Current Distortion Evaluation in Traction 4 Q Constant Switching Frequency Converters

The paper deals with the power quality analysis of interlaced four quadrant (4Q) converters with constant switching frequency. These are in fact the input stages of the locomotives and high speed trains supplied by 25 kV, 50 Hz and 15 kV, 16.7 Hz lines. Due to the high power needed for the trains circulation, the 4Q converter can absorb distorted currents, whose harmonic content can affect the signaling systems and communication devices. The presence of more converters gives the opportunity, using dedicated commutation strategy, to interlace them in order to reduce the harmonic content in the absorbed current. In the paper a suitable model of more 4Q converters is developed. The control logic implemented in the model allows the evaluation of the harmonic contribution of both single converter and the interlaced configuration. The analysis is carried out through electromagnetic transient simulations.


Introduction
The four quadrant converter (4Q) is the actual best choice to supply the DC voltage link from AC power contact line.A typology of distributed power high speed electric trains having as input stages more 4Q converters [1] has been considered.
The presence of more converters is necessary to guarantee a good redundancy in case of failure and gives the opportunity, using dedicated control logics, to interlace them in order to reduce the harmonic content of the absorbed current.The high power requested by the train for its acceleration in starting phase and for the auxiliary services needs high power converters that do not allow high switching frequencies.Consequently the absorbed current presents a high ripple value characterized by high harmonic current components that cannot be tolerated by the system.Indeed the track circuit used for signaling and communication for the traffic management and safety employs signal currents overlapped with the power ones.These currents can have low frequencies (50 Hz and 178 Hz) in the traditional signaling system, or they are in the audio frequencies range for the new European ERTMS/ETCS one.Therefore, the harmonics produced by the 4Q converter can disturb the communications of the track circuit, degrading the safety of the trains circu-lation.On the other hand the 4Q converter has the main benefit to give a nearly sinusoidal line current in both directions of energy flow and the mitigation of reactive power drawn from the line.In fact the 4Q converter is based on the use of forced commutation switches (GTO, IGBT) and presents a sinusoidal current absorption in phase with the contact line voltage.Moreover, this converter is intrinsically bidirectional and then it can be used both for traction and regenerative braking phases.
The aim of this paper is an analysis of the current absorbed by the high speed trains through a suitable model of more 4Q converters.Thanks to a control logic applied in this work, it is possible to interlace two or more converters in order to evaluate the harmonic contribution of both single converter and the interlaced configuration.The simulation results obtained with an electromagnetic analysis will be presented.

Mathematical Model of the 4Q Converter
The principle scheme reported in Figure 1 shows how the four-quadrant converter is structurally equal to a single phase voltage source inverter and employs the same switches used for the motor drives.This is an advantage because the various branches can have a modular construction [2].The system includes:  he transformer secondary side;  four switches (GTO or IGBT), T 1 ÷ T 4 ;  four freewheeling diodes D 1 ÷ D 4 ;  a DC link with capacitive middle circuit C between the terminals AK and working at the imposed voltage V d =cost;  a second harmonic filter L 2 -C 2 placed downstream the main bridge, tuned to a frequency f 2 =2•f 1 double to the line one.The system has two main purposes.The first one is the absorption from the contact line, at a voltage e 1 and frequency f 1 , of a current having the fundamental harmonic i 1 in phase with e 1 and with low harmonic content, in order to respect the following conditions: The second purpose is the absorption from the line of a power with a mean value P 1 , pulsing at the frequency 2•f 1 and the supply of the three-phase motor drive inverters connected to the dc link with a continuous power P d .
In order to study the behaviour of this converter, the modelling process starts by a mathematical representation of the discrete operation modes of the converter.The discrete model describes each working mode through separate equations.Figure 2 shows a simplified representtation of the two-level converter from which the discrete model is derived.In the equivalent circuit of Figure 2, the transformer secondary side is represented by:  an ideal voltage generator (e 2 =e 1 /h, where h is the transformer ratio);  an inductance L, equivalent to the leakage transformer one;  a resistances R s mainly due to the switches;  load resistances R 0 ;  DC link capacitance C. Considering the state variables i s and v c , the converter state equations are the following: 2 sin( ) where ω is the angular frequency.Rewriting these equations in matricial format, it comes out: or where x is the state vector and is its time derivative.
x  Taking into account that the converter is a two level type, there is another possible operation mode, described by the following matrix A: Looking at the matrix A, it is possible to note that in these two operation modes the first element of the first row and the second one of the second row are the same.Therefore, the state equation can be rewritten to: where  depends from the operation mode and can assumes the value 1 or -1.
In this analysis, the switches resistance R s has been neglected.
The transformer leakage reactance depends by the contact line frequency: and it causes a lagging phase shift of an angle ψ between the converter voltage 2 V and the supply one 2 E .Therefore the first harmonic component of voltages and current become: where the rms value of the AC voltage v 2 is related to th 7) e DC one V d through a proportional coefficient: (8) The a whe DC link voltage of the single-phase converter has significant ripple component at twice the supply frequency.In fact, as it is possible to note in Figure 1 and reported in [3], the current i given by the 4Q converter to the middle circuit is composed by two components: is the direct component of the i(t) r    esponsible of the power absorption, and is an harmonic at the frequency f 2 =2•f 1 , with the following rms value: The i s (t) is absorbed by the d at orption the input current edicated filter L 2 -C 2 tuned frequency f 2 so that into the dc section that feeds the motor inverter flows the only continuous component I d .
Referring to the control, a smart modulation [4] has been applied, where the input current follows a suitable sinusoidal reference in order to have a sinusoidal absorption, as explained in the following paragraph.

Current Modulation
In order to have a sinusoidal abs follows a suitable sinusoidal reference.
The AC reference current is obtained by multiplying the AC line voltage with a suitable equivalent conductance, in accordance with where i is the phasor of the AC phase current and 1 v is the fun amental component of the AC contact line voltage.
The AC voltage contains the fundamental component an d d the component v" corresponding to the perturbations present in the AC line (i.e.harmonics), the Park vector of the AC voltage being Thus, the instantan ve eous real power, expressed as Park ctor, is given by where is the complex conjugated value of id th e single phase converter, the AC -DC power balance is given by Cons ering null power losses associated wi th and the DC current, neglecti pressed with ng the ripple, can be ex- Considering the curren a non-linear relation between the DC voltage v , the contro ts flowing through the DC bus, dc l variable G and the load current i load is obtained Since V 1 can be considered constant, the (17) can be linearized as follows: d represents a perturbation, due to the AC line disturbances and the DC ripple.

Its transfer function is
For the DC voltage control, a proportional-integral (PI) controller has been chosen.

 
where F(s) is the transfer functio required for reducing the DC bus ripple.

puting the value n of the low pass filter
The Equations ( 18) and (19) constitute the closed loop of the DC voltage control that allows com s of the PI parameters.In a first approximation the transfer function F(s) can be neglected, obtaining The denominator, considering a damping ratio of 0.707, constrains the PI controller parameters to respect the following relations:

Model of the System
retical analysis above dehe system has been imple-In order to validate the theo scribed, a suitable model of t mented in the EMTP-ATP dynamic simulation tool.The data employed for the modelization refer to a real High Speed Train operating in Italy in 25kV-50Hz lines.This Each 4Q converter is sized for a rated power equal to 900kW and a maxim ese are relevant power values for a switching converter that has to be small and light enough to be installed onboard.
The 4Q converters are constituted by a power part and a contro e circuital elements already available in EMTP-ATP.The MODELS language has been used for implementing the proposed converter control.
The final schematic model is reported in Figure 4, where it is shown only one of nverters.
In the block diagram represented in Figure 4 three control loop k voltage control loop constituted by the voltage measurement, the low-pass filter explained in the previ-ous paragraph, the comparison with the reference value V 0ref and the PI controller.Its output is the value of the equivalent conductance G that keeps the DC link voltage constant varying the power requested or injected by the traction motors and auxiliary services.The second loop is the reference current generator.It considers the input voltage measurement followed by a filter dedicated to the high frequency disturbances.The obtained value multiplied with the equivalent conductance G gives the reference current that the converter, through the switching modulation, has to generate in order to balance the input and output powers.The third loop is related to the DC component compensation in the AC input current.In fact its output value is a constant current that, algebraically added to the reference one, allows to cancel the DC component avoiding the saturation of the input transformer.

The
The use of modulation techniques at constant f the different 4Q converters onboard the train.
The final goal is to interlace more converters in order to diminish the harmonic content in the tota rrent.The interlacing operation gives a shifting through the current waves coming out from the converters that have to be the one that minimize the harmonic content.
In order to preliminary study the benefit of the interlacing practice, two boost converters are considered.A ported in Figure 5, their current waves are shifted of a generic angle φ.
The two waves have the same duty cycle δ and they are only shifted o s f and f' of the two converters are the following: ) Applying the known trigonometric formula, the f' ca e rewritten as: n Considering that the current absorbed by the train is th  ) e sum of the currents absorbed by the two converters, the resulting function to consider for the THD calculus is given by the sum of the two functions, that means: . his case the ab HD is define following: In t solute T d as s been carried out.The (22) has been considered, varying the duty cycle between 0.1 and 0.9 and the shifting between 0 and 2π and supposing the current ripple amplitude equal to the 1% of the continuous component.The result is reported in Figure 6.
From Figure 6 it is possible to make some considerations.First of all, as expected, the and φ=π, because it is the only case in which the two triangle waves are symmetric and in opposite phase, therefore all harmonics are cancelled.
From the optimal point, the THD initially grow fast, both varying δ and φ, underlining the fact that also small dissymmetry in the two converters switching diminish the advantages given by the interlacing.The optimal shifting φ is always equal to π and in this case the THD is always minor than the case of not interlaced converters (φ=0).
Finally, the THD is directly proportional to the ripple amplitude respect to the DC component, and for this t '( ) operable locomotives and electro trains nor higher than the ones reported in Figure 6, considering the relevant current undulation due to the low switching frequencies.
Taking into the railway lines, it is possible to note that after each input transformer there are 2 or 4 4Q converters.Each converter is supplied by a dedicated winding of the main transformer.It comes out that, in order to have a good interlacing, the shift of ea spectively equal to ½ or ¼ of the switching period.
In this way the ripple of the current absorbed by the transformer due to all the 4Q converters has the m equency equal to two or four times the switching one with the advantages of a minor amplitude and easier filtration.
In Figure 7 is possible to note the shifting of the four switchin Analyzing the smart modulation here applied, it is possible to note as the com arting from points on the reference current that have the same time distance.The interlacing can be easily obtained intercalating these time instants referred to the different converters.

4Q Converters
In order to evaluate the harmonic contribution of rters in single operation and in interlaced ration, some numerical simulations are carried out, using ATP/EMTP program.
These converters are often supplied by a distorted input voltage, due to the ay lines and a greater harmonic content allowed by the dedicated standards.A typical waveform measured on the secondary side of the transformer for auxiliary services ins reported in Figure 8.
The most significant components are reported in Table 1.This distortion is mainly ong the line, also enhanced by the harmonic current incted by the trains.In order to have a correct operation of the 4Q converters it is necessary to have a cleaner input voltage obtained installing LC filters in the input stage.Moreover, the greater components (>50 th ) are carefully In the controller are implemented two control loops.
The first one gives the value of conductance G that multiplied by the input voltage deter nt reference.The value of the equivalent conductance G obtained through PI controller comparing the measure of the DC link voltage with its reference in order to keep the DC voltage constant at its nominal value of 1800 V.The function of this controller is to guarantee the equivalence between the input power and the one absorbed by the load.In order to assure a uniform power absorption among the various converters, there is only one regulator for all the converters.
The second loop is necessary to cancel the DC component in the AC current that can be generated by the low switching frequency o nts can be dangerous for the onboard transformer because can saturate the magnetic core.Indeed this transformer is not oversized due to the need to reduce its weight and volume.Therefore it is really sensible to the direct current component.Because the four 4Q converters are supplied by four independent transformer windings, it is necessary to adopt one of these regulators for each converter.In Figure 9 the current absorbed by one 4Q converter and its Fourier analysis are reported.It is possible to note a high ten order components corresponding to the switching frequency, but also an appreciable third harmonic.This last one is due to the difficulty to follow up the sinusoidal reference having a so low switching frequency.
The interlacing of the four converters has really reduced the harmonic content, as it is well represented in Figure 10, where the current absorbed at pantograph and Fourier analysis are reported.In particular there are no more harmonics at the switching frequency, but there is still the third harmonic, due, as told before, to the low switching frequencies.
What told above is confirmed by Figure 11, where it is possible to note that there is a difficulty of the converter to switch at the end of e In fact, in the In order to better evaluate the contribution of the four converters interla th is condition can occur during the train operation.
The current absorbed at pantograph in this second case with only three 4Q converters working and its Fourier analysis are reported in Figure 12.
It is possible to note the worst harmonic content respect to the previous case and, most of all, the contribution of the switching frequency com evident the presence of the ten order component.Regarding the third harmonic, the situation is not changed, always due to the low switching frequency.
In Figure 13 is reported the shifting of the three switching frequencies and also in this case it is evident the difficulty of the converter to switch at th lf period.

Conclusions
The paper de the input stage of trains constituted by m In fact, the use of the 4Q converter allows to overcome the limits imposed by traditional rectifier.However, the high power needed for the train acceleration, in the order of 6000 kW per locomotive, does not allow to have high switching frequencies.

Figure 1 .Figure 2 .
Figure 1.Principle scheme of a four-quadrant converter.e 1 =line voltage at a frequency f 1 ; i 1 =line current

Figure 3 .Figure 4 .
Figure 3. Traction circuit schematic diagram of the Italian high speed train

Figure 5 .Figure 6 .
Figure 5. Waves shifting of the two interlaced boost converters currents frequency, because all the ripple is an undesirable component.The THD so calculated is function of the shifting angle φ between the two waves and of the duty e supposed the same for the two converters.In order to underline the THD dependence from the parameters δ and φ, a simulation in Matlab environment 

Figure 7 .Figure 8 .
Figure 7. Shifting of the four switching frequencies

Figure 9 .
Figure 9. Current absorbed by one 4Q converter and its Fourier analysis

Figure 10 .Figure 11 .Figure 12 .Figure 13 .
Figure 10.Current at pantograph in case of four 4Q converters working and its Fourier analysis