Parallel Three-Phase Interfacing Converters

Parallel‎Three-Phase‎Interfacing‎Converters‎Operation‎ under‎Unbalanced‎Voltage‎in‎Hybrid‎AC/DC‎Microgrid Farzam Nejabatkhah, Student Member, IEEE, Yun Wei Li, Senior Member, IEEE, and Kai Sun, Member IEEE  Abstract-- Parallel interfacing converters (IFCs) with the same DC link are commonly used to handle higher power and currents. In hybrid AC/DC microgrids, they are used to connect AC and DC subsystems, called as interlinking interfacing converters. However, researches on parallel IFCs under unbalanced voltage are quite limited. Considering the adverse effects of unbalanced voltage on single IFC operation (such as output power oscillations, DC link ripples, and output current increase) and possibility of aggregation of these effects for the parallel IFCs with common DC link, a novel control strategy for parallel IFCs with various power factors is proposed in this paper. The proposed control strategy cancels out active power oscillation of parallel IFCs, which results in oscillation-free DC link/subsystem voltage. In this paper, a thorough study on the peak current of individual and parallel IFCs is conducted, and their relationship with active power oscillation mitigation is analyzed. Based on analysis, it is proven that under zero active power oscillation the collective peak current of parallel IFCs is constant under fixed average active and reactive powers. The proposed control strategy keeps the individual IFCs' peak currents in the same phase with collective peak current of parallel IFCs, and thus ensures reduced peak current for redundant IFC. Index Terms-- Hybrid AC/DC microgrids, parallel interfacing converters, power oscillations, peak current, unbalanced voltage.

I. INTRODUCTION N recent years, interests on hybrid AC/DC microgrids are growing rapidly due to increasing penetration of DC power sources and power electronic loads. Since hybrid microgrids integrate ever-increasing power electronics interfaced distributed generations (DGs), storage elements (SEs) and the loads together with nonlinear loads ‎[1], power quality will be an important topic. Especially, unbalanced voltage is one of the important power quality issues. The unbalanced voltage causes adverse effects on the power system equipment ‎[2]-‎[4]. In addition, the unbalanced voltage affects power electronic interfacing converter (IFC) operation. It causes double-frequency power oscillations at the output of IFC. These oscillations are reflected as ripple in the DC link voltage (especially in three-phase power system with small DC link capacitor). Moreover, the peak current of IFC will increase under unbalanced voltage with the same active and reactive power output ‎[5]. In literature, adverse effects of

I

F. Nejabatkhah and Y. W. Li are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4 Canada (e-mail: [email protected]; and [email protected]). K. Sun is with the Department of Electrical Engineering, Tsinghua University, 100084, P. R. China (e-mail: [email protected]).

unbalanced voltage on IFC's operation is addressed by cancelling out IFC output power oscillations and DC link ripple, and controlling the peak current of IFC ‎[6]-‎[12]. In addition to operate under unbalanced voltage, IFC's active and reactive powers can be controlled for unbalanced voltage compensation ‎[13]-‎[15]. However, all aforementioned control strategies are for single IFC. Researches on the control of parallel IFCs under unbalanced voltage conditions are quite limited. In literature, complementary controllers mainly based on negative sequence reactive power control ‎[4], ‎[16], IFCs negative sequence impedance/current control ‎[17]-‎[21] or unbalanced voltage level factor control ‎[22], ‎[23] are added to conventional voltage controllers for unbalanced voltage compensation purposes. However, these control strategies are focusing on parallel DGs/SEs interfacing converters without common DC link. Moreover, they are compensating unbalanced voltage with less attention to its adverse effects on IFCs operation such as output power oscillations and peak current increase. In hybrid AC/DC microgrids, when parallel IFCs with common DC link are employed under the unbalanced voltage, the adverse effects could be amplified. But if controlled properly, the adverse effects could be cancelled out, resulting in enhanced power quality in both the AC and DC subsystems. As a result, study about parallel IFCs with common DC link under unbalanced voltage is necessary. In ‎[24], authors have proposed control strategy for parallel IFCs with common DC link under unbalanced voltage to cancel out active power oscillations. However, only unity power factor (PF) operation mode has been considered in ‎[24]. Moreover, relationship between parallel IFCs' collective peak current and individual IFCs' peak currents under zero active power oscillations have not been addressed in previous study. In this paper, a novel control strategy for parallel IFCs with zero active power oscillation under various PFs is proposed. In this paper, PF is defined based on average active and reactive powers. In the proposed control strategy, a redundant IFC cancels out collective active power oscillations of other IFCs, which results in DC link/subsystem voltage ripple cancellation. Moreover, thorough study on the peak current of individual and parallel IFCs under unbalanced voltage is carried out. Based on the analysis, it is proven that the collective peak current of parallel IFCs under zero active power oscillation is constant in the fixed average active and reactive powers output. Under constant collective peak current, individual IFCs' peak currents can be controlled to be in the same phase with collective peak current of parallel

active and reactive powers are described as:

Utility Grid

AC Subsystem

Hybrid AC/DC Microgrid

Storage Element

SE’s Interfacing Converter

AC Distributed Generation

AC DG’s Interfacing Converter

DC Distributed Generation

Localized AC Loads

DC DG’s Interfacing Converter

𝑝𝑖 = 𝑣 ∙ 𝑖𝑖 & 𝑞𝑖 = 𝑣⊥ ∙ 𝑖𝑖  where 𝑣 = [ 𝑣𝑎 , 𝑣𝑏 , 𝑣𝑐 ] and 𝑖𝑖 = [𝑖𝑎𝑖 , 𝑖𝑏𝑖 , 𝑖𝑐𝑖 ]𝑇 are three-phase PCC voltage vector and the 𝑖 𝑡ℎ -IFC's output current vector, and 𝑣⊥ lags 𝑣 by 90°. Considering symmetric-sequence component of the PCC voltage vector and the 𝑖 𝑡ℎ -IFC's output current vector, (1) is described as: +

AC DG’s Interfacing Converter

DC DG’s Interfacing Converter

DC Distributed Generation

AC Distributed Generation

SE’s Interfacing Converter

𝑞𝑖 =

Localized DC Loads

Storage Element

DC Subsystem

Fig. 1 Typical hybrid AC/DC microgrid.

IFCs, providing reduced peak current for redundant IFC. This paper is organized as following. Instantaneous power analysis of individual and parallel IFCs and constraints for active power oscillation cancellation of parallel IFCs are provided in Section II. In Section III, individual and parallel IFCs' peak currents are studied. Boundary conditions for IFCs' peak currents control under unity and non-unity power factor operations are addressed in Section IV. The proposed control strategy for parallel IFCs under unbalanced voltage is presented in Section V. The simulation results for evaluation of the proposed control strategy's performance are provided in Section VI. Finally, Section VII concludes the main parts of this work. II. INTERFACING CONVERTERS ACTIVE-REACTIVE POWERS ‎Fig. 1 shows a typical hybrid AC/DC microgrid. Here, parallel IFCs between AC and DC subsystems and parallel IFCs of DGs/SEs connected to the AC subsystem can be used to achieve high power and current ratings. In following, active and reactive powers of parallel IFCs under unbalanced voltage are presented, and constraints for active power oscillations cancellation in parallel IFCs are discussed. A. Parallel Interfacing Converters ‎Fig. 2 shows n-parallel IFCs with common DC and AC links. The IFCs are connected to the point of common coupling (PCC) with filters. To derive instantaneous active and reactive powers and collective current of n-parallel IFCs, individual IFC's relations are used. According to instantaneous power theory ‎[5], ‎[25], ‎[26], 𝑖 𝑡ℎ -IFC's instantaneous output Parallel Interfacing Converters IFC1 i1

DC Subsystem/Bus

Output Filter

PCC

in

Output Filter

AC Subsystem/Bus

...

Output Filter

+

where vectors of three-phase PCC voltage vector and 𝑖 -IFC's output current vector, 𝑃𝑖 and ∆𝑃𝑖 are average and oscillatory terms of instantaneous active power, 𝑃𝑖+ and 𝑃𝑖− are the positive and negative sequences of IFC's average active power, 𝑄𝑖 and ∆𝑄𝑖 are average and oscillatory terms of instantaneous reactive power, and 𝑄𝑖+ and 𝑄𝑖− are the positive and negative sequences of IFC's average reactive power. Here, it is assumed that zero sequence of current doesn't exist, and the current vector is only composed of positive and negative sequences. Moreover, the system doesn't contain high-order harmonics and just fundamental component exists. Considering (2) and (3), in unity PF where 𝑄𝑖 = 0, since − + − + 𝑣 + ∙ 𝑖𝑖 and 𝑣 − ∙ 𝑖𝑖 , and 𝑣⊥+ ∙ 𝑖𝑖 and 𝑣⊥− ∙ 𝑖𝑖 are in-phase quantities, the active or reactive power oscillations of 𝑖 𝑡ℎ -IFC can be compensated using scalar coefficient 𝑘𝑝𝑖 as following ‎[10], ‎[27]: ∆𝑃𝑖 = 0 ∆𝑄𝑖 = 0

⇒

− + 𝑣 + . 𝑖𝑝𝑖 = −𝑘𝑝𝑖 𝑣 − . 𝑖𝑝𝑖

𝑘𝑝𝑖 ≥ 0

(4)

⇒

− 𝑣⊥+ . 𝑖𝑝𝑖

𝑘𝑝𝑖 ≥ 0

(5)

=

+ −𝑘𝑝𝑖 𝑣⊥− . 𝑖𝑝𝑖

where subscript "p" is related to active power control. From (2)-(5), the 𝑖 𝑡ℎ -IFC's reference current vector under unity PF can be obtained as following: ∗ + − 𝑖𝑝𝑖 = 𝑖𝑝𝑖 + 𝑖𝑝𝑖 = |𝑣+ |2

𝑃 +𝑘𝑝𝑖 |𝑣− |2

𝑣 + + |𝑣+ |2

𝑃𝑘𝑝𝑖

𝑣−

+𝑘𝑝𝑖 |𝑣− |2

(6)

Similarly, under zero PF operation mode (𝑃𝑖 = 0), the active or reactive power oscillations can be compensated using scalar coefficient 𝑘𝑞𝑖 as following ‎[10], ‎[27]: ∆𝑃𝑖 = 0 ∆𝑄𝑖 = 0

⇒

− + 𝑣 + . 𝑖𝑞𝑖 = −𝑘𝑞𝑖 𝑣 − . 𝑖𝑞𝑖

𝑘𝑞𝑖 ≥ 0

(7)

⇒

− 𝑣⊥+ . 𝑖𝑞𝑖

𝑘𝑞𝑖 ≥ 0

(8)

=

+ −𝑘𝑞𝑖 𝑣⊥− . 𝑖𝑞𝑖

where subscript "q" is related to reactive power control. Considering (2)-(3) and (7)-(8), the 𝑖 𝑡ℎ -IFC's reference current vector under zero PF can be derived as in (9). ∗ + − 𝑖𝑞𝑖 = 𝑖𝑞𝑖 + 𝑖𝑞𝑖 = |𝑣 +|2

𝑄 𝑣+ +𝑘𝑞𝑖 |𝑣− |2 ⊥

𝑄𝑘𝑞𝑖

+ |𝑣 +|2

+𝑘𝑞𝑖 |𝑣 − |2

𝑣⊥−

(9)

Combining (6) and (9), the 𝑖 𝑡ℎ -IFC's reference current vector is obtained as following: 𝑃𝑖 𝑘𝑝𝑖

𝑃𝑖 +𝑘𝑝𝑖 |𝑣− |2

𝑣 + + |𝑣 +|2

+𝑘𝑝𝑖 |𝑣− |2

𝑄𝑖 𝑘𝑞𝑖

𝑄𝑖 +𝑘𝑞𝑖 |𝑣 − |2

𝑣⊥+ + |𝑣+ |2

+𝑘𝑞𝑖 |𝑣− |2

𝑣 −) +

𝑣⊥− )

(10)

Considering (2)-(3) and (10), the 𝑖 -IFC's instantaneous active and reactive powers can be achieved as: 𝑡ℎ

IFCn

Fig. 2 Parallel interfacing converters with common DC and AC links.



+ − − + ∙ 𝑖𝑖 + 𝑣⊥− ∙ 𝑖𝑖 ) + (𝑣⊥+ ∙ 𝑖𝑖 + 𝑣⊥− ∙ 𝑖𝑖 ) = 𝑄𝑖 + ∆𝑄𝑖  + − + − 𝑣 , 𝑣 , 𝑖𝑖 , 𝑖𝑖 are positive and negative sequence 𝑡ℎ

(|𝑣 +|2 v

−

(𝑣⊥+

∗ ∗ 𝑖𝑖∗ = 𝑖𝑝𝑖 + 𝑖𝑞𝑖 = (|𝑣 +|2

iIFCs

IFC2 i2

−

𝑝𝑖 = (𝑣 + ∙ 𝑖𝑖 + 𝑣 − ∙ 𝑖𝑖 ) + (𝑣 + ∙ 𝑖𝑖 + 𝑣 − ∙ 𝑖𝑖 ) = 𝑃𝑖 + ∆𝑃𝑖

AC-DC Subsystem Interlinking Interfacing Converter

𝑝𝑖 = 𝑃𝑖 +

𝑃𝑖 (1+𝑘𝑝𝑖 )(𝑣 + .𝑣− )

⏟|𝑣 +|2+𝑘𝑝𝑖|𝑣− |2 ∆𝑃𝑝𝑖

+

+ .𝑣− ) 𝑄𝑖 (1−𝑘𝑞𝑖 )(𝑣⊥

⏟|𝑣 +|2+𝑘𝑞𝑖|𝑣− |2 ∆𝑃𝑞𝑖

(11)

𝑞𝑖 = 𝑄𝑖 +

𝑃𝑖 (1−𝑘𝑝𝑖 )(𝑣 + .𝑣⊥− ) |𝑣 + |2 +𝑘

⏟

𝑝𝑖

|𝑣− |2

+ .𝑣− ) 𝑄𝑖 (1+𝑘𝑞𝑖 )(𝑣⊥ ⊥

+

(12)

⏟|𝑣 +|2+𝑘𝑞𝑖|𝑣− |2

∆𝑄𝑝𝑖

∆𝑄𝑞𝑖

From (10)-(12), it can be concluded that under unbalanced voltage, 𝑖 𝑡ℎ -IFC's current will increase and the output active and reactive powers will oscillate. The IFC's output current will be balanced under 𝑘𝑝 = 𝑘𝑞 = 0. Moreover, the two components of active power oscillations (∆𝑃𝑝𝑖 and ∆𝑃𝑞𝑖 ) are orthogonal, and 𝑘𝑝𝑖 = −1 and 𝑘𝑞𝑖 = 1 result in zero active power oscillations. It is worth mentioning that aforementioned operating points are independent from average active and reactive powers flow directions. Using individual IFC's relations in (10)-(12), the reference current vector and instantaneous active and reactive powers of n-parallel IFCs in ‎Fig. 2 can be achieved as follows: ∗ 𝑖𝐼𝐹𝐶𝑠 = ∑𝑛𝑖=1 𝑖𝑖∗ = ∑𝑛𝑖=1 |𝑣 +|2

∑𝑛𝑖=1 |𝑣 + |2

𝑄𝑖 +𝑘𝑞

𝑝 = ∑𝑛𝑖=1 𝑃𝑖 + ∑𝑛𝑖=1 ⏟

𝑖

𝑃𝑖 𝑘𝑝

𝑃𝑖 +𝑘𝑝

𝑖

𝑣 + + ∑𝑛𝑖=1 |𝑣 +|2 |𝑣− |2 𝑄𝑖 𝑘𝑞

𝑣 + + ∑𝑛𝑖=1 |𝑣 + |2 |𝑣− |2 ⊥

𝑃𝑖 (1+𝑘𝑝 )(𝑣 + .𝑣 − ) 𝑖

|𝑣 + |2 +𝑘𝑝 |𝑣− |2 𝑖

+𝑘𝑞

+ ∑𝑛𝑖=1 ⏟

𝑖

𝑖

|𝑣 + |2 +𝑘𝑝 |𝑣− |2

𝑖

(13)

+ −) 𝑄𝑖 (1−𝑘𝑞 )(𝑣⊥ .𝑣 𝑖

|𝑣 + |2 +𝑘𝑞 |𝑣− |2

(14)

𝑖

𝑖

+ ∑𝑛𝑖=1 ⏟

+ .𝑣− ) 𝑄𝑖 (1+𝑘𝑞 )(𝑣⊥ ⊥ 𝑖

|𝑣+ |2 +𝑘𝑞 |𝑣− |2

𝑃 |𝑣 + | +𝑘𝑝𝑖 |𝑣− |2 𝑃

𝐼𝑝𝑆𝑖 = |𝑣 +|2 𝑖

Since the two terms of (16) are orthogonal, each term should be equal to zero, which results in following constraints: 𝑃𝑖 (1+𝑘𝑝 )(𝑣+ .𝑣 − ) 𝑖

|𝑣+ |2 +𝑘𝑝 |𝑣− |2

=0

⇒

𝑖

∑𝑛 𝑖=1 𝑃𝑖

𝑃𝑖 +𝑘𝑝 |𝑣− |2 𝑖

∑𝑛𝑖=1 ∆𝑃𝑞 𝑖 = ∑𝑛𝑖=1

= |𝑣 + |2

𝑄𝑖 (1−𝑘𝑞 )(𝑣⊥+ .𝑣 − )

𝑄𝑖 +𝑘𝑞 |𝑣− |2 𝑖

𝑖

|𝑣 + |2 +𝑘𝑞 |𝑣− |2 𝑖 ∑𝑛 𝑖=1 𝑄𝑖

= |𝑣 +|2

(17)

−|𝑣− |2

=0

⇒

Considering (13)-(15) and (17)-(18), the reference current vector, and instantaneous active-reactive powers of n-parallel IFCs under zero active power oscillations are obtained as: ∑𝑛 𝑖=1 𝑃𝑖

− ∑𝑛 𝑃 = |𝑣 +|2 −|2 + |𝑣 +|2𝑖=1 −𝑖|2 𝑣 − −|𝑣 −|𝑣 𝑛 ∑𝑛 𝑖=1 𝑄𝑖 + + ∑𝑖=1 𝑄𝑖 𝑣 − 𝑣 |𝑣 + |2 +|𝑣− |2 ⊥ |𝑣 + |2 +|𝑣− |2 ⊥

𝑣+

+

2(𝑣 + .𝑣− ) ∑𝑛𝑖=1 𝑄𝑖 + |𝑣 +|2 ⊥−|2 ∑𝑛𝑖=1 𝑃𝑖 −|𝑣

+

(19)  

𝑝 = ∑𝑛𝑖=1 𝑃𝑖 + .𝑣 − ) 2(𝑣⊥ ⊥

∑𝑛 𝑄  |𝑣+ |2 +|𝑣− |2 𝑖=1 𝑖

𝑄𝑖 𝑘𝑞𝑖

+ |𝑣 +|2

+𝑘𝑞𝑖 |𝑣 − |2

𝑣−

(22)

𝑣⊥−

(23) −

𝑄 |𝑣 + | +𝑘𝑞𝑖 |𝑣 − |2

𝐼𝑞𝐿𝑖 = |𝑣 +|2 𝑖 𝑄

𝐼𝑞𝑆𝑖 = |𝑣 +|2 𝑖

|𝑣 + |

+𝑘𝑞𝑖 |𝑣 − |2

𝑃𝑖 𝑘𝑝𝑖 |𝑣 − |

+ |𝑣 +|2



+𝑘𝑝𝑖 |𝑣− |2

𝑃𝑖 𝑘𝑝𝑖 |𝑣− |

− |𝑣 +|2

𝑄𝑖 𝑘𝑞𝑖

+ |𝑣 +|2



+𝑘𝑝𝑖 |𝑣− |2 |𝑣− |



+𝑘𝑞𝑖 |𝑣− |2

𝑄𝑖 𝑘𝑞𝑖 |𝑣− |

− |𝑣 +|2



+𝑘𝑞𝑖 |𝑣 − |2

The maximum current at each phase of 𝑖 𝑡ℎ -IFC is the maximum projection of the current ellipse on the 𝑎𝑏𝑐 axis ‎[25]. From (22)-(27), the projection of current ellipse on the 𝑎𝑏𝑐 axis can be derived as ′

∗ 𝑖𝑥𝑖 = (𝐼𝑝𝐿𝑖 𝑐𝑜𝑠 𝛾 − 𝐼𝑞𝐿𝑖 𝑠𝑖𝑛 𝛾) 𝑐𝑜𝑠(𝜔𝑡) + (−𝐼𝑞𝑆𝑖 𝑐𝑜𝑠 𝛾 − 𝐼𝑝𝑆𝑖 𝑠𝑖𝑛 𝛾) 𝑠𝑖𝑛(𝜔𝑡) 𝑥 = 𝑎, 𝑏, 𝑐 (28)

where 𝛾 is rotation angle which is equal to 𝜌, 𝜌 + 𝜋/3 and 𝜌 − 𝜋/3 for abc axis, respectively where 𝜌 is defined as 𝜌 = (𝜃 + − 𝜃 − )/2 . Using (24)-(28), the maximum current at each phase of 𝑖 𝑡ℎ -IFC can be expressed as: 𝑚𝑎𝑥 𝐼𝑥𝑖 =

(18)

+|𝑣− |2

|𝑣 + |

+𝑘𝑝𝑖 |𝑣− |2

∆𝑄𝑞

∑𝑛𝑖=1 ∆𝑃𝑝 𝑖 = ∑𝑛𝑖=1

𝑄𝑖 𝑣+ +𝑘𝑞𝑖 |𝑣− |2 ⊥

+𝑘𝑝𝑖 |𝑣− |2

where 𝑣+ = 𝑉 𝑒 and 𝑣 − = 𝑉 − 𝑒 𝑗(−𝜔𝑡−𝜃 ). Considering (22) and (23), the loci of 𝑖 𝑡ℎ -IFC's active and reactive reference current vectors are ellipses in which their semi major and semi minor axis' lengths can be achieved as:

(15)

(16)

𝑃𝑖 𝑘𝑝𝑖

𝑣 + + |𝑣+ |2

+ 𝑗(𝜔𝑡+𝜃 + )

𝑖

∑𝑛𝑖=1 ∆𝑃𝑝 𝑖 + ∑𝑛𝑖=1 ∆𝑃𝑞 𝑖 = 0

𝑞=

∗ + − 𝑖𝑞𝑖 = 𝑖𝑞𝑖 + 𝑖𝑞𝑖 = |𝑣 +|2

𝑃𝑖 +𝑘𝑝𝑖 |𝑣− |2

𝐼𝑝𝐿𝑖 = |𝑣 +|2 𝑖

B. Active Power Oscillation Cancellation Constraints in Parallel IFCs From (14) in order to cancel out the collective active power oscillations of n-parallel IFCs in proposed control strategy, the following constraint should be satisfied:

∗ 𝑖𝐼𝐹𝐶𝑠 |∆𝑃=0

∗ + − 𝑖𝑝𝑖 = 𝑖𝑝𝑖 + 𝑖𝑝𝑖 = |𝑣+ |2

𝑣− + |𝑣− |2

𝑣− |𝑣 − |2 ⊥

∆𝑄𝑝

∑𝑛𝑖=1 |𝑣 + |2

From (10), 𝑖 𝑡ℎ -IFC's active and reactive reference current vectors can be rewritten as follows:

∆𝑃𝑞

−) 𝑃𝑖 (1−𝑘𝑝 )(𝑣+ .𝑣⊥

∑𝑛𝑖=1 |𝑣 + |2

A. Individual Interfacing Converter

𝑖

𝑖

∆𝑃𝑝

𝑞 = ∑𝑛𝑖=1 𝑄𝑖 + ∑𝑛𝑖=1 ⏟

+𝑘𝑝

III. IFCS' PEAK CURRENTS UNDER UNBALANCED VOLTAGE Under unbalanced voltage, individual IFCs' peak currents should be controlled not to exceed their rating current limits. Moreover, the relation between parallel IFCs' active power oscillation and the IFCs' peak currents should be studied to analyze the influence of active power oscillation cancellation on IFCs' peak currents. As will be shown in parallel IFCs under zero active power oscillations, the collective peak current is constant in the fixed average active-reactive powers.

 

From (19), it is observed that under zero active power oscillations, the current is independent from 𝑘𝑝𝑖 and 𝑘𝑞𝑖 , and is affected by ∑𝑛𝑖=1 𝑃𝑖 and ∑𝑛𝑖=1 𝑄𝑖 variations. Moreover, from (21) it can be concluded that the reactive power oscillations are independent from 𝑘𝑝𝑖 and 𝑘𝑞𝑖 under ∆𝑃 = 0.

𝑃𝑖 2 2 + 2 (|𝑣 | +𝑘𝑝𝑖 |𝑣− |2 ) 𝑄𝑖 2 2 (|𝑣 + |2 +𝑘𝑞𝑖 |𝑣− |2 )

2 + 2|𝑣 + ||𝑣 − |𝑘 cos(2𝛾)) + (|𝑣 + |2 + |𝑣 − |2𝑘𝑝𝑖 𝑝𝑖 2 − 2|𝑣 + ||𝑣 − |𝑘 cos(2𝛾)) − (|𝑣 + |2 + |𝑣 − |2𝑘𝑞𝑖 𝑞𝑖 2𝑃𝑖 𝑄𝑖 |𝑣 + ||𝑣 − |

√

(|𝑣 + |2 +𝑘𝑝𝑖 |𝑣− |2 )(|𝑣 + |2 +𝑘𝑞𝑖 |𝑣− |2 )

(𝑘𝑝𝑖 + 𝑘𝑞𝑖 ) sin(2𝛾)

𝑥 = 𝑎, 𝑏, 𝑐 (29) 𝑡ℎ From (29), the peak current of individual 𝑖 -IFC can be achieved as follows: (30) Moreover, considering (24)-(28), the phase angle of 𝑖 𝑡ℎ IFC's peak current can be derived as 𝑚𝑎𝑥 𝑚𝑎𝑥 𝑚𝑎𝑥 ) 𝐼𝑖𝑚𝑎𝑥 = 𝑚𝑎𝑥(𝐼𝑎𝑖 , 𝐼𝑏𝑖 , 𝐼𝑐𝑖

(

𝛿|𝐼𝑖𝑚𝑎𝑥 = tan −1 (

(

−𝑄 (|𝑣+ |+𝑘𝑞𝑖 |𝑣− |) 𝑃𝑖 (|𝑣+ |+𝑘𝑃𝑖 |𝑣− |) ) cos 𝛾+( 𝑖 2 ) sin 𝛾 2 |𝑣+ | +𝑘𝑃𝑖 |𝑣− |2 |𝑣+ | +𝑘𝑞𝑖 |𝑣− |2

−𝑄𝑖 (|𝑣+ |−𝑘𝑞𝑖 |𝑣− |) −𝑃 (|𝑣+ |−𝑘𝑃𝑖 |𝑣− |) ) sin 𝛾 ) cos 𝛾+( 𝑖 2 2 |𝑣+ | +𝑘𝑞𝑖 |𝑣− |2 |𝑣+ | +𝑘𝑃𝑖 |𝑣− |2

) (31)

Considering (29)-(31), it can be understood that the

amplitude and phase angle of individual IFC's peak current depend on IFC output average active and reactive powers, the PCC positive and negative sequence voltages, and 𝑘𝑃𝑖 and 𝑘𝑞𝑖 . B. Parallel Interfacing Converters Considering (24)-(28), the projection of n-parallel IFCs' collective current ellipse on each phase can be expressed as: ′

′

∗ 𝑖𝑥−𝐼𝐹𝐶𝑠 = ∑𝑛𝑖=1 𝑖𝑥∗ = 𝑃

(∑𝑛𝑖=1 |𝑣+ |2 𝑖

|𝑣+ |

+𝑘𝑝 |𝑣− |2

𝑄

(∑𝑛𝑖=1 |𝑣+ |2 𝑖

𝑖 |𝑣+ |

+𝑘𝑞 |𝑣− |2

𝑄

(∑𝑛𝑖=1 |𝑣+ |2 𝑖

𝑖 |𝑣+ |

+𝑘𝑞 |𝑣− |2

𝑃𝑖

(∑𝑛𝑖=1 |𝑣+ |2

𝑖 |𝑣+ |

+𝑘𝑝 |𝑣− |2 𝑖

𝑃𝑖 𝑘𝑝 |𝑣− |

+ ∑𝑛𝑖=1 |𝑣 +|2

𝑖

+ ∑𝑛𝑖=1 |𝑣 +|2

𝑖

− ∑𝑛𝑖=1 |𝑣 +|2

𝑖

+𝑘𝑝 |𝑣− |2 𝑖 𝑄𝑖 𝑘𝑞 |𝑣 − | +𝑘𝑞 |𝑣− |2 𝑖 𝑄𝑖 𝑘𝑞 |𝑣 − | +𝑘𝑞 |𝑣− |2 𝑖 𝑃𝑖 𝑘𝑝 |𝑣− |

− ∑𝑛𝑖=1 |𝑣 +|2

) cos 𝛾 cos 𝜔𝑡 −

) sin 𝛾 cos 𝜔𝑡 − ) cos 𝛾 sin 𝜔𝑡 −

𝑖

+𝑘𝑝 |𝑣− |2

) sin 𝛾 sin 𝜔𝑡

𝑖

𝑥 = 𝑎, 𝑏, 𝑐 (32) From (32), it is clear that the amplitude and phase angle of the collective current projection on each phase depend on average active and reactive powers, positive and negative sequence of PCC voltage, and 𝑘𝑃𝑖 and 𝑘𝑞𝑖 . In (32), applying the active power oscillation cancellation constraints in (17) and (18), the maximum collective current amplitude at each phase, the peak current, and the peak current phase angle are: 𝑚𝑎𝑥 𝐼𝑥−𝐼𝐹𝐶𝑠 |∆𝑃=0

∑𝑛 𝑃𝑖

((|𝑣+ |𝑖=1 2

2

∑𝑛 𝑄

2

𝑖 ) + (|𝑣 +|𝑖=1 2 +|𝑣− |2 ) ) ×

=√ (|𝑣 + |2 + |𝑣 − |2 − 2|𝑣 + ||𝑣 − | cos(2𝛾)) −|𝑣− |2

𝑥 = 𝑎, 𝑏, 𝑐 𝑚𝑎𝑥 𝐼𝐼𝐹𝐶𝑠 |∆𝑃=0

=

𝑚𝑎𝑥 𝑚𝑎𝑥 𝑚𝑎𝑥 𝑚𝑎𝑥(𝐼𝑎−𝐼𝐹𝐶𝑠 |∆𝑃=0 , 𝐼𝑏−𝐼𝐹𝐶𝑠 |∆𝑃=0, 𝐼𝑐−𝐼𝐹𝐶𝑠 |∆𝑃=0)

(33) (34)

𝑛 + − (|𝑣+ |−|𝑣− |) ∑𝑛 𝑖=1 𝑃𝑖 ) cos 𝛾+(−(|𝑣 |+|𝑣 |) ∑𝑖=1 𝑄𝑖 ) sin 𝛾 2 2 |𝑣+ | −|𝑣− |2 |𝑣+ | +|𝑣− |2 (|𝑣+ |−|𝑣− |) ∑𝑛 𝑄𝑖 (|𝑣+ |+|𝑣− |) ∑𝑛 𝑃𝑖 𝑖=1 ) cos 𝛾−( 𝑖=1 ) sin 𝛾 (− 2 2 |𝑣+ | +|𝑣− |2 |𝑣+ | −|𝑣− |2

( 𝑚𝑎𝑥 | 𝛿|𝐼𝐼𝐹𝐶𝑠 = tan−1 ( ∆𝑃=0

)

(35) From (33)-(35), it can be seen that under zero active power oscillations of parallel IFCs, the collective peak current amplitude and phase angle of parallel IFCs are independent from 𝑘𝑃𝑖 and 𝑘𝑞𝑖 , and are constant values under fixed average active and reactive powers output. C. Discussions Under zero active power oscillation, the collective peak current of parallel IFCs is independent from 𝑘𝑝𝑖 and 𝑘𝑞𝑖 , and it is a constant value under fixed values of active and reactive powers (see (33)-(34)). If all IFCs' peak currents are in the same phase with collective peak current of parallel IFCs, the summation of their peak currents' amplitudes will be reduced. However, considering (29) and (33), under ∆𝑃 = 0 the peak currents of individual IFCs can be in the same phase with collective peak current of parallel IFCs or in different phases, depending on 𝑘𝑝𝑖 and 𝑘𝑞𝑖 values with given 𝑃𝑖 and 𝑄𝑖 . For example in two-parallel IFCs with 𝑃1 = 4𝑘𝑊, 𝑄1 = 7𝑘𝑉𝑎𝑟, 𝑃2 = 5𝑘𝑊 and 𝑄2 = 0.5𝑘𝑉𝑎𝑟, under 𝑘𝑝1 = −0.74, 𝑘𝑞1 = 0.74, 𝑘𝑝2 = −1.20, and 𝑘𝑞2 = 6.27, the system has ∆𝑃 = 0, and the peak currents of two IFCs and the collective peak current of parallel IFCs are in phase 𝑏. In this operating point, 𝑚𝑎𝑥 | the collective peak current is 𝐼𝐼𝐹𝐶𝑠 ∆𝑃=0 = 49.02𝐴 which is

shared between the two IFCs as 𝐼1𝑚𝑎𝑥 = 30𝐴 and 𝐼2𝑚𝑎𝑥 = 𝑚𝑎𝑥 | 23.79𝐴 (the difference between 𝐼1𝑚𝑎𝑥 + 𝐼2𝑚𝑎𝑥 and 𝐼𝐼𝐹𝐶𝑠 ∆𝑃=0 is due to phase angle difference between 𝛿|𝐼1𝑚𝑎𝑥 and 𝛿|𝐼2𝑚𝑎𝑥 ). Under 𝑘𝑝1 = −1.81, 𝑘𝑞1 = −1.39, 𝑘𝑝2 = −0.24, and 𝑘𝑞2 = −15.85 operating point, the collective active power oscillations is zero again. However, the peak current of first, second and collective peak current are in phase 𝑐, 𝑎, and 𝑏, respectively. In this operating point, the collective peak current is similar to previous operating point and equal to 𝑚𝑎𝑥 | 𝐼𝐼𝐹𝐶𝑠 ∆𝑃=0 = 49.02𝐴 (independent from 𝑘𝑃𝑖 and 𝑘𝑞𝑖 under ∆𝑃 = 0) while the first and second IFCs' peak currents are 𝐼1𝑚𝑎𝑥 = 42.05𝐴 and 𝐼2𝑚𝑎𝑥 = 33.36𝐴. As clear from the example, under ∆𝑃 = 0, when all IFCs' peak currents are in the same phase with collective peak current of parallel IFCs, the summation of their peak currents' amplitudes are reduced. Considering aforementioned discussions, 𝑘𝑝𝑖 and 𝑘𝑞𝑖 of individual IFCs can be controlled to lead their peak currents in the same phase with collective peak current of parallel IFCs. IV. BOUNDARY CONDITIONS FOR IFCS' PEAK CURRENTS CONTROL In this section, different conditions in which the peak currents of individual IFCs and collective peak current of parallel IFCs are in the same phase are studied. Considering these conditions, appropriate boundaries are proposed for coefficient factors to keep the peak currents of individual IFCs and collective peak current in the same phase, which leads to smaller peak currents' amplitudes summation of parallel IFCs. Considering (33), among three phases, the collective peak current of parallel IFCs is in the phase where cos(2𝛾) has its minimum value. For individual 𝑖 𝑡ℎ -IFC, the maximum current expression at each phase in (29) can be rewritten as follows: 𝑚𝑎𝑥 𝐼𝑥𝑖 = √𝐹1𝑖 − √(𝐹2𝑖 )2 + (𝐹3𝑖 )2 cos(2𝛾 + 𝛽𝑖 )

(36)

𝑥 = 𝑎, 𝑏, 𝑐

where 𝐹1𝑖 =

𝑃𝑖2

2

2 (|𝑣 + |2 + |𝑣 − |2𝑘𝑝𝑖 )+

2

2 (|𝑣 + |2 + |𝑣 − |2𝑘𝑞𝑖 )

(|𝑣 + |2 +𝑘𝑝𝑖 |𝑣− |2 ) 𝑄𝑖2

(|𝑣 + |2 +𝑘𝑞𝑖 |𝑣− |2 )

(37)

2

𝐹2𝑖 =

2

2|𝑣 + ||𝑣− |×(𝑃𝑖2 𝑘𝑝𝑖 (|𝑣+ |2 +𝑘𝑞𝑖 |𝑣− |2 ) −𝑄𝑖2 𝑘𝑞𝑖 (|𝑣+ |2 +𝑘𝑝𝑖 |𝑣− |2 ) ) (|𝑣 + |2 +𝑘

𝑝𝑖

2 |𝑣− |2 ) (|𝑣 + |2 +𝑘

2𝑃𝑖 𝑄𝑖

𝐹3𝑖 = − (|𝑣 + |2

|𝑣 + ||𝑣− |(𝑘

𝑞𝑖

𝑝𝑖 +𝑘𝑞𝑖 )

(39)

+𝑘𝑝𝑖 |𝑣− |2 )(|𝑣+ |2 +𝑘𝑞𝑖 |𝑣 − |2 )

2 2 𝑃𝑖2 𝑘𝑝𝑖 (|𝑣+ |2 +𝑘𝑞𝑖 |𝑣− |2 ) −𝑄𝑖2 𝑘𝑞𝑖 (|𝑣+ |2 +𝑘𝑝𝑖 |𝑣− |2 ) + 2 − 2 + 2 − 2 −𝑃𝑖 𝑄𝑖 (𝑘𝑝𝑖 +𝑘𝑞𝑖 )(|𝑣 | +𝑘𝑝𝑖 |𝑣 | )(|𝑣 | +𝑘𝑞𝑖 |𝑣 | )

𝛽𝑖 = tan −1 (

(38)

2 |𝑣− |2 )

) +

𝜋 2

(40)

Considering (36)-(40), for individual IFC, the phase with minimum value of cos(2𝛾 + 𝛽𝑖 ) will have maximum current. Therefore, the phase of 𝑖 𝑡ℎ -IFC's peak current depends on 𝛾 (or in other words, 𝜌) and 𝛽𝑖 values. Since under zero active power oscillations, |𝑣 + |, |𝑣 − | and 𝜌 are constant values under fixed average active and reactive powers (see (19)), 𝛽𝑖 (or in other words, 𝑘𝑝𝑖 and 𝑘𝑞𝑖 ) can determine the phase of IFC's peak current. In ‎Fig. 3, conditions in which the peak current of individual 𝑖 𝑡ℎ -IFC and collective peak current of parallel IFCs

under ∆𝑃 = 0 are in different phases are shown with dashed area while in the undashed area these peak currents are in the same phase. From ‎Fig. 3, if 𝛽𝑖 of 𝑖 𝑡ℎ -IFC is in the boundary of 0 < 𝛽𝑖 < 2𝜋⁄3 or 4𝜋⁄3 < 𝛽𝑖 < 2𝜋 and 𝜌 is in the undashed areas, the peak current of that 𝑖 𝑡ℎ -IFC will be in the same phase with collective peak current of parallel IFCs. On the other hand, in the condition that 𝛽𝑖 of individual IFC is in the boundary of 2𝜋⁄3 < 𝛽𝑖 < 4𝜋⁄3, the peak current of that individual IFC and the collective peak current of parallel IFCs will always be in different phases, which is not shown in ‎Fig. 3. It is worth mentioning that in ‎Fig. 3, dashed and undashed areas are controlled by 𝛽𝑖 values (or in other words, by 𝑘𝑝𝑖 and 𝑘𝑞𝑖 values). Since 𝜌 is a constant value under fixed values of average active and reactive powers, 𝛽𝑖 can be controlled to lead individual IFC's peak current to the same phase with collective peak current of parallel IFCs. In the previous example of two-parallel IFCs in Section III with 𝑃1 = 4𝑘𝑊, 𝑄1 = 7𝑘𝑉𝑎𝑟, 𝑃2 = 5𝑘𝑊 and 𝑄2 = 0.5𝑘𝑉𝑎𝑟 where 𝜌 = 35.65° , under 𝑘𝑝1 = −0.74, 𝑘𝑞1 = 0.74, 𝑘𝑝2 = −1.20, and 𝑘𝑞2 = 6.27 operating point, 𝛽1 and 𝛽2 are 0° and 346.7° . Therefore, considering ‎Fig. 3, both IFCs' peak currents are in the same phase with the collective peak current of parallel IFCs, and they are in phase 𝑏. However, under 𝑘𝑝1 = −1.81, 𝑘𝑞1 = −1.39, 𝑘𝑝2 = −0.24, and 𝑘𝑞2 = −15.85 operating point, 𝛽1 and 𝛽2 are 111.51° and 261.09° respectively, which lead peak currents of parallel IFCs to different phases considering ‎Fig. 3. In following, detailed analyses about boundary conditions are presented: A. IFCs under Unity Power Factor Operation Mode Considering (40), under unity PF operation mode (𝑄𝑖 = 0), if 𝑘𝑃𝑖 < 0, 𝛽𝑖 = 0, and if 𝑘𝑃𝑖 > 0, 𝛽𝑖 = 𝜋. Therefore, 𝑘𝑃𝑖 < 0, the peak current of individual 𝑖 𝑡ℎ -IFC will be in the same phase with collective peak current of parallel IFC, regardless the value of 𝜌 and active power flow direction (under 𝛽𝑖 = 0, the dashed areas are not exist in ‎Fig. 3). Considering aforementioned discussions, in the proposed control strategy under unity PF operation mode, 𝑘𝑝𝑖 will be controlled to be less than zero to keep all individual IFCs' peak currents and collective peak current of parallel IFCs in the same phase, leading to reduced peak currents' summation of parallel IFCs. B. IFCs under Non-Unity Power Factor Operation Mode Under non-unity PF operation mode considering (40), determination of boundaries in which the individual IFCs' peak currents are in the same phase with collective peak current of parallel IFCs is challenging, and they should be updated under average active or reactive powers' variations. However, (40) can be simplified and boundaries can be determined under specific relation between 𝑘𝑝𝑖 and 𝑘𝑞𝑖 as: 𝑘𝑝𝑖 + 𝑘𝑞𝑖 = 0

𝑖 = 1, … , 𝑛 − 1

(41)

In this paper since redundant IFC is utilized for active power oscillation cancellation using (17) and (18), (41) may not be applicable for redundant IFC. The relation between 𝑘𝑝𝑛 and 𝑘𝑞𝑛 of redundant IFC will be discussed later.

Phase c   i 2

2 3   i 2 Phase a 2 3  3  i  3 2 Phase b

 Phase b

2 

4  i  3 2

Phase a

4 3   i 2 2 3  3

5  i  3 2

Phase b





Phase c

i 2

5 3 Phase c 4 3 Phase a 5 3   i 2

2 

i 2

Phase b

i 2

4 3

Phase c 8  i  3 2

5 3 Phase a 7 3   i 2

(a) (b) Fig. 3 Relation between the phase of individual 𝑖 𝑡ℎ -IFC peak current and parallel IFCs' collective peak current (peak currents are in the same phase in undashed areas and in different phases in dashed areas); (a) 0 < 𝛽𝑖 < 2𝜋 ⁄3 , and (b) 4𝜋 ⁄3 < 𝛽𝑖 < 2𝜋.

Applying (41) for all IFCs except redundant one, if 𝑘𝑝𝑖 < 0; 𝑖 = 1, … , 𝑛 − 1 (or 𝑘𝑞𝑖 > 0; 𝑖 = 1, … , 𝑛 − 1), the peak current of individual 𝑖 𝑡ℎ -IFC and collective peak current of parallel IFCs will be in the same phase since 𝛽𝑖 = 0, regardless of value of 𝜌 and average active and reactive powers flow directions. Moreover, 𝑘𝑝𝑖 > 0; 𝑖 = 1, … , 𝑛 − 1 (or 𝑘𝑞𝑖 < 0; 𝑖 = 1, … , 𝑛 − 1) will lead peak currents to different phases since 𝛽𝑖 = 𝜋. Considering individual IFC's power oscillation and its peak current in (11) and (29), (41) could also satisfy the active power oscillation-free operation of individual IFCs under 𝑘𝑝𝑖 = −1 and 𝑘𝑞𝑖 = 1, and provide minimum peak current of individual IFC under 𝑘𝑝𝑖 = 𝑘𝑞𝑖 = 0. Therefore, in the proposed control strategy under non-unity PF operation mode, (41) will be applied to all individual IFCs except redundant one, and their 𝑘𝑝𝑖 will be controlled to be less than zero to keep their peak currents in the same phase with collective peak current of parallel IFCs. As mentioned, the redundant IFC cancels out active power oscillations using (17) and (18). As a result, the relation between 𝑘𝑝𝑛 and 𝑘𝑞𝑛 of redundant IFC can be achieved as following: 𝐺𝑛 =

𝑘𝑞𝑛 𝑘𝑝𝑛

=

∑𝑛 ∑𝑛 𝑃𝑖 𝑄𝑖 𝑃𝑖 𝑖=1 𝑄𝑖 −∑𝑛−1 ] )]×[ 𝑖=1 −∑𝑛−1 2 2 𝑖=1 + 2 𝑖=1 + 2 |𝑣+ | +|𝑣− |2 |𝑣 | −𝑘𝑝 |𝑣− |2 |𝑣+ | −|𝑣− |2 |𝑣 | +𝑘𝑝 |𝑣− |2 𝑖 𝑖 ∑𝑛 𝑃𝑖 ∑𝑛 𝑄 𝑖 𝑃𝑖 𝑄𝑖 [𝑃𝑛 −|𝑣+ |2 ( 𝑖=1 ] −∑𝑛−1 )]×[ 𝑖=1 −∑𝑛−1 2 2 𝑖=1 |𝑣+ |2 𝑖=1 |𝑣+ |2 |𝑣+ | −|𝑣− |2 |𝑣+ | +|𝑣− |2 +𝑘𝑝 |𝑣− |2 −𝑘𝑝 |𝑣− |2 𝑖 𝑖

[𝑄𝑛 −|𝑣+ |2 (

(42)

In the proposed control strategy, all IFCs will work under the same power factor. Thus, following relation can be considered between average active-reactive powers of IFCs: 𝑃𝑖 𝑄𝑖

∑𝑛 𝑃𝑖

= ∑𝑛𝑖=1

𝑖=1 𝑄𝑖

=

1 𝑈

𝑖 = 1, … , 𝑛

(43)

where 𝑈 is a number. Assuming |𝑣 − | = 𝑀 × |𝑣 + | in which 𝑀 is the unbalanced ratio and 0 ≤ 𝑀 ≤ 1, (42) can be rewritten as following: 𝐺𝑛 =

𝑘𝑞𝑛 𝑘𝑝𝑛

∑𝑛 ∑𝑛 𝑃𝑖 𝑃𝑖 𝑛−1 𝑖=1 𝑃𝑖 𝑖=1 𝑃𝑖 −∑𝑛−1 𝑖=1 1−𝑘 𝑀2 )]×[ 1−𝑀2 −∑𝑖=1 1+𝑘 𝑀2 ] 1+𝑀2 𝑝𝑖 𝑝𝑖 ∑𝑛 𝑃 ∑𝑛 𝑃𝑖 𝑃𝑖 𝑖=1 𝑃𝑖 −∑𝑛−1 [𝑃𝑛 −( 𝑖=1 2𝑖 −∑𝑛−1 )]×[ 𝑖=1 1+𝑘𝑝 𝑀2 𝑖=1 1−𝑘𝑝 𝑀2 ] 1−𝑀 1+𝑀2 𝑖 𝑖

[𝑃𝑛 −(

=

(44)

In practical power system, 𝑀 is a small value (based on IEEE Standard, typical value of 𝑀 in a three-phase power system under steady state operation is less than 3% ‎[28]). Assuming that |𝑣 + | and |𝑣 − | are constant values and since

k pi  1 Pq Pqi 

Qi (1  kqi ) v  v  2

v   kqi v 

Pi (1  k pi ) v  v 

i 1

v   k pi v 

n 1

Qi (1  kqi ) v  v 

i 1

v   kqi v 

Pqn  

Fig. 4 The variations of 𝐵3 (for redundant IFC) under 𝑘𝑝2 = −𝑘𝑞2 = −0.9, 𝑆3 = 0.5𝑆𝑇 , 𝑈 = 1⁄15 , 𝑀 = 0.3, and different IFCs apparent powers: Case#1: 𝑃1 = 7𝑘𝑊, 𝑄1 = 466.6𝑉𝑎𝑟, 𝑃2 = 2.979𝑘𝑊, 𝑄2 = 198.52𝑉𝑎𝑟 Case#2: 𝑃1 = 5𝑘𝑊, 𝑄1 = 333.3𝑉𝑎𝑟, 𝑃2 = 4.977𝑘𝑊, 𝑄2 = 331.85𝑉𝑎𝑟 Case#3: 𝑃1 = 3𝑘𝑊, 𝑄1 = 200𝑉𝑎𝑟, 𝑃2 = 2.979𝑘𝑊, 𝑄2 = 198.52𝑉𝑎𝑟 Case#4: 𝑃1 = 1𝑘𝑊, 𝑄1 = 66.67𝑉𝑎𝑟, 𝑃2 = 8.977𝑘𝑊, 𝑄2 = 598.52𝑉𝑎𝑟

−1 ≤ 𝑘𝑝𝑖 ≤ 0; 𝑖 = 1, … , 𝑛 − 1, 𝑘𝑝𝑖 𝑀 2 will be small enough to be neglected in (44). Therefore, (44) can be simplified as following: 𝐺𝑛 =

𝑘𝑞𝑛 𝑘𝑝𝑛

=

𝑛 𝑀2 (𝑃𝑛 +𝑀2 ∑𝑛−1 𝑖=1 𝑃𝑖 ) ∑𝑖=1 𝑃𝑖 𝑛 −𝑀2 (𝑃𝑛 −𝑀2 ∑𝑛−1 𝑖=1 𝑃𝑖 ) ∑𝑖=1 𝑃𝑖

=−

𝑃𝑛 +𝑀2 ∑𝑛−1 𝑖=1 𝑃𝑖 𝑃𝑛 −𝑀2 ∑𝑛−1 𝑖=1 𝑃𝑖

(45)

Considering (45), 𝐺𝑛 will be close to −1 depend on the number of parallel IFCs, the power rating of redundant IFC in comparison to other IFCs, and the value of 𝑀. As a result, considering (40), 𝛽𝑛 will be very small value close to zero degree, which results in small shaded area in ‎Fig. 3. In this case, even though we fall in this small area (this area is the transition that the peak current is switched from one phase to another phase), the peak current basically doesn't change that much. In other words, the peak currents of two phases are almost the same. Therefore, it doesn't matter in which phase the peak current is. As a numerical example, three-parallel IFCs have been simulated under the same PF (𝑈 = 1⁄15) and different apparent powers and coefficient factors. In this example, 𝑀 = 0.3, 𝑆3 = 0.5𝑆𝑇 (𝑆𝑇 is total apparent power), 𝑘𝑝1 = −𝑘𝑞1 : −0.9 ⟶ −0.1, 𝑘𝑝2 = −𝑘𝑞2 = −0.9, and the third IFC (redundant one) cancels out active power oscillations. The variations of 𝛽3 under different operating conditions are shown in ‎Fig. 4. From the figure, 𝛽3 is change within 0.4 degree under different operating conditions, leading to small shaded areas. Considering aforementioned discussions, 𝑘𝑝𝑖 ; 𝑖 = 1, … , 𝑛 will be controlled to be less than zero to provide reduced peak currents' summation of parallel IFCs. V. PROPOSED CONTROL STRATEGY FOR PARALLEL IFCS' OPERATION UNDER UNBALANCED VOLTAGE

2

2

2

2

Pp1 Pp2 Pp 3

Ppi 

Pp

Pi (1  k pi ) v  v  2

v   k pi v 

2

Pqn

Fig. 5 Vector representation of ∆𝑃 cancellation using redundant IFC.

voltage oscillation cancellation. In ‎Fig. 5, vector representation of active power oscillations cancellation using redundant IFC is shown. In this figure, it is assumed that 𝑃𝑖 > 0 and 𝑄𝑖 > 0 𝑖 = 1, … , 𝑛, 𝜃 = 𝜃 + − 𝜃 − = 0, and 𝑘𝑝𝑖 ≥ −1 and 𝑘𝑞𝑖 ≤ 1 𝑖 = 1, … , 𝑛 − 1 (except redundant IFC). As a result, redundant IFC works under 𝑘𝑝𝑛 ≤ −1 and 𝑘𝑞𝑛 ≥ 1 to produce 180-degree out-of-phase power oscillation to cancel out active power oscillations. A. IFCs under Unity Power Factor Operation Mode In this control strategy, the redundant IFC is controlled based on active power oscillation cancellation constraints in (17) to cancel out ∆𝑃𝑃𝑖 part of active power oscillations of parallel IFCs, and the other IFCs are controlled based on their peak currents rating limits. The block diagram of the proposed control strategy is shown in ‎Fig. 6. In this control strategy, 𝑚𝑎𝑥 𝐼𝑃𝑖 can be measured or calculated using (29)-(30), and the system is started under 𝑘𝑝𝑖 = −1; 𝑖 = 1,2, … , 𝑛 where ∆𝑃𝑝 𝑖 = 0. If the peak current of each 𝑛 − 1 IFC exceeds its

rating current limit (𝐼𝑖𝑟𝑎𝑡𝑒 ), its 𝑘𝑝𝑖 will move toward zero to limit its peak current on the rating value. As a result, the peak currents of IFCs except the redundant one are constant values under fixed average active powers in different operation conditions (under 𝑘𝑝𝑖 = −1 or under their rating values). All information of 𝑛 − 1 IFCs (𝑃𝑖 and 𝑘𝑝𝑖 ) is sent to redundant IFC's controller. Based on information, 𝑘𝑝𝑛 of the redundant converter is determined using (17) to cancel active power oscillations ‎[24]. In this control strategy, IFCs operate under 𝑘𝑃𝑖 ≤ 0, so IFCs' peak currents will be in the same phase with collective peak current of parallel IFCs as discussed in Section IV-A, which leads to reduced peak currents' summation of IFCs (see Section III-C). Since all IFCs' peak currents except the redundant one are constant values under fixed average active -1 PI

+

I Pmax i

+

A novel control strategy is proposed for parallel IFCs with various PFs under unbalanced voltage, which reduces the adverse effects of unbalanced voltage on IFCs' operation. In the proposed control strategy, one interfacing converter which has the largest power rating among parallel IFCs, named as redundant IFC, cancels out active power oscillations produced by other parallel IFCs, which results in DC subsystem/link

Pq2

n 1

Ppn  

Pq3 Pq1

Ppn

kqi  1

kqi  1

2

k Pi  1

+ k Pi i  1,..., n  1

I irate

Pi i  1,..., n

kpn Calculation using (17)

Fig. 6 The proposed control strategy under unity PF operation mode.

k Pn

powers in different operation conditions (see ‎Fig. 6), the redundant IFC's peak current will be reduced. This capability of control strategy is important since the redundant IFC cancels out other IFCs active power oscillations and then its peak current will be much higher. The variation of IFCs' average powers flow direction and values will not affect the proposed control strategy performance although the operating point will be changed. B. IFCs under Non-Unity Power Factor Operation Mode The proposed control strategy block diagram under nonunity PF mode is shown in ‎Fig. 7. Considering the block diagram, the performance of control scheme under non-unity PF mode is similar to unity PF operation mode in which all 𝑛 − 1 IFCs are controlled based on their current rating limits, and redundant IFC is controlled based on information communicated from other IFCs (𝑃𝑖 , 𝑘𝑃𝑖 , 𝑄𝑖 , 𝑘𝑞𝑖 ) and using (17)-(18) to cancel out ∆𝑃𝑃𝑖 and ∆𝑃𝑞𝑖 parts of active power oscillations of parallel IFCs. From ‎Fig. 7, the system is started under 𝑘𝑝𝑖 = −1; 𝑖 = 1,2, … , 𝑛 (result in 𝑘𝑞𝑖 = 1 considering (41)) which provides zero active power oscillation of individual IFCs. This operating point will be changed if each IFC peak current exceeds its rating limit (𝐼𝑖𝑟𝑎𝑡𝑒 ). Thus, 𝑘𝑝𝑖 of those IFCs that hit the current limits will move toward zero (𝑘𝑞𝑖 will move toward zero, too), and the redundant IFC will cancel out active power oscillation produced by peak currents control. Here, 𝐼𝑖𝑚𝑎𝑥 can be measured or calculated using (29). In the proposed control strategy, 𝑘𝑝𝑖 ≤ 0; 𝑖 = 1, … , 𝑛 − 1, and relations between coefficient factors of 𝑘𝑝𝑖 and 𝑘𝑞𝑖 for all IFCs except redundant one are controlled using (41), which lead those IFCs peak currents to the same phase with collective peak current of parallel IFCs, as discussed in Section IV-B. For redundant IFC as mentioned in Section IVB, since relation between coefficient factors of 𝑘𝑝𝑛 and 𝑘𝑞𝑛 is very close to −1, 𝑘𝑝𝑛 ≤ 0 may lead the redundant IFC's peak current to the same phase with collective peak current. In case that they are not in the same phase, the peak current basically doesn't change that much due to small value of 𝛽𝑛 . As a result, 𝑘𝑝𝑖 ≤ 0; 𝑖 = 1, … , 𝑛 provides reduced peak currents summation of parallel IFCs (see Section III-C). Due to constant peak current values of IFCs except redundant one under fixed average active-reactive powers (see ‎Fig. 7), reduced redundant IFC's peak current is achieved. Similarly, average active and reactive power flow direction will not affected the performance of proposed control strategy.

-1 PI

+

+

I imax

+

k Pi

-1

I irate

Pi Qi i  1,..., n

kpn and kqn Calculations using (17) and (18)

kqi i  1,..., n  1 k Pn kqn

Fig. 7 The proposed control strategy under non-unity PF operation mode.

Qi

Pi k pi kqi

    ii* Pk Qi kqi P Q i pi ii*  i*pi  iqi*   2 i v  v    2 i v  v   2  2   2  2  2  v  k v 2    v  k pi v v  kqi v pi    v  kqi v  i  1,..., n

i  1,..., n

v

v

Sequence Extractor

v

Fig. 8 Individual IFCs' reference current production scheme.

C. Control Scheme In the proposed control strategy, distributed control structure ‎[29] can be used for parallel IFCs' control. The 𝑘𝑝𝑖 and 𝑘𝑞𝑖 of IFCs are generated in the outer control layer of IFCs' local controllers by the proposed control strategy (see ‎Fig. 6 and ‎Fig. 7). These parameters are used in the inner control loops for individual IFCs' reference currents production, which is shown in ‎Fig. 8. In the control system, it is assumed that IFCs average active-reactive powers have been determined in high-level control layer, and their flow can be bidirectional in each IFC. In this paper, all parallel IFCs operate on power/current control mode in which the output currents of individual IFCs are controlled to regulate output average powers on their reference values ‎[1], ‎[5]. Among different current control methods, closed-loop current control with practical proportional-resonant (PR) controller in the stationary 𝛼𝛽 reference frame is adopted ‎[30]. Frequencylocked loop (FLL) based sequence extractor in ‎[31] is used to separate the PCC voltage into positive and negative sequences among different sequence extractors ‎[31]-‎[33]. This extractor is utilized to update the resonant frequency of PR in the feedback loop. In addition, due to utilization of practical PR controllers, the controllers are robust enough under slight frequency variation in a typical utility grid ‎[34]. It is worth mentioning again that in this paper, just the fundamental positive and negative sequence components are exist and controlled. Therefore, some frequency change doesn't affect the control gain that much. For harmonics, it is a different scenario, and adaptive resonant control should be used ‎[34]-‎[36]. For non-linear phenomena such as limit cycle, proper current tracking controller can be adopted to address it ‎[37]-‎[38]. For example, limit cycle oscillations are addressed in ‎[39]-‎[42] where these oscillations are suppressed. Since the main focus of this work is producing reference current vectors for individual IFCs to reduce the adverse effects of unbalanced voltage on parallel IFCs operation (cancelling out active power oscillations, controlling the peak currents of IFCs not to exceed their rating limits, and reducing the redundant IFC's peak current), more detailed design of the current controller is out of the scope of this work. VI. SIMULATION VERIFICATIONS Three parallel IFCs have been simulated using the proposed control strategy in MATLAB/Simulink. The simulated system parameters are shown in ‎TABLE I. In the simulations, the third IFC with largest power/current rating is considered as the redundant converter. Two-phase

Fig. 9 𝑘𝑃𝑖 coefficient factors of the first, second, and third IFCs.

Fig. 10 Peak currents' of IFCs.

Fig. 11 First IFC's output active power.

Fig. 12 Second IFC's output active power.

Fig. 13 Third IFC's output active power.

Fig. 14 Parallel IFCs' collective active power.

Fig. 15 DC link voltage.

unbalance fault with the fault resistance of 2Ω is applied to the system at 𝑡 = 0.15𝑠 as a source of unbalanced voltage. During 0.15 < 𝑡 < 0.3, 𝑘𝑝𝑖 and 𝑘𝑞𝑖 of all IFCs are set on zero. At 𝑡 = 0.3𝑠, the proposed control strategy is applied to cancel out active power oscillations, which results in oscillation-free DC link voltage. A. IFCs under Unity Power Factor Operation Mode In this simulation, during 0 < 𝑡 < 0.75, the average active powers of IFCs are 𝑃1 = 6𝑘𝑊, 𝑃2 = 2.7𝑘𝑊, and 𝑃3 = 3𝑘𝑊. At 𝑡 = 0.75, average active powers of IFCs are modified into 𝑃1 = 7𝑘𝑊, 𝑃2 = 3𝑘𝑊, and 𝑃3 = 2.5𝑘𝑊. The simulation results are shown in ‎Fig. 9 to ‎Fig. 16. As mentioned, the proposed control strategy is applied at 𝑡 = 0.3𝑠. During 0.3 < 𝑡 < 0.5, the proposed control strategy initially sets 𝑘𝑝1 , 𝑘𝑝2 and 𝑘𝑝3 on −1 (see ‎Fig. 9). Although the initial set point provides zero active power oscillations for IFCs (see ‎Fig. 11 to ‎Fig. 13), the first and second IFCs' peak currents exceed their rating limits (see ‎Fig. 10). Therefore, at 𝑡 = 0.5𝑠, the proposed control strategy sets the first and second IFCs' peak currents on their rating limits (see ‎Fig. 10) by moving 𝑘𝑝1 and 𝑘𝑝2 toward zero (see ‎Fig. 9), and cancels out active power oscillations produced by the IFCs' peak currents control using redundant IFC (see ‎Fig. 11 to ‎Fig. 14). The cancellation of parallel IFCs' active power oscillations provides oscillation-free DC link voltage which is shown

Fig. 16 Parallel IFCs' collective reactive power.

in ‎Fig. 15 (𝑡 > 0.3𝑠, ∆𝑉𝑝𝑒𝑎𝑘−𝑝𝑒𝑎𝑘 |𝑚𝑎𝑥 = 0.4𝑉). After average powers variations at 𝑡 = 0.75, the proposed control again sets the first and second IFCs’‎peak‎currents‎on‎their‎rating‎limits since they hit limits, and provides zero collective active power and DC link voltage oscillations by redundant IFC. According to the results, the collective peak current of parallel IFCs is independent from 𝑘𝑝𝑖 variations under ∆𝑃 = 0, and it is just affected by average active powers' variations. In this simulation, under 0.5 < 𝑡 < 0.75𝑠, 𝜌 = 45.26° and 𝛽1 = 𝛽2 = 𝛽3 = 0°, and under 0.75 < 𝑡 < 1𝑠, 𝜌 = 46.25° and 𝛽1 = 𝛽2 = 𝛽3 = 0°, so all IFCs' peak currents and the collective peak current of parallel IFCs are in phase 𝑏, as expected from ‎Fig. 3. As a result, the redundant IFC has its minimum possible peak current value. The collective reactive power of parallel IFCs is shown in ‎Fig. 16 which is constant TABLE I.

SYSTEM PARAMETERS FOR SIMULATIONS. Symbols

DC link voltage IFCs' power rating IFCs' current ratings Grid voltage (rms) and frequency Grid coupling impedance

𝑣𝑑𝑐 𝑆𝑖 𝐼𝑖𝑟𝑎𝑡𝑒

Parameters 800𝑉 𝑆1 = 9𝑘𝑉𝐴; 𝑆2 = 4𝑘𝑉𝐴; 𝑆3 = 10𝑘𝑉𝐴 𝐼1𝑟𝑎𝑡𝑒 = 30𝐴; 𝐼2𝑟𝑎𝑡𝑒 = 13𝐴 𝐼3𝑟𝑎𝑡𝑒 = 34𝐴

𝑣𝑔 - 𝑓𝑔

240𝑉 , 60𝐻𝑧

𝑍𝐺𝑟𝑖𝑑

𝑅𝐺𝑟𝑖𝑑 : 0.2𝛺; 𝑋𝐺𝑟𝑖𝑑 : 1.88𝛺

Fig. 17 𝑘𝑃𝑖 coefficient factors of the first, second, and third IFCs.

Fig. 18 𝑘𝑞𝑖 coefficient factors of the first, second, and third IFCs.

Fig. 20 First IFC's output active power.

Fig. 21 Second IFC's output active power.

Fig. 22 Third IFC's output active power.

Fig. 24 DC link voltage.

Fig. 25 Parallel IFCs' collective reactive power.

Fig. 23 Parallel IFCs' collective active power.

under ∆𝑃 = 0 and just affected by average active powers' variations. B. IFCs under Non-Unity Power Factor Operation Mode In this operation mode, average active and reactive powers of IFCs are set on 𝑃1 = 7.5𝑘𝑊, 𝑄1 = 3.17𝑘𝑉𝑎𝑟, 𝑃2 = 1.5𝑘𝑊, 𝑄2 = 634.6𝑉𝑎𝑟, 𝑃3 = 4𝑘𝑊, and 𝑄3 = 1.69𝑘𝑉𝑎𝑟 during 𝑡 < 0.75. At 𝑡 = 0.75, the average active and reactive powers of IFCs are changed into 𝑃1 = 8𝑘𝑊, 𝑄1 = 2.66𝑘𝑉𝑎𝑟, 𝑃2 = 2𝑘𝑊, 𝑄2 = 666.6𝑘𝑉𝑎𝑟, 𝑃3 = 3.5𝑘𝑊, and 𝑄3 = 1.16𝑘𝑉𝑎𝑟. The simulation results are shown in ‎Fig. 17 to ‎Fig. 25. The proposed control strategy is applied at 𝑡 = 0.3𝑠, and it initially sets 𝑘𝑝1 , 𝑘𝑝2 and 𝑘𝑝3 on −1 during 0.3 < 𝑡 < 0.5 (see ‎Fig. 17), which results in 𝑘𝑞1 = 𝑘𝑞2 = 𝑘𝑞3 = 1 (see ‎Fig. 18). This initial set point provides zero active power oscillations of IFCs (see ‎Fig. 20 to ‎Fig. 22), however the first IFC’s‎peak‎current‎exceeds‎its‎rating‎limit‎(see ‎Fig. 19). As a result, at 𝑡 = 0.5𝑠, control system moves 𝑘𝑝1 toward zero (consequently 𝑘𝑞1 moves toward zero, too) to reduce the first‎IFC’s‎peak‎current‎and‎set‎it‎on‎rating‎ value‎(see‎ ‎Fig. 17 to ‎Fig. 19). The produced active power oscillation by the first IFC’s‎ current‎ control‎ (see‎ ‎Fig. 20) is cancelled out by redundant IFC (see ‎Fig. 22 and ‎Fig. 23). Since the second IFC’s‎ peak‎ current‎ doesn’t‎ hit‎ the‎ rating‎ limit,‎ it‎ operates‎ under 𝑘𝑝2 = −1 and 𝑘𝑞2 = 1 with oscillation-free output

Fig. 19 Peak currents' of IFCs

active power (see ‎Fig. 21). After average powers variations at 𝑡 = 0.75𝑠, the control system has similar performance since the‎ first‎ IFC’s‎ peak‎ current‎ exceeds‎ the‎ rating‎ limit,‎ and‎ the‎ second‎ IFC‎ peak‎ current‎ doesn’t hit the rating limit. In this simulation, cancellation of collective active power oscillations by redundant IFC provides oscillation-free DC link voltage as shown in ‎Fig. 24 (𝑡 > 0.3𝑠, ∆𝑉𝑝𝑒𝑎𝑘−𝑝𝑒𝑎𝑘 |𝑚𝑎𝑥 = 0.6𝑉). In this simulation, during 0.5 < 𝑡 < 0.75𝑠, 𝜌 = 39.85°, and 𝛽1 = 0°, 𝛽2 = 0°, 𝛽3 = 357.23° and under 0.75 < 𝑡 < 1𝑠, 𝜌 = 41.1°, and 𝛽1 = 0°, 𝛽2 = 0°, 𝛽3 = 356.09°. Therefore, as expected, the dashed areas in ‎Fig. 3 are very narrow. Furthermore, all‎ IFCs’‎ peak‎ currents‎ and‎ collective‎ peak current of parallel IFCs are in phase 𝑏 (see ‎Fig. 3), which provides reduced peak current of redundant IFC. In this operation mode, the collective peak current of parallel IFCs under ∆𝑃 = 0 is independent from 𝑘𝑝𝑖 and 𝑘𝑞𝑖 . However, each IFC average active and reactive powers variation will affect its value. Moreover, the collective reactive power oscillation is constant under ∆𝑃 = 0 and depends on average active-reactive powers (see ‎Fig. 25). VII. EXPERIMENTAL VERIFICATIONS The proposed control strategy is applied into two-parallel IFCs' experimental setup to verify its performance under different operating conditions. The experimental setup

EXPERIMENTAL SETUP SPECIFICATIONS. Symbols

Parameters

DC link voltage

𝑣𝑑𝑐

400𝑉

IFCs' current ratings

𝐼𝑖𝑟𝑎𝑡𝑒

Unbalance grid voltage IFCs' output filters Switching frequency

IFCs output three-phase currents (A)

0

-14.26

IFC#2 P1  P2  600W ; Q1  Q2  300Var kP1  0.6; kP 2  1.4; kq1  0.6; kq 2  1.4

P1  P2  600W ; Q1  Q2  300Var k P1  k P 2  1

Individual IFCs and parallel IFCs active powers (W)

Fig. 26 First and second IFCs' output three-phase currents under non-unity PF, 2A/div; (50ms/div). 2000 1800 1600

Parallel IFCs

1400 1200 1000

IFC#1

800 600 400

IFC#2

200 0 0.46

0.48

0.5

P1  P2  600W ; Q1  Q2  300Var k P1  k P 2  1

0.52

= 4𝐴;

𝐼2𝑟𝑎𝑡𝑒

= 5𝐴

𝑣𝑔 𝐿𝑓 -𝐶𝑓

3.6𝑚𝐻-4𝜇𝐹

𝑓𝑠

10𝑘𝐻𝑧

0.56

IFC#1 5.74

0

IFC#2 P1  P2  600W ; Q1  Q2  300Var kP1  0.6; kP 2  1.4; kq1  0.6; kq 2  1.4

P1  600W ; P2  400W ; Q1  Q2  300Var k P1  0.6; k P 2  2.3; kq1  0.6; kq 2  1.4

Fig. 28 First and second IFCs' output three-phase currents under non-unity PF, 2A/div; (50ms/div). 2000 1800 1600 1400

Parallel IFCs

1200 1000

IFC#1

800 600 400 200 0 0.86

𝑉𝐴 = 55∠0°; 𝑉𝐵 = 83.8∠250.9°; 𝑉𝐶 = 83.8∠109.1°

0.54

P1  P2  600W ; Q1  Q2  300Var kP1  0.6; kP 2  1.4; kq1  0.6; kq 2  1.4

Fig. 27 First and second IFCs' output active powers and collective active power of parallel IFCs under non-unity PF.

Individual IFCs and parallel IFCs active powers (W)

In this paper, a control strategy for parallel IFCs operation under unbalanced voltage is proposed. The proposed control strategy cancels out collective active power oscillations of parallel IFCs using one IFC with largest power rating among parallel IFCs, called redundant IFC. The proposed control strategy has the capability to be applied to parallel IFCs with various PFs and different average active-reactive powers in terms of flow directions and values. In this paper, based on thorough analysis on the peak currents of IFCs, it is proven

𝐼1𝑟𝑎𝑡𝑒

5.74

-14.26

VIII. CONCLUSION

TABLE II.

IFC#1

IFCs output three-phase currents (A)

specifications are listed in ‎TABLE II. Here, the results of IFCs under non-unity PF operation mode are shown and discussed, which can be extended to unity PF operation mode. In the experiments, the second IFC is considered as the redundant converter. Moreover, two-phase to ground fault is applied as a source of unbalance condition. In this experiment, the output currents of IFCs and grid voltage waveforms are achieved by scopecorder (YOKOGAWA DL850E), and their saved data is used in MATLAB/Simulink to achieve active and reactive powers waveforms. Under non-unity PF operation mode, at the beginning, the average active and reactive powers of IFCs are set on 𝑃1 = 600𝑊, 𝑃2 = 600𝑊, 𝑄1 = 300𝑉𝑎𝑟, 𝑄2 = 300𝑉𝑎𝑟, and power coefficients are adjusted on 𝑘𝑝1 = 𝑘𝑝2 = −1 and 𝑘𝑞1 = 𝑘𝑞2 = 1 (power oscillation of IFCs are zero). Since the first IFC's peak current exceeds its rating limit, the proposed control strategy adjusts the power coefficients on 𝑘𝑝1 = −0.6, 𝑘𝑝2 = −1.4, 𝑘𝑞1 = 0.6, and 𝑘𝑞2 = 1.4 to set the first IFC's peak current on its rating limit and provide zero collective active power oscillation of parallel IFCs. The first and second IFCs three-phase currents are shown in ‎Fig. 26. In ‎Fig. 27, the first and second IFCs' output active powers and parallel IFCs collective active power are shown. As seen from this figure, the active power oscillation of first IFC is cancelled out by redundant IFC (second one). In this figure, small errors in active power oscillation cancellation are due to errors in voltage and current measurements. In order to evaluate the performance of proposed control strategy under the variation of average active-reactive powers, these powers are modified into 𝑃1 = 600𝑊, 𝑃2 = 400𝑊, 𝑄1 = 300𝑉𝑎𝑟, 𝑄2 = 300𝑉𝑎𝑟. The proposed control strategy archives the control targets by adjusting the power coefficients on 𝑘𝑝1 = −0.6, 𝑘𝑝2 = −2.3, 𝑘𝑞1 = 0.6, and 𝑘𝑞2 = 1.4. The results are shown in ‎Fig. 28 and ‎Fig. 29.

IFC#2 0.88

0.9

P1  P2  600W ; Q1  Q2  300Var kP1  0.6; kP 2  1.4; kq1  0.6; kq 2  1.4

0.92

0.94

0.96

P1  600W ; P2  400W ; Q1  Q2  300Var k P1  0.6; k P 2  2.3; kq1  0.6; kq 2  1.4

Fig. 29 First and second IFCs' output active powers and collective active power of parallel IFCs under non-unity PF.

that under zero active power oscillations, (1) the collective peak current of parallel IFCs is constant in the fixed average

active and reactive powers, and (2) individual IFCs' peak currents can be either in the same phase or different phases with collective peak current. The proposed control strategy keeps all IFCs peak currents in the same phase with collective peak current of parallel IFCs, which provide reduced peak current for redundant IFC. ACKNOWLEDGMENT The authors gratefully acknowledge the contributions of Ms. Song Qiong for her assistance on the experimental setup. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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Yun Wei Li (S’04–M’05–SM’11)‎received‎the‎B.Sc.‎ in Engineering degree in electrical engineering from Tianjin University, Tianjin, China, in 2002, and the Ph.D. degree from Nanyang Technological University, Singapore, in 2006. In 2005, Dr. Li was a Visiting Scholar with Aalborg University, Denmark. From 2006 to 2007, he was a Postdoctoral Research Fellow at Ryerson University, Canada. In 2007, he also worked at Rockwell Automation Canada before he joined University of Alberta, Canada in the same year. Since then, Dr. Li has been with University of Alberta, where he is a Professor now. His research interests include distributed generation, microgrid, renewable energy, high power converters and electric motor drives. Dr. Li serves as an Associate Editor for IEEE Transactions on Power Electronics, IEEE Transactions on Industrial Electronics, IEEE Transactions on Smart Grid, and IEEE Journal of Emerging and Selected Topics in Power Electronics. Dr. Li received the Richard M. Bass Outstanding Young Power Electronics Engineer Award from IEEE Power Electronics Society in 2013 and the second prize paper award of IEEE Transactions on Power Electronics in 2014. Kai Sun (M'12-SM'16) received the B.E., M.E., and Ph.D. degrees in electrical engineering from Tsinghua University, Beijing, China, in 2000, 2002, and 2006, respectively. He joined the faculty of Electrical Engineering, Tsinghua University, in 2006, where he is currently an Associate Professor. From 2009 to 2010, he was a Visiting Scholar of Electrical Engineering with Department of Energy Technology, Aalborg University, Aalborg, Denmark. His current research interests include power electronics for renewable generation systems, microgrids, and active distribution networks. Dr. Sun is a member of IEEE Industrial Electronics Society Renewable Energy Systems Technical Committee, a member of IEEE Power Electronics Society Technical Committee of Sustainable Energy Systems, and a member of Awards Sub-committee of IEEE Industry Application Society Industrial Drive Committee. He is an Associate Editor for the Journal of POWER ELECTRONICS. He was a recipient of the Delta Young Scholar Award in 2013.