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The dc link voltage in a grid tied ac-dc inverter is needed to be regulated to control the power flow between the grid and the system on the other side of the dc link [41–47]. The three phase PET is similar in aspect where the two level VSI is connected to the grid through the transformer and the dc link of the VSI is to be controlled to regulate power flow. In a grid tied inverter, the power flow is regulated in a cascade control scheme, where the inner loop consists of grid current controllers, which regulate currents by generating appropriate voltages at the ac terminals of the inverter. The outer loop consists of the active and reactive power control, where the dc link voltage controller regulates the active power and the reactive power controller regulates the reactive power by controlling the grid currents. The dc link voltage control scheme is shown in Fig. 3.14. The dc link voltage controller Cv(s) ensures to keep the dc link voltage Vdc close

to the reference V∗

dc. It issues the current command i∗ and the current controller Ci(s)

ensures that the actual grid current i follows this value.

There is however, a key difference between the three phase PET and a grid tied inverter. In a grid tied inverter dc link control, the grid currents controlled and then the inverter voltages are controlled in the inner control loop. This is because the active power is related to the grid currents, which are related to the inverter voltages. In the three phase PET however, while the active power is inherently dependent on grid currents, it is also directly dependent on the parameter δ, given the grid voltage is fixed, as seen in equations (3.48)-(3.50). Thus, by directly issuing a command for δ, the power flow can be regulated and thus the dc link voltage can be regulated. The VSI voltages

72

G(s) Cv(s) δ

V∗

dc Vdc

Figure 3.15: Three phase PET dc link voltage control scheme

to be generated are determined by knowing the grid voltage, since the VSI voltage is to be equal to the grid voltage (reflected on transformer secondary) in a switching period. The control scheme is demonstrated in Fig. 3.15. In the figure, the function G(s) is the transfer function of dc link voltage with respect to the parameter δ.

The power transfer function for the three phase PET is a non-linear function, as seen in (3.48)-(3.50). Therefore, in order to design a controller for the PET, the power transfer function is to be linearized about an operating point and then the approximated linear transfer function G(s) is used for deriving the controller parameters. The rate of change of energy stored in the DC link capacitor is a difference of the power coming in from the grid and power absorbed by the load, which is assumed resistive. This can be written mathematically as:

d(0.5CdcVdc2) dt = Pgrid− Pload =⇒ CdcVdc dVdc dt = f (δ, Vgrid, Vdc) − V_dc2 Rload (3.112) The grid power is a function of grid voltage Vgrid, phase shift parameter δ and DC link

voltage Vdc. The grid voltage is assumed to be fixed. A Taylor’s series expansion is

done on this function about a given initial operating point (where δ0, Rload and ˆVdc are

given) and the first order terms are kept while higher order terms are neglected. In addition, the grid voltage is considered fixed, so the term corresponding to disturbance in grid voltage is zero.

Cdc( ˆVdc+ ˜vdc) d( ˆVdc+ ˜vdc) dt = f (δ0, Vgrid, ˆVdc) + ∂f ∂δ|δ=δ0δ +˜ ∂f ∂Vdc|Vdc= ˆVdc ˜ vdc− ( ˆVdc+ ˜vdc)2 Rload =⇒ Cdc d˜vdc dt = p1˜δ + p2v˜dc− 2 ˆVdc Rload ˜ vdc (3.113) where p1 = ∂f_∂δ|δ=δ0 and p2 = ∂f ∂Vdc|Vdc= ˆVdc

73 It should be noted that in the equation (3.113), the quantities ˜vdc and ˜δ repre-

sent small signal disturbances in the dc link voltage vdc and phase shift parameter δ

respectively. Taking Laplace transform, the equation (3.113) becomes

(sCdc+ 2 ˆVdc Rload − p2 )˜vdc(s) = p1δ(s)˜ =⇒ G(s) = ˜v_˜dc(s) δ(s) = p1 (sCdc+ _R2 ˆV_loaddc − p2) (3.114)

The transfer function to use for designing the DC link voltage controller is given by equation (3.114). In order to determine the quantities p1 and p2, the partial derivatives

of power Po wrt Vdc and δ need to be determined around an operating point. For

illustration purposes, the plots of ∂P_∂δ are shown in Fig. 3.16(a) for different values of δ and m. As with all the quantities plotted so far, this is plotted in per unit (pu) with the base values given in (3.69). In these plots, the dc link voltage Vdc is kept fixed, while δ

is varied along x-axis and different plots are for different values of modulation index m. The plots of ∂P

∂Vdc for a given value of the dc link voltage are illustrated in Fig. 3.16(b).

In the figure, the various plots are for different values of δ. In these plots, instead of varying m directly, it is varied indirectly as Vdc changes along the x-axis while keeping

the grid voltage fixed throughout. It should be noted that since the dc link voltage Vdc

is changing in these plots, the per unit system is not used here. The plots are obtained for peak grid voltage ˆVgrid = 61V , secondary side inductance L = 480µH and a range

of dc link voltages, ranging from Vˆgrid

0.55,

ˆ Vgrid

0.05



. The turns ratio of the transformer is assumed to be unity. The chosen range of Vdc here essentially means that modulation

index m varies between (0.05, 0.55), which covers most of the full range between (0,√1 3).