flying-capacitor topology used in an Active Power Filter (APF) is proposed. ... The performance of the inverter system largely depends on the quality of the ... response, simple implementation and operation, high accuracy, inherent current peak ...
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Zare, Firuz and Zabihi, Sasan and Ledwich, Gerard F. (2007) An adaptive hysteresis current control for a multilevel inverter used in an active power filter. In Proceedings 2007 European Conference on Power Electronics and Applications, Aalborg, Denmark.
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An adaptive hysteresis current control for a multilevel inverter used in an active power filter Firuz Zare1
Sasan Zabihi2
School of Engineering Systems Queensland University of Technology Brisbane, GPO Box 2434, QLD, Australia 1
Gerard Ledwich1 School of Electrical Engineering Mazandaran University Babol, Iran 2
Keywords Active filter, Adaptive control, Multilevel converters, Pulse Width Modulation (PWM)
Abstract In most high-performance application of voltage source pulse-width modulation inverters, current control is an essential part of the overall control system. In this paper a novel adaptive hysteresis current control technique based on time and magnitude error for a single phase five-level inverter with flying-capacitor topology used in an Active Power Filter (APF) is proposed.
Introduction Due to crucial problems such as power losses and heat in power transformers and voltage distortions, caused by harmonics in the grid, harmonic elimination seems to be vital and APF plays an effective role in distortion recognition and elimination. Voltage source converters have several configurations such as full/half bridge, reduced topologies and multilevel inverters; each one has its own control strategy. Fig.1 shows a block diagram of a single phase multilevel inverter with an adaptive hysteresis current control to compensate a distorted current. During the last decade, multilevel inverters have attracted much attention due to reduced harmonic content and voltage stress in high power electronic applications. Switching of power electronic devices is controlled in multilevel inverters to synthesis a desired output waveform using several levels of voltages based on pulse-width modulation (PWM) techniques. As the number of voltage levels increases in the multilevel converters, the output voltage waveform becomes a staircase wave which approaches a sine waveform with minimum harmonic distortion. One of the most popular structures proposed as a voltage source multilevel inverter is the flying-capacitor topology [1] applied in the present investigation as shown in Fig.1. Generalized structures and applications of the multilevel inverters have been reported [2], [3]. The performance of the inverter system largely depends on the quality of the control strategy. Amongst different current control techniques considered by several authors, the hysteresis current control is preferred due to its unassailable advantages such as: appropriate stability, fast transient response, simple implementation and operation, high accuracy, inherent current peak limitation, overload rejection, compensation of effects due to load parameters changes and semiconductor voltage drops of the inverters. By comparing a reference current and a load current, the current controller can generate switching states for the power electronic devices, which decrease the current error and provide the desired current waveform for a load. In this method, the reference current is surrounded by several bands. While the load current is between upper and lower bands, no switching occurs and when the load current crosses one of the bands to pass the upper limit (lower limit) the output voltage is decreased (increased). An important issue in implementing the hysteresis control in power converters is a variable switching pattern which causes vast range of switching frequency variations. Although variable switching
frequency has been recognized as solution for motor drive systems to minimize mechanical noise [4,5]; but it is not recommended for power system applications due to generation of sub and low order harmonics which affects the quality of the power system. So providing the ways to limit the switching frequency variations is worthwhile. The adaptive hysteresis current control is a classical method to control the switching frequency [6]. This paper proposes and analyses a new adaptive current control for a single-phase five-level inverter, which can be expanded to multilevel inverters, based on magnitude and time errors in association with a band limiter to reduce the current ripples and to control the frequency variations.
Fig.1: Adaptive hysteresis current control for an APF using a single phase five-level inverter based on flying-capacitor topology
An adaptive hysteresis current control for a multilevel inverter Different applications and modulation techniques have been considered in relation to the multilevel control strategies. Reducing switching losses through switch adjacency and capacitor voltage control are reported in [7-9] which are very important and should be considered in most of the multilevel converters. The hysteresis current control based on the magnitude error for an “n” level inverter can be associated with a number of bands around the reference current, in such a manner that each band does not belong to a specific voltage level [9]. The tolerance bands for an “n” level inverter are of two different natures. The first band consists of a main zone and the load current always has to be inside the main zone to minimize the harmonic distortion. The second set of switching bands has different zones surrounding the main zone in order to provide a reliable and a robust control for an “n” level inverter. A key issue is the change of switch voltage when the derivative of load current is not high enough to follow the reference current and five-level controller may select one of these voltage levels ( 2Vdc , Vdc , 0 , −Vdc and −2Vdc ). Indeed each band is not associated with a specific voltage level but the voltage level pair is changed in the process of tracing the reference current and each crossing of the additional bands will trigger a level change [9]. Different solutions have been proposed to provide a constant switching frequency for three-phase inverters with mathematical analysis describing a relationship between the switching frequency and the system parameters .The main concept is shown in Fig.2 where the derivation of the load current and the reference current determine the switching time and frequency. As the output voltage of utilized multilevel inverter has five levels, the relation between the load and the reference current are found according to Fig.2 where the ascendant and the descendant slopes of the load current generated by four different voltage groups (+2Vdc ,+Vdc ) & (+Vdc ,0) & (0,−Vdc ) & (−Vdc ,−2Vdc ) imposed on an inductor which connects the inverter to the grid . We can drive the following equations from Fig.2 for these voltage levels (+Vdc ,0) : + 1 diCa = (Vdc − Vs ) dt L − diCa 1 = − Vs dt L
(1) (2)
Geometric analyses of the left two triangles in Fig.2 yields: + diCa di* − t1 ⋅ Ca dt dt − di di * 2HB = C + D = −t 2 ⋅ tan γ + t 2 ⋅ tan α = −t 2 ⋅ Ca + t 2 ⋅ Ca dt dt
2HB = A − B = t1 ⋅ tan β − t1 ⋅ tan α = t1 ⋅
(3) (4)
Fig.2: Schematic analyses of current and voltage waveforms with a hysteresis band current control And the switching frequency can be obtained as follow: t1 + t 2 = TC =
1 fc
(5)
Adding and subtracting Eq.3 and Eq.4 yield: * diCa di + di − + t1 Ca − t 2 Ca dt dt dt * + − diCa diCa diCa (t1 + t 2 ) − t1 − t2 =0 dt dt dt
4 HB = (t 2 − t1 )
(6) (7)
Eq.6 and Eq.7 show a relationship between a Harmonic Band (HB) and (t1+t2) which is completely independent from (t1&t2) & (t1-t2): HB =
Vdc ( Lm + Vs ) − ( Lm + Vs ) 2 di* where m = Ca 2 f c ⋅ Vdc ⋅ L dt
(8)
For the voltage group of (0, −Vdc ) : + diCa 1 = − Vs dt L − 1 diCa = − (Vdc + Vs ) dt L
(9) (10)
Assuming: tb − t a = t3
&
tc − tb = t 4
(11)
Geometric analyses of the right two triangles in Fig.2 yield: + di * diCa − t3 ⋅ Ca dt dt − di * di 2HB = H − G = −t 4 ⋅ tan φ + t 4 ⋅ tan ω = −t 4 ⋅ Ca + t 4 ⋅ Ca dt dt
2 HB = E + F = t3 ⋅ tan θ − t3 ⋅ tan ω = t3 ⋅
(12) (13)
As: t3 + t 4 = TC =
1 fc
(14)
Adding and subtracting Eq.12 and Eq.13 results: * di − di + diCa + t3 Ca − t 4 Ca dt dt dt + − * di di di (t3 + t 4 ) Ca − t3 Ca − t 4 Ca = 0 dt dt dt
(15)
4HB = (t 4 − t3 )
(16)
Eq.15 and Eq.16 result a relationship between the HB and (t1+t2). HB = −
Vdc ( Lm + Vs ) + ( Lm + Vs ) 2 di* , m = Ca 2 f c ⋅ Vdc ⋅ L dt
(17)
Similar to the above analyses, the HB calculation is possible for the voltage groups of (+2Vdc ,+Vdc ) and (−Vdc ,−2Vdc ) : + 1 diCa = (2Vdc − Vs ) dt L − diCa 1 = (Vdc − Vs ) dt L Vdc [5( Lm + Vs ) − Vdc ] − ( Lm + Vdc + Vs ) 2 HB = 2 f c ⋅ Vdc ⋅ L
(18) (19) (20)
+ 1 diCa = − (Vdc + Vs ) dt L − 1 diCa = − (2Vdc + Vs ) dt L V ( Lm + Vs + Vdc ) − ( Lm + Vdc + Vs ) 2 HB = − dc 2 f c ⋅ Vdc ⋅ L
(21) (22) (23)
The slop of the reference current, the DC link voltage, the grid voltage and the inductance value are definite in these equations. The hysteresis band size can be determined by a calculation block using Eq.8, Eq.17, Eq.20 and Eq.23 in order to fine a desired switching frequency.
Simulation results: Simulations have been carried out using appropriate codes in M-file platform, implementing a similar condition for a multilevel inverter operation. Table I shows the load and the system parameters used in the simulations. Table I. System Parameters Parameters Appropriate switching frequency Fundamental frequency Supply voltage AC Inverter DC bus voltage Inverter inductance Cdc capacitor
Quantity 10 kHz 50 Hz 30 V 80 V 5 mH 1000 µF
Switching frequency variations for the fixed and the adaptive bands are shown in Fig.3 and Fig.4, respectively; considering two different cases; a pure sine wave and a distorted current as two different reference currents with 10 kHz switching frequency. As shown in Fig.3 the switching frequency varies in a broad band (13 kHz for sinusoidal & 15 kHz for distorted current) for a traditional hysteresis current control (fixed band). An adaptive hysteresis current control for a single phase five level inverter was simulated and the results are shown in Fig.4 and Fig.5. A five-level inverter output voltage and two different output currents, sinusoidal and distorted current waveforms are shown in Fig.5.
At zero crossing points, there are problems associated with the error magnitude which changes the switching frequency. According to the multilevel inverter control strategy shown in Fig.6, when the current error is close to zero, the load current exceeds the first band and the switching happens and the load current crosses the second band in order to change the voltage level. In this case, the switching time depends on the size of the second band and other parameters such as the voltage across the load, the grid voltage and the reference current. At this time if the load current exceeds the first band, the inverter and the grid voltages imposed on the inductance, may not increase the current slope significantly to control the load current and keep it among the main zone which is shown observable in detail in Fig.6. This issue can affect the output current ripples and causes the switching frequency reduction. This is one of the problems in implementing the adaptive band current control for the multilevel inverter which can be solved by including a time error simultaneously with the magnitude error. The time error can be defined based on 10kHz switching frequency where the load current exceeds the first band, the controller waits for a certain time=1/(10kHz) and then changes the output voltage one level.
(a) (b) Fig.3: Variations of switching frequency with constant hysteresis bands (HB1=0.3A & HB2=0.4A), for (a) a sine wave (b) a distorted current as reference currents for the inverter
Fig.4: Variations of switching frequency with the adaptive hysteresis bands
(a) (b) Fig.5: Five-level inverter output voltage levels for (a) a sine waveform (b) a distorted currents as reference currents for the inverter In a hysteresis band PWM current control method for a five-level inverter, two complementary bands are used to accomplish a reference current. As shown in Fig.6 the second band is utilized to reduce the current error when it exceeds the first band. The hysteresis band convergence in multilevel current control happens according to Eq.8, Eq.17 and Eq.23 which is described as follow: Vdc ( Lm + Vs ) − ( Lm + Vs ) = 0 (24) or: −Vdc ( Lm + Vs ) − ( Lm + Vs ) = 0 (25) which means: ( Lm + Vs )(±Vdc − Lm − Vs ) = 0 (26) According to the system specifications, the last term of the above equation never equal to zero while the first term may equal to zero at least two times over each cycle. This issue can be controlled by considering a minimum value for the first band (band limiter), preventing the band to become less than a specific value, and applying the time error when the load current is between the first and the second bands in order to control the switching frequency.
Fig.6: Hysteresis bands limitations and current errors
The controller has been modified based on the time and magnitude errors and the simulation result is shown in Fig.7(a) where the switching frequency is kept constant around 10 kHz. In this case, the first band never becomes less than 0.07A and the simulation result confirms the proposed method. Finally, simulation results in Fig.8 show suitable filter capability for design objectives which generates an acceptable reference current for compensation and harmonic elimination.
(a) (b) Fig.7: (a) Variations of switching frequency for an adaptive hysteresis band limiter with the time error control (b) current error and the first upper and lower bands
Fig.8: (a) Load current (b) compensated current (c) supply current waveforms after harmonic compensation
Conclusion: Low order harmonic contents of an output current of an inverter generated by a hysteresis current control method with a variable switching frequency is known as a drawback of the current controller. An adaptive hysteresis band current control is an effective solution which limits ripples frequency variations and allows passive filters to eliminate such ripples. This paper presents a new adaptive band height current control for a multilevel inverter based on time and a band limiter to avoid switching frequency variation. Geometric analyses with simulation results prove the validity of the combined controller to achieve a high quality output current.
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