A New Simplified Approach for Capacitor Voltage Balancing of Flying Capacitor Multilevel Converters using Space Vector Modulation Mehdi Narimani1, Venkata Yaramasu1, Bin Wu1, Navid Reza Zargari2 1
2
Ryerson University, Toronto, Canada Medium Voltage R&D Department, Rockwell Automation, Cambridge, Canada
E-Mail:
[email protected],
[email protected],
[email protected],
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Keywords «Voltage Source Inverters (VSI) », «Multilevel converters », «Modulation strategy ».
Abstract This paper presents a new, simple and generalized approach to calculate flying capacitor currents for an n-level flying capacitor converter without the use of look-up tables. The performance of the proposed method has been studied for a three-, four- and five-level flying capacitor converter in MATLAB/Simulink environment.
Introduction Multilevel voltage source converters are the most attractive converters for medium-voltage (MV), high power applications. The multilevel structure has some features including; high voltage capability, low dv/dt’s, smaller output filters, and reduced common mode voltages [1]-[6]. The most commercialized multilevel converter topologies are the diode-clamped, flying capacitor, and cascaded H-bridge converters [7]. Compared with the other two configurations, flying capacitor (FC) converter does not require clamping diodes and isolated dc sources. DC capacitor voltages balancing of an n-level diode-clamped converter has practical limitations. To ensure DC-link capacitor voltages balancing during all operating conditions, an additional power circuitry is required [8]. Compared with an n–level diode-clamped converter, the n-level FC converter constitutes simple structure and control [7]. For FC multilevel inverter topologies, several modulation strategies have been developed [8]-[14]. Generally, the multilevel converters are intended to be used in high-power and medium -voltage applications; therefore two major challenges in the selection of modulation strategies should be considered; power quality and switching frequency. The switching frequency should be minimized as much as possible to reduce the switching losses in the converter which is almost about 600Hz for medium-voltage applications. One of the modulation strategies that are the most preferred for high power multilevel converters is space vector modulation (SVM). The SVM method benefits from redundancy for a given switching states. This redundancy makes the flexibility to select the best switching states to satisfy different criteria including, output harmonic content, common mode voltage, and regulating voltages of capacitors in multilevel converters [10]-[14]. Various types of SVM algorithms have been proposed in the technical literatures [10]-[11]. In order to implement SVM-based modulation scheme, the following steps should be followed; transform the reference voltage from three-phase system to two-phase system, identify the sector and the triangle where the reference vector is located, determine the three adjacent switching vectors, calculation of duty-cycle corresponding to the switching states, investigate the redundant switching state, select the best switching states based on minimization of the defined cost function and finally generate and apply the gating signals to the converter.
The flying capacitor voltages should be maintained at their desired nominal values to ensure proper operation of an FC converter. The capacitor voltages can diverge/drift in different conditions [15][16]. There have been proposed some techniques to balance and regulate the capacitor voltages [15]– [21]. The methods proposed in [15]-[19] use open-loop control technique based on the phase-shiftedPWM technique, which benefits from natural self-balancing property. However, in practice, the switching frequency should be high or an external control algorithm is required to balance the capacitor voltages due to non-idealities, system imbalance, and disturbances [20]. Another method is using extra power circuitry to regulate the capacitor voltages [21]. The approach proposed in [21] shows a closed-loop space vector modulation (SVM)-based capacitor voltage balancing method. This method uses the SVM switching state redundancy to regulate the capacitor voltages at their desired values. The proposed strategy in [23] does not have any deteriorating impact on the ac-side waveforms and is general and applicable to an n-level FC converter with any number of levels. The work in [22] demonstrate the balancing of the FC voltages for a fourlevel diode clamped converter by selecting suitable cost function and using SVM. In order to minimize the cost function to balance flying capacitors, the current of flying capacitors should be measured. However, to reduce the cost and complexity associated with the current sensors, the relationship between the flying capacitor currents and output currents can be obtained based on a table defined for different switching states [22]. When the number of levels increases the dimension of the table will increase and consequently the complexity of the calculation of the flying capacitor currents will increase significantly. In this paper, a new, simple and generalized method is proposed to calculate flying capacitor currents for an n-level FC converter. The proposed approach eliminates the use of look-up tables, and thus greatly reduces the complexity of the control algorithm. The performance of the proposed method has been studied for a three-level, four-level and five-level FC converter in MATLAB/Simulink environment. In the following sections, the SVM strategy for an n-level multilevel converter is summarized and the proposed approach to calculate flying capacitor currents based on output currents will be explained and finally simulation results validate the performance of the proposed technique.
Review of SVM for an n-Level Flying Capacitor Multilevel Converter The procedure for the implementation the SVM strategy for an n-level FC multilevel converter, shown in Fig. 1, is summarized based on [22]-[25]. This procedure has the following steps;
Identify the sector and triangle where reference vector is located in α-β coordinate system; Determine the adjacent switching vectors; Duty-cycle calculation; Determine of redundant switching state combinations; Calculate the average capacitor currents; Select the best switching states based on minimization of the defined cost function. Generate the gating signals for the n-level converter.
This procedure is explained in [25] in details. In this paper, a new, simple and generalized approach to calculate the average flying capacitor currents for an n-level flying capacitor converter is proposed. This approach facilitates and helps to implement SVM for the multilevel flying capacitor converter in order to regulate voltage of flying capacitors at nominal values.
Fig. 1. A Three-Phase n-Level Flying Capacitor Converter.
Capacitor voltage balancing strategy Based on the SVM strategy, there are redundant switching states for a given switching state. This redundancy can help to regulate voltage of flying capacitors in a multilevel FC converter. In order to regulate voltage of FCs, the best switching states should be selected among the available redundant switching state to minimize the voltage deviation of the flying capacitors from their nominal values. In order to regulate voltage of flying capacitors, the cost function J can be defined as; 1 2
,
(1)
x=a,b,c. Where is the reference voltage value of the flying capacitor , the following condition should be satisfied;
. To minimize cost function J,
0
,
(2)
, , where; (3) and
, is the current of capacitor
,
, ,
, x=a,b,c , i=1,2,…,n-2. Eq. (2) can be rewritten as; 0
(4)
The best switching states should be find to minimize Eq. (4). If averaging operator applies to (4) over a one sampling period; 1
0
,
(5)
, ,
If the capacitor voltages assumed to be constant over one
0
,
, then;
(6)
, ,
And consequently;
0
,
(7)
, ,
is the average value of capacitor current. For different switching states, the average where, currents and therefore the cost function can be calculated and thus the switching states which minimizes the cost function J , should be selected to apply to the converter. The current can be calculated based on the switching states and their relationship to ac-side currents ia, ib and ic. [22] shows that these current can be calculated based on a table which shows the contribution of switching states to the capacitor currents and their association with the three-phase currents ia, ib and ic; however, when the number of levels increases the dimension of the table will increase and consequently the complexity of the calculation of the flying capacitor currents will increase significantly. This paper proposes a simple generalized method to calculate flying capacitor currents without the use of trigonometric calculations or tables for an n-level flying capacitor converter. It should be noted that if the flying capacitor currents can be calculated for different combinations of adjacent redundant switching states over a sampling period, the best combination which minimize (7) should be chosen to regulate flying capacitor voltages in desired values. In the next Section, calculation of dc-link capacitor currents will be explained.
Calculation of Average Flying Capacitor Currents for an n-Level FC Converter Assume a three-level FC inverter as shown in Fig. 2. The relationship between the capacitor current in can be calculated from Table I. As can be seen from Table I, the phase x (x=a,b,c) and current , and switching states for a three-level FC relationship between the capacitor current in phase x, inverter can be written as; .
=
(8) , ,
Fig. 2. Schematic of a three-level flying capacitor inverter.
Table I: Relationship Between Voltage Level, Switching States and Capacitor Current for a Three-Level FC Inverter Voltage Level Vx (x=a,b,c) 2
1 1 0 0
1 0
1 0 1 0
0 ix -ix 0
In order to calculate the average value of capacitor currents, the averaging operator over one sampling period should be applied to (8) results in; .
=
.
.
.
.
. (9)
, , is the average value of capacitor current for phase x and where, of three adjacent switching vectors , , and respectively.
,
and
are the duty cycles
For a four-level FC inverter, which has two capacitors in each phase, Table II shows the relationship and and phase current x. If the procedure repeats for the fourbetween the capacitor currents and for a four-level FC inverter can be level FC inverter, the relationship between currents written as; =
.
.
.
.
=
.
.
.
.
.
.
.
.
(10)
, , This procedure can be generalized for an n-level flying capacitor converter and the relationship between capacitor currents and output currents can be written as; .
= 1,2, … , , ,
2
.
.
.
.
.
(11)
Table II: Relationship Between Voltage Level, Switching States and Capacitor Current for a Four-Level FC Inverter Voltage Level Vx (x=a,b,c) 3 2 1 0
1 1 1 0 1 0 0 0
1 1 0 1 0 1 0 0
1 0 1 1 0 0 1 0
0 0 ix -ix ix -ix 0 0
0 ix -ix 0 0 ix - ix 0
Equation (11) shows that how to calculate the average current of the flying capacitors based on the switching states, duty cycles and output current. Therefore, in each sampling time, the best switching states which minimize (6) can be chosen and apply to the converter.
Simulation Results In order to show the performance of the proposed method, simulation studies have been done in MATLAB/Simulink environment. The simulation studies demonstrate the effectiveness of the developed SVM strategy to regulate the voltages of flying capacitors. The performance of the proposed method has been studied for a three-level, four-level and five-level flying capacitor converter with parameters shown in Table III. Fig. 3, 4 and 5 show the simulated waveforms of flying capacitors with and without activating controllers.The simulation results validate the proposed strategy.
Table III: Parameters of the Study System Converter Parameters Converter Rating DC-Link Capacitors Input DC voltage Output frequency
Values 25 KVA 500 µF 1200 V 60 Hz
Inverter Line Voltage (V)
1500 1000 500 0 -500 -1000 -1500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
(a) inverter output voltage (Vab) Capacitor Voltages (V)
750 700 650 600 550 500 450
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
(b) voltage of flying capacitors (VCa1, VCb1, VCc1) Fig. 3. Three-level FC converter.
0.2
Inverter Line Voltage (V)
1500 1000 500 0 -500 -1000 -1500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.16
0.18
0.2
(a) inverter output voltage (Vab)
Capacitor Voltages (V)
900 800 700 600 500 400 300 0.02
0.04
0.06
0.08
0.1
0.12
0.14
(b) voltage of flying capacitors (VCa1, VCa2, VCb1, VCb2, VCc1, VCc2) Fig. 4. Four-level FC converter.
(a) inverter output voltage (Vab)
Capacitor Voltages (V)
1000 800 600 400 200 0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
(b) voltage of flying capacitors (VCa1, VCa2, VCa3, VCb1, VCb2, VCb3, VCc1, VCc2, VCc3) Fig. 5. Five-level FC converter.
Conclusion This paper presents a new and simple method to regulate capacitor voltages of an n-level flying capacitor multilevel converters using space vector modulation (SVM). The proposed method introduces a generalized approach to calculate flying capacitor currents without the use of look-up tables. The performance of the proposed method has been studied for three, four, and five-level FC inverter in MATLAB/Simulink environment. The proposed strategy can be easily extended to any nlevel diode-clamped multilevel inverter.
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