DC traction load flow including AC distribution network

www.ietdl.org Published in IET Electric Power Applications Received on 30th June 2008 Revised on 13th November 2008 doi: 10.1049/iet-epa.2008.0147

ISSN 1751-8660

DC traction load flow including AC distribution network C.L. Pires S.I. Nabeta J.R. Cardoso LMAG – Laborato´rio de Eletromagnetismo Aplicado, PEA – Departamento de Engenharia de Energia e Automac¸a˜o Ele´tricas, Escola Polite´cnica da Universidade de Sa˜o Paulo, Av. Prof. Luciano Gualberto, tr.3, no 158, Pre´dio da Engenharia Ele´trica, Sala A2-17 CEP 05508-900 Sa˜o Paulo, SP, Brazil E-mail: [email protected]

Abstract: In the first part some information and characterisation about an AC distribution network that feeds traction substations and their possible influences on the DC traction load flow are presented. Those influences are investigated and mathematically modelled. To corroborate the mathematical model, an example is presented and their results are confronted with real measurements.

Nomenclature G

DC nodal conductance, S

Id

DC nodal current, A

IS Ntps

per-phase AC busbar current, A ratio of transformation, 1

Pd PS

DC power, kW per-phase busbar active power, kW

QS

per-phase busbar reactive power, kVAr

R RSS

resistance, V equivalent rectifier substation resistance, V

SS Ud

per-phase busbar apparent power, kVA DC nodal voltage, V

Ud0 US

rectifier no-load voltage, V per-phase AC busbar voltage, V

Y

AC busbar admittance, S

u l

firing angle, degree total power factor, 1

1

Introduction

Power is transmitted to electric railway and trains using DC or single-phase AC networks. The parallel development of traction technology has led to a plethora of different electrification systems [1]. IET Electr. Power Appl., 2009, Vol. 3, Iss. 4, pp. 289– 297 doi: 10.1049/iet-epa.2008.0147

The fundamental choice between AC and DC transmission to the trains is related to safety, transmission efficiency and economics of the conductor system. For street running, safety requires voltages of around 750 V, and also in underground tunnels it is not possible to provide the safe clearances needed to operate at higher voltages, as required with AC systems [2]. If voltage is constrained, the current becomes proportionally large and the series impedance of the conductor system becomes important. The AC impedance is considerably higher than the DC impedance. Therefore the voltage loss is higher in a low-voltage AC system than in a DC one [2]. All electric traction power supplies whether they use AC or DC for the final supply to the trains via some form of conductor system along the track, supply power to the track from regularly spaced track-side substations. In almost all cases, this power will be delivered from the industrial supply at a level within the grid system that is appropriated to the power being drawn [2]. On DC networks, power is derived by trackside rectifier substations, from the AC three-phase utility supply (most common) or railwaygenerated supply. In the first case, utility supplied energy, usually at distribution voltage level, is transformed, rectified and connected to the contact line and track at specific points [1]. 289

& The Institution of Engineering and Technology 2009

Authorized licensed use limited to: UNIVERSIDADE DE SAO PAULO. Downloaded on July 10, 2009 at 09:09 from IEEE Xplore. Restrictions apply.

www.ietdl.org However, on urban mass transit systems, the distance between rectifier substations is relatively short and the connection to the AC three-phase utility supply is not always available. It is therefore necessary in most of the cases for the mass transit system authority to provide its own high-voltage AC distribution network [3]. In some cases, this distribution network also supplies the passenger stations and other loads. The presence of this distribution network interfacing the AC three-phase utility supply and the DC traction network causes two basic difficulties to AC load flow and DC traction simulation: † For AC load flow, rectifier substations are constantly changing their loads because of trains; † For DC traction simulation, the presence of a distribution network implies a variable voltage at the AC side of rectifier busbars. In the first case, AC load flow is necessary not only for system design or expansion. It is a powerful tool to predict the impact of changes on the operation of trains (e.g. headway, number of trains, maximum train speed), on energy consumption and on peak demand at AC threephase utility connection points. In the second case, a variable voltage at the AC side of rectifier busbars indicates different output voltages of each rectifier, even if they have the same load and connection. This output voltage difference among rectifiers will modify the load distribution on the DC traction network and will also interfere on system receptivity during the regenerative braking of trains. Obviously, the unified AC/DC load flow for traction system simulation has been the main objective of some researches. Talukdar and Koo [4] published a stationary equivalent method for moving loads, which results in a set of nonlinear algebraic equations. In the same manner, Tzeng et al. [5, 6] proposed unified calculation method where the AC and the DC system equations are combined together in one set of equations to be solved simultaneously. Out of traction applications, the AC/DC load flow procedures presented in [7 – 9] are worthy of mention. In the network equation solver, standard load flow procedures using the conventional algorithms, such as Gauss – Seidel and Newton– Raphson methods, need to be modified to cater to the nonlinearities of the traction systems, such as regenerative trains and system nonreceptivity [10]. However, the introduction of some refinements on the DC traction network model capable of handling multi-branched networks, and incorporating a detailed model of the return 290 & The Institution of Engineering and Technology 2009

circuit [11 – 15] may cause a problem to previously described unified AC/DC load flow techniques. The traction return circuit is modelled by dividing the track into a number of finite cells that are generally equal in length. Each cell forms a pi-circuit for two-earth case [11 – 13] and a double pi-circuit for three-earth case [14, 15], according to EN 50122-1 standard [16]. Those cells are composed of series resistances and shunt leakage conductances to represent the distributed parameters. By introducing this refinement, the quantity of circuit nodes is greatly increased and the resistances between circuit nodes span too wide a range of magnitudes, which would cause the convergence failure of modified load flow algorithms [11, 12]. Powell [17] also reminds us that ill-conditioned systems whose admittance matrix is sparsely populated and the magnitude of diagonal elements is reduced will affect the various techniques of load flow analysis by way of slow convergence, slow oscillation of successive solutions or even divergence. For those reasons, the best solution for the unified AC/ DC load flow for traction system simulation would be performing a standard AC load flow separated from DC traction simulation and exchange the results iteratively at each step until convergence. This approach would prevent convergence failure of the AC load flow and could maintain a detailed model of the DC traction network. Such an idea was mentioned by Shao et al. [18] and briefly described by Hofmann et al. [19] and it is the purpose of this paper. After all those considerations, it can be observed that the DC traction simulation results may be more accurate when the presence of the AC distribution network is considered. However, it can also be observed that the earlier developed models for a unified AC/DC load flow do not take into account some details of the DC traction network. This paper proposes a novel approach for the unified AC/ DC load flow where the entire system is solved together and the details of the DC traction network are not lost. Before the development of this approach, it is interesting to recall the DC traction network model as well the AC load flow.

2

DC traction network

The DC traction network can be modelled by defining nodal voltage equations relating the node conductances and voltages to node currents. The components of this network are modelled in terms of resistances and current sources, and the interconnection of these components constitutes a nodal electric circuit [11]. The trains do not move rapidly enough to induce pronounced electrical transients and the DC traction IET Electr. Power Appl., 2009, Vol. 3, Iss. 4, pp. 289– 297 doi: 10.1049/iet-epa.2008.0147

Authorized licensed use limited to: UNIVERSIDADE DE SAO PAULO. Downloaded on July 10, 2009 at 09:09 from IEEE Xplore. Restrictions apply.

www.ietdl.org network can be assumed to move slowly from one state to another as the locations and input power of the trains vary. Therefore a comprehensive diagnosis of a DC traction network performance can be obtained by running a series of simulations, one for every instant of time [4].

21], the L-U factorisation technique [18], the Zollenkopf bifactorisation [11, 12] and the ICCG (incomplete Cholesky conjugate gradient) [15, 22].

A rectifier substation is represented by a The´venin equivalent voltage source (constant voltage equal to the noload substation voltage and a resistance representing the slope of the rectifier Ud(Id) characteristics), which is converted within the model into a Norton equivalent current source [11, 20].

3

According to Goodman and Siu [20], the model representation of the trains may be a pure voltage source in series with a resistor or a constant current source in both the powering and braking modes. However, the relationships between the terminal voltage and input current make the problem nonlinear and the solutions must be iterative in nature. Cai et al. [12] concluded that the constant current model is preferred because it takes less time to solve power and voltage constraints and can more easily incorporate train regenerative braking. As described earlier, the traction return circuit can be divided into a number of sections longitudinally called finite cells. An equivalent pi or double pi circuit is derived to represent the distributed parameters for each finite cell and a network consisting of lumped parameters is constructed [13 – 15]. Fig. 1 shows an example of a typical traction circuit. A solution of DC traction network to find all nodal voltages essentially involves the solution of the following linear algebraic equation for each i node of a network composed by n nodes 9 8 n = X 1 < Idi  Gik  Udk Ud i ¼ ; Gii : k¼1

(1)

k=i

A number of methods are available to solve systems of linear algebraic equations. Examples of tested methods for solving linear systems are the Cholesky matrix decomposition [20,

AC load flow

The load flow problem consists of the calculation of power flows and voltages of a network for specified terminal or busbar conditions. Associated with each busbar are four quantities: the real and the reactive power, the voltage magnitude and the phase angle. At each busbar, two of four quantities are specified. The combination of those quantities produces three types of busbars: generator busbar, load busbar and infinite busbar [17, 23]. Associated with each busbar are four quantities: the perphase active and reactive powers, per-phase voltage magnitude and phase angle. The per-phase active power and per-phase voltage magnitude are specified at generator busbars. Per-phase active and reactive powers are specified at a load busbar. Finally, per-phase voltage and magnitude are specified at infinite busbars [17, 23]. In the load flow calculation of an AC distribution network feeding a DC traction network, load busbars and infinite busbars may be recognised. Load busbars may be divided into two types: constant and variable load busbars. Constant load busbars have a fixed load during the simulation process and may represent passenger station loads. On the other hand, variable busbars change their loads at each instant and may represent rectifier loads which are not constant because of the position of trains and power variation. The AC three-phase utility connection points are represented by infinite busbars. Their voltage magnitude and phase angle are specified to serve as an origin for all the other busbars and those quantities remain fixed during the simulation process. After the solution of the load flow problem has been obtained, the power at AC three-phase utility connection points can be calculated for each instant. Energy and peak demand can be obtained through postprocessing and further result analysis can be drawn from those series of simulations. The fundamental equations for the ith busbar of a network composed of n busbars are [17, 23]

I Si ¼

n X

Yik  USk

(2)

k¼1

and Figure 1 Example of a double-track DC traction network with two-earth return circuit IET Electr. Power Appl., 2009, Vol. 3, Iss. 4, pp. 289– 297 doi: 10.1049/iet-epa.2008.0147

SSi ¼ USi  ISi

(3) 291

& The Institution of Engineering and Technology 2009

Authorized licensed use limited to: UNIVERSIDADE DE SAO PAULO. Downloaded on July 10, 2009 at 09:09 from IEEE Xplore. Restrictions apply.

www.ietdl.org drops are very small in comparison with the reactive voltage drop [24, 26].

whose combination results in the equation 8 9  n = 1 < SSi X USi ¼  Yik  USk ; Yii :USi k¼1

(4)

k=i

Ud ¼ Ud0  (u)  Id  RSS

where SSi ¼ PSi þ j  QSi

(5)

for load busbars. It may be observed in the previous sections that the only common elements in both AC and DC networks are the rectifier substations. In fact, rectifiers play an important role in developing the unified AC/DC load flow and their model for the AC and DC sides will be detailed.

4

The basic relationship between the no-load voltage and the output voltage of the rectifier substation is given by [26, 27]

Rectifier substations

In Section 2, it was pointed out that a rectifier substation is represented on a DC network by a The´venin equivalent voltage source and a series resistance. This equivalent circuit of a rectifier substation is used to describe the characteristic of rectifiers as they are loaded, that is, it describes the drop in output voltage with load relative to the no-load condition. The output voltage of a rectifier is reduced by the reactive and resistive voltage drop (disregarding phase control). There are three main sources contributing to the output voltage drop [24, 25]: † the voltage drop across the diodes and/or thyristors R1 (resistive), † the resistance of the AC supply source and conductors R2 (resistive) and † the AC supply source inductance R3 (reactive). These three voltage drops are represented by a series resistance (RSS) in the equivalent DC circuit shown on Fig. 2. The voltage drop because of AC supply inductance is the overlap effect and it is the most important resistance. That is because during commutation, the rates of change of the thyristor currents are so great that the resistive voltage

(6)

It is important to note that this equivalent DC circuit is valid only for an overlap angle smaller than 608, which excludes short-circuit studies. For those cases, the relationship between the no-load voltage and the output voltage of the rectifier substation is no longer governed by (6), but by other equations [24, 27]. The instantaneous direct voltage is composed of arcs of alternating line-to-line voltages whose duration depends on the rectifier connection. The average direct voltage Ud0 is found by integrating the instantaneous voltages over such a period [27]. If the transform is assumed to be ideal with a turns ratio, that is secondary windings turns to primary windings turns per phase, of Ntps , and the voltages on the AC side are balanced and sinusoidal, then the following relationship holds Ud0 ¼ K  US  Ntps

(7)

where K is a function of the rectifier connection and its value is shown on Table 1. It was observed that a rectifier connection represents a special kind of load to an AC supply system. According to Schaefer [24], an extremely useful method to establish a relation between AC and DC quantities is to assume the condition that input and output power must be in balance at any instant. For deriving this relation, it is also assumed that the rectifier connection is not able to absorb or store any amount of energy and still, according to Schaefer [24], this is true under the following conditions: † No heat is dissipated and no energy is stored in the leakage field of the transformer or the magnetic field of the busbars; † All the connections with unbalanced ampe`re-turns are excluded so that no energy is stored in the magnetic field; † The transformer exciting current is disregarded so that no energy is stored in the core of the transformer. Further, assumptions are that the direct current is without ripple (regular operation), and the alternating voltages are precisely defined [24]. With losses neglected, the active AC power equals the DC power as

Figure 2 Equivalent DC circuit of rectifier substation 292 & The Institution of Engineering and Technology 2009

PS ¼ Pd

(8)

IET Electr. Power Appl., 2009, Vol. 3, Iss. 4, pp. 289– 297 doi: 10.1049/iet-epa.2008.0147

Authorized licensed use limited to: UNIVERSIDADE DE SAO PAULO. Downloaded on July 10, 2009 at 09:09 from IEEE Xplore. Restrictions apply.

www.ietdl.org Table 1 Rectifier connections Connection

K pffiffiffi 3 2=p pffiffiffi 3 6=2  p pffiffiffi 3 6=p pffiffiffi 6 6=p pffiffiffi 3 6=p

diametrical six-phase double-Wye with inter-phase transformer three-phase bridge twelve-pulse (cascade connection) twelve-pulse (parallel connection)

3=p 3=p 3=p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1=( 1=18 þ 3=36  p) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1=( 1=18 þ 3=36  p)

of (1) for every node. This first solution has then to be checked for violation of nonlinear constraints [21].

where PS ¼ 3  U S  I S  l

(9)

and Pd ¼ U d  I d

(10)

The rectifier AC power may be calculated through the total power factor l which, according to IEC 60146-1-1 standard [28], is the ratio of the active power to the apparent power

l¼

PS SS

(11)

Values of the total power factor l for each rectifier connection commonly used in traction applications is also shown on Table 1, whose values were extracted from [24 – 27]. It may be observed in Table 1 that the first three connections may be found in older electrification systems. In newer projects, twelve-pulse connections are preferred. Once the equations for the AC and DC network and rectifier substation models for both the AC and DC sides are derived, the next section will describe the integration process between the AC and DC load flows.

5

Total power factor l

These constraints are associated either with the rectifier substations or the trains. Rectifier substations cannot accept power (current) and thus if, after a circuit solution this situation arises, the solution has to be declared invalid and a new solution obtained with the rectifier substations switched off. Also, the regenerating train voltages have to be examined for exceeding prescribed limits, and if found to be so, the train model must be changed to represent this non-receptivity condition [12, 21]. When an accepted circuit solution is obtained, the instantaneous DC power for every rectifier substation is calculated using (10). Instantaneous active and reactive powers at the AC side of rectifiers are also calculated through (8), (11) and (5). AC load flow calculation can then be performed. The solution of (4) for each rectifier busbar is applied on (7) and the new value for the no-load voltage of the rectifier updates the rectifier substation model for the DC traction load flow using (6). This iterative process between the AC and DC load flows is continued until changes in all AC side rectifier busbar voltages are negligible. After the AC voltage solution has been obtained, the instantaneous power at the infinite

Integration process

During the simulation process, it is necessary to model the running of the trains and also to establish the sequential states of the DC traction network. The simulation is timebased and it is assumed that time proceeds in discrete steps. At the start of each update period, the positions and consumed or generated power of all trains are known, according to a previously calculated performance and traffic simulation. Also, the AC side voltages of rectifier substations busbars are set to their nominal values. Once all train positions and currents are known, it is possible to begin to assemble a linear equivalent circuit representation of a DC traction network. The solution of this circuit is given through nodal analysis and the solution IET Electr. Power Appl., 2009, Vol. 3, Iss. 4, pp. 289– 297 doi: 10.1049/iet-epa.2008.0147

Figure 3 Flowchart 293

& The Institution of Engineering and Technology 2009

Authorized licensed use limited to: UNIVERSIDADE DE SAO PAULO. Downloaded on July 10, 2009 at 09:09 from IEEE Xplore. Restrictions apply.

www.ietdl.org

Figure 5 Detail of ‘station’ blocks Figure 4 Simplified AC and DC network busbars are calculated using (4), the time step is incremented and a new update period is started. An outer loop can be performed to update the voltage values along the tracks and the train performance can be recalculated. The integration process may be summarised in the algorithm in Fig. 3.

6

Application

To illustrate the previous theoretical considerations, the AC/ DC load flow of Line 3 (Red) of the Sa˜o Paulo Metro, Brazil, will be calculated.

6.1 Input data The Line 3 of the Sa˜o Paulo Metro is the most loaded line in the system. It has a length of 46.4 km, 18 passenger stations, and the maximum number of trains is 42 during rush hours (6:30–8:30 am and 5:30–7:30 pm). Those 42 trains intend to provide 57 400 seats/hour with a minimum headway of 101 s. The average weight of the trains’ is 344 metric tons, their maximum acceleration is 1.12 m/s2 and their maximum deceleration is 1.2 m/s2. However, for comfort reasons, the real deceleration value is between 0.67 and 0.85 m/s2. Those trains are driven by a chopper-controlled system whose nominal power is about 6600 kW distributed on 24 motors installed on all train axes.

The nominal voltage of the DC system is 750 V and it is fed by 18 twelve-pulse (parallel connection) rectifier substations placed close to passenger stations. The nominal DC power of those substations is 4250 kW and their noload voltage is 825 V. The rectifier substations are fed by an AC 22 kV network that also feeds the passenger stations. This network is supplied by four primary substations, which have two transformers of 20 MVA and which receive energy from the public utility at the level of 88 kV. Fig. 4 shows a simplified diagram of the AC and DC network. The connections in grey are activated only when one primary substation is out of operation. Fig. 5 shows the details of three ‘station’ blocks of Fig. 4 connected to a YVP primary substation.

6.2 Results The previously described system was simulated on a software for DC traction simulation developed in LMAG (Applied Electromagnetism Laboratory) [22] and written on Cþþ language. The simulation scenario was the morning rush hours (6:30– 8:30 am), 42 trains and headway of 101 s. To validate the model for primary substations and the AC network, a comparison was made a between simulated and measured values of the mean active power during the morning rush hours. Those measures were taken from the registered values during morning rush hours at all primary substations in

Table 2 Simulated and measured values

total power factor at YPS, 1 active power at YBF, kW

a

Measured

Simulated

Error, %

0.99

0.98

21.41

5672.29

210.35

6259.63a

active power at YPS, kW

13 551.48

14 730.80

8.01

active power at YTA, kW

12 808.44

14 273.20

10.26

active power at YVP, kW

10 333.80

11 677.10

11.50

Estimated value for this line

294 & The Institution of Engineering and Technology 2009

IET Electr. Power Appl., 2009, Vol. 3, Iss. 4, pp. 289– 297 doi: 10.1049/iet-epa.2008.0147

Authorized licensed use limited to: UNIVERSIDADE DE SAO PAULO. Downloaded on July 10, 2009 at 09:09 from IEEE Xplore. Restrictions apply.

www.ietdl.org

Figure 6 Results for train performance simulation, track 1

Figure 7 Results for train performance simulation, track 2

the month of March 2008. The measures at primary substation YBF were estimated because it feeds Line 3 and also Line 2 (Green) of the Sa˜o Paulo Metro, and the measurements show the global value. This estimation was calculated according to several records of load distribution between those two lines. Results of this comparison are shown on Table 2.

It should be observed that the mean load for each passenger station, the value of which were taken from registers during the last year (2007) it was considered. Those values are important because the estimated values for station loads would increase the error between the measured and simulated values on Table 2.

Figure 8 Minimum voltage over the train along track 1 IET Electr. Power Appl., 2009, Vol. 3, Iss. 4, pp. 289– 297 doi: 10.1049/iet-epa.2008.0147

295

& The Institution of Engineering and Technology 2009

Authorized licensed use limited to: UNIVERSIDADE DE SAO PAULO. Downloaded on July 10, 2009 at 09:09 from IEEE Xplore. Restrictions apply.

www.ietdl.org

Figure 9 Minimum voltage over the train along track 2

It is important to note that the error encountered in Table 2 is within the 10– 15% of accuracy mentioned by Hetherington and White. This occurs because on multitrain simulations, there is less control on the real-world activities and the accuracy slips to the region of 10 – 15% as real-life variances start to take effect [29]. Figs. 6 and 7 show the train performance simulation for both tracks of Line 3. They also show the voltage over the train during its running. This voltage was calculated, as mentioned at the end of Section 5, as an outer loop of the AC/DC load flow and the train performance was calculated considering a variable voltage on the contact line. To demonstrate the influence of the AC distribution network on the DC traction network, two simulations, besides that of the morning rush hours, were performed. The first one was performed taking the primary substation YPS out of operation. The second simulation was run taking the primary substation YVP out of operation. The results for a minimum voltage over the train along the line for both tracks are shown in Figs. 8 and 9.

7

Conclusion

This paper presented a method to include the AC distribution network that feeds the rectifier substations on a DC traction load flow. This method, different from earlier approaches, performs the AC/DC unified load flow by separating both systems and exchanging information between them iteratively. The results presented in an example of application have shown that the voltage differences among the AC side rectifier busbars reflect on the DC side of the rectifiers. This influence affects the results of the DC traction load flow and should be taken into account whenever it is possible. 296 & The Institution of Engineering and Technology 2009

8

Acknowledgments

The authors would like to thank Prof. Dr. Jose´ Augusto Pereira da Silva from the Sa˜o Paulo Metro for providing the input data and the measurements at the substations. The work reported in this paper was supported by the National Council for Scientific and Technological Development (CNPq) – Brazil.

9

References

[1] HILL R.J.: ‘Electric railway traction part 3: traction power supplies’, Power Eng. J., 1994, 8, (6), pp. 275– 286 [2] GOODMAN C.J.: ‘Overview of electric railway systems and the calculation of train performance’. The 9th Institution of Engineering and Technology Professional Development Course on Electric Traction Systems, November 2006, pp. 1–24 [3] WHITE R.D.: ‘AC/DC railway electrification and protection’. The 9th Institution of Engineering and Technology Professional Development Course on Electric Traction Systems, November 2006, pp. 281– 322 [4] TALUKDAR S.N., KOO R.L.: ‘The analysis of electrified ground transportation networks’, IEEE Trans. Power Appar. Syst., 1977, 96, (1), pp. 240– 247 [5] TZENG Y.-S., WU R.-N., CHEN N.: ‘Unified AC/DC power flow for system simulation in DC electrified transit railways’, IEE Proc., Electr. Power Appl., 1995, 142, (6), pp. 345 – 354 [6] TZENG Y.-S., CHEN N., WU R.-N.: ‘A detailed R-L fed bridge converter model for power flow studies in industrial AC/ DC power systems’, IEEE Trans. Ind. Electron., 1995, 42, (5), pp. 531– 538 [7] BRAUNAGEL A.D., KRAFT L.A., WHYSONG J.L.: ‘Inclusion of DC converter and transmission equations directly in a IET Electr. Power Appl., 2009, Vol. 3, Iss. 4, pp. 289– 297 doi: 10.1049/iet-epa.2008.0147

Authorized licensed use limited to: UNIVERSIDADE DE SAO PAULO. Downloaded on July 10, 2009 at 09:09 from IEEE Xplore. Restrictions apply.

www.ietdl.org Newton power flow’, IEEE Trans. Power Appar. Syst., 1976, PAS-95, (1), pp. 76– 88

and Operation in the Railway and Other Mass Transit Systems, 1994, pp. 551– 558

[8] TYLAVSKY D.J., TRUTT F.C.: ‘The Newton – Raphson load flow applied to AC/DC systems with commutation impedance’, IEEE Trans. Ind. Appl., 1983, IA-19, (6), pp. 940 – 948

[19] HOFMANN G. , RO¨ HLIG S. , MOM A.: ‘Berechnung von DCBahnnetzen mit einbezogenen AC-Versorgungsnetz’, Elektr. Bahnen, 2000, 98, (7), pp. 254– 258

[9] TYLAVSKY D.J.: ‘A simple approach to the solution of the ac-dc power flow problem’, IEEE Trans. Educ., 1984, E-27, (1), pp. 31– 40

[20] GOODMAN C.J., SIU L.K. : ‘DC railway power network solutions by Diakoptics’. Proc. IEEE/ASME Joint Railroad Conf., 1994, pp. 103 – 110

[10] GOODMAN C.J., SIU L.K., HO T.K.: ‘A review of simulation models for railway systems’. IEE In. Conf. Develop. Mass Trans. Syst., 1998, pp. 80 – 85

[21] GOODMAN C.J., MELLITT B., RAMBUKWELLA N.B.: ‘CAE for the electrical design of urban rail transit systems’. Proc. Int. Conf. Computer Aided Design, Manufacture and Operation in the Railway and Other Mass Transit Systems, 1987, pp. 173 – 193

[11] CAI Y., IRVING M.R., CASE S.H.: ‘Modelling and numerical solution of multibranched DC rail traction power systems’, IEE Proc., Electr. Power Appl., 1995, 142, (5), p. 323 – 328 [12] CAI Y., IRVING M.R., CASE S.H.: ‘Iterative techniques for the solution of complex DC-rail-traction systems including regenerative braking’, IEE Proc., Gener. Transm. Distrib., 1995, 142, (5), pp. 445– 452 [13] YU J.G., GOODMAN C.J.: ‘Modelling of rail potential rise and leakage current in DC rail transit systems’. IEE Colloquium on Stray Current Effects of DC Railway and Tramway, 1990, pp. 2/2/1 – 2/2/6 [14] SILVA J.A.P., CARDOSO J.R., ROSSI L.N.: ‘A fourth differentialintegral formulation applied to the simulation of subway grounding systems’, Electr. Power Compon. Syst., 2002, 30, (4), pp. 331– 343 [15] PIRES C.L., NABETA S.I., CARDOSO J.R.: ‘ICCG method applied to solve DC traction load flow including earthing models’, IET Proc., Electr. Power Appl., 2007, 1, (2), pp. 193– 198 [16] EN 50122-1: ‘Railway applications – fixed installations Part 1’, ‘Protective provisions relating to electrical safety and earthing’ (CENELEC, Brussels, 1998) [17] POWELL L.: ‘Power system load flow analysis’ (McGrawHill, 2004) [18] SHAO Z.Y., CHAN W.S., ALLAN J., MELLITT B.: ‘A new method of DC power supply modelling for rapid transit railway system simulation’. Int. Conf. Computer Aided Design, Manufacture

IET Electr. Power Appl., 2009, Vol. 3, Iss. 4, pp. 289– 297 doi: 10.1049/iet-epa.2008.0147

[22] PIRES C.L. : ‘Railway and subway electric traction system simulation (in Portuguese)’. PhD Thesis, Escola Polite´cnica da Universidade de Sa˜o Paulo, 2006 (http:// www.teses.usp.br/teses/disponiveis/3/3143/tde-22042007212920/) [23] STAGG G.W., EL-ABIAD A.H.: ‘Computer methods in power system analysis’ (McGraw-Hill Kogakusha, Tokyo, 1968) [24] SCHAEFER J.: ‘Rectifier circuits: theory and design’ (John Wiley and Sons, NY, 1965) [25] LANDER C.W.: ‘Power electronics’ (McGraw-Hill, Maidenhead, 1987, 2nd edn.) [26] BARTON T.H.: ‘Rectifiers, cycloconverters, and AC controllers’ (Oxford University Press, Oxford, 1994) [27] KIMBARK E.W.: ‘Direct current transmission’ (John Wiley and Sons, NY, 1971) [28] IEC 60146-1-1: ‘Semiconductor convertors – general requirements and line commuted convertors Part 1-1’, ‘Specifications of basic requirements’ (IEC, Gene`ve, 1991) [29] HETHERINGTON D., WHITE G.: ‘Understanding power supply from a railway operating company’s perspective’. The 3rd IET Professional Development Course on Railway Electrification Infrastructure and Systems, May 2007, pp. 25– 34

297

& The Institution of Engineering and Technology 2009

Authorized licensed use limited to: UNIVERSIDADE DE SAO PAULO. Downloaded on July 10, 2009 at 09:09 from IEEE Xplore. Restrictions apply.