Sinusoidal Voltage Shaping of Inverter-Equipped Stand ... - IEEE Xplore

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 6, JUNE 2015

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Sinusoidal Voltage Shaping of Inverter-Equipped Stand-Alone Generating Units Alessandro Lidozzi, Member, IEEE, Luca Solero, Member, IEEE, Stefano Bifaretti, Member, IEEE, and Fabio Crescimbini, Member, IEEE

Abstract—Voltage-source inverters in microgrid standalone applications need the accurate control of the ﬁlter output voltages. Most of the time, supplied loads are not constant along daily based period; they change the total power consumption, the power factor, the current harmonic content of nonlinear equipment, and the power balancing among the phases. Consequently, the control dynamic behavior is affected, and the effectiveness of harmonic controllers is reduced. In this paper, the very sinusoidal shaping of inverter output voltage is achieved through a multiple resonant controller, which is adaptively tuned online according to the output power ﬁlter and load behavior. Resonant compensators are placed in the loop to achieve low total harmonic distortion of output voltage waveforms. Index Terms—Closed-form control tuning, four-leg inverter, harmonics compensators, phase compensation, power generating systems, resonant controllers (RCs).

I. I NTRODUCTION

E

LECTRICAL energy is a product, and like any other product, it should satisfy the proper quality requirements. Electrical equipment requires electrical energy to be supplied at a voltage that is within a specified range around the rated value. A significant part of the equipment in use today, particularly electronic and computer devices, require good power quality. However, the same apparatus often causes distortion of the supply voltage waveform, because of its nonlinear characteristics. Reference standards for grid requirements are the European Standards EN-50160 and EMC EN-61000, which specify the quality limits that the utilities must satisfy. Even if the European Union has its own regulation, each country can apply some restrictive mandatory requirements. Considering the voltage total harmonic distortion (THDV ) EN-50160 is referred up to the 25th harmonic with respect to the fundamental component. On the other hand, when stand-alone power generation is considered, the uninterruptible power supply (UPS) systems reference standard is the IEC 62040-3, which denotes that

Manuscript received March 24, 2014; revised April 1, 2014, July 8, 2014, and September 24, 2014; accepted October 22, 2014. Date of publication November 20, 2014; date of current version May 8, 2015. A. Lidozzi, L. Solero, and F. Crescimbini are with the Department of Engineering, Roma Tre University, 00146 Rome, Italy (e-mail: lidozzi@ uniroma3.it; [email protected]; [email protected]). S. Bifaretti is with the Department of Industrial Engineering, University of Rome Tor Vergata, 00133 Rome, Italy (e-mail: bifaretti@ing. uniroma2.it). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2014.2370939

the output voltages harmonic content should be within IEC 61000-2-2 limits under linear and nonlinear load conditions. In UPS power generating units, usually, an output transformer is used to provide load impedance adaptation and to block thirdorder harmonic and its multiples. However, due to incremental costs and reduced overall efficiency, the decoupling transformer should be avoided in new installation. According to that, a voltage-controlled front-end inverter operating in off-grid applications should be able to comply with the aforementioned regulations [1]. Resonant controllers (RCs) have attracted several researchers working on different applications mainly concerning active power filters and active front-end and stand-alone inverters. RCs allow zero steady-state error for a sinusoidal reference. They can be effectively combined together to achieve a multiple RC (MRC), which allows performing selective harmonic control. The conventional control techniques for the load voltage control of four-leg inverters employ proportional–integral (PI) regulators, which present very poor dynamic and power quality performances in case of output voltage control [2]. A better dynamic behavior can be obtained with linear controllers based on poles placement [3] for the same requirements on power quality. A significant improvement can be obtained using model predictive control strategies, which have been recently proposed for stand-alone generation units based on four-leg inverters [4]–[6]. Such kind of control allows eliminating the PI voltage control loop, resulting in better dynamic performances than conventional controllers. However, due to the variable switching frequency, the LC output filter design cannot be minimized, producing an oversizing of the converter; moreover, due also to the mismatch between model and real parameters, in some cases, the THD of the output voltage overcomes the 3% limit imposed by some European countries. Within the model predictive control strategies to be used in power electronics applications, the MRC main competitor is the repetitive control [7], [8], which is able to provide theoretically infinite RCs. However, it does not allow regulating the gain of each harmonic independently, requiring a stabilizing filter to cut off high-frequency behaviors. Moreover, in the equivalent transfer function obtained with the repetitive controller, the width and the phase of each resonance cannot be regulated autonomously. Furthermore, the repetitive control is not able to stabilize the system, demanding for an additional controller to accomplish that task [7]. In fact, differing from the traditional approaches, a repetitive controller is intended to augment an existing control system for removal of the periodic error signals.

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Moreover, when the control is engaged, it takes at least one fundamental period to provide output compensation, which means delay in the waveform reconstruction. In this paper, the aforementioned MRC is investigated with the purpose of providing sinusoidal shape to the output voltage of four-leg inverters for stand-alone generating units. In grid-tied applications, the possibility to force the injected harmonic to zero has been shown in several papers [9]–[15], whereas control strategies for voltage-source inverters (VSIs) have been presented in [16]–[19], where RCs, dq-axes architecture, and mixed linear-resonant algorithms are proposed for the regulation action. For inverters with rated power higher than tens of kilowatt, the moderately low switching frequency increases the difficulties in removing the switching component from the output voltage and current waveforms, due to the interaction between the output power filter and the required control bandwidth. In practical applications, the system can result in even undamped oscillations in the outputs when high-order RCs are implemented in the loop to compensate for the harmonics introduced by the nonlinear loads. This behavior is strongly related to the output power filter phase lagging, which heavily depends on the load conditions. However, bounded-input and bounded-output stability performance imposes a ceiling on the number of harmonic controllers that can be used in a multiresonant control system; as a result, significant harmonics content cannot be compensated, and the filter–load interaction deteriorates the overall control accuracy. Some solutions for RC phase control have been investigated in [20] for single-phase inverters, where the RC ideal transfer function (TF) is extended, including phase compensation, in order to allow using higher order RCs. However, the proposed control structure does not take into account the influence of the load variation in the tuning process and during the inverter operation. Moreover, in industrial applications where high-performance control platform is usually too expensive, the nonideal RC is preferred due to its more predictable behaviors (finite and selectable gain) from discretization and quantization point of view. Harmonic controllers tuning process using the Nyquist diagram have been proposed in [21]. In [22], different control structures have been compared for selective harmonic compensation as multiple rotating integrators and vector PI regulators. However, performances degradation and load adaptation control behavior were not taken into account. In this paper, the full nonideal form of the resonant compensator has been preferred due to its controllable gain and more suitable digital implementation [23], being the employed control board based on an industrial-grade digital signal processor (DSP) with single-precision floating-point unit. Section II depicts the detailed analytical and mathematical behavior of the full nonapproximated RC being at the basis of the proposed load-adaptive tuning strategy. In Section III, interaction between VSI output power filter and loads is described, highlighting the issues from the control algorithm point of view. Section IV shows the proposed magnitude and phase compensation strategy applied to a multiresonant control structure. Finally, Section V deals with system implementation and experimental activities to verify the proposed control algorithm.

II. F ULL RC W ITH P HASE C OMPENSATION C APABILITY Several topologies of RCs have been described in the literature [9]–[11]; each RC configuration has its own tracking, discretization, and tuning procedures; as a result, the preference among the RC structures would affect the entire control system behavior. The full form of the RC has been selected to be used in the investigated application. RC configuration as in (1) exhibits two degrees of freedom, allowing to regulate almost independently the controller gain kir and width ωcr . Moreover, the shown form is the simplest to be discretized. In fact, wellknown issues arise when ideal RCs are used, particularly when implemented in digital form on industrial-grade DSPs [23]. Equation (1) is as follows: Gf rc (s) =

2 ) 2kir (ωcr s + ωcr . 2 + ω2 ) s2 + 2ωcr s + (ωcr 0

(1)

As it results from (1), it is possible to control both the gain and the width of the RC, but phase is locked to zero at the resonance frequency. In order to introduce the possibility to select the controller phase at the resonant frequency, RC structure needs to be modified. In fact, direct phase control can be achieved by adding two terms to the RC nonideal form, both related to the desired angle ϑ at the resonant frequency. The angle ϑ is the phase, either lead or lag, that will be provided by the RC. Phase-adjustable RC (PA-RC) TF is depicted as Grf h (s) = 2kir ωcr

s cos(ϑ) + ωcr − ω0 sin(ϑ) . 2 + ω2 ) s2 + 2ωcr s + (ωcr 0

(2)

A. Controller Phase The controller phase can be evaluated considering separately the numerator and the denominator, as follows: NRC (s) = 2kir ωcr [s cos(ϑ) + ωcr − ω0 sin(ϑ)] 2 DRC (s) = s2 + 2ωcr s + ωcr + ω02 .

(3)

Complete expression for the RC phase is calculated according to ϕRC = ∠NRC (s) − ∠DRC (s).

(4)

Substituting s = jω, the final expressions for the RC angle are ω cos(ϑ) ∠NRC (jω) = atan (5) ωcr − ω0 sin(ϑ) 2ω ωcr . (6) ∠DRC (jω) = atan 2 + ω2 − ω2 ωcr 0 Equations (5) and (6) can be plotted to show the behavior of the phase as a function of frequency and angle ϑ. Fig. 1 is obtained with ωcr = 0.003 rad/s and ω0 ≈ 314 rad/s. The dashed line is related to the denominator’s phase, whereas the numerator’s phase is shown with solid lines and plotted for four values of the angle ϑ : −π/6, 0, π/6, and π/3.

LIDOZZI et al.: SINUSOIDAL VOLTAGE SHAPING OF INVERTER-EQUIPPED STAND-ALONE GENERATING UNITS

Fig. 1. Numerator and denominator phases for a real harmonic controller as a function of the angle ϑ.

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Fig. 3. Position of the controller zero on the complex plane for different values of the angle ϑ at the resonance frequency.

case the selected angle ϑ is negative, the corresponding RC acts as phase lagging controller, being the numerator phase smaller than π/2. Finally, as the angle ϑ is set at a value greater than zero, as shown in Fig. 3, the numerator of the RC TF exhibits a phase greater than π/2, which results in positive complete RC phase. This behavior depicts phase leading compensation at the resonance frequency. Looking at the pole–zero map of the PA-RC, it can be seen how the phase disposition only acts on the positioning of the RC TF zero, as shown in Fig. 3. In this figure, the colored circles are related to the position of the zero for a single RC for different values of ϑ : −π/6, 0, π/6, and π/3. Of course, it can be noted the modification of the zero moving from negative real part to positive real part when a phase boost has to be provided. Fig. 2. Bode plots for the nonideal RC at different values of the angle ϑ.

As it can be verified in Fig. 1, the numerator’s phase at the resonance frequency can be written as π ∠NRC (s)|s=jω0 = ϑ + . (7) 2 The denominator phase contribution can be summarized as follows: ⎧ ⎨ ∠DRC (jω)|ωω0 Thus, the phase (4) is calculated at the resonance frequency as π π ϕRC = ϑ + − = ϑ. (9) 2 2 ∠N

∠D

This proves the effectiveness of the phase control when it is extended to the nonideal RCs. From (9), it is clear how the phase value can be selected without changes to the RC width and gain. Fig. 2 shows the Bode plots of (2) at different values of the angle ϑ. Even if this method provides the ability to control the RC phase, it also presents some drawbacks: in Fig. 2, it is noticeable how the larger is ϑ, the higher is the asymmetry in the magnitude function of the RC. The zero of the controller TF provides phase compensation capabilities. In a conventional RC, phase ϑ is forced to be zero because the controller zero is placed directly on the imaginary axis. The resulting complete controller phase is zero, being the denominator phase equal to π/2 at the resonance frequency. In

B. Controller Magnitude Starting from (3), the magnitude function of the RC can be evaluated in a similar way as the one used to demonstrate the phase control capability. The magnitude at the resonance frequency is expressed as in (10). Usually, in RCs, the width parameter ωcr is selected in the range from hundreds of microradians per seconds to less than hundreds of milliradians per second. Thus

2 − 2ω ω sin(ϑ) + ω 2 ωcr cr 0 0 . (10) |Grf h (jω0 )| = 2kir 2 4ω02 + ωcr Being ωcr smaller than ω0 and lower than one, (10) can be simplified as

−2ωcr sin(ϑ) + ω0 |Grf h (jω0 )| = kir . (11) ω0 Expression (11) shows that magnitude is directly affected by the desired amount of phase to be compensated. Fig. 4 shows the magnitude (10) plotted versus the controller width ωcr for different values of the angle ϑ. In stand-alone applications, the front-end inverter imposes the output frequency; this allows using narrow-bandwidth RCs (i.e., controllers having a small ωcr value). On the contrary, in grid-connected systems, due to the fluctuations of the electrical frequency, it would be better to have wide-bandwidth RCs. Fig. 4 shows how it is possible to approximate the magnitude of the RC with the value of kir , which has been set equal to 200, being the error very small in the selected ωcr range and leading to |Grf h (jω0 )| ≈ kir .

(12)

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Fig. 4. Magnitude of the nonapproximated full RC at the resonance frequency for different values of ϑ; kir = 200, ω0 = 314 rad/s.

Fig. 5.

Fig. 6.

Filter Bode plots at different loads.

Scheme of the considered output power filter.

III. F ILTER –L OAD I NTERACTION A NALYSIS Inverter output filters are necessary to remove the switching components from the output voltage and current waveforms. In this paper, it is considered the single-phase equivalent filter structure, as shown in Fig. 5, where the conventional LC second-order filter is connected to two tuned RLC branches: the trap filter and the selective damper [24]. The selective damper is centered at the frequency of about 1.2 times the LC resonance frequency in order to damp the LC resonance peak, making the Rd resistor visible to the rest of the circuit only in a restricted range of frequency. The trap filter is instead tuned to resonate at the switching frequency (Rt is the sum of Lt and Ct equivalent series resistances), in order to short circuit the switching fundamental component. A. Unitary PF Loads With reference to stand-alone applications, filter magnitude and phase behaviors are strongly affected by the actual inverter output power. In fact, modeling the connected loads as an equivalent resistive load [i.e., power factor (PF) equal to one], as the output power increases, the filter equivalent TF becomes more damped, and its amplitude and phase responses are modified. Fig. 6 shows the Bode plots of the filter TF at four different output powers. It can be noticed how the LC resonance, already attenuated by the selective damper circuit, becomes further damped as the output power increases. When high-order harmonics need to be compensated, the variation in the filter gain has to be taken into account to properly design the controller. This is true also for the filter phase decrease, making necessary a strong phase compensation action at high frequencies.

Fig. 7. Filter magnitude as a function of the output power and harmonic order for purely resistive loads.

The variation of the complete filter magnitude as a function of the output power and harmonic order is depicted in Fig. 7. At the inverter rated power and considering the fifteenth harmonic, the corresponding filter gain is close to 0.4. This means that, in the gain of the open-loop TF, the effect of the fifteenth harmonic RC at rated output power is reduced by 60% with respect to no-load operation. Fig. 8 depicts the filter phase variation versus the output power and the harmonic order. It can be noticed that, also for the phase, the implementation of high-order harmonic compensator becomes difficult as the load approaches the inverter rated power. In fact, phase losses introduced by the filter directly influence the harmonics compensators and then the whole system behavior. However, when changes in filter magnitude and phase are known as a function of the equivalent load, they can be used in the control tuning process and taken into account in the online parameters adaptation. This allows continuously tuning the MRC in order to maintain a constant behavior in the entire range of the inverter output power. B. Inductive Loads In stand-alone applications, load PF can be lower than one, particularly when three-phase and single-phase induction machines need to be fed. In order to properly design and tune the


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Fig. 8. Filter phase as a function of the output power and harmonic order for purely resistive loads.

Fig. 10. Bode plots for resistive–capacitive load. (a) PF = 0.8. (b) PF = 0.1.

Fig. 9. Bode plots for resistive–inductive load. (a) PF = 0.8. (b) PF = 0.1.

controller, filter behavior with inductive loads is then analyzed. Fig. 9(a) shows the filter Bode plots when inductive–resistive loads having a PF of 0.8 are connected to the output. Power requested by loads is increased from 1 to 35 kVA with the same PF.

The figure shows how inductive–resistive loads weakly change filter–load equivalent TF, and then, they only marginally affect the system dynamic behavior, as pure resistive loads instead do. It can be noticed that the resonance is almost undamped and the characteristic frequency tends to move to higher frequency with load power increasing. However, during the start-up transients of induction motors or when they are operated at light loads, the total PF is very low, and motors appear as high reactive loads. Fig. 9(b) shows the filter behavior when an inductive load having PF = 0.1 is connected. Hence, at light load, there is no critical effect even if the current is lagging the regulated output voltage. The main issue is the low damping due to the interaction between the filter and the reactive load. However, a properly tuned selective damper is able to limit the resonance gain to acceptable values. C. Capacitive Loads When leading loads are considered, the equivalent effect is to increase both the filter damping and the filtering action, as depicted in Fig. 10(a), for a PF equal to 0.8 leading. In

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Fig. 11.


Block scheme of the selected MRC architecture.

fact, the reactive resistive–capacitive load acts as an additional filter due to the additional capacitance and as a damper due to the required load active power. This happens when the leading PF is close to one. On the other hand, when the PF is very small, the whole filter cutoff frequency is significantly lowered, as shown in Fig. 10(b) for PF = 0.1. This behavior can usually introduce oscillations in the inverter outputs of voltage and current controlled VSI, because of the strong interaction between filter and control algorithm. However, as described in the following sections, the proposed adaptive tuning control algorithm is theoretically able to sufficiently boost the phase at the required value compensating the TF phase lagging due to the capacitive load, thus giving the system the appropriate operating conditions. IV. P HASE AND M AGNITUDE C OMPENSATION A LGORITHM In order to overcome the illustrated drawback introduced by the interaction between the inverter output filter and the loads, the adaptive control structure has been selected for this particular application. Control algorithm is required to compensate for both filter magnitude and phase variations. Consequently, the tuning strategy requires knowing the inverter output operating conditions, which means output power and PF. Moreover, being the loads can be either single and/or three phases, the control has to regulate the output voltage of each phase independently, operating the three-phase VSI as three completely independent singlephase inverters. The proposed application has a four-leg topology. This configuration allows controlling the voltage of the neutral connection, providing a three-phase balanced voltage system even in case of nonlinear and unbalanced loads. The selected control architecture is based on the MRC, as depicted in Fig. 11. A single control loop strategy is chosen due to the fast compensation of voltage unbalances and load transients. As previously described, load–filter interaction causes the decreasing of RCs magnitude and phase, which should be compensated by the control algorithm. The phase variation causes the reduction of the system phase margin. In order to adapt the MRC structure to the load conditions, it is necessary to evaluate the inverter operating point in terms of active (Pout ) and reactive (Qout ) power supplied to the loads. Several algorithms have been proposed so far to calculate instantaneous P and Q power [25]–[28], but they all require a significant increase of the computational burden. Algorithms based on dq-axes power calculation should be avoided even

Fig. 12.

Multiple resonant compensator load-adaptive tuning scheme.

if they require low computational effort, being the four-leg inverter devoted to supply both single-phase and three-phase unbalanced loads. For these reasons, it was chosen to employ an off-the-shelf integrated circuit (IC), as ADE7953 from Analog Devices, to perform power-related calculations. The selected IC is able to acquire and elaborate up to the 24th harmonic of the output phase voltages and currents. Then, the evaluation of the output power takes into account the effects of distortive loads, also during transient load variations. With reference to Fig. 12, the Filter Analysis block receives the estimated active and reactive power supplied to the load, and it provides, for each harmonic compensator, the magnitude and the phase that need to be compensated. The MRC Parameter Evaluation block receives the outputs of the Filter Analysis block, which are the filter–load magnitude and phase evaluated at the resonance frequencies. According to that, the MRC Parameter Evaluation section is in charge of online calculation of the correction coefficients used to modify each RC magnitude and phase from the initial tuning, performed at no load. This requires that the filter TF has to be known as a function of the filter outputs as in the following: Gpwf L (jω) = F (Pout , Qout , ωn )

(13)

where ωn is the frequency of the n harmonic: first, third, and so on. Starting from the filter parameters, in order to reduce the required computational load, filter magnitude and phase can be evaluated online from a precomputed look-up table with a suitable fine mesh. Once filter magnitude and phase are known for each harmonic of interest, this information is forwarded to the MRC Parameter Evaluation block, which is devoted to calculate the required gain (kir ) and phase (ϑ) for each resonant compensator. Finally, the complete TF of the RC is discretized to be implemented on the digital control platform based on an industrial-grade DSP. Proposed adaptive tuning control algorithm and its implementation strategy allow including, in the curve fitting equations, the gain and delay modifications introduced by the inverter and the low-pass filter, usually inserted in the feedback chain. According to Fig. 13, RCTOT (s) represents the complete TF of the MRC, which is defined as Grf h1 (s) + · · · + Grf hn (s). VSI is modeled as a gain plus a delay equal to 1.5 times the switching period, and it is represented by


Fig. 13.

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Block scheme of the inverter control loop.

the TF G4-leg (s). A second-order low-pass Butterworth-type filter accomplishes the output voltage filtering for measuring purpose, i.e., Glpf (s). Finally, Gpwf (s) is the output power filter and load equivalent TF. When the inverter switching frequency is far from the highest ω0 employed in the RCTOT (s) block, the delay introduced by the inverter can be neglected. Concerning the signal conditioning chain, if it is properly designed, the resulting gain and phase lagging effects are very small with respect to the power filter gain and phase lagging effects. The effect of the phase compensation is to force to zero the total phase at the resonance frequency where the considered RC is centered. According to that, Fig. 14(a) shows the complete system open-loop Bode plots achieved at 30-kW output power with unitary PF in case of control adaptation (solid line) and conventional static tuning (dashed line). The zoom shown in Fig. 14(b) depicts in detail the Bode plot centered on the fifteenth harmonic controller. It can be noticed how the phase has been boosted to cross 0◦ at the resonance frequency instead of −77.8◦ , which is related to the conventional nonadaptive tuning. Moreover, Fig. 14(b) highlights the magnitude variation due to control–filter interaction. Without adaptation, the fifteenth RC gain is reduced by −6.85 dB (more than 30%), drastically changing the overall system capability to compensate for the load harmonics. A. Algorithm Implementation According to the previously depicted algorithm, the filter TF online evaluation requires a curve fitting analysis to be performed or, alternatively, a preevaluation of the filter magnitude and phase to be corrected as a function of the operating conditions. Depending on the physical hardware characteristics as available memory to store data and computational performance, one of the two methods can be preferred. According to the available hardware control platform for the investigated application, the algorithm has been implemented considering a curve fitting procedure. These values are evaluated for each RC being part of the control structure. Second-order polynomial fitting has been used for both magnitude and phase. A small approximation error is tolerated with the benefits of low computational effort requested to the DSP. With reference to the filter scheme shown in Fig. 5, values of the filter components are listed in Table I [25]. In the selected application, the fundamental RC together with both even and odd higher RCs compose the MRC. According to the proposed implementation, each RC has its own curve concerning the filter magnitude and phase versus estimated output power for different PFs. Of course, filter curves do not need to be evaluated during each control sampling interval. In fact, output power estimation takes more than one switching period to be achieved as

Fig. 14. (a) Open-loop Bode plots of the whole system at 30-kW output power with and without magnitude and phase adaptation. (b) Zoom centered on the fifteenth RC. TABLE I O UTPUT F ILTER PARAMETERS

it requires the knowledge of the output voltage and current RMS values. Hence, a proper undersampling scheme has been digitally implemented in the DSP working at an integer lower frequency Fus (250 Hz) with respect to the main task sample frequency Fsw (12 kHz). Once the filter characteristics are known, each RC can change its gain and phase, whereas the controller width is kept constant. Moreover, an online

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Fig. 15. System curve fitting up to fifteenth harmonic at different operating conditions.

implementation requires complicated arithmetical calculations, which cannot be performed in real time by an industrialgrade DSP. Hence, MRC variations with respect to filter–load TF modification are preevaluated offline within the complete operating conditions of the stand-alone power generating unit. Considering a general implementation, VSI and measuring filter have been included in the polynomial system approximation. Such a case can represent a conventional high-power lowswitching-frequency inverter without the output power transformer. Curve fitting analysis was performed through a properly designed MATLAB code, whose results are shown in Fig. 15, where the x-axis represents the estimated output power and the y-axis the magnitude and the phase required to achieve full compensation, respectively. Dots are the operating point used for curve fitting, whereas solid lines show the second-order approximation. Each line in Fig. 15 is related to one specific harmonic order showing for clearness only odd RCs from third to fifteenth; without loss of generality, the curve fitting can include odd and even higher order RCs. As described, larger compensation in terms of magnitude and phase is required for higher order RCs, being the effects introduced by the filter–load interaction more pronounced with respect to light load operation. The achieved polynomial approximation can be directly used into the algorithm starting from the estimated output power and then for each harmonic can be straightforwardly obtained the magnitude and the phase to be compensated online. As can be noticed in Fig. 15, compensation ranges of phase and magnitude strongly depend on the maximum considered harmonic order and equivalent output power. When the analysis is extended to higher harmonics, it is found that the practical range of compensation is within 135◦ for the 50th harmonic and full-load operating condition. RC discretization has been accomplished by Tustin approximation with frequency prewarp to not alter the resonance frequency in z-domain. Single-precision floating-point 32-bit arithmetic is used for evaluation with gain scaling to reduce computational errors.

Fig. 16. Phase A output phase-to-neutral and line-to-line voltage waveforms at no load with power estimation circuit and fitting functions completely enabled (200 V/div).

V. E XPERIMENTAL R ESULTS The prototype of the four-leg inverter has been built to perform the experimental validation of the proposed RC with phase compensation capability. The control unit is based on the TMS320F28335 digital signal controller from Texas Instruments mounted on a purpose-designed board. VSI switching frequency is set to 12 kHz. According to the previously described phase compensation load-adaptive control strategy, a proper MRC has been implemented following the shown guidelines, to compensate up to the 49th harmonic to comply with the IEC 61000-2-4 Class 1 standard. According to the MRC starting parameters, the no-load test is performed to evaluate the zero threshold of the output power estimation circuit. Fig. 16 depicts the inverter output phase-toneutral and line-to-line voltages related to phase A when the inverter is operated at no load. A very sinusoidal voltage waveform is achieved, and the resulting THD is close to 0.45% when it is measured up to the 99th harmonic order by the Yokogawa PZ4000 power meter. In order to validate the correctness of the proposed load-adaptive phase compensation tuning algorithm, some specific tests have been accomplished during transient operating conditions. Fig. 17 shows the transition when the adaptive phase compensation algorithm is switched OFF and ON. Results are achieved using the signal explorer in real-time recording mode, where it is able to save waveforms data as ASCII file during acquisition. After some tests, ASCII data have been postprocessed to find the exact transition, which is shown in Fig. 17. Results are obtained having phase A loaded with 13-kW single-phase linear load, whereas phase B and phase C are at no load. The amplitude of phase A voltage oscillations increases during time evolution; if the phase compensation algorithm is not timely turned on, the system overvoltage protection would be tripped, and the inverter control turns the modulator off. In the picture, occurrence of undesired oscillatory condition is not related to the phase compensation turn ON/OFF transition. In fact, the experimental test is achieved through the following steps:


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Fig. 17. Output voltage behavior after the activation of the phase compensation algorithm. Fig. 19. Voltage (200 V/div) and current (50 A/div) with a single-phase 4.5-kW diode rectifier load. Phase B and phase C are at no load. Loadadaptive control and phase compensation are enabled.

Fig. 18. Voltage (200 V/div) and current (20 A/div) with a three-phase 9-kW diode rectifier load. Load-adaptive control and phase compensation are enabled.

system start-up at no load and phase compensation algorithm ON; load is applied to the inverter phase A, and then, steadystate condition is reached; phase compensation algorithm is turned OFF; after some fundamental periods, evident oscillatory condition occurs; phase compensation algorithm is again turned ON, and the oscillations are consequently damped. It can be noticed that the oscillatory condition occurs only to the phase that is loaded, due to the described filter–load interaction. The proposed adaptive control strategy has been verified also when nonlinear loads are fed. In such a case, output power estimation and monitoring of the load condition are performed according to the RMS value related to the output voltage harmonics. The selected IC is able to take into account these effects up to the 24th harmonic for the evaluation of both active and reactive power. Fig. 18 shows the achieved experimental results with a 9-kW nonlinear three-phase load, which has been arranged through a three-phase diode rectifier with output inductive–capacitive filter and resistive load. As it can be noticed, even if the requested current presents a high harmonic content and a crest factor of about 3, autocompensated MRC is able to produce the output

Fig. 20. Phase A output phase-to-neutral and line-to-line voltage waveforms at cos(ϕ) = 0.80 (200 V/div, 10 A/div).

voltage waveform with very low distortion, proving performance capabilities of the load-adaptive proposed control algorithm. Output voltage THDV achievable with full adaptive compensation is around 1.7%, whereas when the tuning algorithm is disabled, high-order RCs must be removed from the MRC structure in order to avoid output voltage undesired oscillations; as a consequence, the resulting THDV increases up to 3.2%. Nonlinear unbalanced load test is performed using a singlephase diode rectifier load connected to phase A, whereas phase B and phase C are at no-load condition. Inverter output phase voltages and currents are shown in Fig. 19, where the achieved THDV is lower than 2.3% with control adaptation and around 3.8% with conventional static tuning. In fact, in the case without load-dependent control, RCs higher than eleventh have to be disabled in order to avoid undesired oscillations related to phase delay and digital implementation. Alternatively, high-order RCs gains must be reduced, making their contribution negligible. In order to validate the proposed adaptive control approach also in case of reactive power, the specific test in Fig. 20 has

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Fig. 21. Output phase voltages, adaptive algorithm versus standard tuning for a 16-kW load step from no-load condition. Three-phase diode rectifier load (200 V/div).

been accomplished. A three-phase RL load with cos ϕ = 0.8 is supplied by the inverter; line-to-line voltage, phase voltage, and current waveforms have the sought sinusoidal shape. Transient tests have been also performed to verify the correctness of the proposed load-adaptive tuning strategy. A proper switch to the output of the VSI filter connects the previously described three-phase diode rectifier. Concerning dynamic behavior, as the demanded output power increases, conventional static MRC takes several fundamental periods to compensate harmonics, exhibiting the typical second-order behavior in the inverter output voltage waveform. As shown in Fig. 21, in case of a power load step from no load to 16 kW, as soon as the three-phase diode rectifier load is applied, both adaptive and standard control strategies exhibit a quite similar behavior during the first half-fundamental period. However, within this time distance, the load-adaptive control is performing the gain scheduling and phase adaptation, which results in a different dynamic evolution of the output voltage waveforms with respect to the static MRC. After about one and a half of the fundamental period, main voltage harmonics are balanced, and the output THDV is strongly reduced. Steadystate condition is reached after almost ten fundamental periods, where the value of the THDV is around 1.9% with adaptive RC, whereas without the adaptation algorithm, it is around 5.75%, as can be observed in Fig. 22. In some European countries, the maximum allowed THDV is set to 3%, which would bring the system out of the standards when nonadaptive control is used. Both the proposed control strategy and the conventional multiresonant structure have been evaluated with respect to the IEC 61000-2-4 Class 1 Standard, which defines the maximum allowed harmonic content of the inverter output voltages up to the 50th harmonic. Analysis has been performed by the PZ4000 power meter from Yokogawa. Dashed line shown in Fig. 23 represents the IEC 61000 mask, whereas red left and blue right bars are related to the amplitude of each harmonic considering the classical and proposed control strategies, respectively, being the VSI loaded by the 16-kW three-phase diode rectifier. It can be

Fig. 22. Output phase voltages, adaptive algorithm versus standard tuning for a 16-kW steady-state condition three-phase diode rectifier load (200 V/div).

Fig. 23. Comparison of output voltage harmonic content with the proposed control strategy and using conventional nonadaptive MRC, with reference to the (dashed line) IEC 61000-2-4 Class 1 standard, 16-kW three-phase diode rectifier load.

noticed that, without load adaptation and phase compensation capabilities, the system fails to comply with the IEC 61000-2-4 Class 1 standard, being some harmonics above the imposed limits. VI. C ONCLUSION The load-adaptive MRC structure has been presented and tested in inverter-based stand-alone generating applications. Control architecture is able to change each controller gain and phase according to the load condition; regulation with high performance is assured, and the system is provided with very sinusoidal output voltages also for high distorting and unbalanced loads. Moreover, the proposed algorithm allows better usage of RCs from the point of view of system dynamic performance and output voltages harmonic content removal. In fact, RC gain scheduling and phase compensation capabilities jointly with load-adaptive properties significantly improve the dynamic behavior of the controlled system at any of the potential operating conditions. The investigated RC structure has been selected due


to its finite and tunable gain and reliable digital implementation, particularly when industrial-grade DSPs are used. The adaptive capabilities of the MRC are useful in transformerless high-power converter, having low switching frequency and thus having low output power filter cutoff frequency. Additionally, the achieved phase compensation allows increasing the RCs order, obtaining significant improvements in output voltage THD. The described filter–load-adaptive online tuning procedure can be applied to any inverter-based power generating system. In fact, the MRC parameters adaptation law can easily include the main filter TF, through the polynomial fitting method. In conclusion, RC online parameters adaptation allows obtaining lower output voltage THD jointly with a better response to load variation. Furthermore, possible implementation on grid-connected system is straightforward when grid impedance estimation methods are used. R EFERENCES [1] Q. C. Zhong and T. Hornik, Control of Power Inverters in Renewable Energy and Smart Grid Integration. Hoboken, NJ, USA: Wiley-IEEE Press, 2013. [2] J. Liang, T. Green, C. Feng, and G. Weiss, “Increasing voltage utilization in split-link, four-wire inverters,” IEEE Trans. Power Electron., vol. 24, no. 6, pp. 1562–1569, Jun. 2009. [3] R. Nasiri and A. Radan, “Pole-placement control strategy for 4-leg voltagesource inverters,” in Proc. PEDSTC, Feb. 17/18, 2010, pp. 74–79. [4] V. Yaramasu, M. Rivera, W. Bin, and J. Rodriguez, “Model predictive current control of two-level four-leg inverters—Part I: Concept, algorithm, simulation analysis,” IEEE Trans. Power Electron., vol. 28, no. 7, pp. 3459–3468, Jul. 2013. [5] M. Rivera, V. Yaramasu, J. Rodriguez, and W. Bin, “Model predictive current control of two-level four-leg inverters—Part II: Experimental implementation and validation,” IEEE Trans. Power Electron., vol. 28, no. 7, pp. 3469–3478, Jul. 2013. [6] V. Yaramasu, M. Rivera, M. Narimani, W. Bin, and J. Rodriguez, “Model predictive approach for a simple and effective load voltage control of fourleg inverter with an output LC filter,” IEEE Trans. Ind. Electron., vol. 61, no. 10, pp. 5259–5270, Oct. 2014. [7] K. Zhou and D. Wang, “Digital repetitive controlled three-phase PWM rectifier,” IEEE Trans. Power Electron., vol. 18, no. 1, pp. 309–316, Jan. 2003. [8] S. Chen, Y. M. Lai, S. C. Tan, and C. K. Tse, “Analysis and design of repetitive controller for harmonic elimination in PWM voltage source inverter systems,” IET Power Electron., vol. 1, no. 4, pp. 497–506, Dec. 2008. [9] R. Teodorescu, F. Blaabjerg, M. Liserre, and P. C. Loh, “Proportional– resonant controllers and filters for grid-connected voltage-source converters,” Proc. Inst. Elect. Eng.—Elect. Power Appl., vol. 153, no. 5, pp. 750– 762, Sep. 2006. [10] M. Liserre, R. Teodorescu, and F. Blaabjerg, “Multiple harmonics control for three-phase grid converter systems with the use of PI-RES current controller in a rotating frame,” IEEE Trans. Power Electron., vol. 21, no. 3, pp. 836–841, May 2006. [11] A. Timbus, M. Liserre, R. Teodorescu, P. Rodriguez, and F. Blaabjerg, “Evaluation of current controllers for distributed power generation systems,” IEEE Trans. Power Electron., vol. 24, no. 3, pp. 654–664, Mar. 2009. [12] G. Shen, X. Zhu, J. Zhang, and D. Xu, “A new feedback method for PR current control of LCL-filter-based grid-connected inverter,” IEEE Trans. Ind. Electron., vol. 57, no. 6, pp. 2033–2041, Jun. 2010. [13] S. Eren, M. Pahlevaninezhad, A. Bakhshai, and P. K. Jain, “Composite nonlinear feedback control and stability analysis of a grid-connected voltage source inverter with LCL filter,” IEEE Trans. Ind. Electron., vol. 60, no. 11, pp. 5059–5074, Nov. 2013. [14] W.-L. Chen and J.-S. Lin, “One-dimensional optimization for proportional–resonant controller design against the change in source impedance and solar irradiation in PV systems,” IEEE Trans. Ind. Electron., vol. 61, no. 4, pp. 1845–1854, Apr. 2014. [15] C. Xia, Z. Wang, T. Shi, and X. He, “An improved control strategy of triple line-voltage cascaded voltage source converter based on proportional–resonant controller,” IEEE Trans. Ind. Electron., vol. 60, no. 7, pp. 2894–2908, Jul. 2013.

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[16] T. Dragicevic, J. M. Guerrero, and J. C. Vasquez, “A distributed control strategy for coordination of an autonomous LVDC microgrid based on power-line signaling,” IEEE Trans. Ind. Electron., vol. 61, no. 7, pp. 3313–3326, Jul. 2014. [17] A. F. Zobaa and A. Lecci, “Particle swarm optimization of resonant controller parameters for power converters,” IET Power Electron., vol. 4, no. 2, pp. 235–241, Feb. 2011. [18] D. Dong, T. Thacker, R. Burgos, W. Fei, and D. Boroyevich, “On zero steady-state error voltage control of single-phase PWM inverters with different load types,” IEEE Trans. Power Electron., vol. 26, no. 11, pp. 3285– 3297, Nov. 2011. [19] I. Vechiu, O. Curea, and H. Camblong, “Transient operation of a four-leg inverter for autonomous applications with unbalanced load,” IEEE Trans. Power Electron., vol. 25, no. 2, pp. 399–407, Feb. 2010. [20] Y. Yang, K. Zhou, and M. Cheng, “Phase compensation resonant controller for PWM converters,” IEEE Trans. Ind. Informat., vol. 9, no. 2, pp. 957–964, May 2013. [21] A. G. Yepes, F. D. Freijedo, O. Lopez, and J. Doval-Gandoy, “Analysis and design of resonant current controllers for voltage-source converters by means of Nyquist diagrams and sensitivity function,” IEEE Trans. Ind. Electron., vol. 58, no. 11, pp. 5231–5250, Nov. 2011. [22] C. Lascu, L. Asiminoaei, I. Boldea, and F. Blaabjerg, “Frequency response analysis of current controllers for selective harmonic compensation in active power filters,” IEEE Trans. Ind. Electron., vol. 56, no. 2, pp. 337– 347, Feb. 2009. [23] A. G. Yepes, F. D. Freijedo, O. Lopez, and J. Doval-Gandoy, “Highperformance digital resonant controllers implemented with two integrators,” IEEE Trans. Power Electron., vol. 26, no. 2, pp. 563–576, Feb. 2011. [24] A. Girgis, “Reactive power calculations using quadratic phase coupling estimation,” in Conf. Rec. 8th IEEE IMTC, May 1991, pp. 598–601. [25] G. Lo. Calzo, A. Lidozzi, L. Solero, and F. Crescimbini, “LC filter design for on-grid and off-grid distributed generating units,” IEEE Trans. Ind. Appl., vol. 51, no. 2, pp. 1639–1650, Mar./Apr. 2015. [26] J. L. J. Driesen and R. J. M. Belmans, “Wavelet-based power quantification approaches,” IEEE Trans. Instrum. Meas., vol. 52, no. 4, pp. 1232– 1238, Aug. 2003. [27] K. Hyosung and H. Akagi, “The instantaneous power theory based on mapping matrices in three-phase four-wire systems,” in Proc. Power Convers. Conf.—Nagaoka, Aug. 1997, vol. 1, pp. 361–366. [28] H. Akagi and E. H. Watanabe, Instantaneous Power Theory and Applications to Power Conditioning. Hoboken, NJ, USA: Wiley-IEEE Press, 2007. Alessandro Lidozzi (S’06–M’08) received the Laurea degree in electronic engineering and the Ph.D. degree from Roma Tre University, Rome, Italy, in 2003 and 2007, respectively. During 2005–2006, he was a Visiting Scholar with the Center for Power Electronics Systems, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA. Since 2010, he has been a Researcher with the Department of Engineering, Roma Tre University. His research interests are mainly focused on multiconverterbased applications, dc–dc power converter modeling and control, control of permanent-magnet motor drives, and control aspects for power electronics in diesel–electric generating units. Dr. Lidozzi was a recipient of a Student Award and a Travel Grant at the International Symposium on Industrial Electronics in 2004. Luca Solero (M’98) received the Laurea degree in electrical engineering from the University of Rome “La Sapienza,” Rome, Italy, in 1994. Since 1996, he has been with the Department of Engineering, Roma Tre University, Rome, where he is currently an Associate Professor in charge of teaching courses in the fields of power electronics and industrial electric applications. He has authored or coauthored over 130 published technical papers. His current research interests include power electronic applications to electric and hybrid vehicles and to distributed power and renewable energy generation units. Prof. Solero is a member of the IEEE Industrial Electronics, IEEE Industry Applications, and IEEE Power Electronics Societies. Since 2013, he has been serving as a European Liaison Officer for the IEEE Industry Applications Society Industrial Power Converter Committee.

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Stefano Bifaretti (M’07) received the Laurea degree and the Ph.D. degree in electronic engineering from the University of Rome Tor Vergata, Rome, Italy, in 1999 and 2003, respectively. In 2004, he became an Assistant Professor with the University of Rome Tor Vergata, where he is currently a Lecturer in power electronics. In 2007, he was with the Power Electronics, Machines and Control Group, University of Nottingham, Nottingham, U.K., collaborating on the UNIFLEX-PM European project. He has authored or coauthored over 80 papers published in international journals and conference proceedings. His research interests include power electronics converters, industrial drives, and future electricity networks. Dr. Bifaretti is a member of the IEEE Industry Applications, IEEE Industrial Electronics, and IEEE Power Electronics Societies.

Fabio Crescimbini (M’90) received the Laurea degree in electrical engineering and the Ph.D. degree from the University of Rome “La Sapienza,” Rome, Italy, in 1982 and 1987, respectively. From 1989 to 1998, he was the Director of the Electrical Machines and Drives Laboratory with the Department of Electrical Engineering, University of Rome “La Sapienza.” In 1998, he joined the newly established Roma Tre University, Rome, where he is currently a Full Professor of power electronics and electrical machines and drives in the Department of Engineering. His research interests include newly conceived permanent-magnet machines and novel topologies of power electronic converters for emerging applications such as electric and hybrid vehicles and electric energy systems for distributed generation and storage. Prof. Crescimbini served as a member of the Executive Board of the IEEE Industry Applications Society (IAS) from 2001 to 2004. In 2000, he served as a Cochairman for the IEEE-IAS World Conference on Industrial Applications of Electric Energy, and in 2010, he served as a Cochairman for the 2010 International Conference on Electrical Machines. He was a recipient of awards from the IEEE IAS Electric Machines Committee, including the Third Prize Paper in 2000 and the First Prize Paper in 2004.