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 Malmb...
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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.

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Sammanfattning I ett flertal tidigare undersökningar och rapporter har konstaterats att spänningsvariationer, och andra förändringar, hos strömförsörjningen till tåg kan påverka dess prestanda och dess färd längs rälsen. Som en del av ett modulärt 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.

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Power supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3

Rail current return systems . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.3.1

Booster transformer . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3.4

3.3.2 Auto transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-voltage transmission system . . . . . . . . . . . . . . . . . . . . . . . .

13 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

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii

33

4.4

4.5

4.6

Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

4.4.1

IORE locomotives . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

4.4.2

Thyristor based locomotives . . . . . . . . . . . . . . . . . . . . . . .

38

Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

4.5.1

Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

4.5.2

Equations and constraints . . . . . . . . . . . . . . . . . . . . . . . .

40

4.5.3

Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

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

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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.

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• 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.

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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.

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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 = Fv

(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 y=

1 z

(2.4)

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

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

Uk

ykm

Ikm

k

m

yk0

ym0

Um

0 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 = Uk Ikm = Uk ykm (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: Pkm =

gkm Uk2 − Uk Um (gkm cos θkm + bkm sin θkm )

(2.7)

Qkm = −

bkm Uk2

(2.8)

+ Uk Um (bkm cos θkm − gkm sin θkm )

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 Ploss,km = Pkm + Pmk = gkm (Uk2 + Um ) − 2gkm Uk Um cos θkm .

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(2.10)

The total system losses can be calculated as: Ploss =

X

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: Ik =

n X

Ikm =

m=0

n X

ykm (Uk − Um ).

(2.14)

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:

Ik = Uk

n X

ykm −

m=0

n X

ykm Um ,

m=1

where the number of nodes in the system, excluding the reference node, is n.

8

(2.15)

Expanding the sums for an arbitrary node k for all values of m results in: Ik = Uk (yk0 + yk1 + yk2 + ykk + · · · + ykn ) − U1 yk1 − U2 yk2 − Uk ykk − · · · − Un ykn (2.16) and eliminating Uk ykk gives Ik = Uk (yk0 + yk1 + yk2 + · · · + ykn ) − U1 yk1 − U2 yk2 − · · · − Un ykn .

(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]. Ykk =

n X

ykm

k 6= m

(2.18)

m=0

Ykm = −ykm

9

(2.19)

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.

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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]. k

Uk

Ikm

r · dx

jx · dx

jbk0 · dx 2

m

jbm0 · dx 2

Um

dx 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 Magnitude [Ω] − Phase/10 [°]

Magnitude [Ω] − Phase/10 [°]

6 Phase

5 4 3 2

Magnitude

1 0 0

20

40 60 Distance[km]

80

Phase 5 4 3

1 0 0

100

(a) Single-fed AT-system.

Magnitude

2

20

40 60 Distance[km]

80

100

(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 autotransformers 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

162⁄3 Hz

50 Hz

Frequency

25 MVA

FC

High-voltage trasmission line

Catenary

16 MVA

25 MVA

FC 16 MVA

25 MVA

FC Generator voltage

Grid voltage

15 kV

132 kV

Nominal 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 T1

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 E1 N1 I2 = = E2 N2 I1 16

(3.1)

and is dependent on the turn-ratio N1 :N2 , where N1 and N2 represents the number of turns on the primary and secondary coil. Transforming an impedance from the secondary side, Z2 , to the primary side, Z20 , using (3.1) gives the relationship: Z20

 =

N1 N2

2 · Z2

(3.2)

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 Zk1 = Z1 + Z20 ,

(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

U1

Rm

jXk1

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

δ Id

φ

U0

I

U

jXq Iq

U sin(δ) RI jXd Id

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  δ = arctan

Xq · P U 2 + Xq · Q

 (3.9)

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 voltages, Q50 and QG represent the reactive power flow and Xqm and Xqg represent the quadrature-axis synchronous reactance, at the motor side and generator side of the converter unit respectively. To simplify the equations, Xqg 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.

PG − jQ50

PG + jQG

M

Um

G

Ug

Xqm

Xqg

Figure 3.9. A rotary converter unit.

The total phase difference between the motor and generator voltages for a single rotary converter, ψ, is 1 ψ = − arctan 3

!

Xqm · PG

 − arctan

U m 2 + Xqm · Q50

Xqg · PG



U g 2 + Xqg · QG

,

(3.10)

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 θ0 = θ50 −

1 arctan 3



X50 · P50 U m 2 + X50 · Q50

 (3.11)

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

Catenary

G

50 Hz

162/3 Hz

θ50

θ0 X50

M

G

..

..

M

G

θ

..

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

U =

g Uidle



QG · 1 + Cf · SG

 .

(3.12)

g U g 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 θ = θ0 + 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 selfcommutated 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 the values of the phase angle, θ0 , or motor constants, Xqm and Xqg , 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: λ=

P U · I1 · cos φ = = g · cos φ S U ·I

(3.14)

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 QAC

PDC ∼

Ptraction Qtraction

= =

3∼

AC Motor

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 Pd2 + Q2d ≤ Sd,max .

24

(3.17)

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 transformer

Converter 2

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

25

Power factor for thyristor based trains

1

1

0.8

0.8

0.6

0.6

cosφ

cosφ

Power factor IORE

0.4

0.4

0.2 0 13

0.2

Motoring Braking 14

15 16 Voltage [kV]

17

Motoring 0 0

18

0.2

0.4

0.6 0.8 Speed [p.u.]

1

1.2

(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)

Pd

P

Pd (b) Thyristor based train.

(a) Asynchronous 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

a · Unom

Umin2

U

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].

Unom a a · Unom Umin2

AC 15 kV 25 kV 0.95 0.9 14.25 kV 22.5 kV 11 kV 17.5 kV

27

3000 V 0.9 2700 V 2000 V

DC 1500 V 0.9 1350 V 1000 V

750 V 0.8 600 V 500 V

Id,max

  Imax ,       Utrain − Umin2 = Imax · ,  a · Unom − Umin2     0,

Utrain ≥ a · Unom Umin2 y √ x 2 + 2 + x max(x, 0) = 2

(4.8)

(4.9)

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 (φp (U ) − π)2 + 2 − (φp (U ) − π) φ(U ) = π − , 2

(4.10)

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 181

180

180

170

179 φ(U) [°]

φ(U) [°]

160 150 140 130

178 177 176 175

Original Approximation

120 10

12

174

14 Voltage [kV]

16

173 14.7

18

Original Approximation 14.75

14.8 14.85 14.9 Voltage [kV]

14.95

15

(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 (Utrain − 7) 8 1 ≤ PD,train,max · (Utrain − 9) 4

PD,train ≤ PD,train,max ·

(4.12)

PD,train

(4.13)

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

Supply system

1

1 0.8 Pbrake [p.u.]

P [p.u.]

0.8 0.6 IORE train 0.4

0.4 0.2

0

0 10

11 12 13 14 Catenary voltage [kV]

15

15

(a) Tractive power limitation. The power limit for the supply system is dependent on the power factor of the train and is here plotted with cos φ = 1.

Supply system

0.6

0.2

9

IORE train

16 17 18 Catenary voltage [kV]

19

(b) Regenerative braking limitation.

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 Programming 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 CONOPT Ipopt MINOS PATHNLP SNOPT

Time [p.u.] 1.00 1.18 1.04 1.20 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) subject to gL ≤ g(x) ≤ gU xL ≤ x ≤ xU

(4.15) (4.16) (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. PG,k − PD,k = Uk ·

X

(Um · (Gkm · cos θkm + Bkm · sin θkm ))

(4.18)

(Um · (Gkm · sin θkm − Bkm · cos θkm ))

(4.19)

m

QG,k − QD,k = Uk ·

X 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 PD,iore + Q2D,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), 0 θconv

=

50 θconv

1 − arctan 3



X50,conv · PG,conv m Uconv 2 + X50,conv · Q50,conv

 ,

(4.27)

followed by the converter equation (4.28), cf. equation (4.4), ψconv

m Xq,n · PG,n

1 = − arctan 3

!

m 2 + Xm · Q Uconv 50,conv · Sp,n q,n ! g Xq,n · PG,n − arctan , 2 g g Uconv + Xq,n · QG,n

(4.28)

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) Sp,n =

Sg,n , St,conv

(4.29)

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 Uconv

=

g Uidle

  QG,conv · 1 + Cf · . SG,conv

(4.31)

50 m Constants in equation (4.27), (4.28), (4.30), (4.29) and (4.31) are θconv , X50,conv , Uconv , g g m 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 )

42

(4.32)

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

Id,max

  Imax ,       Uiore − Umin2 = Imax · ,  a · Unom − Umin2     0,

Uiore ≥ a · Unom Umin2

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