position of the electrical angles θ, θ0 and θ50.

20 For the case where all the converters in a converter station are of the same type, the total power ﬂow through the station is divided equally among the converters. In stations with other combinations of converters, the individual converter rating relative to the stations total rating could be used as a distribution factor. Equation (3.10) can be used for calculating the phase diﬀerence for a converter station with multiple converters of the same type, as in

ﬁgure 3.10, if PG, QG and Q50 are all divided by the number of converters in the converter station [10].

The output voltage level for a converter, U g, is in Sweden controlled as a function of the reactive power [24] and is given by:

g g QG U = Uidle · 1 + Cf · . (3.12) SG

g g U depends on the no-load catenary voltage at a converter station, Uidle, the reactive power output from the generator, QG, the rated apparent power of the generator, SG, and the parameter Cf , which represent the amplitude compounding factor of the converter. The same equation is applicable for parallel operation of rotary converters if QG and SG represent the total reactive power and apparent power of the converter station.

The total phase diﬀerence, θ, for a converter station, from the 50 Hz grid to the 162⁄3 Hz catenary is 0 θ = θ + ψ(PG,QG,Q50). (3.13)

Rotary converters are capable of withstanding a large overloading of the converter for shorter periods of time, but eventually they shuts down for either an over current or if temperature thresholds are reached [1]. This could eﬀect the voltage level of the catenary as the voltage drop increases if a converter station is oﬄine.

3.5.2 Static converters

Several diﬀerent types of static converter designs are in use around the world. The three techniques used in Sweden are: direct converters, self-commutated converters with an intermediate DC-link, and since 2012, modular multilevel converters. All, except the self- commutated converters from before 2012, are constructed to be able to transfer power in any direction [25].

Static converters are in Sweden and Norway used in such a way that they mimic the behavior of the rotary converters for power loading levels within rating [8,24]. But one of the important diﬀerences is that the static converter can not be overloaded in the same way as a rotary converter, and therefore some method of protecting the converter is needed. Most of the

21 self-commutated converters used in Sweden, are not capable of transferring power from the railway power system back to the public grid [25].

Using the same function for the angle ψ, (3.10), as for the rotary converter and changing 0 m g the values of the phase angle, θ , or motor constants, Xq and Xq , gives the possibility to either shift or tilt the voltage angle function [8]. In this way the power ﬂow from diﬀerent static converter stations can be changed.

In the event of the converter reaching the power limit, a change in voltage angle should be used to limit the current flowing through the converter [9]. Parts of the control of voltage angle as a function of active power is shown in figure 3.11. The limit of a converter is based on the current, therefore the apparent power is influencing at which level the derivative of the angle starts to change, and not just the active power. An alternative is to change the voltage setting on the railway side of the converter station transformer to decrease the output voltage, and thereby decrease the power transfer [1]. For converter stations without the possibility to transfer power back to the public grid, or when the maximum level of power is reached, the same type of change in the output angle is used but in the other direction.

ψ −

Imax I sgn(P ) · Figure 3.11. Voltage angle change at static converter to limit the output current. Depending on the possibility to transfer power to the public grid, the angle either continues linearly with the current or follows along the axis of I = 0. For illustrative purposes, a linear relation is shown in the ﬁgure.

When a static converter has changed the voltage angle to divert the power ﬂow to other converters, it is still providing power to the trains at its maximum rating.

3.6 Trains

Several types of propulsion systems for trains has seen the light over the time of railway history. As the railways in Sweden is mostly electrified, the share of diesel trains is limited compared to the electrified ones, and also compared to the rest of the world. The development of trains has followed the evolution of more efficient electrical machines and converters. As new trains are developed, new techniques are implemented to increase their performance.

22 The different systems used for trains can be divided into different groups depending on the type of power supply system, the type of on the train converter and as to what type of traction motor they use. Combinations of either an AC or DC catenary system together with either an AC or a DC motor gives several traction systems. Some trains are also equipped with two, or more, types of converters to be able to travel on several railway systems, for example the railway of two different countries [15].

The two most common types of traction systems used nowadays is the half bridge, thyristor based, phase-controlled rectifier and the voltage source converter, VSI [15]. One of the important differences is the possibility to use regenerative braking when using a VSI, as power can flow in either direction. For AC railways, the VSI is combined with a rectifier to become a voltage source converter. For DC railways the VSI can either be connected directly to the catenary or to an input DC-DC converter.

The half-bridge, phase-controlled thyristor trains is in the following text also referred to as thyristor based trains. A VSI based traction system with the use of an asynchronous motor in the train, will be referred to as an asynchronous train.

Seen as a train set, there is no diﬀerence between an electric locomotive with unpowered wagons and a train consisting of several motorized wagons, an electrical multiple unit, in this thesis as the inﬂuence of, for example, couple forces is not of interest in this study.

The power factor, λ, and the phase factor, cos φ, of a train is usually deﬁned as:

P U · I · cos φ λ = = 1 = g · cos φ (3.14) S U · I

In this study only the fundamental current, I1 is studied, while the harmonic content of I is neglected, therefore λ and cos φ is treated as equal [26].

3.6.1 Asynchronous trains

The asynchronous trains for an AC railway power system are built upon the design where two AC/DC converters are connected to a common DC-link, see figure 3.12, where the power can flow in any direction. Depending on the control-scheme the converters are using, both voltage and frequency can vary freely to the traction motor, within the limits of the converter [26]. For a pulse width modulated scheme where the frequency of the switching is higher than the fundamental frequency, the voltage output could be created so that after filtering it almost reassemble a sinusoidal curve, with a small harmonic content [15].

The power factor of the converter at the catenary side can be controlled by changing the phase angle, φ, to be able to generate or consume reactive power from the catenary, or using a power factor of unity to minimize the current ﬂowing into the train. The power factor

23 PAC PDC Ptraction Q = Q AC ∼ traction AC Motor = 3 ∼ Figure 3.12. Principle layout of a voltage source converter with the active power from the catenary, PAC , the active power ﬂow on the DC-link, PDC , and the traction power, Ptraction. of the train type IORE is shown in ﬁgure 3.14(a) to visualize an example of power factor control.

The tractive power Ptraction could be limited by the converter’s current rating if the power factor is diﬀerent from one as

Ptraction ≤ PDC ≤ PAC = UAC IAC cos φAC . (3.15)

The losses in each part of the converter increase the power drawn from the catenary for a demanded tractive power, and therefore is Ptraction in reality less than PAC when motoring.

Regenerative braking of an asynchronous train gives a similar relation as in (3.15),

UAC IAC cos φAC = PAC ≤ PDC ≤ Ptraction (3.16) however, the losses in the conversion limits the amount of available traction power delivered back to the catenary.

In both (3.15) and (3.16) the amount of power is dependent of the eﬃciency of the converters. If neglecting converter losses in the train models, the equal sign applies to both equation.

If the catenary voltage is lower than its nominal value, a production of reactive power could be used to increase the voltage level. When the train use regenerative braking, the voltage at the catenary rises, and changing the phase angle to increase the reactive power demand, could be used to lower the catenary voltage.

Any deviation from a power factor of unity would here increase both the current through the pantograph and the losses in the converter. From a systems point of view, a change in the power factor could both increase or decrease the transmission losses.

In ﬁgure 3.15(a), the active power, the reactive power and the phase angle is visualized. The phase angle can vary as long as the active power demand is satisﬁed and the apparent power is not exceeding the rating, Sd,max of the train

2 2 2 Pd + Qd ≤ Sd,max . (3.17)

24 3.6.2 Thyristor based trains

Before the introduction of asynchronous traction converters, the development of semiconductors in the 1960s made it possible to also use DC-machines for AC railways, and many of the trains in use today still uses a DC-machine for it’s propulsion. A DC-machine has a rotational speed proportional to its armature voltage, but can use an even higher speed if a separate machine excitation circuit is used to vary the magnetization and flux. One way of varying the train speed is by a tap changing traction transformer connected to a diode rectifier, where different outputs changes the turn ratio. A tap changer can only give distinct voltage magnitude levels, and therefore the rectified voltage to the DC-machine could only be varied in a limited number of steps. Using a thyristor based rectifier to control the voltage introduced both a much smoother voltage level variation, compared to the previous tap changer control, and a larger demand for reactive power, as the power factor changed to the worse [15].

For a thyristor based train the reactive power is dependent on the active power demand, the speed, and voltage on the catenary. In ﬁgure 3.15(b), the active power, the reactive power and the phase angle is visualized.

A common thyristor based train in Sweden is the Rc-type, which uses two half controlled thyristor bridges in series, see ﬁgure 3.13. An example of the power factor for a thyristor based converter can be seen in ﬁgure 3.14(b). One of the reasons for connecting two bridges in series is to improve the power factor [15].

Converter 1

M =

Traction Converter 2 transformer Figure 3.13. Principle schematic of a series connected thyristor based converter for a DC- motor using two traction transformer windings. The upper rectiﬁer leg in each converter is using diods, whereas the lower uses thyristors [15].

25 Power factor IORE Power factor for thyristor based trains

1 1

0.8 0.8

φ 0.6 φ 0.6 cos cos 0.4 0.4

0.2 0.2 Motoring Braking Motoring 0 0 13 14 15 16 17 18 0 0.2 0.4 0.6 0.8 1 1.2 Voltage [kV] Speed [p.u.] (a) An example of power factor control for an asyn- (b) Power factor for a thyristor based train with chronous train, here the IORE locomotive. The two series connected thyristor bridges, based on power factor while braking increases the reactive ﬁgure in [27]. power demand when the catenary voltage rises, and is plotted as -cos φ to use a common axis.

Figure 3.14. Fundamental power factor as function of the catenary voltage for the IORE-train, and as function of per-unit of base speed, for a thyristor based train.

Q Q

Sd,max

Qd Qd Sd Sd

φ P φ(u, v) P

Pd Pd (a) Asynchronous train. (b) Thyristor based train.

Figure 3.15. Possible values of the reactive power, Qd, for a given tractive power demand, Pd. The dotted curve represent the rated apparent power, Sd,max, whereas the fat dashed line for the asynchronous train represent the possible values of the reactive power demand for a given active power and phase angle in the ﬁrst quadrant. Depending on the control system used, the beahavior could be diﬀerent for an actual train.

26 3.6.3 Regulation for motoring

A limitation for the trains tractive current demand is stipulated in the standard for the trains in a railway power system [28], and is a system parameter that needs to be implemented for all trains following the standard. For Sweden, the value of the maximum current in tractive mode, Imax, is 900 A for a train set [3]. This is used to be able to maintain stable operation for weak power systems or if the system is experiencing some type of degraded state. The maximum current a train is allowed to draw from the contact line is decreasing linearly as a function of the voltage at the current collector, as shown in ﬁgure 3.16. Below the system value of Umin2, the lowest non permanent voltage, only auxiliary power should be drawn from the catenary whereas no tractive power is allowed [28].

Depending on the railway power system type there is a coefficient, a, as for which voltage at the current collector the limitation of the maximum current is in effect. In table 3.1 the values of both a and the voltages, Unom and Umin2, for different power systems are shown. The equations for calculating the maximum allowed current and power drawn from the catenary at a given catenary voltage, Id,max and Pd,max, can be seen in equations (3.18) and (3.19).

Id,max

Imax

0 U a U U min2 · nom Figure 3.16. Tractive current limitation accourding to EN-50388 [28]. Numerical values can be found in table 3.1.

Table 3.1. System coeﬃcient and voltages for diﬀerent system types [17, 28]. AC DC Unom 15 kV 25 kV 3000 V 1500 V 750 V a 0.95 0.9 0.9 0.9 0.8 a · Unom 14.25 kV 22.5 kV 2700 V 1350 V 600 V Umin2 11 kV 17.5 kV 2000 V 1000 V 500 V

27   Imax,Utrain ≥ a · Unom   U − U I = I · train min2 ,U

Pd,max = Utrain · Id,max · cos(φ) (3.19)

Depending on the year of production of a train, it is not always the case that this limitation is included in the traction control as the standard is quite new. It can be assumed that any train produced after the standard was implemented is limited as described above. Older trains may lack the control to limit the current according to EN-50388, however, the train drivers are instructed to reduce the train’s power demand when the catenary voltage is low [7].

3.6.4 Regulation for regenerative braking

Another part of the standard [28] is when trains are restricted to use regenerative braking. For this project, only the situation when there is a too high voltage level on the catenary is of interest. Detection for loss of catenary voltage, short circuit between catenary and rail/earth or if the catenary fails to absorb the energy, is not part of this thesis.

When the voltage on the catenary reaches Umax2 during regenerative braking, trains should cease to use the regenerative brake [28]. Trains are however allowed to constantly regenerate power upto the voltage level limit of Umax2 [3]. The eﬀects of this is that the amount of regenerated power could be limited, and therefore less energy is fed back to the catenary than mechanical calculations would indicate, if the voltage limit is reached.

The voltage level Umax2 for diﬀerent railway power systems are shown in table 3.2. In Sweden the maximum voltage level has been lowered to 17.5 kV instead of the European standard of 18 kV as the vehicles has not been designed to withstand that voltage level [3].

Table 3.2. Maximum voltage during regenerative braking for diﬀerent system types [3, 17]. The voltage level for Sweden is marked with a (S). AC DC Unom 15 kV 25 kV 3000 V 1500 V 750 V Umax2 (S) 17.5 kV, 18 kV 29 kV 3900 V 1950 V 1000 V

28 4| Computer model implementation and calculations

4.1 Program layout

The implementation of the power flow calculations was divided into three parts, precalculation of train movements, initialization of simulation, and a time step loop, see figure 4.1. The layout of the railway power supply infrastructure and converter configurations are for most of the time static and only needs to be evaluated at the start of the program. Trains and the length of the catenary between the nodes are dynamic and changing over time and needs to be handled at every time step. Simulation data is saved into memory for every calculation, and depending on the intended receiver of the result, it can be saved to file or sent back to the calling module.

Figure 4.1. Structure of the program ﬂow. Yellow boxes represents reading of system data and precalculated train movements. Green boxes represents initialization of simulation, and blue boxes represents the calculation loop for each time step.

29 For this project precalculated train movements and power demands were used, which here is referred as a static time table. When a previous calculation of the allowed power demand is used to generate the next position for a train, it is here called a dynamic time table.

A static time table disregards the eﬀect of a limited tractive power from a previous time step, however, the results indicates if the time table used is feasible or if the power demand has been limited. Calculations with static time tables are commonly used in the industry, and it is only in recent time limitations of the railway power system has been taken into account [29].

4.2 A modular standard node

One of the goals of the programming part of this thesis was to implement a simple and modular design of the individual electrical power system nodes. Dividing the electrical parts of a railway power system into static and dynamic parts, where the moving trains and the changing catenary lengths are dynamic, makes it possible to use a simple interface to other parts of the program and separate the power system construction.

The proposed design uses a common structure for the static parts of the power system nodes, and depending on it’s function, four different types, as shown in figure 4.2, can be derived. The original idea of the node design were proposed during the initial thesis meetings with Mario Lagos [30], but refined as:

• Only one up- and one down-track catenary connection for each standard node. Multiple parallel tracks are electrically connected at each node and any number of trains are allowed to occupy the same point along the track.

• Addition of the possibility of a high-voltage transmission system.

• Return currents and ground currents are not studied in detail for AT-systems and BT-systems as a simpler length based impedance is used.

The number, and type of converters in the converter station model could be of any combination of rotary or static converters, as with the number, and size, of the transformers. The hiding of the internal structure of the nodes is intended to give an easier overview of the power system during modeling.

With this modular design it is possible to give each component a standardized interface to the rest of the program, but still, with a possibility to maintain individual parameter settings.

30 High-voltage feeder High-voltage feeder

3 3 ∼ ∼ 1 1 ∼ ∼

Catenary Catenary

(a) Node with both a converter station and a (b) Transformer only node. high-voltage transformer.

3 3 ∼ ∼ 1 1 ∼ ∼

Catenary Catenary

Figure 4.2. The four possible conﬁgurations of the modular standard node. Each modular standard node is for aesthetically reasons here shown as an outspread electrical grid, but is for all calculations seen as a single point, to which the converter, transformer and catenary are all connected to.

4.3 Converters

4.3.1 Converter losses

The rotary converter losses influence the total system losses quite significant as their efficiency is on average not better than 85 % [9]. In the time available for this thesis, no converter commitment, or on/off control, was implemented, such an implementation would effect the total calculated losses. Connecting an extra converter gives an additional no-load loss, and a reduced current dependent loss, as more converters share the load.

In the operation of the Swedish railway power supply system, the aim is to control the converter stations configuration, of online converters, in such a way that the converter losses are minimized for the planned traffic [1]. Therefore an approximation of the losses following the configuration that gives the minimum possible converter losses seems appropriate. Assuming a constant catenary voltage of 16.5 kV, and a converter power factor of 0.8 as

31 in [9], the different loss curves are plotted in figure 4.3.1, which are based on converter loss coefficients from [9]. The dashed line represent a least square fit of the minimum possible loss of any combination possible and should in theory be the optimal converter configuration for a specific power demand.

Converter losses Converter losses 3000 3000 1xQ48 1xQ48 2xQ48 2xQ48 2500 1xQ38 2500 1xQ38 1xQ38 1xQ48 1xQ38 1xQ48 2000 1xQ38 2xQ48 2000 1xQ38 2xQ48 2xQ38 2xQ38 1xQ48 1500 2xQ38 2xQ48 1500

Losses [kW] 1000 Losses [kW] 1000

500 500

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Output power [MW] Output power [MW] (a) Converter losses in Kiruna and Torne- (b) Converter losses in Stenbacken. hamn.

Converter losses

2000 1xQ48 2xQ48

1500

1000 Losses [kW]

500

0 0 5 10 15 20 Output power [MW] (c) Converter losses in Rombak.

Figure 4.3. Converter losses and approximation of the minimum possible loss conﬁguration, where the approximation is plotted using a dashed line. Some parts of the plots have been removed for visibility.

The least square ﬁt of the dashed lines are

2 Ploss,conv =135.74 + 48.71 · PG,conv + 1.36 · PG,conv, Kiruna and Tornehamn (4.1) 2 Ploss,conv =118.85 + 46.88 · PG,conv + 2.28 · PG,conv, Stenbacken (4.2) 2 Ploss,conv =205.68 + 37.32 · PG,conv + 1.82 · PG,conv, Rombak (4.3) where Ploss,conv is given in kW when PG,conv is given in MW.

The approximation is supposed to better follow a theoretical control scheme for minimizing the losses, than if adding each individual converters loss as no converter on/oﬀ control is implemented, even though the power factor most likely deviates from 0.8 under normal operation. This would especially eﬀect the calculated losses for converter stations during low

32 power demand. The used loss minimization calculation do not effect the angle stiffness of the converter stations, as all converters are treated as online during load flow calculations.

From the plotted loss curves it can be seen that some of the conﬁgurations always gives a higher loss than the others, this implies that this mode of operation should be avoided if the losses is to be minimized.

4.3.2 Parallel converters

The implemented converter equations are based on equations formulated in [10] and [8], but are here extended to handle converters of diﬀerent ratings as compared to the previous models. The converters in the converter stations along Malmbanan uses a setup of mixed ratings and this gives a change in both voltage and angle behavior as compared to a station with the same type of converters.

The change to equation (3.10) for multiple diﬀerent converters in a converter station used in this implementation is that every converter phase change is calculated individually as

m ! g 1 X · PG,n X · PG,n ψ = − arctan q,n − arctan q,n , (4.4) m2 m g2 g 3 U + Xq,n · Q50 · Sp,n U + Xq,n · QG,n

m where PG,n and QG,n represent the power ﬂow through an individual converter, Xq,n and g Xq,n represent the quadrature synchronous reactances of an individual converter, and Sp,n represent the individual converters apparent power rating, Sg,n, as a fraction of the total apparent power rating for the converter station, St,conv.

Sg,n Sp,n = (4.5) St,conv

The phase change over a converter in a converter station is the same as for all the other converters, as the converters are connected in parallel, but the individual power through the converters can be diﬀerent, and is dependent on the quadrature-axis synchronous reactance in the motor and the converter.

If the values of the reactive power to the grid, Q50, is not known during a simulation, it is assumed to be zero.

33 4.4 Trains

4.4.1 IORE locomotives

The trains traveling along Malmbanan, in the northern part of Sweden, with iron ore from the mines, called IORE, are of special interest in this thesis. Their heavy weight and a hilly landscape gives a load variation that potentially could inﬂuence the railway power system, in contrast to a more modest power demand and regenerative braking.

Each train set uses two connected IORE locomotives to haul the heavy wagons, where the driver controls them both as a single unit. The IORE locomotive have two boogies, with three powered wheel axles on each. The locomotive’s traction transformer have two windings connected to each of the two traction converter units in the locomotive, and each traction converter unit powers three asynchronous motors in a boogie. A single traction converter unit is constructed by four AC/DC converters, a dc-link, and three DC/AC converters, as seen in ﬁgure 4.4 where the schematic of one of the traction converter units and its main components is shown [31]. This design gives some backup if any of the traction converter units break down so that the train can continue to travel, however, with limited power [15].

AC-DC DC-AC

= ∼ = ∼ M 3 ∼ ∼ = = M 3 ∼ ∼ ∼ = M 3 = ∼ ∼ = ∼ Traction DC-link transformer with ﬁlters Figure 4.4. Principle schematic of a traction converter unit for an induction motor drive system with two traction transformer windings on the motor side, and three asynchronous motors in parallel.

Power factor control

The IORE locomotives use a power factor control to give a unity power factor when motoring, and a voltage level dependent power factor when braking. The focus in this part is the modeling of the power factor control, as this is used during normal operation of the IORE

34 locomotives in the following chapter. Both calculations of the maximally allowed regenerative braking power, which is going to vary depending on the power system state, and the eﬀects of using a non unity power factor when motoring or braking is therefore of interest.

While braking the locomotive, the power factor follows a curve, as seen in figure 3.14(a), where the desired value is unity at voltage levels, U, below 14.8 kV. Above 14.8 kV, the locomotive starts to consume reactive power in order to moderate the voltage level [19]. The increased reactive power flow along the catenary changes the catenary voltage so that it is lowered compared to what the effect should have been if braking with a unity power factor.

The power factor behavior is mathematically  −1 U ≤ 14.8 kV  cos(φ(U)) = −1 , (4.6) q U ≥ 14.8 kV  (U−14.8)2  1 + 2.25 where for the cos φ control during regenerative breaking has a discontinuous derivative. A numerically more appropriate function representation could for example help the solver to ﬁnd a solution if the voltage is near the discontinuity. An alternative could also be to use a solver capable of mixed integer problems, however, that would increase the complexity of the mathematical model, and probably increase the solution time. The implementation of the power factor control in this thesis uses the angle φ(U) as a variable, and therefore the IORE power factor angle φ(U) is approximated instead of cos φ(U).

The approximation of φ(U) used is constructed in two steps, an approximation of the two distinct parts of the curve, and the combination of them both. The ﬁrst part of the curve, for voltages below 14.8 kV, is replaced by a constant angle of 180◦, and the second part, for voltages between 14.8 kV and 18.5 kV, is approximated with a polynomial function φp(U) that follows the original φ(U). As the original values of φ(U) is always less than 180◦ during the polynomial part, a min(x, y) function as shown in equation (4.8), is here used as the ﬁrst step of a two-step approximation between the curves to give the wanted angle of φ(U).

A least squares approximation was performed on the non constant part of the function in equation (4.6), to which a third degree polynomial was chosen. The coeﬃcients were calculated to give the equation

3 2 φp(U) = −0.01537636 · U + 0.8558433 · U − 15.98191 · U + 102.0657 , (4.7) which gives the angle φp(U) in radians for a voltage U in kV.

The min(x, y) function also has a discontinuous derivative, but using the approximation of the maximum function used in [32], equation (4.9), in combination with the relationship that min(x, 0) = − max(−x, 0), a function with a smooth derivative can be formulated as

35 the second step in this approximation. This approximation only works when comparing a value with zero, therefore the polynomial needs to be shifted by π radians before and after the approximation.

 x , x ≤ y min(x, y) = (4.8) y , x > y √ x2 + 2 + x max(x, 0) = (4.9) 2

In equation (4.9), represent a small number, which aﬀects the balance between smoothness of the function, and size of the error. The error at the non-derivative point of the original function (4.6) is with this approximation /2.

The resulting approximation of φ(U),

p(φ (U) − π)2 + 2 − (φ (U) − π) φ(U) = π − p p , (4.10) 2 and the original function is compared in ﬁgure 4.5(b) to show the resulting diﬀerence near the point of the discontinuous derivative.

Combining both steps of the approximations and choosing

= 0.01 (4.11) gives an error of 0.13◦ at 14.8 kV, which seems acceptable as the range of values for φ(U) is between 112◦ and 180◦.

Voltage level limitations on active power

For voltage levels on the catenary outside of the normal range, the tractive power is often limited. The two kind of limits presented in the following two sections are either caused by a limit in the trains traction converter or a limit stipulated in the specifications for a railway power supply system. The limits in the specification is only in effect if the train is programmed to follow it.

Under voltage

In the event of a low voltage at the catenary, where the maximum allowed tractive power used by the locomotive is limited by the regulations of the railway power system, the

36 Power factor angle IORE Power factor angle IORE 180 181 180 170 179 160

] ] 178 ° 150 ° 177 (U) [ (U) [ φ 140 φ 176 130 175 120 Original 174 Original Approximation Approximation 173 10 12 14 16 18 14.7 14.75 14.8 14.85 14.9 14.95 15 Voltage [kV] Voltage [kV] (a) φ(U) for values of U where regenerative braking (b) Comparison of the approximation of the power is allowed by the IORE tractive converter. factor angle and the original function near the point of the discontinuous derivative for the IORE locomotive.

Figure 4.5. The used approximation of φ during regenerative breaking compared with the original function. equations (3.18) and (3.19), based on ﬁgure 3.16, should be used as an upper limit, even if the trains are capable of operating at a lower voltage.

In this study, the only train model capable of handling voltage level limitations is the IORE, therefore, the power limitation is based on the speciﬁcation for that train [19]. A combined plot of both the limit from the power system regulations and the power limit of the IORE can be seen in ﬁgure 4.6(a).

The limits used for modeling the IORE tractive power limitations are

1 P ≤ P · (U − 7) (4.12) D,train D,train,max 8 train 1 P ≤ P · (U − 9) (4.13) D,train D,train,max 4 train where PD,train,max is the maximum power that can be drawn by the IORE-locomotive and

Utrain is the catenary voltage in kV. As can be seen in ﬁgure 4.6(a), the limits of the tractive converter unit in the IORE is for low voltages higher than the allowed tractive power drawn from the railway power system.

Over voltage

Over voltages at the catenary can be caused by power being fed from the train back to the railway power system, regenerative braking. The regenerated power is limited by both the standard for railway power systems and the control system in the train. The standard allow for trains to feed back power if the catenary voltage is below 17.5 kV, and no regenerative braking is allowed if the voltage is higher.

37 The traction controller of the IORE starts to limit the amount of regenerative brake power above 16.5 kV, and no power is allowed above 17 kV [19]. This is here modeled as

PD,train ≤ PD,train,max · 2 (17 − Utrain) (4.14) where PD,train,max is the maximum regenerative brake power that the IORE can deliver.

The maximum allowed regenerative power for diﬀerent catenary voltages can be seen in ﬁgure 4.6(b).

Tractive power limit Regenerative brake power limit

1 Supply system 1 Supply 0.8 0.8 IORE train system

0.6 0.6

IORE train [p.u.]

P [p.u.] 0.4 0.4 brake P 0.2 0.2

0 0 9 10 11 12 13 14 15 15 16 17 18 19 Catenary voltage [kV] Catenary voltage [kV] (a) Tractive power limitation. The power limit for (b) Regenerative braking limitation. the supply system is dependent on the power factor of the train and is here plotted with cos φ = 1.

Figure 4.6. Power limitations for asynchronous trains used during calculations. Permitted area of operation is below the both curves.

4.4.2 Thyristor based locomotives

Two approximations have been used for the implemented model of thyristor based trains, as a more detailed model was not part of the thesis aim. This thesis uses a simplified model for their reactive power demand. The first approximation is that the power factor has been set to cos φ = 0.8, regardless of speed or any other factor, instead of the curve presented in figure 3.14(b). The second simplification is that thyristor based trains are modeled to not use a voltage dependent maximum power, were a more accurate model would limit the maximum power as the voltage decreases [9–11].

4.5 Mathematical model

General Algebraic Modeling System, or GAMS, is used as a high level model interpreter with several diﬀerent solvers available [5]. The use of GAMS is chosen for this project to save development time, instead of creating a new solver or implementing an available solver

38 without a high level interface. The modular properties of an object oriented programming language, like Java, the program language used for TRAINS, still makes it possible to quite easily change the used solver, as long as a common solver interface is used between the modules.

4.5.1 Solver

Several diﬀerent solvers are available in the GAMS environment to solve Nonlinear Pro- gramming problems, NLP [33]. In a previous Master’s Thesis with similar calculations using GAMS [11], the model classes Constrained Nonlinear System, CNS, and Nonlinear Programming with Discontinuous Derivatives, DNLP, was used. CNS is used to solve an equation system with equal number of variables as equations, and without any binding constraints, whereas DNLP can solve non linear problems, as power ﬂows, where the functions are non continuous and non smooth. Reformulation of a DNLP model into a NLP model is sometimes possible, but any non smooth function needs to be approximated.

For this thesis, one of the goals is to minimize the voltage drop induced reduction of the tractive power to the trains, within the speciﬁed limits. In case that the tractive power demand would be locked at a given value, the system of power ﬂow equations could be solved with a CNS solver, as all equations and loads are known, but if there is one ore more unknown variables that are solved for, a NLP solver is used instead.

A test was performed to evaluate the performance of diﬀerent NLP-sovers where a small test system gave the results in table 4.1. All of the solvers gave the same solution and only the calculation time diﬀered.

Table 4.1. Calculation time for small test case. Solver Time [p.u.] CONOPT 1.00 Ipopt 1.18 MINOS 1.04 PATHNLP 1.20 SNOPT 0.96

As not all of the above solvers were available for the size of the actual system simulated, the fastest available was CONOPT, which therefore was chosen as the solver for all calculations.

39 4.5.2 Equations and constraints

The NLP-model is formulated as the minimization of an objective function z, under the constraints of function vector g(x) with the limited variables of vector x, (4.15)-(4.17).

minimize z = f(x) (4.15)

subject to gL ≤ g(x) ≤ gU (4.16)

xL ≤ x ≤ xU (4.17)

All calculations made in GAMS in this thesis uses per-unit values, see appendix A.1, as the CONOPT solver is optimized to handle results with a magnitude of about one, therefore also all the equations in this chapter uses per-unit notation [34]. As the case-study in chapter 5 is limited to a part of the Swedish railway system where not all of the models from chapter 3 are represented, only relevant models are shown here.

All nodes in the system are subdivided into diﬀerent data sets in the model, and are categorized as either converters, trains or others, and uses the indices of either conv, train or other in the following equations. The nodes within the set others are those which are neither converters or trains. A subset of the data set trains is iore which can be used for equations only applicable to IORE-locomotives. The static, conv and other, and dynamic, train, parts of the model in the used implementation are separated by a distance of at least 1 meter when calculating the admittance matrix.

For PD and QD are positive values consumption and negative values generation, for PG and

QG the opposite applies.

Power ﬂow

For every node in the system, the power flow equations (4.18) and (4.19) must hold. The generated power minus the consumed power in a node must equal the power flowing to other nodes. Here Gkm and Bkm represents the real and imaginary part of the admittance-matrix element from node k to node m and θkm is the difference in voltage angle between node k and node m.

X PG,k − PD,k = Uk · (Um · (Gkm · cos θkm + Bkm · sin θkm)) (4.18) m X QG,k − QD,k = Uk · (Um · (Gkm · sin θkm − Bkm · cos θkm)) (4.19) m

Calculations of reactive power demand for the trains use the power factor cos φtrain at the

40 catenary side of the trains, either speciﬁed as a constant or given as a result from another calculation.

During a traction demand from the TRAINS software, the value of PD,traction is given and the active power PD,train, and reactive power QD,train, are calculated as

PD,train = btrain · PD,traction,PD,traction ≥ 0 (4.20)

QD,train = btrain · PD,traction · tan φtrain,PD,traction ≥ 0 , (4.21) where the result is the actual power allowed to be consumed or produced in the current time step, and depending on if the train type is IORE or other, btrain is deﬁned as

0 ≤ btrain ≤ 1, IORE (4.22)

btrain = 1, Other (4.23) where the variable btrain is introduced to be able to limit the amount of power for the trains in the optimization equation (4.59), as their maximum allowed power could be limited when the catenary voltage varies.

During a braking operation of a train capable of regenerative braking, as the IORE, equations (4.20) and (4.21) are changed into

PD,train = btrain · PD,traction · cos φtrain,PD,traction < 0 (4.24)

QD,train = btrain · PD,traction · sin φtrain,PD,traction < 0 . (4.25)

During the calculation of the active and reactive power limits for the IORE trains, equations (4.20), (4.21), (4.24) and (4.25), are limited by

2 2 2 PD,iore + QD,iore ≤ Sd,max,iore , (4.26) so that the rated apparent power of the train, Sd,max,iore, is not exceeded.

For thyristor based trains this limitation of Sd,max,train is not used as the calculation of the reactive power demand is using the simpliﬁed model in section 4.4.2, and could therefore exceed the rated apparent power even under normal conditions.

Converters

To show the used equation for the optimization and following the power ﬂow from the public grid to the trains, it starts with the phase shift of the grid transformer in a converter station

41 which is given by equation (4.27), cf. equation (3.11),

1 X · P θ0 = θ50 − arctan 50,conv G,conv , (4.27) conv conv m 2 3 Uconv + X50,conv · Q50,conv followed by the converter equation (4.28), cf. equation (4.4),

m ! 1 X · PG,n ψ = − arctan q,n conv m 2 m 3 Uconv + Xq,n · Q50,conv · Sp,n (4.28) g ! Xq,n · PG,n − arctan g 2 g , Uconv + Xq,n · QG,n where the introduced variables PG,n and QG,n represent the active and reactive power through an individual converter in a converter station, and the introduced variable Sp,n represent the fraction of the individual converter apparent power rating compared to the converter station total apparent power rating, with the equation (4.29), cf. equation (4.5)

Sg,n Sp,n = , (4.29) St,conv and the total phase shift over a converter station, θconv is, with the equation (4.30), cf. equation (3.13), 0 θconv = θconv + ψconv . (4.30)

The voltage at the catenary side of a converter station is given by equation (4.31), cf. equation (3.12), g g QG,conv Uconv = Uidle · 1 + Cf · . (4.31) SG,conv

50 m Constants in equation (4.27), (4.28), (4.30), (4.29) and (4.31) are θconv, X50,conv, Uconv, g m g g Uconv, Q50,conv, Xq,n, Xq,n, Sg,n, St,conv, Uidle, Cf and SG,conv which are numerical values as speciﬁed in appendix A, or calculated from those values.

Train power limits and power factor

Equations based on regulatory demands that are used to limit the maximum allowed tractive power for IORE-locomotives is given by equation (4.32), cf. equation (3.19),

PD,iore ≤ Uiore · Id,max · cos(φiore) (4.32)

42 where Id,max is given by equation (4.33), cf. equation (3.18),

  Imax,Uiore ≥ a · Unom   U − U I = I · iore min2 ,U

Limited tractive power based on limits of the traction converter in the IORE-locmotive are given by equation (4.34) and (4.35), cf. equation (4.12) and (4.13),

1 P ≤ P · (15 · U − 7) (4.34) D,iore D,iore,max 8 iore 1 P ≤ P · (15 · U − 9) (4.35) D,iore D,iore,max 4 iore and limited regenerative braking is given by equation (4.36), cf. equation (4.14),

PD,iore ≤ PD,iore,max · 2 (17 − 15 · Uiore) (4.36)

Constants in equation (4.32), (4.33), (4.34), (4.35) and (4.36) are Imax, Unom, Umin2 and

PD,iore,max which are numerical values as speciﬁed in section 3.6.3 and appendix A.

The voltage angle function controlling the power factor while, braking and used during normal operation of the IORE-locomotive is given by equation (4.37), cf. equation (4.10),

p(φ − π)2 + 2 − (φ − π) φ = π − p p , (4.37) iore 2 where is a user-deﬁned constant and φp is given by equation (4.38), cf. equation (4.7),

3 2 φp = −0.01537636 · Uiore + 0.8558433 · Uiore − 15.98191 · Uiore + 102.0657 . (4.38)

Variable bounds

A global limit for the voltage magnitude, U, and voltage angle, θ, for every node in the system is used to help GAMS to ﬁnd a solution, but also to avoid unreasonable solutions if a problem is not feasible. The upper limit of the voltage magnitude was chosen just above the maximum allowed short time voltage level, Umax2, of 18 kV [17].

0.5 ≤ Uk ≤ 1.25 (4.39) 89 89 − π ≤ θ ≤ π (4.40) 180 k 180

To give constraints for the power ﬂow through the converter station nodes and also individual

43 converters, but at the same time not hinder the calculation, a large value has been chosen as three times the total amount of installed apparent power in the converter station, St, and three times the individual converter apparent power rating, Sg, for each converter.

−3 · St,conv ≤ PG,conv ≤ 3 · St,conv (4.41)

−3 · St,conv ≤ QG,conv ≤ 3 · St,conv (4.42)

−3 · Sg,n ≤ PG,n ≤ 3 · Sg,n (4.43)

−3 · Sg,n ≤ QG,n ≤ 3 · Sg,n (4.44)

Other nodes than converter station nodes have been assigned a ﬁxed value of zero as the generated power. Likewise, the converter station nodes have been assigned a ﬁxed value of zero for their demand.

PD,conv = 0 (4.45)

QD,conv = 0 (4.46)

PG,train = 0 (4.47)

QG,train = 0 (4.48)

PG,other = 0 (4.49)

QG,other = 0 (4.50)

The value of φtrain in equation (4.21), (4.24) and (4.25) is dependent on both train type, with the associated control, and if either motoring or braking. During normal operation with the use of power factor control as speciﬁed in chapter 4.4, the limits are speciﬁed for the IORE as

φiore = 0,PD,iore ≥ 0 (4.51) 91 269 π ≤ φ ≤ π, P < 0 , (4.52) 180 iore 180 D,iore and for other types of trains as 36.87 φ = π (4.53) train 180

In the case where the motoring power factor for the IORE-trains is optimized, as in section 5.4.3, equation (4.51) is changed into

45 − π ≤ φ ≤ 0,P ≥ 0 , (4.54) 180 iore D,iore and in the case where the regenerative braking power factor for the IORE-trains is optimized,

44 as in section 5.4.3, equation (4.52) is changed into

135 π ≤ φ ≤ π, P < 0 . (4.55) iore 180 D,iore

Initial values

To give initial values for the different nodes, the last calculation results of the voltages and angles are kept in a file and read as input for the solver. If this file does not exist, voltages will be given a starting value of 1 p.u. and the angles 0.01 p.u. It is up to the Java program to delete this file if deemed necessary, as when starting a new simulation or when a, not so small, change has been done on the Y-matrix in the program loop of figure 4.1. The variable btrain is given an initial value of 1, as the power into or from the trains are assumed to be unlimited for most of the time.

0 Un = 1 (4.56) 0 θn = 0.01 (4.57) 0 btrain = 1 (4.58)

All remaining variables used are given an initial value of zero.

4.5.3 Optimization

A common optimization objective in ordinary power systems is to minimize the transmission losses, or the cost of power production [12]. The losses in transmission and power conversion is also of interest in loss minimization of a railway power system. However, it is more common that power can ﬂow in both directions compared to regular power systems where producer and consumer of power are often unidirectional and stationary. Minimizing the total injected power into a system would not limit the regenerated power, unless it would cause additional production in the system to increase the regeneration further.

The optimization in this project has been divided into two steps. First the maximum possible tractive power is calculated, within the limits of the trains and the system. Thereafter any other optimization is performed with the tractive power values fixed at the previously calculated results. In this way it is possible to avoid a multiple objective problem, and finding the optimal objective coefficients is not necessary. The use of two separate optimization problems could possible increases the computation time, however, this projects generic program structure would make it harder to find, or calculate, multiple objective coefficients that are optimal during the hole simulation period, and for any railway system. The coefficient problem and that it is for train operators more important to optimize for demanded traction power than loss minimization, this seems to be a better solution to use.

45 To use the two-step optimization, a variable, btrain, was introduced in (4.20), (4.21) and (4.22), as a per train tractive power coefficient that represent the possible portion of demanded power, to or from a train. The first optimization step is calculated with the optimization problem defined as

X min (1 − btrain) , (4.59) train under the general constraint of equations (4.20), (4.21), (4.23), (4.27), (4.28), (4.29), (4.30), (4.31), (4.39), (4.40), (4.41), (4.42), (4.43), (4.44), (4.45), (4.46), (4.47), (4.48), (4.49), (4.50), (4.53), and the power demand for IORE-locomotives are limited by the catenary voltage as given by equations (4.32), (4.33), (4.34), (4.35), (4.36), and IORE-locomotives are also constrained by equations (4.22), (4.24), (4.25), (4.26), (4.37), (4.38), (4.51), (4.52).

The sum of btrain in (4.59) is hereby maximized to give the most possible tractive power for all the trains, regardless of the direction of the power ﬂow. In preparation for the second 2 1 step, the constant btrain is given the values of variable btrain, which is the resulting power limit calculated in step one: 2 1 btrain = btrain (4.60)

The second step use another objective function where the goal is to minimize the system losses and potentially optimize any controllable parameters in the trains. In order to minimize the injected active power into the railway power system’s converter stations, PG,in,conv, the objective function used is X min PG,in,conv , (4.61) conv for which

PG,in,conv = Ploss,conv + PG,conv . (4.62) and is under the same constraints as in the ﬁrst step of the optimization with the addition of equations (4.60), (4.62), (4.1), (4.2) and (4.3).

The total injected power into a converter station is a combination of converter losses,

Ploss,conv, and the power sent out on the catenary system, PG,conv, as shown in (4.62).

The generator losses Ploss,conv are in this thesis calculated from the approximation in chapter 4.3. The second step in the two-step optimization will also implicitly increase the amount of regenerative power from the trains, if their power factor can be varied and there is a margin to the power limits of the train.

For optimization with a reactive power generation while motoring, in section 5.4.3, the two- step optimization is used with the same constraints as previous, except for equations (4.37), (4.38) and (4.51) which are exchanged with equation (4.54). The two-step optimization

46 constraints are also used for optimization of regenerative braking with relaxed power factor, in section 5.4.3, but instead is equation (4.52) exchanged with equation (4.55).

4.6 Java-GAMS interaction

The standard input and output for the GAMS environment is in text-files. They are easy to read and write for humans, but for computers there are other means of communication for sources and destinations of data. In this case, most of the evaluation of the calculated data is done in the Java environment, and the results are to be handed over in an efficient and easy way. In this project only text-files are used for the communication with GAMS to limit work effort and ease of use, however, the time of a simulation step could be decreased if direct use of the solver library was implemented as no disk access and file creation is needed [35, 36].

47 5| Case-study and simulation

5.1 The iron ore line

The railway line between Kiruna and Narvik has been electrified for almost a century, and the iron ore transports are still representing a majority of the trains. The part that runs through Norway, between Riksgränsen and Narvik, is officially named Ofotbanen. The figure below, 5.1, shows the route from Riksgränsen to Luleå, and the placement of the converter stations along the Swedish part of the track.

100 2Å Fö Riksgränsen (23) 100 2Å Fö (10) 100 2Å Fö Riksgränsen (23) 100 2Å Fö (10) Rombak omformarstation Björnfjell ML (14,8) + 80 2Å Fö (6) + 80 2Å Fö (6) Björnfjell - Riksgränsen 100 2Å Fö ML (1,4) 80 2Å Fö (34) Tor ne VF 10,5 träsk 2 x Q38/Q39 Abisko 80 2Å Fö 2 x Q48/Q49 VF 7,5 (34) 2 x Q38/Q39 AT 1 x Q48/Q49 (49) Tor Kiruna Råtsi, VF 9,3 n 2 x Q38/Q39 e t Krokvik 80 2Å Fö 2 x Q48/Q49 VF 10,5 (10) Tor 100 2AT Fö ne älv r (43) ä ML Råtsi - Råtsi sk Krokvik (8) Svappavaara 2 x Q38/Q39 Abisko K al ix ä lv

100 2AT Fö 2 x Q48/Q49 (97)

Koskullskulle

80 Å J (7) VF 7,5 ML Gällivare - mot Murjek (2) VF 8,1 4 x Q38/Q39 2 x Q38/Q39 Porjus AT Harsprånget AT (78) 1 x Q48/Q49 (49)

Jokkmokk 100 2Å Fö (4) VF 7,0 2 x Q38/Q39 100 2Å Fö (4) L u leäl

v AT en (74) AT (125) Kiruna Råtsi, VF 9,3 Morjärv ML Boden – mot Haparanda Murjek (9) Kalix ML Boden - DLC Brännberg (6) Boden 2 x Q38/Q39 100 2Å Fö (3) JL2:S1 Karlsborg Boden Klx-Ben AT (4) Piteälven Ben-Kbbg100 2Å Fö (5) Krokvik 100 2Å Fö (46) Hornavan Sävast VF 26,0 2 x Q48/Q49 V 80 2Å Fö in AT (30) 4 x 15 MVA, MEGAMACS, 1997 del älve JL2:S2 Notviken (1 omriktare matar norrut) n Nyfors Älvsbyn 80 Å (10) (10) 100 2Å Fö Nya To Storavan r (56) n - e ä malm- 100 Å (7) 100 2AT Fö lv Arvidsjaur + hamnen AT (50) (43) Långträsk Råtsi - S ML ke JL2:S3 Piteå llef Råtsi teä lven 100 2Å Fö (41) Krokvik (8) Svappavaara Jörn

Storuman 100 2Å Fö JL2:S4 (33) Skellefteå Skelleftehamn 100 2Å Fö (63) Rönnskärsverken VF 16,5 K JL3:S1 2 x 15 MVA, MEGAMACS, 1999 Lycksele a l 100 2Å Fö (47) Vilhelmina U m eäl ix ve n Yttersjön ä Hällnäs lv JL3:S2 100 2Å Fö Dorotea (63)

n e

lv Hörnsjö - Högbränna (6) ä Högbränna - Öre Älv (6) Hoting n a Öre Älv - Oxmyran (6)

m Öre Älv - Oxmyran via Nyåker (8) Vännäs r 100 2Å Fö e g Umeå n JL3:S3 Å Öre 100 2Å Fö AT 100 2AT Fö (23) 100 2Å Fö Älv Hörnsjö (37) Gimonäs (14) Högbränna 100 Å (10) Strömsund JL3:S4 Oxmyran (97) Ulriksfors 120 AT Holmsund Backe Norrfors (69) VF 21,0 100 2Å Fö (61) JL3:S8 Hörnefors 3 x 14 MVA, TGTO, 1990 Nordmaling JL3:S5 Saluböle JL3:S6 100 2Å Fö (41) 120 AT (42) 100 2Å Fö (43) (29) Husum Åre Häggenås Holmån Koskullskulle (59) JL3:S7 Ånn (58) Forsmo Örnsköldsvik Österås Storlien (52) VF 5,5 Mattmar 120 AT JL4:S1 Långsele (70) 2 x Q38/Q39 Storsjön (50) Kabel (3) + 80 2Å (5) +80 Å (56) Odensvik, VF 5,0 100 2Å Fö Västeraspby (54) JL3:S9 80 Å J 1 x Q24/Q25 Brunflo Bollstabruk Nyland 2 x Q38/Q39 (69) Bispgården 100 2Å Fö Kramfors (7) Pilgrimstad (36) Dockmyr AT (57) JL4:S2 Bräcke In Svenstavik da lsä (30) Gällivare - mot lve ML n Härnösand VF 27,0 AT Åsarna 1 x 14 MVA, TGTO, 1991 Timrå (65) DLC 2 x 15 MVA, Areva, 2009 Murjek (2) VF 8,1 Ånge (45) Viskan (44) Sundsvall

(54) JL5:S76 JL5:S6 JL4:S3 JL5:S7 (53) 4 x Q38/Q39

JL5:S5 Ytterhogdal Ramsjö

Lj JL5:S54 Gnarp Sveg us na (55) n JL4:S4 JL5:S4 (42)

Ljusdal Delsbo Näsviken Porjus Hudiksvall Ö JL5:S43 s ter D JL5:S3 alä JL4:S5 l ven (58) JL5:S32 (53?) AT Edsbyn Bollnäs Harsprånget (13) JL5:S2 Söderhamn 80 Å (78) Granbo (36) Kilafors (43) JL5:S1 (21) Holm- Furudal JL4:S6 Orsa sveden (23) Axmarby V

äs

t e FT 2,2 Mo grindar FT 25,0 r Dal 2 x Q24/Q25 3 x 13 MVA, YOQC, 1988 100 Å (58) JL9:S1 Ockelbo äl (47)

v Rättvik Mog – Sv 80 Å Mog - Gä e Siljan 100 Å n (43) Gävle (44) DLC - Gävle har överordnat JL9:S2 120 2Å DLC Malungsfors 100 Å (11?) ansvar för 132 kV matar- (61) Hagaström Furuvik Södra Grycksbo JL6:S1 Gävle ledningssystemet Sandviken (3) Turkiet Malung 100 2Å Fö Hofors 120 2Å Skutskär JL10:S1 (10) (41) Falun (6?) Älvkarleö 2 x 16 MVA 100 2Å Fö (40) Storvik K 120 2Å l Vansbro Ryggen a JL7:S1 r (28) (35?) ä l v Björbo JL7:S12 e Äppelbo Tierp n (59) BE 19,0 n JL9:S3 2 x 15 MVA, MEGAMACS, 1999 JL6:S2 e JL7:S2 Örbyhus - lälv Jokkmokk (46?) (62) a D JL7:S23 Ludvika 120 2Å Torsby Krylbo (55?) 80 Å Hallstavik Grängesberg (15?) (44) JL7:S3 (35) Cg - Kvg (33) 100 2Å Fö (20?) (29) Samnan (44) Fagersta Sala 120 2Å JL9:S4 (18?) (5?) VF 7,0 Charlottenberg JL6:S3 (20) Uppsala Lesjöfors (34) (4) Ställdalen Skinnskatte- Vittinge (34) (23) JL7:S4 (31) Sunne (26) berg Hällefors (61) Finnslätten, VF 19,0 MK VF 5,3 Bäckhaga JL6:S4 Ramnäs Odensala (51) 2 x 15 MVA, PWM-MEGAMACS, 1996 Häggvik - Solna 2 st M1, M2 (12) 2 x Q38/Q39 Filipstad AT 120 2Å 120 2Å Häggvik - Solnatågvärme via Hagalund (10) 2 x Q38/Q39 JL8:S1 JL7:S5 (139) Grythyttan 100 Å ML (56) (26) (32) Solna - Älvsjö 2 st M3, M4 (13) Arvika Valskog - Daglösen (37) Arlanda (60) Hornkullen Jädersbruk 120 2Å Enköping 120 2Å VF 34,0 Nykroppa VF 5,0 (11) (25) Bålsta 100 2Å Fö Mälaren (22) VF 8,5 Provningsstation (24) Kolbäck 5 x Q48/Q49 Nora (18) Kungsängen Storfors JL9:S5 Jäders- ML So – Hgv dsp 120 Å (12) 1 x Q24/Q25 (27) Valskog Rekarne Kallhäll AT (18) bruk Eskilstuna - So – Hgv dsp 100 Å (12) 3 x Q38/Q39 (38) Nässundet JL6:S5 Solna DLC (4) Skattkärr 120 2Å Rekarne (11) Årjäng (29) Ervalla Ökna (11?) (34) Stockholm (29) (21?) (5?) (46) Åker L 120 2Å Arboga Stockholm C Hovsta Alväng (10) Äs- Flb fsp120 2Å (10) Karlstad 120 2Å (2?) Läggesta 100 2Å 100 2Å Fö Bofors Gredby, VF 8,3 (25) (50) (58) Örebro (10?) ATBT (35) u 2 x 6,1 MVA, MEGAMACS, 1998 (10) (15) Älvsjö 200, VF 46,0 Kornsjö - Sarpsborg Strömtorp (32) 100 2Å (62) Svanskog DLC Ryssjöbrink Flemingsberg (44) 3 x 15 MVA, YOQC, 1988 l VF 6,1 Hallsberg Hjälmaren Södertälje 120 2Å Västerhaninge Halden (35) 1 x 14 MVA, MEGAMACS, 1995 e 2 x Q38/Q39 Flen Säffle Endast åt ena hållet Hallsberg Katrine- (7) (22) 2 x 15 MVA, Areva, 2009 Laxå-Kristinehamn 2 x 16 MVA (62) holm (17) (58) ML ä Bengtsfors Åmål Järna – VF 5,5 (25) Vänern Gullspång Laxå Södertälje (14) (12) 2 x Q48/Q49 80 Å ML l AT Ström- (7) 100 2Å Fö (35) Hallsberg 2, VF 17,5 Kornsjö Tälle – VF 12,0 v (59) 2 x 15 MVA, YOQC,1976 (55) Nynäshamn stad (8) Ed Hallsberg 3 x Q38/Q39 80 Å Gårdsjö 2 st (7) 2 x 15 MVA, Areva, 2009 (47) e Varp VF 21,7 (74) Mon ML Torved (41) (56) Mellerud - 3 x 14 MVA, TGTO Återmatning, 1992 n (57) ML ML 32kV Dals Rostok (4) Mariestad VF 7,0 (10)? Eksund - Åby (50) 80 Å Varp - Mon Dals Rostok Åby (12) (14) VF 5,6 1 x Q38/Q39 Degerön 100 J VF 6,3 (14) Oxelösund AT (23) 2 x Q38/Q39 80 Å 18 1 x Q48/Q49 (67) 2 x Q38/Q39 DLC Norrköping Skälebol Norrköping Smedberg (42) (125) VF 15,5 Motala VF 6,7 Karlsborg 2 x 15 MVA, YOQC,1978 (20)? (37) VF 10,6 2 x Q38/Q39 (32) 80 Å 1 x Q38/Q39 (27) Tibro 2 x 15 MVA, YOQC, 1981 (35) 100 2Å Fö Öxnered n Skänninge 80 Å (23) r (7) Linköping (24) Skövde e Lysekil Vänersborg ätt Morjärv Trollhättan Håkantorp (32) Bjärka Säby (68) V 120 2Å Fö 100 2Å Fö (15?) VF 16,0 (42) Velanda (36) Åtvidaberg (16?) 2 x Q38/Q39 Flunbo (33) VF 11,6 2 x Q48/Q49 120 2Å Fö 2 x Q38/Q39 (11?) Stenungsund Nygård (36) 1x Q38/Q39 HOG Tranås Boden – mot (9?) Älekärr ML Herrljunga 100 2Å Fö (46) Älvängen (59) 100 2Å Fö Ha VF 24,0 120 2Å Fö VF 15,5 (54) Kalix (13?) Gamleby 2 x 15 MVA, Cegelec, 2000 Bohus Murjek (9) 2 x 14 MVA, TGTO Återmatning, 1991 Bankeryd 1 x 15 MVA, Areva, 2009 (16?) (41) 1 x Q38/Q39 Eon 19,5/19,0 0 50 100 km (49) Huskvarna Västervik Göteborg Hindås Jönköping 2 x 15 MVA, YOQC, 1979 DLC Borås (37) 1 x 14 MVA, PWM-TGTO, 1991 r Visby Dalhem 100 2Å Fö å Boden - Almedal p ML (54) Vimmerby DLC (29) Ekeryd ls Göteborg Mölndal Malmbäck a VF 5,1 m (37) Lindome 80 Å S Eken Boden (75) 4 x Q24/Q25 Limmared Brännberg (6) 107 2Å Fö Pauliström Hultsfred Kungsbacka Fjärås Kinna (71) Vaggeryd Vetlanda n 120 2Å Fö Järnforsen 100 2Å Fö (3) 120 2Å a n Lekarekulle (39) k a Nissafors (43) Kullfors is V iss Karls b Veddige N Stockaryd 80 2Å Oskarshamn JL2:S1 Värö (14) Trönninge VF 8,0 Ohs 107 2Å Fö n Berga 1 x Q38/Q39 a Värnamo (90) bruk tr Mönsterås Klx-Ben AT (4) 1 xQ48/Q49 Ä 120 2Å Fö Åseda (48) bruk Boden Hamra Landeryd Bor Omformarstationer: 46 st (inklusive utbildningsstationen i Frövi) (6)? 120 2Å Transformatorstationer: 33 st (endast stationer utan omriktare och omformare) P Ben-Kbbg100 2Å Fö (5) (40?) iteä Torebo Hyltebruk Blomstermåla lv Kopplingscentraler: 33 st en Heberg Torup Växjö 100 2Å Fö (46) Ljungby Rockneby Sektioneringsstationer: 19 st 120 2Å (30) Åryd (43) Nerspår 120 2Å Fö (47) Nybro Uppspår (47) Eon 16,5/16,5 Furet 2 x Q48/Q49 107 2Å Fö Kalmar Sammanställning över omformare och omriktare i drift vid kartans utgivningsdatum: (44) H (58) ornavan Halmstad Eon 10,3/10,3 2 x Q38/Q39 A ntal Effekt/st Tot.effekt/typ 2 x 15 MVA, MEGAMACS, 1996 Eon 7,0/6,4 Sävast VF 26,0 Strömsnäs bruk Tiaryd 2 x Q38/Q39 Kontinuerligt överlast max [M VA] 120 2Å Älmhult 1 x Q24/Q25 (22) gan Båstad norra La (42) [MVA] (A) [MVA] (A) [MVA] (A) Markaryd M ö AT (30) (19) (51) r Båstad södra r Q24/Q25 11 st 3,2 (195) 4,8 (300) 6,8 (450) 35,2 u 4 x 15 MVA, MEGAMACS, 1997 (37) m Förslöv s Skånes å Bräkne 80 2AT Fö (53) + Eon 17,0/17,0 Vejbyslätt n 107 2AT Fö (46) Q38/Q39 44 st 5,8 (350) 8 (500) 9 (600) 255,2 (7) Fagerhult Hoby Ronneby 3 x 7,3 MVA, YRLA, 1986 Ängelholm Olofström Q38/Q39 HOG 1 st 5,8 (350) 9 5,8 ä 2xQ24/Q25 Notviken (15) (14) l Perstorp Hanaskog v JL2:S2 (1 omriktare matar norrut) Höganäs 107 2AT Fö Q48/Q49 20 st 10 (625) 14 (875) 18 (1175) 200,0 Kattarp (25) Karlskrona e (12) (108) (16) (28) Karlshamn Kristianstad Roterande omformare: 76 st 496,2 n Helsingborg 100 Å (34) Billesholm Nyfors (21) [M VA] Ramlösa Teckoma- Eon 22,0/22,0 (30) torp 80 Å 2 x 14 MVA, TGTO Återmatning, 1994 YOQC DIR 16 st 13-15 234,0 Älvsbyn Landskrona (16) (67) 1 x Q48/Q49 80 Å (10) 120 2Å 100 Å Eslöv YRLA PWM 3 st 8,2 24,6 (48) (9) Kävlinge Brösarp TGTO PWM 12 st 14,0 168,0 100 2Å Fö Nya Eon 25,0/25,0 100 Å (24) Lund M egamacs PWM 13 st 15,0 195,0 2x15 MVA, YOQC, 1983 (18) S Staffanstorp toravan (56) 1x15 MVA, (18 MVA) MEGAMACS, 1996 Högestad 100 2Å Fö M egamacs PWM 1 st 15,0 15 min: 18 M VA 15,0 1xQ48/Q49 (42) Simrishamn (6) malm- - Lernacken Tomelilla M egamacs-6 PWM 2 st 6,1 12,2 DLC (Eon stn) (10) (32) (30) Gärsnäs Malmö Fosieby M egamacs-6 PWM 2 st 7,5 15 min: 9 M VA 15,0 100 Å (7) (27) Skurup Cegelec PWM 2 st 15,0 5 min: 17 M VA 30,0 Eon 5,2/5,2 + hamnen Trelleborg 2x7,5 MVA, (9 MVA) MEGAMACS-6, 1996 Areva PWM 7 st 15,0 105,0 Arvidsjaur Statiska omriktare: 58 st 798,8 AT (50) TOTALT: 134 st 1295 Figure 5.1. Malmbanan.

48 5.2 Railway line model

For this study, only the part between Kiruna and Riksgränsen is traﬃcked during the simulations. To give relevant transmission losses in the simulations, only that part of the Swedish railway power system is included. As the last converter station in Sweden is located in Tornehamn, 22 km from the Norwegian border, this part of the track would be given a too large voltage drop if it was modelled to only be single-fed from Sweden. Therefore the converter station in Rombak, 19.53 km further away into Norway from Riksgränsen, is added and connected to Riksgränsen using a catenary type of 100 2Å FÖ, with the impedance as shown in table A.2. The converter station in Rombak is only given two converters in this study as the power demand is lower than usual when no trains are simulated on the Norwegian part of the track. Other parts of the electrical layout of the railway power system of Malmbanan can be seen in table A.6.

The simplified electrical layout in figure 5.2 is used under normal operation, with all converter stations available and with no interruption of the catenary. In figure 5.3, the gradients and the altitude above sea level that has been used by TRAINS during simulation is shown.

Rombak Tornehamn Stenbacken Kiruna 48 48 48 48 38 38 48 38 38 48 48 38 38

M M M M M M M M M M M M M

G G G G G G G G G G G G G 42 km 50 km 65 km

Riksgr¨ansen Figure 5.2. Simple schematic of the electrical layout used for simulations, 38 and 48 represent the converter types Q38/Q39 and Q48/Q49 respectively.

The gradient in ﬁgure 5.3 is collected from the Swedish Transport Administration and their system for track information, the accuracy of the data is to some extent uncertain but seems to follow the landscape proﬁle.

The train schedule used in the simulations, and shown in figure 5.4, is based on the actual train movements from the train schedule [37] during a Monday in January of 2013, with the difference that a more aggressive driving, where the maximum possible traction power is used, while still obeying the speed limit, is used. This gives the minimum running time possible, neglecting possible power system limitations. Thus, the trains in the schedule might use a higher level of traction power than in reality. Therefore, some of the train movements are faster than in the real schedule from that day as the timetables time of arrival is not honored in the static model computation. This deviation from the real schedule might make some of the trains meet or overtake during some of the time steps, this is however only regarded as a small source of error in the calculations as focus is on the railway power system and not on the traffic schedule.

49 Gradients KMB−RGN 20

−10 Gradient [per mille] −20 1420 1440 1460 1480 1500 1520 1540 Position [km] Elevation curve KMB−RGN 550 Kiruna Riksgränsen 500 Tornehamn 450 Stenbacken Altitude [m] 400

350 1420 1440 1460 1480 1500 1520 1540 Position [km]

Figure 5.3. Gradient and elevation curve from station ’Kiruna Malmbangård’, KMB, to Riksgränsen, ’RGN’ used for static timetable simulations.

A more gentle maneuvering of the trains would probably give a somewhat lower power demand on the railway power system than for that of a minimum running time. Instead of problems with a power limited locomotive, an increased travel time in the schedule would allow the drivers to keep a lower power demand in order to arrive at the scheduled time. This could be used to avoid costly upgrades of the power system while still keeping the timetable.

5.3 Train models

The choice of train types used for the simulations have been based on the real traffic on Malmbanan, but somewhat simplified as to the selection of locomotives modeled in the simulation software. The power demand for the IORE trains are calculated under the assumption that they are being fully loaded with iron ore when traveling from Kiruna towards Narvik, and empty going the other way. Passenger trains and non-IORE freight trains are assumed half full in both directions, and both are using thyristor based locomotives of type Rc4 and Rc6, respectively. The difference of the Rc4 and Rc6 locomotives is here only the gearing of the motor which effects the top speed and traction force when calculating the train movements, see table A.6.

50 Train schedule RGN 1540 41905 9901 9903 9905 9907 95

9921 1520 THA 41904 1500

1480 9919 SBK 1460 Track position [km] 1440

9926 9904 9906 45902 9908 9910 9912 1420 KMB 0 2 4 6 8 10 12 Time since midnight [h] (a) Train schedule for the time interval 00:00 to 12:00.

Train schedule RGN 1540 9909 45903 9911 9913 9915 9917 9919 9921 93 1520 THA 95 1500

9912 1480 SBK 1460 Track position [km]

1440 9907 9916 9926

94 9914 96 9918 9920 9922 41904 1420 KMB 12 14 16 18 20 22 Time since midnight [h] (b) Train schedule for the time interval 12:00 to 24:00.

Figure 5.4. Train schedule used for simulations. Train specifications, and type, for different train numbers used can be seen in appendix A.5. Approximate positions of the converter stations Tornehamn, THA, and Stenbacken, SBK, togheter with the limits of the simulated traffic from Kiruna Malmbangård, KMB, to Riksgränsen, RGN, is marked by the dashed lines.

To increase the loading of the power system, the freight trains modeled in TRAINS are hauled by a double Rc4-locomotive conﬁguration.

The power factor modeling of the Rc-locomotives were not part of this thesis and they are therefore modeled to be using a constant power factor of cos φ = 0.8 inductive, which is relatively close to that of ﬁgure 3.14(b), as the speed of the train is for most of the time at the upper part of the curve. None of the RC-locomotives modeled here are capable of regenerative braking.

There exist several diﬀerent models to describe the running resistance of a train, and it can include the physical impact of friction, airﬂow and gravity. Commonly used parts of the total running resistance are rolling resistance, curve resistance, aerodynamic resistance and gradient resistance. The model of the total running resistance, FR, used here is formulated

51 as: 2 FR = A + B · v + C · v (5.1) where v represent the trains speed in m⁄s, and the coeﬃcients for the diﬀerent train types can be found in A.7.

Train numbering in Sweden commonly uses a scheme were trains traveling from south to north use an even number, whereas trains traveling from north to south use an odd number. Train numbers used here and their corresponding train type can be found in table A.5.

5.4 Simulations

Simulations are performed with the modular power system model to study the railway power system, during both a normal state of the grid and during loss of a converter station. An optimized power factor control for either motoring or braking is studied, where the IORE trains are controlled to minimize the amount of injected power into the converter stations, whereas other train types are operated as normal. Several diﬀerent combinations of the state of the grid and reactive control of the trains are studied, and for all of them are the tractive power and regenerative braking power demand ﬁrst maximized, equation (4.59), and thereafter is the total power input to the converter stations minimized, equation(4.61).

The state of the grid is operated in the following two states:

• Normal operation - Grid is operating as normal with all converter stations connected and the equations for the power ﬂow are solved.

• Power outage - Loss of a converter station in order to weaken the grid, and to show the eﬀects on traﬃc caused by a power outage.

The power factor control for IORE-locomotives are changed as in the following cases:

• Normal power factor control - No change of the constant power factor while motoring or the voltage dependent power factor while braking as modeled in equation (4.10).

• Optimized power factor control during motoring - Allow a reactive power generation from trains to raise the catenary voltage and supply other trains with their reactive power demand.

• Optimized power factor control during regenerative braking - Change of the power factor to increase regenerated active power and decrease the reactive power.

To rate the performance of the system with some sort of measurable quantity, the change of energy delivered, either to or from the trains, the transmission and generator losses, and the total power input to the converter are used as a comparison. To limit the amount of resulting data from each simulation, only some of the results of all the daily trains are shown

52 in the following simulations. The number of time steps is also changed from TRAINS one second interval into an averaged 30 second time step. The time step selected can give a somewhat lower value of the calculated losses, as the power peaks are averaged out, and a less severe voltage situation.

Two types of data plots are mainly used to present results, a plot of active power versus the catenary voltage, and a plot of the catenary voltage versus the position on the track, as can bee seen in ﬁgure 5.5. The power plot shows the trains voltage-dependent power limits, if a power demand has been limited during a low catenary voltage, and the equivalent for regenerated power and a high catenary voltage.

5.4.1 Normal operation

During normal operation, the voltage level should not be a hinder for a properly designed power system and train schedule, where the trains power demand should not be limited by the conditions of the electric grid. To be able to analyze the effects of different changes in the power system or trains power control, the power flow during normal operation is calculated to give information about the trains maximal possible power demands, their maximal possible regenerated power, and the voltage levels in the grid. The trains selected from the train schedule in figure 5.4 for this base performance are train number 9909, 9914, 9915 and 9920. Here 9914 and 9920 are fully loaded iron ore trains without any stops along the track, 9909 and 9915 are empty iron ore trains with several stops. 9915 only runs together with other IORE trains on the tracks, while 9908, 9909 and 9914 are running in a mixed locomotive environment.

The energy flow and losses for a 24 hour simulation under normal operation is summarized in table 5.1. From the table it can be seen that the amount of energy needed, WD, for the trains to follow the intended traffic schedule is calculated to be 169124 kWh. Any simulation result with less than this amount of energy delivered into the train has reached a limit, and consequently, the trains tractive power have been restricted. The low efficiency of the converter stations is dependent on their no-load losses, and that at least one converter in each converter station is connected to the catenary, regardless of the traffic density and power demand.

The four trains, 9909, 9914, 9915 and 9920, were chosen to represent different traffic situations within the train schedule for the different simulation scenarios, and the accumulated energy for the journey during 24 hour can be seen in table 5.2 for which none were limited in their tractive power demand or regenerative braking. These numbers depends on the number of stops at stations along the track and the driving style of the locomotive driver. As the train schedule used here tries to achieve the fastest possible train movements, the minimum run time, the possibility for an even lower energy demand is likely.

53 Table 5.1. Energy ﬂow and losses in kWh during normal operation on the 162⁄3 Hz side of the railway power system. Consumed energy is denominated as WD, produced energy as WG, and the values below WD−G represent the diﬀerence between energy consumed and produced by the trains and the public grid.

Energy ﬂow WD WG WD−G Trains 169124 51709 117415 Public grid 30632 185122 -154489

Losses Wloss Generators 27537 Transmission 9537 Total 37074

Table 5.2. Energy ﬂow in kWh during normal operation for all the IORE train movements during 24 hour and unlimited tractive power demands. In the column for WG,max is the calculated regenerative braking energi from TRAINS without consideration of the used power factor or other limits.

Train number To From ∆W WG,max 9901 3265 240 3025 352 9903 3564 201 3363 297 9904 10138 3074 7064 4469 9905 3619 222 3397 325 9906 9787 3125 6662 4498 9907 3401 221 3181 324 9908 9787 3095 6692 4498 9909 3516 228 3288 337 9910 9787 3097 6690 4498 9911 3596 222 3374 324 9912 10243 3098 7144 4487 9913 3521 219 3302 324 9914 9787 3145 6642 4498 9915 3306 230 3076 338 9916 9787 3186 6601 338 9917 3642 232 3410 4498 9918 9787 3096 6691 4498 9919 3621 210 3411 307 9920 9787 3092 6695 4498 9921 3578 215 3364 305 9922 9787 3094 6693 4498 9926 9238 2461 6776 4278 Total IORE 146542 36003 110540 52484

54 The voltage to power plot for train 9914 and 9920 in ﬁgure 5.5 shows a distinct slope of the regenerative braking power and there is some distance between the shown power system limit and resulting values. This is caused by the IORE-locomotives power factor control, which is modeled in equation (4.10), that decrease the power factor for voltages higher than 14.8 kV and therefore limits the regeneration of active power. The distinct edge of the slope is for values where the rated regenerative braking power of the trains has been produced at diﬀerent voltages and a varying power factor.

The total energy demand during the 24 hour simulation for the IORE trains is calculated to be 146542 kWh, and the regenerated energy for the same time is 36003 kWh as shown in table 5.2.

5.4.2 Converter station outages

To investigate the impact of an outage of a whole converter station, for example if the transformer connected to the public grid breaks down, either the stations in Tornehamn or the one in Stenbacken is disconnected. Only these stations can be removed in the simulation to give a relevant result as the eﬀects of the other stations are already approximated with the lack of converter stations and traﬃc on the other side of Riksgränsen or Kiruna.

Both Tornehamn and Stenbacken converter stations are placed at positions where there is a noticeable altitude change in the landscape, and the power demand and regeneration could be assumed high and a loss of a converter station should therefore likely impact on the performance of the system.

Outage in Tornehamn

Tornehamn is located in the middle of the ascent towards Riksgränsen from one of the lowest points of the simulated section.

The energy table 5.3 for the simulation shows a signiﬁcant decrease in the amount of energy delivered to the trains. The transmission losses has also increased, even with less amount of energy actually reaching the trains.

The simulation shows that the amount of energy delivered to the IORE trains is reduced by 1.5 % on average, with a 25 % peak value, while the transmission losses increased to 180 % from their normal values. The reduction of delivered energy is here calculated as the sum of the reduced energy in each time step. The converter losses have by numbers fallen, but are actually higher if seen as losses per online converter, as the no-load losses of Tornehamn should have been almost 3300 kWh for the duration of the simulation using the loss approximation equation (4.1) from chapter 4.3.

55 9909 − Power profile 9909 − Catenary voltage 17 17

16 16

15 15 Voltage [kV] Voltage [kV] 14 14

13 13 −10 −5 0 5 10 1420 1440 1460 1480 1500 1520 1540 Active power [MW] Position [km] (a) (b)

9914 − Power profile 9914 − Catenary voltage 17 17

16 16

15 15 Voltage [kV] Voltage [kV] 14 14

13 13 −10 −5 0 5 10 1420 1440 1460 1480 1500 1520 1540 Active power [MW] Position [km] (c) (d)

9915 − Power profile 9915 − Catenary voltage 17 17

16 16

15 15 Voltage [kV] Voltage [kV] 14 14

13 13 −10 −5 0 5 10 1420 1440 1460 1480 1500 1520 1540 Active power [MW] Position [km] (e) (f)

9920 − Power profile 9920 − Catenary voltage 17 17

16 16

15 15 Voltage [kV] Voltage [kV] 14 14

13 13 −10 −5 0 5 10 1420 1440 1460 1480 1500 1520 1540 Active power [MW] Position [km] (g) (h)

Figure 5.5. Power proﬁle and catenary voltage for train 9909, 9914, 9915 and 9920 during normal operation of the supply system. The black, sloping lines in the power proﬁle plots shows the voltage limits of the traction converter. The maximum value of the power demand for the IORE of about 13 MW is a combination of the train types maximum tracive power, as seen in table A.6, and its converter losses.

56 Table 5.3. Energy flow and losses in kWh during a power outage at Tornehamn. In the table, ∆WD represent the difference in consumed energy compared to normal operation, and ∆WG is the difference in produced energy compared to normal operation.

Energy ﬂow WD WG WD−G ∆WD ∆WG Trains 166682 52036 114646 -2442 327 Public grid 29276 185716 -156441 -1357 595

rel Losses Wloss ∆Wloss ∆Wloss Generators 24602 -2935 89.3 % Transmission 17193 7655 180.3 % Total 41795 4721 112.7 %

Inspecting the power profiles of the selected trains, only the fully loaded IORE trains traveling up towards Riksgränsen have been limited in their power demand and affected by the lower catenary voltage, as seen in the figures 5.6 and 5.7. Thus implies that the IORE trains are only affected by the limited energy demand in the uphill slopes on the way towards Riksgränsen, and their reduced energy consumption is therefore instead 2.2 % averaged out over those trains. From the figures can be seen that train number 9920 is less limited than train 9914. This could be explained by two nearby trains, 45903 and 9911, traveling south at the same time and increasing the total power demand for the track section.

A converter station outage at Tornehamn would probably disturb the traffic as the power demand can not be satisfied with an aggressive driving style. If enough meeting time is scheduled at train stations nearby Riksgränsen, the delay of the trains might just affect the fully loaded IORE trains and not interfere with the schedule of other trains.

Outage in Stenbacken

The converter station Stenbacken, as with Tornehamn, is placed nearby a steep ascent. The diﬀerence here is that the landscape descent in the direction of the fully loaded IORE trains and that would most likely decrease the power demand and instead increase the amount of energy regenerated. In table 5.4 the energy statistics for the simulation can be seen.

During this simulation, none of the four selected trains that are studied in more detail are affected in their power demand. The total decrease in delivered energy is very small, about 0.1 % with peaks of 22 %, compared to the total demand and should probably not affect the traffic in any greater extent.

Regenerated power

One interesting result for both cases of a power outage is that the total amount of regenerated energy has actually increased compared to normal operation. The lower catenary voltage

57 9914 − Power profile − ∆W: −237.3 kWh 9914 − Catenary voltage 17 17

16 16

15 15

Voltage [kV] 14 Voltage [kV] 14

13 13 −10 −5 0 5 10 1420 1440 1460 1480 1500 1520 1540 Active power [MW] Position [km] Positions of limited traction power 550 Kiruna Riksgränsen 500 Tornehamn 450 Stenbacken Altitude [m] 400

350 1420 1440 1460 1480 1500 1520 1540 Position [km]

Figure 5.6. Limited IORE train 9914 during power outage at Tornehamn. Blue dots in the power profile shows the calculated power demands that could be satisfied, red dots indicates unsatisfied power demands which have been lowered to the black limit line. Red dots in the position plot indicate locations where the power demand has been limited.

9920 − Power profile − ∆W: −85.2 kWh 9920 − Catenary voltage 17 17

16 16

15 15

Voltage [kV] 14 Voltage [kV] 14

13 13 −10 −5 0 5 10 1420 1440 1460 1480 1500 1520 1540 Active power [MW] Position [km] Positions of limited traction power 550 Kiruna Riksgränsen 500 Tornehamn 450 Stenbacken Altitude [m] 400

350 1420 1440 1460 1480 1500 1520 1540 Position [km]

Figure 5.7. Limited IORE train 9920 during power outage at Tornehamn. Blue dots in the power profile shows the calculated power demands that could be satisfied, red dots indicates unsatisfied power demands which have been lowered to the black limit line. Red dots in the position plot indicate locations where the power demand has been limited.

58 Table 5.4. Energy ﬂow and losses in kWh during a power outage at Stenbacken.

Energy ﬂow WD WG WD−G ∆WD ∆WG Trains 168918 52376 116542 -206 667 Public grid 29132 184686 -155554 -1500 -435

rel Losses Wloss ∆Wloss ∆Wloss Generators 24304 -3233 88.3 % Transmission 14708 5170 154.2 % Total 39012 1938 105.2 % has most likely inﬂuenced the power factor for the IORE trains into producing more active power, and therefore helped in supplying the demanded energy.

5.4.3 Eﬀects of alternative power factor control

Reactive power generation for normal state of grid

In the standard for electrical railway interoperability [28], the use of a reactive power generation while motoring is allowed in order to control the voltage level of the catenary. The amount of reactive power that trains are allowed to produce is limited by the catenary voltage, which is not allowed to increase above Umax,1, or 17.25 kV on a 15 kV system. Injecting reactive power at the trains increases both the catenary voltage and could raise the power transfer capability of the power system, compared to normal operation [15].

Optimizing the power factor with the knowledge of all trains power demand and regeneration, which in [2] is called a centralized control, is used in an attempt to decrease the probability for a limited tractive power and to reduce the injected power into the railway power system. This type of control could possible minimize the amount of reactive power from the converter stations, as trains capable of varying its power factor helps to supply other trains with their reactive power demand, especially for those with a poor power factor.

A simulation was performed with the IORE trains as moving reactive power generators while motoring. The two step optimization minimization of section 4.5.3, to maximize the tractive power and minimize the injected power into the railway power system, was still used. The IORE trains use their normal power factor control, equation (4.37) and (4.38), while braking. Limiting the power factor to cos(45◦) leading, as used in [2], changed the ﬁxed value of φiore in equation (4.51) into a free variable with limits given by equation (4.54).

From the results of using trains with reactive power generation while motoring, it can be seen that the total losses has decreased with about 2 %, even with a small increase in transmission losses. The energy saving, for the used train schedule, could be proﬁtable for the railway grid owner if investments to implement the control is less than the cost for

59 Table 5.5. Energy ﬂow and losses in kWh during a simulation with centralized control of reactive power generation while motoring.

Energy ﬂow WD WG WD−G ∆WD ∆WG Trains 169124 51709 117415 0 0 Public grid 30625 184465 -153840 -8 -657

rel Losses Wloss ∆Wloss ∆Wloss Generators 26710 -827 97.0 % Transmission 9715 178 101.9 % Total 36425 -649 98.2 % the saved energy through the converter stations. From a train operators view, this kind of control gives no direct beneﬁts with today’s tariﬀ, and the losses from the increased current generates more heat in the traction converter that needs to be managed.

The high ratio of IORE trains in this simulation has probably inﬂuenced the results as their reactive power demand is non-existent as compared to the older thyristor controlled trains, and a more varying mix of trains could give an even more positive result if reactive power were also supplied from the trains.

Reactive power generation during converter station outage

Reactive generation while motoring the trains has the property of increasing the catenary voltage, therefore a simulation of its eﬀect on a power system with a high voltage drop, as during a power outage of a converter station, is studied. The same stations as in section 5.4.2 are here disconnected and all IORE trains are allowed to use a power factor for reactive power generation of up to cos(45◦), with the limit of equation (4.54) used.

Table 5.6. Energy ﬂow and losses in kWh with reactive power generation while motoring during a power outage at Tornehamn.

Energy ﬂow WD WG WD−G ∆WD ∆WG Trains 168982 52034 116948 -142 326 Public grid 29232 188384 -159153 -1401 3263

rel Losses Wloss ∆Wloss ∆Wloss Generators 23897 -3640 86.8 % Transmission 18308 8770 192.0 % Total 42205 5131 113.8 %

In table 5.6 and 5.7 it can be seen that almost all of the demanded energy has been delivered to the trains compared with the normal power factor of unity, as shown in table 5.3 and 5.4. The losses are still high, but if the trains performance is prioritized over energy consumption,

60 Table 5.7. Energy ﬂow and losses in kWh with reactive power generation while motoring during a power outage at Stenbacken.

Energy ﬂow WD WG WD−G ∆WD ∆WG Trains 169114 52376 116739 -9 667 Public grid 29189 184114 -154925 -1443 -1008

rel Losses Wloss ∆Wloss ∆Wloss Generators 23607 -3930 85.7 % Transmission 14579 5042 152.9 % Total 38186 1112 103.0 % this type of operation could help to ensure that the timetable is followed, even during a power outage in a converter station.

Regenerative braking with relaxed power factor and normal state of grid

As seen in ﬁgure 5.5(c) and 5.5(g), the power factor control for regenerative braking of the IORE trains keeps a margin up to the voltage limit of the train, for a given power. If the power factor during regenerative braking were changed so that the active power sent back to the grid were not limited by the original power factor control, from equation (4.37) and (4.38), while still obeying the traction converter voltage dependent limit of equation (4.36), the amount of regenerated power could possible be increased. An adjustable power factor initially maximizing train power, secondly minimizing power input to the system would also decrease the amount of reactive power produced by the train, compared to the normal power factor control.

All IORE trains are in this simulation allowed to use a power factor during regenerative braking between cos(135◦) and cos(180◦), which is the same angular freedom as used in section 5.4.3, with the limit of equation (4.55) used while optimizing.

Table 5.8. Energy ﬂow and losses in kWh when using a relaxed power factor control during regenerative braking of IORE trains for minimizing power input to the railway power system.

Energy ﬂow WD WG WD−G ∆WD ∆WG Trains 169124 52786 116338 0 1077 Public grid 32019 183731 -151712 1387 -1390

rel Losses Wloss ∆Wloss ∆Wloss Generators 26346 -1190 95.7 % Transmission 9027 -510 94.6 % Total 35374 -1701 95.4 %

The regenerated power from the IORE trains is now about 2 % higher for a 24 h simulation, compared to the case of normal power factor control. A decrease in both converter losses

61 and transmission losses can also be noted while using the changed power factor control. The decreased losses in the converter stations indicate that the current through them also has decreased, and that trains with a reactive power demand, like the older Rc-trains, are partly being supplied by the reduced consumption in IORE-locomotives.

9920 − Power profile 9920 − Power profile 17 17

16 16

15 15 Voltage [kV] Voltage [kV] 14 14

13 13 −10 −5 0 5 10 −5 0 5 10 Active power [MW] Active power [MW] (a) Power proﬁle during normal power factor control. (b) Power proﬁle during a maximized regenerated active power control of the power factor.

Figure 5.8. Power proﬁle comparison for IORE train 9920.

In ﬁgure 5.8(b) it can be seen that the regenerated power has moved upwards in the plot showing that the power is being regenerated at a higher voltage level.

62 6| Conclusions and future work

6.1 Conclusions

A program structure and a new modular standard node was developed in this thesis that can be used for simulations of a generic railway power system. Several simulations were performed to visualize the usability of the simulation software and used models. A two-step optimization was also introduced, where ﬁrst the tractive power and regenerative braking power is maximized and secondly the injected power into the power system is minimized, to also eliminate the need for a multiple variable optimization routine.

The simulations of the northern part of Malmbanan indicates that under normal conditions, with the traffic in the presented train schedule, the trains performance is not limited by the power system. Even with a more aggressive style of driving used than scheduled, the limits of the system were not reached, which could imply that the power system is well dimensioned for the traffic, or the trains are scheduled with a large margin to limit their power demand. Worth noting is that the calculated regenerative braking power received from TRAINS is limited by the used power factor control in the IORE trains, and the pure traffic simulator in TRAINS overestimate the amount of active power that can be sent to the power system.

It can also be seen from the results that the railway power systems sensitivity to a power outage is a little bit higher if it happens in Tornehamn than in Stenbacken. This is most likely caused by the larger power demand at the steep ascent towards Riksgränsen from Tornehamn, and that the IORE trains are traveling empty towards Stenbacken. The power system connected to Rombak on the Norwegian side of Malmbanan is in reality also a bit more powerful than with no connection at all which was used in this study, and could possibly have increased the sensitivity to a power outage.

The low eﬃciency of the converter stations that was seen in the results for normal operation is dependent on their no-load losses, and that at least one converter in each converter station is connected to the catenary, regardless of the traﬃc density and power demand.

63 An alternative to invest in new or upgraded converter stations, or one way to limit the sensitivity of a converter station failure, was also modeled and studied, where the trains were given the ability to use an optimized control of reactive power generation while motoring. A moderately improved system performance can be noted as the converter station losses decreased more than the transmission losses increased. The most noticeable improvement were in the case of a power outage, where almost all demanded energy could be delivered. An optimized power factor control is shown to keep voltage levels higher, and could possibly be used as an alternative to improve the infrastructure for an increased traﬃc, or to keep the same infrastructure for an increased traﬃc.

A train conﬁguration with more reactive power consumers, like older Rc-trains, could possible show a larger performance beneﬁt of power factor control as most of the trains in this simulation were IORE based, and used a unity power factor when motoring, and therefore only a limited amount of reactive power was needed.

6.2 Future work

Implementing a feedback to TRAINS, where the dynamic electrical behavior is used in the mechanical calculations of the trains movement would increase the usability of the software as a support for decisions during train operation.

The averaging of the data from TRAINS in 30 seconds interval to keep the amount of data to a manageable level could unfortunately hide potential voltage and tractive power problems. The use of a diﬀerent ﬁle format or other ways to handle all produced data could give the potential to see more time steps were the traction power have been limited.

To give a more versatile software, an implementation of DC calculations and models would make it possible to increase the types of traction systems to simulate.

One of the limitations of the program is the lack of a converter commitment control, that could give a more accurate calculation of the power ﬂow, as the electrical angles are dependent on the number of converters that are operating in a converter station. The calculation of converter losses would also beneﬁt from the use of commitment control.

Other types of future work in this area could be a more detailed research, and ﬁeld experiments, of the eﬀects of a changed power factor control, both for motoring and during regenerative braking.

64 A| Numerical data used

A.1 Per-unit system

In electrical power systems, the nominal voltage levels in different parts of the network can vary, typically on either side of a transformer. To limit the need for tedious conversion between voltage levels, a conversion to a common base value would make it possible to reduce the power system into a network of impedances, and to easily compare different values of voltages and currents [12]. Numerical calculations are also simplified when using a per-unit system, and it is possible to easily determine a values deviation from the used base value.

A per-unit value is calculated as [12]:

actual quantity Quantity in per-unit = (A.1) base value of quantity

In an electrical calculation, the selection of a common base is chosen as an apparent power for the entire network, and then one of either the voltage, current or impedance, for one point in the power system is selected. The two other quantities are then calculated from the selected bases.

To calculate the impedance base, ZB, and current base, IB, for a given 3-phase apparent power, SB, and line-to-line voltage, VB, the following equations are used:

2 VB ZB = (A.2) SB SB IB = √ (A.3) 3 · VB

When bases for a point in the system is set, the bases for the other parts of the system are now dependent, and for either side of a transformer the base-values changes with the voltage-ratio.

65 The voltage VB for three-phase grids are the line-to-line voltage, while for railway power systems the line-to-ground is more applicable, which changes the base values to their single-phase equivalent and equation (A.3) to:

SB IB = . (A.4) VB

A common combination for power systems is to use an apparent power in MVA and voltage levels in kV.

A.2 Converters and grid-connection

For any newly ordered railway converter, after the publishing of the standard for railway converters in Sweden [24], the compounding factor should be able to be set at any value between 0 % and -15 %. The compounding factor, Cf , is in this thesis set to -5 % for all converters during simulations:

Cf = −0.05. (A.5)

The parameters used for short circuit reactance of the public grid, X50, and the delivered reactive power to the public grid from the converter station, Q50, is assumed in both [10] and [8] to the reasonable values:

X50 = 15 % (A.6) and

Q50 = 0. (A.7)

Speciﬁcations for standard rotary converters and their calculated reactances is located in table A.1. The static converters are set to, within its rating, mimic the rotary converter Q48/Q49 and the values of the reactances used is therefore the same.

If neglecting the magnetization and any losses of the transformer connected to the catenary, the reactances of the generator and the transformer could be added together which simpliﬁes the formulation of the generator side of the converter into equation (3.10) [9].

A.3 Catenaries

The impedances used for simulation of catenaries in this thesis are shown in table A.2, unknown or not applicable values are marked with a dash. The most common variant used on northern Malmbanan is the 80 mm2 catenary with two return conductors and a single

66 Table A.1. Ratings and speciﬁcations of mobile rotary converters [9, 10, 24, 38]. Parameter Unit Q24/Q25 Q38/Q39 Q48/Q49 Motor Rated power MVA 3.2 4.4 10.7 Rated voltage kV 6.3 6.3 6.3 m Xq,% % 47 40 49 m Xq,Ω Ω 5.8294 3.6082 1.8176 Generator

Rated power, Sg MVA 2.4 4.0 10.0 Cont. power MVA 3.1 5.8 10.0 Max power MVA 6.8 9.0 18.0 Rated voltage kV 3.0 4.0 5.2 g Xq,% % 37 47 53 g Xq,Ω Ω 1.3875 1.8800 1.4331 Catenary transformer Rated power MVA 2.4 4.0 10.0 Rated voltage kV 3/16 4/16.6 5.2/17 t Xc,% % 4.7 3.4 4.2 t Xc,Ω Ω 5.0132 2.3423 1.0895 Grid transformer Rated power MVA 2.17 3.6 8.5 Rated voltage kV 6.3/2.23 6.3/2.6 6.2/2.6 t X50,% % 16.4 9.0 7.8 t X50,Ω Ω 2.9996 0.9923 0.3527 support wire, and the 100 mm2 auto transformer catenary with two return conductors and a single support wire, 80 mm2 2Å FÖ and 100 mm2 2AT FÖ in table A.2.

A.4 High-Voltage transmission lines

At least three types of high-voltage transmission lines are used in the Swedish railway power grid. The nominal voltage level used, Un, is 132 kV, 30 kV and 15 kV. Only the 15 kV between Luleå and Boden could be of interest if the entire Malmbanan would be studied as the other types of transmission lines are not used for this part of the track, and the eﬀects on power ﬂow calculations are assumed small as long as only Malmbanan is simulated. In this thesis, only the track between Rombak and Kiruna is studied. Even though, the transmission lines are included here for completeness.

67 Table A.2. Catenary impedances [22].

Catenary type Z[Ω/km] C [nF/km] Z0 [Ω] 80 mm2, 1Å 0.30 + j 0.23 - - 100 mm2, 1Å 0.28 + j 0.23 - - 109 mm2, 1Å 0.27 + j 0.23 - - 120 mm2, 1Å 0.26 + j 0.23 - - 80 mm2, 2Å 0.22 + j 0.20 - - 100 mm2, 2Å 0.21 + j 0.20 8.8 - 109 mm2, 2Å 0.21 + j 0.20 - - 120 mm2, 2Å 0.20 + j 0.20 - - 80 mm2, 2Å, FÖ 0.15 + j 0.16 - - 100 mm2, 2Å, FÖ 0.14 + j 0.16 11.1 - 109 mm2, 2Å, FÖ 0.14 + j 0.16 - - 120 mm2, 2Å, FÖ 0.13 + j 0.16 - - 100 mm2, 1Å, 1AT 0.069 + j 0.059 - 0.503 + j 0.774 100 mm2, 2Å, 1AT 0.069 + j 0.060 - 0.411 + j 0.635 100 mm2, 2AT, FÖ 0.034 + j 0.032 11.9 0.215 + j 0.343 120 mm2, 2AT, FÖ 0.0335+ j 0.031 13.5 0.189 + j 0.293 80 mm2, 100J 0.20 + j 0.18 - -

Table A.3. High-Voltage transmission line impedances [22].

Un [kV] Z[Ω/km] C [nF/km] 15 0.30 + j 0.23 - 30 0.67 + j 0.242 6.7 132 0.051 + j 0.063 4.79

Transformer values used in the models for the connection from the high-voltage transmission line to the catenary is based on Swedish requirements speciﬁed in [22] by its apparent power

Sn, and relative voltage drop magnitude |ek|, as shown in table A.4. The voltage drop magnitude for a transformer is mainly reactive, but as calculations of both losses and the leakage inductance in the transformer is of interest, the transformer short-circuit impedance,

Zk1 = Rk1 + jXk1, needs to be determined.

From the report [39] the no-load losses and the rated-load loss are speciﬁed for an estimated eﬃciency of 0.995 for the high-voltage transformers connected to the 132 kV transmission lines. The magnetizing resistance, Rm and short-circuit resistance, Rk1, is calculated as

2 U0 Rm = (A.8) P0

68 and Pk Rk1 = 2 (A.9) Ik with the no-load loss P0, full-load loss Pk and the full-load current Ik for the speciﬁed transformer [40].

The short-circuit impedance magnitude |Zk1|, as seen from the 16.5 kV side, with rated power Sn in MVA, is calculated as

2 |ek| 16.5 |Zk1| = · (A.10) 100 Sn

From the results of equation (A.9) and equation (A.10) it is possible to determine the transformer inductance Xk1 as q 2 2 Xk1 = |Zk1| − Rk1 (A.11)

Table A.4. High-voltage transformer data requirements for Sn and |ek| from [22], and calculated values from measurements in [39] and using (A.8), (A.9), (A.10) and (A.11). Values for ek are calculated from the resulting Zk1

Ratio [kV] Sn [MVA] |ek| [%] ek [%] Zk1 [Ω] Rm [Ω] 32/16 6 8.49 0.652 + j 8.46 0.296 + j 3.84 - 2x66/16.5 16 5 0.385 + j 4.99 0.0655 + j 0.848 29592 2x66/16.5 25 5 0.318 + j 4.99 0.0346 + j 0.543 26432

The same short-circuit impedance angle is used for the 6 MVA transformer calculations as for the larger 16 MVA transformer in table A.4.

A.5 Trains

Trains are numbered even for trains traveling north and with odd numbers for trains moving south. The trains used during simulations are divided into four groups, IORE loaded, IORE empty, Rc freight and Rc passenger. Passenger trains use a single locomotive of Rc6 type whereas freight trains use double Rc4 locomotives.

The maximum tractive power for the Rc-trains are based on figures in [11], and limited to 5 MW at an efficiency of 90 %. The IORE locomotives are always connected in pairs, with an available tractive power for a train set of 10.8 MW [31]. The apparent power rating, Sn, used in this study is 10.8 MVA at the traction motors, and with a current limit of 900 A at the catenary. With a converter efficiency of about 85 %1, including auxiliary power, the

1The eﬃciency model for the traction converter is proprietary software, but is at average about 85 % in this study.

69 IORE draws nearly 13 MW from the power system while motoring [19, 30, 41]. The used eﬃciency of the converter while regenerative breaking is also about 85 %.

Train numbering can be seen in table A.5, train speciﬁcations in table A.6 and their running resistance coeﬃcients in table A.7.

Table A.5. Train numbers for the diﬀerent train types used during simulation of the trains schedule. Type Train numbers IORE loaded 9904, 9906, 9908, 9910, 9912, 9914, 9916, 9918, 9920, 9922, 9926 IORE empty 9901, 9903, 9905, 9907, 9909, 9911, 9913, 9915, 9917, 9919, 9921 Rc4 double 41904, 41905, 45902, 45903 Rc6 93, 94, 95, 96

Table A.6. Train specifications used during simulation for the different train types. Power is specified as maximum traction power at rail. The maximal allowed speed for the IORE train is specified by the manufacturer as the maximum for the specified train set, and not the highest possible speed for the locomotive, which is higher. Type Power [MW] Speed [km/h] Wagons Weight [tons] Length [m] IORE loaded 10.8 60 68 8520 747 IORE empty 10.8 70 68 1720 747 Rc4 double 10 135 30 1056 631 Rc6 5 160 5 178 116

Table A.7. The used running resistance for the diﬀerent train types, where the total running 2 resistance for a train is FR = A + B · v + C · v for a given speed v in m/s. Type A [N] B [Ns/m] C [Ns2/m2] IORE loaded 50000 120 70 IORE empty 26000 120 73 Rc4 double 17866 126 77 Rc6 4202 23 19

A.6 Electrical layout

In the table A.6 are the catenary types used between the nodes, and converter configuration in the converter stations along Malmbanan. The discrepancy between the catenary length and the position difference depends on a changed track route, where the length between the converter stations has changed from their initial length. The difference is small and the influence is assumed negligible, therefore the position of the converter stations is used to calculate the length of the catenaries.

70 Table A.8. Electrical layout of the Swedish part of Malmbanan, from the most northern station Riksgränsen to Nya Malmhamnen in Luleå, with approximate catenary lengths and positions in kilometers. The route from Kiruna to Svappavaara is not included in this table as it is not trafficked during simulations. The specifications of the different catenaries can be found in table A.2 [22]. Full name Sign Converters Catenary Length[km] Position[km] Riksgränsen RGN - - - 1542 - 100 2Å FÖ 23 - 2xQ38/Q39 Tornehamn THA - - 1520 2xQ48/Q49 - 100 2Å FÖ 10 - - 80 2Å FÖ 6 - Abisko AK - - - 1504 - 80 2Å FÖ 34 - 2xQ38/Q39 Stenbacken SBK - - 1470 1xQ48/Q49 - 100 2AT FÖ 49 - Krokvik KV - - - 1417 - 80 2Å FÖ 10 - 2xQ38/Q39 Kiruna KRA - - 1405 2xQ48/Q49 - 100 2AT FÖ 97 - Gällivare GV 4xQ38/Q39 - - 1308 - 100 2AT FÖ 78 - - 100 2Å FÖ 4 - Murjek MK 2xQ38/Q39 - - 1230 - 100 2Å FÖ 4 - - 100 2AT FÖ 74 - - 100 2Å FÖ 3 - 4xMegamacs Boden BDN - - 1150 PWM - 100 2AT FÖ 30 - Notviken NVN - - - 1120 - 80 1Å 10 - Nya Malmhamnen - - - - 1110

71 Bibliography

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