Dynamic Modeling of Wound-Rotor Slip-Ring Induction Generator with Switched-Excitation Capacitance and Chopper Resistance Across Bridge Rectifier in the Rotor Circuit

The proposed scheme of WRSRIG and its steady-state equivalent circuit is shown in Figure 1. The machine can be operated as an induction generator if the rotor is driven over synchronous speed by wind or hydro turbine.

The excitation capacitance connected across H-bridge inverter in the rotor circuit to control the stator terminal voltage by varying the duty ratio Ɣ_inv of the inverter switches. The effective external rotor excitation capacitance can be calculated as [17]:

$C_{r e f}=C_{r} /\left(2.Ɣ_{i n v}-1\right)^{2}$               (1)

where, C_r is the switched excitation capacitance in the rotor circuit. This capacitance appears in series with the external chopper resistance across the diode bridge rectifier in the rotor circuit. The frequency of terminal voltage can be controlled by varying the external resistance (R_ex) in the rotor circuit. The chopper-thyristor switch is connected across the resistance. The effective - resistance appears across the bridge rectifier in the rotor circuit depending on the duty ratio Ɣ_ch of the chopper and can be calculated as [18]:

$Ŕ _{e f f}=\left(1Ɣ_{c h}\right)^{2} * Ŕ _{e x}$                 (2)

In this method, the terminal voltage and its frequency can be controlled with a wide range of load variations.

Due to the presence of leakage reactance's in the rotor windings of the WRSRIG, the commutation of currents among the rotor phases between the terminals of the bridge rectifier does not occur instantaneously. There is a period of current overlap between the phases. This leads to a reduction in the rectified voltage. The reduction voltage due to the over-lap can be obtained as:

$V_{d r}=3 * \acute{X}_{\ell r} * I_{d} / \pi$                   (3)

where, $X_{\ell r}$ is the leakage reactance of the rotor per phase reflected onto the stator.

I_d is the rectified DC output current of the bridge rectifier. The current I_d can be obtained in terms of the rotor phase current referred to the stator side (ĺ_r) as:

$I_{d}=\sqrt{3} * ĺ_{r} / \sqrt{2}$                   (4)

Then from (3) and (4), the reduction voltage due to the over-lap can be obtained as:

$V_{d r}=1.17 * \cdot \acute{X}_{\ell r}$               (5)

Then the over-lap reactance ($X_{o \ell}$) in the rotor circuit can be represented as:

$\acute{X}_{o \ell}=1.17 *  \acute{X}_{\ell r}$                   (6)

Due to the rectification process of the diode bridge rectifier, a harmonic loss is generated in the rotor circuit and can be represented by a resistance (Ŕ_h ) per phase of the rotor circuit as [18]:

$Ŕ_{h}=\left[Ŕ_{r}+\left(\frac{1}{2}\right) * Ŕ_{e f f}\right] *\left(\frac{\pi^{2}}{9}-1\right)$                (7)

where, Ŕ_r is the rotor resistance per phase reflected onto the stator circuit.

Stray load loss resistances $\left(R_{s s s \ell}^{*} \& Ŕ_{r s s \ell}\right)$ of the stator and rotor circuits in the steady-state equivalent circuit of the WRSRIG can be calculated as [19]:

$R_{s s s \ell}=\left(\omega_{s} \cdot L_{\ell s}\right)^{2} * R_{s} /\left[R_{s}^{2}+\left(\omega_{s} . L_{\ell s}\right)^{2}\right]$                   (8)

$R_{r s s \ell}=\frac{S \cdot\left(\omega_{s \ell} \cdot \acute{L}_{\ell r}\right)^{2} * \acute{R}_{r}}{\left(\acute{R}_{r}\right)^{2}}=\frac{s^{2} \cdot\left(\omega_{s} \cdot \acute{L}_{\ell r}\right)^{2}}{\acute{R}_{r}}$                     (9)

where, $L_{\ell s}$ and $L_{\ell r}^{\prime}$=stator and rotor leakage inductances. ω_s is the angular speed of the stator terminal voltage. R_s and Ŕ_r are stator and rotor winding resistances respectively. If the machine operates near the synchronous speed the slip (s) is very small and the rotor circuit stray load loss resistance ($Ŕ_{r s s \ell}$) can be neglected.

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(a) Schematic diagram of wound- rotor slip-ring induction generator

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(b) Steady-state equivalent circuit of WRSRIG

Figure 1. The proposed scheme of WRSRIG and steady-state equivalent circuit

2a.jpg

(a) D-axis model

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(b) Q-axis model

Figure 2. Modified equivalent circuit of SEIG in D-Q axes of synchronously rotating reference frame

In the proposed D-Q axes equivalent circuit shown in Figure 2, the modified series equivalent of stator and rotor iron core resistances (R_ssic & R_rsic) can be reflected as voltage drops in the stator and rotor circuits to take the effect of iron core losses without increasing the order of the system differential equations. These resistances as well as the equivalent dynamic inductance (L_m) can be calculated as [20]:

$R_{s s i c}=R_{s i c} \cdot\left(\omega_{S} \cdot L_{M}\right) /\left[R_{\text {sic }}^{2}+\left(\omega_{S} \cdot L_{M}\right)^{2}\right]$                 (10)

$R_{r s i c}=s . R_{s s i c}$              (11)

$L_{m}=L_{M} \cdot R_{\text {sic }}^{2} /\left[R_{\text {sic }}^{2}+\left(\omega_{s} \cdot L_{M}\right)^{2}\right]$              (12)

where, R_sic is the stator iron core resistance which can be calculated from the no-load test of the machine. L_m is the dynamic magnetizing inductance which can be found from the no-load test in terms of magnetizing current (I_m), represented by a polynomial curve-fitting technique as:

$L_{M}=0.04765 * I_{m}^{5}-0.5129 * I_{m}^{4}+2.18 * I_{m}^{3}-4.57 * I_{m}^{2}+4.636 * I_{m}+0.25$               (13)

The parameters of stator and rotor voltages of the WRSRIG with excitation capacitors derived from the proposed circuit are shown in Figure 2 as:

$v_{s d}=\left(R_{s}+R_{s s s \ell}\right) \cdot i_{s d}+\frac{d \psi_{s d}}{d t}+R_{s s i c} \cdot i_{m d}-\omega_{s} \cdot \psi_{s d}$               (14)

$v_{s q}=\left(R_{s}+R_{s s s \ell}\right) \cdot i_{s q}+\frac{d \psi_{s q}}{d t}+R_{s s i c} . i_{m q}+\omega_{s} . \psi_{s q}$                   (15)

$v_{c d}=\left(Ŕ_{r}+Ŕ_{r s s \ell}+Ŕ_{h}+Ŕ_{e f f}\right) í_{r d}+R_{r s i c} \cdot i_{m d}+\frac{d' \psi_{r d}}{t d}-\omega_{s \ell} \cdot \acute{\psi}_{r q}-\omega_{s \ell} \cdot \acute{\psi}_{r q o}$                  (16)

$v_{c q}=\left(Ŕ_{r}+Ŕ_{r s s \ell}+Ŕ_{h}+Ŕ_{e f f}\right) \cdot í_{r q}+R_{r s i c} \cdot i_{m q}+\frac{\acute{d} \psi_{r q}}{t d}+\omega_{s \ell} \cdot \acute{\psi}_{r d}+\omega_{s \ell} \cdot\acute{\psi}_{r d o}$                 (17)

where $\omega_{s \ell}$ is the slip speed $=\left(\omega_{s}-\omega_{r}\right) \mathrm{rad} . / \mathrm{sec} .$               (18)

where, ω_r is the rotor speed of the generator in electrical. rad./sec. ψ_sd, ψ_sq, $\acute{\psi}_ {rd}$and $\acute{\psi}_ {rq}$ are the stator and rotor flux linkage in d-q axes, and defined as:

$\psi_{s d}=L_{\ell s} \cdot i_{s d}+L_{m} \cdot i_{m d}$                    (19)

where $\omega_{s \ell}$ is the slip speed $=\left(\omega_{s }-\omega_{r}\right) \mathrm{rad} . / \mathrm{sec} .$                      (20)

$\acute{\psi}_{r d}=\acute{L}_{\ell r} \cdot i_{r d}+\acute{L}_{o \ell}\acute{i}_{r d}+L_{m} \cdot i_{m d}+\acute{\psi}_{r d o}$                     (21)

$\acute{\psi}_{r q}=\acute{L}_{\ell r} \cdot i_{r q}+\acute{L}_{o \ell} \cdot i_{r q}+L_{m} \cdot i_{m q}+\acute{\psi}_{r q o}$                     (22)

$i_{m d}=i_{s d}+\acute{l}_{r d}$                 (23)

$i_{m q}=i_{s q}+\acute{l}_{r q}$                 (24)

where, $\acute{\psi}_ {rdo}$ and $\acute{\psi}_ {rqo}$=the residual rotor linkage fluxes in the D and Q axes, respectively.

$\acute{v}_{c d}=\int\left(\frac{\acute{l}_{r d}}{C_{r e f}}+\omega_{s l} \cdot v_{c q}\right) d t+V_{c d o}$                (25)

$\acute{v}_{c q}=\int\left(\frac{\acute{l}_{r q}}{C_{r e f}}-\omega_{s l} \cdot v_{c d}\right) d t+V_{c q o}$                  (26)

where, $\acute{V}_ {cdo}$ and $\acute{V}_ {cqo}$ are the initial voltages of the rotor excitation capacitor in the D-Q synchronously rotating frame?

$i_{m}=\sqrt{i_{m d}^{2}+i_{m q}^{2}}$               (27)

$v_{c m}=\sqrt{\acute{v}_{c d}^{2}+\acute{v}_{c q}^{2}}$               (28)

where, i_m is the maximum magnetizing current and v_cm is the maximum capacitance voltage.

The electromagnetic torque generated by WRSRIG is calculated as:

$T_{e}=\left(\frac{3}{2}\right) \cdot\left(P_{p}\right) \cdot\left(L_{m}\right) \cdot\left(i_{s d} \cdot \acute{l}_{r q}-{i}_{s q} \cdot \acute{l}_{r d}\right)$                   (29)

The active output power of WRSRIG is calculated as:

$P_{t}=\frac{3}{2}\left[v_{s d} \cdot i_{s d}+v_{s q} \cdot i_{s q}\right]$                 (30)

The reactive output power can be calculated as:

$Q_{t}=\frac{3}{2}\left[v_{s q} \cdot i_{s d}-v_{s d} \cdot i_{s q}\right]$                (31)

The machine power factor can be calculated as:

$P \cdot F=\cos \phi=\frac{P_{t}}{\sqrt{P_{t}^{2}+Q_{t}^{2}}}$                   (32)

The minimum capacitance in the external rotor circuit is needed to generate the rated terminal voltage at no-load and rated machine speed conditions derived from the steady-state equivalent circuit of the WRSRIG in Figure 1 as:

$C_{r e f(\max )}=1 /\left[\omega_{r c}^{2}\left(L_{m}+L_{\ell r}+\acute{L}_{o \ell}\right)\right]$                (33)

$C_{r e f(\min )}=1 /\left[\omega_{r m}^{2} \cdot\left(L_{m}+L_{\ell r}+\acute{L}_{o \ell}\right)\right]$                   (34)

where, ω_rc=the machine cut-off or minimum rotor speed in (electrical rad./second).

ω_rm is the maximum (above synchronous speed) allowable machine speed in electrical radian/second.

The minimum, or maximum excitation capacitance depends on the maximum or minimum generator rotor speed (ω_r) in electrical rad./sec.

$\omega_{r c}=\omega_{s} \cdot\left(1-S_{m}\right)$                 (35)

$\omega_{r m}=\omega_{s} \cdot\left(1+S_{m}\right)$                 (36)

where, ω_s=the machine synchronous-speed or angular-frequency in electrical. rad./esc.

S_m=the machine slip at maximum torque or maximum power. The slip is positive when the machine rotates under synchronous speed, and it is negative when the machine rotates above synchronous speed.