An Adaptive Control for UPS to Compensate ... - Semantic Scholar

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 2, APRIL 2007

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An Adaptive Control for UPS to Compensate Unbalance and Harmonic Distortion Using a Combined Capacitor/Load Current Sensing G. Escobar, Member, IEEE, P. Mattavelli, Member, IEEE, A. M. Stanković, Fellow, IEEE, Andrés A. Valdez, Student Member, IEEE, and Jesus Leyva-Ramos, Member, IEEE

Abstract—This paper investigates the control of an uninterruptible power supply (UPS) using a combined measurement of capacitor and load currents in the same current sensor arrangement. The purpose of this combined measurement is, on one hand, to reach a similar performance as that obtained in the inductor current controller with load current feedforward and, on the other hand, to easily obtain an estimate of the inductor current for overcurrent protection capability. Based on this combined current measurement, a voltage controller based on resonant harmonic filters is investigated in order to compensate for unbalance and harmonic distortion on the load. Adaptation is included to cope with uncertainties in the system parameters. It is shown that after transformations the proposed controller gets a simple and practical form that includes a bank of resonant filters, which is in agreement with the internal model principle and corresponds to similar approaches proposed recently. The controller is based on a frequency-domain description of the periodic disturbances, which include both symmetric components, namely, the negative and positive sequence. Experimental results on the output stage of a three-phase three-wire UPS are presented to assess the performance of the proposed algorithm. Index Terms—Adaptive control, nonlinear systems, uninterruptible power supply (UPS) systems.

I. I NTRODUCTION

T

HE PROBLEM of designing an appropriate uninterruptible power supply (UPS) control strategy that fulfills requirements such as voltage regulation, total harmonic distortion, output impedance, transient response, operation with nonlinear/distorted unbalanced loads, and robustness to parametric uncertainties is challenging. The growing importance of UPS systems has motivated a flourishing development of different Manuscript received May 6, 2005; revised September 29, 2005. Abstract published on the Internet January 14, 2007. This work was supported in part by the National Council of Science and Technology of Mexico (CONACYT) under Grant SEP-2003-C02-42643 and in part by the Bilateral Laboratory France–Mexico of Applied Control (LAFMAA) under Grant MOPOFA-2004. G. Escobar, A. A. Valdez, and J. Leyva-Ramos are with the Division of Applied Mathematics, Research Institute of Science and Technology of San Luis Potosí (IPICYT), San Luis Potosí 78216, Mexico (e-mail: gescobar@ ipicyt.edu.mx; [email protected]; [email protected]). P. Mattavelli is with the Department of Electrical, Mechanical and Management Engineering (DIEGM), University of Udine, 33100 Udine, Italy (e-mail: [email protected]). A. M. Stanković is with the Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115 USA (e-mail: astankov@ ece.neu.edu). 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.2007.891998

control schemes found in the literature [1]–[10]. Some controllers rely on single voltage loop using proportional–integral (PI), dead-beat [2], or sliding-mode controllers as compensators (see [3] and [4] for a brief survey on conventional control techniques for UPS). Other solutions proposed in the literature include a nested connection of output voltage and inductor current control loops, usually two PIs or possibly a PI plus a highgain controller like a sliding-mode controller [5]. Although these techniques are able to ensure a good transient response, the distortion on the output voltage due to nonlinear loads is typically not compensated completely. In general terms, we can say that the implementation of the proposed controllers usually involves measurements of the output voltage and either the capacitor current or the inductor current, and in some cases, they even require measurements of the current load with the idea of attenuating the effect of disturbances in the load. The capacitor voltage is introduced in those controllers in a second voltage loop to alleviate imperfections in the response due to parameter uncertainties and load disturbances as well. Therefore, based upon the current signals used in the feedback control, we can distinguish two types of controllers, namely: 1) those based on the inductor current and 2) those based in the capacitor current sensing. Excellent results using the latter approach have been reported in [6]–[8]. These works show that the performance of a UPS can be considerably improved if the capacitor current is effectively controlled. This is clear from the fact that while the output voltage is typically the controlled output, its time derivative is proportional to the capacitor current. It has been shown also that a capacitor current feedback topology will exhibit better dynamic stiffness (inverse of the output impedance), which is a key metric in UPS’s performance, than a controller based on inductor current feedback. Moreover, since the capacitor current is small and alternate in nature, it may be sensed with a small and inexpensive current transformer, and thus it can be considered a low-cost alternative that has a potential to exhibit outstanding performance. However, as neither the load current nor the inductance current is measured, this controller is unable to detect any anomaly arising on the load side. For instance, if a short circuit appears, the capacitor current is maintained at the desired reference theoretically, while the inductance current will grow unlimited, with the unavoidable destructive effects. A solution for this protection issue consists in adding low-cost sensors on the load side, with the unavoidable intrinsic circuitry.

0278-0046/$25.00 © 2007 IEEE

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Fig. 1. Current sensor array to obtain the combined current.

In this paper, an alternative solution is proposed for the control of a UPS system to overcome such a disadvantage while preserving a good performance and guarantee balanced sinusoidal output voltages despite of the presence of nonlinear and unbalanced loads. This solution involves the use of a combined measurement of both load and capacitor currents, which is performed by a single current sensor arrange for each phase, as seen in Fig. 1, thus preserving the number of current sensors as in other approaches. This measurement provides a linear combination, i.e., a weighted sum, of both currents, which is then used as the feedback variable in the control loop. Notice that, in the particular case where both weights are unitary, this combined current is simply the inductor current. On the other hand, assigning a zero weight to the load current leads to the capacitor current case. So it is expected that a good performance, which is close to the capacitor current-based controller, can be obtained if a smaller weight is assigned to the load current compared to the one for the capacitor current. In addition, an estimate for the inductor current can now be easily reconstructed using the information contained in the combined current measurement, which can now be used for overcurrent protection. Adaptive refinements have also been added to the controller to cope with parametric uncertainties. By using the frequencydomain descriptions of some unknown signals (disturbance), the solution presented here is able to perform precise voltage tracking (even with nonlinear loads). It reduces the effects of unbalance and harmonic distortion, which is similar to other frequency-domain techniques, such as synchronous frame harmonic control [9], [10]. For such purpose, the system dynamics is modeled using stationary frame quantities and the load currents (disturbance) with slowly varying phasors. Both sequence components, i.e., positive and negative, are considered so that the unbalanced case can be treated. The proposed controller realizes a partial inversion of the system and adds the needed damping. The resulting system contains a disturbance term due to uncertainties in the system parameters, which is addressed via adaptation. Due to the complexity of this controller, a simple rotational transformation is proposed so that the computation complexity can be significantly reduced. Similar to other frequency-domain techniques, a group of selected harmonics is taken into account for parameter adaptation and voltage regulation, and thus the proposed approach can be classified as selective, since only a selected set of harmonics is targeted for compensation. The resulting scheme is directly connected to the previous work [13], [14] where the inductor current is used instead. A key observation here is that the resulting

Fig. 2.

Three-phase three-wire UPS inverter system.

compensator has a very similar structure as those controllers presented in [15] and [16], which include a bank of resonant filters as the main harmonic compensation element and were derived following other approaches. The solution proposed here is based on a new more rigorous theoretical framework following nonlinear control design techniques and based on the frequency-domain representation of the disturbances. Finally, the proposed control scheme has been implemented and tested in a 1.5-kVA three-phase inverter. The experimental results are presented here. II. S YSTEM C ONFIGURATION AND P ROBLEM F ORMULATION The basic setup for the three-phase three-wire inverter discussed in this paper is shown in Fig. 2. The system dynamics in fixed frame coordinates are described by di0 E diC = −vC + u − L − r(iC + i0 ) dt 2 dt dvC = iL − i0 = iC C dt im = αiC + βi0 L

(1) (2) (3)

where L inductance; C capacitance; E voltage source; r parasitic inductor resistance; α, β weights for iC and i0 , respectively; capacitor voltages ∈ IR2 ; vC u control vector ∈ IR2 ; inductor currents ∈ IR2 ; iL load currents ∈ IR2 ; i0 capacitor currents ∈ IR2 ; iC combined currents ∈ IR2 . im Parameters L, C, r, and E are all assumed unknown constants. The continuous signal u represents the actual control input and is used to generate the switching sequence to control the

ESCOBAR et al.: ADAPTIVE CONTROL FOR UPS TO COMPENSATE UNBALANCE AND HARMONIC DISTORTION

switching devices using a sinusoidal pulsewidth-modulation (PWM) technique at a relatively high frequency. Parameters α and β are the known weights to form the combined current im = αiC + βi0 . Current i0 is an unbalanced periodic signal which can be expressed as the combination of a fundamental component (at a fixed frequency w0 ) and its harmonics of higher order, that is, i0 can be represented as p n + e−J kw0 t I0,k i0 = eJ kw0 t I0,k k∈H J w0 t

e

=

cos(w0 t) − sin(w0 t) , sin(w0 t) cos(w0 t)

0 −1 J = 1 0

where w0 represents the fundamental frequency, and vectors p n , I0,k ∈ IR2 are the kth harmonic coefficients for the positive I0,k and negative sequence representation, they are also assumed unknown constants (or slowly varying); H = {1, 3, 5, 7, 11, . . .} is the set of multiples of the harmonic components considered. Its time derivative, used later in the control derivation, is given by di0 p n . = J kw0 eJ kw0 t I0,k − e−J kw0 t I0,k dt

The controller design is based on the following expression for the error model: E d˜im ∗ =α u − v˜C − vC − r˜im − rim L dt 2 Ldi0 Ldim − (α − β) ri0 + − dt dt αC

(4)

(5)

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d˜ vC ˜ = im dt

(8)

∆ ∆ ∗ where the increments are defined as v˜C = vC − vC and ˜im = ∗ im − im , with vC and im as defined before. In the known parameters case, and following the ideas of the energy shaping plus damping injection design technique [11], the following controller that guarantees perfect tracking, i.e., v˜C → 0 and ˜im → 0, is proposed:

αE ∗ u = −R1˜im − R2 v˜C + αvC + rim 2 di0 dim + (α − β) ri0 + L +L dt dt

k∈H

The control objective is to track a balanced voltage reference, i.e., ∗ vC = eJ w0 t [Vd , 0]

which is a purely balanced sinusoidal vector signal, i.e., it contains only fundamental component, in spite of the presence of harmonic disturbances. Notice that its time derivative is ∗ ∗ = J w0 vC . Here, and in what follows, (·)∗ will be simply v˙ C used to denote references and (·) will be used for values in the equilibrium. Thus, the control objective implicitly includes two problems, namely: 1) reference tracking in the fundamental harmonic and 2) disturbance attenuation of the output voltage response to higher harmonics mainly introduced by the load current. The equilibrium point of the overall system by forcing ∗ = eJ w0 t [Vd , 0] is given by vC ∗ im = αJ w0 CvC + βi0

∗ v C = vC .

III. P ROPOSED C ONTROLLER Let us write the system dynamics (1) and (2) in terms of the combined current im , this yields E dim di0 =α u − vC − rim − (α − β) ri0 + L L dt 2 dt dvC = im − βi0 . dt

αE ∗ u = −R1 (im − i∗m ) − R2 v˜C + αvC + rî∗m 2 ∗ î0 d ˆ dim ˆ +L + (α − β) rî0 + L dt dt

is used to represent the estimated value of (·), and where (·) ∆ ˆ ∗ + βî0 . Notice that, in im has been replaced by i∗m = αJ wCv C this controller, all unknown terms have been replaced by their estimates. The main idea behind the controller design consists in lumping all periodic uncertainties in a single term φˆ as follows:

(6)

Note that, to guarantee perfect voltage tracking, it suffices to force the combined current im to follow a reference signal im , as given by (6), which, unfortunately, depends on the unavailable signal i0 and unknown parameter C.

αC

where R1 and R2 are two design parameters to add the required damping. Notice that most terms of the above controller are intended to cancel the corresponding terms in the error model. Now, based on the structure of the above controller, the following controller is proposed:

(7)

αE ∗ u = −R1 im − R2 v˜C + αvC + φˆ 2

(9)

∆ ˆ î0 /dt)+ rî∗ + Ldi ˆ ∗ /dt. where φˆ = R1 i∗m +(α − β)(ˆ ri0 + Ld m m The error dynamics, after applying controller (9) to the error model (8), reduces to

d˜im = −(α + R2 )˜ vC − (r + R1 )˜im + φ˜ dt d˜ vC ˜ = im αC dt L

(10) (11)

∆ ˆ ∆ where φ˜ = (φ−φ) and φ = R1 im +(α−β)(ri0 + Ldi0 /dt) + rim + Ldim /dt. Notice that, this is a linear time-invariant ˜ system perturbed by a periodic disturbance φ.

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According to (4), the periodic signals φ and φˆ can be described as follows: p ∆ φ= φk + φnk = eJ kw0 t Φpk + e−J kw0 t Φnk k∈H ∆ φˆ =

k∈H

φˆpk + φˆnk =

k∈H

ˆ p + e−J kw0 t Φ ˆn eJ kw0 t Φ k k

k∈H

where vectors Φpk , Φnk ∈ IR2 are the kth harmonic coefficients for the positive and negative sequence representation of the ˆ p, Φ ˆ n are their corresponding estimates. disturbance φ, and Φ k k Out of which the error signal φ˜ = (φˆ − φ) are expressed as p ∆ ˜ p + e−J kw0 t Φ ˜ nk eJ kw0 t Φ φ˜ = φ˜k + φñk = k k∈H

k∈H ∆

ˆ p − Φp ) and ˜ p = (Φ where the parameter errors are defined as Φ k k k ∆ ˆn n ñ = Φ ( Φ − Φ ). k k k The adaptive laws are obtained by following a Lyapunov approach where the energy storage function is W =

(α + R2 )|˜ vC |2 αLC|v˜˙ C |2 + 2 2 1 p 2 2 ˜ ñ Φ + Φ + k k 2γk

Block diagram of the proposed controller.

together with the disturbance rejection objective since it is proven that φ˜ → 0, i.e., the harmonic compensation issue is solved. To facilitate the implementation of the adaptive laws, the following transformations are proposed: ˆp φˆpk = eJ kw0 t Φ k

(12)

k∈H

where γk , k ∈ H are positive design constants, and | · | represents the module of a vector, thus |X|2 = X X. Its time derivative along the trajectories of (10) gives ˙ = −α(r + R1 )C v˜˙ 2 W C ˜ p + e−J kw0 t Φ ˜ nk + v˜˙ C eJ kw0 t Φ k k∈H

1 ˙ p p ˙ n ˜k Φ ˜ + Φ ˜ nk ˜k Φ Φ + k γk k∈H

which is made negative semidefinite by proposing the following adaptation laws: p ˆ˙ k = −γk e−J kw0 t v˜˙ C Φ

Fig. 3.

n ˆ˙ k = −γk eJ kw0 t v˜˙ C Φ

p n p n ˆ˙ k = Φ ˜˙ k and Φ ˆ˙ k = Φ ˜˙ k are appealed. Since where the facts Φ ˙ = −α(r + R1 )C v˜˙ 2 , as a first conclusion we have that W C v˜˙ C → 0 and is bounded. Then invoking standard LaSalle’s theorem arguments [12] assuming v˜˙ C ≡ 0, we obtain an in˜ in addition Φ ˆ p, Φ ˆn vC = φ, variant set described by (α + R2 )˜ k k are constant ∀k ∈ H, which implies in its turn that φ˜ is a timevarying bounded signal. However, v˜C is a constant and bounded signal; therefore, the only possible solution is v˜C → 0 which ˜ n → 0. ˜ p → 0 and Φ in its turn implies φ˜ → 0, and moreover Φ k k p n p n ˜˙ k = Φ ˆ˙ k and Φ ˜˙ k = Φ ˆ˙ k . Notice that we have used the fact that Φ Remark 1: The proposed controller thus guarantees stable perfect tracking of the output voltage toward its sinusoidal reference since v˜C → 0 as t → ∞, this objective is fulfilled

ˆ n. φˆnk = e−J kw0 t Φ k

(13)

The adaptation laws can thus be written as ˙p φˆk = −γk v˜˙ C + J kw0 φˆpk

(14)

n

˙ φˆk = −γk v˜˙ C − J kw0 φˆnk .

(15)

The transfer function expressions of the above adaptation laws are −γk s(s + J kw0 ) v˜C φˆpk = s2 + k 2 w02 −γk s(s − J kw0 ) v˜C φˆnk = s2 + k 2 w02 where s is the complex variable out of which the adaptations are reduced to −2γk s2 v˜C φˆk = φˆpk + φˆnk = 2 s + k 2 w02 = −2γk v˜C +

2γk k 2 w02 v˜C . s2 + k 2 w02

(16)

The final expression for the controller is αE u = −R1 im − R2 + 2 γk v˜C 2 k∈H

∗ + αvC +

2γk k 2 w2 0 v˜C . s2 + k 2 w02

(17)

k∈H

Fig. 3 presents the block diagram of the proposed controller (17). Notice that this structure is composed of two proportional terms acting over the voltage error v˜C and the combined current


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im as expected, plus a bank of resonant filters of the form 2γk k 2 w02 /(s2 + k 2 w02 ), ∀k ∈ H acting over the voltage error v˜C dedicated to the harmonic compensation. Remark 2: Notice that this form of the resonant filters corresponds to the “cosine wave transfer function” and not to the usual “sine wave transfer function,” as reported in previous works [15], [16]. This is due to the fact that the time derivative of the capacitor voltage was not available, nor the capacitor current, instead the capacitor voltage has been considered as the input signal to the resonant filters, which introduces a phase shift of 90◦ . Remark 3: Notice that the same result is obtained if the inductor parasitic resistance is neglected. This is due to the fact that the resonant filters introduce an infinite gain at the harmonics under compensation, thus zeroing the steady state error caused by the parasitic resistance at those harmonic frequencies.

IV. E STIMATION OF iL The estimation of iL is carried out indirectly by estimating the capacitor current iC first. Then, based on the knowledge of the estimated îC , the combined current im and the weights used, namely, α and β, it is possible to solve for iL from (2) and (3). Since the capacitor current iC is defined as C times the time derivative of the capacitor voltage vC , an estimate for iC can be obtained by using a limited bandwidth time derivative as follows: îC =

Cs vC τs + 1

(18)

where 1/τ represents the bandwidth of the filter, which is selected big enough, to guarantee a good tracking of iC . The estimate îL is solved from (2) and (3), yielding ˆ ˆ îL = im + (β − α)iC β

(19)

where îm is a filtered version of im using the following lowpass filter: îm =

1 im . τs + 1

(20)

Since the higher order harmonics in iC have been truncated in îC due to the limited bandwidth of the estimator, then the higher order harmonics of im should be also eliminated before computation of îL in (19) to avoid unnecessary distortion. The estimate of iL can now be used in a surveillance block to guarantee a safer operation of the inverter system. It is clear that a much smaller β compared to α would produce a faster response, as im is dominated mainly by iC ; however, notice from (19) that extremely low values of β make the above estimator very sensitive to variations on the parameter C. Thus, values for parameters α and β should

Fig. 4. Load current (only one phase). (Top) Current signal i01 in time domain (x-axis: 200 ms/div and y-axis: 5 A/div) and (bottom) its corresponding frequency spectrum (x-axis: 62.5 Hz/div and y-axis: 20 dB/div).

be carefully selected to establish a tradeoff between speed response and sensitivity. V. E XPERIMENTAL R ESULTS A three-phase three-wire prototype has been built using the following parameters: L = 1 mH, C = 25 µF, E = 320 V, switching frequency fsw = 10 kHz, and output voltage amplitude 110 V with a fundamental frequency w0 = 377 r/s (f0 = 60 Hz). Notice that the LC filter has been tuned at approximately 1 kHz, i.e., to filter the switching effects. Therefore, the compensation is limited to harmonic components of the fundamental lower than 1 kHz. The controller is implemented in the dSPACE card model ACE1103 with a sampling rate fixed to 14.28 kHz. The sampling instant has been synchronized with the PWM so that the current im is sampled at the middle of switch-on time. Thus, the average value of im is obtained without low-pass antialiasing filters in the loop. The sensors used to build im are the closed-loop hall-effect CLN-50 current sensors from LEM. In each CLN-50, both conductors, for iC and i0 , have been wired as to obtain α = 10 and β = 1. A voltage source composed by a three-phase diode rectifier with a dc capacitor of 235 µF feeding a resistor of 100 Ω is connected to the inverter output as a nonlinear load. A resistor of 150 Ω is connected in between two phases to produce unbalance. Fig. 4 shows (top plot) the time response for one of the load currents and (bottom plot) its corresponding frequency spectrum. Notice that the load current is composed mainly by odd harmonics of the fundamental f0 , namely the first, third, fifth, and seventh components. These harmonics are precisely the components considered for compensation, that is, the bank of resonant filters includes filters tuned at the first, third, fifth, and seventh harmonics of the fundamental f0 . The proposed controller (17) has been implemented. However, to guarantee a safer operation, bandpass filters (BPFs) have been used instead of resonant filters. The latter have infinite gains at the resonant frequency, whereas the BPFs

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Fig. 5. Bode plot of the error dynamics −φ → v˜C using the proposed controller (black) with harmonic compensation and (gray) without the harmonic compensation. (Top) Magnitude (x-axis in hertz and y-axis in decibels) and (bottom) phase shift (x-axis in hertz and y-axis in degrees).

have limited gains due to a damping term introduced in the denominator of the transfer function written as follows: vo Ak k 2 w02 /Qk = 2 vi s + skw0 /Qk + k 2 w02

∀k ∈ H

Fig. 6. Output voltage steady-state response (only one phase). (From top to ∗ (t), the actual output voltage v (t), and the bottom) The reference voltage vC C error v˜C (t) (x-axis: 4 ms/div and y-axis: 100 V/div).

(21)

where Ak and Qk are positive design parameters representing the desired gain and the quality factor of the kth BPF, respectively; vi and vo are the input and output signals of the filter. Notice that, in the case of an ideal resonant filter, Ak → ∞. Further, notice that the original gain γk in the resonant filters has been replaced by Ak /Qk . The controller design parameters were selected as follows: R1 = 0.5, R2 = 0.5, A1 = 40, Q1 = 35, A3 = 10, Q3 = 20, A5 = 10, Q5 = 12, A7 = 10, and Q7 = 12. Fig. 5 shows the Bode plot of the closed-loop error dynamics established from the periodic disturbance −φ to the output voltage error v˜C , i.e., formed by (10), (11) and (16), and for the above parameters. Notice that the controller introduces notches, with a given phase shift, centered at the selected harmonics, that is, at the odd harmonics, namely, first, third, fifth, and seventh. It is important to remark here that the closed-loop system remains stable. All this means that, if the disturbance contains components at those selected harmonics, the response to those components is practically eliminated on the output error, and the remaining part, if any, will have the phase shift shown in the phase plot of the Bode plot. For comparison, the response of the proposed controller without the bank of resonant filters (or BPFs), i.e., without the harmonic compensation has been included as well (in gray). Fig. 6 shows, for one phase, (from top to bottom) the time ∗ , the actual output responses of the output voltage reference vC voltage vC (t), and the tracking error v˜C (t), using the proposed controller based on im and with harmonic compensation. Notice that the actual voltage vC (t) (middle plot) is almost a sinusoidal signal and has an excellent tracking over its reference ∗ (top plot), and thus the error v˜C (t) is made relatively small. vC The responses of the other two phases are very similar and are omitted here for the sake of space limitation.

Fig. 7. Output voltage steady-state response vC (t) in black and correspond∗ (t) in gray (only one phase). (Top) Conventional controller ing reference vC based on iC measurements and (bottom) proposed controller with harmonic compensation (x-axis: 4 ms/div and y-axis: 100 V/div).

Fig. 7 shows a comparison between the responses of the actual output voltage using the proposed controller (bottom plot), and a conventional controller using the measurement of iC and without harmonic compensation (top plot). Notice that the performance of the proposed controller significantly exceeds to the conventional one, even though for the latter, the gains have been adjusted to reach the best possible response. In the same plots, the corresponding voltage references (in gray) have been included to put in evidence the better tracking reached with the proposed controller. Fig. 8 shows the frequency spectrum of (top plot) the conventional controller and (bottom plot) the proposed controller. Notice that, in the proposed controller,


Fig. 8. Frequency spectrum of the output voltage vC (t). (Top) Conventional controller based on iC measurements and (bottom) proposed controller with harmonic compensation (x-axis: 62.5 Hz/div and y-axis: 20 dB/div).

Fig. 9. (Top plot) Steady-state response of the three output voltages vC1 , vC2 , and vC3 (x-axis: 4 ms/div and y-axis: 100 V/div) and (three bottom plots) distorted and unbalance load currents i01 , i02 , and i03 (x-axis: 4 ms/div and y-axis: 10 A/div).

the third, fifth, and seventh have been eliminated almost completely due to the harmonic compensator, leading to an output voltage vC mainly composed by a fundamental component despite of the highly distorted load current. Fig. 9 shows, for the proposed controller, the steady-state response of the output voltages for the three phases vC1 , vC2 , and vC3 (top plot), which are balanced and almost sinusoidal, despite of the distorted and unbalance load current. The last three plots represent the three-phase load currents i01 , i02 , and i03 . Fig. 10 shows the transient response of the output voltage vC (upper plot) when the nonlinear unbalanced load is connected

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Fig. 10. Transient responses during the connection of the nonlinear and unbalanced load. (Top) The output voltage vC (x-axis: 20 ms/div and 100 V/div) and (bottom) load current i0 (x-axis: 20 ms/div and y-axis: 5 A/div).

Fig. 11. Transient responses during the disconnection of the nonlinear and unbalanced load current. (Top) The output voltage vC (x-axis: 20 ms/div and 100 V/div) and (bottom) load current i0 (x-axis: 20 ms/div and y-axis: 5 A/div).

to the inverter under the proposed controller. The corresponding current load i0 is shown in the bottom plot (only one phase is shown for the sake of space limitations). Notice that, after a relatively small transient, the voltage keeps the desired sinusoidal shape and amplitude. Fig. 11 shows the corresponding transient response after disconnecting the load. Fig. 12 shows a detail of the current iL and its estimate îL in the steady state (only one phase) using the estimator (19). Notice that the estimate îL is very close to the actual current iL , and, thus, it can be especially useful for overcurrent protection purposes. To test the usefulness of the estimate îL , a load is connected to the inverter demanding a current peak exceeding the upper limit fixed at 8 A. Once the controller realizes that the estimate îL has gone beyond the 8 A, the

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Fig. 12. (Top) Actual inductor current iL (x-axis: 4 ms/div and y-axis: 5 A/div) and (bottom) its estimate îL (x-axis: 20 ms/div and y-axis: 5 V/div) (only one phase).

monic distortion. In other words, a controller is designed to allow the inverter to deliver an almost sinusoidal and balanced voltage, despite of a distorted and unbalanced load current. The controller is based on a weighted measurement of load and capacitor currents on the same current sensor, thus maintaining the number of sensors as in a conventional controller. In these current sensors, a larger weight is associated to the capacitor currents; hence, the performance was very close to that of a controller based on capacitor current measurements, but with the advantage that information about the load current was also available. Therefore, it was possible to design a simple estimator for the inductor current, which could then be used for protection purposes. Another interesting observation was that, due to suitable transformations, the proposed controller is reduced to a controller with structure very close to the conventional one plus a bank of resonators, thus facilitating the implementation. Finally, experimental results are presented to exhibit the improved performance of the proposed controller in comparison with the conventional one.

R EFERENCES

Fig. 13. Turn-off process due to a load current amplitude exceeding the upper limit fixed at 8 A. (From top to bottom) The output voltage vC (x-axis: 40 ms/div and 100 V/div), actual inductor current iL (x-axis: 40 ms/div and y-axis: 10 A/div), and its estimate îL (x-axis: 40 ms/div and y-axis: 10 V/div) (only one phase).

controller disables the switching process. Due to this action, the semiconductor devices in the inverter are protected, and, moreover, the output voltage and currents are zeroed. Fig. 13 shows the transient responses of (from top to bottom) the output voltage vC , the actual inductor current iL , and its estimate îL during the turn-off process of the inverter after the controller detects an inadmissible load condition. VI. C ONCLUDING R EMARKS In this paper, an adaptive controller is proposed for a threephase inverter, which compensates for unbalance and har-

[1] S. M. Ali and M. P. Kazmierkowski, “PWM voltage and current control of four-leg VSI,” in Proc. IEEE IECON, Aachen, Germany, 1998, pp. 196–201. [2] T. Haneyoshi, A. Kawamura, and R. G. Hoft, “Waveform compensation of PWM inverter with cyclic fluctuating loads,” IEEE Trans. Ind. Appl., vol. 24, no. 4, pp. 582–589, Jul./Aug. 1988. [3] H. L. Jou and J. C. Wu, “A new parallel processing UPS with the performance of harmonic supression and reactive power compensation,” in Proc. IEEE PESC, 1994, pp. 1443–1450. [4] C. D. Manning, “Control of ups inverters,” in IEE Colloq., 1994, pp. 3/1–3/5. [5] S. L. Jung and Y. Y. Tzou, “Discrete feedforward sliding mode control of a PWM inverter for sinusoidal output waveform synthesis,” in Proc. IEEE PESC, 1994, pp. 552–559. [6] M. J. Ryan and R. D. Lorenz, “High performance sine wave inverter with capacitor current feedback and back-EMF decoupling,” in Proc. IEEE PESC, Jun. 1995, vol. 1, pp. 507–513. [7] M. J. Ryan, W. E. Brumsickle, and R. D. Lorenz, “Control topology options for single-phase UPS inverters,” IEEE Trans. Ind. Appl., vol. 33, no. 2, pp. 493–501, Mar./Apr. 1997. [8] N. R. Zargari, P. D. Ziogas, and G. Joós, “A two switch high-performance current regulated dc/ac converter module,” IEEE Trans. Ind. Appl., vol. 31, no. 3, pp. 583–589, May/Jun. 1995. [9] P. Mattavelli, “Synchronous frame harmonic control for high-performance AC power supplies,” IEEE Trans. Ind. Appl., vol. 37, no. 3, pp. 864–872, Jun. 2001. [10] A. Von Jouanne, P. N. Enjeti, and D. J. Lucas, “DSP control of highpower UPS systems feeding nonlinear loads,” IEEE Trans. Ind. Electron., vol. 43, no. 1, pp. 121–125, Feb. 1996. [11] R. Ortega, A. Loria, P. J. Nicklasson, and H. Sira-Ramirez, PassivityBased Control of Euler–Lagrange Systems. New York: Springer-Verlag, 1998. [12] H. K. Khalil, Nonlinear Systems, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [13] G. Escobar, A. M. Stanković, and P. Mattavelli, “Dissipativity-based adaptive and robust control of UPS,” IEEE Trans. Ind. Electron., vol. 48, no. 2, pp. 334–343, Apr. 2001. [14] ——, “Dissipative-based adaptive and robust control of UPS under unbalanced operation,” IEEE Trans. Power Electron., vol. 18, no. 4, pp. 1056–1062, Jul. 2003. [15] D. N. Zmood and D. G. Holmes, “Stationary frame current regulation of PWM inverters with zero steady-state error,” IEEE Trans. Power Electron., vol. 18, no. 3, pp. 814–822, May 2003. [16] S. Fukuda and R. Imamura, “Application of a sinusoidal internal model to current control of three phase utility-interface-converters,” IEEE Trans. Ind. Electron., vol. 52, no. 2, pp. 420–426, Apr. 2005.


G. Escobar (M’00) received the Ph.D. degree from the Laboratory of Signals and Systems (L2S), Supélec, Paris, France, in 1999. From August 1999 to June 2002, he was a Visiting Researcher at Northeastern University, Boston, MA. In July 2002, he joined the Division of Applied Mathematics, Research Institute of Science and Technology of San Luis Potosí (IPICYT), San Luis Potosí, Mexico, where he holds a Professor– Researcher position. His research interests include modeling and control of power electronic systems, especially active filters, inverters, and electric drives.

P. Mattavelli (S’95–A’96–M’00) received the Dr. (with honors) and Ph.D. degrees in electrical engineering from the University of Padova, Padova, Italy, in 1992 and 1995, respectively. From 1995 to 2001, he was a Researcher at the University of Padova. In 2001, he joined the Department of Electrical, Mechanical and Management Engineering (DIEGM), University of Udine, Udine, Italy, where he has been an Associate Professor of electronics since 2002. His research interests include analysis, modeling, and control of power converters, digital control techniques for power electronic circuits, active power filters, and high-power converters. Dr. Mattavelli is a member of the IEEE Power Electronics, IEEE Industry Applications, and IEEE Industrial Electronics Societies, and the Italian Association of Electrical and Electronic Engineers (AEI).

A. M. Stanković (S’88–M’91–SM’02–F’05) received the Dipl.Ing. and M.S. degrees from the University of Belgrade, Belgrade, Yugoslavia, in 1982 and 1986, respectively, and the Ph.D. degree from Massachusetts Institute of Technology, Cambridge, in 1993, all in electrical engineering. Since 1993, he has been with the Department of Electrical and Computer Engineering, Northeastern University, Boston, MA, where he is currently a Professor. His research interests include modeling, analysis, estimation, and control of energy processing systems. Dr. Stanković is a member of the IEEE Power Engineering, IEEE Power Electronics, IEEE Control Systems, IEEE Circuits and Systems, IEEE Industry Applications, and IEEE Industrial Electronics Societies. He was an Associate Editor of the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY during 1997–2001 and presently serves the IEEE TRANSACTIONS ON POWER SYSTEMS in the same capacity.

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Andrés A. Valdez (S’05) was born in San Luis Potosí, Mexico, in 1978. He received the B.S. degree (with honors) in electronic engineering from the Technological Institute of San Luis Potosí, San Luis Potosí, in 2003, and the M.S. degree in control and dynamical systems from the Research Institute of Science and Technology of San Luis Potosí (IPICYT), San Luis Potosí. He is currently working toward the Ph.D. degree in control and dynamical systems in the Division of Applied Mathematics, IPICYT. His main research interests include control of power electronic systems.

Jesus Leyva-Ramos (S’78–M’82) received the M.S. degree in electrical engineering from the California Institute of Technology, Pasadena, in 1978, and the Ph.D. degree in electrical engineering from the University of Houston, Houston, TX, in 1982. He was an Associate Professor at the Iberoamericana University, Lomas de Santa Fe, Mexico, a Radiofrequency and Microwave Engineer at the Jet Propulsion Laboratory, a Teaching Fellow at the University of Houston, the Dean of Professional Studies and Engineering at the ITESM-SLP, Mexico, and a Professor at the Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico. Currently, he is the Head of the Division of Applied Mathematics, Research Institute of Science and Technology of San Luis Potosí (IPICYT), San Luis Potosí. His research interests include modeling of switch-mode dc–dc converters, robust control, and linear systems. Dr. Leyva-Ramos is a member of Eta Kappa Nu, Tau Beta Pi, Sigma Xi, the Mexican Academy of Sciences, and the Mexican Academy of Engineering.