Hysteresis Current Control in Three-Phase Voltage Source Inverter

Transcription

1 Hysteresis Current Control in Three-Phase Voltage Source Inverter Mirjana Milošević Abstract The current control methods play an important role in power electronic circuits, particulary in current regulated PWM inverters which are widely applied in ac motor drives and continuous ac power supplies where the objective is to produce a sinusoidal ac output. The main task of the control systems in current regulated inverters is to force the current vector in the three phase load according to a reference trajectory. In this paper, two hysteresis current control methods (hexagon and square hysteresis based controls) of three-phase voltage source inverter (VSI) have been implemented. Both controllers work with current components represented in stationary (α, β) 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 []. A short review of the available current control techniques for the threephase systems is presented in []. Among the various PWM technique, the hysteresis band current control is used very often because of its simplicity of implementation. Also, besides fast response current loop, the method does not need any knowledge of load parameters. However, the current control with a fixed hysteresis band has the disadvantage that the PWM frequency varies within a band because peak-topeak current ripple is required to be controlled at all points of the fundamental frequency wave. The method of adaptive hysteresis-band current control PWM technique where the band can be programmed as a function of load to optimize the PWM performance is described in []. The basic implementation of hysteresis current control is based on deriving the switching signals from the comparison of the current error with a fixed tolerance band. This control is based on the comparison of the actual phase current with 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 phase current interactions which is typical in three-phase systems. This is mainly due to the interference between the commutations of the three phases, since each phase current not only depends on the corresponding phase voltage but is also affected by the voltage of the other two phases. Depending on load conditions switching frequency may vary during the the fundamental period, resulting in irregular inverter operation. In [4] the authors proposed a new method that

2 minimize the effect of interference between phases while maintaining the advantages of the hysteresis methods by using phase-locked loop (PLL) technique to constrain the inverter switching at a fixed predetermined frequency. In this paper, the current control of PWM-VSI has been implemented in the stationary (α, β) reference frame. One method is based on space vector control using multilevel hysteresis comparators where the hysteresis band appear as a hysteresis square. The second method is based on predictive current control where 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. The load model is consisting of a sinusoidal inner voltage e and an inductance (L). i dc S a i a L + e a + U dc C S b i b L + e b - S c i c L + e c Figure : VSI power topology To describe inverter output voltage and the analysis of the current control methods the concept of a complex space vector is applied. This concept gives the possibility to represent three phase quantities (currents or voltages) with one space vector. Eight conduction modes of inverter are possible, i.e. the inverter can apply six nonzero voltage vectors u k (k = to 6) and two zero voltage vectors (k =, 7) to the load. The state of switches in inverter legs a, b, c denoted as S k (S a, S b, S c ) corresponds to each vector u k, where for S a,b,c = the upper switch is on and for S a,b,c = the lower switch is on. The switching rules are as following: due to the DC-link capacitance the DC voltage must never be interrupted and the distribution of the DC-voltage U dc into the three line-to-line voltages must not depend on the load. According to these rules, exact one of the upper and one of the lower switches must be closed all the time. There are eight possible combinations of on and off switching states. The combinations and the corresponding phase and line-to-line voltages for each state are given in table in terms of supplying DC voltage U dc. If we use the transformation from three-phase (a,b,c) into stationary (α, β) coordinate system:

3 [ uα u β ] = [ ] u a u b u c () this results in eight allowed switching states that are given in table and figure. State S a S b S c u a /U dc u b /U dc u c /U dc u ab u bc u ca u α /U dc u β /U dc u u 5 / / / - / / u / / / - / / u 4 / / / - / u / / / - / u 6 / / / - / / u / / / - / / u 7 Table : On and Off states and corresponding outputs of a three-phase VSI U DC u () S u () S S u ref u 4 () u () u 7 () u () S 4 S 6 - U DC - U DC u 5 () - U DC S 5 u 6 () U DC U DC Figure : Switching states of the VSI output voltage

4 Hexagon Hysteresis Based Control Three hysteresis bands of the width δ are defined around each reference value of the phase currents (i a, i b, i c ) (figure ). i a,b,c time Figure : Hysteresis bands around the reference currents i a, i b, i c The goal is to keep the actual value of the currents within their hysteresis bands all the time. As the three currents are not independent from each other, the system is transformed into (α, β) coordinate system. With the transformation of the three hysteresis bands into this coordinate system, they result in an hysteresis hexagon area. The reference current vector i ref points toward the center of the hysteresis what can be seen in figure 4. In steady state, the tip of the reference current moves on circle around the origin of the coordinate system (figure 4). Therefore, the hexagon moves on this circle too. bc S III S II i e S IV i e S I i ref i ca S V S VI ab Figure 4: Hysteresis hexagon in α, β 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 of the hexagon, the inverter has to be switched in order to force the current into 4

5 the hexagon area. The current error is defined as: i e = i i ref () The error of each phase current is controlled by a two level hysteresis comparator, which is shown in figure 5. A switching logic is necessary because of the coupling of three phases. i a i a,ref S I S IV i b i b,ref S VI S III Switching logic Switches states i c i c,ref S V S II Figure 5: Structure of hysteresis control When the current error vector i e touches the edge of the hysteresis hexagon, the switch logic has to choose next, the most optimal switching state with respect to the following: ) the current difference i e should be moved back towards the middle of the hysteresis hexagon as slowly as possible to achieve a low switching frequency; ) if the tip of the current error i e is outside of the hexagon, it should be returned in hexagon as fast as possible (important for dynamic processes). In order to explain the control method the mathematical equations should be introduced (figure 6). i L + e u k Figure 6: The load presentation di dt = L (u k e) () 5

6 According to equation, the current error deviation is given by: di e dt = di dt di ref dt (4) From equations () and (4) we have: di e dt = L (u k u ref ) (5) where the reference voltage u ref is defined by: u ref = e + L di ref dt (6) The reference voltage u ref is the voltage which would allow that the actual current i is identical with its reference value i ref. In [5] the authors explained why the decisive voltage for the current control is the sum of the inner voltage and the voltage across the inductance of the load. The switching logic for the switches has to select the most optimal out of eight switching states according to the mentioned criteria. For the optimal choice of the switching state, only two pieces of information are required: ) the sector S, S,..., S 6 (figure ) of the reference voltage, ) the sector S I, S II,..., S V I (figure 4) in which the current error vector touches the border of the hexagon. For the derivation of the stationary switching table one example would be discussed. Let reference voltage vector u ref be somewhere in sector S (figure ). According to equation 5 the current error deviation is somewhere in one of the hatched areas in figure 7. These seven areas describe direction and speed with which the current error deviation can move.. If i e touches the border of hexagon in sector S I : To get back towards the middle of the hexagon, i e must move in direction of a negative α component. It means that vector u k u ref must have a negative α component. The hatched areas A, A, A 4 and A 5 corresponding in full to this 6

7 bc u u u-uref u-uref A A u ref u4-uref u 4 u u u -u - ref 7 ref u u7 u-uref u A 4 A,7 A ca u 5 u5-uref u u 6- ref u 6 ab A 5 A 6 Figure 7: Corresponding areas for u k u ref criterion are those that suit for states u, u, u 4 and u 5. The second criterion for the choice of the next optimal state is the length of vector u k u ref, which is proportional to the speed of i e. The speed should be as small as possible, which implies that the length of vector u k u ref must be the shortest. It can be seen from figure 7 that state u is the optimal choice because vector u u ref has the minimum length.. If i e touches the border of hexagon in sector S II : To get back i e towards the middle of hexagon, vector u k u ref must be below the ab axis. Hatched areas A, A 4, A 5 and A 6 fulfil this condition (figure 7). Vector u u ref has the shortest length among vectors u k u ref (k=, 4, 5, 6). Therefore, state u is the optimal choice.. If i e touches the border of hexagon in sector S III : To get back i e towards the middle of hexagon, vector u k u ref must be below ca axis (figure 7). Areas A, A 5 and A 6 satisfy this condition in full and state u has the shortest length of vector u u ref and this is the optimal choice. 4. If i e touches the border of hexagon in sector S IV : To get back towards the middle of hexagon, vector i e must move in direction of a positive α component (figure 7). Only state u satisfies this condition fully and therefore, this is the optimal choice. 7

8 5. If i e touches the border of hexagon in sector S V : To get back i e towards the middle of hexagon, vector u k u ref must be beyond ab axis. Only state u fulfils this condition and this is the optimal choice (figure 7). 6. If i e touches the border of hexagon in sector S V I : To get back i e towards the middle of hexagon, vector u k u ref must be beyond ca axis. Areas A, A and A 4 (figure 7) fulfil this condition, but state u has the shortest length of the corresponding vector u u ref and this is the optimal choice. Similarly, the optimal switching states for all other reference voltage sectors S, S,..., S 6 can be determined. Table gives the complete logic for all sectors. Sectors S I S II S III S IV S V S V I S u,7 u,7 u u u u S u u,7 u,7 u u u S u 4 u 4 u,7 u,7 u u S 4 u 4 u 5 u 5 u,7 u,7 u 4 S 5 u 5 u 5 u 6 u 6 u,7 u,7 S 6 u,7 u 6 u 6 u u u,7 Table : Stationary switching table The switching table for stationary behavior is derived for a movement of the current error i e as slowly as possible [6,7]. Due to the fast changes of current reference value i ref, i e can be situated far outside of the hexagon region. In this case, it must be returned as fast as possible back into the hexagon. For the detection of dynamic processes, an additional larger hysteresis hexagon is placed around the existing one (figure 8). In dynamic processes the information about the sector of reference voltage u ref is not needed. If vector i e touches one of the borders of the dynamic hexagon in any sector (S I, S II,..., S V I ) by choosing an inverter voltage u k which directs straight opposite of the direction of current error i e, the speed of getting back i e towards the middle of hexagon will be maximum. For the derivation of the dynamic switching table one example would be discussed. Let current error i e hit the dynamic hysteresis band in sector S I. The state u 4 directs straight to the opposite of sector S I and it should be applied in order to get back i e as fast as possible in the inner hysteresis area. There are similar explanations for all other sectors (S II,..., S V I ), which gives a very simple dynamic table (table ). 8

9 S III S II Dynamic hysteresis hexagon Stationary hysteresis hexagon S IV S I +h S V SVI Figure 8: Stationary and dynamic hysteresis hexagon Sector S I S II S III S IV S V S V I Voltage u 4 u 5 u 6 u u u Table : Dynamic switching table Simulation Results for Hexagon Hysteresis Control The VSI is simulated in MATLAB using PLECS. The simulation result for the explained hexagon hysteresis control is given in figure 9 (steady state). From that figure it can be seen that the vector current error stays within the hexagon area. If we apply step change in reference current that we have that the current error goes outside of the hexagon, because the current changing causes the change in the radius of the circle where the reference current moves on (figure 4), but the hexagon tolerance surface remains the same. The simulation result is presented in figure and the step change can be seen in figure. Hexagon Current Control (steady state).5 beta component of current error alpha component of current error Figure 9: The current error movement in α, β plane (steady state) 9

10 Hexagon Current Control (with Iref step change).5 beta component of current error alpha component of current error Figure : The current error movement in α, β plane (with step change) 4 Three phase current (reference value with step change after. sec) current Three phase current (measured value) current time (sec) Figure : Three-phase VSI current with step change in reference current after. sec (hexagon control) Square Hysteresis Based Control The employed current control method based on square hysteresis band working in (α, β) plane is show in figure. i ref i ref Hysteresis comprator Hysteresis comprator d d Switching table Sa Sb Sc VSI-PWM inverter Load i i Figure : Block diagram of the used method for current vector control From equation () it can be seen that the current vector moves in direction of the voltage across the load inductance, which is the difference between inverter voltage u k and inner voltage of the load e.

11 In this method we have only two tolerance bands (for α and β current components). Therefore, the hysteresis surface is a tolerance square for the current error which is shown in figure. i e i ref i Figure : Square hysteresis area Whenever the current vector touches the border of the surface, another voltage state is applied to force it back within the square. Similarly, as in the case of the hexagon hysteresis control method, here the square tolerance band moves together with the reference current such that the current vector points always in the center of the square. For this purpose two hysteresis comparators for the α and β components are employed. A simple consideration makes it possible to control the current without any information about the load inner voltage. If the current reaches, for example, the right border of the tolerance square, then another voltage state has to be applied which has the smaller α component then the actual state. In this case, regardless of the position of the load inner voltage, the α component of the voltage across the load inductance and therefore the current deviation in direction of α can be reversed. The complex (α, β) plane can be divided into different sectors as defined by the dotted lines in figure. d h/ h/ d h/ i e i e Figure 4: Multilevel hysteresis comparators for α and β components In α-axis it is possible to apply four different voltage levels of u k ( U DC,

12 U DC, U DC and U DC). In β-axis there are three voltage levels of u k (, and ). The exact selection of the appropriate voltage vector u k is determined by structure of the α and β hysteresis comparators and a corresponding switching table (table 4). Hysteresis comparators are depicted in figure 4, where because of the simplicity hysteresis levels are denotes as,, and. For α comparator, level corresponds to level U DC, to U DC, to U DC and to U DC. For β comparator level corresponds to level, level to and level to. The control scheme of this method uses one four level hysteresis comparator for the α component and three level hysteresis comparator for the β component of the current vector error. Digital outputs of the comparators (d α, d β ) select the state of the inverter switches S a, S b, S c using the switching table: α,β levels u 5 u 5 u 6 u 6 u 4 u,7 u,7 u u u u u Table 4: Switch logic table for square hysteresis control The practical implementation of three-level hysteresis comparator is given in figure 5. The implementation of four-level comparator is very similar. h/ + d h Figure 5: Practical implementation of multilevel hysteresis comparator

13 Simulation Results for Square Hysteresis Control The VSI is simulated in MATLAB using PLECS. The simulation result for the explained square hysteresis control is given in figure 6. It can be seen that the current error vector is in the square area (steady state). The simulation result for the step change in reference current is given in figure 7. It can be seen that the current error vector goes outside of the square (similarly as for hexagon control) due to the step change in reference current because the current changing causes the change in the radius of the circle where the reference current moves on (figure ) but the square tolerance surface remains the same. Square Current Control (steady state).5 beta component of current error alpha component of current error Figure 6: The current error movement in α, β plane Square Current Control (with Iref step change).5 beta component of current error alpha component of current error Figure 7: The current error movement in α, β plane (with step change)

14 Comments Current regulator techniques based on the hysteresis control together with switch logic are presented. The hysteresis hexagon control requires knowledge of the parameters of the load, while the square method does not require that. The simulation are done for the following data: E=5V, L=mH, U dc =4V, I ref =A, (after t=5ms, I ref =A). The simulation time is ms, δ =.8A, h=.4a for both control techniques. Averaged switching frequency is higher for square method than for hexagon method. The switching frequencies are different for different phase switches (even for one type of control either for square or for hexagon control). With given parameters of the circuit, for square control, the averaged switching frequency (frequency during the simulation time) for S a is 7kHz, for S b is 7.5kHz and for S c is.7khz and for hexagon control, the averaged switching frequencies are, for S a 94.7kHz, for S b 94.kHz and for S c 8.4kHz. The hexagon method has smaller switching frequencies because this method is based on the rule to get back current error towards the middle of the hexagon area as slowly as possible, which is not the case for square method. Also, the switching frequencies are different for different phases because when one switch changes the state (either from to or from to ) that does not mean that other switches are changing their states, too (for example, if state u () is applied after state u () then only switch S b has to change state from to ). 4

15 References. S. Buso, L. Malesani, P. Mattavelli, Comparison of Current Control Techniques for Active Filter Applications, IEEE Transaction on Industrial Electronics, Vol. 45, No.5, October 998., pp M.P. Kazmierkowski, L. Malesani, Current control techniques for threephase voltage-source PWM converters: a survey, Industrial Electronics, IEEE Transactions on Industrial Electronics, Vol. 45, No. 5, Oct.998., pp B.K. Bose, An Adaptive Hysteresis-Band Current Control Technique of a Voltage-Fed PWM Inverter for Machine Drive System,IEEE Transactions on Industrial Electronics, Vol. 7, No.5, October 99., pp L. Malesani, P. Mattavelli, Novel Hysteresis Control Method for Current- Controlled Voltage-Source PWM Inverters with Constant Modulation Frequency, IEEE Transaction on Industry Applications, Vol. 6, No., January/February 99., pp A. Ackva, H. Reinold, R. Olesinski, A Simple and Self-Adapting High- Performance Current Scheme for Three Phase Voltage Source Inverter PECS Toledo F. Jenni, D. Wuest, Steuerverfahren für selbstgeführte Stromrichter 7. P. Eichenberger, M. Junger, Predictive Vector Control of the Stator Voltages for an Induction Machine Drive with Current Source Inverter, Power Electronics Specialists Conference, 997. PESC 97 Record., 8th Annual IEEE, Vol., -7 June 997., pp. 95-5