## Institutionen för systemteknik

### Department of Electrical Engineering

**Examensarbete**

**Load flow control and optimization of Banverket’s**

**132 kV 16 2/3 Hz high voltage grid**

Examensarbete utfört i Reglerteknik vid Tekniska högskolan i Linköping

av

**Annica Lindén and Anna Ågren**

LiTH-ISY-EX–05/3727–SE Linköping 2005

Department of Electrical Engineering Linköpings tekniska högskola

Linköpings universitet Linköpings universitet

**Load flow control and optimization of Banverket’s**

**132 kV 16 2/3 Hz high voltage grid**

### Examensarbete utfört i Reglerteknik

### vid Tekniska högskolan i Linköping

### av

**Annica Lindén and Anna Ågren**

LiTH-ISY-EX–05/3727–SE

Handledare: **Markus Gerdin**

ISY, Linköpigs universitet

**Thorsten Schütte**

Rejlers Engineering AB
Examinator: **Svante Gunnarsson**

ISY, Linköpigs universitet Linköping, 20 August, 2005

**Avdelning, Institution**

Division, Department

Division of Automatic Control
Department of Electrical Engineering
Linköpings universitet
S-581 83 Linköping, Sweden
**Datum**
Date
2005-08-20
**Språk**
Language
Svenska/Swedish
Engelska/English
**Rapporttyp**
Report category
Licentiatavhandling
Examensarbete
C-uppsats
D-uppsats
Övrig rapport

**URL för elektronisk version**

http://www.control.isy.liu.se

**ISBN**

—

**ISRN**

LiTH-ISY-EX–05/3727–SE

**Serietitel och serienummer**

Title of series, numbering

**ISSN**

—

**Titel**

Title

Lastflödesreglering och optimering i Banverkets 132 kV 16 2/3 Hz högspänningsnät Load flow control and optimization of Banverket’s 132 kV 16 2/3 Hz high voltage grid

**Författare**

Author

Annica Lindén and Anna Ågren

**Sammanfattning**

Abstract

The purpose of this thesis was to investigate the possibility of power flow control, on a section of a railway grid fed by rotary converters, using an extra feeding line. Two possible solutions for the power flow control were examined. The first using a series reactance in connection to each converter station and the second by changing the tap changer level of the transformer between the converter station and the feeding line.

In the two models a distance, comparable to the distance between Boden and Häggvik, in Stockholm, was used. The simulations were performed using the soft-ware SIMPOW.

The results from the performed simulations show that series reactances, under the stated conditions, can essentially improve the power flow. To implement this air coils with inductances in the approximate size of 10 to 45 mH could be used. Further, the tap changer levels of the transformer may be used, for individual converter stations, as a way to control the reactive power flow.

**Nyckelord**

Keywords 132 kV, rotary converter, load flow, power flow, feeding line, catenary, tap changer, control system, railway, electrification

**Abstract**

The purpose of this thesis was to investigate the possibility of power flow control, on a section of a railway grid fed by rotary converters, using an extra feeding line. Two possible solutions for the power flow control were examined. The first using a series reactance in connection to each converter station and the second by changing the tap changer level of the transformer between the converter station and the feeding line.

In the two models a distance, comparable to the distance between Boden and Häggvik, in Stockholm, was used. The simulations were performed using the software SIMPOW.

The results from the performed simulations show that series reactances, under the stated conditions, can essentially improve the power flow. To implement this air coils with inductances in the approximate size of 10 to 45 mH could be used. Further, the tap changer levels of the transformer may be used, for individual converter stations, as a way to control the reactive power flow.

**Acknowledgements**

We would like to thank Thomas Jobenius and Thorsten Schütte, at Rejler’s En-gineering AB, for the opportunity to do our Master Thesis at the company. All the support and dedication has been invaluable. We also want to give an extra thanks to the employees in Linköping, whom have made us feel very welcome and included.

An extra thanks to Banverket and particularly to Sam Berggren, for all the help with material and answers to our questions and to Jonas Persson and Mikael Ström at STRI AB for their fast, good and detailed replies to our endless questions concerning our model and SIMPOW.

Finally we would like to thank our professor Svante Gunnarsson and tutor Markus Gerdin at LiTH.

**Contents**

**1** **Introduction** **1**

1.1 Background . . . 1

1.2 Purpose . . . 1

**2** **Electrical traction system** **3**
2.1 History . . . 3

2.2 Power supply . . . 4

2.3 Rotary converters . . . 6

2.3.1 General . . . 6

2.3.2 Phase shift angle . . . 8

2.4 Static converters . . . 9

2.4.1 Cyclo converter . . . 9

2.4.2 PWM converter . . . 9

2.5 Loads . . . 11

**3** **Problem and possible solutions** **13**
3.1 Problem . . . 13

3.2 Possible solutions . . . 15

3.2.1 Solution one - series reactances . . . 15

3.2.2 Solution two - transformer tap changer . . . 17

3.2.3 Solution three - rotary converter, static converter or para-metric transformer . . . 18

3.2.4 Solution four - combination . . . 19

**4** **Modeling and simulations** **21**
4.1 Restrictions and general simplifications . . . 21

4.2 Original models . . . 22

4.2.1 Small model . . . 22

4.2.2 Large model . . . 22

4.3 Simulations for the influence of α . . . 24

4.4 Solution one - series reactances . . . 25

4.4.1 Additions to the general model in SIMPOW . . . 25

4.4.2 Simulations with variable inductances . . . 25

4.4.3 Model for control system . . . 28

4.4.4 Complementary simulations . . . 28 ix

4.5.1 Additions to the general model in SIMPOW . . . 29

4.5.2 Simulations with different tap changer levels . . . 29

4.5.3 Model for control system . . . 31

**5** **Results and conclusions** **33**
5.1 Results using series reactances . . . 33

5.2 Results for complementary simulations using series reactances . . . 37

5.3 Result using the tap changers on the transformers . . . 38

5.4 Conclusions . . . 39

**6** **Future work** **41**
**Bibliography** **43**
**A Blondel transformation for power flow** **45**
A.1 The basic machine parameters . . . 45

A.2 The general power equation . . . 48

A.3 The Blondel transformation . . . 48

A.4 The machine under symmetrical conditions . . . 51

A.4.1 Derivation of voltage and current relations . . . 51

A.4.2 Phasor diagram . . . 53

**B The general model of the simulated system** **55**
B.1 Short description of SIMPOW . . . 55

B.2 The data used in SIMPOW . . . 56

B.3 Figure of the general system . . . 58

B.4 Figure of the system with reactances . . . 59

B.5 Phase difference α during 24-hour period . . . 60

**Chapter 1**

**Introduction**

**1.1**

**Background**

The Swedish railway system is a low frequency grid fed from the public 50 Hz grid to the railway catenary by rotary and static converters. Rotary converters are synchronous-synchronous machines, which means that the railway system is synchronous with the public 50 Hz grid. This implies that all phase displacement on the public 50 Hz grid, will be transmitted directly to the railway grid. Static converters are constructed so that the phase can be chosen arbitrarily.

For efficient load flow separate 132 kV feeding lines, parallel with the railway, are used. The existing 132 kV feeding line supplies only parts of the Swedish railway, since this technique is not efficient in systems with more than one rotary converter, where phase differences appear. Today only static converters can be adjusted to generate voltage with adequate phase.

To further optimize the load flow an extension of the 132 kV feeding grid, to include a larger part of the railway system, would be favorable. One option could be to replace all rotary converters with static converters and then extend the 132 kV feeding grid. This is not an economical option, since today approximately half of the generated power comes from rotary converters.

**1.2**

**Purpose**

The purpose of the thesis is to investigate the possibility of feeding the 132 kV feeding line with several rotary converters. In other words, the goal is to examine the possibility of power flow control, on the 132 kV feeding line, independent of what kind of converters that are used.

Some or all of the ideas stated below will be evaluated:

1. Vary the voltage angle using series rectances, inductances or capacitances, between the converter station and transformer.

2. Vary the voltage fed to the feeding line by adjusting the circuit coupler of the transformer.

3. Add a 90 degree phase shifted voltage component generated by a smaller rotary or static converter or a parametric transformer.

4. A combination of some or all of the ideas.

To compensate for the variation in phase displacement between the rotary con-verters, a control system is demanded. The ideas are evaluated with the simulation program SIMPOW.

**Chapter 2**

**Electrical traction system**

This chapter gives a short introduction of Swedish railway history and a description of how it works today.

**2.1**

**History**

The kind of electricity used on the railway grids differ between countries. There exists no given standard for electrical traction systems. Some countries use DC-voltage and other AC-DC-voltage of varying frequency and amplitude, see Figure 2.1.

**Figure 2.1. Electrical traction in Europe [14]**

The electrification of the Swedish railway system started in the beginning of the 20th century, using 16 2/3 Hz single phase voltage fed by converters from

the public 50 Hz grid. The voltage used is 16.5 kV, nominal voltage 15 kV plus 10 %. The only converters available at this time were rotary converters. As the technology progressed static converters became a good alternative to the rotary converters. This contributes to the existence of both rotary and static converters in the railway system.

As the load on the Swedish railway increased the need for more efficient power supply grew. The options were to install new converter stations or to build a feeding line parallel to the railway. In the late 1980s a 132 kV feeding line was built from Hallsberg to Jörn [7, 12]. Figure 2.2 shows the Swedish feeding grid in 1997, extended to Boden in the north. Today some further extensions have been made in Mälardalen and in the south of Norrland. All rotary converters in the parts with feeding line, except for the converter station in Häggvik, have been replaced by static converters.

**Figure 2.2. The extension of the Swedish feeding grid in 1997 [13]**

**2.2**

**Power supply**

The train is supplied with power from the catenary, through a pantograph on the locomotive, as described in Figure 2.3. This power is fed from the three phase public 50 Hz grid through a transformer to a converter station, with either rotary or static converters. These are described below in chapter 2.3 and 2.4. The converted voltage passes through a transformer to the catenary. If a feeding line exists it is fed as well. The voltage on the catenary has a nominal voltage of 15 kV. Normally an extra 10 procent is added which gives a voltage of 16.5 kV.

**2.2 Power supply** **5**

**Figure 2.3. Description of the power supply [2]**

The thickness of the catenary and the return conductors is limited by its weight [18]. Further, the catenary also has inductance which leads to the occurance of power drops when the distance between two feeding points is too far. This sets an upper limit for the distance between two converter stations.

As mentioned in chapter 2.1 a separate 132 kV feeding line, parallel with the catenary, is used for a sufficient load flow. The feeding line is built up by two 66 kV lines in counterphase to one another. Between the converter stations, the power flow is transferred through transformers from the feeding line to the catenary, See Figure 2.4. The catenary receives power directly from the converter stations and indirectly from the feeding line.

**2.3**

**Rotary converters**

A converter station usually consists of several converter units in parallel. Each converter unit comprises one rotary converter and one step-up transformer to 16.5 kV. In this chapter the principal function of a rotary converter is described.

**2.3.1**

**General**

A rotary converter is a synchronous-synchronous machine, which consists of an electrical motor, a generator and a common shaft. The motor is fed by the three phase public 50 Hz grid and the generator generates a single phase voltage, see Figure 2.5.

**Figure 2.5. The principle of voltage transmission from the 50 Hz grid to the contact**

line through rotary converters [6].

The rotation speed, ns, of a synchronous motor depends on the frequency of

the feeding grid [1], in this case the public 50 Hz grid. The rotation speed, in rounds per minute, is given by

ns =

120 · f

**2.3 Rotary converters** **7**

where f is the frequency of the feeding grid and p is the number of poles of the motor [4]. The rotation speed is unchanged as long as the feeding grid delivers a constant frequency to the motor.

A synchronous generator works as a reversed synchronous motor. In a rotary
converter the motor and generator are linked together by a shaft. This means that
the rotation speed is the same in both parts. For a rotary converter the outgoing
frequency is given by
nsg = 120·f_{p} g
g
nsm = 120·f_{p} m
m
nsg = nsm
⇒ fg =
pg
pm
· fm (2.2)

where indexes m and g indicate motor and generator respectively. In Sweden the rotary converters have a motor with 12 poles and a generator with 4 poles [12]. This results in a generated frequency, fg, of a third of the frequency entering the

converter. For the rotary converter this means a frequency of 16 2/3 Hz.

The magnitude, of the generated voltage, depends on the magnetizing current in the generator and the power flow from the motor [1]. In the Swedish railway grid, three different sizes of rotary converters are used. The small one, Q24, generates 3000 V and the two bigger ones Q38 and Q48 generate 4000 V and 5000 V respectively [5]. Q38 can be seen in Figure 2.6. The nominal power, delivered by the rotary converters, is 2.4 MVA, 4 MVA and 10 MVA respectively [11].

**Figure 2.6. A rotary converter of medium size, Q38 [12].**

One big advantage with rotary converters is that the load always is symmetric and non oscillating on the 50 Hz grid [16]. Since the synchronous motor of the rotary converter produces adjustable reactive power, it is possible to compensate for voltage drops that occur on the 50 Hz grid. Further, the voltage from the converter station can be held constant by regulating the magnetizing current of the generator. The disadvantage of rotary converters is the long start up time, ap-proximately 2 minutes, and complicated synchronization, see also Chapter 2.3.2.

**2.3.2**

**Phase shift angle**

Any phase displacements that occur on the 50 Hz grid will be transmitted to the railway grid by a third, just like the frequency mentioned earlier.

**Figure 2.7. The phasor diagram for a phase in a synchronous machine with salient**

poles [12]

For a salient pole synchronous machine, the phasor diagram in Figure 2.7 can be drawn [3]. δ, the electrical angle, is defined as the angle between E, the open circuit voltage, and U , the outgoing voltage. Xd and Xq are the reactances in

d-and q-directions. The diagram is drawn per phase, d-and the d- d-and q-axes represent the direct- and quadrature-axis respectively in the mathematical Blondel trans-formation. For an explicit explanation see Appendix A. The following equations can be extracted from Figure 2.7

U sin δ = Xq· Iq (2.3)

Iq = I cos (δ + ϕ) (2.4)

From Equation 2.3 and 2.4 the electrical angle δ = arctan Xq· I cos ϕ

U + Xq· I sin ϕ

(2.5)

is given. The active power, P , and reactive power, Q, are defined by

P = U · I cos ϕ (2.6)
Q = U · I sin ϕ (2.7)
This gives
δ = arctan Xq· P
U2_{+ X}
q· Q
(2.8)

**2.4 Static converters** **9**

The angle turning function for the rotary converter is

θvar = − (δm+ δg) (2.9)
θvar = −
1
3arctan
Xm
q · P
(Um_{)}2_{+ X}m
q · Q50
− arctan X
g
q · P
(Ug_{)}2_{+ X}g
q · Q
(2.10)
The total output voltage angle is

θ = θ0+ θvar (2.11)

where θ0 is the no load angle [12].

For converter stations with several parallel rotary converters, it is important to decrease the outgoing voltage from the generator as the reactive power flow increases. This is a way to avoid negative reactive currents and to keep the oper-ation stable [5]. It is important that the output voltage angles, for all converters in the converter station, are approximately the same [12].

To add a feeding line parallel to the catenary, to a part of the railway grid fed by rotary converters, is possible only if the output voltage angles from the different converter stations are almost equal.

**2.4**

**Static converters**

The technique used in static converters is based on power electronics. There exist two types of static converters, cyclo-converters and PWM-converters [12].

**2.4.1**

**Cyclo converter**

The main component of a cyclo converter is a thyristor bridge. By igniting the thyristors in a suitable order, a one phase low frequency voltage can be created from the three phase public 50 Hz grid. See Figure 2.8 for a principal scheme. To prevent influences on both grids and to increase the generated power a 12-pulse connection, with two thyristor bridges connected in series, is used [5, 16]. One of the bridges is fed from the 50 Hz grid by a Y-connected transformer and the other by a D-connected transformer, see Figure 2.9. A cyclo converter puts high demands on the transformer and creates a lot of reactive power, which must be compensated for.

**2.4.2**

**PWM converter**

The intermediate PWM-converter, Pulse Width Modulated converter, converts the voltage in three steps. For a principal scheme of the PWM-converter see Figure 2.10.

The three phase voltage from the public 50 Hz grid is rectified in a 12-pulse rectifier after a step down transformer [5]. In the intermediate DC link the voltage is filtered and in the last step it is modulated in pulses so that the average for the pulses constitutes the one phase voltage.

**Figure 2.8. Principal scheme of a cyclo converter [12]**

**2.5 Loads** **11**

**Figure 2.10. Principal scheme of a PWM converter [12]**

**2.5**

**Loads**

In railway systems the loads consist of trains. Their power consumption is complex since it depends on a lot of factors, for example speed, acceleration, weight, track conditions etc. Some common trains in Sweden and Norway are presented in Table 2.1.

Locomotive Type Power Weight

MW tonne

IORE double Ore transport Malmbanan 10.8 300

EL16 Conveyance of goods 4.44 129

EG3100 Ore transport Malmbanan 6.50 132

EL15 Conveyance of goods 5.40 132

RC Conveyance of goods 3.60 77

X2000 Passenger train 3.26 73

**Chapter 3**

**Problem and possible**

**solutions**

**3.1**

**Problem**

As described in Chapter 1.1, it would be favorable to add a feeding grid parallel to the railway grid. The rotary converters transmit phase differences from the 50 Hz grid to the railway grid according to Chapter 2.3.2. In order to connect a feeding grid, the phase in the whole grid needs to be approximately the same. Today, for rotary converters, there exists no method for this kind of compensation.

**Figure 3.1. The angles for ¯**U1and ¯U2

The transmitted phase difference, γ, mentioned above, is defined as the angle between the outgoing voltage from two different rotary converter stations, as shown in Figure 3.1.

γ = θ2− θ1 (3.1)

With notation from Figure 3.1 and using complex numbers the two voltages can 13

be written as

¯

U1 = U · ejθ1 (3.2)

¯

U2 = U · ejθ2 = U · ej(θ1+γ) (3.3)

The apparent power, S, and Ohms law give ¯ S = ¯U · ¯I ¯ U = ¯Z · ¯I ⇒ S =¯ ¯ U2 ¯ Z (3.4) where ¯ Z = Z · ejϕ (3.5)

is the impedance from the whole system felt by the rotary converter. This leads to the following expressions for S1and S2

¯
S1 =
¯
U2
¯
Z1
= U
2
Z1
· ej(2θ1−ϕ1) _{(3.6)}
¯
S2 =
¯
U2
¯
Z2
= U
2
Z2
· ej(2θ1+2γ−ϕ2) _{(3.7)}

If both rotary converters experience the same load from the system, the impedances

¯

Z1 = ¯Z2= ¯Z (3.8)

and the apparent power S can be described as
¯
S1 =
U2
Z · e
j(2θ1−ϕ) _{(3.9)}
¯
S2 =
U2
Z · e
j(2θ1+2γ−ϕ)_{=} U
2
Z · e
j(2θ−ϕ)_{· e}j2γ _{(3.10)}

When the voltages out of the rotary converters are the same and the angle γ is nonzero the active and reactive power from the converter stations will differ, see Figure 3.2. To be able to connect an extra feeding line to the system the effect of γ needs to be eliminated, or at least decreased. In the coming sections some ideas for how this could be done are described.

**3.2 Possible solutions** **15**

**Figure 3.2. The apparent power ¯**S1 and ¯S2

**3.2**

**Possible solutions**

In Chapter 1.2 four ideas, that might solve the problem, are presented. These possible solutions are described in this section.

**3.2.1**

**Solution one - series reactances**

One possible solution is to use series reactances to compensate for the phase shift γ between the outgoing voltage from the converters. This can not eliminate the problem totally but reduce its influence. The idea is to reduce the effect from γ, described in Chapter 3.1, by connecting an adjustable reactance, X, between the converter station and the transformer to the feeding grid. Theoretically, the reactance used can be either an inductance, L, or a capacitance, C. The difference between using a capacitance or an inductance is that a capacitance has a negative contribution and an inductance a positive.

¯ U = ¯X · ¯I (3.11) ¯ XL = jωL (3.12) ¯ XC = 1 jωC = −j 1 ωC (3.13)

In phasors this implies, as shown in Figure 3.3, that the voltage is phase shifted with reference to the current.

Returning to the notation in Chapter 3.1 the apparent power
¯
S1 =
U2
Z · e
j(2θ1−ϕ) _{(3.14)}
¯
S2 =
U2
Z · e
j(2θ1+2γ−ϕ)_{=}U
2
Z · e
j(2θ1−ϕ)_{· e}j2γ _{(3.15)}

To compensate for γ a series inductance connected to rotary converter number one may be used. The first rotary converter will now experience another load from the system ¯ ZnL = ¯Z + ¯XL= ZnL· ej(ϕ+∆ϕ) (3.16) which implies ¯ U1n = ¯U1· ej∆ϕ= ¯U · ej(θ1+∆ϕ) (3.17)

and the apparent power generated is
¯
S1n =
U2
ZnL
· ej(2θ1−ϕ)_{· e}j∆ϕ _{(3.18)}
¯
S2 =
U2
2
Z · e
j(2θ1−ϕ)_{· e}j2γ _{(3.19)}

The difference in angle for the apparent power from rotary converter one and two, and the magnitude of S1have decreased. Both things lead to a decreased influence

from γ on the system, see Figure 3.4.

**Figure 3.4. The apparent power under influence of an inductance connected to rotary**

converter number one.

Instead of an inductance a series capacitance connected to rotary converter number two may be used. Now rotary converter number two experiences another load from the system.

¯

ZnC = ¯Z + ¯XC= ZnC· ej(ϕ−∆ϕ) (3.20)

which implies

¯

**3.2 Possible solutions** **17**

and the apparent power generated is
¯
S1 =
U2
Z · e
j(2θ1−ϕ) _{(3.22)}
¯
S2n =
U2
ZnC
· ej(2θ1−ϕ)_{· e}j(2γ−∆ϕ) _{(3.23)}

This can decrease the angular difference between the voltages as much as using

**Figure 3.5. The apparent power under influence of a capacitance connected to rotary**

converter number two.

an inductance would do. Just as before, the magnitude of the apparent power is decreased. The first part contributes to a decrease of the influence from γ, but the second part does not. See Figure 3.5. This leads to the conclusion that it might be better to use inductances for compensation than capacitances.

**3.2.2**

**Solution two - transformer tap changer**

The second idea is to vary the voltage fed to the feeding line by adjusting the tap changer of the transformer. If the voltage fed to the feeding line increases the power flow will also increase and vice versa. The transformers used in Sweden have five different levels, 0, ±2.5, ±5 percent, to chose between. The level used today is zero.

Just like the solution described in Chapter 3.2.1 this solution will not give a perfect result, but the difference in power can be decreased. Returning to the notation in Chapter 3.1 the angular difference γ between ¯U1 and ¯U2 contributes

to a difference in active and reactive power, ¯P and ¯Q. The active power out of converter one is greater than the active power out of converter two, and the other way around for the reactive power. When adjusting the tap changer to decrease the difference in active power, the difference in reactive power will rise. The adjustments made to minimize the difference in P can be accomplished in two ways. The solution would either be to rise the tap changer to a higher level for transformer number two or to lower the tap changer for transformer number one. As illustrated in Figure 3.6 and as mentioned earlier this leads to an increase in the difference in reactive power ∆Q. The change in transformer number one leads to a smaller increase in ∆Q than a change in number two does.

**Figure 3.6. Comparision of the changes in apparent power, at the left when the tap**

changer is lifted one level at transformer two and and at the right when the tap changer is lowered one level at transformer one.

**3.2.3**

**Solution three - rotary converter, static converter or**

**parametric transformer**

The third possible solution, as mentioned earlier in Chapter 1.2, is to add a 90 degree phase shifted voltage component generated by a smaller rotary or static converter or a parametric transformer. This is another way to shift the phase for the voltage. Using the same notation as in Chapter 3.1, and connecting a smaller rotary converter in series with rotary converter number one, the voltage component can be described as

¯

Ucomp = Ucomp· ej(θ1+

π

2) _{(3.24)}

and the new voltage will be
¯
U1n = Ucomp· ej(θ1+
π
2)+ U · ejθ1 _{= U}
1n· ej(θ1+∆θ1) (3.25)
as illustrated in figure 3.7.

**Figure 3.7. The changes in ¯**U1 using a rotary converter to compensate for γ.

Since this phase shift is a total phase shift, i.e. both the voltage and the current are shifted in the same manner, the angle ϕ will be preserved. The apparent power

**3.2 Possible solutions** **19**
can be described as
¯
S1n =
U2
1n
Z · e
j(2θ1−ϕ)_{· e}j2∆θ1 _{(3.26)}
¯
S2 =
U2
ZnC
· ej(2θ1−ϕ)_{· e}j2γ _{(3.27)}

If Ucomp is chosen wisely a total compensation for the difference in phase, γ,

between U1and U2can be achieved. This occurs when

∆θ1 = γ (3.28)

Even though the phase displacement is gone, there still exists a difference in power. As seen to the left in Figure 3.8 the apparent power, Sn1, is bigger in magnitude

than S2. By using the technique described in Chapter 3.2.2 the problem can be

eliminated. As shown to the right in Figure 3.8 the compensation for γ might as

**Figure 3.8. The apparent power when using a rotary converter for compensation of γ**

well be done by connecting a smaller rotary converter to converter number two. This works the same way as stated above with the exception for the phase shift in

¯

Ucomp, which will switch sign and be negative.

**3.2.4**

**Solution four - combination**

The fourth possible solution is to combine one or several of the described solutions above. From an economical aspect the best solution could be to combine serie inductances with a good level for the tap changers of the transformers.

**Chapter 4**

**Modeling and simulations**

In this chapter the used models and simulations are described. The modeling and simulation are based on idea one and two from chapter 3. The third idea was not investigated as the work grew in size with the modeling of the first idea. For implementation and simulations the program SIMPOW, see Appendix B.1, was used.

**4.1**

**Restrictions and general simplifications**

The Swedish railway grid is a very complex system with a lot of factors to account for. Already when studying a smaller part of the system it shows large complexity. The purpose of this thesis is to study the influences of α, the phase difference between the voltages fed to the converter stations on the public 50 Hz grid, on a railway grid fed by rotary converters and evaluate the possibility of power flow control. To create the model some simplifications and restrictions were made.

1. The system is symmetric and this means that the distances between all converter stations and transformers are equal throughout the whole model. 2. All converter stations use only one type of rotary converters, the Q48s.

Fur-ther, the rotary converters in the stations are represented by one rotary converter with three times as big power as one Q48, instead of three Q48s in parallel with each other.

3. The trains are looked upon as a constant load and are placed symmetrically. Their movements are considered as disturbances, i.e. on average the load is constant. This simplification can be made since the power flow studied is the one depending on the angular difference α.

4. The change in angular difference α has a large time constant, and it does not occur instantaneously.

5. The 50 Hz net is modeled as a number of strong constant voltage sources with a specified phase angle for each feeding point.

A detailed description of the model can be found in Chapter 4.2

**4.2**

**Original models**

During the work with the thesis two different models have been used. A small prototype model for initial tests and a large one for the simulations.

**4.2.1**

**Small model**

The purpose of the small model was to understand the power flow and to learn the simulation program SIMPOW. The small model consists of two rotary converters, one train, feeding line, catenary and three transformers. The distance between the two converter stations in the model was 100 km and a transformer, called T 3 as showed in Figure 4.1, was placed half-way. The first calculations of the power flow were made in MATLAB, using very simple models of transformers and converter stations. The next step was to use SIMPOW which provides more sophisticated models for the components. The small model made it easier to identify how the different components in SIMPOW work.

**Figure 4.1. The small model**

**4.2.2**

**Large model**

The large system can be seen as a simplified model of the railway between Boden and Häggvik in Stockholm. The total length of the track is chosen to 1000 km and there are six converter stations symmetrically distributed, with one converter station at each end point. The converter stations consisting of three Q48s with one transformer each are modeled as one rotary converter and one transformer, marked T 1 in Appendix B.3. Each converter station produces at maximum 30 MVA in apparent power. There are two kinds of transformers used to connect the catenary and the feeding line, marked T 2 and T 3. They differ in nominal

**4.2 Original models** **23**

power, 25 MVA in apparent power for T 2 and only 16 MVA for T 3 [9]. The transformers are placed symmetrically with a distance of 50 km in between. A total number of 27 transformers are used. To model an even load from the trains 20 RC-trains, using maximal power, are placed symmetrically half-way between every transformer along the catenary. The train’s movements along the catenary are looked upon as negligible disturbances, as mentioned in Chapter 4.1. The phase difference, α, on the three phase public 50 Hz grid, was modeled as α degrees at rotary converter one with a linear change down to zero degrees phase difference at rotary converter number six. All data used for the model can be found in Appendix B.

**Figure 4.2. The power flow, out of rotary converter station number one through six,**

with no phase difference transferred from the public 50 Hz grid.

Models in SIMPOW are built up by nodes. To simplify the notation, in the following sections, the nodes connecting the rotary converter stations and the transformers, T 2, are marked Inlow1 through Inlow6, see Appendix B.3. Since

the model is symmetric and consists of a finite number of rotary converter stations, the ones in the middle experience more load than those at each end. These edge effects lead to a smaller generation of power in the outer converter stations. Figure 4.2 shows the active, reactive and apparent power in Inlow1 through Inlow6 for

**4.3**

**Simulations for the influence of α**

When adding a phase difference α, on the 50 Hz side, a phase difference γ = α

3

is transferred to the railway grid, as described in Chapter 2.3.2. According to Chapter 3.1 a rotary converter which lags in phase compared to another converter produces more active power. In order to find an approximation for how the changes in power production depend on the phase difference α several simulations were made. The active power flow vector

¯ P = P1 P2 .. . P6

shows the active power flow in Inlow1 through Inlow6, see Appendix B.3, were

studied for α = 0, 15, 30 and 45 degrees. The results for P4, P5 and P6 can be

viewed in Figure 4.3. P1, P2and P3show the same tendencies but with a decrease

in active power with an increased α. From the simulations a linear approximation for how the active power depends on α was estimated.

¯

P (α) = ¯Kα + ¯P0 (4.1)

where ¯K is the vector containing the gradients and ¯P0 is the power flow vector

when α is zero.

**4.4 Solution one - series reactances** **25**

**4.4**

**Solution one - series reactances**

**4.4.1**

**Additions to the general model in SIMPOW**

The first possible solution investigated was the use of series reactances to com-pensate for the induced phase differences. The idea is chosen for its simplicity and because it is probably the most economical solution. The possible solution would be, as described in Chapter 3.2.1, to place a variable reactance between the converter station and the transformer feeding the feeding line. To do this the large model in Chapter 4.2.2 was modified. One extra node, placed between the outgoing node at the converter station and the transformer to the feeding line, was inserted for each converter station. These new nodes are called Contr1 through

Contr6, see Appendix B.4. With no reactance used the two nodes, Inlowi and

Contri were short circuited by an ideal line. The railway system is of inductive

nature, so from that point of view it would be favorable to use capacitances. Ac-cording to Chapter 3.2.1 and because they are easier to construct of suitable size the reactances used were inductive. From now on the notation

X = XL (4.2)

will be used and the reactance connected between Inlowi and Contri will be

referred to as Xi.

**4.4.2**

**Simulations with variable inductances**

The possibility to change the power flow using reactances was obvious after a few simulations with different values of Xi. The next step was to find an expression

for the estimated active power which included Xi. In Chapter 4.3 the linear

dependence of α for the active power vector ¯P is described. To expand equation 4.1 to include the dependence of the reactances ¯x, where

¯ x = X1 .. . X6 (4.3)

the assumption was made that ¯P depends independently on ¯x and α without correlation.

¯

P (¯x, α) = ¯P0+ ¯K (α) + M (¯x) (4.4)

To study the dependence simulations were made with all reactances equal, Xi =

0, 1, 2 and 3. The active power flow was studied for the same α as in section 4.3, α = 0, 15, 30 and 45 degrees phase shift. To define how a change in reactance Xi affects the power flow in all measured points further simulations were made.

In these simulations one reactance at the time were changed, in steps of 0.5 Ω, and the other were held constant. This to define how a change in for example

X1 affects the power flow P3. The test runs showed that the dependence was

non-linear, of the natural logarithm type, so the assumption

M (¯x) = A ln (¯x + 1) (4.5)

was made. It is important to note that the assumption ln (¯x + 1) is possible to make in this case, due to the approximate size of the reactances. If the reactances had been of another approximate size the vector would have had to be normalized with a reference reactance, xref. In this case it can be looked upon as if xref = 1Ω.

The one in Equation 4.5 represents a vector of ones of appropriate size. A is a square matrix with positive elements except for the diagonal elements which represents the influence of Xi on Pi. This assumption was based on the fact that

the reactances are of the inductive kind and that it should be possible to set a reactance to zero, i.e. disconnect it. In order to determine the elements in matrix A the cross influences were assumed to be additive.

M (¯x) = a11 a12 · · · a16 a21 . .. ... .. . . .. ... a61 · · · a66 ln X1+ 1 X2+ 1 .. . X6+ 1 (4.6)

where aij represents the influence from Xj on Pi. By using the assumption in

Equation 4.4 with M(¯x) as in Equation 4.6 and studying the simulations performed in SIMPOW the elements in ¯K and in ¯A were estimated. When comparing the estimated Pivalues to the simulated values for Pia satisfactory result was achieved

as long as the reactances used were approximately the same. This is illustrated in Figure 4.4. In Inlow3 through Inlow6 a small deviation can be seen, but it is

small enough to disregard.

**Figure 4.4. The simulated and estimated values of the active power in Inlow**1through

Inlow6 when all reactances Xi = 2 Ω and α = 30 degrees.

The values of Xineeded to compensate for α differ between the converter stations,

**4.4 Solution one - series reactances** **27**

4.5 shows the estimated and simulated values for Pi when α is 30 degrees and

appropriate ¯x. The estimated and simulated values for P5and P6differ too much.

Further investigations gave that matrix A depends on both ¯x and α. This gives

**Figure 4.5. The simulated and estimated values for the active power in Inlow**1through

Inlow6 when α = 30 degree and the reactances chosen as in simulation number 23.

the following expression for M (¯x, α)

M (¯x, α) = A(¯x, α) ln (¯x + 1) (4.7) First the dependence of α was examined and a small variation could be detected. The estimated values for Pi, using A(α), were compared to the earlier estimations

for Pi. The differences were so small that it is possible to overlook the influence

of α on A. The problem is the dependence of ¯x. In the previous model of ¯P the influences of Xi are assumed to be additive, which apparently is incorrect. A lot

of different approximations have been made, where, for example, the influences from the nearest converter stations are taken into account and other where the mean reactance felt by the system has been used in different compositions. No one of the assumed approximations of ¯P have given a good enough agreement with the simulated values from SIMPOW. The estimated model for the active power generates a good approximation only when the elements of ¯X are close in size. Since the values of ¯x, for a good compensation of α varies, the decision was made not to do any further attempts to estimate a model for the active power flow. The approach now was to find optimal values for ¯x for each α= 0, 15, 30 and 45 degrees.

To find the appropriate values of Xithe knowledge gained during the estimation

process of ¯P was used. The goal was to minimize the differences in Pi with

acceptable differences in reactive power, Qi, for the converter stations. As help

in this selection Figure 4.6 was used. This figure shows the maximal difference in active and reactive power for a number of simulations, when α = 45 degrees. From simulation number 22 and up, all the simulations have an acceptable value for the difference in active power. The reactive power has a large variation between different simulations. The choice of ¯x is a balance between the differences in Piand

**Figure 4.6. The maximal difference in ¯**P and ¯Q for a number of simulations, that was
used as a guideline to determine a suitable choice of ¯x when α = 45 degrees.

Qi, however the size of the reactances need to be taken into account. Simulations

number 23 and 28 have almost the same values for the maximal difference in Qi.

Even though simulation number 28 has a smaller difference in Pi, the reactances

used in simulation number 23 were significantly smaller and therefore the values of ¯x for simulation number 23 were chosen.

**4.4.3**

**Model for control system**

The idea was to use a feedback control system, but as mentioned in Chapter 4.4.2 the approximated active power flows were not good enough for this purpose. The choice then was to use a feed forward control system, that uses the phase difference α as input signal and four possible levels for the reactances as output. Another choice could be to set the reactances for a phase difference α = 30 degrees and be satisfied with this simple solution. A more detailed discussion is presented in Chapter 5.

**4.4.4**

**Complementary simulations**

A few complementary simulations were made for some special cases. These cases were simulated to see which deviation would occure if some basic conditions were changed. This was done in order to get an indication of how robust the solution is. In the first case rotary converter one through three have 30 degrees lag on the 50 Hz public grid and rotary converter four through six has a linear change, in steps of 10 degrees, down to zero. In the second case the load from the trains were halved. A discussion of the result is presented in Chapter 5.

**4.5 Solution two - transformer tap changer** **29**

**4.5**

**Solution two - transformer tap changer**

**4.5.1**

**Additions to the general model in SIMPOW**

To do test runs, for the idea described in Chapter 3.2.2, the original large system was used. The tap changer on the transformer has five different levels, which are 0, ±2.5, ±5%. The transformers used in SIMPOW has no tap changing function so to simulate this the transformers outgoing voltages were changed in steps shown in Table 4.1. Change in Outgoing % voltage in kV -5 125.4 -2.5 130.4 0 132 2.5 135.3 5 138.6

**Table 4.1. Tap changer levels with corresponding outgoing voltage**

The tap changer level is referred to, in the same way as for the reactances in Chapter 4.4.2, as a vector, ¯ V = V1 V2 .. . V6 (4.8)

where Vi can have the values shown in Table 4.1 in Volt.

**4.5.2**

**Simulations with different tap changer levels**

According to the conclusions in Chapter 4.4.2 no attempt was made to estimate a state space model for the large model. Instead an ad hoc approach for this idea was used. Simulations were made for phase differences of α= 0, 15, 30 and 45 degrees. As mentioned in Chapter 3.2.2, it would be favorable to change the tap changer on the transformer that is in lag. This implies that a first approach should be to lower the tap changer connected to Inlow1. If this is not enough the next transformer that would be changed is the one at Inlow2. The decision strategy was based on minimizing the maximal difference in Pi and then minimize

the maximal difference in the reactive power, Qi, as in Chapter 4.4.2. As seen in

Figure 4.7, these two differences vary a lot. To learn how the system responded to different choices of ¯V a large number of simulations were made. Figure 4.8 shows how the power flow changes when V3 is lowered one level, from −2.5 to −5. As

seen in the figure the effect on the reactive power is stronger than on the active power. To change the tap changer one level generates a large difference in the power flow.

**Figure 4.7. The maximal difference in ¯**P and ¯Q for a number of simulations, that was
used as a guideline to determine a suitable choice of ¯V when α = 45 degrees.

**Figure 4.8. The active, reactive and apparent power when tap changer 3 is changed**

**4.5 Solution two - transformer tap changer** **31**

**4.5.3**

**Model for control system**

According to section 4.4.3 it is not possible to use a feed back system, so the choice would be either a feed forward system or a static setting for the tap changer. A more detailed discussion is presented in Chapter 5.

**Chapter 5**

**Results and conclusions**

This chapter presents the results obtained with the methods described in Chapter 4, discussions concerning the results and the conclusions.

**5.1**

**Results using series reactances**

Under the conditions stated in Chapter 4, the simulations performed with the large model show some positive results. It can be seen that under these conditions the power flow can be controlled. For each angular difference α = 0, 15, 30 and 45 degrees, according to Chapter 4.2.2, a specific combination of reactances ¯x, that contributes to an obvious improvement of the power flow, was found.

In Figures 5.1 and 5.2 the power flow in Inlow1through Inlow6, when α is 30

degrees and when α is 45 degrees, are shown. To illustrate the improvements in the power flow, both the power flow for a suitable choice of reactances and the power flow without compensation for α are included. As seen in the Figures 5.1 and 5.2, a large improvement can be detected in both the active power, P , reactive power, Q and the apparent power, S. As desired the difference in the power production, between the converter stations, has decreased.

The reactances used in these cases are, as stated in Chapter 4.4.1, of the inductive kind. The size of the inductances varies between 10 mH and 45 mH. In Appendix C an approximate dimensioning for a 10 mH air coil is included. When studying the simulations the conclusion were drawn that it would not be necessary to have a continuous compensation for changes in α. As mentioned in Chapter 4.4.3 one alternative could be to do the same compensation for an interval of α, for example all α between 7 and 22 degrees would be controlled with the same set of inductances. This would give four intervals for α.

**Figure 5.1. The active, reactive and apparent power, with and without series reactances,**

in Inlow1 through Inlow6 with α = 30 degrees.

**Figure 5.2. The active, reactive and apparent power, with and without series reactances,**

**5.1 Results using series reactances** **35**

Since there are four intervals for α the principal arrangement for the series inductances would be four different levels for each converter station. For converter station number one this would mean a series connection like the one in Figure 5.3. For small α, where no compensation is needed in converter station number one, both circuit breaker B1 and B2 would be closed. Depending on what interval α

is in, one or both of the circuit breakers can be opened. This means that four different levels of compensation can be achieved. In some of the cases it might be necessary to have more than two inductances in series. An alternative solution

**Figure 5.3. An example of circuit connection using series inductances at rotary converter**

number one.

would be, as mentioned in Chapter 4.4.3, to use the same reactances regardless of α. In this solution an average compensation regarding to the fluctuation of α during a 12-hour or 24-hour period could be chosen. As seen in Appendix B.5, α has small fluctuations during long periods of time and then makes a large change, at the beginning and end of the day. According to this it could be possible to use an adjustment with, for example, α = 30 degrees during the whole day and get a good enough improvement. The simulations made for α = 15 and 45 degrees with the suitable choice of reactances for α = 30 degrees, ¯x30, can be studied in Figure

5.4 and 5.5. As expected the compensation is not as good as when the specific reactances for each α are used, also included in Figure 5.4 and 5.5, but it is still clearly an improvement compared to no compensation at all, see Figure 5.1 and 5.2.

**Figure 5.4. The active, reactive and apparent power flow when α = 15 degrees and**

reactances for α = 15 and 30 degrees respectively are used.

**Figure 5.5. The active, reactive and apparent power flow when α = 45 degrees and**

**5.2 Results for complementary simulations using series reactances** **37**

**5.2**

**Results for complementary simulations using**

**series reactances**

The complementary simulations mentioned in Chapter 4.4.4 show that the system is rather robust, when exposed to small changes.

Changing α in the large model, according to Chapter 4.4.4, changes the power flow in Inlow1 through Inlow6. Simulating the angular difference, on the 50 Hz

grid, as 30 degrees for the three first converter stations and then as a linear decrease down to zero at converter station number six and without any compensation will contribute to a power flow according to Figure 5.6. The second set of values in the figure is the power flow when the same reactances are used as for α = 30 degrees in Chapter 5.1. Even though these reactances are not the optimal choice for compensation an improvement is detected when the two power flows in Figure 5.6 are compared.

**Figure 5.6. The active, reactive and apparent power flow in Inlow**1 through Inlow6,

with and without compensation, when the angular difference is unevenly distributed. In the next set of simulations the large model, described in Chapter 4.2.2, was changed by halving the loads from each train, as described in Chapter 4.4.4. In Figure 5.7 the power flow with and without compensating reactances are shown. The angular difference α = 30 degrees and the reactances used are the same as for α = 30 degrees in Chapter 5.1. As seen in Figure 5.7 the used reactances do not give a perfect compensation, but using reactances suitable for a load twice the size still give a good improvement in power flow.

These two cases give indications that the system is robust and for small changes the same compensation can be used.

**Figure 5.7. The active, reactive and apparent power flow in Inlow**1 through Inlow6,

with and without compensation, when the load from the trains is halved compared to earlier.

**5.3**

**Result using the tap changers on the **

**trans-formers**

From the simulations made under the conditions stated in Chapter 4.5.1 it is obvious that it is possible to affect the power flow using the tap changer, see Figure 5.8. As seen in the figure and according to Chapter 4.5.2 the change will affect the reactive power more than the active power.

Several strategies for how to set the tap changers were used with varying results. For example, attempts were made where the voltage from transformer number six was increased by 2.5% and the voltage from transformer number one was decreased by 2.5%. This showed that the power flow from the converter stations in the middle had no or a only small deviations. In Figure 4.8 in Chapter 4.5.2 a change in the transformer connected to rotary converter number three can be studied. According to this figure the effect of a change in the tap changer level decreases quickly with the distance. This shows that a change in tap changer level almost only affect the nearest converter station and its neighbors.

All the simulations show that it is possible to control and improve the power flow, but that the reactive power, as stated earlier, tends to be more sensitive to the changes than the active power. Since the influences on the active power are small and the simulations indicate that steps between available tap changer levels are too large, when only small adjustments are required, to use only the tap changer levels for a control system is not a good enough solution.

**5.4 Conclusions** **39**

**Figure 5.8. Power flows when changing only the tap changer on the transformer **

con-nected to converter station number 6 by +5%

On the other hand, changing the tap changer level could be useful in the attempt to control individual converter stations which produces too high quantities of reactive power. It might be possible to combine a change in one or some tap changer levels with a control system using series reactances.

**5.4**

**Conclusions**

As stated in the purpose, in Chapter 1.2, the goal of this thesis was to examine the possibility of power flow control on the railway grid, using a feeding line and rotary converter stations. Two of the ideas were examined with varying results.

Under the conditions stated in Chapters 4.1 and Chapter 4.2.2, the first method using series reactances showed positive results. As described, in Chapter 5.1, suitable values for the inductances were found for given phase differences on the 50 Hz grid and two different solutions were presented. One solution uses a feed forward control system with four possible combinations of inductances for each converter station. In the other, a static value, based on the mean phase differences during a 12-hour or 24-hour period of time, was used for the inductances. To achieve inductances of suitable size, between 10 and 45 mH, a combination of air coils in series can be used. In Appendix C the calculations for the approximate number of windings necessary for an air coil with an inductance of 10 mH and

realistic size can be found. Further, the results from Chapter 5.2 indicate that the system is robust and small changes to the system does not need to be accounted for.

The second method, using the tap changer levels of the transformers, connected to the converter stations, did not show as good result as expected. Under the conditions given in Chapters 4.1 and 4.2.2, this would not be a good method, used by itself, to control the power flow. As described closer in 5.3, the steps between the tap changer levels are too coarse and the influence on the active power too small. This does not mean that the tap changer levels can not be used as a way to control the power flow. For individual converter stations that produce a lot of reactive power this method could be a perfect complementary solution.

As stated in this chapter the thesis work has shown that the power flow, under the used conditions, can be controlled. Using the examined methods can lead to a more evenly distributed power production and a better use of the converter stations. In the Chapter 6 some ideas about future work are presented.

**Chapter 6**

**Future work**

The result presented in Chapter 5 indicates the possibility of adjusting the power flow from rotary converters with series reactances. These results are based on the simulations performed on the large model, described in Chapter 4. This model is a simplification and an idealized model of the real system. The next step could be to model an existing section of the railway system, that today is fed by rotary converters, taking into account the varying size of the converters, real distances between converter station, the load from the trains etc. This would be done in order to get more exact values for the series reactances needed and to establish what kind of control system that could be used. Furthermore, the load from the trains needs to be studied more explicitly. It has to be investigated how large the actual load is and how the load varies in time. A study over a 24-hour period or maybe even a week could be of interest. It might also be of interest to further extend the model of the 50 Hz public grid and to closer investigate its effect on the system.

Studying a more realistic model of an existing section of the railway, the solu-tions with series reactances and tap changer levels need to be further examined. The two suggested solutions for series reactances, see Chapter 5, using one or for example four levels of different inductances might work very well. If these re-sults are not satisfactory, it can be an idea to further investigate the possibility of finding a simple estimate of the power flow for a feed back control system. The results for the change in tap changer level in the transformer, see Chapter 5, show that this solution can be used, as a way of controlling feeding points with high production of reactive power. This could also be examined closer. Since a control system consisting of both series reactances and tap changer levels has not been investigated, it can be an area to examine further.

As mentioned in Chapter 5.3 the steps of the tap changer are too large for controlling the influence of α. One suggestion could be to investigate if there exist transformers with smaller and/or more levels. If these exist, the change in tap changer levels might be used in a more extended way than indicated in the result in Chapter 5.

Idea number three, described in Chapter 3.2.3, using an extra rotary or static 41

converter or parametric transformer has not been investigated, due to lack of time. Using one of these three ways to compensate for the transfered phase differences, might be most exact solutions of all.

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**Appendix A**

**Blondel transformation for**

**power flow**

The Blondel transformation, BT , for a three phase system. In this case, it is applied to a one phase system and therefore disregard phase two and three.

**A.1**

**The basic machine parameters**

For a three phase synchronous machine, as in Figure A.1, the Kirchoff’s voltage equations for the four separate circuits are [3]

va= −iars− laadi_{dt}a − labdi_{dt}b− lacdi_{dt}c + lardi_{dt}r

vb= −ibrs− lbadi_{dt}a − lbbdi_{dt}b− lbcdi_{dt}c + lbrdi_{dt}r

vc= −icrs− lcadi_{dt}a − lcbdi_{dt}b − lccdi_{dt}c+ lcrdi_{dt}r

vr= −irrr− lradi_{dt}a − lrbdi_{dt}b − lrcdi_{dt}c+ lrrdi_{dt}r

(A.1)

and in short matrix notation

vs ∆ = va vb vc ⇒ v ∆ = vs vr (A.2)

The stator currents are defined as

is ∆ = ia ib ic ⇒ i ∆ = is −ir (A.3) 45

**Figure A.1. Winding inductance and resistance parameters [3]**

For the three phase synchronous machine all three windings have the same char-acteristic and the resistance and inductance matrices are

R=∆ rs 0 0 0 0 rs 0 0 0 0 rs 0 0 0 0 rr L=∆ laa lab lac lar lba lbb lbc lbr lca lcb lcc lcr lra lrb lrc lrr (A.4)

In the case of a round rotor, all stator self- and mutual-inductance elements are

*independent of the rotor angle, α. This means that the inductance coefficient, L*2,

i.e. the fluctuation in inductance caused by an unsymmetrical rotor, is zero. See Figure A.2. Under this assumption the inductance matrix is

L =
L1 −L3 −L3 L5cos α
−L3 L1 −L3 L5cos α −2π_{3}
−L3 −L3 L1 L5cos α +2π_{3}

L5cos α L5cos α −2π_{3} L5cos α +2π_{3} L4

**A.1 The basic machine parameters** **47**

**Figure A.2. Variation in inductance parameters with angular rotor position [3]**

The voltage Equation A.1 can then be written in a shorter matrix form v = −Ri − d

dt(Li) (A.6)

A few remarks: The second term in Equation A.6 can not be simplified d

dt(Li) 6= L di dt

since the L matrix, as mentioned earlier, depends on α, and therefore the Laplace transform can not be used. Consider a typical term in Equation A.1, for example

d dt(laaia)

By performing the differentiation, following terms are obtained
iadα_{dt} sin 2α

dia

dt cos 2α

(A.7)

In this case the rotor angular velocity, ω, is constant, since the machine has large inertia so that

dα

dt = ω ⇒ α = ωt + α0 (A.8)

The terms in Equation A.7 with Equation A.8 are ω sin 2 (ωt + α0) ia

cos 2 (ωt + α0)

dia

dt

*This gives a linear differential equation with time-varying coefficients. If the speed,*
*ω, would change, the equation would be of the non-linear kind.*

**A.2**

**The general power equation**

The total power from the stator windings is
p = iava+ ibvb+ icvc [W ] (A.9)

**where p is defined positive in a generator sense [3]. If v and u are defined as in****Equations A.2 and A.3 the R and the L matrices are defined as**

R = Rs 0 0 rr L = Ls Irs ITrs lrr (A.10) with components defined in Equation A.4. The power equation is then written

p = iT_{s}vs (A.11)

From here the equation A.6 will be expressed as vs vr = − Rs 0 0 rr is −ir − d dt Ls Irs IT rs lrr is −ir (A.12) This equation can be written in component–form

vs= −Rsis−_{dt}d (Lsis− lrsir)

vr= rrir−_{dt}d lTrsis− lrrirr

(A.13)

It is easy to prove that Equation A.13 equals A.1 just by performing the matrix operations.

**A.3**

**The Blondel transformation**

Earlier it has been indicated that it is possible to obtain analytical solutions to the system of the differential equations in Equation A.1, using the assumption that the speed, ω, is constant [3]. It is possible to simplify these equations to a great extent by means of a special technique, the Blondel Transformation, BT . In the Blondel transformation three new variables or components are introduced, the direct axis, d−, quadrature axis, q−, and the zero-sequence, 0−, component. These Blondel currents are defined

id ∆

= 2_{3}cos αia+2_{3}cos α −2π_{3} ib+2_{3}cos α +2π_{3} ic

iq ∆

= −2_{3}sin αia−2_{3}sin α −2π_{3} ib−2_{3}cos α +2π_{3} ic

i0 ∆

= −1_{3}ia+1_{3}ib+1_{3}ic

(A.14)

and in compact matrix notation iB

∆

**A.3 The Blondel transformation** **49**

*where the Blondel currents are*

iB ∆ = id iq i0 (A.16) B=∆ 2 3 cos α cos α −2π 3 cos α +2π 3 − sin α − sin α − 2π 3 − sin α +2π 3 0.5 0.5 0.5 (A.17)

*Equation A.15 is called the direct Blondel transformation. For the stator voltages*
the direct Blondel transformations are

vB ∆

= Bvs (A.18)

*The associated inverse Blondel transformations for stator currents and voltages*
are

is = B−1iB (A.19)

vs= B−1vB (A.20)

By using methods for matrix inversion the inverse B matrix is

B−1 ∆=
cos α − sin α 1
cos α −2π_{3} − sin α − 2π
3
1
cos α +2π_{3}
− sin α + 2π
3
1
(A.21)

Substitutions of the stator voltages and currents in Equation A.13 give the
follow-ing equations
B−1vB = −RsB−1iB−_{dt}d LsB−1iB− lrsir
vr = rrir−_{dt}d lTrsB−1iB− lrrir
(A.22)

Multiplying Equation A.22 with B, the equations can be written vB = −BRsB−1iB− Bdtd LsB−1iB− lrsir

vr = rrir−_{dt}d lTrsB−1iB− lrrir

(A.23)

It is not obvious that the new Equations A.23 are simplifications of the old
Equa-tion A.13. If the matrix operaEqua-tions are performed in EquaEqua-tion A.23 it is reduced
to
vd= −rsid− Lddi_{dt}d + L5di_{dt}r + Lqiqdα_{dt}
vq = −rsiq− Lq
diq
dt − (Ldid− L5ir)
dα
dt
v0= −rsi0− L0di_{dt}0
vr= rrir+ L4di_{dt}r −3_{2}L5di_{dt}d
(A.24)

**The new parameters for the L matrix are**
Ld
∆
= L1+ L3+3_{2}L2
Lq
∆
= L1+ L3−3_{2}L2
L0
∆
= L1− 2L3
(A.25)

*These are referred to, in analogy with currents and voltages, as the quadrature*

*synchronous inductance etc. In this specific case only the d- and q-components*

are of interest. With a constant rotor speed the differential equations in Equation
*A.24 are linear with constant coefficients. In matrix form*

vd vq v0 vr = − rs −ωLq 0 0 ωLd rs 0 ωL5 0 0 rs 0 0 0 0 rr id iq i0 −ir − Ld 0 0 L5 0 Lq 0 0 0 0 L0 0 3 2L5 0 0 L4 d dt id iq i0 −ir (A.26)

The Blondel transformation has reduced the complexity of Equation A.1 on two points:

*1. it has eliminated the time dependency and the equations have now only *
con-stant parameters. This makes it possible to use Laplace transform analysis
technique.

2. the transformed equations contain less terms than the original. This implies that the parameter matrices, in Equation A.26, have many elements equal to zero. The physical stator currents isare strongly coupled to each other,

but the Blondel currents, iB are only weakly coupled.

The Blondel transformation will also have effect on the power equation, Equa-tion A.9, since

is= B−1iB

By using matrix operations

iT_{s} = iT_{B} B−1T

(A.27) used in Equation A.9

p = iT_{B} B−1T

B−1vB (A.28)

and by performing the matrix operation

B−1T
B−1=
3
2 0 0
0 3_{2} 0
0 0 3
(A.29)