Design of an Internal Model Control strategy for single-phase grid-connected PWM inverters and its performance analysis with a non-linear local load and weak grid

ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Design of an Internal Model Control strategy for single-phase grid-connected PWM inverters and its performance analysis with a non-linear local load and weak grid Eric N. Chaves n,1, Ernane A.A. Coelho, Henrique T.M. Carvalho, Luiz C.G. Freitas, João B.V. Júnior, Luiz C. Freitas Universidade Federal de Uberlândia, Uberlândia, MG, Brazil

art ic l e i nf o

a b s t r a c t

Article history: Received 5 February 2016 Received in revised form 12 April 2016 Accepted 3 May 2016 This paper was recommended for publication by Dr. Jeff Pieper.

This paper presents the design of a controller based on Internal Model Control (IMC) applied to a gridconnected single-phase PWM inverter. The mathematical modeling of the inverter and the LCL output filter, used to project the 1-DOF IMC controller, is presented and the decoupling of grid voltage by a Feedforward strategy is analyzed. A Proportional – Resonant Controller (Pþ Res) was used for the control of the same plant in the running of experimental results, thus moving towards the discussion of differences regarding IMC and Pþ Res performances, which arrived at the evaluation of the proposed control strategy. The results are presented for typical conditions, for weak-grid and for non-linear local load, in order to verify the behavior of the controller against such situations. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Grid-connected single-phase systems Internal Model Control – IMC Proportional – Resonant Distributed generation

1. Introduction In grid-connected distributed generation systems, issues involving the design criteria or topology have attracted great interest within the scientific community, leading to an increased number of publications, among which [1,2] respectively stand out. The topology adopted for this work was the single-phase PWM sinusoidal voltage-source inverter (VSI) with an LCL filter, connected to a single-phase grid. The structure for which is presented in Fig. 1. In this system, the DC-Bus voltage (VDC) control is performed by an external loop with a ProportionalþIntegral (PI) compensator, where the reference for the current to be injected is defined. The state diagram in Fig. 2 is a representation of this structure. Through this representation one notes that a grid voltage sample (ES) is measured at the connection point, this sample is that used by the synchronism algorithm, in this case a Phase Locked Loop (PLL), and by the grid voltage decoupling strategy.

n

Corresponding author. E-mail address: [email protected] (E.N. Chaves). 1 Instituto Federal de Educação, Ciência e Tecnologia de Goiás (IFG), Itumbiara, Brazil.

When it comes to control strategies for current injection into the grid, the earlier developments used in VSIs were mostly based on hysteresis and PI controllers. These controllers are capable of compensating these injection systems, if used on three-phase topology and synchronous reference [3,4]. Otherwise, these controllers cannot compensate steady-state errors, in terms of amplitude and phase of the injected current [5]. However, for the stationary reference frame, concerning the case of single-phase systems, significant advances were described in [5] through the Proportional Resonant (Pþ Res) controller, along with the Repetitive Controller [6,7]. The P þRes controllers, implemented to control voltage-source PWM inverters for current injection, are typically band-pass filters. In such, the center frequency is tuned to the nominal grid frequency thus attaining a high gain at this point; these are ideally infinite in the case of Type I Pþ Res. The principal features for such are a narrow pass band and an abrupt changing in the gain at the edge frequencies of the pass band [8]. This characteristic, the gain at grid frequency and the inclusion of sinusoidal reference model in the open-loop system, leads to a zero steady-state error for a stationary reference frame. In other words, the system impeccably follows the sinusoidal reference for the injected current. However, if the grid frequency oscillates at around 60 Hz, there exists the possibility of undesired effects on

http://dx.doi.org/10.1016/j.isatra.2016.05.002 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Chaves EN, et al. Design of an Internal Model Control strategy for single-phase grid-connected PWM inverters and its performance analysis with a non-linear.... ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.05.002i

E.N. Chaves et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

Fulll Bri Bridge r dg d e

Q1

Q2

a

C1

VDC

_

L1

eC

L2

R2

ES i2

C

ES

b

Q3

Z GRID RGR LGR GRID G RID GRID G RID

LCL

R1

Q4

i2

iGRID

RL

iL

EGRID

127V/ 60Hz

Trafo 1:1

Gate Driver Q1 Q2 Q3 Q4

SPWM

i*

uCONT

IMC or PLL P+Res

e2

* iref

i iref

PI PLL

v ref

Feedforward PLL PLL

ES

i2

VDC

ES

Fig. 1. Structure of the grid-connected system used.

Fig. 2. Block diagram for the DC-bus voltage control and current injection.

this current. This effect is usually mitigated by adding a damping factor (ζ) to the transfer function, making it a Type II PþRes [8]. In the Type II case, the gain at the center frequency is reduced. However, the bandwidth is increased, alleviating the gain deviation at the border frequencies, allowing the system to run into small variations on the grid frequency. The gain and the bandwidth can be appropriately adjusted using the parameters Kp, Ki and ζ present in the transfer function of the controller, aiming to meet the necessities of an application and/or a desired operating point. Therefore, the Type II P þRes performance is related to these parameter values, and they are normally set to the inverter's nominal power. Thus, this paper presents a control strategy for grid-connected current injection that has a similar performance to the PþRes Type II, when it comes to steady-state error, although without presenting the disadvantage of the gain variation outside the nominal frequency point. This is accomplished from a monotonic frequency response characteristic, in order to maintain the gain not only at the nominal frequency, but also throughout the pass band. This performance was possible through the development of a controller based on internal model (Internal Model Control – IMC),

which has 1 degree of freedom (1-DOF IMC) in conjunction with the grid voltage decoupling using a Feedforward strategy. One advantage of this control strategy is that although it will react quickly to disturbances by rejecting them, it exerts less control effort to achieve the result [9–13]. Thus, this paper presents a new technique based on control with the internal model coupled with the Feedforward strategy. The main contribution of this work is the understanding that the two techniques together allow for the preparation of an internal model based only on the LCL filter dynamics. These are summarized as active damping of the system with the inversion of the frequency response exerted by the controller using internal model and immunity to variations of the electrical grid, due to the effect caused by Feedforward decoupling. To validate the presented control strategy, a Type II P þRes controller [5,8,14] was designed and tested to compare its results with the proposed IMC 1-DOF controller. The results are obtained from an experimental platform where, at first, the system is connected to a linear local load and to a normal grid. Later, a nonlinear local load and a weak grid (high series impedance and distorted voltage) are incorporated.

Please cite this article as: Chaves EN, et al. Design of an Internal Model Control strategy for single-phase grid-connected PWM inverters and its performance analysis with a non-linear.... ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.05.002i

E.N. Chaves et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2. Feedforward strategy The Feedforward strategy used here, also known as Back-EMF due to its being analogous to the counter-electromotive force feedback of the DC Motors [15], leads to the cancellation of the grid voltage negative feedback, intrinsic to the inverter system. This method, already well explored in [15–17], has the effect of decoupling the voltage at the local-load (RL), which is also the grid connection point, by taking a sample of this voltage (ES) and

3

summing it to the current injection control action (uCONT) before the actuator (the PWM generator, in this case). The diagram in Fig. 3(a) represents the process model, which goes from the output of the IGBT bridge (Vab) to the local-load and connection point, passing through the LCL filter, considering that ZGRID (ZGRID ¼s  LGRID þRGRID) tends to zero and ES tends to EGRID. (See Fig. 2). From this model, one notes that the inverter output current (i2) causes and suffers interference on the connection point voltage (ES) in a feedback process intrinsic to the model (and topology).

Fig. 3. (a) State diagram of the inverter connected to the grid. (b) State diagram highlighting the feedback intrinsic to the model with ES, ZGRID and RL decoupling through the Feedforward strategy. (c) Resulting diagram after the Feedforward.

Fig. 4. Inverter operation steps.

Please cite this article as: Chaves EN, et al. Design of an Internal Model Control strategy for single-phase grid-connected PWM inverters and its performance analysis with a non-linear.... ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.05.002i

E.N. Chaves et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

Moreover, ES is fed back to the system as a disturbance. As the quality of (ES) is strongly dependent on the load characteristics, and since the local load RL is parallel to the grid, preponderating the second one (ZGRID is many times smaller than RL and tends to zero). Therefore, the decoupling of this voltage has a double effect in terms of facilitating the dynamics of current injection into the grid and thus overcoming the effect of loads having unknown dynamics (cannot be well modeled). The flow diagram in Fig. 3(b) shows that ES is decoupled from the LCL filter, and this leads to the decoupling of load (RL//ZGRID). This is due to the addition of a sample of ES to the control action (uCONT). Thus, the dynamics to be considered and controlled are reduced only to the LCL filter, without the necessity for the inclusion of the grid parameters, allowing the formulation of the internal model and controller for 1-DOF – IMC. This strategy can be equally applied to both controllers (PþRes and IMC). The resulting system is illustrated in Fig. 3(c).

From the average model of the inverter at steady state:

therefore: 2 3 2 UV ab =L1 6 7 0 J¼4 5 0

ð9Þ

thus: 2

3. Modeling of the voltage-source inverter and filter The modeling is performed by using the state-space average technique [18,19]. Two events associated with the states of the switches, with the correspondent resulting circuit, and the decoupling effect due to the Feedforward are considered. The illustration in Fig. 4(a) shows the first event, in which the switches Q1 and Q4 are closed and Q2 and Q3 are open. The other event, illustrated in Fig. 4(b), depicts Q2 and Q3 as closed with Q1 and Q4 as open. In both situations, the current i2, flowing through L2, is the variable to be controlled by this current injection strategy. Thus, the inverter can be described by these linear stateequations: x_ ¼ A1 Uxðt Þ þ B1 U uðt Þ

ð1Þ

x_ ¼ A2 Uxðt Þ þ B2 U uðt Þ

ð2Þ

where 2 3 2 0 3 " # i1 i1 V ab 6 7 6 0 7 x ¼ 4 i2 5; x_ ¼ 4 i2 5; u ¼ ES v0C vC

1=L1

6 B1 ¼ 4 0 0

3

2

7  1=L2 5; 0

6 B2 ¼ 4

0

ð3Þ

 1=L1 0 0

0

3

7  1=L2 5 0

ð5Þ

As Gxd is the small-signal transfer function for the inverter, and considering the variations of the state vector (x) elements, in ^ as: function of the PWM active cycle width (dÞ, Gxd ¼ x^ ðsÞ=d^ ðsÞ ¼ ðs UI  AÞ  1 U J

ð6Þ

and J ¼ ðA1 A2 Þ U X þ ðB1  B2 Þ UU

ab

2

2

C U L1 U L2 U s3 þ ðC U L1 U R2 þ C U L2 U R1 Þ U s2 þ ðC U R1 U R2 þ L1 þ L2 Þ U s þ ðR1 þ R2 Þ

ð10Þ and Gi2=d ¼ ^i2 ðsÞ=d^ ðsÞ ¼

2 U V ab U k C U L1 U L2 U s3 þ ðC UL1 U R2 þ C U L2 U R1 Þ U s2 þ ðC U R1 U R2 þ L1 þ L2 Þ U s þ ðR1 þ R2 Þ

ð11Þ therefore, (Gi2/d) is the small-signal transfer function for considering the output changing as a function of the PWM active cycle ^ The term k is a constant of proportionality in order to width (dÞ. create a unitary gain for the PWM (actuator), and Vab is the average value of the voltage at the point of connection between the fullbridge output and the LCL filter input.

4. The LCL filter

In which ES is zero (as well as the grid impedance) due to the decoupling of this voltage by the Feedforward. In addition: 2 3 0 1=L1  R1 =L1 6 0  R2 =L2 1=L2 7 ð4Þ A1 ¼ A2 ¼ 4 5 1=C  1=C 0 2

3 V ab U ð2 U C U L2 U s2 þ 2 U C U R2 U s þ 2Þ 6 C U L1 U L2 U s3 þ ðC U L1 U R2 þ C U L2 U R1 Þ U s2 þ ðC U R1 U R2 þ L1 þ L2 Þ U s þ ðR1 þ R2 Þ 7 6 7 2 U V ab 7 Gxd ¼ 6 6 C U L1 U L2 U s3 þ ðC U L1 U R2 þ C U L2 U R1 Þ U s2 þ ðC U R1 U R2 þ L1 þ L2 Þ U s þ ðR1 þ R2 Þ 7 4 5 2 U V U ðR þ L U sÞ

ð7Þ

In which A ¼A1 ¼ A2 is the matrix of the average value of the variables, where D is the duty cycle, and I is the identity matrix.

The LCL filter was designed using the criteria discussed in [20] and the details were omitted, as they are not the focus of this work. The resulting parameters for the system are presented in Table 1, with the injected current possessing a maximum ripple of 1%. The bode diagram in Fig. 5 shows the frequency response of the transfer function (11), considering the aforementioned parameters. As noted in Fig. 5, the resonance frequency (fres) is approximately 876 Hz. This parameter was considered during the design Table 1 System parameters. Parameters

Name

Value

Grid frequency PWM frequency Rated power Grid peak voltage DC Bus voltage Filter inductance Connection inductance Leakage resistance 1 Leakage resistance 2 Filter capacitance DC Bus capacitor Constant of proportionality

fg fsw Pn Es VDC L1 L2 R1 R2 C C1 k

60 Hz 10 kHz 2 kW 127 Vrms 350 V 1.1 mH 10.0 mH 0.05 Ω 0.05 Ω 30 mF 2200 mF 1/350

Please cite this article as: Chaves EN, et al. Design of an Internal Model Control strategy for single-phase grid-connected PWM inverters and its performance analysis with a non-linear.... ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.05.002i

E.N. Chaves et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

5

Phase (deg)

Magnitude (dB)

Bode Diagram

c( s )

50

r (s )

0

+_

e(s)

q PLL s,

+ +

-50

Controller

Disturbance

d (s) u(s)

pd ( s ) p( s)

+

+

y(s)

Process

∼

pPLL (s)

-100 0

Model

-90 -180 -270 -2 10

Fig. 7. 1-DOF IMC with rearranged structure. -1

0

10

10

1

2

10 10 Frequency (Hz)

3

10

4

10

and yðsÞ ¼

Fig. 5. Inverter frequency response.

ð1  p~ ðsÞqðsÞÞpd ðsÞdðsÞ 1 þ ðpðsÞ  p~ ðsÞÞqðsÞ

ð14Þ

where pd(s) is the system model due to the disturbance, i.e., its effect flows through the system and adds to the output. Applying the final-value theorem, if Eqs. (13) and (14) are dynamically stable and the controller steady-state gain q(0) is ~ chosen to be the reciprocal of the internal model gain (p(0)q(0) ¼ 1), the gain at the denominator of Eqs. (13) and (14) will be p(0)q (0). Therefore, the direct path gain is unitary, the gain related to the disturbance and y is zero, leading to a zero steady-state error. Therefore, the idea behind this control strategy is to obtain:

∼

yðsÞ ¼ r ðsÞ

and

yðsÞ=dðsÞ ¼ 0

ð15Þ

~ ¼ pðsÞ pðsÞ

ð16Þ

since: Fig. 6. 1-DOF IMC generic structure.

pðsÞqðsÞ ¼ 1

of the 1-DOF IMC controller, when it came to creating the additional poles and establishing the cutoff frequency.

5. The Internal Model Control (IMC) The term Model Based Control (MBC) is used to designate control systems that incorporate explicitly a model of the process in the algorithm, such as the Internal Model Control (IMC) and Model-Predictive Control (MPC) [21]. In Fig. 6, the generic form of a control strategy based on the internal model is shown, which possesses one degree of freedom ~ (1-DOF IMC) where p(s) is the process, pðsÞ is the model of that process, u(s) is the control effort, d(s) is the disturbance and d~ e ðsÞ is the estimated disturbance [21]. Since the model p~ ðsÞ is a perfect representation of a stable process p(s), if the controller gain q(s,ε) is the reciprocal of the model gain, the output y(s) will follow the reference r(s) provided ~ that the p(s) and pðsÞ gains have the same signal and the controller is tuned to ensure stability. For a SISO (Single Input, Single Output) system, the model p~ ðsÞis a linear transfer function (Gi2/d) and the controller is the approximate reciprocal of this function. In order to develop the transfer functions among the inputs d(s) and r(s) and the output y(s) and to comprehend the operation of this control strategy, the scheme was redrawn and presented in Fig. 7 showing a traditional structure of a feedback system, in which: cðsÞ 

uðsÞ qðsÞ ¼ eðsÞ 1  qðsÞp~ ðsÞ

ð12Þ

Being c(s) the control action, u(s) the control effort and e(s) the error. From the input–output relationships, presented in Fig. 7, one can infer that: yðsÞ ¼

pðsÞqðsÞr ðsÞ 1 þ ðpðsÞ  p~ ðsÞÞqðsÞ

ð13Þ

and

Consequently, for the control action to be effective on tracking the reference a perfect model is necessary and from (16), the controller has to completely invert this model [21]. However, it is impossible to have a completely accurate model. In addition, if it has any dynamics (very common), any controller will invert the process perfectly. Thus, how close it gets, depends on its project. 5.1. 1-DOF IMC controller design Using the design criteria presented in [21], when the process transfer function does not have zeros near the imaginary axis neither on the right half of the s-plane, and taking into consideration the minimization method of the Integral Square Error (ISE), defined in (17) as: Z 1 ISE  ðyðtÞ  rðtÞÞ2 dt ð17Þ 0

To reach an optimum choice of controller the 1-DOF IMC controller function can be designed as: qðsÞ ¼

DðsÞ N ðsÞ U ðεs þ 1Þr

ð18Þ

To which N(s) and D(s) are respectively the polynomials of the numerator and denominator, of the transfer function Gi2/d corresponding to the process (p(s)) and described in (11). Expression (19) is part of (18) and represents a filter used to make q(s) causal and physically realizable, where ε is a parameter for setting the cutoff frequency for this filter and r is defined as the difference between the denominator and numerator order of the transfer function. 1 ðεs þ 1Þr

ð19Þ

For this project, the parameter ε can be calculated according to [21], from  1=r DðsÞNð0Þ ε Z lims-1 r ð20Þ 20s NðsÞDð0Þ

Please cite this article as: Chaves EN, et al. Design of an Internal Model Control strategy for single-phase grid-connected PWM inverters and its performance analysis with a non-linear.... ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.05.002i

E.N. Chaves et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

6

Table 2 P þRes parameters.

p(s) 0

-50

Value

Proportional gain Integral gain Damping factor Center frequency

Kp Ki ζ ωc

0.7 3.0 0.03 2π60 rad/s

Bode Diagram 50

10

10

10

10

10

10

Frequency (Hz)

Fig. 8. Frequency response of the inverter, controller (IMC) and direct-path gain.

To which the gain at high frequency is arbitrary limited to 20 times the gain at low frequency. However, when using this criterion the system response, verified from computational simulations, was slow and had a pronounced phase lagging between the injected current and the reference. Decreasing the value of ε, the controller gain at high frequency is increased and the phase shift reduces. As the open-loop system frequency response corresponds to a low-pass filter with a maximally flat magnitude response in the pass band and linear phase, the tuning process consists of increasing the open-loop system pass band beyond 60 Hz (Grid Frequency), but maintains the cutoff frequency below the LCL filter resonant frequency (fres). Thus, if the value of ε is reduced, the pass band is enlarged. In addition, besides the linearization of the phase response, the phase angle is reduced to 60 Hz. A value of 0.00017 was chosen based on the aforementioned aspects, but with this value, a phase shift of 10° is still present at the operation point. If the value of ε is reduced, this angle goes to 0°; however, the pass band exceeds the resonant frequency (fres) and the quality of the response degrades significantly. Therefore, good results were obtained maintaining the value of 0.00017 with the pass band in accordance with the range corresponding to the operating frequency and the resonant frequency, along with adding 10 degrees to the current reference generated by the PLL, in order to compensate the aforementioned phase lagging. Fig. 8 shows the Bode Plot for the inverter (p(s)), the controller (q(s)) and the open-loop gain g(s)¼ p(s)q(s). As can be seen, the gain is unitary (0 dB) inside the pass band of g(s), in accordance with Eqs. (15) and (16). The controller does not completely invert the frequency response due to the filter, in Eq. (19), that maintains a flat response after fres with a gain defined by the value of ε, as described in [21]. The resulting effect of g(s) is a low-pass filter with bandwidth of around 475 Hz (3 dB frequency), without the resonance peak which is canceled by the controller, which leads to the damping of the system. There is also a linearization of the phase angle, as one notes from Fig. 8.

6. PþRES controller design For the PþRes controller strategy used in this study, the controller chosen is the Type II with a classic structure as described in [5], for which the transfer function is: Gres ðsÞ ¼ kp þ

Name

2ki ωc s s2 þ 2ζωc s þ ωc 2

ð21Þ

where Kp and Ki are the proportional and integral gains, respectively, ζ is the damping factor and ωc is the center frequency. The

Magnitude (dB)

-100 135 90 45 0 -45 -90 -135 -180 -225 -270 10

g(s)

q(s)

Parameters

0 -50

60 Hz

-100 -150 0

Phase (deg)

Phase (deg)

Magnitude (dB)

Bode Diagram

-90 -180

60 Hz

-270 -360 10

10

10

10

10

Frequency (Hz)

Fig. 9. Open-loop frequency response for the compensated system using the Type II P þRes.

design choices (type, damping rate and center frequency) for the PþRes are not explained here, they have however, been well explored in [5,8,14]. These in fact are not the focus of this work. With the intention of obtaining the best tuning of P þRes using the grid frequency and the inverter nominal power as the operation point, several parameters were exhaustively tested, and the chosen values are listed in Table 2. Therefore, the controller transfer function, defined in (21) and calculated using Table 2 parameters, has the frequency response of a band-pass filter, possessing the grid frequency as its center frequency. The open-loop frequency response for the system with the PþRes controller is shown in Fig. 9.

7. Experimental results The two controllers were discretized by the Tustin method and the internal model, by using ZOH. The control algorithm was coded in C language and loaded into the Digital Signal Controller TMS320F28335 from Texas Instrumentss. A sample rate and switching frequency of 10 kHz was used. The PþRes controller was tested with and without the Feedforward strategy, producing a better qualitative performance in the first situation in terms of steady-state error. However, it should be emphasized that these results apply to the PþRes controller defined in (21) and the parameters shown in Table 2. These may differ if the design criteria are different from those adopted here. Considering this improvement and the inherent presence of that strategy in the 1-DOF IMC controller, only the results using Feedforward are presented. The discrete transfer functions for the IMC 1-DOF (q(z)) and ~ PþRes (Gres(z)) controllers, along with the internal model (p(z)), when presented in their zero/pole/gain forms, are respectively, qðzÞ ¼

35:6995z3  95:9048z2 þ 95:7467z  35:5318 z3  1:6364z2 þ 0:8926z  0:1623

Gres ðzÞ ¼

0:7999z2 1:3975z þ 0:5986 z2  1:9965z þ 0:9979

ð22Þ

ð23Þ

Please cite this article as: Chaves EN, et al. Design of an Internal Model Control strategy for single-phase grid-connected PWM inverters and its performance analysis with a non-linear.... ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.05.002i

E.N. Chaves et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

I Ref ( z )

qPLL z

+_

u( z)

I2 ( z)

p PLL

Controller

Process

∼

pPLL ( z)

VDC 1 zPLL

Model

-

+

ES

de ( z )

Fig. 10. Analysis of the implementation delay on the estimated disturbance calculation.

~ pðzÞ ¼

7

3:28e 4z3 þ 9:86e 4z2 þ 9:86e 4z þ 3:28e 4 z3  2:686z2 þ 2:682z 0:9953

ð24Þ

The option was made to follow the zero/pole/gain structure for the numeric representation of the discrete transference functions. This due to the familiarity of this representation in projects involving power electronics control and the ease at which the digital controller is embedded, onto a computational platform – DSC, from transfer function coefficients on the z plane. The implementation delay is also analyzed, this is implicit to ~ the discretization of the internal model (p(z)), being obtained from the ZOH method, and the estimated disturbance calculation (d~ e (z)), which is performed in the proceeding sampling period, after bringing into operation the control effort (u(z)). This fact is illustrated in Fig. 10.

I2 Connection

Fig. 12. Results for the current injection using P þ Res – Inverter output; Scale: Ch.1 (I2) 5 A/div; Ch.2 (Es) 60 V/div; Ch.3(VDC) 100 V/div; Time 25 ms/div.

VDC ES

IGRID

7.1. Results for linear local load and normal grid For this test, on both control strategies, the system was adjusted to supply approximately 1200 W to the grid at the connection instant by step input. In Fig. 11, the results (covering the previous state, the connection transient and the injection) are shown for the system using Pþ Res and Feedforward. The DC bus voltage VDC, the grid voltage ES and the grid current IGRID are also shown. It is reputed of the phase inversion of IGRID at the moment of connection, that before the inverter starts up, the grid supplies the local load and after the connection, the grid begins to absorb the exceeding current. There is also a small drop in VDC due to the inverter transference of power from the DC to AC side and the PI compensator regulates this voltage in the form of a cascade in the current injection. IGRID reached a peak of 23 A at the transient and it stabilized at 14 A (peak value) at steady state. Fig. 12 presents the results for the same configuration, but the current shown here is the inverter output current I2 (before the local load), rather than IGRID.

VDC

Connection

Fig. 13. Experimental results for the current injection using 1-DOF IMC; Scale: Ch.1 (IGRID) 5 A/div; Ch.2 (Es) 60 V/div; Ch.3 (VDC) 100 V/div; Time 25 ms/div.

VDC ES

Connection I2

Fig. 14. Results for the current injection using 1-DOF IMC – Inverter output; Scale: Ch1(I2): 5 A/div; Ch2(Es):60 V/div; Ch3(VDC):100 V/div; Time: 10 ms/div.

ES

IGRID Connection

Fig. 11. Experimental results for the current injection using P þ Res; Scale: Ch.1 (IGRID) 5 A/div; Ch.2 (Es) 60 V/div; Ch.3 (VDC) 100 V/div; Time 25 ms/div.

Before the connection instant - the inverter is off - I2 is zero. After the transient, the inverter is injecting I2 to the group Grid þlocal load, with a high power factor of 0.996, validating the synchronization with the grid voltage through the PLL and the PþRes gain at 60 Hz. Presented in Fig. 13 are the results (including the previous state, the connection transient and the injection) for the system using IMC and Feedforward. The DC bus voltage VDC, the grid voltage ES and the grid current IGRID are shown. Similarly, the results for PþRes, IGRID inverts, showing that the grid is supplying the local load beforehand, and after the connection it absorbs the current

Please cite this article as: Chaves EN, et al. Design of an Internal Model Control strategy for single-phase grid-connected PWM inverters and its performance analysis with a non-linear.... ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.05.002i

E.N. Chaves et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

8

3th 5th 7th

Fig. 15. Expanded view and FFT of the injected current. (a) Using P þRES. (b) Using 1-DOF IMC. Scale: Y: 20 dB/Div X: 500 Hz/Div.

11

IMC 1DOF P+Res

10 9

THD% (Total)

8 7 6 5 4 3 2

2

3

4

5

6 7 Corrente (A)

8

9

10

Fig. 16. THD of injected current for both control strategies and different amplitudes.

2.5 Percentual (%)

injected by the inverter. From the illustration, one notes that although it had a higher offset level, the connection transient with IMC was smoother than that using the P þRes. Another important point to be made concerning the DC bus voltage control, when using the 1-DOF IMC, is the sagging encountered on the VDC at the moment of connection was less abrupt than with PþRes. These results in effect confirm that the IMC strategy rejects the disturbances rapidly and exerts less control effort to reach the desired result [9–13,21]. Presented in Fig. 14 are the results for the same configuration, but the current shown is the inverter output current I2 (before the local load), rather than IGRID. One observes that I2 is perfectly in phase with ES, indicating a correct synchronism by the PLL and a good phase response, showing a high power factor of 0.998. The illustration in Fig. 15(a) shows the waveform and FFT (Fast Fourier Transformer) of the injection current (I2) at steady state, for the system using PþRes with Feedforward. Illustrated in Fig. 15(b) is the same situation but for the system using 1-DOF IMC. By analyzing Figs. 15(a) and (b), one notes that the injected currents are sinusoidal with low THD (Total Harmonic Distortion) with the presence of some odd harmonics. Although the results were very similar, at this specific current intensity, the THD using the PþRes controller was lower than that using IMC (2.4% for the first and 2.6% for the second). However, the decay of high order harmonics is faster when the system is controlled by the PþRes, which can be explained by its frequency response mentioned above and illustrated in Fig. 9. It is important to mention that during all the experiments, the grid voltage presented a THD of about 2.3% when the inverter was disconnected (off). Depicted in Fig. 16 is a graph generated from experimental values of the output current (I2) THD, for multiple values using both control strategies. This experiment was carried out to verify the performance of the controller, when faced with variations on the available power at the inverter input and consequently, changing the intensity of the current to be injected. Understood from Fig. 16 is that although the Pþ Res response at the beginning and end of the current range is better (lower THD), the IMC performance in terms of harmonic distortion remained more uniform over that range. The best and the worst THD results for the P þRes were respectively 2.4% and 10.6%, whereas for the 1-DOF IMC they were 2.6% and 7.0%. Thus, at times, the injected current controlled by the

2.4

1.9

2 1.5 1

1.72 0.79 0.96

1.64 1.03

0.85

0.64

ES

0.5 0.4

0.5

0.16

IMC-1DOF P+RES

0 3

5

7

9

Harmonics Fig. 17. Most prominent harmonic components of grid voltage and inverter current.

PþRes presented a THD that exceeded the limits indicated by the standards IEC 61000-3-2 (7% for currents up to 16 A) and IEEE 519 (5%) [22,23]. However, for rated power, both controllers presented similar results and they were in accordance with those standards. The chart in Fig. 17 contains the individual amplitudes of the 3rd, 5th, 7th and 9th harmonics present in the injected current, for PþRes and 1-DOF IMC controllers at 1200 W and the components present on the grid voltage (ES) before the injection. Only odd numbered harmonics are shown, as the even numbered ones were not significant. Noted here is that the 3rd harmonic is more prominent when using the P þRes, while in the case of the 5th, 7th and 9th harmonics the 1-DOF IMC contribution stands out.

Please cite this article as: Chaves EN, et al. Design of an Internal Model Control strategy for single-phase grid-connected PWM inverters and its performance analysis with a non-linear.... ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.05.002i

E.N. Chaves et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

7.2. Results for non-linear local load and weak grid This experiment has the purpose of verifying the performance of both controllers when injecting current onto a grid with high series impedance and non-linear load, and consequently possesses a poor voltage regulation and high harmonic distortion. An inductor (LSERIES) of 10 mH was inserted in series with the grid and a non-linear load constituted by a power resistor in series with a half-wave rectifier (DL) was connected in parallel with the previous resistive local load. This circuit set up is illustrated in Fig. 18 and the effect exerted by such modifications, while the inverter is disconnected, is shown in Fig. 19.

LSerie Inverter and LCL

i2

DL RL1

RL 2

ES

LGRID iGRID

RGRID

VGRID

127V/ 60Hz

Trafo 1:1

Fig. 18. Modifications for tests simulating weak grid and non-linear local load.

9

Shown in Fig. 19(a) is the grid voltage at the connection point (ES) and the grid current (IGRID). In this case, the current is absorbed by both non-linear and linear loads - the second possessing higher impedance than the first. Presented in Fig. 19(b) is the measured THD for ES (9.7%) and a histogram showing the percentages of the harmonic components, which predominate the even ones. The RMS amplitude of ES is 124.5 V (while the rated value is 127 V). The graph in Fig. 20(a) shows the results for this experiment at steady state, for injection using the P þRes controller with Feedforward. ES, I2, and IGRID are shown. The same situation is presented in Fig. 20(b) but for the 1-DOF IMC controller. The THD of the injected current for the Pþ Res case was 2.77% while for 1-DOF IMC it was 2.73%, predominating the 3rd harmonic in both cases, especially predominate when using the second case. The harmonic analysis of ES (grid voltage) is shown in Fig. 21 (a) for the injection using PþRes, and in Fig. 21(b), when using IMC. The ES RMS value and THD appear at the top. At the bottom, a histogram showing the amplitude of the fundamental (Fnd) and harmonic components is presented. Noted from Figs. 19 and 21 is that after the inverter connection, a 3rd harmonic component is added to the grid voltage and this

ES

IGRID

Fig. 19. (a) Grid current and voltage waveforms. (b) Voltage harmonics. Weak grid and non-linear load experiment, before injection.

ES I2

I GRID

Fig. 20. Waveforms of the grid voltage, grid current and inverter current. Non-linear load and weak grid. (a) Using Type II P þ Res. (b) Using 1-DOF IMC.

Please cite this article as: Chaves EN, et al. Design of an Internal Model Control strategy for single-phase grid-connected PWM inverters and its performance analysis with a non-linear.... ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.05.002i

E.N. Chaves et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

10

Fig. 21. Grid voltage harmonics during injection. Non-linear load and weak grid. (a) Using Type II P þRes. (b) Using 1-DOF IMC.

Fig. 22. Harmonic components of inverter output current. Non-linear load and weak grid. (a) Using Type II P þ Res. (b) Using 1-DOF IMC.

Table 3 Controllers performance in terms of I2 THD. Experiment

Pþ Res

1-DOF IMC

Linear load and normal grid Non-linear load and high impedance grid

2.40% 2.77%

2.60% 2.73%

was more prominent with the 1-DOF IMC controller. However, the PþRes contributed more toward the 5th harmonic. There is also some rejection of the even harmonics, especially the 2nd, when using the PþRes controller, which leads to a reduction of the THD, reaching 7.22% for the PþRes and 9.4% for the 1-DOF IMC controller. Therefore, both controllers reject in similar fashion the

influence of the grid voltage harmonic distortion caused by the non-linear load, mostly due to the Feedforward strategy, and the grid voltage is positively affected by the current injection, especially when using the PþRes, reducing the THD. Shown in Fig. 22 is the THD analysis for the inverter output current (I2), the corresponding waveform is shown in Fig. 20, when the injection current is controlled by the P þRes with Feedforward (a) and when its controlled by the 1-DOF IMC (b). The I2 RMS value and THD appear at the top, and at the bottom, a histogram showing the amplitude of the fundamental (Fnd) and harmonic components is presented. The following table (Table 3) presents a summary concerning the performance of both controllers in terms of the injected current THD. Finally, a picture of the experimental platform designed and used for this work is shown below (Fig. 23).

Please cite this article as: Chaves EN, et al. Design of an Internal Model Control strategy for single-phase grid-connected PWM inverters and its performance analysis with a non-linear.... ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.05.002i

E.N. Chaves et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

11

Acknowledgements

1:1 Isolation Trafo

The authors would like to thank Capes, CNPq (process n° 406845/2013-1) and Fapemig (case n° APQ - 01219-13) for the financial support.

References

Inverter

DSC

LCL Filter

Local LOAD

Fig. 23. Experimental platform.

8. Conclusions This paper focused on the design steps of a 1-DOF IMC (1Degree of Freedom - Internal Model Control) based controller, which possesses the application of a Feedforward strategy, destined to control the current injection into a single phase grid, using the PWM-VSI with LCL filter and the performance analysis by experimental results. The proposed system is designed to produce a performance similar to the Type II Proportional and Resonant controller using Feedforward (P þRes) when it comes to steady-state error, without the expressive changes in the gain around the grid frequency, which is a characteristic of the PþRes. Both controllers presented an adequate and similar performance for nominal conditions. Regarding harmonic distortions of the injected current, the Pþ Res reached lower values of THD than the 1-DOF IMC for most injection intensities. However, when varying the current over a wide range, the P þRes showed some notable values of THD, while the 1-DOF IMC values were more uniform and always in accordance with the standards. Therefore, the main objective behind this work was successfully reached, that being presenting a technically viable strategy to control the current injection on grid-connected PWM inverters. The design procedure and the experimental results of the 1-DOF IMC controller using Feedforward were shown, and a Type II Proportional and Resonant with Feedforward was also designed and tested in order to provide the basis for comparison. The results confirmed the quality of both controllers and validated the 1-DOF IMC applicability to the system discussed, including situations as weak and/or distorted grid, non-linear local load and changes in the input power and consequently, in the injected current module. The authors propose as future studies, strategies that involve model-based techniques for this type of application. For example, a controller using state feedback for the internal model (MSF-IMC), used for systems with limitations and/or saturation of the control action, as well as predictive control and the 2DOF- IMC.

[1] IEEE Standard 1547-2003, IEEE Standard for Interconnection Distributed Resource with Electric Power Systems, 2003. [2] Kjaer SB, Pedersen J, Glaabjerg F. A review of single-phase grid-connected inverters for photovoltaic modules. IEEE Trans Ind Appl 2005;41:1292–306. [3] Li S, Haskew TA, Hong Y, Xu L. Direct-current vector control of three-phase grid-connected rectifier–inverter. Electr Power Syst Res 2011;81:357–66. [4] Li S, Fu X, Ramezani M, Sun Y, Won H. A novel direct-current vector control technique for single-phase inverter with L, LC and LCL filters. Electr Power Syst Res 2015;125:235–44. [5] Zmood DN, Holmes DG. Stationary frame current regulation of PWM inverters with zero steady-state error. IEEE Trans Power Electron 2003;18:814–22. [6] Rashed M, Klumpner C, Asher G. Repetitive and resonant control for a singlephase grid-connected hybrid cascaded multilevel converter. IEEE Trans Power Electron 2013;28:2224–34. [7] Montera AR, Buenob EJ, García-Cerradac A, Rodríguez FJ, Sáncheza FM. Detailed analysis of the implementation of frequency-adaptive resonant and repetitive current controllers for grid-connected converters. Electr Power Syst Res 2014;116:231–42. [8] Teodorescu R, Blaabjerg F, Liserre M, Loh PC. Proportional-resonant controllers and filters for grid-connected voltage-source converters. IEE Proc -Electr Power Appl 2006;153(5):750–62. [9] Wenfang D, Qun H, Yingnian X. Study on IMC-PID control for single-phase voltage-source inverters. In: Proceedings of the Power Electronics and Motion Control Conference, IPEMC – Conference IEEE; 2009. p. 1514–1518. [10] de J, Rubio J, Ochoa G, Balcazar R, Pacheco J. Disturbance rejection in two mechatronic systems. IEEE Lat Am Trans 2016;14(2):485–92. [11] Rosas D, Alvarez J, Rosas P, Rascon R. Robust observer for a class of nonlinear siso dynamical systems. Math Probl Control 2016;2016:19. [12] Rubio JJ, Ochoa G, Balcazar R, Pacheco J. Uniform stable observer for the disturbance estimation in two renewable energy systems. ISA Trans 2015;58:155–64. [13] Suleman H, Shah A, Man Z. Reconciliation of outliers in CO2 alkanolamine H2O datasets by robust neural network winsorization. Neural Comput Appl 2016:1–12, http://link.springer.com/article/10.1007%2Fs00521-016-2213-z. [14] Cha H, Vu TK, Kim JE. Design and control of proporcional-ressonant controller base photovoltaic power conditioning system. In: Proceedings of the Energy Conversion Congress and Exposition – Conference IEEE; 2009. p. 2198–2205. [15] Ryan MJ, Brumsickle WE, Lorenz RD. Control topology options for single-phase UPS inverters. IEEE Trans Ind Appl 1997;33:493–501. [16] Li W, Pan D, Ruan X, Wang X. A full-feedforward scheme of grid voltages for a three-phase grid-connected inverter with an LCL Filter. In: Proceedings of the Energy Conversion Congress and Exposition – Conference IEEE; 2011. p. 96– 103. [17] Liu T, Hao X, Yang X, Zhao M, Xiong L. A novel grid voltage feed forward control strategy for three-phase grid-connected VSI with LCCL filter. In: Proceedings of the Industrial Electronics – Conference IEEE International Symposium on; 2012. p. 86–91. [18] Middlebrook RD, Cuk SJA. A general unified approach to modelling switchingconverter power stages. In: Proceedings of the Power Eletronics Specialists Conference, Cleveland; 1976. p. 1–6. [19] Erickson RW, Maksimóvic D. Fundamentals of power electronics, 2nd edition, 2004. [20] Liserre M, Blaabjerg F, Hansen S. Design and control of an LCL-filter-based three-phase active rectifier. IEEE Trans Ind Appl 2005;41:1281–91. [21] Brosilow C, Joseph B. Techniques of model-based control. 1st ed.. Nova Iorque: Prentice Hall; 2002. [22] International Electrotechnical Commission, IEC 61000-3-2, Electromagnetic Compatibility (EMC) – Limits for Harmonic Current Emissions, 1998 e Emenda A14; 2001. [23] IEEE 519-1992, IEEE Recommended Practice and Requirements for Harmonic Control in Electric Power Systems; 1992.

Please cite this article as: Chaves EN, et al. Design of an Internal Model Control strategy for single-phase grid-connected PWM inverters and its performance analysis with a non-linear.... ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.05.002i