Complete Model

Solving the closed-loop transfer functions for a cascaded control scheme, such as in Fig. 2.13, is recommended to be done step by step by first computing the closed-loop transfer functions when only the output-current control is active and then treating the obtained model as an “open-loop” system for the input voltage control. Solving the transfer functions directly from Fig. 2.13 would be too laborious.

Fig. 2.14 presents a control block diagram for an output-current controlled CF in-verter. The model includes all the cross-coupling terms (G^H_cr-qd, G^H_cr-dq, G^H_co-qd and G^H_co-dq). According to Fig 2.14, thed andq-channel duty ratios can be given by

dˆd=− 1

G^H_co-dLout-dˆiod+ 1

Req-dG^H_co-dLout-duˆ^iod_ref, (2.246) dˆq=− 1

G^H_co-qLout-qˆioq+ 1

Req-qG^H_co-qLout-qˆu^ioq_ref, (2.247) where

Lout-d=Req-dGcc-dGaG^H_co-d, (2.248)

Lout-q=Req-qGcc-qGaG^H_co-q, (2.249)

are the current control loop gains, Req-(d,q) are the equivalent current sensing resis-tors,Gcc-(d,q) current controller transfer functions,Gamodulator gain, andG^H_co-(d,q)the control-to-output-current transfer functions.

Substituting (2.246) and (2.247) in the nominal open-loop dynamics represented by (2.102)–(2.104) and solving for ˆuin, ˆiod and ˆioqyields

ˆ

uin=Z_in^outˆiin+T_oi-d^outuˆod+T_oi-q^outuˆoq+G^out_ci-duˆ^iod_ref +G^out_ci-quˆ^ioq_ref, (2.250) ˆiod=G^out_io-dˆiin−Y_o-d^outuˆod+G^out_cr-qduˆoq+G^out_co-duˆ^iod_ref +G^out_co-qduˆ^ioq_ref, (2.251) ˆioq=G^out_io-qˆiin+G^out_cr-dquˆod−Y_o-q^outuˆoq+G^out_co-dquˆ^iod_ref +G^out_co-quˆ^ioq_ref, (2.252) where the superscript ‘out’ indicates that the output current control loops are closed, and the above closed-loop transfer functions can be given by

Z_in^out=uˆin

ˆiin

=Z_in^H− Lout-d

1 +Lout-d

G^H_io-dG^H_ci-d

G^H_co-d − Lout-q

1 +Lout-q

G^H_io-qG^H_ci-q

G^H_co-q , (2.253) T_oi-d^out = uˆin

ˆ uod

=T_oi-d^H +Y_o-d^outG^H_ci-d

G^H_co-dLout-d−G^out_cr-dqG^H_ci-q

G^H_co-qLout-q, (2.254)

H cr-qd

G

H

Yo-d H

Gio-d

H

Gco-d

H co-qd

G

Ga G_cc-d Req-d

H

Gio-q H cr-dq

G

H

Yo-q

H co-dq

G

H

Gco-q

Ga G_cc-q Req-q

ˆin

i ˆod

u ˆoq

u

ˆod

i

iod

ˆref

u

ˆoq

i ˆd

d

ˆq

d

H

Zin H

Toi-d H

Toi-q

ˆin

u

H

Gci-d

H

Gci-q

ˆq

d

ˆd

d

q

dˆ ˆd Open-loop d

input dynamics

Open-loop -channel dynamics d

Open-loop -channel dynamics q

ioq

ˆref

u

Fig. 2.14: Control-block diagram for an output-current-controlled grid-connected CF inverter.

2.5. Closed-Loop Transfer Functions for a Grid-Connected CF Inverter

T_oi-q^out = ˆuin

ˆ uoq

=T_oi-q^H −G^out_cr-qdG^H_ci-d

G^H_co-dLout-d+Y_o-q^outG^H_ci-q

G^H_co-qLout-q, (2.255) G^out_ci-d= uˆin

ˆ

u^iod_ref = G^H_ci-d

R^H_eq-dG^H_co-dLout-d−G^out_co-dG^H_ci-d

G^H_co-dLout-d−G^out_co-qdG^H_ci-q

G^H_co-qLout-q, (2.256) G^out_ci-q= uˆin

ˆ

u^ioq_ref = G^H_ci-q

R^H_eq-qG^H_co-qLout-q−G^out_co-qdG^H_ci-d

G^H_co-dLout-d−G^out_co-qG^H_ci-q

G^H_co-qLout-q. (2.257)

G^out_io-d=ˆiod

ˆiin

= G^H_io-d−G^H_io-q^G

H co-qd

G^H_co-q Lout-q

1+L_out-q

1 +Lout-d

1−^G_G^{Hco-qdH GHco-dq}

co-dG^H_co-q Lout-q

1+Lout-q

, (2.258)

Y_o-d^out =−ˆiod

ˆ uod

= Y_o-d^H +G^H_cr-dq^G

H co-qd

G^H_co-q Lout-q

1+Lout-q

1 +Lout-d

1−^G_G^{Hco-qdH GHco-dq}

co-dG^H_co-q Lout-q

1+Lout-q

, (2.259)

G^out_cr-qd= ˆiod

ˆ uoq

= G^H_cr-qd+Y_o-q^{H G}

H co-qd

G^H_co-q Lout-q

1+Lout-q

1 +Lout-d

1−^G_G^{Hco-qdH GHco-dq}

co-dG^H_co-q Lout-q

1+L_out-q

, (2.260)

G^out_co-d= ˆiod

ˆ

u^iod_ref = Lout-d

Req-d

1−^G

H

co-qdG^H_co-dq G^H_co-dG^H_co-q

Lout-q

1+Lout-q

1 +Lout-d

1−^G_G^Hco-qdH ^GHco-dq co-dG^H_co-q

Lout-q

1+Lout-q

, (2.261)

G^out_co-qd= ˆiod

ˆ

u^ioq_ref = G^H_co-qd G^H_co-q

Lout-q

Req-q

1−_1+L^Lout-q_out-q 1 +Lout-d

1−^G_G^{Hco-qdH GHco-dq}

co-dG^H_co-q Lout-q

1+Lout-q

, (2.262)

G^out_io-q=ˆioq

ˆiin

= G^H_io-q−G^H_io-d^G_G^Hco-dqH co-d

L_out-d 1+Lout-d

1 +Lout-q

1−^G_G^{Hco-qdH GHco-dq}

co-dG^H_co-q Lout-d

1+Lout-d

, (2.263)

G^out_cr-dq= ˆioq

ˆ uod

= G^H_cr-dq+Y_o-d^{H G}

H co-dq

G^H_co-d Lout-d

1+L_out-d

1 +Lout-q

1−^G

H

co-qdG^H_co-dq G^H_co-dG^H_co-q

Lout-d

1+L_out-d

, (2.264)

Y_o-q^out =−ˆioq

ˆ uoq

= Y_o-q^H +G^H_cr-qd^G

H co-dq

G^H_co-d Lout-d

1+Lout-d

1 +Lout-q

1−^G

H

co-qdG^H_co-dq G^H_co-dG^H_co-q

Lout-d

1+Lout-d

, (2.265)

G^out_co-dq= ˆioq

ˆ

u^iod_ref = G^H_co-dq G^H_co-d

Lout-d

Req-d

1−_1+L^Lout-d_out-d 1 +Lout-q

1−^G_G^{Hco-qdH GHco-dq}

co-dG^H_co-q Lout-d

1+Lout-d

, (2.266)

G^out_co-q= ˆioq

ˆ

u^ioq_ref = Lout-q

Req-q

1−^G

H

co-qdG^H_co-dq G^H_co-dG^H_co-q

Lout-d

1+Lout-d

1 +Lout-q

1−^G_G^{Hco-qdH GHco-dq}

co-dG^H_co-q L_out-d 1+Lout-d

, (2.267)

It is worth noting thatLout-din (2.248) andLout-qin (2.249) are not the “complete”d andq-channel current control loop gains because of the cross-coupling terms. According to (2.258)–(2.267), thed and q-channel current control loop gains can be given by

LOUT-D=Lout-d 1−G^H_co-qdG^H_co-dq G^H_co-dG^H_co-q

Lout-q

1 +Lout-q

!

, (2.268)

LOUT-Q=Lout-q 1−G^H_co-qdG^H_co-dq G^H_co-dG^H_co-q

Lout-d

1 +Lout-d

!

, (2.269)

according to which the output current control loops can be designed.

Next the effect of input voltage control can be computed based on Fig. 2.15 and (2.250)–(2.252). According to Fig. 2.15, thed-channel current reference (control voltage ˆ

u^iod_ref) can be given by ˆ

u^iod_ref =Gse-vGvcuˆin−Gvcuˆ^uin_ref, (2.270) whereGse-v is the input voltage sensing gain andGvc is the input voltage controller.

Substituting (2.270) to (2.250) and solving for the ˆuin yields ˆ

uin=Z_in^out-inˆiin+T_oi-d^out-inuˆod+T_oi-q^out-inuˆoq+G^out-in_ci-d uˆ^uin_ref +G^out-in_ci-q uˆ^ioq_ref (2.271) where the superscript ‘out-in’ refers to the cascaded control scheme, i.e. that both the input-voltage and output-current loops are closed, and

Z_in^out-in= uˆin

ˆiin

= Z_in^out 1−Lin

, (2.272)

T_oi-d^out-in= uˆin

ˆ uod

= T_oi-d^out 1−Lin

, (2.273)

T_oi-q^out-in= uˆin

ˆ uoq

= T_oi-q^out 1−Lin

, (2.274)

G^out-in_ci-d = uˆin

ˆ

u^uin_ref =− 1 Gse-v

Lin

1−Lin

, (2.275)

G^out-in_ci-q = uˆin

ˆ

u^ioq_ref = G^out_ci-q 1−Lin

. (2.276)

2.5. Closed-Loop Transfer Functions for a Grid-Connected CF Inverter

out cr-qd

G

out

Yo-d out

Gio-d

out

Gco-d

out co-qd

G

out

Gio-q out cr-dq

G

out

Yo-q

out co-dq

G

out

Gco-q

ˆin

i ˆod

u ˆoq

u

ˆod

i

ioq

ˆref

u

ˆoq

i

out

Zin out

Toi-d out

Toi-q

ˆin

u

out

Gci-d

out

Gci-q

Gvc

Gse-v

uin

ˆref

u ˆin

u

Input dynamics when the output current loops are closed

-channel dynamics when the output-current loops are closed

d

-channel dynamics when the output-current loops are closed

q

ioq

ˆref

u

ioq

ˆref

u

iod

ˆref

u

iod

ˆref

u

iod

ˆref

u

Fig. 2.15: Control-block diagram for a grid-connected CF inverter with a cascaded input-voltage-output-current control scheme.

The input voltage control loop gain can be given by

Lin=Gse-vGvcG^out_ci-d, (2.277)

whereGse-v is the input voltage sensing gain, Gvc is the transfer function for the input voltage controller andG^out_ci-d is the transfer function between the input voltage and the d-channel current reference. The minus sign in front ofLin in (2.272)–(2.276) is caused by the fact that the control error signal between the measured input voltageGse-vuˆinand the reference ˆu^uin_ref has to be inverted for proper inverter operation (Figs. 2.13 and 2.15).

The output dynamics under the cascaded control scheme can be computed by substi-tuting (2.270) and (2.271) in (2.251) and (2.252) yielding

ˆiod=G^out-in_io-d ˆiin−Y_o-d^out-inuˆod+G^out-in_cr-qduˆoq+G^out-in_co-d uˆ^uin_ref +G^out-in_co-qduˆ^ioq_ref, (2.278) ˆioq=G^out-in_io-q ˆiin+G^out-in_cr-dquˆod−Y_o-q^out-inuˆoq+G^out-in_co-dquˆ^uin_ref +G^out-in_co-q uˆ^ioq_ref, (2.279) where

G^out-in_io-d =ˆiod

ˆiin

= G^out_io-d

1−Lin − Lin

1−Lin

G^out_io-d-∞ (2.280)

Y_o-d^out-in=−ˆiod

ˆ uod

= Y_o-d^out

1−Lin − Lin

1−Lin

Y_o-d-∞^out (2.281)

G^out-in_cr-qd = ˆiod

ˆ

uoq = G^out_cr-qd

1−Lin− Lin

1−LinG^out_cr-qd-∞ (2.282)

G^out-in_co-d = ˆiod

ˆ

u^uin_ref =− Lin

1−Lin

G^out_co-d

Gse-vG^out_ci-d (2.283)

G^out-in_co-qd = ˆiod

ˆ

u^ioq_ref = G^out_co-qd

1−Lin − Lin

1−Lin

G^out_co-qd-∞ (2.284)

G^out_io-d-∞=G^out_io-d−Z_in^outG^out_co-d

G^out_ci-d (2.285)

Y_o-d-^out∞=Y_o-d^out+T_oi-d^outG^out_co-d

G^out_ci-d (2.286)

G^out_cr-qd-∞=G^out_cr-qd−T_oi-q^outG^out_co-d

G^out_ci-d (2.287)

G^out_co-qd-∞=G^out_co-qd−G^out_ci-qG^out_co-d

G^out_ci-d (2.288)

2.5. Closed-Loop Transfer Functions for a Grid-Connected CF Inverter

G^out-in_io-q =ˆioq

ˆiin

= G^out_io-q

1−Lin − Lin

1−Lin

G^out_io-q-∞ (2.289)

G^out-in_cr-dq = ˆioq

ˆ uod

= G^out_cr-dq

1−Lin − Lin

1−Lin

G^out_cr-dq-∞ (2.290)

Y_o-q^out-in=−ˆioq

ˆ

uoq = Y_o-q^out

1−Lin − Lin

1−LinY_o-q-^out∞ (2.291)

G^out-in_co-dq = ˆioq

ˆ

u^uin_ref =− Lin

1−Lin

G^out_co-dq

Gse-vG^out_ci-d (2.292)

G^out-in_co-q = ˆioq

ˆ

u^ioq_ref = G^out_co-q

1−Lin − Lin

1−Lin

G^out_co-q-∞ (2.293)

G^out_io-q-∞=G^out_io-q−Z_in^outG^out_co-dq

G^out_ci-d (2.294)

G^out_cr-dq-∞=G^out_cr-dq−T_oi-d^outG^out_co-dq

G^out_ci-d (2.295)

Y_o-q-^out∞=Y_o-q^out+T_oi-q^outG^out_co-dq

G^out_ci-d (2.296)

G^out_co-q-∞=G^out_co-q−G^out_ci-qG^out_co-dq

G^out_ci-d (2.297)

Eqs. (2.271)–(2.297) represent the closed-loop dynamics of a grid-connected input-voltage-output-current controlled CF inverter. However, solving the aforementioned transfer functions with the parasitic elements included, e.g. in Matlab even by using transfer function reduction tools such as‘minreal’ or‘balred’, can become too complex.

Furthermore, there is no point in solving the closed-loop dynamics without the parasitic elements because the phase and gain margins from this kind of a model would not repre-sent the true margins from the actual prototype. The major cause for the complexity of the transfer functions in a three-phase inverter origins from the cross-coupling transfer functions G^H_cr-qd, G^H_cr-dq, G^H_co-qd and G^H_co-dq. A reduced-order model was used in Sec-tion 2.4.2 to simplify the load-affected three-phase inverter model. Similar methods will be used in the following subsection to simplify the presented closed-loop model.

In document Issues on Dynamic Modeling and Design of Grid-Connected Three-Phase VSIs in Photovoltaic Applications (sivua 70-76)