Single phase and three phase P+ Resonant based grid connected ...

Single Phase and Three Phase P+Resonant Based Grid Connected Inverters with Reactive Power and Harmonic Compensation Capabilities Ali Maknouninejad University of Central Florida

M. Godoy Simoes Colorado School of Mines

Matthew Zolot UQM Technologies

Abstract- Designing optimized and more efficient controllers for grid connected inverters is a challenge. Conventionally such controllers were established in dq or rotational frame; however, recently stationary or alpha-beta frame has found special interest due to some unique features that it can offer for grid connected inverters. On dq frame variables turn into constants after passing through transform matrixes, so PI controllers that have infinite gain on DC quantities are widely used. But PI controllers are unable to achieve zero steady state error in stationary frame and P+Resonant controllers which have infinite gain on their resonant frequency have been implemented instead. This paper will explain step by step P+Resonant based inverter controllers design for grid connected single and three phase inverters with reactive power compensation and then a novel approach for grid harmonics compensation will be implemented.

I. Introduction Controllers used in inverters are generally based on reference frame with Proportional + Integral (PI) controllers which have infinity gain on DC variables. But sometimes it is desirable to establish controllers on the stationary frame as its variable transformation is much easier, especially for single phase inverters, and also it is possible to use Space Vector Modulation (SVM) directly on stationary frame. Besides it is possible to establish harmonic compensation controllers in the main frame, so it will be an ideal choice for controlling inverters.

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Since PI controllers cannot achieve zero steady state error on stationary frame P+Resonant controllers should be used instead [1]. P+Resonant controllers have infinite gain on the resonant frequency and so they can get zero steady state error on the stationary frame. The general concepts of P+Resonant controller for single phase and three phase inverters are the same and the difference is how to perform the variable transformation to the alpha-beta frame and then back to the main domain, if required. Here after explaining the general P+Resonant controller strategy, it will be shown how to calculate the current reference and do the required transformations for three phase and single phase inverters. Then a novel approach for grid harmonic compensation will be introduced along with simulation results. II. General Control Strategy

Assuming that H = K  + in stationary frame will be [1]:

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  =  +

  

   





in dq frame, its equivalent

(1)

The current controller block is shown in fig. 1.

+

Iα*

-

K +

2 K> s s  + ω, 

K +

2 K> s s  + ω, 

αβ S P W M

Iα Iβ*

+ -

B R I D G E

a-b-c

Iβ

Grid Fig. 1. Inverter current mode controller block diagram.

Where Iα* and Iβ* are the reference currents and Iα and Iβ are the inverter instantaneous output feedback currents, all in stationary frame.

As the three phase system is balanced, I0 = 0 in (7) and V0 = 0 in (8). IV. Single Phase Inverter

III. Three Phase Inverter Assuming that the inverter has to feed P and Q active and reactive powers into the grid for each phase (P and Q are constants), the reference currents can be calculated as follows: I =

 

φ = tan%&

(2)

' (

(3)

Ia = Im sin (ωt – φ)

(4)

Ib = Im sin (ωt - 2π/3 – φ)

(5)

Ic = Im sin (ωt + 2π/3 – φ)

(6)

Then these variables are taken into the stationary frame using (7). 0 I*  I ) + - = /1 . 5 I, 

1√3  15  5 

√3  15  5 

I8 6 7I9 ; I:

(7)

When using Sinusoidal PWM (SPWM) switching scheme, as shown on fig. 1, the controller outputs, which are voltage, should be taken back to the a-b-c frame using (8). However, if Space Vector Modulation (SVM) is used no reverse transform is required [2]. 0 V8 %√. 7V9 ; = /  V: √3 

1

15  15 

1

V* 16 )V+ 1 V,

(8)

Equations (2) to (4) can be used to calculate the current reference for single phase inverter; however, careful attention should be paid for transforming variables to the stationary frame and back. Several methods have been suggested for this purpose [3] [5]. However, the best one is to take the single phase quantity as the alpha term and its 900 delayed signal as the beta. Similarly, the alpha component of controller outputs should be taken as the output single phase quantity to be fed to the SPWM unit. This method has two benefits; first of all it does not demand any kind of transformation matrixes, so it is easy to be implemented and also like its three phase counterpart, produces a rotating vector with the constant magnitude in rotational frame when the magnitude of the reference current is constant. V. Harmonic Compensation There have been already some discussions about P+Resonant controller harmonic compensation [5] [6]. But unfortunately, they lack the details and do not cover all the required and essential aspects of taking the job. Here a novel and complete approach on P+Resonant based harmonic compensation will be discussed. The structure of the P+Resonant controller has been shown in (1) where for the main controller ω0 = 2π60. In order to have a controller capable of compensating the harmonics, a transfer function block should be added to the controller as shown in figure 2 [6]. In fig. 2, HPr and HHr are the main frequency and the nth harmonic controller respectively. Here the focus is on the 3rd harmonic compensation. So n = 3 is chosen. However; the technique for other harmonics; such as 5th,7th ,… is pretty much the same.

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Iref

( =

+

-

C

2 A    + B, 

Iout

+

2 A  =   + D E B, 

Bridge

+

Fig. 2. Current control transfer block with harmonic compensation.

There are more important points for controlling the harmonics yet. The generation of the correct current references is a challenge. The final Iref should contain current references for both the main frequency and harmonics. Current harmonics should be extracted from grid and used for calculating the harmonic reference current as shown on fig. 3. “Harmonic Detection” block on fig. 3 may be a band pass filter or a FFT function which will provide the information of the harmonic, such as magnitude and phase. As the inverter starts compensation, the harmonic magnitude on the grid starts decreasing, as shown on fig. 4 in per unit (PU). So it is not possible to use the output of the ‘harmonic detection’ block as the current reference and a controller is required after it.

Fig. 4. Decrease in the third harmonic of the grid as the inverter starts to compensate it.

Another method has been suggested here to calculate the harmonic reference. The idea is that when the inverter starts injecting current harmonic into the grid, the harmonic of the grid decreases, but the sum of the two remains constant as shown in fig. 5. So the sum of the inverter and grid harmonic currents can be used as the harmonic reference current, accompanied with a saturation limit, as needed. The Simulink block diagram of the final harmonic current reference evaluation is shown on fig. 6.

Reference [5] suggests using another resonant controller with unity gain coefficient (Ki) as the controller. This method, however, increases the complexity of the design and needs more calculations for the design and adjustment of the controller.

Main current reference +

Iref

+ Grid current

Harmonic Detection

Controller

Fig. 3. Current reference calculations for harmonic compensation purposes.

Fig. 5. Sum of the third harmonic of grid and inverter.

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VI.

Fig. 6. Simulink block diagram of harmonic current reference calculation.

Here two band pass filters have been used to extract the 3rd harmonic of the grid and inverter. The central frequency of these filters is 180Hz with 150Hz and 210Hz as stop frequencies and 170Hz and 190Hz as pass frequencies. A transport delay is used after the sum of the filters to compensate for the phase difference. This phase difference can be calculated theoretically from the filter’s transfer function or through simulation to find phase difference between the filter outputs and the original grid harmonics. A gain has been used finally to compensate for any attenuation caused by the filters. The final block diagram of the three phase system is shown in fig. 7.

Simulation Results

The simulation results for a three phase inverter which is connected to a grid with 5KVAR inductive load is shown on fig.8 and fig. 9. Fig. 8 shows the grid current and voltage without compensation and fig. 9 shows them when inverter is compensating the reactive power and 5KW of the active power. The simulation results for the single phase inverter with harmonic compensation are shown on figures 10 and 11. On these figures, the upper figure shows the current of the grid and the lower one depicts that of the inverter. On the simulations inverter is connected to the grid until t = 0.4s and compensates the harmonic and then disconnects. Until t = 0.4 it is seen that the inverter is injecting harmonic to the grid to compensate its current distortion and the grid has no harmonic up to this time. After t = 0.4, inverter disconnects and the harmonic of the grid will not be compensated, as it is seen on the figures. In both cases grid has a 15A 3rd harmonic and in the simulations of fig. 11 the harmonic has a 200 lagging phase.

Fig. 7. General block diagram of the proposed three phase inverter.

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Fig. 8. Grid current and voltage without compensation.

Fig. 9. Grid current and voltage with compensation.

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Fig. 10. Currents of inverter and grid with the 3rd harmonic

. Fig. 10. Currents of inverter and grid with the 3rd harmonic with 200 lagging phase

VII.

Conclusion

In this paper, the P+Resonant controller has been discussed. One of the advantages of three phase inverter P+Resonant controller is its simple transformation stage which contains matrixes with constants only; while for the conventional PI controller there are sine and cosine terms in its conversion matrix. Also, if one decides to exploit SVM [2], no further reverse transformation from stationary frame back to

a-b-c will be required. It was shown how easy it is to use P+Resonant controller for single phase inverters as it is enough to assume the current reference itself as the alpha axis and take its 900 delayed signal as the beta. Also no reverse transform is required and the alpha output of the controller should be assumed as the final variable to be fed to the SPWM module. So, absolutely, there are no transform matrixes.

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The last section of this paper discussed harmonic compensation. A new method was suggested for calculating reference current which can automatically compensate for harmonic reactive power as well. Also required modifications to the controller block were presented. It was shown that by modifying the current reference and controller it is possible to enable the P+Resonant based inverter to compensate the grid harmonics. This may be count as one of the advantages of P+Resonant controllers, as for doing the same on DQ frame, one has to establish a new complete frame for each harmonic frequency which will increase the complexity of the system. References: [1]

[2]

[3]

[4]

[5]

[6]

Zmood, D. N.; Holmes, D. G., “Stationary Frame Current Regulation of PWM Inverters With Zero Steady-State Error”, in Proc. of Power Electronics Spec. Conf., vol. 2, 1999, pp 1185-1190 Zhang, R.,” High performance power converter systems for nonlinear and unbalanced load/source”, Ph.D Dissertation, Virginia Polytechnic Institute, 1998 Roshan, A. “A DQ rotating frame controller for single phase fullbridge inverter used in small distributed generation systems”, Master Thesis, Virginia Polytechnic Institute, 2006. Miranda, U.A.; Aredes, M., Rolim, L.G.B. “A DQ synchronous reference frame control for single-phase converters”, in Proc. Of Power Electronics Spec. Conf., 2005, pp. 1377 – 1381 Teodorescu, R.; Blaabjerg, F.; Liserre, M. and Loh, P.C., ”Proportional-resonant controllers and filters for grid-connected voltage-source converters”, IEE Proc. On Electric Power , vol. 153, Issue 5, Sep. 2006, pp 750 - 762 Sera, D.;Kerekes,T.; Lungeanu,M.; Nakhost, P.; Teodorescu, R.; Anderson, G.K.;Liserre ,M. “Low-Cost digital implementation of proportional-resonant current controllers for PV inverter applications using delta operator”, Industrial Electronics Conf., IECON 2005, pp. 2517-2522

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