Milosevic Hysteresis

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    Hysteresis Current Control in Three-Phase

    Voltage Source Inverter

    Mirjana Milosevic

    Abstract

    The current control methods play an important role in power electronic cir-

    cuits, particulary in current regulated PWM inverters which are widely appliedin ac motor drives and continuous ac power supplies where the objective is toproduce a sinusoidal ac output. The main task of the control systems in cur-rent regulated inverters is to force the current vector in the three phase loadaccording to a reference trajectory.

    In this paper, two hysteresis current control methods (hexagon and squarehysteresis based controls) of three-phase voltage source inverter (VSI) have beenimplemented. Both controllers work with current components represented instationary (, ) coordinate system.

    Introduction

    Three major classes of regulators have been developed over last few decades:hysteresis regulators, linear PI regulators and predictive dead-beat regulators[1]. A short review of the available current control techniques for the three-phase systems is presented in [2].

    Among the various PWM technique, the hysteresis band current control isused very often because of its simplicity of implementation. Also, besides fastresponse current loop, the method does not need any knowledge of load pa-rameters. However, the current control with a fixed hysteresis band has thedisadvantage that the PWM frequency varies within a band because peak-to-peak current ripple is required to be controlled at all points of the fundamentalfrequency wave. The method of adaptive hysteresis-band current control PWMtechnique where the band can be programmed as a function of load to optimizethe PWM performance is described in [3].

    The basic implementation of hysteresis current control is based on derivingthe switching signals from the comparison of the current error with a fixed toler-ance band. This control is based on the comparison of the actual phase currentwith the tolerance band around the reference current associated with that phase.On the other hand, this type of band control is negatively affected by the phasecurrent interactions which is typical in three-phase systems. This is mainly dueto the interference between the commutations of the three phases, since eachphase current not only depends on the corresponding phase voltage but is alsoaffected by the voltage of the other two phases. Depending on load conditionsswitching frequency may vary during the the fundamental period, resulting inirregular inverter operation. In [4] the authors proposed a new method that

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    minimize the effect of interference between phases while maintaining the advan-

    tages of the hysteresis methods by using phase-locked loop (PLL) technique toconstrain the inverter switching at a fixed predetermined frequency.In this paper, the current control of PWM-VSI has been implemented in the

    stationary (, ) reference frame. One method is based on space vector controlusing multilevel hysteresis comparators where the hysteresis band appear as ahysteresis square. The second method is based on predictive current controlwhere the three hysteresis bands form a hysteresis hexagon.

    Model of the Three-Phase VSI

    The power circuit of a three-phase VSI is shown in figure 1. The load modelis consisting of a sinusoidal inner voltage e and an inductance (L).

    ec

    eb

    +

    +

    +

    L

    L

    L

    Sa

    Sb

    Sc

    C

    ia

    ib

    ic

    idc

    Udc

    ea

    +

    -

    Figure 1: VSI power topology

    To describe inverter output voltage and the analysis of the current controlmethods the concept of a complex space vector is applied. This concept givesthe possibility to represent three phase quantities (currents or voltages) with onespace vector. Eight conduction modes of inverter are possible, i.e. the invertercan apply six nonzero voltage vectors uk (k = 1 to 6) and two zero voltage vec-tors (k = 0, 7) to the load. The state of switches in inverter legs a,b,c denotedas Sk(Sa, Sb, Sc) corresponds to each vector uk , where for Sa,b,c = 1 the upperswitch is on and for Sa,b,c = 0 the lower switch is on. The switching rules areas following: due to the DC-link capacitance the DC voltage must never be in-

    terrupted and the distribution of the DC-voltage Udc into the three line-to-linevoltages must not depend on the load. According to these rules, exact one ofthe upper and one of the lower switches must be closed all the time.

    There are eight possible combinations of on and off switching states. Thecombinations and the corresponding phase and line-to-line voltages for eachstate are given in table 1 in terms of supplying DC voltage Udc.

    If we use the transformation from three-phase (a,b,c) into stationary (, )coordinate system:

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    uu

    = 2

    313

    13

    0 13

    13

    uaubuc

    (1)

    this results in eight allowed switching states that are given in table 1 and figure 2.

    State Sa Sb Sc ua/Udc ub/Udc uc/Udc uab ubc uca u/Udc u/Udcu0

    0 0 0 0 0 0 0 0 0 0 0

    u5

    0 0 1 1/3 1/3 2/3 0 -1 1 1/3 1/

    3

    u3 0 1 0 1/3 2/3 1/3 -1 1 0 1/3 1/

    3u4

    0 1 1 2/3 1/3 1/3 -1 0 1 2/3 0

    u1 1 0 0 2/3 1/3 1/3 1 0 -1 2/3 0u6 1 0 1 1/3 2/3 1/3 1 -1 0 1/3 1/

    3

    u2 1 1 0 1/3 1/3 2/3 0 1 -1 1/3 1/

    3u7

    1 1 1 0 0 0 0 0 0 0 0

    Table 1: On and Off states and corresponding outputs of a three-phase VSI

    u2(110)

    uref

    S2

    S3

    S1

    S4

    S5

    S6

    u1(100)

    u6(101)u5(001)

    u4(011)

    u3(010)

    u0(000)u7(111)

    a

    b

    1

    3DC

    U2

    3DC

    U1

    3DC

    U-2

    3DC

    U-

    1

    3DC

    U

    1

    3DC

    U-

    Figure 2: Switching states of the VSI output voltage

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    Hexagon Hysteresis Based Control

    Three hysteresis bands of the width are defined around each reference valueof the phase currents (ia, ib, ic) (figure 3).

    d

    ia,b,c

    time

    Figure 3: Hysteresis bands around the reference currents ia, ib, ic

    The goal is to keep the actual value of the currents within their hysteresisbands all the time. As the three currents are not independent from each other,the system is transformed into (, ) coordinate system. With the transforma-tion of the three hysteresis bands into this coordinate system, they result in anhysteresis hexagon area. The reference current vector iref points toward the

    center of the hysteresis what can be seen in figure 4. In steady state, the tip ofthe reference current moves on circle around the origin of the coordinate system(figure 4). Therefore, the hexagon moves on this circle too.

    iref

    i

    ie

    a

    b

    bc,b

    abca

    aSI

    SIISIII

    SIV

    SV SVI

    ie

    Figure 4: Hysteresis hexagon in , plane

    The actual value of the current i has to be kept within the hexagon area.Each time when the tip of the i touches the border of the surface heading out ofthe hexagon, the inverter has to be switched in order to force the current into

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    the hexagon area. The current error is defined as:

    ie = i iref (2)

    The error of each phase current is controlled by a two level hysteresis com-parator, which is shown in figure 5. A switching logic is necessary because ofthe coupling of three phases.

    SI

    SIV

    SIII

    SVI

    SV

    SII

    ia,ref

    ib,ref

    ic,ref

    ia

    ib

    ic

    Switchinglogic

    Switches

    states

    d

    d

    d

    Figure 5: Structure of hysteresis control

    When the current error vector ie

    touches the edge of the hysteresis hexagon,the switch logic has to choose next, the most optimal switching state with re-spect to the following:

    1) the current difference ie should be moved back towards the middle of thehysteresis hexagon as slowly as possible to achieve a low switching frequency;

    2) if the tip of the current error ie is outside of the hexagon, it should bereturned in hexagon as fast as possible (important for dynamic processes).

    In order to explain the control method the mathematical equations shouldbe introduced (figure 6).

    uk

    i L + e

    Figure 6: The load presentation

    di

    dt=

    1

    L(uk e) (3)

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    According to equation 2, the current error deviation is given by:

    diedt

    =di

    dt

    direfdt

    (4)

    From equations (3) and (4) we have:

    diedt

    =1

    L(uk uref) (5)

    where the reference voltage uref is defined by:

    uref = e + Ldirefdt

    (6)

    The reference voltage uref is the voltage which would allow that the actualcurrent i is identical with its reference value iref. In [5] the authors explainedwhy the decisive voltage for the current control is the sum of the inner voltageand the voltage across the inductance of the load.

    The switching logic for the switches has to select the most optimal out ofeight switching states according to the mentioned criteria. For the optimalchoice of the switching state, only two pieces of information are required:

    1) the sector S1, S2,...,S6 (figure 2) of the reference voltage,

    2) the sector SI, SII,...,SV I (figure 4) in which the current error vectortouches the border of the hexagon.

    For the derivation of the stationary switching table one example would bediscussed. Le