Interaction and aggregated modeling of multiple ... - IEEE Xplore

Interaction and Aggregated Modeling of Multiple Paralleled Inverters with LCL Filter Minghui Lu, IEEE Student Member, Xiongfei Wang, IEEE Member, Poh Chiang Loh, and Frede Blaabjerg, IEEE Fellow Department of Energy Technology Aalborg University Pontoppidanstraede 101, Aalborg East DK-9220, Denmark [email protected]; [email protected]; [email protected]; [email protected] Abstract—This paper discusses the dynamic interaction of multi-paralleled inverters within a weak grid. Interactive current and common current models are proposed to explain the interaction among these inverters, which are studied with both open loop and closed loop analysis. An aggregated model is proposed to describe the totality of multi-inverters. Additionally, system stability is explicitly studied and classified as interactively and commonly stable. The study is validated by simulations and experiments.

I. INTRODUCTION The growing penetration of renewable energy in modern power system is conducted by grid-connected renewable plants, which are drawing the market and research attention as an important choice for sustainable development [1]. These plants are expected to convert and deliver clean energy to the grid through groups of power electronics inverters connected in parallel. Fig. 1 shows such a typical PV power plant. To properly filter out the high-frequency harmonics of the PV inverters, LCL filters are usually preferred to the L filters because of their greater high frequency harmonics attenuation ability with smaller reactive elements. However, due to the presence of undesired resonance issues, maintaining the stability of the control system is still an issue in the controller design. An additional difficulty is that paralleled PV inverters are coupled under weak grid conditions [2]. The complex high order electrical networks may aggravate overall network resonance problems and bring new and practical challenges to the design of grid-connected inverters. Reference [3] has discussed the harmonic interaction associated with a Dutch residential PV distributed network, where a large number of power inverters were installed. In addition, power quality integration problems are reported as the penetration of PV inverters increases [4]. These research works have revealed that resonant hazards, classified into parallel and series resonance, may occur even all PV inverters do individually satisfy the standards. Despite an early highlighting of the complex interaction, these papers still lack of some persuasive and technical explanation of the discussed phenomena. In [5], the authors described the resonant behavior and instability problem of N-paralleled inverters in a This work was supported in part by European Research Council (ERC) under the European Union’s Seventh Framework Program (FP/20072013)/ERC Grant Agreement [321149-Harmony].

978-1-4673-7151-3/15/$31.00 ©2015 European Union

Fig. 1. Typical renewable power plant structure with multi-inverters.

PV plant of 1400kW. It is indicated that the coupling effect exists under weak grid condition. An impressive equivalent model, with grid impedance N times larger, is proposed based on the hypothesis that all the current references are the same. However, considering the fact that the reference of each inverter is controlled independently, such as in a PV plant where current references change every several seconds with Maximum Power Point Tracking (MPPT) algorithm, this model loses some essential physical significance, and cannot describe the characteristics of a PV plant comprehensively. Impedance and admittance based models, stemming from dcdc converter analysis, are actively employed to study the interconnection of distributed generators and the grid [6]-[9]. According to the Norton’s Theorem, each inverter is modeled as an ideal current source plus output admittance. In [6], the resonance characteristic in a parallel-inverter based Microgrid is discussed. Similarly, a dynamic interaction analysis of two Active Power Filters (APF) is presented in [8]. In [9], stability analysis based on minor loop gain and Nyquist criterion is executed for two active rectifiers. Although commonly adopted, the impedance and admittance based models have the disadvantage of being computationally intensive and provide little insight into the resonance mechanism. In this paper, the interactive and common currents are proposed to describe the dynamic interaction of multiple gridconnected inverters with LCL filter in a weak grid, as well as their aggregate impact on the network. The system stability is clearly analyzed with interactively and commonly stability. An aggregated model of multi-paralleled inverters has been explored in this paper. II. MODELING AND INTERACTION A. System Description The current control loop regulates the current injected into the utility grid to improve the current waveform quality. Thus,

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C. Interactive Current and Common Current Modeling In this section, interactive current and common current are proposed to explicitly explain the interaction. Different from the complex expanded expressions of G11 and G12, the disassembled expressions are rewritten as follows n −1 1 ­ °°G11 = n Gi 2 + n Gcoupling ® ° G12 = − 1 Gi 2 + 1 Gcoupling °¯ n n Gi 2 =

Fig. 2. Circuit configuration of multiple inverters.

the current loop bandwidth should be high enough in order to achieve the decoupling of the outer loop from the inter loops. The synchronization is typically done through a Phase-Locked Loop (PLL) algorithm [16]. Note that the bandwidth of the PLL is much lower than current loop, so its influence is omitted. B. Multivarible Plant Modeling Fig. 2 shows the main circuit for three-phase gridconnected inverters with PV array as separate dc sources, where ܼଵ௜ and ܼଶ௜ (i=1···n) are inverter and grid-side inductor impedances; ܼଷ௜ are the filter capacitor impedances; and ܼ௚ is the grid impedance. ‫ݒ‬଴௜ are the inverter output voltages; ݅ଶ௜ are the grid-side currents, and ݁௚ is the grid voltage. Assume these inverters are three-phase symmetrical, thus, they can be analyzed as single-phase configuration [10]. Another assumption is that grid voltage ݁௚ contains only fundamental component, which means the grid can be regarded as short circuit while modeling and discussing the interaction and stability. The dynamics of the multi-paralleled inverters system can be described with multivariable system theory [11] in which each inverter output voltage ‫ ݅Ͳݒ‬represents an input and each inverter current ݅ʹ݅ represents an output to the multiinverter system

I(s) = G(s) ⋅ U(s) ⇔ § i21 · § v01 · § G11 G12 ¨ ¸ ¨ ¸ ¨ i ¨ 22 ¸ = G(s) ⋅ ¨ v02 ¸ = ¨ G21 G22 ¨"¸ ¨"¸ ¨ " " ¨ ¸ ¨ ¸ ¨ © i2n ¹ © v0n ¹ © Gn1 Gn 2

" G1n · § v01 · ¸ ¨ ¸ " G2n ¸ ¨ v02 ¸ ⋅ " " ¸ ¨"¸ ¸ ¨ ¸ " Gnn ¹ © v0n ¹

(2)

Z3 Z3 (3) , Gcoupling = Z1Z2 + Z2Z3 + Z3Z1 Z1Z2 + Z2Z3 + Z3Z1 + nZg (Z3 + Z1 )

where Gi2 represents the plant feature of individual LCL filter, neither depends on the number of inverters n nor on the grid impedance ܼ௚ ; whereas, Gcoupling illustrates the plant feature considering the coupling effects of grid impedance, closely related to n and ܼ௚ . Moving next to the expression of overall grid current݅ଶ௜ , e.g.݅ଶଵ , can be written as (4) according to (1). Notice that ݅ʹͳ has n independent current terms and each term has its own physical significance. N

i21 = G11 ⋅ v01 + ¦ G12 ⋅ v0k k =2

1 1 1 = Gi 2 (v01 − v02 ) + " + Gi 2 (v01 − v0n ) + Gcoupling ( v01 + " + v0n ) n n n

(4)

Specifically, the first n-1 terms have the similar form, representing the currents flowing from 1# inverter to all the remaining inverters, the first term means the current component between 1# and 2# inverters. These currents are defined as interactive currents because of circulating through paralleled inverter instead of being injected to the grid. On the other hand, the last term depicts the current to grid, which is defined as common current since it exists in all ݅ʹ݅ and flows to the grid. Fig. 2 illustrates the current component distribution.

1 1 ⋅ Gi 2 ⋅ ( voi − voj ), si = ⋅ Gcoupling ⋅ ( vo1 + " + von ) (5) n n As (5), ܿ௜௝ represents the interactive current circulating i# and j# inverters; and ‫ݏ‬௜ is the expression of i# common current. From (5), interactive current only relates to Gi2, whereas common current only links with Gcoupling. Hence, it is natural to suppose that they could be discussed separately. These two divided parts with closed-loop control are presented. cij =

(1)

where I(s) is the grid current vector; U(s) is the inverter output voltage vector, acting as the control variable; G(s) is ݊ ൈ ݊ admittance transfer matrix, depicts the influence of each ‫ ݅Ͳݒ‬on the ݅ʹ݅ . G(s) possesses the symmetry characteristic because these inverters are usually of the same product type with small parameters mismatch due to strict industrial standards. So, the diagonal elements of G(s) are equal since each v0i has identical impact on its own current i2i, i.e. Gii =G11 (i=1…n); similarly, all non-diagonal elements are also equal since each v0i has the same effect on other i2j, i.e. Gij =G12 (ij). The elements G11 and G12 are obtained using the principle of linear superposition, whose expressions are given in Appendix A.

D. Resonance Characteristic This section investigates and explains the resonance behavior among the multiple paralleled inverters with an LCL filter. For a single inverter with LCL filter, there is a resonance peak caused by zero impedance seen by some higher order harmonics of the current; while there are multiple resonance peaks for the paralleled case. From (4), the element G11 can be regarded as the transfer function between the grid side current ݅ʹ݅ and its own inverter output voltage ‫ ݅Ͳݒ‬. The Bode diagram is

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Magnitude (dB)

Fig. 4. Control configuration for current control.

Phase (deg)

the control loop into consideration, the expression of ݆ܿ݅ is written in (6), the mathematical derivation is presented in Appendix B.

cij =

Fig. 3. Resonance Characteristic of multiple inverters in parallel.

plotted in Fig. 3, where different numbers of inverters are installed. Because of the two current components, one resonance wres1 is introduced by interactive current; the other one wres2 is caused by common current. As it can be seen from Fig. 3, wres1is a fixed point, namely the LCL filter resonance frequency; and the resonance frequency wres2is a changeable point (lower than the fixed resonance frequency wres1). These frequency characteristics are quite different from the ones of single inverter with LCL filter. Compared with a single inverter, the paralleled inverters system deteriorates the system resonance situation: multiple resonance peaks are introduced while only LCL resonance peak exists in single inverter. The resonance behavior is divided into two categories: (1) interactive resonance among these paralleled inverters; and (2) common resonance between the grid and paralleled system. III. CONTROL AND CLOSED LOOP CURRENT One simplifying assumption for completely decentralized control is that interaction can be treated as a form of disturbance. This approach is, of course, strictly incorrect, since the essence of true disturbances is that they are independent inputs [11]. In this section, interactive current and common current under closed loop control are presented. The dynamics of the studied inverters are coupled due to the grid impedance. However, it may be difficult to adopt decoupling control scheme, because each inverter has its own decentralized controller. Stable operation of grid-connected inverter without any additional damping has been proved to be effective in [12]-[14]. Without loss of generality, a simple and effective single loop control with typical PI or PR controller is chosen. Digital delays [15], [17], including one-sample computation delay z-1 and PWM update delay, are taken into account. Fig. 4 presents the control loop diagram of current i2, in which the inverter is simplified as a linear amplifier with gain ‫ܭ‬௣௪௠ , Gc is a linear controller, and Gi2(s) is the control plant. With proper design (e.g., design method [14], [15]), the system can obtain good output current waveform and sufficient control stability.

Gi 2 H 1 ⋅ ⋅ (i2*i − i2*j ), H = Gc ⋅ z −1 ⋅ K pwm n (1 + Gi 2 H )

(6)

‫כ‬ ‫כ‬ From (6), if the current references ݅ଶ௜ and ݅ଶ௝ are the same, the interactive current ܿ௜௝ is zero and there will be no current circulating between these two inverters. Based on this assumption, [5] proposed an equivalent inverter that describes the totality of the paralleled inverters. However, differences in irradiation, partial shadings, and mismatches in MPPT algorithm will affect the current reference value. It is difficult to guarantee a non-interaction case.

For the common current ‫ݏ‬௜ , it is obvious from (5) that all the ‫ݏ‬௜ are identical and they are injected into the grid. Fig. 5 shows the distribution of common current. In this configuration, all the inverter voltages are same, as (7)

vo1 = voi = von =

1 ⋅ ( vo1 + " + von ) n

(7)

The common current is the result of all the current references acting together. Equation (8) is the analytic closedloop expression of ‫ݏ‬௜ , given in Appendix C. The Gcoupling includes the impacts of grid impedance. So far, the expression of overall current ݅ଶ௜ is shown in Appendix (A.3). The reference tracking characteristic and impacts of other inverter ‫כ‬ current reference ݅ଶ௝ are also shown.

si =

Gcoupling H 1 ⋅ ⋅ (i21* + " + i2*n ) n (1 + Gcoupling H )

(8)

IV. AGGREGATED MODEL AND STABILITY ANALYSIS A. Aggregated Model of Multiple Inverters Following the previous discussion, this part develops an aggregated model to simplify the N paralleled inverters system. So far, the expression of the overall current ݅ଶ௜ is shown in Appendix (A.3). The reference tracking characteristic and impacts of other inverter current reference ‫כ‬ ݅ଶ௝ are also shown. The closed loop relationship between the grid current reference and grid current is shown

For the interactive current ݆ܿ݅ , it circulates between i# and j# inverters, so only two inverters are needed to model. Taking

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§ i21ref § i21 · ¨ ¨ ¸ i ¨ i22 ¸ = T(s) ⋅ ¨ 22ref ¨ " ¨"¸ ¨¨ ¨ ¸ © i2n ¹ © i2nref

· §T11 T12 ¸ ¨ ¸ = ¨T12 T11 ¸ ¨" " ¸¸ ¨ ¹ ©T12 T12

" " " "

T12 · § i21ref ¸ ¨ T12 ¸ ¨ i22ref ⋅ "¸ ¨ " ¸ ¨ T11 ¹ ¨© i2nref

· ¸ ¸ ¸ ¸¸ ¹

(9)

TABLE I VALUES OF PARAMETERS Symbol

Quantity

Value

L1 L2 C Kpwm fsw Lg

inverter side inductance grid side inductance filter capacitance inverter gain switching frequency grid impedance

1.5 mH 1.5 mH 4.7uF 400 10 kHz 1mH

The closed loop transfer matrix T(s) preserves the symmetry characteristic based on the assumption that all the installed inverters are equal. The diagonal element T11 represents the reference tracking characteristic; whereas the non-diagonal elements T12 represents the coupling effects, shown in (10) and (11). T11 =

Gcoupling H Gi 2 H n −1 1 ⋅ + ⋅ n (1 + Gi 2 H ) n (1 + Gcoupling H )

(10)

Gcoupling H Gi 2 H 1 1 ⋅ + ⋅ n (1 + Gi 2 H ) n (1 + Gcoupling H )

(11)

T12 = −

(a)

From (10) and (11), both T11 and T12 consist of two factors in (12), the first factor is the interactive component and the second factor is the common component. In the stable steady state, the two factors are equal to 1. Therefore, the element T11 is equal to 1, and T12 is equal to zero in the stable steady state. It indicates that the coupling effect will be controlled to zero when the system is stable, meanwhile, each grid current ݅ʹ݅ will track the current controller. lim s →0

lim s →0

(b) Fig. 5. Root locus and stability regions of (a) interactive and common current control (b).

Gi 2 H = 1 Ÿ interactive component (1 + Gi 2 H ) Gcoupling H

(1 + Gcoupling H )

Ÿ common component

(12)

The aggregate contribution of these inverters on the grid can be modeled as a single inverter. It is obvious that n identical paralleled LCL filters can be modeled as single set of LCL filter with parameter (L1/n, L2/n, nC); Multiplying T(s) rows in (9), the row sum of the matrix can be derived n

¦i

2i

1

n

Gcoupling ⋅ GcGd K pwm

1

1 + Gcoupling ⋅ Gc Gd K pwm

= (T11 + (n− 1) ⋅T12 ) ⋅ (¦ i2iref ) =

be classified as interactive current stability and common current stability. If and only if both of these current components are stable, the system will be stable. From (6) and (8), the open loop forward-path expressions of ݆ܿ݅ and ‫ ݅ݏ‬are clearly described, Gi2·H and Gcoupling·H, respectively. Therefore, the control system analysis techniques, such as frequency response and root locus can be applied. Note that Gi2·H neither depends on inverter number n nor on the grid impedance ܼ݃ .

n

⋅ (¦ i2iref ) 1

(13) Substituting Gcoupling into (13), the relationship between the sum of current reference and the sum of grid current can be obtained in Appendix (A.4). The controller G’c =1/n·Gc. Gd and KPWM remain the same. An equivalent inverter with LCL filter (Z’1= L1/nƒs, Z’2= L2/nƒs, Z’3=1/(nCƒs)) and a linear controller G’c can represent the paralleled inverters. B. Stability Analysis To explore the stability of the paralleled inverters, the closed loop poles of the multivariable system should be analyzed [5]. However, this method requires tedious calculations. More intuitively, the whole system stability can

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Fig. 6. Simulation results of three paralleled inverters.

Fig. 7. Case I: Stable experimental results.

Fig. 8. Interactively unstable experimental results.

V. SIMULATION AND EXPERIMENTAL VERIFICATION In this part, three inverters are connected to the same point in parallel, some results are presented in order to validate the theoretical analysis of the previous sections. The LCL filter and inverter gain ‫ ݉ݓ݌ܭ‬value are introduced in Table I. Fig. 5 shows the root locus of the interactive current and common current control. Stable regions can be identified as

K p Range

0 < K p < 20.1, interactively stable 0 < K p < 27.1, commonly stable

(14)

If the proportional gain‫ܭ‬௣ satisfies both stable regions; the system will be stable; if ʹͲǤͳ ൏ ‫ܭ‬௣ ൏ ʹ͹ , the system will be interactively unstable, but commonly stable; if ‫ܭ‬௣ ൐ ʹ͹, system is neither interactively nor commonly stable. It is noticeable for the interactively unstable and commonly stable case, the oscillation current circulates between inverters, but without oscillation at the PCC. Fig. 6 reveals this interesting phenomenon: the control gain ‫ܭ‬௣ ൌ ʹ͵, at t = 0ms, three current references are the same ݅ଶ௜̴௥௘௙ ൌ ͵ and the system is stable because the interactive current ܿଵଶ ,ܿଵଷ ,ܿଶଷ are zero and there is no interaction between these inverters. At the time t = 25ms, there is step change for݅ଶଵ̴௥௘௙ , from 3A to 1A, the process triggers the interactive current instability and the current ݅ଶ௜ waveforms diverge, because the system is interactively unstable. Still, the sum of ݅ଶ௜ is stable, because system is commonly stable. Using the parameters in given Table I, experiments are implemented in three 5 kW paralleled Danfoss inverters with LCL filter. Three cases are presented. Case I: ‫ܭ‬௣ satisfies both interactive current and common current stable regions; case II: ‫ܭ‬௣ satisfies common current stable region, but system is interactively unstable; case III: ‫ܭ‬௣ satisfies neither the interactive current nor the common current stable regions, the whole system is unstable. In case I, the paralleled system is stable and a step change of 1# inverter grid current occur, where the system remains stable, the results are shown in Fig. 7. In case II, ‫ܭ‬௣ ൌ ʹ͵, the system is internally unstable where the variables of the N independent inverters will diverge and the duty cycles will saturate. The current ݅ଶ௜ has two components. The first component, equal for the N independent inverters, is at 50Hz.

Fig. 9. Interactively unstable and commonly unstable experimental results.

This component appears in the sum ݅ଶଵ + ݅ଶଶ + ݅ଶଷ and injected in the grid as a result. The second component is at the resonance frequency between inverters that diverges due to interactive instability. This last component goes from the reactive elements of one inverter to the reactive elements of another inverter and it does not circulate through the grid. That is why it does not appear in the sum. In case III, two inverters operate in parallel with grid inductance ‫ܮ‬௚ ൌ ͳǤͷ, proportional gain ‫ܭ‬௣ goes even larger, the system is neither interactively stable nor commonly stable, ݅ଶଵ , ݅ଶଶ , and the sum ݅ଶଵ + ݅ଶଶ are unstable. VI. CONCLUSION The dynamic interactions among paralleled inverters connected to the weak grid are explicitly studied. The interactive current and common current are proposed to better explain the interaction between these inverters, which are studied in not only open loop, but also in closed loop analysis. An aggregated model is proposed to describe the totality of the multiple inverters. The system stability, which is divided into interactively stable and commonly stable, is clearly discussed. Simulation and experiments verify the technical analysis. REFERENCES [1] [2]

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F. Blaabjerg, Z. Chen, S.B. Kjaer, "Power electronics as efficient interface in dispersed power generation systems," IEEE Trans. Power Electron., vol. 19, no. 5, pp. 1184-1194, Sept. 2004. B.Y. Liu, S.X. Duan, T. Cai, “Photovoltaic DC-building-module-based BIPV system-concept and design considerations,” IEEE Trans. Power Electron., vol. 26, no. 5, pp. 1418-1429, May. 2011.

[3] [4] [5]

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J.H.R. Enslin, P.J.M. Heskes, "Harmonic interaction between a large number of distributed power inverters and the distribution network," IEEE Trans. Power Electron., vol. 19, no. 6, pp. 1586-1593, Nov. 2004. D.G. Infield, P. Onions, A.D. Simmons, and G.A. Smith, “Power quality from multiple grid-connected single-phase inverters,” IEEE Trans. Power Del., vol. 19, no. 4, pp. 1983-1989, Oct. 2004. J. Agorreta, M. Borrega, J. Lopez, and L. Marroyo, "Modeling and control of N-paralleled grid-connected inverters with LCL filters coupled due to grid impedance in PV plants," IEEE Trans. Power Electron., vol. 26, no. 3, pp. 770-1194, Mar. 2011. J. He, Y. Li, D. Bosnjak, "Investigation and active damping of multiple resonances in a parallel-inverter-based microgrid," IEEE Trans. Power Electron., vol. 28, no. 1, pp. 234-246, Jan. 2013. M. Armstrong, D. J. Atkinson, C. M. Johnson, and T. D. Abeyasekera, “Low order harmonic cancellation in a grid connected multiple inverter system via current control parameter randomization,” IEEE Trans. Power Electron., vol. 20, no. 4, pp. 885–892, Apr. 2005. P. Dang, T. Ellinger and J. Petzoldt, “Dynamic interaction analysis of APF systems,” IEEE Transactions on Industrial Electronics, vol. 26, no. 9, pp. 4467- 4473, Sep. 2014. X. Wang, F. Blaabjerg, M. Liserre, Z. Chen, "An active damper for stabilizing power-electronics-based AC systems," IEEE Trans. Power Electron., vol. 29, no. 7, pp. 3318-3328, July. 2014.

[10] T. Yi, P. C. Loh, W. Peng, C. Fook Hoong, and G. Feng, “Exploring inherent damping characteristic of LCL-filters for three-phase gridconnected voltage source inverters,” IEEE Trans. Power Electron., vol. 27, no. 3, pp. 1433–1443, Mar. 2012. [11] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design. New York: Wiley, 1997. [12] C. Zou, B. Liu, S. Duan, R. Li, "Influence of delay on system stability and delay optimization of grid-connected inverters with LCL Filter," IEEE Trans. Ind. Inf., vol. 10, no. 3, pp. 1175-1784, Aug. 2014. [13] J. Yin, S. Duan, B. Liu, "Stability analysis of grid-connected inverter with LCL filter adopting a digital single-Loop controller with inherent damping characteristic," IEEE Trans. Ind. Inf., vol. 9, no. 2, pp. 11041112, May. 2013. [14] S.G. Parker, B. P. McGrath, D.G. Holmes, "Regions of active damping control for LCL filters," IEEE Trans. Ind. Appl., vol. 50, no. 1, pp. 424432, Jan./Feb. 2014. [15] D. G. Holmes, T. A. Lipo, B. P. McGrath, and W. Y. Kong, "Optimized design of stationary frame three phase AC current regulators, " IEEE Trans. Power Electron., vol. 24, no. 11, pp. 2417-2426, Nov. 2009. [16] C.-C. Hsieh and J. Hung, “Phase-locked loop techniques—A survery,” IEEE Trans. Ind. Electron., vol. 43, no. 6, pp. 608–615, Dec. 1996. [17] S. Buso and P. Mattavelli, Digital Control in Power Electronics. San Rafael, CA, USA: Morgan and Claypool, 2006.

APPENDIX A The expanded expression of G11 and G12, shown in (A.1) and (A.2) i21 v01

G11 =

G12 =

i21 v02

i2i = si + =

v0 i = 0, ( i ≠ 2)

n

¦

j =1, j ≠ i

v0 i = 0, ( i ≠ 1)

=−

=

Z3 ⋅ (Z1 ⋅ Z2 + Z2 ⋅ Z3 + Z3 ⋅ Z1 ) + (n − 1) ⋅ Z3 ⋅ Z g ⋅ (Z3 + Z1 )

Z3 ⋅ Z g ⋅ (Z3 + Z1 )

(A.2)

(Z1 ⋅ Z2 + Z2 ⋅ Z3 + Z3 ⋅ Z1 ) ⋅ (Z1 ⋅ Z2 + Z2 ⋅ Z3 + Z3 ⋅ Z1 + n ⋅ Z g ⋅ (Z3 + Z1 ))

cij

n ªn −1 Gcoupling H Gcoupling H º * Gi 2 H Gi 2 H 1 1 1 ⋅ ⋅ (i21* + " + i2*n ) + ¦ ⋅ ⋅ (i2*i − i2*j ) = « ⋅ + ⋅ » ⋅ i2i + n (1 + Gcoupling H ) j =1, j ≠ i n (1 + Gi 2 H ) ¬« n (1 + Gi 2 H ) n (1 + Gcoupling H ) ¼»

ig =

(A.1)

(Z1 ⋅ Z2 + Z2 ⋅ Z3 + Z3 ⋅ Z1 ) ⋅ (Z1 ⋅ Z2 + Z2 ⋅ Z3 + Z3 ⋅ Z1 + n ⋅ Z g ⋅ (Z3 + Z1 ))

Z 3 ⋅ G c G d K pwm Z1 Z 2 + Z 2 Z 3 + Z 3 Z1 + nZ g ( Z 3 + Z1 ) + Z 3 ⋅ G c G d K pwm

⋅ igref =

n

¦

j =1, j ≠ i

ª 1 Gcoupling H º * Gi 2 H 1 + ⋅ «− ⋅ » ⋅ i2 j ¬« n (1 + Gi 2 H ) n (1 + Gcoupling H ) ¼»

Z 3' G 'c G d K pwm ' 1

Z ( Z + Z g ) + Z ( Z 2' + Z g ) + Z1' Z 3' + Z 3' G 'c G d K pwm

APPENDIX B

' 2

' 3

⋅ igref

(A.3)

(A.4)

APPENDIX C

Referring to Fig.3, the expression of inverter output voltage ‫כ‬ ‫ݒ‬଴௜ ൌ ሺ݅ଶ௜ െ  ݅ଶ௜ ሻ ȉ ‫ܪ‬ǡ‫ ܪ‬ൌ ‫ܩ‬௖ ȉ ‫ି ݖ‬ଵ ȉ ‫ܭ‬௣௪௠

Referring to (B.1) and (5), the common current ଵ

(B.1)

‫כ‬ ‫ݏ‬௜ ൌ ȉ ‫ܩ‬௖௢௨௣௟௜௡௚ ‫ ܪ‬ȉ ሺσ௡௜ୀଵ ݅ଶ௜ െ σ௡௝ୀଵ ݅ଶ௝ ሻ ௡

Substituting (B.1) to (5) yield ଵ

‫כ‬ ‫כ‬ ܿ௜௝ ൌ  ȉ ‫ܩ‬௜ଶ ‫ ܪ‬ȉ ሺ݅ଶ௜ െ ݅ଶ௝ െ ሺ݅ଶ௜ െ  ݅ଶ௝ ሻሻ ௡

According to (4) and (5), yields (B.2)

σ௡௝ୀଵ ݅ଶ௝ ൌ ݊ ȉ ‫ݏ‬௜

From equation (4), it can be obtained that ݅ଶ௜ െ  ݅ଶ௝ ൌ ‫ܩ‬௜ଶ ȉ ൫‫ݒ‬଴௜ െ ‫ݒ‬଴௝ ൯ ൌ ݊ ȉ ܿ௜௝

ଵ

ீ೔మ ு

௡

ଵାீ೔మ ு

‫כ‬ ‫כ‬ ȉ ሺ݅ଶ௜ െ ݅ଶ௝ ሻ

(C.2)

Combine (C.2) and (C.3), the relationship of ‫ݏ‬௜ and current reference can be obtained

(B.3)

ଵ

ீ೎೚ೠ೛೗೔೙೒ ு

௡

ଵାீ೎೚ೠ೛೗೔೙೒ ு

‫ݏ‬௜ ൌ  ȉ

Combine (B.2) and (B.3), the relationship of ܿ௜௝  and current reference can be obtained ܿ௜௝ ൌ  ȉ

(C.1)

(B.4)

1959

‫כ‬ ‫כ‬ ȉ ሺ݅ଶଵ ൅ ‫ ڮ‬൅ ݅ଶ௡ ሻ

(C.3)