Predictive Current control of Grid Connected Inverterto inject constant power under unbalanced voltages

Gustavo HunterUniversidad de Concepcion

Jaime Torrent #044Concepcion, CHILE

Email: gushunter@udec.cl

Ramon Blasco-GimenezUniversitat Politecnica de Valencia

Camino de Vera, S/NValencia, Spain

Email: r.blasco@ieee.org

Ivan AndradeUniversidad de Magallanes

Av. Bulnes #01855Punta Arenas, CHILE

Email: i.andrade.a@ieee.org

Ruben PenaUniversidad de Concepcion

Edmundo Larenas #215Concepcion, CHILE

Email: rupena@udec.cl

AbstractThis paper presents a theoretical analysis for gridconnected Inverter supplying constant active and reactive powerto the electric system during unsymmetrical voltage fault con-ditions. It is shown that active/reactive power injected into thegrid will depend on voltage unbalanced level. For a given activeand reactive power and supply voltage, the strategy generatesreference currents. A predictive current control is then usedto impose those currents by the inverter. Results showing theperformance of the strategy are presented during unbalancedvoltage sag and swells.

Keywords Solar power generation,Wind power generation,Current control, Power generation control, Voltage unbalance

I. INTRODUCTION

Given the random nature of wind and solar farm powergenerated by these power plants, power electronic interfacesare connected between the energy sources and the grid.Usually the grid side converter is a DC/AC inverter whichmust keep good performance during grid faults.

Unbalanced transient voltage fault in the network couldlead to power oscillations which could be reflected directlyor indirectly to solar or wind energy sources. These poweroscillations are undesirable because they could harm of themechanical stresses and/or reduce the energy capture fromthe wind and solar radiation [1]-[2].

This paper presents a predictive current control strategy tosupply constant active and reactive power to the grid duringunbalanced voltage sag and swells. The main contribution ofthe work is to demonstrate that a restriction arises for theamount of active and reactive power to be injected into thegrid. This restriction depends on the magnitude of the zerosequence voltage unbalanced. A predictive current control hasbeen implemented to impose the reference currents generatedby the strategy in order to inject a desired active and reactivepower. Results for unbalances sag and swells are presented.

II. SISTEM MODELING

Fig. 1 shows the power converter connected to the threephase supply voltages through filter inductance L and corre-sponding resistance R. The modelling of the system in abccoordinates is given in [1]-[3].

VaN (t) = Ldia(t)

dt+Ria(t) + ea(t) + VnN (t) (1)

VbN (t) = Ldib(t)

dt+Rib(t) + eb(t) + VnN (t) (2)

VcN (t) = Ldic(t)

dt+Ric(t) + ec(t) + VnN (t) (3)

Fig. 1: Grid connected inverter.

Using (4), the system model is transformed into an alfa-beta gamma coordinates, where the gamma component reflectsthe zero sequence voltage.

Tabc =2

3

1 12 120 32 3212

12

12

(4)A. Active Power

In coordenates abc considering [vabc] = [ va vb vc ]t

and [iabc] = [ ia ib ic ]t, then the active power is obtained

as p(t) = [vabc]t [iabc]. Using [vabc] = T1[v ] and [iabc] =T1[i ] the active power in coordinates is given by:

p(t) = (T1[v ])t (T1[i ]) (5)

p(t) = [v ]t(T1)t (T1)[i ] (6)

Because ia + ib + ic = 0, then the instantaneous power iscalculated by (7).

p(t) =3

2vi +

3

2vi (7)

This is a well-known expression and it is verified thatbecause the zero sequence current is null, then the zerosequence voltage presents in the faults does not affect theevaluation of the active power.

B. Reactive Power

The instantaneous reactive power is obtained as: |qabc(t)| =|[vabc] [iabc]| and using [vabc] = T1[v ] and [iabc] =T1[i ], yields to:

|[qabc(t)]| =(T1[v ]) (T1[i ]) (8)

Using (M A)(M B) = det(M)(M1)t(AB), wheredim(M) = 3, is obtained the instantaneous reactive power of[qabc(t)] by (9).

|[qabc(t)]| =|T1|((T1)1)t ([v ] [i ]) (9)

On the other hand if the grid voltage is balanced v = 0,but if there a fault in one or two phases v 6= 0, then thereactive instantaneous power is calculated by (10).

|[qabc(t)]| = 32

2v2(i

2 + i

2) + (vi vi)2 (10)

III. CURRENT REFERENCES

Considering (7), (10), P = 23Pref and Q =23Qref . It can

be shown that the reference currents iand i to be imposed

by the inverter, for balanced or unbalanced grid voltage, canbe calculted as: [3], [4], [5], [6].

i1 =P

v

(Pv+v

2P2v2+Q2v2+Q2v2

2v2+v2+v

2

)v2+v

2

v(11)

i1 =Pv + v

2P 2v2+Q2v2+Q2v2

2v2+v2+v

2

v2 + v2

(12)

i2 =P

v

(Pvv

2P2v2+Q2v2+Q2v2

2v2+v2+v

2

)v2+v

2

v(13)

i2 =Pv v

2P 2v2+Q2v2+Q2v2

2v2+v2+v

2

v2 + v2

(14)

There are two posibles set of reference currents, the firstone i1, i

1, corresponds to a lagging power factor, while

the second solution i2, i2 corresponds to a leading power

factor. Both solutions mentioned above, have the restrictionthat

2P 2v2+Q2v2+Q2v22v2+v

2+v

2

0 so there is a relationship betweenPref and Qref expressed in (15) to obtained valid referencecurrents.

Qref

2vv2 + v

2

Pref (15)

To ilustrate the situation in (15) let us consider the valueof sag or swell in one phase only as (1 )vphase where1 1. Fig. 2 shows the |Q|/P ratio and the powerfactor relationship to be acomplished, in order to obtain realvalues of reference currents as a function of .

0 0.05 0.1 0.15 0.2 0.25 0.3400

200

0

200

400

Vpcc

[V]

0 0.05 0.1 0.15 0.2 0.25 0.310

5

0

5

10

Iabc

[A]

0 0.05 0.1 0.15 0.2 0.25 0.3

0

1000

2000

3000

P[W

],Q[V

Ar]

time [s]

P|Q|

Fig. 2: Relationship between P and Q

IV. CONTROL

In order to imposed the reference currents by the invertera predictive current control is used. In order to do so it isnecessary to obtain a discrete model in coordenates forestimating the currents in the instant k+1. These currents willdepend on the inverter vector voltage and the actual current andgrid voltage values. [7], [8].

A. Currents Discrete Model

Using the Euler Aproximation for calculating didt it can beobtained a discrete model of the system in , as given by(16).

i(k + 1) =

(1 RTs

L

)i(k) +

TsL

(v (k) e(k))(16)

where:

i : Current of the system in v : Voltage applied by the inverter in e : Voltage grid in Ts : Sample TimeL : Line InductanceR : Line Resistence

B. Predictive Control

Fig. 3 shows the control algorithm proposed. In eachsample time the optimal state is applied, calculated in theprevious sample time. To calculate this vector voltage it isnecessary to minimize a cost function that depends on theabsolute error between the reference current i and theestimate current of the system ip(k+ 1). Note the referencescurrent used for k + 1 it is the same current calculated for k,because i(k + 1) i(k).

Start

Apply optimal v (k)

Measure i(k) and e(k)

Calculate references i and i

x=0

x=x+1

ip(k + 1) =(1 RTs

L

)i(k)

+TsL(vx (k) e(k))

g = |i ip|-|i ip |

Storage optimal value

x=7?

Wait nextSample time

no

yes

Fig. 3: Control algorithm proposed

V. SIMULATION RESULTS

The proposed system and control technique have beenvalidated using Simulink/Matlab simulations. The system pa-rameters are described in Table I. The sample frequency is set20 kHz. The filter impedance connected between the inverterand the grid is R = 0.5[] and L = 20[mH]. Simulations fornormal and grid fault operation, for a sag of 40% and swell40% with constant reference active power and reactive powerare shown.

Note that in the sag condition of line voltage is erms =380[V ] and for the swell is erms = 220[V ]. This is because theDC link voltage inverter is set to Vdc = 540[V ], and enoughDC voltage is needed to imposed the current when that is aswell of 40% in one phase.

A. Similutation for balance voltage operation

Fig. 4 shows the result considering initial reference ofPref = 0[W ] and Qref = 0[V Ar]. Then at t = 0.05[s] a

TABLE I: Systems Parameters

L 20[mH]

R 0.5[]

Ts 50 [s]

erms 380 [V] (sag)

erms 220 [V] (swell)

Vdc 540[V]

fe 50 [Hz]

step of Pref = 3000[W ] is applied. The reference power isset to zero at t = 0.15[s]. The reference active power is set toQref = 3000[V Ar] at t = 0.2[s].

0 0.05 0.1 0.15 0.2 0.25 0.3400

200

0

200

400

Vpcc

[V]

0 0.05 0.1 0.15 0.2 0.25 0.310

5

0

5

10

Iabc

[A]

0 0.05 0.1 0.15 0.2 0.25 0.30

1000

2000

3000

4000

P[W

],|Q|[V

Ar]

time[s]

P |Q|

Fig. 4: Simulated with references P=3000[W] andQ=3000[VAr]

Good tracking is observed for step changes for active andreactive power.

B. Simulation result for a Sag=0.4[pu] in one phase

Fig. 5 shows the results when a sag=0.4[pu] in one phaseis applied. Initially Pref = 0[W ] and Qref = 800[V Ar].Then at t = 0.05[s] a step change in Pref of 3000[W ] isapplied, to ensure that real reference currents are obtainedfor the magnitude of voltage sag consider. The sag fault isgenerated during t = 0.1[s] and t = 0.2[s].

In order to track the reference powers the control systeminjects current harmonics. This is shown in the current spec-trum in Fig. 6.

Fig. 7 shows simultation results when the sag is appliedand the reference reactive power is set to Qref = 3000[V Ar].Initially Pref = 0[W ] and at t = 0.05 the reference reactivepower is changed to Qref = 3000[V Ar]. These referencesvalues ensure that the restriction in (15) is valid for the voltagesag of 40%.

Again, during the fault, the control strategy injects har-monic currents in order to track the references powers, asshown in Fig. 8.

0 0.05 0.1 0.15 0.2 0.25 0.3400

200

0

200

400Vp

cc [V

]

0 0.05 0.1 0.15 0.2 0.25 0.310

5

0

5

10

Iabc

[A]

0 0.05 0.1 0.15 0.2 0.25 0.30

1000

2000

3000

4000

P[W

],|Q|[V

Ar]

time[s]

P |Q|

Fig. 5: Sag 40% with P=3000[W] and Q=800[VAr]

0 100 200 300 400 500 600 700 8000

50

100

Frequency [Hz]

Mag

nitu

d [%

]

Ia

0 100 200 300 400 500 600 700 8000

50

100

Frequency [Hz]

Mag

nitu

d [%

]

Ib

0 100 200 300 400 500 600 700 8000

50

100

Frequency [Hz]

Mag

nitu

d [%

]

Ic

Fig. 6: FFT during Sag 40% with P=3000[W] andQ=800[VAr]

C. Simulation result for a Swell=0.4[pu] in one phase

As it was mentioned before, the grid voltage was set to220[V ] in this case, in order to have enough DC link voltage (Vdc = 540[V ] ) to impose the reference currents. Fig. 9 showsa simultation results when a swell=0.4[pu] is applied. At thebeginning the reference power is set to Pref = 0[W ] andQref = 800[V Ar]. Then at t = 0.05 a step change in Qrefof 3000[V Ar] is applied. The swell fault is generated duringt = 0.1[s] and t = 0.2[s]. As it can be seen in Fig. 10 duringthe swell there are current harmonics. The third harmonic hasthe highest value.

Fig. 11 shows simultation result when a swell=0.4[pu] isapplied. Initially reference powers are set to Pref = 0[W ]and Qref = 0[V Ar]. A step change in Qref = 3000[V Ar] isapplied at t = 0.05[s] . These power references again ensurereal reference currents for this fault condition. As it can be seen

0 0.05 0.1 0.15 0.2 0.25 0.3400

200

0

200

400

Vpcc

[V]

0 0.05 0.1 0.15 0.2 0.25 0.310

5

0

5

10

Iabc

[A]

0 0.05 0.1 0.15 0.2 0.25 0.30

1000

2000

3000

4000

P[W

],|Q|[V

Ar]

time[s]

P |Q|

Fig. 7: Sag 40% with P=0[W] and Q=3000[VAr]

0 100 200 300 400 500 600 700 8000

50

100

Frequency [Hz]

Mag

nitu

d [%

]

Ia

0 100 200 300 400 500 600 700 8000

50

100

Frequency [Hz]

Mag

nitu

d [%

]

Ib

0 100 200 300 400 500 600 700 8000

50

100

Frequency [Hz]

Mag

nitu

d [%

]

Ic

Fig. 8: FFT during Sag 40% with P=0[W] and Q=3000[W]

in Fig. 10, during the swell there are also currents harmonics.The third harmonic is the dominant.

D. Simulation result for a two levels of sag in one phase

The simulation results shown before are obtained withreference powers that ensure that restriccion in (15) is satisfied.The following results show that the system cannot regulated thecurrent when this restriction is not fulfilled. In Fig. 13 the ini-tial reference powers are set to P = 0[W ] and Q = 800[V Ar].These reference powers ensure that real reference currents fora sag of 0.4[pu] are obtain. During t = 0.1[s] and t = 0.15[s]the voltage sag is 0.4[pu] and during t = 0.15[s] and t = 0.2[s]the voltage sag is 0.5[pu]. For the second sag the restrictionin (15) is not obeyed and the reference currents obtained from(11) -(14) are not real.

0 0.05 0.1 0.15 0.2 0.25 0.3400

200

0

200

400Vp

cc [V

]

0 0.05 0.1 0.15 0.2 0.25 0.320

10

0

10

20

Iabc

[A]

0 0.05 0.1 0.15 0.2 0.25 0.30

1000

2000

3000

4000

P[W

],|Q|[V

Ar]

time[s]

P |Q|

Fig. 9: Swell 40% with P=3000[W] and Q=600[VAr]

0 100 200 300 400 500 600 700 8000

50

100

Frequency [Hz]

Mag

nitu

d [%

]

Ia

0 100 200 300 400 500 600 700 8000

50

100

Frequency [Hz]

Mag

nitu

d [%

]

Ib

0 100 200 300 400 500 600 700 8000

50

100

Frequency [Hz]

Mag

nitu

d [%

]

Ic

Fig. 10: FFT during Swell 40% with P=3000[W] andQ=600[VAr]

VI. DISCUSSION AND CONCLUSION

This paper has shown a theoretical analysis on the re-quiered reference currents to supply a given active and reactivepower to the grid during unbalance faults. It has been shownthat active and reactive power to be injected must satisfied acertain restriction depend on the magnitude of the grid zerosequence voltage. Several results are been obtained for gridvoltage sags and swells showing a good performance of thestrategy when the restriction in (15) is satisfied. The analysishas been shown that is not posible inject constant active powerfor fp = 1 during a sag or swell.

ACKNOWLEDGMENT

This work was funded by Fondecyt Chileunder Contract 1121104. The financial support ofCONICYT/FONDAP/15110019 is also acknowledged

0 0.05 0.1 0.15 0.2 0.25 0.3400

200

0

200

400

Vpcc

[V]

0 0.05 0.1 0.15 0.2 0.25 0.320

10

0

10

20

Iabc

[A]

0 0.05 0.1 0.15 0.2 0.25 0.30

1000

2000

3000

4000

P[W

],|Q|[V

Ar]

time[s]

P |Q|

Fig. 11: Swell 40% with P=0[W] and Q=3000[VAr]

0 100 200 300 400 500 600 700 8000

50

100

Frequency [Hz]

Mag

nitu

d [%

]

Ia

0 100 200 300 400 500 600 700 8000

50

100

Frequency [Hz]

Mag

nitu

d [%

]

Ib

0 100 200 300 400 500 600 700 8000

50

100

Frequency [Hz]

Mag

nitu

d [%

]

Ic

Fig. 12: FFT during Swell 40% with P=0[W] andQ=3000[VAr]

and by the Spanish Ministry of Science and Technology fundsunder Grant DPI2010-16714.

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0 0.05 0.1 0.15 0.2 0.25 0.3400

200

0

200

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0 0.05 0.1 0.15 0.2 0.25 0.350

0

50

Iabc

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0 0.05 0.1 0.15 0.2 0.25 0.30

1

2

3x 104

P[W

],|Q|[V

Ar]

time[s]

P |Q|

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