Damping Torque Analysis of Small-Signal Angular Stability of a Power System Affected by Grid-Connected Wind Power Induction Generators

Abstract

Induction generator based wind power generation has been dominating the wind market since the rise of wind power industry at the end of last century and will be continuously in a favorable position for large-scale grid connection given its lower cost and more mature technology compared with other wind generation for the foreseeable future [1]. Fixed-speed induction generator (FSIG-Type 1 Wind Gen Model) and doubly-fed induction generator (DFIG-Type 3 Wind Gen Model) are two main types of induction generator adopted for wind power generation especially considering the fact that DFIG is the most frequently-used technology to date.

Appendices

Appendix 3.1: A Typical Example of a SMIB System with Interface Equations of a WPIG

From Fig. 3.11, it can obtain

$$ {\overline{\mathrm{I}}}_{\mathrm{L}\mathrm{b}}={\overline{\mathrm{I}}}_{\mathrm{t}\mathrm{L}}+{\overline{\mathrm{I}}}_{\mathrm{w}}={\overline{\mathrm{I}}}_{\mathrm{t}\mathrm{L}}+\frac{{\overline{\mathrm{V}}}_{\mathrm{w}}-{\overline{\mathrm{V}}}_{\mathrm{L}}}{{\mathrm{jX}}_{\mathrm{w}\mathrm{L}}}={\overline{\mathrm{I}}}_{\mathrm{t}\mathrm{L}}+\frac{{\overline{\mathrm{V}}}_{\mathrm{w}}-{\overline{\mathrm{V}}}_{\mathrm{t}}+{\mathrm{jX}}_{\mathrm{t}\mathrm{L}}{\overline{\mathrm{I}}}_{\mathrm{t}\mathrm{L}}}{{\mathrm{jX}}_{\mathrm{w}\mathrm{L}}} $$

(3.34)

$$ {\overline{\mathrm{V}}}_{\mathrm{t}}={\mathrm{jX}}_{\mathrm{t}\mathrm{L}}{\overline{\mathrm{I}}}_{\mathrm{t}\mathrm{L}}+{\mathrm{jX}}_{\mathrm{Lb}}{\overline{\mathrm{I}}}_{\mathrm{Lb}}+{\overline{\mathrm{V}}}_{\mathrm{b}} $$

(3.35)

Fig. 3.11

Diagram of a SMIB power system connected with a WPIG

Substituting (3.34) into (3.35) gives

$$ {\displaystyle \begin{array}{l}{\overline{\mathrm{I}}}_{\mathrm{t}\mathrm{L}}=-\frac{{\mathrm{jX}}_{\mathrm{w}\mathrm{L}}}{{\mathrm{X}}_{\mathrm{t}\mathrm{L}}{\mathrm{X}}_{\mathrm{Lb}}+{\mathrm{X}}_{\mathrm{Lb}}{\mathrm{X}}_{\mathrm{w}\mathrm{L}}+{\mathrm{X}}_{\mathrm{w}\mathrm{L}}{\mathrm{X}}_{\mathrm{t}\mathrm{L}}}\left[\left(1+\frac{{\mathrm{X}}_{\mathrm{Lb}}}{{\mathrm{X}}_{\mathrm{w}\mathrm{L}}}\right){\overline{\mathrm{V}}}_{\mathrm{t}}-\frac{{\mathrm{X}}_{\mathrm{Lb}}}{{\mathrm{X}}_{\mathrm{w}\mathrm{L}}}{\overline{\mathrm{V}}}_{\mathrm{w}}-{\overline{\mathrm{V}}}_{\mathrm{b}}\right]\\ {}=-\frac{{\mathrm{jX}}_{\mathrm{w}\mathrm{L}}}{{\mathrm{X}}_{\mathrm{t}\mathrm{L}}{\mathrm{X}}_{\mathrm{Lb}}+{\mathrm{X}}_{\mathrm{Lb}}{\mathrm{X}}_{\mathrm{w}\mathrm{L}}+{\mathrm{X}}_{\mathrm{w}\mathrm{L}}{\mathrm{X}}_{\mathrm{t}\mathrm{L}}}\left[\left(1+\frac{{\mathrm{X}}_{\mathrm{Lb}}}{{\mathrm{X}}_{\mathrm{w}\mathrm{L}}}\right){\overline{\mathrm{V}}}_{\mathrm{t}}-\frac{{\mathrm{X}}_{\mathrm{Lb}}}{{\mathrm{X}}_{\mathrm{w}\mathrm{L}}}{\overline{\mathrm{V}}}_{\mathrm{w}}-1\right]\end{array}} $$

(3.36)

As ${\overline{\mathrm{I}}}_{\mathrm{w}}=-\left({\overline{\mathrm{I}}}_{\mathrm{tL}}+{\overline{\mathrm{I}}}_{\mathrm{Lb}}\right)$ and ${\overline{\mathrm{V}}}_{\mathrm{L}}={\overline{\mathrm{V}}}_{\mathrm{w}}-{\mathrm{jX}}_{\mathrm{w}\mathrm{L}}{\overline{\mathrm{I}}}_{\mathrm{w}}$, it can have

$$ {\overline{\mathrm{I}}}_{\mathrm{w}}={\overline{\mathrm{Y}}}_1{\overline{\mathrm{V}}}_{\mathrm{w}}+{\overline{\mathrm{Y}}}_2{\overline{\mathrm{V}}}_{\mathrm{t}}+\frac{{\mathrm{jX}}_{\mathrm{t}\mathrm{L}}}{{\mathrm{X}}_{\mathrm{t}\mathrm{L}}{\mathrm{X}}_{\mathrm{Lb}}+{\mathrm{X}}_{\mathrm{Lb}}{\mathrm{X}}_{\mathrm{w}\mathrm{L}}+{\mathrm{X}}_{\mathrm{w}\mathrm{L}}{\mathrm{X}}_{\mathrm{t}\mathrm{L}}} $$

(3.37)

where ${\overline{\mathrm{Y}}}_1=-\frac{\mathrm{j}\left({\mathrm{X}}_{\mathrm{tL}}+{\mathrm{X}}_{\mathrm{Lb}}\right)}{{\mathrm{X}}_{\mathrm{tL}}{\mathrm{X}}_{\mathrm{Lb}}+{\mathrm{X}}_{\mathrm{Lb}}{\mathrm{X}}_{\mathrm{wL}}+{\mathrm{X}}_{\mathrm{wL}}{\mathrm{X}}_{\mathrm{tL}}}$, ${\overline{\mathrm{Y}}}_2=-\frac{{\mathrm{jX}}_{\mathrm{Lb}}}{{\mathrm{X}}_{\mathrm{tL}}{\mathrm{X}}_{\mathrm{Lb}}+{\mathrm{X}}_{\mathrm{Lb}}{\mathrm{X}}_{\mathrm{wL}}+{\mathrm{X}}_{\mathrm{wL}}{\mathrm{X}}_{\mathrm{tL}}}$.

(3.37) can be linearized to be

$$ \Delta {\mathbf{I}}_{\mathrm{w}}={\mathbf{Y}}_1\Delta {\mathbf{V}}_{\mathrm{w}}+{\mathbf{Y}}_2\Delta {\mathbf{V}}_{\mathrm{t}} $$

(3.38)

As the standard algebraic linearized model of a WPIG can be written as ΔI _w = C _wΔX _w + D _wΔV _w, by eliminating ΔI _w, (3.38) becomes

$$ \Delta {\mathbf{V}}_{\mathrm{w}}={\left({\mathbf{Y}}_1-{\mathbf{D}}_{\mathrm{w}}\right)}^{-1}{\mathbf{C}}_{\mathrm{w}}\Delta {\mathbf{X}}_{\mathrm{w}}-{\left({\mathbf{Y}}_1-{\mathbf{D}}_{\mathrm{w}}\right)}^{-1}{\mathbf{Y}}_2\Delta {\mathbf{V}}_{\mathrm{t}} $$

(3.39)

Substituting (3.39) into the linearized equation of (3.36) to eliminate ΔV _w gives

$$ \Delta {\mathbf{I}}_{\mathrm{t}\mathrm{L}}={\mathbf{R}}_{1\mathrm{IB}}\Delta {\mathbf{V}}_{\mathrm{t}}+{\mathbf{R}}_{2\mathrm{IB}}\Delta {\mathbf{X}}_{\mathrm{w}} $$

(3.40)

Transforming (3.40) from the Infinite Bus reference frame to d-q reference frame by introducing Δδ, (3.40) becomes

$$ \Delta {\mathbf{I}}_{\mathrm{t}\mathrm{L}}={\mathbf{R}}_1\Delta {\mathbf{V}}_{\mathrm{t}}+{\mathbf{R}}_2\Delta {\mathbf{X}}_{\mathrm{w}}+{\mathbf{R}}_3\Delta \updelta $$

(3.41)

Substituting the linearized SG equation ΔV_td = X_qΔI_tLq and $\Delta {\mathrm{V}}_{\mathrm{tq}}=\Delta {\mathrm{E}}_{\mathrm{q}}^{\prime }-{\mathrm{X}}_{\mathrm{d}}^{\prime}\Delta {\mathrm{I}}_{\mathrm{tLq}}$ into (3.41) and decomposing ΔI _tL to ΔI_tLd and ΔI_tLq gives

$$ {\displaystyle \begin{array}{l}\Delta {\mathrm{I}}_{\mathrm{tLd}}={\mathrm{R}}_{\updelta d}\Delta \updelta +{\mathrm{R}}_{{\mathrm{E}}_{\mathrm{q}}^{\prime}\mathrm{d}}\Delta {\mathrm{E}}_{\mathrm{q}}^{\prime }+{\mathrm{R}}_{{\mathrm{I}}_{\mathrm{q}}\mathrm{d}}\Delta {\mathrm{I}}_{\mathrm{tLq}}+{\mathbf{R}}_{{\mathrm{X}}_{\mathrm{w}}\mathrm{d}}\Delta {\mathbf{X}}_{\mathrm{w}}\\ {}\Delta {\mathrm{I}}_{\mathrm{tLq}}={\mathrm{R}}_{\updelta q}\Delta \updelta +{\mathrm{R}}_{{\mathrm{E}}_{\mathrm{q}}^{\prime}\mathrm{q}}\Delta {\mathrm{E}}_{\mathrm{q}}^{\prime }+{\mathrm{R}}_{{\mathrm{I}}_{\mathrm{q}}\mathrm{q}}\Delta {\mathrm{I}}_{\mathrm{tLq}}+{\mathbf{R}}_{{\mathrm{X}}_{\mathrm{w}}\mathrm{q}}\Delta {\mathbf{X}}_{\mathrm{w}}\end{array}} $$

(3.42)

Then (3.42) is substituted into the SMIB system linearized model in (3.43) and the Phillips-Heffron model of a SMIB system with interface equations of a WPIG is derived.

$$ {\displaystyle \begin{array}{l}\Delta \dot{\updelta}={\upomega}_0\Delta \upomega \\ {}\Delta \dot{\upomega}={\mathrm{M}}^{-1}\left(\Delta {\mathrm{T}}_{\mathrm{E}}-\mathrm{D}\Delta \upomega \right)\\ {}\Delta {\dot{{\mathbf{E}}^{\prime}}}_{\mathrm{q}}={\mathrm{T}}_{\mathrm{d}0}^{-1}\left(-\Delta {\mathrm{E}}_{\mathrm{Q}}+\Delta {\mathrm{E}}_{\mathrm{fd}}\right)\\ {}\Delta {\dot{\mathrm{E}}}_{\mathrm{fd}}=\left(-\Delta {\mathrm{E}}_{\mathrm{fd}}-{\mathrm{K}}_{\mathrm{A}}\Delta {\mathrm{V}}_{\mathrm{t}}\right)\Delta {\mathrm{T}}_{\mathrm{A}}^{-1}\end{array}} $$

(3.43)

where

$$ \Delta {\mathrm{T}}_{\mathrm{E}}=\Delta {\mathrm{I}}_{\mathrm{tLq}}{\mathrm{E}}_{\mathrm{q}0}^{\prime }+{\mathrm{I}}_{\mathrm{q}0}\Delta {\mathrm{E}}_{\mathrm{q}}^{\prime }+\Delta {\mathrm{I}}_{\mathrm{tLq}}\left({\mathrm{X}}_{\mathrm{q}}-{\mathrm{X}}_{\mathrm{d}}^{\prime}\right){\mathrm{I}}_{\mathrm{tLd}0}+{\mathrm{I}}_{\mathrm{tLq}0}\left({\mathrm{X}}_{\mathrm{q}}-{\mathrm{X}}_{\mathrm{d}}^{\prime}\right)\Delta {\mathrm{I}}_{\mathrm{tLd}}, $$

$\Delta {\mathrm{E}}_{\mathrm{Q}}=\Delta {\mathrm{E}}_{\mathrm{q}}^{\prime }-\left({\mathrm{X}}_{\mathrm{d}}-{\mathrm{X}}_{\mathrm{d}}^{\prime}\right)\Delta {\mathrm{I}}_{\mathrm{tLd}}$, ΔV_td = X_qΔI_tLq and $\Delta {\mathrm{V}}_{\mathrm{tq}}=\Delta {\mathrm{E}}_{\mathrm{q}}^{\prime }-{\mathrm{X}}_{\mathrm{d}}^{\prime}\Delta {\mathrm{I}}_{\mathrm{tLd}}$.

Appendix 3.2: Data of Examples 3.1 and 3.2

3.1.1 Example 16-Machine 68-Bus New York and New England Power System [36] (Tables 3.4, 3.5 and 3.6)

All the synchronous generators employ sixth-order detailed model with damping D = 0.0. The structure of first-order excitation system model is shown by Fig. 3.12.

Table 3.4 Bus data

Table 3.5 Line data

Table 3.6 Machine data

Fig. 3.12

The first-order excitation system model of synchronous generator

The parameters of excitation system model are K_A = 3.95, T_A = 0.1s, e_fdmax = 10.0, e_fdmin =  − 10.0.

3.1.2 Data of DFIG and FSIG

3.1.2.1 Induction Generator Parameters

M_w = 3.4s, D_w = 0, R_r = 0.0007, X_s = 0.0878, X_r = 0.0373, X_m = 1.3246, X_r3 = 0.05, X_ss = X_s + X_m, X_rr = X_r + X_m, P_w = 2.0 p. u., V_w = 1.015 p. u.

3.1.2.2 Control Parameters of RSC

K_psp1 = K_qsp1 = 0.2, K_psp2 = K_qsp2 = 1,

K_psI1 = K_qsI1 = 12.56 s⁻¹, K_psI2 = K_qsI2 = 62.5 s⁻¹.