Predictive Control of Four-Leg Power Converters - IEEE Xplore

Predictive Control of Four-Leg Power Converters Venkata Yaramasu ∗, Marco Rivera†, Bin Wu‡ and Jose Rodriguez § ∗ Department

of Electrical Engineering and Computer Science, Northern Arizona University, Flagstaff, AZ, USA † Department of Industrial Technologies, Universidad de Talca, Curic´ o, Chile ‡ Department of Electrical and Computer Engineering, Ryerson University, Toronto, Canada § Department of Electronics Engineering, Universidad Tecnica Federico Santa Maria, Valparaiso, Chile Email: [email protected], [email protected], [email protected], [email protected]

Abstract—In this paper, finite control-set model predictive control of two-level and three-level four-leg voltage source converters (VSCs) is presented. The predictive current control (PCC) and predictive voltage control (PVC) schemes are presented considering grid-connected and standalone distributed generation applications respectively. The discrete-time model of load currents, load voltages, DC-link capacitors voltage and VSC terminal voltages is formulated in terms of four-leg converter switching states. A cost function is defined with the PCC scheme to minimize the error between reference and predicted load currents. Similarly, in PVC scheme, the cost function deals with the minimization of error between reference and predicted load voltages. The balancing of split DC-link capacitors voltage is considered with the three-level four-leg VSCs. The optimal switching states which minimize the cost function are chosen and applied to the four-leg VSC directly without involving linear regulators and modulation stage. The proposed PCC and PVC strategies are verified through simulation results considering single-/threephase, balanced/unbalanced and linear/nonlinear loads.

I. I NTRODUCTION In three-phase four-wire systems, the unbalanced and nonlinear loads cause large amount of neutral current. The conventional three-phase power converters with three legs are not suitable for such applications. The four-leg two-level and three-level (also known most popularly as neutral-point clamped, NPC) converters provide best solution in providing a path to the neutral currents in low-power and high-power applications respectively [1], [2]. Many applications such as gridconnected distributed generation (DG), universal power quality conditioners, electric drives, shunt active power filters, and active front-end rectifiers which utilize the four-leg converters demand precise current control [3]–[7]. Some applications such as uninterruptible power supplies, off-grid (standalone) DG systems, and dynamic voltage restorers demand tight output voltage control. Nowadays, with technological advances, the implementation of new and more complex modulation and control strategies is possible. One of the new control schemes for power converters is finite control-set model predictive control (FCS-MPC) as described in [8]–[10]. The FCS-MPC is a nonlinear control technique and an attractive alternative to the classical control methods, due to its fast dynamic response and simple concept. It uses model of the system, as the name implies, and this requirement has prompted many investigations of the system model used in different applications.

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The predictive current control (PCC) of four-leg two-level and NPC converters with output L filter are analyzed in [11]–[14] considering the grid-connected/active power filter application. The predictive voltage control (PVC) with an output LC filter is analyzed in [15], [16] for four-leg twolevel and NPC converters. The questions about the generalized implementation of PCC and PVC control schemes for both two-level and NPC converters is not answered in literature. In this paper, a generalized approach for the predictive control of four-leg converters is proposed. The discrete-time model of load currents, load voltages, DC-link capacitors voltage and converter terminal voltages is formulated in terms of converter switching states. Four different control solutions are described in this paper which use a simple, intuitive and generalized discrete-time model: • PCC of four-leg two-level converter with output L filter and balanced/unbalanced linear loads •

PCC of four-leg NPC converter with output L filter and balanced/unbalanced linear loads

•

PVC of four-leg two-level converter with output LC filter and balanced/unbalanced linear/nonlinear loads

PVC of four-leg NPC converter with output LC filter and balanced/unbalanced linear/nonlinear loads The cost function definition for above four cases is described such that the error between the reference and predicted control variable is minimized. In case of NPC converters, the regulation of DC-link capacitors voltage is also included in cost function definition. The simulation results based on MATLAB/Simulink software are presented to support the theory and proposed generalized approach. •

II. M ODEL OF P OWER C ONVERSION S YSTEMS The configuration of power conversion system with fourleg converters is shown in Fig. 1. The power converters include three-phase two-level and NPC converters with fourthleg, output L and LC harmonic filters, three-phase grid, and arbitrary single-phase or three-phase, balanced or unbalanced, linear or nonlinear loads. The two-level converter is composed of 8 IGBTs and the NPC converter is realized by 16 IGBTs and 8 clamping diodes. The neutral inductor (L n ) helps to mitigate the neutral converter-side current ripple [15].

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Three-Phase Voltage Source Converter

Fourth-Leg

Output Harmonic Filter

Arbitrary Loads

Fig. 1. Power circuit of two-level and three-level four-leg voltage source converters with output harmonic filters and arbitrary loads.

The following voltage and current vectors are defined in order to simplify the mathematical modeling: • • • • • •

Leq

T

Output voltage vector: v o = [vao vbo vco ] Output current vector: i o = [iao ibo ico ]T T Converter voltage vector: v i = [van vbn vcn ] T Converter current vector: i i = [ia ib ic ] T NPC converter capacitor voltage vector: vC = [vC1 vC2 ] T NPC converter capacitor current vector: iC = [iC1 iC2 ]

By applying Kirchhoff voltage and current laws to the LC filter nodes in Fig. 1, the state-space model of output voltages and converter currents can be obtained as, ! " ! " ! " d vo vo v =A +B i (1) i i io dt i i where A=

and

!

0 −L−1 eq

and Req

" I Cf , −L−1 eq Req 6×6 ⎡ rf + rn = ⎣ rn rn

B=

rn rf + rn rn

!

0 L−1 eq

⎤ rn rn ⎦ rf + rn

−I " Cf

0

6×6

⎡ Lf + Ln = ⎣ Ln Ln

Ln Lf + Ln Ln

⎤ Ln Ln ⎦ Lf + Ln

where 0 and I are third-order null and identity matrices, respectively. The digital implementation of the proposed control algorithm requires discrete-time model. The continuous-time statespace system in (1) can be converted to discrete-form as " ! " ! " ! vo (k) vi (k) vo (k + 1) =Φ +Γ (4) ii (k + 1) ii (k) io (k) and Φ = eATs ,

Γ = A−1 (Φ − I)B

(5)

where Ts is the sampling time. The discrete-time model in (4) can be used for PVC realiza(2) tion with both two-level and NPC converter. The calculation of converter AC-side output voltage v i will be different with two-level and NPC converter. v i (k) for two-Level VSC is: ⎡ ⎤ ⎤ ⎡ Sa1 − Sn1 van (k) ⎣ vbn (k) ⎦ = vC1 (k) ⎣ Sb1 − Sn1 ⎦ (3) (6) vcn (k) Sc1 − Sn1 122

and vi (k) for three-Level VSC is: ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ Sa1 − Sn1 Sa2 − Sn2 van (k) ⎣ vbn (k) ⎦ = vC1 (k) ⎣ Sb1 − Sn1 ⎦+vC2 (k) ⎣ Sb2 − Sn2 ⎦ . vcn (k) Sc1 − Sn1 Sc2 − Sn2

(7)

For the realization of PVC scheme, only first three rows of (4) are computed. In other words, only v o (k + 1) is computed for PVC scheme. To implement the PCC scheme, last three rows of (4) can be used, because with L filter i i becomes equal to io . In the PCC scheme, the load values R a , Rb and Rc should be properly treated, and thus the matrix R eq is re-written as (only for PCC case), Req

⎡ rf + rn + Ra rn =⎣ rn

rn rf + rn + Rb rn

⎤ rn ⎦. rn rf + rn + Rc

(8)

In case of NPC converter, the DC-link capacitors voltage must be maintained at equal values to ensure less stress on the switching devices and also to achieve good output power quality. The discrete-time model of the DC-link capacitors voltage is given as [14], [16]: Ts vC1 (k + 1) = vC1 (k) + i (k) C1 C1 Ts vC2 (k + 1) = vC2 (k) + i (k) C2 C2

where λdc is weighting factor for DC-link capacitors voltage balancing. Superscript p denotes the predicted variable. The switching state corresponding to the minimum cost function is chosen and applied to the converter directly at next sampling instant. Additional constraints such as switching frequency reduction, current limitation and spectrum shaping can be included just by adding those conditions in the cost function [9]. IV. S IMULATION R ESULTS The simulation models of four-leg two-level and NPC converters are developed with the parameters shown in Table I. To validate the PCC and PVC schemes, the transient results with unbalanced loads are presented. The results with twolevel converter are similar to NPC converter, except that the two-level converter need not to control the DC-link capacitors voltage. TABLE I PARAMETERS OF THE F OUR -L EG P OWER C ONVERTERS

(9)

where iC1 (k) and iC2 (k) are predicted DC-link capacitors current, and they can be predicted based on all the possible switching states and measured converter currents i i . III. M ODEL P REDICTIVE C ONTROL OF F OUR -L EG C ONVERTERS The PCC and PVC schemes for four-leg two-level converter are shown in Figs. 2a and 2b respectively. Similarly, the PCC and PVC schemes for four-leg NPC converter are shown in Figs. 3a and 3b respectively. The reference current and voltages are calculated according to the specific application. The extrapolation block calculates the future value of reference control variables (voltages or currents) based on the present and past samples of the reference control variables. The predictive model uses power conversion system discrete-time model discussed in Section II to predict the future behavior of the variables to be controlled. The NPC converter PCC and PVC schemes use additional prediction for DC-link capacitors voltage in comparison to the two-level converter. The cost function uses the reference and predicted control variables. The cost functions for all the four predictive control schemes are defined by, 2 ∗ p • PCC of 2L-VSC: g(k) = [io (k + 1) − io (k + 1)] • •

•

PVC of 2L-VSC: g(k) = [vo∗ (k + 1) − vop (k + 1)]2

2

PCC 'of 3L-VSC: g(k) = [i∗o((k + 1) − ip o (k + 1)] 2 p p λdc ∗ vC1 (k + 1) − vC2 (k + 1)

2

+

PVC 'of 3L-VSC: g(k) = [vo∗((k + 1) − vop (k + 1)] + 2 p p (k + 1) − vC2 (k + 1) λdc ∗ vC1

Variable

Description

Value

vdc C1 , C 2 rf , r n Lf , Ln Cf i∗o vo∗ fo∗ R Ts λdc

DC-link voltage DC-link capacitors Filter leakage resistance Filter inductance Filter capacitance Nominal reference load current Nominal reference load voltage Nominal reference frequency Nominal load resistance Sampling time Weighting factor

300 [V] 1000 [µF] 0.045 [Ω] 5 [mH] 60 [µF] 10 [A rms] 120 [V rms] 60 [Hz] 12 [Ω] 50 [µs] 0.5

In Fig. 3a, PCC scheme simulation results are presented with four-leg NPC converter. A step-change in reference currents from balanced 10 A (rms) to unbalanced case (i ∗ao = 11A, i∗bo = 10A, i∗co = 9A) is given. This is the typical application for four-wire systems, where the load demand on each phase is different. The controller handles each phase current independently, and thus the load currents track to their references with less steady-state error. The neutral-current, which is sum of the three-phase currents, flows through the fourth NPC leg. It is important to note that unlike in the threeleg NPC converters, the neutral current does not circulate through the DC-link capacitors. Due to this, the DC-link capacitors voltage remain balanced as shown in Fig. 3a. The PVC test results with step-change in load are presented in Fig. 3b. This test considers balanced load voltage references ∗ ∗ ∗ which are given as v ao = vbo = vco =120V (rms) @ 60Hz. A step change from no load to unbalanced loads (R a = 24 Ω, Rb = 12 Ω, Rc = 6 Ω) is applied as shown in Fig. 3b, where a very fast dynamic response without any overshot is observed. The load voltages are slightly affected by this step change, but this duration is very small (a few microseconds). The fast dynamic response is possible due to the elimination of PI controllers and modulation stage. Similar to the previous operating condition, the capacitors voltage are well balanced with a very small error between them.

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vdc (k) vo (k) ii (k) io (k)

Digital Control System

Φ, Γ


(a) Predictive Current Control (PCC)

(b) Predictive Voltage Control (PVC)

Fig. 2. Block diagram of PCC and PVC schemes for four-leg two-level power converter.

Four-Leg NPC Converter

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vc (k)

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i∗o (k)

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Predictive Model

Sxj , x ∈ {a, b, c, n} , j ∈ {1, 2} Extrapolation of Reference Voltages

vo∗ (k)

vCp (k + 1)

Predictive Model

Reference Voltages Calculation

vc (k) vo (k) ii (k) io (k)



Fig. 3. Block diagram of PCC and PVC schemes for four-leg NPC power converter.

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Fig. 4. Simulation results with PCC and PVC schemes for four-leg NPC power converter.

V. C ONCLUSIONS A generalized approach for the model predictive control of four-leg two-level and NPC converters is discussed in this paper. The dynamics of the load currents/voltages and DClink capacitors voltage are formulated in terms of converter switching states. The control algorithm evaluates each of the possible switching states and then chooses a switching state that minimizes the cost function. As well it can never be achieved, because this control strategy only ensures the minimal error, the ideal minimum of the cost function is zero, which represents perfect achievement of the control objectives. With the proposed predictive control the load currents/voltages are tracked to their references with acceptable error, and the DC-link capacitors voltage are balanced during all the operating conditions. It is possible to merge the four cost functions in only one by choosing the appropriate weighting factors in order to have a generalized cost function. R EFERENCES [1] D. Fernandes, F. Costa, and E. dos Santos Jr., “Digital-scalar PWM approaches applied to four-leg voltage-source inverters,” IEEE Trans. Ind. Electron., vol. 60, no. 5, pp. 2022–2030, May. 2013. [2] Z. Liu, J. Liu, and J. Li, “Modeling, analysis and mitigation of load neutral point voltage for three-phase four-leg inverter,” IEEE Trans. Ind. Electron., vol. 60, no. 5, pp. 2010–2021, May. 2013. [3] X. Wang, F. Zhuo, J. Li, L. Wang, and S. Ni, “Modeling and control of dual-stage high-power multifunctional PV system in d-q-o coordinate,” IEEE Trans. Ind. Electron., vol. 60, no. 4, pp. 1556–1570, Apr. 2013. [4] N. Prabhakar and M. Mishra, “Dynamic hysteresis current control to minimize switching for three-phase four-leg VSI topology to compensate nonlinear load,” IEEE Power Electron. Lett., vol. 25, no. 8, pp. 1935– 1942, Aug. 2010. [5] B. Singh, P. Jayaprakash, S. Kumar, and D. Kothari, “Implementation of neural-network-controlled three-leg VSC and a transformer as threephase four-wire DSTATCOM,” IEEE Trans. Ind. Appl., vol. 47, no. 4, pp. 1892–1901, Jul./Aug. 2011.

[6] S. Karanki, N. Geddada, M. Mishra, and K. Boddeti, “A modified threephase four wire UPQC topology with reduced DC-link voltage rating,” IEEE Trans. Ind. Electron., vol. 60, no. 9, pp. 3555–3566, Sep. 2013. [7] Y. Kumsuwan, S. Premrudeepreechacharn, and V. Kinnares, “A carrierbased unbalanced PWM method for four-leg voltage source inverter fed unsymmetrical two-phase induction motor,” IEEE Trans. Ind. Electron., vol. 60, no. 5, pp. 2031–2041, May. 2013. [8] J. Rodr´ıguez, M. P. Kazmierkowski, J. R. Espinoza, P. Zanchetta, H. Abu-Rub, H. A. Young, and C. A. Rojas, “State of the art of finite control set model predictive control in power electronics,” IEEE Trans. Ind. Informat., vol. 9, no. 2, pp. 1003–1016, May. 2013. [9] S. Kouro, P. Co´rtes, R. Vargas, U. Ammann, and J. Rodr´ıguez, “Model predictive control-A simple and powerful method to control power converters,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1826–1838, Jun. 2009. [10] P. Co´rtes, M. Kazmierkowski, R. Kennel, D. Quevedo, and J. Rodr´ıguez, “Predictive control in power electronics and drives,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4312–4324, Dec. 2008. [11] V. Yaramasu, M. Rivera, B. Wu, and J. Rodriguez, “Model predictive current control of two-level four-leg inverters – Part I: Concept, algorithm and simulation analysis,” IEEE Trans. Power Electron., vol. 28, no. 7, pp. 3459–3468, Jul. 2013. [12] M. Rivera, V. Yaramasu, J. Rodriguez, and B. Wu, “Model predictive current control of two-level four-leg inverters – Part II: Experimental implementation and validation,” IEEE Trans. Power Electron., vol. 28, no. 7, pp. 3469–3478, Jul. 2013. [13] M. Rivera, V. Yaramasu, A. Llor, J. Rodriguez, B. Wu, and M. Fadel, “Digital predictive current control of a three-phase four-leg inverter,” IEEE Trans. Ind. Electron., vol. 60, no. 11, pp. 4903–4912, Nov. 2013. [14] V. Yaramasu, M. Rivera, M. Narimani, B. Wu, and J. Rodriguez, “Finite state model-based predictive current control with two-step horizon for four-leg NPC converters,” Journal of Power Electronics, vol. 14, no. 6, pp. 1178–1188, 2014. [15] V. Yaramasu, M. Rivera, M. Narimani, B. Wu, and J. Rodriguez, “Model predictive approach for a simple and effective load voltage control of four-leg inverter with an output LC filter,” IEEE Trans. Ind. Electron., vol. 61, no. 10, pp. 5259–5270, Oct. 2014. [16] V. Yaramasu, M. Rivera, M. Narimani, B. Wu, and J. Rodriguez, “High performance operation for a four-leg NPC inverter with twosample-ahead predictive control strategy,” Int. J. Electric Power Systems Research, vol. 123, pp. 31–39, 2015.

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