Symmetrical Tuning for Resonant Controllers in Inverter based Micro-Grid Applications A. Lidozzi, G. Lo Calzo, L. Solero, F. Crescimbini University of ROMA TRE, Department of Engineering Via della Vasca Navale 79, 00146 Roma (Italy).
[email protected]
Abstract – Voltage source inverters in micro-grid applications need the effective control of the filter output voltages. Harmonic compensation is a mandatory request to eliminate voltage distortion due to non-linear and unbalanced loads. The use of resonant controllers allows achieving high quality voltage regulation. The proposed paper investigates a closed form tuning strategy for controllers’ parameters selection based on the symmetrical frequency behavior of the whole resonant controller structure. Symmetrically tuned controllers allow achieving the largest bandwidth around the resonance frequency in a multiresonance controller structure. Index Terms – Resonant controllers, integral resonant control, harmonic compensator, four-leg inverter, closed form control tuning, power generation systems.
I. INTRODUCTION Over the years, power inverters have found several applications in numerous either grid-tied or stand-alone systems. More stringent regulations on harmonic distortion and power quality require even more complex control architectures, that conventional PI and PID controllers are not able to effectively provide. Compensation of distorted loads is mandatory in both grid connected and off-grid systems. In micro-grid applications the regulation of the inverter output voltage is required to provide constant voltage constant frequency to the loads. As in grid-tied applications, also in stand-alone systems the ability of the inverter control structure to regulate to zero the injected DC-component is a mandatory requirement. However, the feedback signals to control the DCcomponent are different in grid-tied configurations with respect to off-grid applications. In grid-connected systems, the feedback is usually closed on the inverter current, either grid side or inverter side, and its DC-current component is forced to be zero. In stand-alone application, main control loop is closed on the filtered output voltages and the DC-voltage component must be forced to be zero. The ability of resonant controllers to track sinusoidal references and saving computational load have made this controller topology widely used when sinusoidal current or voltage must be regulated at inverter output. However, they are totally blind from DC-component point of view (definitely unsuitable for regulating to zero the injected DC-component), as it is shown in Section III.A of the present paper. Several studies have been proposed so far concerning the use of Resonant Controller (RC) structures in both grid-tied and off-grid applications. The benefits in adopting this kind of controllers are well known and deeply explained as in [1-5]. Moreover, RCs have been combined in both single loop and
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double loop control schemes to accomplish voltage and current regulation when needed [6-9]. Usually in a single control loop, the control action is performed only on the inverter output voltage or current. When a double control loop is implemented, the outer loop is the VSI output voltage whereas the inner loop regulates the supplied current. Different combinations of transfer functions as proportionalresonant and integral-resonant have been analyzed in [8]. In [8] a single loop control structure with integral compensation and current limiting feature is finally proposed for a 50 Hz stand-alone generation system. Also in [9] the single loop architecture is preferred and applied to a 400 Hz ground power unit. Resonant controllers’ application has been extended also to electric drives. Both induction motor and permanent-magnet base drives have been considered to be controlled by RCs as shown in [13- 15]. Some basic criteria for RCs tuning have been preliminary depicted in [17] where the resonant control structure is compared with the H-infinity algorithm. Controllers gain and width are mainly selected by the designer using a trial and error procedure which is usually verified through root-locus and Bode diagrams. After that, the controller is digitally implemented and the performances are verified by simulations and experimental tests. In [18] a different approach based on the Nyquist diagram is proposed for Proportional+Resonant (PR) and Vector Proportional+Integral (VPI) controllers. However, the mathematical-analytical tuning approach for RCs is usually missing; for this reason, the aim of the paper is to provide a closed form tuning procedure to be used to select the parameters of each controller being part of a Multi Resonant Control System. According to the proposed approach, the control designer is able to select the desired controller gain with respect to the harmonic to be regulated, and the controller width will be automatically selected by the tuning algorithm. When two or more resonant controllers are combined together, the achieved tuned controller structure results in a symmetrical form in terms of magnitude and phase distribution around each controller resonance frequency. The symmetrical tuning assures the maximum distance in term of frequency between two adjacent resonant controllers reducing their reciprocal interaction. The extension of the tuning procedure from the pure Resonant (R) structure to the more suitable Integral+Resonant (I+R) scheme, which is able to compensate the DC-component, is also shown and experimentally verified.
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II. RESONANT CONTROLLERS Resonant controllers can be synthesized through different analytical expressions, the ideal form, the approximated form and the full real form. The RC ideal form is very simple and it is shown in (1) k s G frci ( s) = 2 ir 2 (1) s + ω0 where kir is the controller gain and ω0 the resonance frequency. However, the ideal form should be avoided due to its infinite gain at the resonance frequency which makes the transfer function difficult to be digitally implemented. The approximated form of the RC is expressed as (2) 2kir ωcr s G frca ( s) = 2 (2) s + 2ωcr s + ω02 where ωcr is the controller width. It could be used in the proposed analysis making the tuning expressions quite simple to be achieved. However, the validity of the approximation must be verified every time the controller gains are changed. In order to provide a more general procedure, which can be absolutely applied to the previously mentioned RCs, the full real resonant controller having the well-known transfer function reported in (3) is considered. G frc ( s ) =
2kir (ωcr s + ωcr2 )
(3) s 2 + 2ωcr s + ωcr2 + ω02 From (3), the controller gain at the resonant frequency can be simply achieved through few arithmetical operations and it is shown as follows
2kir ω02 + ωcr2 4ω02 + ωcr2
(4)
It can be noticed that when ω02 >> ωcr2 the gain can be directly set by the selection of the variable kir . This approximation is often verified being ω0 the frequency where the controller is centered, whereas ωcr is usually lower than 1rad/s. For the European grid, ω0 values are shown in (5) for the I, III and V compensator ω01 50 Hz → 314rad / s
ω03 150 Hz → 942rad / s
Transfer function magnitude at the resonant frequency can be plotted versus the ωcr parameter for a constant kir = 200 as
shown in Fig. 1. The parameter ωcr sets the frequency width symmetrically at -3dB from the resonant peak value. It can be noted that the magnitude is almost constant for some reasonable values of ωcr in [rad/s]. Fig. 2 depicts the magnitude and phase behavior nearby the resonant frequency when the ωcr parameter changes. It can be noticed the reduction of the selectivity of the controller as well a smoother phase transition when ωcr increases.
Fig. 2. Magnitude and phase of the resonant controller at the resonance frequency for increasing controller width ωcr [rad/s].
When a single resonant controller, usually at the fundamental frequency, is combined with higher harmonic compensators (HCs), the interaction causes the notch resonance frequencies moving as depicted in Fig. 3. The dashed lines show each single resonant controller, whereas the I+III solid trace shows the combination of I and III HCs. When two HCs are added, a negative resonance occurs due to the presence of two complex conjugate zeros in the resulting transfer function. This particular resonance is located at the frequency for which the two independent HCs magnitudes have the same value and in this particular point the phase shift of the resulting controller is equal to zero. These effects are used in the tuning procedure to symmetrically locate the negative resonance between two HCs.
(5)
ω05 250 Hz → 1570rad / s
Fig. 3. Resonant controllers interaction. Dashed lines are related to each single HC whereas the solid lines are related to resonant controllers combinations.
Fig. 1. Magnitude of the resonant controller as a function of ωcr [rad/s].
When one more HC is added, the previously mentioned negative resonant peak between I and III harmonic changes its frequency, as shown in figure (I+III+V solid trace). Moreover, due to the addition of the V harmonic a new resonance is
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present between the III and the V harmonic controller. In fact, when three HCs are combined, two pairs of complex conjugate zeroes are present in the whole controller transfer function. Then, when two or more HCs interact, the shape of each HC must be chosen also on the basis of the selected parameter values for the others controllers. Controllers interaction can be viewed also considering a polezero map (PZmap) of the compensators as shown in Fig. 4, where circle and cross represent respectively zeros and poles position. Fig. 4a shows the PZmap of the first HC and the third HC being considered separately. When these two HCs are combined together in the control loop, the result is depicted in Fig. 4b, where the controller poles do not change their location. On the contrary, controller zeroes change from two to three and two complex conjugate zeros appear introducing the negative resonance peak which is used for controller tuning as discussed in Section III.
Fig. 5. Multiple resonant compensator behavior when a single harmonic controller width is changed.
III. PROPOSED TUNING STRATEGY
Fig. 4. Pole-zero map of I and III harmonic controllers considered separately (a) and combined together (b).
Nowadays there are not specific criteria to select the frequency width of a resonant controller. In grid connected applications the HC must be designed to be quite large to still compensate electrical system small frequency changes. Alternatively, controllers’ parameters are continuously evaluated to adapt the resonant frequency to the estimated grid frequency. On the contrary, in isolated grid applications the HC width can be reduced being the frequency directly controlled by the output power stage. However, selection of the characteristics for a resonant controller which is usually part of multiple harmonic system, strongly affects the performances of the others controllers, with particular reference to the closest ones. Fig. 5 shows the behavior of the multiple resonant system composed by I, III and V harmonics when the V harmonic width is changed from ωcrV1=3 mrad/s up to ωcrV2=30 mrad/s and then finally down to ωcrV3=150 μrad/s. Once the controller gain is chosen, through the variable kir value, according to the desired requirements; controller width selection is the challenging part in the tuning process. Moreover, being the HC chain implemented in digital form on either a microcontroller (μC) or a Digital Signal Processor (DSP), the resonance frequency can change due to the unavoidable arithmetical approximation.
In order to achieve symmetrical resonant controllers in the complete controller structure, a closed form expression is proposed to find the HC width considering the controllers interaction. It is assumed that the negative resonance peak, which is present between each pair of HCs, is placed in the middle frequency of HC(j) and HC(j-1). If the fundamental frequency is 50 Hz, the first negative resonance is placed between the first and the closest higher HC, for instance the III HC. Due to that, the first peak must be placed at 100 Hz. According to this procedure, if the control structure includes more HCs, each negative resonance frequency is located in the middle frequency of two adjacent HCs. Fig. 6 depicts the proposed spatial frequency for a typical multiple resonant controller, where the subscript x,y,z stands for the generic higher harmonic with respect to the fundamental.
Fig. 6. Notch frequency locations with respect the HCs resonance frequency selection.
First of all, in order to retrieve a closed form design procedure, it can be noticed that when only two HCs are considered, the negative resonance peak is located exactly at the intersection of the magnitude of each HC considered alone as shown in Fig. 7.
Fig. 7. Bode plots of two resonant controllers (dashed lines) and the resulting multi resonant controller (solid line).
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The magnitude of each HC can be simply evaluated by solving (6) for the width of one of the two HC, and then the symmetrical tuning is achieved. 2 2kirI (ωcrI s + ωcrI )
s
2
2 + 2ωcrI s + ωcrI
+ ω02I
=
2 2kirIII (ωcrIII s + ωcrIII ) 2
2 s + 2ωcrIII s + ωcrIII + ω02III
(6)
When the fundamental HC is considered, it is preferred to completely select the controller parameters being these values very dependent on the desired system behavior. According to that, the designer chooses the values of kirI , ωcrI for the main HC and then finally select the desired gain for the next harmonic to be compensated (i.e. third harmonic order), kirIII . Equation (6) can now be simply solved to find the unknown term ωcrIII forcing the Laplace variable to be s = sI − III = jωresI − III = j 2π ⋅100
(7)
For two HCs, equation (7) causes the solution ωcrIII to be exactly the necessary controller width to locate the negative resonant peak at the desired frequency ωresI − III which, for symmetrical tuning, is 2π ⋅100 Hz. However, a generic multiple harmonic controller is composed of more than two HCs. According to the shown procedure, when three HCs are considered, for instance I, III and V harmonics as shown in Fig. 8, the unknown quantities to be determined are ωcrIII and ωcrV .
Fig. 8. Bode plots of three single resonant controllers (dashed lines) and the resulting multi resonant controller (solid line).
Hence, a general expression useful for the Symmetrical Tuning (ST) becomes very difficult to be achieved through the magnitude approach. The multiple mathematical equations to be solved increase the complexity as the number of HCs increases. A slightly different approach is then considered. Instead of using the magnitude of the system transfer function (TF), it is more convenient to evaluate the phase of the whole system. It can be noted that when either a positive or negative resonance occurs, system phase crosses zero degrees. First of all, the designer should select the maximum order for the harmonic to be compensated according to the system constraints, i.e. the output filter resonance frequency and the sampling frequency for the digital implementation. After that, the complete HC structure can be defined as shown in Fig. 9.
Fig. 9. Block scheme of the multiple resonant controller.
Hence, the complete TF to be solved is evaluated as 2k (ω s + ω 2 ) 2k (ω s + ω 2 ) HCtot ( s ) = 2 irI crI 2 crI 2 + 2 irN crN 2 crN 2 s + 2ωcrI s + ωcrI + ω0 I s + 2ωcrN s + ωcrN + ω0 N (8) where N stands for the system last harmonic to be compensated. Equation (8) must be solved as shown in (9) ∠ HCtot ( s) s = s* = 0 (9)
where s* indicates the array of frequencies for each negative resonance peak in the system. The size of the array s* is the total number of HCs minus one. s* = [ s I − III
... s M − N ] = [ jω resI − III T
jω resM − N ] (10) T
...
The array s* yields a system which is composed of HCs-1 equation in HCs-1 unknown terms (11), which are the widths of each HCs having the first harmonic controller freely designed.
[ωcrIII
(11) ωcrV ... ω crN ] When the symmetrical tuning is used, considering the fundamental frequency of 50 Hz, the presence of only odd HCs, and assuming the VII harmonic as the highest to be compensated, the resulting s* vector is as in (12) ⎡ j 2π ⋅100 ⎤ ⎢ j 2π ⋅ 200 ⎥ ⎥ (12) s* = ⎢ ⎢ j 2π ⋅ 300 ⎥ ⎢ ⎥ ⎣ j 2π ⋅ 400 ⎦ Solving for the phase of (9) is quite simple when complex number representation is used. Moreover (9) can be rewritten as in (13) where the phase of numerator and denominator of the TF are considered separately. T
(
NUM DEN ∠ HCtot ( s) s = s* = ∠HCtot ( s) − ∠HCtot (s)
)
s = s*
= 0 (13)
Avoiding the atg (arctangent) function which makes the system equations very difficult to be solved, being the equation (13) forced to be zero, instead of the angle, the tg (tangent) can be used as in (14)
(
) (
tg ϕ NUM − tg ϕ DEN
)
s = s*
=0
(14)
From (14), eq. (15) can be directly derived NUM DEN Im( HCtot ( jωres )) Im( HCtot ( jωres )) (15) − =0 NUM DEN Re( HCtot ( jωres )) Re( HCtot ( jωres )) where Im and Re are respectively the imaginary and real operators. Finally (15) can be simplified as NUM DEN DEN NUM Im( HCtot ) Re( HCtot ) − Im( HCtot ) Re( HCtot ) NUM DEN Re( HCtot ) Re( HCtot )
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= 0 (16)
The final solution can be obtained by solving only for the numerator. System equation to be solved will be similar to NUM DEN DEN NUM ⎧Im( HCtot =0 ) Re( HCtot ) − Im( HCtot ) Re( HCtot ) ωresI −M ⎪ ⎪ (17) # ⎨ ⎪ NUM DEN DEN NUM =0 ⎪Im( HCtot ) Re( HCtot ) − Im( HCtot ) Re( HCtot ) ω crK − N ⎩
where M is the HC next to the fundamental, N is the last HC and K is the resonant controller located immediately before N. The proposed tuning strategy can be used to achieve the parameters for a very general multiple resonant structure (MRS) which could be devoted to control a power inverter. Fig. 10 shows the tuned MRS compensator where the solid line represents the symmetrically tuned multi resonant controller, whereas the dashed lines are related to the single resonant controllers centered on each frequency to be compensated.
Complete controller poles can be viewed as a simple multiplication of the denominators of the single resonant controller transfer function. On the other hand, controller zeroes depend on a more complicated combination of numerator and denominator of each HC. However, in this case an elliptical zeroes distribution is also found. A. Integral-Resonant controller tuning A pure resonant control action however is not often used and recommended because of its inability to compensate DCcomponents [8]. For these reasons the proposed tuning strategy has been extended also when the Integral-Resonant regulation is used. In fact, in off-grid power generation systems the DC-component on the output voltages has to be reduced and controlled. A pure resonant control action is unsuitable for regulating to zero the injected DC-component whereas the integral action provides the necessary gain at lowfrequency and even at DC to regulate the offset of the output voltages. Using the same tuning steps as previously shown, the addition of the integral term is very intuitive. Starting from (8) the complete controller transfer function can be now rewritten as in (18) k IHCtot ( s ) = ii + HCtot ( s ) (18) s where kii is the desired integral gain.
Fig. 10. Symmetrically tuned MRS harmonics controller.
Fig. 12. Symmetrically tuned resonant controller with the addition of the integral action.
Fig. 11. Pole-zero map of the symmetrically tuned MRS controller.
The symmetrical tuning procedure provides a geometric placement of the poles (Ps) and zeroes (Zs) of the final controller. The evaluated controllers’ width, which is related to the selected gains, is able to arrange the Ps and Zs on two separates elliptical curves as shown in Fig. 11.
Fig. 13. Controller transfer function of I+R harmonic compensator versus the integral gain variation.
The symmetrically tuned I+R control, achieved for the integral gain kii=0.10s-1, is depicted in Fig. 12. When the I+R controller structure is used, the first negative resonance peak (Ipeak), highlighted in Fig. 12, cannot be symmetrically placed.
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In fact, the addition of the integral action imposes to select the desired integral gain kii and the slope of the integral action as shown by the dotted line in Fig. 12. This behavior moves the frequency position of the first negative resonance. When kii is increased, the resonance peak moves closer to the fundamental harmonic compensator as shown in Fig. 13, where the integral gain is changed from 0.02 s-1 to 1 s-1. When the Pole-Zero map is plotted for the symmetrically tuned I+R controller, a pole in the origin is added because of the presence of the integral term as shown in Fig. 14. Further, one extra zero is added to the TF; hence, the real zero of the integral controller is combined with this new zero to form two complex conjugate zeros. A fully symmetrically tuned I+R structure would require the placement of the first notch at half the fundamental frequency (i.e. in the between of the DCcomponent and the first harmonic). When the I+R controller is selected, the fully symmetrical tuning is not recommended to let the designer free to select the integral gain which is related on how fast the DC-component must be suppressed. As a consequence, the proposed tuning algorithm does not require as input the frequency position of the first negative resonance, (i.e. the resonance just before the fundamental harmonic controller). This behavior is shown is Fig. 14, where the highlighted pole and zeroes are not on their respective elliptical curves.
Fig. 15. Block scheme of the 4-leg inverter output voltage control loop.
From the control point of view, each block of the system can be represented through its characteristic transfer function as shown in Fig. 15. The HCtot(s) represents the complete transfer function of the resonant controller as in (8) and (18). Power inverter is modeled by its first order approximation, which is very simple to manage. The inverter is seen from the control algorithm mainly as a gain with a delay due to the discretization caused by the PWM unit. Hence K mVdc G4 −leg ( s) = (19) s 1+ 2π Fsw where Km is the gain related to the modulation strategy, Vdc and Fsw are respectively the input DC-link voltage and the inverter switching frequency. Output phase voltages are filtered by means of the second order low-pass Butterworth filter having the transfer function as in (20) ⎛ ⎞ ω2 f ⎟ Glpf ( s ) = ⎜ 2 (20) ⎜ s + 2ω f s + ω 2 f ⎟ ⎝ ⎠ where ωf is the filter cut-off frequency. The power filter has been accurately designed for this application and its structure is shown in Fig. 16.
Fig. 16. Single phase circuit of the output power filter.
Fig. 14. Pole-zero map of the symmetrically tuned I+R controller.
IV. CONTROL LOOP DESIGN In this section the I+R controller and the suggested symmetrical tuning procedure are proposed for a 4-leg voltage source inverter being intended for both 3-ph and single phase distorted loads. The control loop transfer function is investigated to verify system stability and interaction between components, as main filter, measure filter, trap filter and dampers which will be described as follow. Output voltage control in four-leg inverter is obtained considering each phase as a single phase inverter, and using the HCs to regulate the phase-to-neutral voltage. Inverter fourth leg, which is devoted to provide the neutral connection, is modulated without direct voltage regulation as suggested in [19].
Main LkCf1 filter is connected to a selective damper which is accomplished by the RdLdCd leg and to the LtCt switching trap having its resonant frequency as close as possible to the inverter switching frequency. Filter transfer function can be simply achieved considering the impedance of each part of the filter in the s domain as follows 1 Ztrap ( s) = sLt + (21) sCt
1 + Rd (22) sCd where Ztrap(s) and Zdump(s) are respectively the TF of the trap filter and the TF of the selective damper. Then the filter transfer function Vxno(s)/Vxni(s) can be obtained through the combination of the impedance of each leg as 1 Z dump ( s ) / / Z trap ( s ) / / sC f 1 (23) G pwf ( s ) = ⎛ 1 ⎞ sLk + ⎜ Z dump ( s ) / / Z trap ( s ) / / ⎟ ⎜ sC f 1 ⎟⎠ ⎝
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Z dump ( s) = sLd +
Complete open-loop system transfer function GCol(s) is shown in Fig. 17 (solid trace). On the same figure are also depicted the Butterworth type low-pass filter Glpf (dashed trace), the output power filter Gpwf (dashed trace) and finally the inverter which is approximated with the transfer function G4-leg(s) (dashed trace). It can be noticed that the inverter gain KmVdc causes an upper shift of the whole TF as well instability because of the remaining resonance of the output LC filter.
Fig. 17. Complete system open-loop transfer function at first tentative tuning using symmetrical method.
The discussed effect can be solved through scaling the desired HC gain with the inverter total gain as in (24). kirM =
DESIRED kirM
K mVdc
(24)
where M stands for the generic M-harmonic compensator. According to (24) the complete system TF yields a stable system as shown in Fig. 18. As a result, using the proposed tuning procedure, the controllers dynamic can be adjusted by changing their gains without affecting the symmetrical tuning.
Fig. 18. Open-loop transfers function of the whole system after the scaling gain kirM.
Considering the Integral+Resonant controller, which is suggested for this application due to the capability of DCvoltage compensation, the comparison with a conventional resonant controller is performed. In Fig. 19 the tuned openloop complete transfer function is plotted when respectively a pure R (dashed trace) and an I+R (solid trace) controller is used. It can be observed the symmetrical behavior of the resonant controllers for both R and I+R structure. At the same
manner, the proposed tuning method can be simply extended to a proportional-integral controller. In Fig. 19 are also highlighted the frequency position of the filter elements: LC element, selective damper and switching trap.
Fig. 19. Open-loop Bode diagram of R and I+R complete system transfer functions with the HCs symmetrical tuning final procedure.
V. EXPERIMENTAL RESULTS In order to verify the effectiveness of the proposed tuning procedure, a complete test-bed has been realized with a 40kVA VSI 4-leg inverter previously developed for microgrid applications. The VSI is intended to supply a 3P+N electrical system at rated voltage and frequency of 380V, 50Hz. With reference to the equivalent circuit of Fig. 16, output power filter components values are Lk=800μH, Cf1= 5μF, Lt=172μH, Ct=1.12μF, Ld =1mH, Cd =2.56μF and Rd=15Ω. Fig. 20 depicts the output phase voltage and current when a balanced resistive load is connected to the output power filter.
Fig. 20. Experimental results: 28 kW 3-phase balanced resistive load, I (40 A/div), V (100 V/div).
In order to test the harmonic compensators, a very unbalanced and distorted load condition is proposed. A single phase diode rectifier with a DC resistor Rdc=16Ω is supplied between phase A and neutral N of the 4-leg VSI, a resistive load RB=18Ω is directly connected between phase B and neutral N. Phase C is at no load. Fig. 21 shows the phase A and B both output voltages and currents with the previously mentioned load. Achieved waveforms attest the effectiveness of the proposed control algorithm and tuning strategy.
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REFERENCES [1] [2]
[3]
Fig. 21. Experimental results: unbalanced and distorted load, phase A and B VSI output voltages (200 V/div) and currents (20 A/div).
[4]
[5]
[6]
[7] Fig. 22. Experimental results: 9kW 3-phase induction motor running at light load, output phase voltage (200 V/div) and output current (5 A/div).
A critical test to validate the behavior of the control architecture is performed through the direct starting of an induction motor, when the occurring very low power factor operating condition is critical for off-grid inverter applications. The worst load condition occurs when the induction motor is run at light load; in this case the power factor is very low, mostly even lower than 0.1. Fig. 22 shows the output phase voltage when a 9 kW induction machine is operated at light load. It can be noticed that machine current lags the output phase voltage of an angle which is slightly less than 90°. The proposed symmetrical tuning for resonant controllers assures the required stability, even during the transients of the induction motor starting as well in operating conditions of very low power factor. It can be noticed how the proposed control structure can operate with extremely variable power factor loads without the need of any additional control block or active-reactive power estimation algorithms. CONCLUSIONS
[8]
[9]
[10] [11] [12]
[13]
[14]
A newly tuning method for integral-resonant control architecture is investigated and experimentally verified. The proposed symmetrical tuning allows achieving a symmetrical behavior of the controller gain around the resonance frequency. The symmetrically tuned multiple-resonant control assures the maximum distance between two adjacent compensators, minimizing the reciprocal interaction and increasing their selectivity. The proposed strategy has been extended to a multi-controller system with and without the integral action. When the symmetrical tuning process is applied to the integral-resonant control structure, the maximum frequency width between two adjacent resonant controllers is achieved together with the benefits of using the integral term to regulate to zero the output voltage DCcomponent. Without loss of generality, this approach can be extended to the proportional-resonant or even more complex resonant-based control structures.
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