6.2.1 RLC Circuit Impedance Model

Assume that the series-compensated network has no mutual inductance and the three phases are symmetrical. For such a three-phase RLC circuit, the impedance model observed in space vector can be expressed as

$\begin{array}{l}\hfill Z_{\text{net}}=\frac{\vec{V}}{\vec{I}}=\frac{v_a}{i_a}=R+sL+\frac{1}{sC}\end{array}$

6.2.2 Induction Machine Impedance Model

To find out the impedance model of a DFIG, we start to find out the relationship between the stator voltage and the stator current. Essentially, it is just to find the equivalent Thevenin impedance and the equivalent Thevenin voltage for the induction machine circuit in space vector (shown in Fig. 6.4).

A few steps can lead us to the stator voltage expression in terms of stator current and rotor voltage solely.

Observe the dynamic model in space vector:

$\begin{array}{ll}\hfill \overrightarrow{v_{\text{s}}}& =R_{\text{s}}\overrightarrow{i_{\text{s}}}+\frac{d\overrightarrow{{\psi }_{\text{s}}}}{dt}\hfill \end{array}$

$\begin{array}{ll}\hfill \overrightarrow{v_{\text{r}}}& =R_{\text{r}}\overrightarrow{i_{\text{r}}}+\frac{\text{d}}{\text{d}t}\overrightarrow{{\psi }_{\text{r}}}-j{\omega }_{\text{m}}\overrightarrow{{\psi }_{\text{r}}}\hfill \end{array}$

$\begin{array}{ll}\hfill \overrightarrow{{\psi }_{\text{s}}}& =L_{\text{s}}\overrightarrow{i_{\text{s}}}+L_{\text{m}}\overrightarrow{i_{\text{r}}}\hfill \end{array}$

$\begin{array}{ll}\hfill \overrightarrow{{\psi }_{\text{r}}}& =L_{\text{m}}\overrightarrow{i_{\text{s}}}+L_{\text{r}}\overrightarrow{i_{\text{r}}}.\hfill \end{array}$

We will first find the expression of $\overrightarrow{i_{\text{r}}}$ by $\overrightarrow{i_{\text{s}}}$ and $\overrightarrow{v_{\text{r}}}$ from the rotor voltage and rotor flux relationship, shown in (6.9). Then in the stator voltage and stator flux relationship, the rotor current will be replaced by the following expression:

$\begin{array}{ll}\hfill \overrightarrow{i_{\text{r}}}& =\frac{\overrightarrow{v_{\text{r}}}-(s-j{\omega }_{\text{m}})L_{\text{m}}\overrightarrow{i_{\text{s}}}}{R_{\text{r}}+(s-j{\omega }_{\text{m}})L_{\text{r}}}\hfill \end{array}$

The stator voltage can be expressed as follows:

$\begin{array}{l}\hfill \overrightarrow{v_{\text{s}}}(s)=\underset{Z}{\underbrace{\left[R_{\text{s}}+s\left(L_{\text{s}}-\frac{L_{\text{m}}^2[s-j{\omega }_{\text{m}})]}{R_{\text{r}}+[s-j{\omega }_{\text{m}})L_r]}\right)\right]}}{\vec{i}}_{\text{s}}+\frac{s{L}_{\text{m}}}{R_{\text{r}}+(s-j{\omega }_{\text{m}})L_r}{\vec{v}}_{\text{r}}\end{array}$

Note a space vector based on the stationary frame is related with a complex vector based on the qd reference frame (rotating speed is ω) is as follows:

$\begin{array}{l}\hfill \vec{f}={\stackrel{-}{F}}_{qds}{\text{e}}^{\text{j}\omega t}.\end{array}$

In the frequency domain, their relation becomes

$\begin{array}{l}\hfill \vec{f}(s)={\stackrel{-}{F}}_{qds}(s-j\omega )\vec{f}(s+j\omega )={\stackrel{-}{F}}_{qds}(s).\end{array}$

Therefore, in complex vector, the stator voltage is expressed by replacing s in (6.10) with s + jω.

$\begin{array}{ll}\hfill {\stackrel{-}{V}}_{\text{s}}& =\underset{Z}{\underbrace{\left[R_{\text{s}}+(s+j\omega )\left(L_{\text{s}}-\frac{L_{\text{m}}^2[s+j(\omega -{\omega }_{\text{m}})]}{R_{\text{r}}+[s+j(\omega -{\omega }_{\text{m}})L_r]}\right)\right]}}{Ī}_{\text{s}}\hfill \\ \hfill & +\frac{(s+j\omega )L_{\text{m}}}{R_{\text{r}}+[s+j(\omega -{\omega }_{\text{m}})]L_r}{\stackrel{-}{V}}_{\text{r}}\hfill \end{array}$

If L_m is very large compared to the other parameters, then the above equation can be simplified as

$\begin{array}{l}\hfill Z_{\text{s}}(s)=R_{\text{s}}+s(L_{\text{ls}}+L_{\text{lr}})+\frac{s}{s-j{\omega }_{\text{m}}}{R}_{\text{r}}.\end{array}$

The above derivation corroborates with the following analysis on slip. Slip is related to the rotating speed ω_m and the stator frequency ω. For the per-phase induction machine circuit shown in Fig. 6.5, the rotating speed can be assumed to be constant since mechanical dynamics is much slower than the electric dynamics. Slip can be expressed as 1 − ω_m/ω. In Laplace domain, slip can be expressed as

$\begin{array}{l}\hfill {}_{\text{slip}}(s)=\frac{s-j{\omega }_{\text{m}}}{s}.\end{array}$

Hence, the impedance of the DFIG seen from its terminal can be represented by

$\begin{array}{l}\hfill Z_{\text{DFIG}}(s)=R_{\text{r}}{/}_{\text{slip}}(s)+R_{\text{s}}+(L_{\text{ls}}+L_{\text{lr}})s.\end{array}$

The above derivation also proves that the impedance model can be developed using the per-phase circuit of a DFIG. The simplified DFIG impedance in (6.15) ignoring RSC and GSC is adopted in [4].

6.2.3 Converter Impedance Model

Cascaded control loops are used in converters in wind generators. The control loops consist of the fast inner current control loops and the slow outer power/voltage control loops. The current control loops usually have bandwidths at or greater than 100 Hz while the outer control loops usually have bandwidths less than several Hz. For SSR studies, the dynamics of interest are much faster than the outer control loops. Thus, the outer control is considered to be constant and will not be included in the impedance model.

For the vector current control scheme in Fig. 6.6, the qd-axis voltage and current relationship is

$\begin{array}{ll}\hfill v_{\text{q}}& =(i_{\text{q}}^{*}-i_{\text{q}})H_i(s)-K_{\text{d}}{i}_{\text{d}}\hfill \end{array}$

$\begin{array}{ll}\hfill v_{\text{d}}& =(i_{\text{d}}^{*}-i_{\text{d}})H_i(s)+K_{\text{d}}{i}_{\text{q}}\hfill \end{array}$

This leads to the expression in complex vector

$\begin{array}{l}\hfill {\stackrel{-}{V}}_{qds}={Ī}_{qds}^{*}H(s)-(H(s)-j{K}_{\text{d}}){Ī}_{qds}.\end{array}$

To view the current controlled converter from space vector, then

$\begin{array}{l}\hfill \vec{V}={\vec{I}}^{*}H(s-j\omega )-(H(s-j\omega )-j{K}_{\text{d}})\vec{I}\end{array}$

For the GSC at positive sequence scenarios, it can be represented by a voltage source ${\vec{I}}_{\text{g}}^{*}{H}_{\text{g}}(s-j\omega )$ behind an impedance Z_GSC, where Z_GSC = H_g(s − jω) − jK_dg.

For the RSC at positive sequence scenarios, it can also be represented by a voltage source ${\vec{I}}_{\text{r}}^{*}{H}_{\text{r}}(s-j\omega )$ behind an impedance Z_RSC where Z_RSC = H_r(s − jω) − jK_dr.

6.2.4 Overall Circuit

The overall circuit is now shown in Fig. 6.7. The circuit consists of two impedances: the network impedance Z_net and the DFIG impedance Z_DFIG.

When the gains of the RSC controller K_pK_i and the feed forward gain K_dr are assumed to be zero, the RSC output voltage will no longer vary even there exists error in measured currents and the reference currents. The RSC can be viewed as a constant voltage source in this case.

6.2.5 Comparison of Z_sr and Z_g

The DFIG impedance consists of two parallel components: the stator and rotor (including RSC) Z_sr and the GSC branch including the GSC and the filter: Z_g = Z_GSC + sL_tg. Bode plots of the RLC network impedances, Z_sr and Z_g at 95% rotating speed (or 57 Hz) and 50% compensation level are plotted in Fig. 6.8. Z_sr is plotted for two scenarios: with and without current PI controllers. When the PI controller gain is not zero, Z_sr has two peaks: one at 57 Hz and the other at 60 Hz. When the gain is zero, Z_sr however has just one peak at 57 Hz. For Z_g, the resonant frequency is 60 Hz. The network impedance has a resonant frequency at 37.5 Hz, which correspond to the 50% compensation level. The network resonant frequency is expressed as:

$\begin{array}{l}\hfill f_{\text{n}}=60\sqrt{\frac{X_c}{X_{\text{line}}+X_{\text{T}}}}\end{array}$

where X_c is the series capacitor reactance, X_line and X_T are the line reactance and the transformer reactance.

It is observed that at the interested SSR frequency region (<37.5 Hz), the magnitude of Z_g is much larger than Z_sr. Therefore, Z_sr is dominant. For the SSR analysis, the impact of Z_g will be ignored (Z_DFIG = Z_sr).

6.3.1 RLC Circuit

For a three-phase RLC circuit, the impedance model observed in space vector can be expressed as:

$\begin{array}{l}\hfill Z_{\text{line,p}}=Z_{\text{line,n}}=R+sL+\frac{1}{sC}\end{array}$

The above expression also fits (6.26). Since there is no imaginary part for the positive sequence expression, the negative sequence and the positive sequence expressions are the same.

6.3.2 Induction Machine Impedance Model

We already have the relationship between the stator voltage and stator current, rotor voltage in dq reference frame:

$\begin{array}{ll}\hfill {\stackrel{-}{V}}_{\text{s}}=& \underset{Z}{\underbrace{\left[R_{\text{s}}+(s+j\omega )\left(L_{\text{s}}-\frac{L_{\text{m}}^2[s+j(\omega -{\omega }_{\text{m}})]}{R_{\text{r}}+[s+j(\omega -{\omega }_{\text{m}})L_r]}\right)\right]}}{Ī}_{\text{s}}\hfill \\ \hfill & +\frac{(s+j\omega )L_{\text{m}}}{R_{\text{r}}+[s+j(\omega -{\omega }_{\text{m}})]L_r}{\stackrel{-}{V}}_{\text{r}}.\hfill \end{array}$

Therefore, a Thevenin equivalent circuit for the induction machine can be developed. Notice that the impedance Z(s) is based on the dq-reference frame. Since the RLC circuit is expressed in the abc reference frame, it will be convenient that there is also an impedance model based on the abc reference frame.

In space vector the stator voltage can be expressed as follows by replacing s by s − jω for positive sequence.

Positive sequence:

$\begin{array}{ll}\hfill \vec{V}(s)& ={\stackrel{-}{V}}_{\text{s}}(s-j\omega )\hfill \\ \hfill & =\underset{Z}{\underbrace{\left[R_{\text{s}}+s\left(L_{\text{s}}-\frac{L_{\text{m}}^2(s-j{\omega }_{\text{m}})}{R_{\text{r}}+(s-j{\omega }_{\text{m}})L_r]}\right)\right]}}\overrightarrow{I_{\text{s}}}+\frac{s{L}_{\text{m}}}{R_{\text{r}}+(s-j{\omega }_{\text{m}})L_r}\overrightarrow{V_{\text{r}}}\hfill \end{array}$

Therefore, the positive sequence impedance model is

$\begin{array}{ll}\hfill Z_{\text{DFIG,p}}& =R_{\text{s}}+s\left(L_{\text{ls}}+L_{\text{m}}-\frac{L_{\text{m}}^2}{R_{\text{r}}s-j{\omega }_{\text{m}}+L_{\text{lr}}+L_{\text{m}}}\right)\hfill \\ \hfill & =R_{\text{s}}+s{L}_{\text{ls}}+s{L}_{\text{m}}\frac{\frac{R_{\text{r}}}{s-j{\omega }_{\text{m}}}+L_{\text{lr}}}{R_{\text{r}}s-j{\omega }_{\text{m}}+L_{\text{lr}}+L_{\text{m}}}\hfill \end{array}$

The negative sequence impedance model is

$\begin{array}{ll}\hfill Z_{\text{DFIG,n}}=Z_{\text{DFIG,p}}^{*}& =R_{\text{s}}+s\left(L_{\text{ls}}+L_{\text{m}}-\frac{L_{\text{m}}^2}{R_{\text{r}}s+j{\omega }_{\text{m}}+L_{\text{lr}}+L_{\text{m}}}\right)\hfill \\ \hfill & =R_{\text{s}}+s{L}_{\text{ls}}+s{L}_{\text{m}}\frac{\frac{R_{\text{r}}}{s+j{\omega }_{\text{m}}}+L_{\text{lr}}}{R_{\text{r}}s+j{\omega }_{\text{m}}+L_{\text{lr}}+L_{\text{m}}}\hfill \end{array}$

6.3.3 Converter Impedance Model

Cascaded control loops are used in converters in wind generators. The control loops consist of the fast inner current control loops and the slow outer power/voltage control loops. The current control loops usually have bandwidths at or greater than 100 Hz while the outer control loops usually have bandwidths less than several Hz. For SSR studies, the study dynamics is considered faster than the outer control loops. Thus, the outer control is considered to be constant and will not be included in the impedance model.

For the vector current control scheme in Fig. 6.6, the qd-axis voltage and current relationship is

$\begin{array}{ll}\hfill v_{\text{q}}& =(i_{\text{q}}^{*}-i_{\text{q}})H_i(s)-K_{\text{d}}{i}_{\text{d}}\hfill \end{array}$

$\begin{array}{ll}\hfill v_{\text{d}}& =(i_{\text{d}}^{*}-i_{\text{d}})H_i(s)+K_{\text{d}}{i}_{\text{q}}\hfill \end{array}$

This leads to the expression in complex vector:

$\begin{array}{l}\hfill {\stackrel{-}{V}}_{qd}={Ī}_{qd}^{*}{H}_i(s)-(H_i(s)-j{K}_{\text{d}}){Ī}_{qd}.\end{array}$

To view the current controlled converter from space vector, then:

$\begin{array}{l}\hfill \left\{\begin{array}{l}\text{Positive:}\vec{V}={\vec{I}}^{*}{H}_i(s-j\omega )-(H_i(s-j\omega )-j{K}_{\text{d}})\vec{I}\\ \text{Negative:}\vec{V}={\vec{I}}^{*}{H}_i(s+j\omega )-(H_i(s+j\omega )+j{K}_{\text{d}})\vec{I}\end{array}\right.\end{array}$

For the GSC converter at positive sequence scenarios, it can be represented by a voltage source ${\vec{I}}_{\text{g}}^{*}{H}_{\text{g}}(s-j\omega )$ behind an impedance Z_GSC where Z_GSC = H_g(s − jω) − jK_dg. At negative sequence scenarios, the GSC converter can be represented by a voltage source ${\vec{I}}_{\text{g}}^{*}{H}_{\text{g}}(s+j\omega )$ behind an impedance Z_GSC where Z_GSC = H_g(s + jω) + jK_dg.

For the RSC converter at positive sequence scenarios, it can also be represented by a voltage source ${\vec{I}}_{\text{r}}^{*}{H}_{\text{r}}(s-j\omega )$ behind an impedance Z_RSC where Z_RSC = H_r(s − jω) − jK_dr. The negative sequence impedance of the RSC is the conjugate of Z_RSC and can be expressed as H_r(s + jω) + jK_dr.

6.3.4 Overall Circuit

The overall circuit for negative sequence is now shown in Fig. 6.9.

6.4.1 Mechanical System

A two-mass system [6, 7] is used to the represent torsional dynamics given by

$\begin{array}{l}\hfill \left[\begin{array}{l}\mathrm{\Delta }\dot{{\omega }_t}\\ \mathrm{\Delta }\dot{{\omega }_{\text{m}}}\\ \dot{T_{12}}\end{array}\right]=\left[\begin{array}{lll}\frac{-D_t-D_{tg}}{2H_t}& \frac{D_{tg}}{2H_t}& \frac{-1}{2H_t}\\ \frac{D_{tg}}{2H_{\text{g}}}& \frac{-D_{\text{g}}-D_{tg}}{2H_{\text{g}}}& \frac{1}{2H_{\text{g}}}\\ K_{tg}{\omega }_{\text{e}}& -K_{tg}{\omega }_{\text{e}}& 0\end{array}\right]\left[\begin{array}{l}\mathrm{\Delta }{\omega }_t\\ \mathrm{\Delta }{\omega }_{\text{m}}\\ T_{12}\end{array}\right]+\left[\begin{array}{l}\frac{T_{\text{m}}}{2H_t}\\ \frac{-T_{\text{e}}}{2H_{\text{g}}}\\ 0\end{array}\right].\end{array}$

where ω_t and ω_r are the turbine and generator rotor speed, respectively; P_m and P_e are the mechanical power of the turbine and the electrical power of the generator, respectively; T₁₂ is an internal torque of the model; H_t and H_g are the inertia constants of the turbine and the generator, respectively; D_t and D_g are the mechanical damping coefficients of the turbine and the generator, respectively; D_tg is the damping coefficient of the flexible coupling (shaft) between the two masses; K_tg is the shaft stiffness.

From the two-mass mechanical system model, the transfer functions $G_{m1}=\frac{\mathrm{\Delta }{\omega }_{\text{m}}}{\mathrm{\Delta }{T}_{\text{m}}}$ and $G_{\text{m}}=\frac{\mathrm{\Delta }{\omega }_{\text{m}}}{\mathrm{\Delta }{T}_{\text{e}}}$ can be found.

6.4.2 Electrical System

Torque-speed transfer function has been used in [8] to examine torsional interactions. The voltage and current relationship for an induction machine (2.53) in the dq reference frame are linearized. These equations are presented again here.

$\begin{array}{ll}\hfill & \left[\begin{array}{l}{v}_{qs}\\ v_{ds}\\ v_{0s}\\ v_{qr}\\ v_{dr}\\ v_{0r}\end{array}\right]=\hfill \\ \hfill & \left[\begin{array}{llllll}{R}_{\text{s}}+\frac{p}{{\omega }_{\text{b}}}{X}_{\text{s}}& \frac{{\omega }_{\text{s}}}{{\omega }_{\text{b}}}{X}_{\text{s}}& 0& \frac{p}{{\omega }_{\text{b}}}{X}_{\text{m}}& \frac{{\omega }_{\text{s}}}{{\omega }_{\text{b}}}{X}_{\text{m}}& 0\\ -\frac{{\omega }_{\text{s}}}{{\omega }_{\text{b}}}{X}_{\text{s}}& R_{\text{s}}+\frac{p}{{\omega }_{\text{b}}}{X}_{\text{s}}& 0& -\frac{{\omega }_{\text{s}}}{{\omega }_{\text{b}}}{X}_{\text{m}}& \frac{p}{{\omega }_{\text{b}}}{X}_{\text{m}}& 0\\ 0& 0& R_{\text{s}}+\frac{p}{{\omega }_{\text{b}}}{X}_{\text{ls}}& 0& 0& 0\\ -\frac{p}{{\omega }_{\text{b}}}{X}_{\text{m}}& \frac{{\omega }_{\text{s}}-{\omega }_{\text{m}}}{{\omega }_{\text{b}}}{X}_{\text{m}}& 0& R_{\text{r}}+\frac{p}{{\omega }_{\text{b}}}{X}_r& \frac{{\omega }_{\text{s}}-{\omega }_{\text{m}}}{{\omega }_{\text{b}}}{X}_r& 0\\ -\frac{{\omega }_{\text{s}}-{\omega }_{\text{m}}}{{\omega }_{\text{b}}}{X}_{\text{m}}& \frac{p}{{\omega }_{\text{b}}}{X}_{\text{m}}& 0& -\frac{{\omega }_{\text{s}}-{\omega }_{\text{m}}}{{\omega }_{\text{b}}}{X}_r& R_{\text{r}}+\frac{p}{{\omega }_{\text{b}}}{X}_r& 0\\ 0& 0& 0& 0& 0& R_{\text{r}}+\frac{p}{{\omega }_{\text{b}}}{X}_{\text{lr}}\end{array}\right]\hfill \\ \hfill & \left[\begin{array}{l}{i}_{ds}\\ i_{qs}\\ i_{0s}\\ i_{qr}\\ i_{dr}\\ i_{0r}\end{array}\right]\hfill \end{array}$

where ω_b = 377 rad/s and the rest variables are in per unit.

The resulting linearized differential equations are as follows:

$\begin{array}{l}\hfill \frac{\text{d}}{\text{d}t}\mathrm{\Delta }{\text{i}}_{sr}=A_1\mathrm{\Delta }{\text{i}}_{sr}+A_2\mathrm{\Delta }{\omega }_{\text{m}}+B_1\mathrm{\Delta }{\text{v}}_{\text{s}}+B_2\mathrm{\Delta }{\text{v}}_{\text{r}}\end{array}$

where i_sr = [i_qs,i_ds,i_qr,i_dr]^T, v_s = [v_qs,v_ds]^T and v_r = [v_qr,v_dr]^T,

$\begin{array}{l}\hfill A_1=-{\left(\frac{1}{{\omega }_{\text{b}}}\left[\begin{array}{llll}\hfill X_{\text{s}}\hfill & \hfill 0\hfill & \hfill X_{\text{m}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill X_{\text{s}}\hfill & \hfill 0\hfill & \hfill X_{\text{m}}\hfill \\ \hfill X_{\text{m}}\hfill & \hfill 0\hfill & \hfill X_{\text{r}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill X_{\text{m}}\hfill & \hfill 0\hfill & \hfill X_{\text{r}}\hfill \\ \hfill \hfill \end{array}\right]\right)}^{-1}\left[\begin{array}{llll}\hfill R_{\text{s}}\hfill & \hfill X_{\text{s}}\hfill & \hfill 0\hfill & \hfill X_{\text{m}}\hfill \\ \hfill -X_{\text{s}}\hfill & \hfill R_{\text{s}}\hfill & \hfill -X_{\text{m}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill s{X}_{\text{m}}\hfill & \hfill R_{\text{r}}\hfill & \hfill s{X}_{\text{r}}\hfill \\ \hfill -s{X}_{\text{m}}\hfill & \hfill 0\hfill & \hfill -s{X}_{\text{r}}\hfill & \hfill R_{\text{r}}\hfill \end{array}\right],\end{array}$

$\begin{array}{l}\hfill A_2=-{\left(\frac{1}{{\omega }_{\text{b}}}\left[\begin{array}{llll}\hfill X_{\text{s}}\hfill & \hfill 0\hfill & \hfill X_{\text{m}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill X_{\text{s}}\hfill & \hfill 0\hfill & \hfill X_{\text{m}}\hfill \\ \hfill X_{\text{m}}\hfill & \hfill 0\hfill & \hfill X_{\text{r}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill X_{\text{m}}\hfill & \hfill 0\hfill & \hfill X_{\text{r}}\hfill \\ \hfill \hfill \end{array}\right]\right)}^{-1}\left[\begin{array}{llll}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -X_{\text{m}}\hfill & \hfill 0\hfill & \hfill -X_{\text{r}}\hfill \\ \hfill X_{\text{m}}\hfill & \hfill 0\hfill & \hfill X_{\text{r}}\hfill & \hfill 0\hfill \end{array}\right],\end{array}$

$\begin{array}{l}\hfill B_1=\frac{{\omega }_{\text{b}}}{\sigma X_{\text{s}}{X}_{\text{r}}}\left[\begin{array}{ll}\hfill X_{\text{r}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill X_{\text{r}}\hfill \\ \hfill -X_{\text{m}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -X_{\text{m}}\hfill \end{array}\right]\end{array}$

where $\sigma =1-\frac{X_{\text{m}}^2}{X_{\text{s}}{X}_{\text{r}}}$,

$\begin{array}{l}\hfill B_2=\frac{{\omega }_{\text{b}}}{\sigma X_{\text{s}}{X}_{\text{r}}}\left[\begin{array}{ll}\hfill -X_{\text{m}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -X_{\text{m}}\hfill \\ \hfill X_{\text{s}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill X_{\text{s}}\hfill \end{array}\right].\end{array}$

In Laplace domain, the above equation becomes:

$\begin{array}{l}\hfill (sI-A_1)\mathrm{\Delta }{\text{i}}_{sr}=A_2\mathrm{\Delta }{\omega }_{\text{m}}+B_1\mathrm{\Delta }{\text{v}}_{\text{s}}+B_2\mathrm{\Delta }{\text{v}}_{\text{r}}\end{array}$

Consider the current control loops in the RSC, the RSC voltage can be expressed as:

$\begin{array}{l}\hfill \mathrm{\Delta }{\text{v}}_{\text{r}}=H_{\text{RSC}}(s)\mathrm{\Delta }{\text{i}}_{\text{r}}^{*}-Z_{\text{RSC}}\mathrm{\Delta }{\text{i}}_{\text{r}}\end{array}$

If we assume that the converter outer control loops can be ignored, then the reference signal ${\text{i}}_{\text{r}}^{*}$ are constants and its deviation is zero.

Substituting Δv_r′ in (6.43) by (6.44) leads to:

$\begin{array}{ll}\hfill \mathrm{\Delta }{\text{i}}_{sr}& ={(sI-A_1+B_2\left[\begin{array}{ll}\hfill \text{0}\hfill & \hfill Z_{\text{RSC}}\hfill \end{array}\right])}^{-1}(A_2\mathrm{\Delta }{\omega }_{\text{m}}+B_1\mathrm{\Delta }{v}_{\text{s}})\hfill \\ \hfill & =G_{vi}\mathrm{\Delta }{v}_{\text{s}}+G_{\omega i}\mathrm{\Delta }{\omega }_{\text{m}}.\hfill \end{array}$

It is obvious to find the expression of Δi_s in terms of the rotating speed and the stator voltage:

$\begin{array}{l}\hfill \mathrm{\Delta }{\text{i}}_{\text{s}}=G_{vis}\mathrm{\Delta }{\text{v}}_{\text{s}}+G_{\omega is}\mathrm{\Delta }{\omega }_{\text{m}}.\end{array}$

The expressions can be developed for both positive and negative sequences. For positive sequence, the DQ reference frame (dq⁺) is rotating counter clockwise at nominal frequency ω. While for negative sequence, the DQ reference frame is rotating clockwise at the nominal frequency ω. The major differences in expressions are A₁ and Z_RSC. For positive sequence, since the vector control is based on the same reference frame, $Z_{\text{RSC}}^{+}=H(s)=K_{\text{p}}+K_i/s$. However, for negative sequence, since the vector control is based on dq⁺ while the model is based on dq⁻, the impedance of the RSC should be converted to qd⁻. Hence $Z_{\text{RSC}}^{-}=H(s-j2\omega )=K_{\text{p}}+K_i/(s-j2\omega )$.

$\begin{array}{l}\hfill \left\{\begin{array}{l}\mathrm{\Delta }{i}_{s,p}^{+}=G_{vis}^{+}\mathrm{\Delta }{v}_{s,p}+G_{\omega is,p}^{+}\mathrm{\Delta }{\omega }_{\text{m}}\\ \mathrm{\Delta }{i}_{s,n}^{-}=G_{vis}^{-}\mathrm{\Delta }{v}_{s,n}+G_{\omega is,n}^{-}\mathrm{\Delta }{\omega }_{\text{m}}\end{array}\right.\end{array}$

where the superscripts “+” and “−” represent the dq + and dq − rotating reference frames.

The electromagnetic torque of an induction machine can be expressed by the stator and the rotor currents.

$\begin{array}{l}\hfill T_{\text{e}}=\frac{X_M}{2}(i_{qs}{i}_{dr}-i_{ds}{i}_{qr}).\end{array}$

When there is only positive sequence, T_e is a DC variable. When there is only negative sequence, T_e is also a DC variable. To evaluate the impact of positive and negative sequence components on the DC component of T_e, the following definitions are made:

$\begin{array}{l}\hfill \left\{\begin{array}{l}{T}_{e,dc1}=\frac{X_M}{2}(i_{qs,p}^{+}{i}_{dr,p}^{+}-i_{ds,p}^{+}{i}_{qr,p}^{+})\\ T_{e,dc2}=\frac{X_M}{2}(i_{qs,n}^{-}{i}_{dr,n}^{-}-i_{ds,n}^{-}{i}_{qr,n}^{-})\end{array}\right.\end{array}$

Note that the positive sequence components in the dq + frame are DC at steady-state while the negative-sequence components in the dq − frame are DC at steady-state.

The linearized expressions are as follows:

$\begin{array}{ll}\hfill \mathrm{\Delta }{T}_{e,dc1}& =\frac{X_M}{2}\left[\begin{array}{llll}\hfill i_{dr,p}^{+}\hfill & \hfill -i_{qr,p}^{+}\hfill & \hfill -i_{ds,p}^{+}\hfill & \hfill i_{qs,p}^{+}\hfill \end{array}\right]\mathrm{\Delta }{\text{i}}_{sr,p}^{+}=G_{vt}^{+}\mathrm{\Delta }{v}_{s,p}^{+}+G_{\omega t}^{+}\mathrm{\Delta }{\omega }_{\text{m}}\hfill \end{array}$

$\begin{array}{ll}\hfill \mathrm{\Delta }{T}_{e,dc2}& =\frac{X_M}{2}\left[\begin{array}{llll}\hfill i_{dr,n}^{-}\hfill & \hfill -i_{qr,n}^{-}\hfill & \hfill -i_{ds,n}^{-}\hfill & \hfill i_{qs,n}^{-}\hfill \end{array}\right]\mathrm{\Delta }{\text{i}}_{sr,n}^{-}=G_{vt}^{-}\mathrm{\Delta }{v}_{s,n}^{-}+G_{\omega t}^{-}\mathrm{\Delta }{\omega }_{\text{m}}\hfill \end{array}$

In addition, the stator voltage can be expressed by the stator current through the network equation if the current through the GSC is ignored:

$\begin{array}{l}\hfill \left\{\begin{array}{l}\mathrm{\Delta }{v}_{s,p}^{+}=-Z_{\text{line}}^{+}\mathrm{\Delta }{i}_{s,p}^{+}\\ \mathrm{\Delta }{v}_{s,n}^{-}=-Z_{\text{line}}^{-}\mathrm{\Delta }{i}_{s,n}^{-}\end{array}\right.\end{array}$

where

$\begin{array}{l}\hfill \left\{\begin{array}{l}{Z}_{\text{line}}^{+}=\left[\begin{array}{ll}R+sL+\frac{s}{(s^2+{\omega }^2)C}& \omega L-\frac{\omega }{(s^2+{\omega }^2)C}\\ -\omega L+\frac{\omega }{(s^2+{\omega }^2)C}& R+sL+\frac{s}{(s^2+{\omega }^2)C}\end{array}\right]\\ Z_{\text{line}}^{-}=\left[\begin{array}{ll}R+sL+\frac{s}{(s^2+{\omega }^2)C}& -\omega L+\frac{\omega }{(s^2+{\omega }^2)C}\\ \omega L-\frac{\omega }{(s^2+{\omega }^2)C}& R+sL+\frac{s}{(s^2+{\omega }^2)C}\end{array}\right]\end{array}\right.\end{array}$

According to (6.51), (6.47) and (6.52), the torque versus the speed transfer functions G_e(s) can be found:

$\begin{array}{ll}\hfill G_e^{+}(s)& =-G_{vt}^{+}{(I+Z_{\text{line}}^{+}{G}_{vis}^{+})}^{-1}{Z}_{\text{line}}^{+}{G}_{wis}^{+}+G_{wt}^{+}\hfill \\ \hfill G_e^{-}(s)& =-G_{vt}^{-}{(I+Z_{\text{line}}^{-}{G}_{vis}^{-})}^{-1}{Z}_{\text{line}}^{-}{G}_{wis}^{-}+G_{wt}^{-}\hfill \end{array}$

So far, we have given the torque/speed diagram’s transfer functions block by block. The model will be useful to investigate torsional interactions. The derived model has included the effect of DFIG and its rotor converter control.

In this example, Nyquist criterion based stability analysis will be carried out to study the effect on SSR due to wind speed, compensation degree and RSC PI controller gain. The expression of Z_sr can be found from the overall circuit shown in Fig. 6.7. The open-loop transfer function to be studied is Z_net/Z_sr and its expression is as follows:

$\begin{array}{l}\frac{Z_{\mathrm{n}\mathrm{e}\mathrm{t}}}{Z_{\mathit{sr}}}=\frac{r+sL+\frac{1}{sC}}{R_s+s{L}_{\mathrm{l}\mathrm{s}}+\frac{s{L}_m\left(s{L}_{\mathrm{l}\mathrm{r}}+\frac{s}{s-j{\omega }_m}\left(R_{\mathrm{r}}+K_{\mathrm{p}}+\frac{K_i}{s-j\omega }\right)-j{K}_{dr}\right)}{s{L}_m(s{L}_{\mathrm{l}r}+\frac{s}{s-j{\omega }_m}\left(R_{\mathrm{r}}+K_{\mathrm{p}}+\frac{K_i}{s-j\omega }\right)-j{K}_{dr}}}\\ \end{array}$

6.5.1.1 Effect of Wind Speed

Using the same study system and same parameters as in [9], two impedances are derived and the Nyquist map of Z_net/Z_sr is plotted in Fig. 6.11. The gains of the RSC controllers are set to zeros. The compensation level is 75%. It is shown that when the rotating speed at 0.75 pu, the Nyquist map will encircle (−1,0) which indicates instability. When the rotating speed is at 0.85 and 0.95 pu, the Nyquist maps will not encircule (−1,0) and the system is marginally stable. The phase margins are 2° and 5° for the 0.85 and 0.95 pu rotating speed. The points when the Nyquist maps intersect with the unit circuit have frequencies of about 40 Hz. This indicates the resonance frequency. It also corresponds to the LC resonant frequency at 75% compensation level.

The Nyquist map demonstrates the effect of wind speed on SSR stability since low wind speed corresponds to low rotating speed. Hence a Type-3 wind generator is more prone to SSR when wind speed is lower.

6.5.1.2 Effect of Compensation Degree

The Bode plots of the network impedance Z_net at different compensation degrees and the DFIG impedance Z_sr are plotted in Fig. 6.12. The rotating speed is assumed to be 0.75 pu. It is observed that a higher compensation degree results in a higher frequency at which Z_sr = Z_net: 251 rad/s (40 Hz) at 75% versus 206 rad/s (32.8 Hz) at 50%. It is also observed that the phase angle of Z_sr at that frequency is 107° at 75% compensation level compared to 95° at 50% compensation level. Therefore, Z_net/Z_sr will have a less phase margin or become unstable when the compensation level is higher.

The Nyquist maps of the network Z_net and the DFIG impedance Z_sr at different compensation degrees are plotted in Fig. 6.13. The points where the Nyquist maps intersects the unit circle are at 33 Hz for 50% compensation level (phase margin (5°) and 40 Hz for 75% compensation level. The Nyquist maps indicate that (-1,0) is encircled at 75% compensation level. Hence, a higher compensation degree results in a less stable system.

6.5.1.3 Effect of RSC PI Controller Gain

In this section, the effect of the RSC PI controller gain is examined. The time constant of the PI controller is fixed at 0.02 s. The wind speed is selected as 9 m/s and the corresponding rotating speed is 0.95 pu. The compensation level is 50%. Two cases are examined.

• Case 1 K_p = 0

• Case 2 K_p = 0.01

The Nyquist plots are shown in Figs. 6.14 and 6.15. It is observed that when the gain is larger or equal 0.01, the Nyquist maps encircle (−1,0) clockwise, which indicates instability. The point when the curve traverses the real axis is at 35 Hz. When the gain is zero, (−1,0) is not encircled and the phase margin is 11° at 33 Hz.

Bode plots of the impedances Z_net and Z_sr are shown in Fig. 6.16. It is shown that a higher current controller gain increases the magnitude of Z_sr and leads to a lower frequency at Z_net = Z_sr. The phase angle of the impedances at unit gain of Z_net/Z_sr are recorded in Table 6.1.

Table 6.1

Phase Angles of the Impedances and Stability Detection

	Kp = 1	Kp = 0.1	Kp = 0.01	Kp = 0
Res. freq. (Hz)	20	32	35	33
$\angle Z_{\text{net}}$	− 88°	− 81.3°	− 77°	− 77°
$\angle Z_{sr}$	161°	149°	99.4°	91.4°
$\angle Z_{\text{net}}/Z_{sr}$	111°	130°	− 178°	− 168.4°
Stability	N	N	Marginal	stable

It is found that a higher gain leads to less phase margin or instability.

6.5.1.4 Validation Via Simulation

The nonlinear model presented in Chapter 5 has been built in Matlab/ Simulink. The impact of wind speed and the RSC current loop control on SSR have also been discussed. It is observed that at lower wind speed, with high compensation level, the system is subject to SSR. In this simulation study, controller interaction between the RSC’s current control and the electrical network will be verified. In [10], such phenomenon is classified as subsynchronous controller interaction (SSCI). The case study scenarios will be setup with wind speed at 9 m/s and the rotating speed at 95% of the nominal. The compensation level at 50%. The RSC control gain will be varied to show the impact of controller interaction.

• Case 1 K_p = 0.0

• Case 2 K_p = 0.01

The system is initially operated at 25% compensation level. At t = 1 s, additional capacitors are switched in to make the compensation level reach 50%. Figures 6.17–6.19 show the dynamic responses of the system with thick lines refer to Case 1 while the thin lines refer to Case 2. The dynamic responses of the electromagnetic torque, the rotating speed of the wind turbine and the terminal voltage of the DFIG wind farm are shown Fig. 6.17. It is found that there is a resonance at about 25 Hz in the voltage magnitude and torque. For Case 1 the resonance can be damped out in the torque and the terminal voltage after 1 s. However, for case 2, the resonance cannot be damped out after 1 s.

Figure 6.18 shows the dynamic responses of the DFIG wind farm output power P_e and Q_e, the voltage across the capacitors V_c and the current through the line I_e. The resonance is damped out in Case 1 while persists in Case 2.

Figure 6.19 shows the dynamic responses of the rotating speeds of two masses ω_t and ω_m, the torque T₁₂ between the two masses of the wind turbine and the DC-link voltage V_dc. The resonance is more obvious in Case 2. In addition, power/torque imbalance is observed that the rotating speed and the DC-link voltage keep decreasing.

According to the Nyquist/Bode analysis in Section III, the resonance is 35 Hz at 50% compensation level and 0.95 pu rotating speed. This resonance reflected in power, torque and voltage/current magnitude will have a frequency of 25 Hz. The analysis in Fig. 6.15 also shows marginal stability or instability when the gain K_p is 0.01. Simulation results demonstrate the effect of current controller gain.

6.5.2 Example 2: Impact of Unbalance Operation on SSR

The bode plots of the negative sequence impedances are shown in Fig. 6.20. Two network impedances are presented: one is at 25% compensation level while the other is at 70% compensation level. Two DFIG impedances are presented: one is at 75% nominal speed while the other is at 95% nominal speed. From the Bode plots, the two Bode plots almost match each other. Therefore wind speed has negligible impact on the DFIG impedance. The network impedance magnitudes and the DFIG impedance magnitudes meet at 150 rad/s (24 Hz, 25% compensation level) and 250 rad/s (40 Hz, 75% compensation level). There are phase margin for Z_line/Z_DFIG. Hence the negative sequence impedance will not cause SSR instability.

A comparison of the positive and negative sequence impedances of the DFIG is shown in Fig. 6.21. From this figure, it is obvious that the positive sequence impedance of the DFIG could be greater than 90° at the gain cross frequencies. This indicates that it is the positive sequence impedance that could cause SSR instability.