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The Influence of Controller Parameters on the Quality of the Train Converter Current

The Influence of Controller Parameters on the Quality of the Train Converter Current Hindawi Publishing Corporation Advances in Power Electronics Volume 2011, Article ID 832737, 10 pages doi:10.1155/2011/832737 Research Article The Influence of Controller Parameters on the Quality of the Train Converter Current M. Brenna, F. Foiadelli, and D. Zaninelli Dipartimento di Energia, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy Correspondence should be addressed to F. Foiadelli, federica.foiadelli@polimi.it Received 15 November 2010; Revised 25 March 2011; Accepted 31 March 2011 Academic Editor: C. M. Liaw Copyright © 2011 M. Brenna et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper presents a stability analysis of train converters in order to evaluate how the controller parameters affect the absorbed current. The new dynamic model presented in this paper is capable of considering the time-variant nature of the system for the correct tuning of the feedback proportional-integral PI controller, applying a current controlled modulation technique never used in high-power traction converters. The reduction of the harmonic content of the current absorbed by a converter employed at the input stage onboard high-speed trains is really important, considering the interaction with the signaling system set up for traffic control. A computer model of the converter, considering both the power and the control structure, has also been implemented in order to deliver a validated tool for the developed theoretical analysis. 1. Introduction braking. The high power value of the input stage does not allow high switching frequencies for the 4Q converter. The main purpose of public transport is to ensure, in every Consequently the absorbed current presents a high ripple operational situation, the safety and regularity of the service value characterized by high harmonic current components provided by rolling stocks. To this purpose it is necessary that that cannot be tolerated by the system. Indeed the track the harmonic disturbances generated by traction units are circuit used for signaling and communication for the traffic conformed to the EMC limits imposed by the compatibility management and safety employs signal currents overlapped with the signalling system set up for traffic control. with the power ones. For this reason it is really important to The propulsion motor drives and auxiliary services of control the current harmonic content in order to prevent the the modern traction units are powered by microprocessor possible interference phenomena and to meet the requests of controlled bidirectional electronic converters. Therefore, the new Technical Specification for the Interoperability [1]. there is a continuous energy exchange with the power supply First of all, it is necessary to perform a stability analysis of the system in large spectral contents, since the new traction 4Q converter. The paper presents a mathematical model for units cannot be considered anymore as simple passive loads the single-phase four-quadrant (4Q) converter, introducing absorbing energy from the line. a method able to consider the time-variant nature of the Nowadays the input stage of the locomotives and high system. The main goal of this algorithm is the analysis of the speed trains supplied in AC at 25 kV, 50 Hz and 15 kV, control parameter effects on the absorbed current in order to 16.7 Hz are constituted by more four-quadrant (4Q) con- perform a correct tuning of the feedback PI controller. The verters. In fact, the high power requested by the traction developed method applies a particular modulation technique instead of the traditional PWM control, in order to better motor drives and the DC/DC converters for auxiliary services request a stable input DC voltage in the range of 1500– avail the converter characteristic, controlling directly the 1800 V to better exploit the modern IGBT switches. This DC absorbed current and maintaining, at the same time, a con- link voltage has to be provided by 4Q converters that allow stant switching frequency. Therefore, the main characteristic its stabilization and the power inversion during regenerative of the proposed algorithm is its application to a single-phase 2 Advances in Power Electronics 25 kV 50 Hz AC Auxiliary services step-down DC/DC converter Double 4Q converter input stage 600 V DC MT1 MT2 1st motor drive Figure 1: Traction circuit schematic diagram of a High Speed Train with double 4Q converters input stage. The 4Q converter function is to absorb/deliver power from/to the AC network according to the traction motors 4Q and auxiliary services power demand [2]. First of all, the converter has to absorb current from the contact line at a S1 S2 voltage e and frequency f with a fundamental harmonic, 1 1 TR R L s i 2 i ,in phase with e and with limited harmonic disturbance, 1 1 e that means a power factor cos ϕ = 1 and a distortion factor 2 1 λ ∼ 1. Moreover, it has to absorb power from the contact line with a mean value, P , pulsing at the frequency 2 · f ,as 1 1 S3 S4 well as to supply a continuous power, P , to the three-phase motor drive inverters connected to the direct current DC link [3]. Figure 2: Schematic diagram of the 4Q converter. The use of a forced switching converter allows for higher DC link voltages than those generated by diode or thyristor converter that employs a current controlled control strategy bridges. In this manner, it is possible to increase the flowing (Smart Modulation). This allows to linearize the system in power at the same converter losses [4]. The maximum value order to apply the traditional methodology for the stability of the DC link voltage is imposed by the presence of the analysis. Other algorithms employ vectorial techniques, Park switches and the voltage smoothing capacitors [5]. Moreover, Transform, and so forth, but they are useful for three- the filtering on the AC side is simplified and is only related to phase converters or other methodologies are applied to the the switching frequencies [6]. traditional PWM modulation. Starting from real cases referred to European High-Speed 3. Mathematical Model of the 4Q Converter Trains, a circuital model of the train input stage has been implemented in EMTP environment, considering both the The 4Q converters used in modern locomotives are two- power and the control structure. Many simulations have been level converters, because they employ the new high voltage performed to validate the theoretical analysis presented. IGBT. Therefore, their stability analysisneedsto consider the four different circuit configurations as a function of the switch status. Figure 2 reports a representation of the two 2. AC/DC Conversion Input Stage converter levels considered in which its equivalent circuit is The input stages of a High-Speed Train (HST) is composed composed of an ideal voltage generator (e = e /h,where 2 1 of dedicated converters that realize the bidirectional AC/DC h is the transformer ratio of the onboard transformer), a conversion. The HST here considered is constituted by two commutation inductance L, an equivalent resistances R ,a locomotives with two motor drives each. Every motor drive DC link capacitance C, and an equivalent current absorbed is supplied by double 4Q converters input stage, in order by the train motor drives I . TR to realize redundancy and then guarantee the continuity of The modeling process starts by representing the discrete the service, through the main transformer as depicted in modes of the converter mathematically. The discrete model Figure 1. of the converter describes each status of the converter by 2nd motor drive Advances in Power Electronics 3 Table 1: State equations of a 4Q converter. State variable i State variable v Matrix format 2 d First configuration S1, S4 ON; S2, S3 OFF L(di /dt) =−R i − v + E sin(ωt) C(dv /dt) = i − I A · x˙ = B · x + K 2 s 2 d 2 d 2 TR Second configuration S1, S4 OFF; S2, S3 ON L(di /dt) =−R i + v + E sin(ωt) C(dv /dt) =−i − I A · x˙ = B · x + K 2 s 2 d 2 d 2 TR Third configuration S1, S2 ON; S3, S4 OFF L(di /dt) =−R i + E sin(ωt) C(dv /dt) =−I A · x˙ = B · x + K 2 s 2 2 d TR Fourth configuration S1, S2 OFF; S3, S4 ON L(di /dt) =−R i + E sin(ωt) C(dv /dt) =−I A · x˙ = B · x + K 2 s 2 2 d TR ω is the supply voltage angular frequency, x = [i v ] is the state vector, x˙ is its time derivative, A contains the conservative elements of the converter, B, 2 d B ,and B are the matrices of the state variables coefficients that represent the specific working conditions, and K is the forcing term vector. The switching operation of the 4Q converters realizes three different state conditions (because the third and fourth configurations bring to the same circuit topology) characterized by different matrices B, B ,and B . separate linear equations. Taking the state variables as i and From (2)a solution for i and v can be 2 2 d v , the four-converter state equations are reported in Table 1. i = i sin ωt + ϕ + I , 2 20 Observing the matrices B, B ,and B , it is possible (3) to note that the working conditions do not influence the v = v + v cos(2ωt + ϑ), d 0 r elements on the diagonal. Therefore, the state equation in generic condition can be reduced to where i has one constant forcing term and another sinu- soidal term at the primary angular frequency, ω,and v ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ has one constant component v and a ripple at an angular ( ) L 0 −R −γ E sin ωt s 2 frequency of 2ω. Both the constant term I and phase ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ x˙ = x + , (1) ϕ are forced to zero by the converter control. The ripple 0 C γ 0 −I TR component, v , is quite low due to an LC filter in the DC A B if γ=1 K link that is tuned at a frequency, f = 2 · f , aswell asahigh 2 1 B if γ=−1 B if γ=0 capacitance and voltage in the DC link. As it will be presented in the next session, it reaches the 0.2% of the DC component, therefore, the voltage v can be approximated with its average where γ depends on the mode of the converter and can be value v . equal to 1, −1, or 0. Deriving (3) and substituting it into (2), a new system The 4Q converter function is to absorb or deliver power of three equations in three unknown terms (v , m, δ)can be from/to the AC line depending on the power value requested obtained equating the cosine and sine terms: by the traction motors and auxiliary services. This objective can be achieved using different control logic [7]. For this di L + R i = E − mv cos δ, s 2 0 application PWM is often used, which is a well-known dt commutation technique. This control has the advantage of Liω = mv sin(δ), (4) having a constant switching frequency, which allows for the sizing of dedicated filters tuned at that frequency; however, dv 1 ( ) this control has a disadvantage in that it has indirect control C = mi cos −δ − I . TR dt 2 of the current through the variation of the modulation index, m, and the load angle, δ, causing possible transient These equations are time invariant, but nonlinear. To be overcurrents. implemented in a control system design they must be Assuming the use of PWM modulation, there are many linearized around their steady-state operating point. To commutations in a period at the main frequency, and, linearize the system, the following substitutions are made, therefore, γ can be replaced with the time-varying function, where capital letters represent the steady-state operating γ = m · sin(ωt − δ), where m is the modulation index and point and the prime symbol represents a small deviation δ is the load angle. The time-varying nature of γ complicates from the operating point: the resolution of (1). (i) m = M + m modulation index; To eliminate the time variance in the equations, instead of transforming the reference frame and applying the Park (ii) v = V + v DC link voltage; 0 0 Transform, a substitution method has been adopted [8]. The (iii) δ = Δ + δ load angle; first step is to rewrite the state equations including γ: (iv) i = I + i absorbed current. The value of E is stiff and imposed by the contact line di ( ) ( ) L + R i =−m sin ωt − δ v + E sin ωt , s 2 d 2 connected to the main line, whereas I is imposed by the TR dt power requested by the train. (2) dv d The first-order approximations from the Taylor Series for C = m sin(ωt − δ)i − I . 2 TR dt the sine and cosine functions are sin(X + x) ≈ sin X + x cos X 4 Advances in Power Electronics and cos(X + x) ≈ cos X − x sin X used to simplify the system determinethe valueof m . Substituting m into the third of (4): equation of (7)we obtain 1 1 sL + R di sCv = Mi cos(−Δ) − I cos(−Δ) i ( ) ( ) L + R I + i = E − M + m V + v cos Δ s 2 0 2 2 V cos Δ 0 0 dt 1 M + (M + m ) V + v δ sin Δ, 0 − I cos(−Δ) v , 0 (8) 2 V L(I + i )ω = (M + m ) V + v sin Δ ((1/2)M cos (Δ) − (1/2)I((sL + R )/V )) s 0 v = i . (5) 0 (sC + (1/2)I cos(Δ)(M/V )) + (M + m ) V + v δ cos Δ, 0 dv 1 0   Equation (8)gives thevalue, v , as a function of i .Itis now C = (M + m )(I + i ) cos(−Δ) dt 2 possible to determine the variations of the DC link voltage for small changes of the current absorbed by the converter − (M + m )(I + i )δ sin Δ − I . TR ∂v ((1/2)M cos(Δ) − (1/2)I((sL + R )/V )) s 0 ∂i (sC + (1/2)I cos(Δ)(M/V )) 4. Stability Analysis of 4Q Converter with −((1/2)L(I/V ))s + (1/2)M cos(Δ) − (1/2)R (I/V ) 0 s 0 Current Controlled Strategy = . ( ) ( ) ( )( ) C s + 1/2 I cos Δ M/V Retaining only the first-order terms, it is possible to obtain (9) the small-signal model, useful to find the frequency response of the converter [9]: The reference current is given by the product between the equivalent conductance, G, and the supply voltage, E (i = G · E ), where G comes from the PI regulator that maintains di L + R i =−Mv cos Δ − m V cos Δ, s 0 the DC link voltage constant: dt Li ω = Mv sin Δ + m V sin Δ, (6) G = k + (V − V ). (10) p 0ref 0 dv 1 1 ( ) ( ) C = Mi cos −Δ + m I cos −Δ . dt 2 2 The block diagram of the DC voltage regulation loop is shown in Figure 3, where only one of the two-input The small-signal model is developed in terms of the Laplace stage 4Q converter is represented. In the block diagram operator. Applying the Laplace transformation to (6)and shown in Figure 3 three control loops can be identified. assuming zero initial conditions, we obtain The first control loop is the DC link voltage control loop comprised of the voltage measurement, the low-pass filter explained above, the comparison with the reference value, ( ) Ls + R i =−Mv cos Δ − m V cos Δ, s 0 V , and the PI controller. Its output is the value of the 0ref equivalent conductance, G, that keeps the DC link voltage Li ω = Mv sin Δ + m V sin Δ, (7) constant while varying the power requested or injected 1 1 by the traction motors and auxiliary services. The second Csv = Mi cos(−Δ) + m I cos(−Δ). 2 2 loop is the reference current generator, which considers the input voltage measurement followed by a filter dedicated to high-frequency disturbances. The obtained value from Instead of the PWM control that generates an AC voltage this loop multiplied with the equivalent conductance, G, trough the modulation index m and its phase δ,in the gives the reference current that the converter, through Smart following a new modulation technique that can directly Modulation, has to generate in order to balance the input control the current value is proposed. and output powers. The third loop is related to the DC Considering the variability of the switching frequency, component compensation in the AC input current. In fact, f ,and thedifficulty defining an adequate current ripple, its output value is a constant current that, when added to the the Smart Modulation, presented in [10, 11]and briefly reference current, allows the DC component to be cancelled described in the appendix, is applied. out in order to avoid saturation of the input transformer. This strategy allows direct control of the current value The DC link voltage variation for small changes of G maintaining, at the same time, a constant value of the given by the regulator is obtained multiplying (9)for the switching frequency. Therefore, the modulation index, m, input voltage E : does not directly appear in the control algorithm, but, assuming a unitary power factor, the converter absorbed ∂v ∂v current is controlled by the equivalent conductance, G.From 0 0 = · E . (11) the second equation of the system in (7) it is possible to ∂G ∂i Advances in Power Electronics 5 Table 2: Main data of the considered High Speed Train input stage Switching Inductance L and DC Link Capacitance C Calcula- converter. tion. Referring to Figure 2, the elements that more influence the control parameters are the switching inductance L and Value M.U. the DC link capacitance C. The inductance value is chosen E 1000 V 2MAX to limit the input current ripple, while the capacitance C V 1800 V influences the DC link oscillation. Therefore, the obtained P 900 kW ass values are L = 1.5mH and C = 200 mF. I 500 A TR I 1800 A MAX Calculation of the Correct Tuning Parameters of the Regulator f 500 Hz sw (Proportional k and the Integral k ). In order to study the p i system stability using the Bode criteria through module and phase diagrams, identified by the open loop transfer function The open-loop function given by the product of (10)and (12), a dedicated code in MATLAB environment has been (11)is implemented. From this program it is possible to calculate FT OL the control parameters k and k , giving as input the specific p i data of the converter considered (Table 2), that have to k ∂v stabilize the converter dynamics. Under these conditions and = k + · s ∂G referring to the studied case, a good compromise to obtain a fast system response and, at the same time, a good phase = k + margin is k = 0.1and k = 3. The Bode diagrams obtained p i from (12) using these values are shown in Figure 4(a).The −((1/2)LI(E /V ))s + (1/2)ME cos(Δ)−(1/2)R I(E /V ) 2 0 2 s 2 0 cut-off frequency that guarantees a good response with a · . (C)s+(1/2)I cos(Δ)(M/V ) 0 sufficient phase margin is about 200 Hz. (12) Calculation of the Tuning Parameters of the Regulator (Pro- The above presented theory leads to a procedure for the portional k and the Integral k )atthe Limit of Stability. The p i converter design. In particular, the following steps can be parameter values for the limit of the stability condition have identified: been investigated using the same Matlab code, obtaining (1) converter parameters identification, such as rated and k = 0.2and k = 3. TherelativeBode diagramsfor these p i maximum powers, AC input and DC output voltages, parameters are shown in Figure 4(b). As can be seen from and consequently input and output currents; this figure, the phase margin is close to zero. (2) switching inductance L and DC link capacitance C calculation. These two parameters are determined Sensitivity Analysis. Applying the same algorithm, it is pos- considering the maximum input current ripple and sible to obtain the effect of changes of the parameters on the the maximum acceptable DC link voltage oscillation; stability of the controller. As it can be shown in Figure 5,an (3) calculation of the correct tuning parameters of the increase of the proportional gain causes significant reduction regulator (proportional k and the integral k )to of the gain margin leading to the loss of stability. On the p i have good stability and at the same time a fast time contrary, the reduction of k improves the stability but it response of the system; degrades the time response performance. Once determined the power system structure, the traction power converters (4) calculation of the tuning parameters of the regulator are mainly subjected to the line voltage variation. In fact, (proportional k and the integral k )atthe limitof p i in traction systems the voltage can vary in the range of stability in order to perform the sensitivity analysis; −33%–+20% of the nominal one. As it can be seen in (12), (5) sensitivity analysis. the line voltage E (reported on the secondary side of the onboard transformer) is common to all the numerator terms, 5. Analysis Application in a Real Case therefore, it can be considered as a proportional gain. The sensibility analysis carried out and presented in Figure 5 The converter design steps previously formulated have been shows that around the optimal value of k (−33%–+20%) applied to a real case as example consisting in the converter p there is enough gain margin to maintain the stability. studied in this paper (Figure 1) employed as input stage onboard High-Speed Railway (HSR) Trains operating on the new Italian HSR lines. 6. Harmonic Absorption due to Single-Phase Supply Converter Parameters Identification. In transportation sys- tems many typologies of power converters are employed that One of the problems that characterize the 4Q converter can have different applications (i.e., front-end converters, and, in general, all of the AC/DC single-phase converters, auxiliary services, UPS, high-speed trains, heavy traction, is the absorption of variable instantaneous power. In fact, naval propulsion) and sizes. The typical size of the converter in contrast to a balanced three-phase system, the flowing and its usual data for this application are reported in Table 2. power in a single-phase system is characterized by a mean 6 Advances in Power Electronics x PI + 0ref ref − DC Smart modulation IGBT IGBT V Figure 3: Block diagram of the system and control loops. Bode diagram Bode diagram 150 150 100 100 50 50 0 0 −50 −50 270 270 225 225 180 180 −2 −1 0 1 2 3 4 −2 −1 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Frequency (rad/s) Frequency (rad/s) (a) (b) Figure 4: Bode diagram of the converter and regulator system for (a) k = 0.1and k = 3and (b) k = 0.2and k = 3. p i p i value equal to the active power requested from the load and leaving the regulator, given as the product between G and the a variable term oscillating at a frequency twice that of the sinusoidal input voltage e = E sin(ωt), is 2 2 supply frequency. This oscillation, in the case of traction converters, causes the generation of a ripple in the DC output I = G · e = [G + G sin(2ωt)] · E sin(ωt) ref 2 0 100 2 voltage. The 4Q converters are equipped with a filter that is 1 1 tuned to a frequency of 100 Hz in 25 kV/50 Hz train systems = E G sin(ωt) + G E cos(ωt) − G E cos (3ωt). 2 0 100 2 100 2 or 33.3 Hz in 15 kV/16.7 Hz train systems. However, it is 2 2 (13) not possible to completely cancel the ripple in the DC link voltage. Due to the regulator response time, the equivalent conductance at the end of the transient state condition The reference current reported in (13) can be defined as the coming out from the regulator can be defined as the sum of sum of the following three terms: its mean theoretic value requested G and a ripple at double frequency G (G = G + G sin(2ωt)). This phenomenon 100 0 100 (i) I = E G sin(ωt) is the theoretical value of the ref 0 2 0 has not been considered in the stability analysis because reference current proportional to the absorption it is not influential; however, it is the cause of harmonic power; it is in phase with the voltage signal and at current absorption from the converter. The reference current the same frequency; Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Advances in Power Electronics 7 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 K K p p (a) (b) Figure 5: Influence of k to the cut frequency and to the gain margin. 2500 1815 0 1800 −625 −1250 −1875 −2500 0.15 0.17 0.19 0.21 0.23 0.25 0.15 0.17 0.19 0.21 0.23 0.25 (s) (s) Figure 7: DC link voltage in end of transient state and stable Figure 6: Current absorbed by one 4Q converter in the traction condition (k = 0.1, k = 3). p i phase in stable condition (k = 0.1, k = 3). p i (ii) I = (1/2)G E cos(ωt) is a term at the same net ref ϕ 100 2 for the model is obtained from a real HSR Train operating frequency, but shifted π/2 radians, which is the source in Italy on a 25 kV/50 Hz line. Each 4Q converter is sized of absorption of the reactive power; for a rated power equal to 900 kW and a maximum peak (iii) I =−(1/2)G E cos(3ωt)is a term at three times ref 3 100 2 power equal to 1500 kW. These are relevant power values for the frequency that determines the presence of a third a switching converter that has to be small and light enough harmonic current. to be installed onboard. The 4Q converters consist of a power circuit and a In order to reduce the G component, it is necessary control system. The power circuit has been modeled with to filter the DC link voltage with a suitable low-pass filter. the circuital elements already available in EMTP-ATP. The This filter tuning is critical because it simultaneously allows MODELS language has been used for implementing the for a fast system response and a low cut-off frequency. It proposed converter control. is impossible to achieve both of these objectives, but for In the following, some simulation results are reported the system stability it is necessary to focus on the system response, which implies a small ripple. in order to verify the outcomes of the performed analysis. The simulation cases considered for the theoretical analysis validation are the following: 7. Model and Simulation Results (A) nominal condition; In order to validate the theoretical analysis described above, (B) loading; a suitable model of the system has been implemented using the EMTP-ATP dynamic simulation tool. The data employed (C) converters regulators working at the stability limit. (A) Cut frequency (Hz) Gain margin (dB) (V) 8 Advances in Power Electronics 4000 3500 2000 1750 0 0 −875 −1000 −2000 −1750 −2625 −3000 −3500 −4000 0.35 0.38 0.41 0.44 0.47 0.5 0.2 0.25 0.3 0.35 0.4 0.45 0.5 (s) (s) Figure 10: DC link current at full load for k = 0.2and k = 3. p i Figure 8: Loading transient of the current absorbed by one 4Q converter in stable condition (k = 0.1, k = 3). p i 2.5 2.3 2.1 1.9 1780 1.7 1.5 0.35 0.38 0.41 0.44 0.47 0.5 1760 (s) 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Figure 11: Equivalent conductance at full load for k = 0.2and (s) k = 3. Figure 9: DC link voltage and filtered signal in loading phase for k = 0.1and k = 3. p i Table 4: Most significant current harmonic components when working at the stability limit. Table 3: Most significant current harmonic components in normal Harm. Amplitude [A] pu operating condition. 1 2038.9 1 Harm. Amplitude [A] pu 3 110.68 0.054 1 1997.5 1 8 87.856 0.043 3 53.6 0.026 10 345.03 0.169 8 79.3 0.039 12 76.157 0.037 10 347.6 0.174 19 68.314 0.033 12 68.8 0.034 21 52.113 0.025 19 71.9 0.036 23 20.975 0.010 21 55.9 0.028 28 32.188 0.015 23 20.0 0.010 28 31.595 0.016 switching frequency of the power GTO due to the high rated power of the converter. Therefore, many harmonic (A) Nominal Condition. The normal operating condition components are generated in the input current. The most considers the traction phase at the nominal power of 3.6 MW significant harmonic components are reported in Table 3. due to four 4Q converters. The parameters k = 0.1and In particular, considerable amplitude is associated to k = 3 obtained for the stability condition are employed the 10th harmonic order, corresponding to the switching in the model. The current absorbed by one 4Q converter is frequency of the converter (17% of the fundamental com- shown in Figure 6. ponent). Moreover, there is the third harmonic component From Figure 6 it is easy to see that the converter (which has an amplitude that is 2.7% of the fundamental commutations are numerically limited because of the low amplitude) and is caused by the interaction between the (A) (A) Advances in Power Electronics 9 ripple at 100 Hz of the DC link voltage and the supply voltage Theequivalent conductance atthe limitof stability at 50 Hz. The THD is equal to 19.07%. assumes, at the end of the transient state condition, a With respect to the DC link voltage, its behavior at the mean value equal to the value at the stability condition end of the transient state condition is shown in Figure 7. because it is correlated to the power transfer requested by the The mean value of the voltage is equal to 1800 V, and it is loads. However, in this last case, there is a highly irregular maintained at a constant value equal to the reference value by oscillation around this mean value that, combined with the the PI regulator. Figure 7 shows the clear oscillation at 100 Hz input sinusoidal voltage, causes the absorption of important due to the oscillation of the power absorbed by the single- current harmonic components (Table 4). phase converter. 8. Conclusions (B) Loading. The loading case, obtained applying a step load to the system, has been simulated in order to stress the In this paper the influence of controller parameters on system with high-power variation for stability test. Indeed, the quality of the absorbed current has been investigated. the transient behaviour has been analyzed in the case of Moreover the application of a current control method to a load change on the DC side from 50 A (presuming that four-quadrant (4Q) input stage converters employed in 10% is dedicated to the auxiliary services functioning) to traction systems has been proposed. In particular, it has 500 A. This means that a power variation of 3.24 MW (from been chosen the Smart Modulation technique with good 360 kW to 3.6 MW) has been simulated. Also in this case, the results, that has been originally developed for low power high parameters k = 0.1and k = 3 obtained for the stability switching frequency converters. The changing of modulation p i condition are employed in the model. technique, from the traditional PWM to the innovative The absorbed current behaviour in the loading transient Smart Modulation, has requested a new and deep stability at t = 0.3 s (instant at which the initial simulation transient analysis due to the changing of the mathematical equations is finished) is shown in Figure 8. that describe the whole system. This analysis has also led to a Moreover, this power variation causes a DC link voltage methodology for setting the regulator parameters. swell equal to 30–35 V (1.6%), as shown in Figure 9 in In order to validate the theoretical analysis above continues line. In fact, in the transient phase, the unbalance described, a circuital model of real cases referred to Euro- between requested/remised power from the traction motors pean High Speed Trains has been implemented in EMTP and absorbed/injected power from the 4Q converter, caused environment. The outputs of the numerous simulations by the delay of the PI regulator, is compensated by the carried out confirm the results of the theoretical analysis energy stored in the smoothing capacitors. The speed of the and furnish important indications for the performance of regulator is an important design parameter because it affects the train and the observance of the European Technical the dynamic of the DC voltage during transient events. A Specification for the Interoperability [1]. In particular, they lower speed can cause high overvoltages that can damage the show how the oscillations in the control loop can cause great electronic components or deep voltage sags that can lead to current harmonic absorption that can interfere with the track loss of converter control. signalling system. The DC link voltage ripple implies that the measured value is not suitable for the regulator because it will cause the Appendix absorption of a high 3 harmonic current as shown in (13). The Smart Modulation (SM) technique is a predictive For this reason, a filter that limits the 100 Hz component, but control that imposes the switching frequency independently is sufficiently fast to follow the DC link voltage transients, is from operating conditions. In addition, the SM allows inserted. The filtered input signal of the converter regulator is shown in Figure 9 by the dashed line. In order to guarantee interlacing of switching instants in case of more converters connected in parallel. Referring to Figures 2 and 3, this algo- the stability of the system, this filtered signal has to limit rithm considers the reference current I and the inductor the 100 Hz component before and after the transient phase ref flux error λ = L · (i − i ) to calculate the commutation maintaining, at the same time, a fast response to follow the ref instants. The expression that defines the ideal voltage V at transient phenomena. 2 the 4Q AC terminals when the reference current is absorbed (C) Converters Regulators Working at the Stability Limit. This is last case has been simulated in order to verify the efficacy di ref of the choice of the regulator parameters. For this reason, v = e + L · + R · i . (A.1) 2 2 ref dt different values of k and k have been inserted in the model. p i In particular, the following examples will demonstrate the Considering that the trajectories of the real current are results obtained using the same values obtained previously linear in a switching period T, it is possible to calculate the for the limit of the stability condition: k = 0.2and k = 3. p i commutation instants as the intersection of such trajectories: The DC link current and the conductance at full load are demonstrated in Figures 10 and 11. For both of these (v + v )(T/2) − λ 2 d Δt = , behaviours it is possible to note a clear increase of an irregular ripple due to the control operation at the limit of (A.2) (v − v )(T/2) + λ d 2 stability. These results confirm the theoretical analysis carried Δt = . out. d 10 Advances in Power Electronics A more detailed description of this algorithm can be found in [10, 11]. References [1] European Technical Specificationfor Interoperability (TSI) of the Railway Lines, 2008. [2] M. Brenna, F. Foiadelli, G. C. Lazaroiu, and D. Zaninelli, “Four quadrant converter analysis for high speed trains,” in Proceedings of the 12th International Conference on Harmonics and Quality of Power, Cascais, Portugal, 2006. [3] S. Ostlund, “Reduction of transformer rated power and line current harmonics in a primary switched converter system for traction applications,” in Proceedings of the 5th European Conference on Power Electronics and Applications (EPE ’93), vol. 7, pp. 112–119, Brighton, UK, september 1993. [4] S. Burdett, J. Allan, B. Mellitt, and J. Taufiq, “A power factor and harmonic comparison of AC railway power electronic traction converter circuits,” in Proceedings of the 5th European Conference on Power Electronics and Applications (EPE ’93),pp. 235–240, Brighton, UK, 1993. [5] W. Runge, “Control of line harmonics due to four-quadrant- converter in AC tractive stock by means of filter and trans- former,” in Proceedings of the 7th European Conference on Power Electronics and Applications (EPE ’97), pp. 3.459–3.464, Trondheim, Norway, 1997. [6] G. W.Chang,H. W.Lin,and S. K. Chen,“Modeling characteristics of harmonic currents generated by high-speed railway traction drive converters,” IEEE Transactions on Power Delivery, vol. 19, no. 2, pp. 766–773, 2004. [7] C.Bachle, H.P.Bauer,and T. Seger, “Requirements on the control of a three-level four quadrant power converter in a traction application,” in Proceedings of the 3rd European Conference on Power Electronics and Applications (EPE ’89), vol. 2, pp. 577–582, Aachen, Germany, 1989. [8] J. Carter, C. J. Goodman, and H. Zelaya, “Analysis of the single- phase four-quadrant PWM converter resulting in steady-state and small-signal dynamic models,” IEE Proceedings Electric Power Applications, vol. 144, no. 4, pp. 241–247, 1997. [9] D. Casadei,J.Clare, L.Empringhamet al.,“Large-signal model for the stability analysis of matrix converters,” IEEE Transactions on Industrial Electronics, vol. 54, no. 2, pp. 939– 950, 2007. [10] M. Brenna, G. C. Lazaroiu, G. Superti-Furga, and E. Tironi, “Bidirectional front end converter for DG with disturbance insensitivity and islanding-detection capability,” IEEE Trans- actions on Power Delivery, vol. 23, no. 2, pp. 907–914, 2008. [11] M. S. Carmeli, F. Castelli Dezza, and G. Superti Furga, “Smart modulation: a new approach to power converter control,” in Proceedings of the 9th European Conference on Power Electronics and Applications (EPE ’01), Graz, Austria, August 2001. 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The Influence of Controller Parameters on the Quality of the Train Converter Current

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Copyright © 2011 M. Brenna et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Advances in Power Electronics Volume 2011, Article ID 832737, 10 pages doi:10.1155/2011/832737 Research Article The Influence of Controller Parameters on the Quality of the Train Converter Current M. Brenna, F. Foiadelli, and D. Zaninelli Dipartimento di Energia, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy Correspondence should be addressed to F. Foiadelli, federica.foiadelli@polimi.it Received 15 November 2010; Revised 25 March 2011; Accepted 31 March 2011 Academic Editor: C. M. Liaw Copyright © 2011 M. Brenna et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper presents a stability analysis of train converters in order to evaluate how the controller parameters affect the absorbed current. The new dynamic model presented in this paper is capable of considering the time-variant nature of the system for the correct tuning of the feedback proportional-integral PI controller, applying a current controlled modulation technique never used in high-power traction converters. The reduction of the harmonic content of the current absorbed by a converter employed at the input stage onboard high-speed trains is really important, considering the interaction with the signaling system set up for traffic control. A computer model of the converter, considering both the power and the control structure, has also been implemented in order to deliver a validated tool for the developed theoretical analysis. 1. Introduction braking. The high power value of the input stage does not allow high switching frequencies for the 4Q converter. The main purpose of public transport is to ensure, in every Consequently the absorbed current presents a high ripple operational situation, the safety and regularity of the service value characterized by high harmonic current components provided by rolling stocks. To this purpose it is necessary that that cannot be tolerated by the system. Indeed the track the harmonic disturbances generated by traction units are circuit used for signaling and communication for the traffic conformed to the EMC limits imposed by the compatibility management and safety employs signal currents overlapped with the signalling system set up for traffic control. with the power ones. For this reason it is really important to The propulsion motor drives and auxiliary services of control the current harmonic content in order to prevent the the modern traction units are powered by microprocessor possible interference phenomena and to meet the requests of controlled bidirectional electronic converters. Therefore, the new Technical Specification for the Interoperability [1]. there is a continuous energy exchange with the power supply First of all, it is necessary to perform a stability analysis of the system in large spectral contents, since the new traction 4Q converter. The paper presents a mathematical model for units cannot be considered anymore as simple passive loads the single-phase four-quadrant (4Q) converter, introducing absorbing energy from the line. a method able to consider the time-variant nature of the Nowadays the input stage of the locomotives and high system. The main goal of this algorithm is the analysis of the speed trains supplied in AC at 25 kV, 50 Hz and 15 kV, control parameter effects on the absorbed current in order to 16.7 Hz are constituted by more four-quadrant (4Q) con- perform a correct tuning of the feedback PI controller. The verters. In fact, the high power requested by the traction developed method applies a particular modulation technique instead of the traditional PWM control, in order to better motor drives and the DC/DC converters for auxiliary services request a stable input DC voltage in the range of 1500– avail the converter characteristic, controlling directly the 1800 V to better exploit the modern IGBT switches. This DC absorbed current and maintaining, at the same time, a con- link voltage has to be provided by 4Q converters that allow stant switching frequency. Therefore, the main characteristic its stabilization and the power inversion during regenerative of the proposed algorithm is its application to a single-phase 2 Advances in Power Electronics 25 kV 50 Hz AC Auxiliary services step-down DC/DC converter Double 4Q converter input stage 600 V DC MT1 MT2 1st motor drive Figure 1: Traction circuit schematic diagram of a High Speed Train with double 4Q converters input stage. The 4Q converter function is to absorb/deliver power from/to the AC network according to the traction motors 4Q and auxiliary services power demand [2]. First of all, the converter has to absorb current from the contact line at a S1 S2 voltage e and frequency f with a fundamental harmonic, 1 1 TR R L s i 2 i ,in phase with e and with limited harmonic disturbance, 1 1 e that means a power factor cos ϕ = 1 and a distortion factor 2 1 λ ∼ 1. Moreover, it has to absorb power from the contact line with a mean value, P , pulsing at the frequency 2 · f ,as 1 1 S3 S4 well as to supply a continuous power, P , to the three-phase motor drive inverters connected to the direct current DC link [3]. Figure 2: Schematic diagram of the 4Q converter. The use of a forced switching converter allows for higher DC link voltages than those generated by diode or thyristor converter that employs a current controlled control strategy bridges. In this manner, it is possible to increase the flowing (Smart Modulation). This allows to linearize the system in power at the same converter losses [4]. The maximum value order to apply the traditional methodology for the stability of the DC link voltage is imposed by the presence of the analysis. Other algorithms employ vectorial techniques, Park switches and the voltage smoothing capacitors [5]. Moreover, Transform, and so forth, but they are useful for three- the filtering on the AC side is simplified and is only related to phase converters or other methodologies are applied to the the switching frequencies [6]. traditional PWM modulation. Starting from real cases referred to European High-Speed 3. Mathematical Model of the 4Q Converter Trains, a circuital model of the train input stage has been implemented in EMTP environment, considering both the The 4Q converters used in modern locomotives are two- power and the control structure. Many simulations have been level converters, because they employ the new high voltage performed to validate the theoretical analysis presented. IGBT. Therefore, their stability analysisneedsto consider the four different circuit configurations as a function of the switch status. Figure 2 reports a representation of the two 2. AC/DC Conversion Input Stage converter levels considered in which its equivalent circuit is The input stages of a High-Speed Train (HST) is composed composed of an ideal voltage generator (e = e /h,where 2 1 of dedicated converters that realize the bidirectional AC/DC h is the transformer ratio of the onboard transformer), a conversion. The HST here considered is constituted by two commutation inductance L, an equivalent resistances R ,a locomotives with two motor drives each. Every motor drive DC link capacitance C, and an equivalent current absorbed is supplied by double 4Q converters input stage, in order by the train motor drives I . TR to realize redundancy and then guarantee the continuity of The modeling process starts by representing the discrete the service, through the main transformer as depicted in modes of the converter mathematically. The discrete model Figure 1. of the converter describes each status of the converter by 2nd motor drive Advances in Power Electronics 3 Table 1: State equations of a 4Q converter. State variable i State variable v Matrix format 2 d First configuration S1, S4 ON; S2, S3 OFF L(di /dt) =−R i − v + E sin(ωt) C(dv /dt) = i − I A · x˙ = B · x + K 2 s 2 d 2 d 2 TR Second configuration S1, S4 OFF; S2, S3 ON L(di /dt) =−R i + v + E sin(ωt) C(dv /dt) =−i − I A · x˙ = B · x + K 2 s 2 d 2 d 2 TR Third configuration S1, S2 ON; S3, S4 OFF L(di /dt) =−R i + E sin(ωt) C(dv /dt) =−I A · x˙ = B · x + K 2 s 2 2 d TR Fourth configuration S1, S2 OFF; S3, S4 ON L(di /dt) =−R i + E sin(ωt) C(dv /dt) =−I A · x˙ = B · x + K 2 s 2 2 d TR ω is the supply voltage angular frequency, x = [i v ] is the state vector, x˙ is its time derivative, A contains the conservative elements of the converter, B, 2 d B ,and B are the matrices of the state variables coefficients that represent the specific working conditions, and K is the forcing term vector. The switching operation of the 4Q converters realizes three different state conditions (because the third and fourth configurations bring to the same circuit topology) characterized by different matrices B, B ,and B . separate linear equations. Taking the state variables as i and From (2)a solution for i and v can be 2 2 d v , the four-converter state equations are reported in Table 1. i = i sin ωt + ϕ + I , 2 20 Observing the matrices B, B ,and B , it is possible (3) to note that the working conditions do not influence the v = v + v cos(2ωt + ϑ), d 0 r elements on the diagonal. Therefore, the state equation in generic condition can be reduced to where i has one constant forcing term and another sinu- soidal term at the primary angular frequency, ω,and v ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ has one constant component v and a ripple at an angular ( ) L 0 −R −γ E sin ωt s 2 frequency of 2ω. Both the constant term I and phase ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ x˙ = x + , (1) ϕ are forced to zero by the converter control. The ripple 0 C γ 0 −I TR component, v , is quite low due to an LC filter in the DC A B if γ=1 K link that is tuned at a frequency, f = 2 · f , aswell asahigh 2 1 B if γ=−1 B if γ=0 capacitance and voltage in the DC link. As it will be presented in the next session, it reaches the 0.2% of the DC component, therefore, the voltage v can be approximated with its average where γ depends on the mode of the converter and can be value v . equal to 1, −1, or 0. Deriving (3) and substituting it into (2), a new system The 4Q converter function is to absorb or deliver power of three equations in three unknown terms (v , m, δ)can be from/to the AC line depending on the power value requested obtained equating the cosine and sine terms: by the traction motors and auxiliary services. This objective can be achieved using different control logic [7]. For this di L + R i = E − mv cos δ, s 2 0 application PWM is often used, which is a well-known dt commutation technique. This control has the advantage of Liω = mv sin(δ), (4) having a constant switching frequency, which allows for the sizing of dedicated filters tuned at that frequency; however, dv 1 ( ) this control has a disadvantage in that it has indirect control C = mi cos −δ − I . TR dt 2 of the current through the variation of the modulation index, m, and the load angle, δ, causing possible transient These equations are time invariant, but nonlinear. To be overcurrents. implemented in a control system design they must be Assuming the use of PWM modulation, there are many linearized around their steady-state operating point. To commutations in a period at the main frequency, and, linearize the system, the following substitutions are made, therefore, γ can be replaced with the time-varying function, where capital letters represent the steady-state operating γ = m · sin(ωt − δ), where m is the modulation index and point and the prime symbol represents a small deviation δ is the load angle. The time-varying nature of γ complicates from the operating point: the resolution of (1). (i) m = M + m modulation index; To eliminate the time variance in the equations, instead of transforming the reference frame and applying the Park (ii) v = V + v DC link voltage; 0 0 Transform, a substitution method has been adopted [8]. The (iii) δ = Δ + δ load angle; first step is to rewrite the state equations including γ: (iv) i = I + i absorbed current. The value of E is stiff and imposed by the contact line di ( ) ( ) L + R i =−m sin ωt − δ v + E sin ωt , s 2 d 2 connected to the main line, whereas I is imposed by the TR dt power requested by the train. (2) dv d The first-order approximations from the Taylor Series for C = m sin(ωt − δ)i − I . 2 TR dt the sine and cosine functions are sin(X + x) ≈ sin X + x cos X 4 Advances in Power Electronics and cos(X + x) ≈ cos X − x sin X used to simplify the system determinethe valueof m . Substituting m into the third of (4): equation of (7)we obtain 1 1 sL + R di sCv = Mi cos(−Δ) − I cos(−Δ) i ( ) ( ) L + R I + i = E − M + m V + v cos Δ s 2 0 2 2 V cos Δ 0 0 dt 1 M + (M + m ) V + v δ sin Δ, 0 − I cos(−Δ) v , 0 (8) 2 V L(I + i )ω = (M + m ) V + v sin Δ ((1/2)M cos (Δ) − (1/2)I((sL + R )/V )) s 0 v = i . (5) 0 (sC + (1/2)I cos(Δ)(M/V )) + (M + m ) V + v δ cos Δ, 0 dv 1 0   Equation (8)gives thevalue, v , as a function of i .Itis now C = (M + m )(I + i ) cos(−Δ) dt 2 possible to determine the variations of the DC link voltage for small changes of the current absorbed by the converter − (M + m )(I + i )δ sin Δ − I . TR ∂v ((1/2)M cos(Δ) − (1/2)I((sL + R )/V )) s 0 ∂i (sC + (1/2)I cos(Δ)(M/V )) 4. Stability Analysis of 4Q Converter with −((1/2)L(I/V ))s + (1/2)M cos(Δ) − (1/2)R (I/V ) 0 s 0 Current Controlled Strategy = . ( ) ( ) ( )( ) C s + 1/2 I cos Δ M/V Retaining only the first-order terms, it is possible to obtain (9) the small-signal model, useful to find the frequency response of the converter [9]: The reference current is given by the product between the equivalent conductance, G, and the supply voltage, E (i = G · E ), where G comes from the PI regulator that maintains di L + R i =−Mv cos Δ − m V cos Δ, s 0 the DC link voltage constant: dt Li ω = Mv sin Δ + m V sin Δ, (6) G = k + (V − V ). (10) p 0ref 0 dv 1 1 ( ) ( ) C = Mi cos −Δ + m I cos −Δ . dt 2 2 The block diagram of the DC voltage regulation loop is shown in Figure 3, where only one of the two-input The small-signal model is developed in terms of the Laplace stage 4Q converter is represented. In the block diagram operator. Applying the Laplace transformation to (6)and shown in Figure 3 three control loops can be identified. assuming zero initial conditions, we obtain The first control loop is the DC link voltage control loop comprised of the voltage measurement, the low-pass filter explained above, the comparison with the reference value, ( ) Ls + R i =−Mv cos Δ − m V cos Δ, s 0 V , and the PI controller. Its output is the value of the 0ref equivalent conductance, G, that keeps the DC link voltage Li ω = Mv sin Δ + m V sin Δ, (7) constant while varying the power requested or injected 1 1 by the traction motors and auxiliary services. The second Csv = Mi cos(−Δ) + m I cos(−Δ). 2 2 loop is the reference current generator, which considers the input voltage measurement followed by a filter dedicated to high-frequency disturbances. The obtained value from Instead of the PWM control that generates an AC voltage this loop multiplied with the equivalent conductance, G, trough the modulation index m and its phase δ,in the gives the reference current that the converter, through Smart following a new modulation technique that can directly Modulation, has to generate in order to balance the input control the current value is proposed. and output powers. The third loop is related to the DC Considering the variability of the switching frequency, component compensation in the AC input current. In fact, f ,and thedifficulty defining an adequate current ripple, its output value is a constant current that, when added to the the Smart Modulation, presented in [10, 11]and briefly reference current, allows the DC component to be cancelled described in the appendix, is applied. out in order to avoid saturation of the input transformer. This strategy allows direct control of the current value The DC link voltage variation for small changes of G maintaining, at the same time, a constant value of the given by the regulator is obtained multiplying (9)for the switching frequency. Therefore, the modulation index, m, input voltage E : does not directly appear in the control algorithm, but, assuming a unitary power factor, the converter absorbed ∂v ∂v current is controlled by the equivalent conductance, G.From 0 0 = · E . (11) the second equation of the system in (7) it is possible to ∂G ∂i Advances in Power Electronics 5 Table 2: Main data of the considered High Speed Train input stage Switching Inductance L and DC Link Capacitance C Calcula- converter. tion. Referring to Figure 2, the elements that more influence the control parameters are the switching inductance L and Value M.U. the DC link capacitance C. The inductance value is chosen E 1000 V 2MAX to limit the input current ripple, while the capacitance C V 1800 V influences the DC link oscillation. Therefore, the obtained P 900 kW ass values are L = 1.5mH and C = 200 mF. I 500 A TR I 1800 A MAX Calculation of the Correct Tuning Parameters of the Regulator f 500 Hz sw (Proportional k and the Integral k ). In order to study the p i system stability using the Bode criteria through module and phase diagrams, identified by the open loop transfer function The open-loop function given by the product of (10)and (12), a dedicated code in MATLAB environment has been (11)is implemented. From this program it is possible to calculate FT OL the control parameters k and k , giving as input the specific p i data of the converter considered (Table 2), that have to k ∂v stabilize the converter dynamics. Under these conditions and = k + · s ∂G referring to the studied case, a good compromise to obtain a fast system response and, at the same time, a good phase = k + margin is k = 0.1and k = 3. The Bode diagrams obtained p i from (12) using these values are shown in Figure 4(a).The −((1/2)LI(E /V ))s + (1/2)ME cos(Δ)−(1/2)R I(E /V ) 2 0 2 s 2 0 cut-off frequency that guarantees a good response with a · . (C)s+(1/2)I cos(Δ)(M/V ) 0 sufficient phase margin is about 200 Hz. (12) Calculation of the Tuning Parameters of the Regulator (Pro- The above presented theory leads to a procedure for the portional k and the Integral k )atthe Limit of Stability. The p i converter design. In particular, the following steps can be parameter values for the limit of the stability condition have identified: been investigated using the same Matlab code, obtaining (1) converter parameters identification, such as rated and k = 0.2and k = 3. TherelativeBode diagramsfor these p i maximum powers, AC input and DC output voltages, parameters are shown in Figure 4(b). As can be seen from and consequently input and output currents; this figure, the phase margin is close to zero. (2) switching inductance L and DC link capacitance C calculation. These two parameters are determined Sensitivity Analysis. Applying the same algorithm, it is pos- considering the maximum input current ripple and sible to obtain the effect of changes of the parameters on the the maximum acceptable DC link voltage oscillation; stability of the controller. As it can be shown in Figure 5,an (3) calculation of the correct tuning parameters of the increase of the proportional gain causes significant reduction regulator (proportional k and the integral k )to of the gain margin leading to the loss of stability. On the p i have good stability and at the same time a fast time contrary, the reduction of k improves the stability but it response of the system; degrades the time response performance. Once determined the power system structure, the traction power converters (4) calculation of the tuning parameters of the regulator are mainly subjected to the line voltage variation. In fact, (proportional k and the integral k )atthe limitof p i in traction systems the voltage can vary in the range of stability in order to perform the sensitivity analysis; −33%–+20% of the nominal one. As it can be seen in (12), (5) sensitivity analysis. the line voltage E (reported on the secondary side of the onboard transformer) is common to all the numerator terms, 5. Analysis Application in a Real Case therefore, it can be considered as a proportional gain. The sensibility analysis carried out and presented in Figure 5 The converter design steps previously formulated have been shows that around the optimal value of k (−33%–+20%) applied to a real case as example consisting in the converter p there is enough gain margin to maintain the stability. studied in this paper (Figure 1) employed as input stage onboard High-Speed Railway (HSR) Trains operating on the new Italian HSR lines. 6. Harmonic Absorption due to Single-Phase Supply Converter Parameters Identification. In transportation sys- tems many typologies of power converters are employed that One of the problems that characterize the 4Q converter can have different applications (i.e., front-end converters, and, in general, all of the AC/DC single-phase converters, auxiliary services, UPS, high-speed trains, heavy traction, is the absorption of variable instantaneous power. In fact, naval propulsion) and sizes. The typical size of the converter in contrast to a balanced three-phase system, the flowing and its usual data for this application are reported in Table 2. power in a single-phase system is characterized by a mean 6 Advances in Power Electronics x PI + 0ref ref − DC Smart modulation IGBT IGBT V Figure 3: Block diagram of the system and control loops. Bode diagram Bode diagram 150 150 100 100 50 50 0 0 −50 −50 270 270 225 225 180 180 −2 −1 0 1 2 3 4 −2 −1 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Frequency (rad/s) Frequency (rad/s) (a) (b) Figure 4: Bode diagram of the converter and regulator system for (a) k = 0.1and k = 3and (b) k = 0.2and k = 3. p i p i value equal to the active power requested from the load and leaving the regulator, given as the product between G and the a variable term oscillating at a frequency twice that of the sinusoidal input voltage e = E sin(ωt), is 2 2 supply frequency. This oscillation, in the case of traction converters, causes the generation of a ripple in the DC output I = G · e = [G + G sin(2ωt)] · E sin(ωt) ref 2 0 100 2 voltage. The 4Q converters are equipped with a filter that is 1 1 tuned to a frequency of 100 Hz in 25 kV/50 Hz train systems = E G sin(ωt) + G E cos(ωt) − G E cos (3ωt). 2 0 100 2 100 2 or 33.3 Hz in 15 kV/16.7 Hz train systems. However, it is 2 2 (13) not possible to completely cancel the ripple in the DC link voltage. Due to the regulator response time, the equivalent conductance at the end of the transient state condition The reference current reported in (13) can be defined as the coming out from the regulator can be defined as the sum of sum of the following three terms: its mean theoretic value requested G and a ripple at double frequency G (G = G + G sin(2ωt)). This phenomenon 100 0 100 (i) I = E G sin(ωt) is the theoretical value of the ref 0 2 0 has not been considered in the stability analysis because reference current proportional to the absorption it is not influential; however, it is the cause of harmonic power; it is in phase with the voltage signal and at current absorption from the converter. The reference current the same frequency; Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Advances in Power Electronics 7 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 K K p p (a) (b) Figure 5: Influence of k to the cut frequency and to the gain margin. 2500 1815 0 1800 −625 −1250 −1875 −2500 0.15 0.17 0.19 0.21 0.23 0.25 0.15 0.17 0.19 0.21 0.23 0.25 (s) (s) Figure 7: DC link voltage in end of transient state and stable Figure 6: Current absorbed by one 4Q converter in the traction condition (k = 0.1, k = 3). p i phase in stable condition (k = 0.1, k = 3). p i (ii) I = (1/2)G E cos(ωt) is a term at the same net ref ϕ 100 2 for the model is obtained from a real HSR Train operating frequency, but shifted π/2 radians, which is the source in Italy on a 25 kV/50 Hz line. Each 4Q converter is sized of absorption of the reactive power; for a rated power equal to 900 kW and a maximum peak (iii) I =−(1/2)G E cos(3ωt)is a term at three times ref 3 100 2 power equal to 1500 kW. These are relevant power values for the frequency that determines the presence of a third a switching converter that has to be small and light enough harmonic current. to be installed onboard. The 4Q converters consist of a power circuit and a In order to reduce the G component, it is necessary control system. The power circuit has been modeled with to filter the DC link voltage with a suitable low-pass filter. the circuital elements already available in EMTP-ATP. The This filter tuning is critical because it simultaneously allows MODELS language has been used for implementing the for a fast system response and a low cut-off frequency. It proposed converter control. is impossible to achieve both of these objectives, but for In the following, some simulation results are reported the system stability it is necessary to focus on the system response, which implies a small ripple. in order to verify the outcomes of the performed analysis. The simulation cases considered for the theoretical analysis validation are the following: 7. Model and Simulation Results (A) nominal condition; In order to validate the theoretical analysis described above, (B) loading; a suitable model of the system has been implemented using the EMTP-ATP dynamic simulation tool. The data employed (C) converters regulators working at the stability limit. (A) Cut frequency (Hz) Gain margin (dB) (V) 8 Advances in Power Electronics 4000 3500 2000 1750 0 0 −875 −1000 −2000 −1750 −2625 −3000 −3500 −4000 0.35 0.38 0.41 0.44 0.47 0.5 0.2 0.25 0.3 0.35 0.4 0.45 0.5 (s) (s) Figure 10: DC link current at full load for k = 0.2and k = 3. p i Figure 8: Loading transient of the current absorbed by one 4Q converter in stable condition (k = 0.1, k = 3). p i 2.5 2.3 2.1 1.9 1780 1.7 1.5 0.35 0.38 0.41 0.44 0.47 0.5 1760 (s) 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Figure 11: Equivalent conductance at full load for k = 0.2and (s) k = 3. Figure 9: DC link voltage and filtered signal in loading phase for k = 0.1and k = 3. p i Table 4: Most significant current harmonic components when working at the stability limit. Table 3: Most significant current harmonic components in normal Harm. Amplitude [A] pu operating condition. 1 2038.9 1 Harm. Amplitude [A] pu 3 110.68 0.054 1 1997.5 1 8 87.856 0.043 3 53.6 0.026 10 345.03 0.169 8 79.3 0.039 12 76.157 0.037 10 347.6 0.174 19 68.314 0.033 12 68.8 0.034 21 52.113 0.025 19 71.9 0.036 23 20.975 0.010 21 55.9 0.028 28 32.188 0.015 23 20.0 0.010 28 31.595 0.016 switching frequency of the power GTO due to the high rated power of the converter. Therefore, many harmonic (A) Nominal Condition. The normal operating condition components are generated in the input current. The most considers the traction phase at the nominal power of 3.6 MW significant harmonic components are reported in Table 3. due to four 4Q converters. The parameters k = 0.1and In particular, considerable amplitude is associated to k = 3 obtained for the stability condition are employed the 10th harmonic order, corresponding to the switching in the model. The current absorbed by one 4Q converter is frequency of the converter (17% of the fundamental com- shown in Figure 6. ponent). Moreover, there is the third harmonic component From Figure 6 it is easy to see that the converter (which has an amplitude that is 2.7% of the fundamental commutations are numerically limited because of the low amplitude) and is caused by the interaction between the (A) (A) Advances in Power Electronics 9 ripple at 100 Hz of the DC link voltage and the supply voltage Theequivalent conductance atthe limitof stability at 50 Hz. The THD is equal to 19.07%. assumes, at the end of the transient state condition, a With respect to the DC link voltage, its behavior at the mean value equal to the value at the stability condition end of the transient state condition is shown in Figure 7. because it is correlated to the power transfer requested by the The mean value of the voltage is equal to 1800 V, and it is loads. However, in this last case, there is a highly irregular maintained at a constant value equal to the reference value by oscillation around this mean value that, combined with the the PI regulator. Figure 7 shows the clear oscillation at 100 Hz input sinusoidal voltage, causes the absorption of important due to the oscillation of the power absorbed by the single- current harmonic components (Table 4). phase converter. 8. Conclusions (B) Loading. The loading case, obtained applying a step load to the system, has been simulated in order to stress the In this paper the influence of controller parameters on system with high-power variation for stability test. Indeed, the quality of the absorbed current has been investigated. the transient behaviour has been analyzed in the case of Moreover the application of a current control method to a load change on the DC side from 50 A (presuming that four-quadrant (4Q) input stage converters employed in 10% is dedicated to the auxiliary services functioning) to traction systems has been proposed. In particular, it has 500 A. This means that a power variation of 3.24 MW (from been chosen the Smart Modulation technique with good 360 kW to 3.6 MW) has been simulated. Also in this case, the results, that has been originally developed for low power high parameters k = 0.1and k = 3 obtained for the stability switching frequency converters. The changing of modulation p i condition are employed in the model. technique, from the traditional PWM to the innovative The absorbed current behaviour in the loading transient Smart Modulation, has requested a new and deep stability at t = 0.3 s (instant at which the initial simulation transient analysis due to the changing of the mathematical equations is finished) is shown in Figure 8. that describe the whole system. This analysis has also led to a Moreover, this power variation causes a DC link voltage methodology for setting the regulator parameters. swell equal to 30–35 V (1.6%), as shown in Figure 9 in In order to validate the theoretical analysis above continues line. In fact, in the transient phase, the unbalance described, a circuital model of real cases referred to Euro- between requested/remised power from the traction motors pean High Speed Trains has been implemented in EMTP and absorbed/injected power from the 4Q converter, caused environment. The outputs of the numerous simulations by the delay of the PI regulator, is compensated by the carried out confirm the results of the theoretical analysis energy stored in the smoothing capacitors. The speed of the and furnish important indications for the performance of regulator is an important design parameter because it affects the train and the observance of the European Technical the dynamic of the DC voltage during transient events. A Specification for the Interoperability [1]. In particular, they lower speed can cause high overvoltages that can damage the show how the oscillations in the control loop can cause great electronic components or deep voltage sags that can lead to current harmonic absorption that can interfere with the track loss of converter control. signalling system. The DC link voltage ripple implies that the measured value is not suitable for the regulator because it will cause the Appendix absorption of a high 3 harmonic current as shown in (13). The Smart Modulation (SM) technique is a predictive For this reason, a filter that limits the 100 Hz component, but control that imposes the switching frequency independently is sufficiently fast to follow the DC link voltage transients, is from operating conditions. In addition, the SM allows inserted. The filtered input signal of the converter regulator is shown in Figure 9 by the dashed line. In order to guarantee interlacing of switching instants in case of more converters connected in parallel. Referring to Figures 2 and 3, this algo- the stability of the system, this filtered signal has to limit rithm considers the reference current I and the inductor the 100 Hz component before and after the transient phase ref flux error λ = L · (i − i ) to calculate the commutation maintaining, at the same time, a fast response to follow the ref instants. The expression that defines the ideal voltage V at transient phenomena. 2 the 4Q AC terminals when the reference current is absorbed (C) Converters Regulators Working at the Stability Limit. This is last case has been simulated in order to verify the efficacy di ref of the choice of the regulator parameters. For this reason, v = e + L · + R · i . (A.1) 2 2 ref dt different values of k and k have been inserted in the model. p i In particular, the following examples will demonstrate the Considering that the trajectories of the real current are results obtained using the same values obtained previously linear in a switching period T, it is possible to calculate the for the limit of the stability condition: k = 0.2and k = 3. p i commutation instants as the intersection of such trajectories: The DC link current and the conductance at full load are demonstrated in Figures 10 and 11. For both of these (v + v )(T/2) − λ 2 d Δt = , behaviours it is possible to note a clear increase of an irregular ripple due to the control operation at the limit of (A.2) (v − v )(T/2) + λ d 2 stability. These results confirm the theoretical analysis carried Δt = . out. d 10 Advances in Power Electronics A more detailed description of this algorithm can be found in [10, 11]. References [1] European Technical Specificationfor Interoperability (TSI) of the Railway Lines, 2008. [2] M. Brenna, F. Foiadelli, G. C. Lazaroiu, and D. Zaninelli, “Four quadrant converter analysis for high speed trains,” in Proceedings of the 12th International Conference on Harmonics and Quality of Power, Cascais, Portugal, 2006. [3] S. Ostlund, “Reduction of transformer rated power and line current harmonics in a primary switched converter system for traction applications,” in Proceedings of the 5th European Conference on Power Electronics and Applications (EPE ’93), vol. 7, pp. 112–119, Brighton, UK, september 1993. [4] S. Burdett, J. Allan, B. Mellitt, and J. Taufiq, “A power factor and harmonic comparison of AC railway power electronic traction converter circuits,” in Proceedings of the 5th European Conference on Power Electronics and Applications (EPE ’93),pp. 235–240, Brighton, UK, 1993. [5] W. Runge, “Control of line harmonics due to four-quadrant- converter in AC tractive stock by means of filter and trans- former,” in Proceedings of the 7th European Conference on Power Electronics and Applications (EPE ’97), pp. 3.459–3.464, Trondheim, Norway, 1997. [6] G. W.Chang,H. W.Lin,and S. K. Chen,“Modeling characteristics of harmonic currents generated by high-speed railway traction drive converters,” IEEE Transactions on Power Delivery, vol. 19, no. 2, pp. 766–773, 2004. [7] C.Bachle, H.P.Bauer,and T. Seger, “Requirements on the control of a three-level four quadrant power converter in a traction application,” in Proceedings of the 3rd European Conference on Power Electronics and Applications (EPE ’89), vol. 2, pp. 577–582, Aachen, Germany, 1989. [8] J. Carter, C. J. Goodman, and H. Zelaya, “Analysis of the single- phase four-quadrant PWM converter resulting in steady-state and small-signal dynamic models,” IEE Proceedings Electric Power Applications, vol. 144, no. 4, pp. 241–247, 1997. [9] D. Casadei,J.Clare, L.Empringhamet al.,“Large-signal model for the stability analysis of matrix converters,” IEEE Transactions on Industrial Electronics, vol. 54, no. 2, pp. 939– 950, 2007. [10] M. Brenna, G. C. Lazaroiu, G. Superti-Furga, and E. Tironi, “Bidirectional front end converter for DG with disturbance insensitivity and islanding-detection capability,” IEEE Trans- actions on Power Delivery, vol. 23, no. 2, pp. 907–914, 2008. [11] M. S. Carmeli, F. Castelli Dezza, and G. Superti Furga, “Smart modulation: a new approach to power converter control,” in Proceedings of the 9th European Conference on Power Electronics and Applications (EPE ’01), Graz, Austria, August 2001. 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