High-Performance Indirect Current Control Scheme for ... - IEEE Xplore

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 12, DECEMBER 2014

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High-Performance Indirect Current Control Scheme for Railway Traction Four-Quadrant Converters Liqun He, Student Member, IEEE, Jian Xiong, Hui Ouyang, Pengju Zhang, and Kai Zhang

Abstract—In the recent development of railway traction drives, four-quadrant converters are widely adopted at the grid side as front-end converters. Since the switching frequency of such megawatt-rated converters is limited to several hundreds of hertz, the bandwidth of the current controller is usually restricted. In addition to feedback control of instantaneous current, indirect current control (ICC) is also an alternative in high-power low-switching-frequency applications. Suffering from the inherent cross coupling between active and reactive power as well as sluggish response of ac current, traditional ICC is not a perfect solution for power conversion. To realize the independent control of active and reactive power, a phasor diagram arranged in an orthogonal coordinate system is introduced. In order to improve the dynamic performance of ICC, a transient-free current control method is proposed after analyzing the characteristic of ac current response in detail. In addition, a large-signal model of dc voltage loop, which is based on the conception of average power control, is set up. Finally, robustness to circuit parameters and adaptability to weak grids are certified for the transient-free controller. Both simulation and experimental results are presented to prove the validity of the proposed control scheme. Index Terms—Four-quadrant converter (4QC), indirect current control (ICC), railway traction drive, transient-free method.

I. I NTRODUCTION

H

IGH-POWER ac/dc/ac traction drive systems fed from single-phase ac power supplies are widely used in railway applications [1], [2]. A typical structure of such a system is shown in Fig. 1. At the grid side, four-quadrant converters (4QCs) are widely applied as front end to provide bidirectional power flow, controllable power factor, and constant dc voltage [1]–[4]. The performance of four-quadrant energy conversion mainly depends on the control of active and reactive power, i.e., active and reactive current. For megawatt-level traction drives, switching frequency is limited to several hundreds of hertz, typically 500 Hz or lower. In regions of low railway grid frequency (e.g., 16.7-Hz ac traction power supply in Central Europe), control methods based on instantaneous current can still be applicable for a Manuscript received March 18, 2013; revised August 9, 2013 and December 19, 2013; accepted February 28, 2014. Date of publication April 9, 2014; date of current version September 12, 2014. L. He, J. Xiong, and K. Zhang are with the State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: heliqun714@ hust.edu.cn; [email protected]; [email protected]). H. Ouyang is with the Wuhan Second Ship Design and Research Institute, Wuhan 430064, China (e-mail: [email protected]). P. Zhang is with Delta Electronics (Shanghai) Company, Ltd., Shanghai 200120, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2014.2316240

Fig. 1. Typical structure of an ac-fed railway traction drives system.

4QC [5], [6]. On the other hand, in regions of relatively high grid frequency (50/60 Hz in most parts of the world), carrier ratio diminishes a lot. As a result, control schemes with a fast current loop inside a slow voltage loop are not effective [7]. Instead, ac current and dc voltage are controlled together by indirect current control (ICC), which is also called “phase and amplitude control” [7]–[13]. In ICC, ac current is indirectly controlled by the magnitude and phase of the voltage phasor at the ac side of a 4QC. Therefore, ac current is essentially open-looped, and its transition from one steady state to another is determined by the natural response of the ac-side circuit. The time constant of acside line impedance is usually as long as several fundamental periods, which leads to long-lasting dc offset and significant overshoot current in a dynamic process. These drawbacks can cause dc magnetization of magnetic elements, jeopardize the safe operation of power devices, and even impact the stability of a 4QC [8], [9]. Therefore, dynamic performance of traditional ICC is often limited. In [12] and [13], a modified ICC method based on zero-pole cancelation effectively improves the dynamic performance and expands the safety operation area of ICC, but its effect depends on the parameter precision of the ac side. Furthermore, when fast current response is demanded, the compensator may output a huge value of control voltage, which draws the 4QC into overmodulation operation, and zero-pole cancelation becomes invalid. To realize independent control of active and reactive power for single-phase converters, several decoupling methods have been proposed [14]–[16]. However, these methods are still based on feedback control of instantaneous current. In the case of designing a dc voltage controller, a smallsignal model is generally adopted since the state equations are nonlinear [17]. However, small-signal modeling is quite complex. Stability of the designed controller has to be verified on each operating point, and good control performance is only guaranteed for small perturbation [18], [19]. In this paper, aforementioned issues are investigated, and corresponding solutions are proposed as follows. First, to achieve

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Fig. 3. Fundamental phasor diagram of 4QC ac side (rearranged in an orthogonal coordinate system).

According to steady-state relation, i.e., Uab_p = Us − ωLIsq − RIsp Uab_q = ωLIsp − RIsq uab (t) = (Us − RIsp − ωLIsq ) sin(ωt) − (ωLIsp − RIsq ) cos(ωt).

Fig. 2. (a) Equivalent circuit of 4QC. (b) Fundamental phasor diagram of 4QC ac side (original).

independent control of active/reactive power and improve the dynamic response of ac current, a transient-free method of ICC is proposed after probing into the mathematic characteristic of instantaneous ac current. By this method, ac current can directly shift from an initial steady state to the demanded one without a transient process. Second, the design of the dc voltage controller is modified by building up the average large-signal model, therefore avoiding a small-signal approximation error and realizing large-signal stability. Finally, parameter insensitivity as well as weak-grid adaptability of the proposed current control scheme is proved. II. M ODIFIED ICC A. Retrospect of Traditional ICC Derived from Fig. 1, the equivalent circuit of a 4QC and the ac-side fundamental phasor diagram are presented in Fig. 2(a) and (b), respectively. Variables in the figure include grid voltage us and current is , ac-side inductance L and resistance R, 4QC ac-side voltage uab , dc-link voltage udc and current id , dc capacitance C, parallel resonant branch consisted of L2 and C2 , and equivalent load resistance RL . The resonant frequency of L2 and C2 branch is double fundamental frequency. According to Fig. 2(b), ac current phasor I s can be determined by amplitude Uab and phase angle δab of voltage phasor U ab . In traditional ICC, Uab is used to control power factor cos ϕ, and δab is used to control udc , but cross coupling exists between the two control loops. Particularly in high-power traction drive systems, δab is not small enough because of the large impedance of L; therefore, the cross-coupling effect is more obvious.

us = Us sin(ωt) is = Isp sin(ωt) − Isq cos(ωt) uab = Uab_p sin(ωt) − Uab_q cos(ωt).

(1) (2) (3)

(6)

Isp and Isq represent active and reactive current. It is indicated that active and reactive power flow are controlled by Uab_p and Uab_q . III. T RANSIENT-F REE C URRENT C ONTROL M ETHOD A. Current Response of Traditional ICC Assume that I s is required to change from I s1 to I s2 , and U ab shifts from U ab1 to U ab2 . Then, ΔI s and ΔU ab are the incremental components. As shown in Fig. 4, the ac-side circuit of the 4QC can be split into an initial steady-state component and an incremental component, i.e., is2 = is1 + Δis uab2 = uab1 + Δuab .

(7) (8)

The command of incremental current is Δi∗s . The superscript ∗ “ ” is used to indicate the command value. Define ΔIsp and ∗ ∗ ΔIsq as active and reactive components of ΔI s , then ∗

∗ ∗ Δi∗s (t) = ΔIsp sin(ωt) − ΔIsq cos(ωt).

(9)

The corresponding Δuab can be expressed in the p−q coordinate system as ∗ ∗ ΔUab_p = −RΔIsp − ωLΔIsq ∗ ∗ ΔUab_q = ωLΔIsp − RΔIsq Δuab (t) = ΔUab_p sin(ωt) − ΔUab_q cos(ωt).

(10) (11) (12)

Similar to (4)–(6), (10)–(12) are also based on steady-state operation of the 4QC. The differential equation obtained from Fig. 4(d) is

B. ICC in Orthogonal Coordinate System Rearrange Fig. 2(b) in an orthogonal coordinate system, as shown in Fig. 3. Suppose that the p-axis is in-phase with grid voltage phasor U s , and the q-axis is 90◦ lagging to the p-axis. Decompose U ab and I s to p- and q-axes components, i.e., Uab_p , Uab_q , Isp , and Isq . Thus

(4) (5)

L

d(Δis ) + RΔis = −Δuab , dt

t > 0.

(13)

Suppose that power change happens at t = t0 . The waveform of is should be continuous; therefore, Δis (t0 ) = 0. Substitute (10)–(12) into (13), and the time-domain solution of Δis is given as follows: ∗ R ∗ Δis (t) = − ΔIsp sin(ωt0 ) − ΔIsq cos(ωt0 ) e− L (t−t0 ) ∗ ∗ + ΔIsp sin(ωt) − ΔIsq cos(ωt),

t ≥ t0 .

(14)

HE et al.: HIGH-PERFORMANCE ICC SCHEME FOR RAILWAY TRACTION FOUR-QUADRANT CONVERTERS

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Fig. 4. (a) Simplified equivalent circuit of 4QC ac side (uab has changed from uab1 to uab2 ). (b) Decompose uab2 to initial component uab1 and incremental component Δuab . (c) Simplified equivalent circuit of initial component. (d) Simplified equivalent circuit of incremental component.

Fig. 6. Complete implementation of the proposed transient-free current control method in block diagram. ∗ updated at ωt = kπ [i.e., the dots in Fig. 5(a)], whereas Isq ∗ keeps unchanged (in other words, ΔIsq is zero). For instance, ∗ ∗ is applied and ΔIsq = 0. at t0p in Fig. 5(b) (ωt0p = 2π), ΔIsp ∗ According to (14) and (15), there will be K = ΔIsp × 0 − ∗ sin(ωt). The control 0 × cos(ωt0p ) = 0 and Δis (t) = ΔIsp voltages should be

Fig. 5. Principle of the proposed transient-free method. (a) Waveform of us ∗ and ΔI ∗ is applied (dots: at which and the specialized moments when ΔIsp sq ∗ are applied; circles: at which ΔI ∗ is applied). (b) Waveform of active ΔIsp sq current isp . (c) Waveform of reactive current isq . (d) Waveform of grid current is as a synthesis of isp and isq .

In (14), the second and third items of the solution are the expected steady-state components, whereas the first item is the undesired transient component. Define its maximum absolute value as K, thus ∗ ∗ (15) sin(ωt0 ) − ΔIsq cos(ωt0 ) . K = ΔIsp In the worst case, K could reach the magnitude of rated current (e.g., from no-load to rated-load), and the transient component slowly decays with the time constant of the acside circuit. If this component is eliminated, transient-free performance will be realized. B. Proposed Transient-Free Current Control Method ∗ ∗ From (15), if Isp and Isq are simultaneously refreshed, K is always larger than zero. This paper proposes a transient-free current control method to reduce K to zero, by staggering the control moments of Δisp and Δisq , which will be introduced in the following. The waveform of us is expressed in Fig. 5(a) according to (1). Taking U s as reference phasor, the phases of isp and isq are determined as shown in Fig. 5(b) and (c). The zero-crossings of us are marked with dots, whereas poles of us are marked ∗ should be with circles. In case of active power regulation, Isp

∗ ΔUab_p = −RΔIsp

(16)

∗ ΔUab_q = ωLΔIsp .

(17)

∗ should Similarly, in case of reactive power regulation, Isq be updated at ωt = π/2 + kπ [i.e., the circles in Fig. 5(a)], ∗ keeps unchanged. For instance, at t0q in whereas Isp ∗ ∗ is applied and ΔIsp = 0. There Fig. 5(c) (ωt0q = 3π/2), ΔIsq ∗ will be K = 0 × sin(ωt0q ) − ΔIsq × 0 = 0 and Δis (t) = ∗ cos(ωt). The control voltages should be −ΔIsq ∗ ΔUab_p = −ωLΔIsq

ΔUab_q =

∗ −RΔIsq .

(18) (19)

The waveform of is synthesized by isp and isq is shown in Fig. 5(d). In consequence, transient components of is are eliminated. Fig. 6 shows the implementation of the proposed current control method in block diagram. Referring to the phase of us obtained from phase-locked loop (PLL), whether active or reactive current satisfies the transient-free condition is judged. ∗ ∗ (or ΔIsq ) is applied, substitute (16) and (17) [or When ΔIsp (18) and (19)] into (12), then control voltage of new steady state uab2 is worked out by (8) and sent to the modulation part. Following the above method, K will always be zero, and the transient component is eliminated, consequently preventing dc offset and overshoot in is . In the proposed transient-free ICC, active and reactive currents are independently controlled not only in logic but also in time. While in traditional ICC, the active and reactive currents are simultaneously controlled, and the regulating moments are not specified, which means either the active current or the reactive current or neither of them satisfies the transientfree condition. Compared with traditional ICC, the proposed

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TABLE I PARAMETERS OF A R EDUCED -S CALE P ROTOTYPE

where T1 = 2π/ω is the fundamental period of grid t voltage. Substitute (1) and (2) into (21), define x ¯ = 2/T1 t−(T1 /2) x(τ )dτ to indicate the period-average variable. Then, (21) can be expressed as du2 u2 1 1 Us Isp = Ceq dc + dc . 2 2 dt RL

(22)

Consequently, the large-signal model of u2dc is established. Based on this linear model, voltage controller can be designed without small-signal approximation. For brevity, leave out the period-average symbol “-” and apply u2dc ≈ (udc )2 since the fluctuation of udc is small. The plant of voltage control is P (s) =

2 (s) Udc = Isp (s)

1 2 Us 1 1 2 Ceq s + RL

.

(23)

The complete control structure of 4QC is shown in Fig. 7. The PI-type controller is adopted as voltage controller, i.e., transient-free ICC also promotes current control frequency since both isp and isq are controlled twice in a fundamental period. Hence the control bandwidth of outer voltage loop can be extended. IV. DC-L INK VOLTAGE R EGULATION L OOP The average large-signal model of dc voltage loop is set up. Based on the conception of average power control, this “squared voltage control” method takes u2dc as a state variable. It was initially introduced in a single-phase power-factorcorrected boost converter , whereas in this paper, it is applied to the dc voltage control of the 4QC. In traction drives, a large dc capacitor and an LC resonant branch is usually installed in dc link to absorb the ripple power pulsating at twice line frequency [21], [22]. In this paper, the parameters of the dc capacitor and the LC resonant branch are provided in Table I. Since C2 also supports dc voltage, the energy stored in it should be taken into consideration [23]. Hence, C2 is combined with C as Ceq = C + C2 . Suppose there is no power loss on switching devices, and secondorder ripple in udc has been filtered out [7], [17]. Neglect R since ωL R in most practical applications, then (20) can be derived from instantaneous power balance, i.e., us (t)is (t) = is (t)L

dudc (t) u2dc (t) dis (t) + udc (t)Ceq + . dt dt RL (20)

The average model of (20) on a time scale of T1 /2 is 2 T1

t

2 us (τ )is (τ )dτ = T1

t−T1 /2

2 + T1

t is (τ )L

dis (τ ) dτ dτ

t−T1 /2

t t−T1 /2

2 dudc (τ ) dτ + udc (τ )Ceq dτ T1

t

u2dc (τ ) dτ RL

t−T1 /2

(21)

P I(s) =

Kp s + Ki . s

(24)

Applying the backward-difference method to (24) yields P I(z) =

(Kp + Ki Ts )z − Kp . z−1

(25)

In the proposed transient-free method, since the 4QC gets to ∗ can be regarded the demanded steady state within T1 /2, Isp as coming out to Isp through a zero-order holder of T1 /2. The input active power of 4QC absorbed from the grid side is Ps . Output load power is PL = u2dc /RL . Different values of RL correspond to various load power. In a complete control scheme of the traction drive system, the inverter output power is known to the 4QC, which can be used to accelerate its dynamic process, as indicated by “Load Feedforward” in Fig. 7. A proof of closed-loop stability is given under Ts = T1 /2. Discretize the control system in Fig. 7 with Ts . The z-domain characteristic function of the closed-loop system is 1 − e−Ts s P (s) = 0. (26) 1 + P I(z)Z s “Z” denotes z-transform. Substitute (23) and (25) into (26), then 2z 2 + [Us RL (1 − E)(Kp + Ki Ts ) − 2(1 + E)] z + 2E − Kp Us RL (1 − E) = 0 (27) where E = e−(2Ts /RL Ceq ) . Applying the Jury stability test to (27) yields ⎧ ⎨ Ki > 0 2 Kp > − Us R L (28) ⎩ Kp + T2s Ki < Us2(1+E) . RL (1−E) In the no-load condition, RL → ∞, and (27) becomes U s Ts U s Ts (Kp + Ki Ts ) − 2 z + 1 − Kp = 0. (29) z2 + Ceq Ceq


Fig. 7.

Fig. 8.

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Complete control structure of the proposed transient-free ICC scheme.

Stability region of the dc-link voltage control system.

Then, the stable condition becomes ⎧ ⎨ Ki > 0 Kp > 0 2C ⎩ Kp + T2s Ki < Us Teqs .

(30)

The stability region in terms of Kp and Ki is drawn in Fig. 8. In Fig. 8, the dotted line is the limitation of Kp and Ki in the rated-load condition. The inside solid line is the limitation in the no-load condition. For other load conditions, the limitations are between these two lines. In order to guarantee system stability in the overall range of load, parameters of the voltage controller are designed under the no-load condition, i.e., in Fig. 8, Kp and Ki inside the gray region are available for stable operation. V. E XPERIMENTAL R ESULTS Based on a real 1.2-MW traction drive, a 4-kW reduced-scale prototype is designed to verify the proposed control scheme. The main parameters are listed in Table I. In the real 1.2-MW system, the per-unit value of ac-side inductance is 48%, which is realized by the leakage inductance of the transformer. Hence, in the 4-kW prototype, L is chosen as 17 mH, and the switching frequency is chosen as 500 Hz. Both active and reactive power experiments are carried out to compare the proposed transient-free ICC with traditional ICC. Experiments of active power are implemented in unity power factor, from no-load to rated-load. Fig. 9 shows the dynamic performance of is and udc when suddenly switching from noload to rated-load. In addition to active power, in some cases, traction drive converters need to provide reactive power such as in a low-gridvoltage situation or ice-melting working mode. Experiments of reactive power are implemented in the no-load condition,

from 2 kvar (lagging) to −2 kvar (leading). In Fig. 10, a sudden change in reactive power is caused by a step command ∗ . in ΔIsq In Figs. 9 and 10, subfigure (a) of all present perfect response of is with the proposed control scheme applied. While in subfigure (b), there is significant dc offset as well as overshoots in current response, and the transient process lasts for several grid periods. Moreover, in subfigure (a), fluctuation in dc voltage is also greatly improved. That is because the oscillating power caused by the ac-side transient current is eliminated, and so is dc voltage improved. In practical applications, reactive power control for compensation purposes does not require high dynamics, whereas the change in the active power command can be both quick and unpredictable. Therefore, there will be a delay when adjusting active power according to the transient-free principle, which varies from 0 to T1 /2 (most serious situation). For the sake of generality, the experiment on active power switching at a nonspecific time is given in Fig. 11. Compare Fig. 11(a) with Fig. 9(a), transient-free current response is achieved as well. Fig. 11(b) is a partially enlarged view of Fig. 11(a). The control delay Δt is 9.8 ms, which is very close to the most serious condition. The voltage drop is around 8.8%. VI. D ISCUSSION ON PARAMETER S ENSITIVITY Ideally, the precise value of ac-side impedance, L and R is required in order to calculate Uab_p and Uab_q . However, R is actually the equivalent series resistance at the ac side of 4QC, and it is difficult to get the accurate value. The following theoretical analysis and experimental results will demonstrate that inaccuracy of R or even neglecting it will not lead to a great impact on the transient-free characteristic as well as steadystate operation. Suppose that in an extreme condition, R is totally neglected in the control. Steady-state value ΔIsp and ΔIsq should be recalculated. If a change in active power is required, the control result is (ωL)2 ∗ ΔIsp = (ωL) 2 +R2 ΔIsp (31) ωLR ∗ ΔIsq = − (ωL) 2 +R2 ΔIsp . Similarly, a required change in reactive power will result in ωLR ∗ ΔIsp = (ωL) 2 +R2 ΔIsq (32) 2 (ωL) ∗ ΔIsq = (ωL) 2 +R2 ΔIsq .

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Fig. 9. Comparison of dynamic performance from no-load to rated-load (experimental results). (a) Isp is controlled at ωt = kπ (proposed). (b) Isp is controlled at ωt = π/2 + kπ.

Fig. 10. Comparison of dynamic performance from 2 kvar (lagging) to −2 kvar (leading) (experimental results). (a) Isq is controlled at ωt = π/2 + kπ (proposed). (b) Isq is controlled at ωt = kπ.

Fig. 11. Dynamic performance of the proposed transient-free current control for load switching at a nonspecific time from no-load to rated-load (experimental results). (a) Overall dynamic process. (b) Partial enlarged view at load switching moment.

The derivation process of (31) and (32) is given in the Appendix. As a consequence of neglecting R, steady-state active (or reactive) current gets a little deviation to the command value. Define quality factor Q = ωL/R. Then, the ratio of actual steady-state current to its command value is 2 ΔIsp (or ΔIsq ) = Q . (33) W1 = ∗ ∗ ) ΔIsp (or ΔIsq Q2 + 1 Accompany with deviation, slight cross coupling appears at the same time, i.e., unexpected reactive current emerges during the control of active current, and vice versa. The ratio of unexpected current to the command value is ΔIsq (or ΔIsp ) = Q . (34) W2 = ∗ ∗ ) ΔIsp (or ΔIsq Q2 + 1

A little transient current is introduced by the unexpected current. The ratio of the transient component to the amplitude of actual current is K 1 . (35) = W3 = 2 2 Q2 + 1 ΔIsp + ΔIsq The tendency of W1 − W3 with Q are respectively shown in Fig. 12(a)–(c). When the value of Q increases, actual current gets close to the command value (i.e., W1 approaches 1), whereas the unexpected current and transient current tend to zero (i.e., W2 and W3 tend to 0). The analysis above proves that when Q is large enough, the influence of giving up R is quite small. Fortunately, although the accurate value of R is difficult to obtain and prone to variation, Q 1 is often satisfied in practical applications.


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Fig. 12. Influence of neglecting R on the proposed transient-free ICC. (a) W1 (ratio of actual steady-state current) with Q. (b) W2 (ratio of unexpected steadystate current) with Q. (c) W3 (ratio of transient current amplitude) with Q.

Fig. 13. Dynamic response of the proposed control scheme when R is neglected (experimental results). (a) From no-load to rated-load. (b) From 2 kvar (lagging) to −2 kvar (leading).

To verify the parameter sensitivity of the proposed control scheme, experiments of both active and reactive power are carried out by neglecting R when calculating uab . Fig. 13(a) and (b) presents the experimental results. Although in this prototype, the value of Q is relatively small (less than five), both active and reactive current response are still acceptable. In megawatt-level traction drive systems, per-unit impedance of L can be as large as 50%, whereas power dissipation on series resistance can hardly exceed 5%. Therefore, Q is large enough, and neglecting R is reasonable in the proposed control scheme.

Fig. 14. Simplified equivalent circuit of a 4QC connected to weak grids.

VII. A DAPTABILITY TO W EAK G RIDS Generally, there will be a substation every tens of kilometers. However, sometimes, the distance between substations is so long that grid impedance is comparative to locomotive ac-side impedance. As a result, the railway grid can no longer be modeled as a stiff voltage source and becomes a “weak grid”. Some literatures [22]–[25] have presented that low-frequency oscillation and instability may occur in weak grids. To examine the adaptability to weak grids for the proposed control scheme, simulations are carried out as follows.

A. Single Traction Converter Operating in Weak Grids Shown in Fig. 14 is the equivalent circuit of a 4QC connecting to weak grids, in which all the parameters have been referred to secondary winding of the traction transformer. ugrid is the voltage of strong grid, and its amplitude is Ugrid . us is the input ac voltage of 4QC, and its amplitude is Us as defined in

Fig. 15. (a) Simplified circuit of a weak railway grid with locomotives processed as load. (b) Phasor diagram derived from (a).

Section II. Zgrid is equivalent internal impedance of the weak grid. R, L, and dc-side parameters can be referred to Fig. 2. In the simplified circuit in Fig. 15(a), all the locomotives are regarded as a load in weak grids. Define the rated power of 4QC as Pe and the amplitude of rated input voltage as Ue . When the power factor is unity, locomotives √ can be considered as a resistive load: Zloco = Re = (Ue / 2)2 /Pe . Assuming that us = ue , and Zloco = 1 (per-unit value). Define the impedance of weak grids as Zgrid = jXg . It is chosen as Zgrid = 20% (p.u.), which means short-circuit capability √ is five times√ that of 4QC rated power: (Ugrid / 2)2 /Xg = (Ue / 2)2 /(20%Re ) = 5 Pe . The fundamental phasor diagram in Fig. 15(a) is illustrated in Fig. 15(b). U grid is the phasor of ugrid , and θ is the phase between ugrid and us .

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Fig. 16. Dynamic performance of a 4QC operating in weak grid (Zgrid = 20% Zloco ) when switching from no-load to rated-load (simulation result). (a) Proportion of power command to rated power Pref /Pe . (b) Waveforms of strong grid voltage ugrid , weak-grid voltage us , and dc voltage of 4QC udc . (c) Waveform of grid current is .

Fig. 18. Dynamic performance of three locomotives operating in weak grid (Z_grid = 20% Z_loco ), respectively switching from no-load to 100% Pe , 90% Pe , and 60% Pe at t = t1 , t = t2 , and t = t3 (simulation results). (a) Waveforms of udc1 , udc2 , and udc3 . (b) Waveform of grid current is .

to Fig. 15(b), the active power transmitted from the grid to locomotives is 1 1 2 1 Ugrid − (Is Xg )2 Is . P = Ugrid Is sin θ = Us Is = 2 2 2 (36)

Fig. 17. Simplified equivalent circuit of three locomotives operating in the same weak-grid section.

Simulations of 4QC in such a weak-grid configuration are carried out based on the parameters in Table I. Switching from no-load to rated-load, the results are presented in Fig. 16(a)–(c). Fig. 16(a) shows the ratio of power command Pref to Pe . At t = t1 , it steps from 0 to 1, which means switching from noload to rated-load. Fig. 16(b) shows the waveforms of ugrid , us , and udc . Obviously, us is distorted and lags ugrid after load switching on. Fig. 16(c) shows the waveforms of is . Simulation results indicate that 4QC is in stable operation and that transient-free performance can still be realized in such a weak grid.

B. Multilocomotives Operating in the Same Railway Grid Section Assume that three locomotives are running in the same supply section with the same weak-grid configuration (Zgrid = 20% Zloco ). In the simplified circuit in Fig. 17, ac input current of three locomotives are is_1 , is_2 , and is_3 , respectively. Grid current is = is_1 + is_2 + is_3 . The stability of the overall system is tested by simulations. First, it should be noted that these three locomotives cannot be simultaneously loaded with rated power, and the limitation of maximum power transmission is explained as follows. Regarding all the locomotives as resistive load, the circuit in Fig. 17 can also be equivalent to that in Fig. 15(a). According

By applying mathematical inequality ab ≤ (a2 + b2 )/2, we can get

2 Ugrid − (Is Xg )2 (Is Xg ) P = 2Xg 2 − (Is Xg )2 + (Is Xg )2 1 Ugrid 2Xg 2 √ 2 1 (Ugrid / 2) = . 2 Xg

≤

(37)

This means that in this condition, the maximum power transmitted from grid to load is half of short-circuit capability. When Zgrid = 20% Zloco , the total power of all the locomotives running in the same supply section can reach 2.5 Pe at most. Next, the system stability in the condition of maximum load power is tested by simulation. The three locomotives are in the no-load condition initially and then switch to 100% Pe , 90% Pe , and 60% Pe at t = t1 , t = t2 , and t = t3 , respectively. After t = t3 , the total load power reaches the maximum value 2.5 Pe . Simulation results are shown in Fig. 18. Fig. 18(a) shows the dynamic response of udc1 , udc2 , and udc3 . Finally, they all reach steady state. Fig. 18(b) shows the waveform of grid current is . It also gets to expected steady state after load switching. VIII. C ONCLUSION In the control of railway traction 4QCs, the generally applied ICC has the drawbacks of distinct cross coupling between active and reactive power, undesired dc offset, and overshoot in current transients. To solve these problems, a high-performance


ICC scheme with transient-free current response and improved voltage controller design is proposed. Adjusting moments of active and reactive current are specified, and grid current can immediately get into expected steady-state operation without a transient process, successfully getting rid of dc offset and overshoot in ac current response. Meanwhile, dc voltage fluctuation is alleviated, and current control frequency is doubled compared with traditional ICC. Squared-voltage regulation is adopted in the design of the dc voltage controller, thereby avoiding the complicated small-signal modeling process and approximation error. Experiments on a 4-kW reduced-scale prototype verify the feasibility and effectiveness of the proposed control scheme. Compared with traditional ICC, the dynamics of ac current as well as dc voltage of 4QC are greatly improved. In further research, this control scheme is robust to circuit parameter drift and has adaptability to weak grids. In consequence, it is practical to high-power low-frequency applications. A PPENDIX Derivation process of (31) and (32): ∗ is When a change in active power is ordered, i.e., ΔIsp ∗ applied, ΔIsq = 0. If R is neglected in (10) and (11), the actual control voltages become ΔUab_p = 0 (38) ∗ . ΔUab_q = ωLΔIsp Assume that the actual incremental current in steady state is ΔIsp and ΔIsq , then ΔUab_p = −RΔIsp − ωLΔIsq (39) ΔUab_q = ωLΔIsp − RΔIsq . Combine (38) with (39) and get the equation set −RΔIsp − ωLΔIsq = 0 ∗ . ωLΔIsp − RΔIsq = ωLΔIsp

(40)

Regard ΔIsp and ΔIsq as variables, their solutions are achieved by solving (40), i.e., (ωL)2 ∗ ΔIsp = (ωL) 2 +R2 ΔIsp (41) ωLR ∗ ΔIsq = − (ωL) 2 +R2 ΔIsp . ∗ When a change in reactive power is ordered, i.e., ΔIsq is ∗ applied, ΔIsp = 0. If R is neglected in (10) and (11), similar to (40), we can get ∗ −RΔIsp − ωLΔIsq = −ωLΔIsq (42) ωLΔIsp − RΔIsq = 0.

The solutions of ΔIsp and ΔIsq are ωLR ∗ ΔIsp = (ωL) 2 +R2 ΔIsq ΔIsq =

(ωL)2 ∗ (ωL)2 +R2 ΔIsq .

(43)

Formulas (41) and (43) correspond to (31) and (32) in the text.

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Liqun He (S’12) was born in Hubei, China, in 1989. She received the B.E. degree from Huazhong University of Science and Technology, Wuhan, China, in 2010, where she is currently working toward the Ph.D. degree in the School of Electrical and Electronic Engineering. Her research interests include design and control of power electronic systems, high-power-factor rectifiers, and modular multilevel converters.

Jian Xiong was born in Hubei, China. He received the B.E. degree from the East China Shipbuilding Institute, Zhenjiang, China, in 1993, and the M.E. and Ph.D. degrees from the Huazhong University of Science and Technology (HUST), Wuhan, China, in 1996 and 1999, respectively. In 1999, he joined HUST as a Lecturer, where he was promoted to Associate Professor in 2003. His research interests include uninterruptible power system, ac drives, switch-mode rectifiers, STATCOM, and the related control techniques.

Hui Ouyang was born in Hubei, China, in 1983. He received the B.E., M.E., and Ph.D. degrees from the Huazhong University of Science and Technology (HUST), Wuhan, China, in 2004, 2007, and 2012, respectively. In 2012, he joined the Wuhan Second Ship Design and Research Institute, Wuhan, China, as an Engineer. His research interests include inverter control and railway traction drives.

Pengju Zhang was born in Hubei, China, in 1986. He received the B.E. and M.E. degrees from the Huazhong University of Science and Technology, Wuhan, China, in 2009 and 2012, respectively. He is currently an Electronic Engineer with Delta Electronics Company, Ltd., Shanghai, China. His research interests include wind power converter systems, pulsewidth modulation rectifiers, and traction converters.

Kai Zhang was born in Henan, China. He received the B.E., M.E., and Ph.D. degrees from the Huazhong University of Science and Technology (HUST), Wuhan, China, in 1993, 1996, and 2001, respectively. In 1996, he joined HUST as an Assistant Lecturer. During 2004–2005, he was a Visiting Scholar with the University of New Brunswick, Fredericton, NB, Canada, where he was promoted to Full Professor in 2006. He is an author of more than 40 technical papers. His research interests include uninterruptible power systems, railway traction drives, modular multilevel converters, and electromagnetic compatibility techniques for power electronic systems.