Читать книгу Smart Grids and Micro-Grids - Umashankar Subramaniam - Страница 43

The equivalent circuit of the boost converter is shown in Figure 2.4. V_B and i_B are the battery voltage and curent, respectively. Applying Kirchhoff’s Voltage Law (KVL) in the inductor loop and Kirchhoff’s current law at the node C_in gives,

(2.1)

The instantaneous state-space variable vector can be written as,

(2.2)

Where I_L and D_L represent the DC values of input current (=I_B) and duty cycle of the lower switch, respectively, and î_L and represent the perturbation values of input current and duty cycle, respectively. During the charge and discharge mode of the battery, the output voltage (V_dc) and battery voltage (V_B) can be assumed to be constant. Equations (2.1) and (2.2) can be used to obtain the transfer function between input current and duty cycle as

Figure 2.4 Equivalent circuit of the converter.

(2.3)

The current compensator is given by

(2.4)

Where k_{p, i} and k_{i, i} are gain of PI controller of current loop. The control system loop gain is

(2.5)

For improving the open-loop frequency response, the pole can be canceled by the zero of the PI compensator. Defining τ_i is the desired time constant of the closed-loop system ( is the bandwidth of the current closed-loop), we can choose

(2.6)

(2.7)

As discussed earlier, sometimes the dc-dc converter is supposed to control the output voltage, i.e., dc microgrid voltage. To derive the transfer function between the output voltage and inductor current which would be the outer control loop of the converter, let the perturbation in output voltage be . Using the perturbations in equation (2.1) and linearizing the resulting expressing, one would obtain

(2.8)

The transfer function from output voltage to inductor current of the outer loop is given by

(2.9)

The voltage compensator is given by

(2.10)

Where kp,v and ki,v are gain of PI controller of voltage loop. The control system loop gain is

(2.11)

As stated earlier, the pole can be eleminated by the zero of the PI compensator. Defining τ_v is the desired time constant of the closed-loop system ( is the bandwidth of the voltage closed loop), we can choose

(2.12)

(2.13)

The current loop is designed with a crossover frequency of less than a tenth of the switching frequency of the converter. The voltage compensator is designed to have the crossover frequency to be one half of the crossover frequency of the current loop [11].