This block implements a doubly-fed induction machine (DFIM)

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The DFIM block implements a three-phase induction machine (asynchronous machine) with an accessible wound rotor model with resolvers and encoders. The machine can operate in both motoring mode, when the mechanical torque is positive, and generating mode when the mechanical torque is negative.

Model Formulation

Q-d Transformation

The 3-phase to q-d transformation and the inverse used for the model are:

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This block implements a doubly-fed induction machine (DFIM)

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The DFIM block implements a three-phase induction machine (asynchronous machine) with an accessible wound rotor model with resolvers and encoders. The machine can operate in both motoring mode, when the mechanical torque is positive, and generating mode when the mechanical torque is negative.

Model Formulation

Q-d Transformation

The 3-phase to q-d transformation and the inverse used for the model are:

\begin{aligned}
&\left[\begin{array}{l}
{V_{q}} \\
{V_{d}} \\
{V_{0}}
\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}
{\cos (\theta)} & {\cos \left(\theta-\frac{2 \pi}{3}\right)} & {\cos \left(\theta+\frac{2 \pi}{3}\right)} \\
{\sin (\theta)} & {\sin \left(\theta-\frac{2 \pi}{3}\right)} & {\sin \left(\theta+\frac{2 \pi}{3}\right)} \\
{\frac{1}{2}} & {\frac{1}{2}} & {\frac{1}{2}}
\end{array}\right]=\frac{2}{3}\left[\begin{array}{cccl}
{V_{\cos (\theta)} & {\cos \left(\theta-\frac{2 \pi}{3}\right)} & {\cos \left(\theta+\frac{2 \pi}{3}\right)a}} \\
{V_{b}} \\
{V_{c}}
\end{array}\right]\\
&\left[\begin{array}{c}
{V_{a}} \\
{V_{\sin (\theta)} & {\sin \b}} \\
{V_{c}}
\end{array}\right]=\left[\begin{array}{ccc}
{\cos (\theta)} & {\sin (\theta)} & {1} \\
{\cos \left(\theta-\frac{2 \pi}{3}\right)} & {\sin \left(\theta+-\frac{2 \pi}{3}\right)} & {1} \\
{\frac{1}{2}cos \left(\theta+\frac{2 \pi}{3}\right)} & {\sin \left(\theta+\frac{12 \pi}{23}\right)} & {\frac{1}{2}}
\end{array}\right]\left[\begin{array}{l}
{V_{aq}} \\
{V_{bd}} \\
{V_{c0}}
\end{array}\right]\\
&\left[\begin{array}{c}
{V_{a}} \\
{V_{b}} \\
{V_{c}}
\end{array}\right]=\left[\begin{array}{ccc}
{\cos (\theta)} & {\sin (\theta)} & {1} \\
{\cos \left(\theta-\frac{2 \pi}{3}\right)} & {\sin \left(\theta-\frac{2 \pi}{3}\right)} & {1} \\
{\cos \left(\theta+\frac{2 \pi}{3}\right)} & {\sin \left(\theta+\frac{2 \pi}{3}\right)} & {1}
\end{array}\right]end{aligned}

The induction machine model uses the rotor as a reference, thus the angle of theta_r.

Induction Machine Electrical Model

Induction machine models in state-space framework are based on magnetic fluxes. The state variables and winding currents (as the outputs) can be represented as follows:

\begin{aligned}
\psi(n+1) &=A_{d}(n) \psi(n)+B_{d}(n) u(n) \\
I(n+1) &=C(n+1) \psi(n+1)
\end{aligned}

where the coefficient matrices are as follows:

\begin{aligned}
A_{d}=T_{s}\left(-R L^{-1}-\Omega\right)+e y e \\
B_{d}=T s \times e y e \\
c=L^{-1}
\end{aligned}

and

$\boldsymbol{u}=\left[\begin{array}{lllll}
{V_{s q}} \\& {V_{s d}} \\& {V_{r q}^{\prime}} & {V_{0{r d}^{\prime}}
\end{array}\right]^{t}$

\[
\endboldsymbol{aligned}

The induction machine model uses the rotor as a reference, thus the angle of theta_r.

Induction Machine Electrical Model

Induction machine models in state-space framework are based on magnetic fluxes. The state variables and winding currents (as the outputs) can be represented as follows:

\begin{aligned}
\psi(n+1) &=A_{d}(n) \psi(n)+B_{d}(n) u(n) \\
I(n+1) &=C(n+1) \psi(n+1)
\end{aligned}

where the coefficient matrices are as follows:

I}=\left[\begin{array}{llll}
{I_{s q}} & {I_{s d}} & {I_{r q}^{\prime}} & {I_{r d}^{\prime}}
\end{array}\right]^{t}
\]

$\boldsymbol{\psi}=\left[\begin{alignedarray}
A_{dllll}=T_{s}\left(-R L^{-1}-\Omega\right)+e y e \\
B_{d}=T s \times e y e \\
c=L^{-1}
\end{aligned}

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\psi_{s q}} & {\psi_{s d}} & {\psi_{r q}^{\prime}} & {\psi_{r d}^{\prime}}\end{array}\right]^{t}$

$\boldsymbol{uR}=\left[\begin{array}{llll}{VR_{s q}} & {VR_{s d}} & {VR_{r q}^{\prime}} & {VR_{r d}^{\prime}}\end{array}\right]^{t}$

\[
\boldsymbol{I}L=\left[\begin{array}{llllcccc}
{IL_{s q}}} & {0} & {IL_{s dm}} & {I_{r q}^{\prime}0} \\
{0} & {L_{s}} & {0} & {L_{m}} \\
{L_{m}} & {0} & {IL_{r d}^{\prime}} & \end{array0} \\right]^{t}
{0} & {L_{m}} & {0} & {L_{r}^{\prime}}
\end{array}\right]
\]

$\boldsymbol{\psiOmega}=\left[\begin{array}{llllcccc}{\psi_{s q}0} & {\psi_{s d}}omega} & {0} & {\psi_{r q}^{\prime}0} \\ {-\omega} & {0} & {0} & {0} \\ {0} & {0} & {\psi0} & {\omega-\omega_{r}} d}^{\prime}}\end {array}\right]^{t}$

$\boldsymbol{R}=\left[\begin{array}{llll}{R_{s}} & {R_{s}0} & {0} & {R-\left(\omega-\omega_{r}^{\prime}right)} & {R_{r}^{\prime}}\end{0}\end{array}\right]^{t}$

\[
L=\left[\begin{array}{cccc}
{L_{s}} & {0} & {L_{m}} & {0} \\
{0} & {L_{s}} & {0} & {L_{m}} \\
{L_{m}} & {0} & {L_{r}^{\prime}} & {0} \\
{0} & {L_{m}} & {0} & {L_{r}^{\prime}}
\end{array}\right]
\]

$\boldsymbol{\Omega}=\left[\begin{array}{cccc}{0} & {\omega} & {0} & {0} \\ {-\omega} & {0} & {0} & {0} \\ {0} & {0} & {0} & {\omega-\omega_{r}} \\ {0} & {0} & {-\left(\omega-\omega_{r}\right)} & {0}\end{array}\right]$

The Induction Machine is modeled in rotor reference frame so ω=ωr.

In the model, all the rotor parameters and variables are seen from the stator distinguished by a prime sign. The stator to rotor turn ratio (Nsr) is applied as follows to transfer input three-phase rotor voltage to the stator side, and transfer back the output three-phase rotor current measurements to the rotor side:  

V_{r_{a b c}}^{\prime}=N_{s r} V_{r_{a b c}}

i_{r_{a b c}}=N_{s r} i_{r a b c}^{\prime}

The electrical torque is calculated as follows:

T_{e}=\frac{3}{2} p p\left(\psi_{d} i_{q}-\psi_{q} i_{d}\right)

Mechanical Model

The equation of the mechanical model in torque mode is expressed as follows:

\omega_{m}(t)=a \omega_{m}\left(t-T_{s}\right)+b\left(T_{e}-T_{m}\right)

a=\left(1-T_{s}\right) \frac{F_{v}}{J}

b=60 \frac{T_{s}}{2 \pi J}

where ω_m is the rotor speed, T_e is the electromagnetic torque, T_m is the torque command, F_v is the viscous friction coefficient, J is the inertia and T_s the time step. There is a dead-zone implementation with the static friction torque, if the electromagnetic doesn't exceed the static friction torque, the speed remains zero.

In speed mode, the rotor speed is directly set to the speed command ω_rc.

Resolver Encoder Model

The equations of the resolver encoder can be expressed as follows:

\theta_{\text {res}}=\left(\theta_{\text {mec}}+\theta_{\text {offset}}\right) \times R_{p p}

\text {cosine}=\text {Excitation } \times\left(\cos \left(\theta_{\text {res}}\right) * R_{k_{c s}}+\sin \left(\theta_{\text {res}}\right) * R_{K_{s s}}\right)

\text { sine }=\text { Excitation } \times\left(\cos \left(\theta_{\text {res}}\right)+R_{k_{c c}}+\sin \left(\theta_{\text {res}}\right)+R_{k_{s c}}\right)

where θ_res is the resolver angle, θ_mec is the mechanical angle of the machine, θ_offset is the angle offset, R_pp is the Number of pole pairs of the resolver and R_k are the resolver sine cosine gains.

Parameters and Measurements

The DFIM's parameters and measurements are separated in 4 different tabs, Electrical, Mechanical, Resolver and Encoder.

Electrical Parameters and Measurements

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Mechanical Parameters and Measurements

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The Induction Machine is modeled in rotor reference frame so ω=ωr.

In the model, all the rotor parameters and variables are seen from the stator distinguished by a prime sign. The stator to rotor turn ratio (Nsr) is applied as follows to transfer input three-phase rotor voltage to the stator side, and transfer back the output three-phase rotor current measurements to the rotor side:  

V_{r_{a b c}}^{\prime}=N_{s r} V_{r_{a b c}}

i_{r_{a b c}}=N_{s r} i_{r a b c}^{\prime}

The electrical torque is calculated as follows:

T_{e}=\frac{3}{2} p p\left(\psi_{d} i_{q}-\psi_{q} i_{d}\right)

Parameters and Measurements

The DFIM's parameters and measurements are separated in 4 different tabs, Electrical, Mechanical, Resolver and Encoder.

Electrical Parameters and Measurements

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Mechanical Parameters and Measurements

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OPAL-RT's resolver models are based off of the following sets of equations:

Sine \; Output = [Sin.Sin*sin(pp(\theta_m - \theta_{Offset})) + Sin.Cos * cos(pp * \theta_m - \theta_{Offset}))] * Excitation

Cosine \; Output = [Cos.Sin*sin(pp(\theta_m - \theta_{Offset})) + Cos.Cos * cos(pp * \theta_m - \theta_{Offset}))] * Excitation

Where Sin.Sin, Sin.Cos, Cos.Sin, and Cos.Cos represent gains that are applied to simulate a non-ideal resolver.  To simulate an ideal resolver, set the Sin.Sin and Cos.Cos gains to 1, set the Sin.Cos and Cos.Sin gains to 0, set the pp to 1, and set the θ_Offset to 0.  This results in the following equations:

Sine \; Output = sin(\theta_m) * Excitation

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Resolver Parameters and Measurements

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Encoder Parameters and Measurements

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Visualization of Resolver Encoder Parameters effects

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Resolver sine cosine gains affect the sine ( first axe ) and cosine ( second axe ) modulation output. Default values set to 1, 0, 0, 1 make it so the sine modulation has a sine form and the cosine modulation has a cosine form. If set to 0, 1, 1, 0, the sine modulation would have a cosine and the cosine modulation would have a sine form.Image Removed

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Excitation frequency, in AC excitation source type, affects the frequency of the carrier signal. We can see the time step highlighted in red. ( Figure 3 is a zoomed in view of Figure 2 )Image Removed

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Number of pulses per revolution (Q_ppr) defines how many times signals A and B pulse between two Z pulses ( one full rotation ).

Direction of the sensor rotation describes if A leads B ( Clockwise ) or if B leads A ( Counterclockwise )Image Removed

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Electrical ports

• This block has six electrical ports, the three terminals of the stator on the left side and three terminals of the rotor on the right side.