Improvement to observability measures of LFO modes in power systems with DFIGs

Abstract

Observation of the low-frequency oscillation (LFO) modes in power systems is important to design the damping scheme. The state equations of the power system with the doubly-fed induction generators (DFIGs) are derived to find the LFO modes related to the synchronous generator (SGs) and the DFIGs. The definition of the observability measure is improved to consider the initial output and the attenuation speed of the modes. The sensitivities of the observability measures to the control parameters are derived. The numerical results from the small and large-disturbance validate the LFO modes caused by the DFIGs, and different observability measures are compared. Adjustment of the control parameters is chosen based on the sensitivity model to improve the observability and damping ratio of the LFO mode, and the stability of the wind power system.

Graphical abstract

Keywords

wind power system / low-frequency oscillation (LFO) / observability measure / sensitivity / doubly-fed induction generator (DFIG)

1 Introduction

After the disturbances, e.g. short circuit or load surge [1,2], the low-frequency oscillation (LFO) is dangerous to the power system [3], yielding the periodic oscillation of the power angles, the bus voltages, and the line currents. When the oscillation center is located on a transmission line, the distance protection of this line may undesirably trip due to the much less apparent impedance seen by the impedance relays. The synchronous generators (SGs) may experience the angular swing and the transient instability when the angular difference of it from any other SG exceeds the critical value (theoretically 180°, practically much less than that). The active and reactive outputs of the wind turbine generators (WTGs) will also change periodically with the LFO, and the unbalanced energy may exert a great pressure on the security of the windings, the converters, and the DC capacitors of the WTGs.

With more wind power, the LFO may be intensified and more complicated [4]. Slootweg and Kling [5] studied the impact of the wind power on the LFOs. With the improved control strategy to the WTGs or the adjusted parameters [6], the local [7–10], interarea [11–14], and wide-area LFOs [15,16], are damped. The LFO mode caused by the doubly-fed induction generators (DFIGs) instead of the SGs is defined [17].

The observability of the LFO modes belongs to the area of state estimation [18]. Göl and Abur [19] quantified the network observability and the measurement criticality. Liao [20] observed the harmonics with sparsity maximization. Kavasseri and Srinivasan [21], Rashidi et al [22], Babu and Bhattacharyya [23], and Saleh et al. [24] decided the optimal location and redundancy of the phasor measure unit (PMU) for network observability. The papers concern with observation of the steady-state configuration and parameters.

For the dynamic purpose, Sanchez-Gasca and Chow [25], Wang et al. [26], and Qi et al. [27] reduced the network with the critical modes. The canonical form and Grammian are popular in judging the observability [28,29]. For power systems, it is difficult to decompose the matrix or judge its rank. The observability measure using the right eigenvectors is more useful [30–32], however as an absolute and static index, it cannot describe the mode attenuation. To improve the observability, the relation of the modes and their observability measures to the control parameters is to be given, which is not found in existing literature. Modeling the wind power instead of the WTGs is inaccurate to describe and observe the LFOs, since the response of the WTGs and their interaction with the SGs may change the existing modes and introduce new modes.

In this paper, the observability of the LFOs in the wind power systems is studied. With the interface equations of the DFIGs, the state equations of the system are derived to find the LFO modes. The definition of the observability measure is extended to consider the initial outputs and the attenuation of modes. The sensitivities of the LFO modes and their observability measures to the control parameters are derived to improve the observability, the damping ratio, and the system stability. The numerical results from small and dynamic give the observability measures and the sensitivities, and show that the LFO caused by the DFIGs is observable from both the SGs and the DFIGs.

2 Dynamic description of wind power systems with DFIGS

2.1 LFO modes of wind power system based on dynamic modeling to DFIG

In a wind power system, the DFIG may be modeled with different accuracies. One may keep the difference of the DFIGs but simplify the control loops, or model the DFIGs as exactly as possible [33] but the DFIGs in the same wind farm of the same type are aggregated to one. The latter is adopted in this paper.

Considering the detailed configuration of the DFIG, the wind power system is initialized by power flow analysis [12]. Based on the Cp function of the wind turbine (WT), the incremental output of the WT is given by

(1)\mathrm{\Delta }{P}_t={\mu }_{Ps}\mathrm{\Delta }{s}_t+{\mu }_{P\beta }\mathrm{\Delta }\beta ,

where P_t is the active power output of the WT, s is the rotor slip, b is the pitch angle, m is the coefficients between P_t and s or b, and D denotes the increment.

The transfer shaft equations are given by

(2)\{\begin{array}{l}\mathrm{2}{H}_{\mathrm{t}}\mathrm{p}\mathrm{\Delta }{s}_{\mathrm{t}}=-\mathrm{\Delta }(P_{\mathrm{t}}/(\mathrm{1}-s_{\mathrm{t}}))-D(\mathrm{\Delta }{s}_{\mathrm{t}}-\mathrm{\Delta }{s}_{\mathrm{r}})+K\mathrm{\Delta }\gamma ,\\ \mathrm{2}{H}_{r}p\mathrm{\Delta }{s}_{\mathrm{r}}=L_{\mathrm{m}}\mathrm{\Delta }(I_{\mathrm{s}d}{I}_{\mathrm{r}q}-I_{\mathrm{s}q}{I}_{\mathrm{r}d})+D(\mathrm{\Delta }{s}_{\mathrm{t}}-\mathrm{\Delta }{s}_{\mathrm{r}})-K\mathrm{\Delta }\gamma ,\\ \mathrm{p}\mathrm{\Delta }\gamma =-\mathrm{\Delta }{s}_{\mathrm{t}}+\mathrm{\Delta }{s}_{\mathrm{r}},\end{array}

where I is the current, L is the inductance, g is the torsional angle, H is the inertia, D is the damping coefficient, K is the stiffness coefficient, d and q are the direct and quadrature axis, and p denotes the derivative. The subscripts s and r denote the stator and the rotor respectively. By substituting Eq. (1) to Eq. (2), DP_t will be eliminated.

With the d axis is orientated by the stator voltage, the voltages of the stator and rotor windings are given by

(3)\{\begin{array}{l}{L}_{\mathrm{ss}}p\mathrm{\Delta }{I}_{\mathrm{s}d}+L_{\mathrm{m}}p\mathrm{\Delta }{I}_{\mathrm{r}d}=\mathrm{\Delta }{V}_d-R_{\mathrm{s}}\mathrm{\Delta }{I}_{\mathrm{s}d}+L_{\mathrm{ss}}\mathrm{\Delta }{I}_{\mathrm{s}q}+L_{\mathrm{m}}\mathrm{\Delta }{I}_{\mathrm{r}q},\\ L_{\mathrm{ss}}p\mathrm{\Delta }{I}_{\mathrm{s}q}+L_{\mathrm{m}}p\mathrm{\Delta }{I}_{\mathrm{r}q}=-L_{\mathrm{ss}}\mathrm{\Delta }{I}_{\mathrm{s}d}-R_{\mathrm{s}}\mathrm{\Delta }{I}_{\mathrm{s}q}-L_{\mathrm{m}}\mathrm{\Delta }{I}_{\mathrm{r}d},\\ L_{\mathrm{m}}p\mathrm{\Delta }{I}_{\mathrm{s}d}+L_{\mathrm{rr}}p\mathrm{\Delta }{I}_{\mathrm{r}d}=\mathrm{\Delta }{V}_{\mathrm{rd}}+L_{\mathrm{m}}\mathrm{\Delta }(s_{\mathrm{r}}{I}_{\mathrm{s}q})-R_{\mathrm{r}}\mathrm{\Delta }{I}_{\mathrm{r}d}+L_{\mathrm{rr}}\mathrm{\Delta }(s_{\mathrm{r}}{I}_{\mathrm{r}q}),\\ L_{\mathrm{m}}p\mathrm{\Delta }{I}_{\mathrm{s}q}+L_{\mathrm{rr}}p\mathrm{\Delta }{I}_{\mathrm{r}q}=\mathrm{\Delta }{V}_{\mathrm{r}q}-L_{\mathrm{m}}\mathrm{\Delta }(s_{\mathrm{r}}{I}_{\mathrm{s}d})-L_{\mathrm{rr}}\mathrm{\Delta }(s_{\mathrm{r}}{I}_{\mathrm{r}d})-R_{\mathrm{r}}\mathrm{\Delta }{I}_{\mathrm{r}q},\end{array}

where V is the voltage, R is the resistance, the subscript m denotes the magnetizing circuit.

The pitch angle is controlled to maintain the fixed rotor speed of the WT, as shown in Eq. (4), where k_p and k_i are the proportional and the integral coefficients, and the superscript * denotes the reference value.

(4)\{\begin{array}{l}{k}_p\mathrm{p}\mathrm{\Delta }{s}_t+p\mathrm{\Delta }{\beta }^{*}=-k_{\mathrm{i}}\mathrm{\Delta }{s}_{\mathrm{t}},\\ \mathrm{p}\mathrm{\Delta }\beta =\mathrm{\Delta }{\beta }^{*}-\mathrm{\Delta }\beta .\end{array}

The rotor-side converter (RSC) controls the active and reactive power outputs injected to the system from the stator, as given by Eqs. (5) and (6),

(5)\{\begin{array}{l}-k_{\mathrm{p}}\mathrm{p}\mathrm{\Delta }(V_{\mathrm{s}}{I}_{\mathrm{s}d})+p\mathrm{\Delta }{I}_{\mathrm{r}d}^{*}=k_{\mathrm{i}}\mathrm{\Delta }(V_{\mathrm{s}}{I}_{\mathrm{s}d}),\\ -k_{\mathrm{p}}\mathrm{p}\mathrm{\Delta }(V_{\mathrm{s}}{I}_{\mathrm{s}q})+p\mathrm{\Delta }{I}_{\mathrm{r}q}^{*}=k_{\mathrm{i}}\mathrm{\Delta }(V_{\mathrm{s}}{I}_{\mathrm{s}q}).\end{array}

(6)\{\begin{array}{l}{L}_{\mathrm{m}}\mathrm{p}\mathrm{\Delta }(s_{\mathrm{r}}{I}_{\mathrm{s}q})+k_{\mathrm{p}}\mathrm{p}\mathrm{\Delta }{I}_{rd}+L_{\mathrm{rr}}\mathrm{p}\mathrm{\Delta }(s_{\mathrm{r}}{I}_{\mathrm{r}q})-k_{\mathrm{p}}\mathrm{p}\mathrm{\Delta }{I}_{\mathrm{r}d}^{*}+p\mathrm{\Delta }{V}_{\mathrm{r}d}\\ =-k_{\mathrm{i}}\mathrm{\Delta }{I}_{\mathrm{r}d}+k_{\mathrm{i}}\mathrm{\Delta }{I}_{\mathrm{r}d}^{*},\\ -L_m\mathrm{p}\mathrm{\Delta }(s_{\mathrm{r}}{I}_{\mathrm{s}d})-L_{\mathrm{rr}}\mathrm{p}\mathrm{\Delta }(s_{\mathrm{r}}{I}_{\mathrm{r}d})+k_{\mathrm{p}}\mathrm{p}\mathrm{\Delta }{I}_{\mathrm{r}q}-k_{\mathrm{p}}\mathrm{p}\mathrm{\Delta }{I}_{\mathrm{r}q}^{*}+p\mathrm{\Delta }{V}_{\mathrm{r}q}\\ =-k_{\mathrm{i}}\mathrm{\Delta }{I}_{\mathrm{r}q}+k_{\mathrm{i}}\mathrm{\Delta }{I}_{\mathrm{r}q}^{*}.\end{array}

The DC voltage is decided by the active power imbalance Eq. (7), where C is the capacitance.

(7)C_{\mathrm{dc}}{V}_{\mathrm{dc}}p\mathrm{\Delta }{V}_{\mathrm{dc}}=-\mathrm{\Delta }(V_{\mathrm{r}d}{I}_{\mathrm{r}d}+V_{\mathrm{r}q}{I}_{\mathrm{r}q})-\mathrm{\Delta }(V_{\mathrm{g}d}{I}_{\mathrm{g}d}+V_{\mathrm{g}q}{I}_{\mathrm{g}q}).

The grid-side converter (GSC) controls the DC voltage and its reactive power injected to the stator from GSC, as given by Eqs. (8) and (9).

(8)\{\begin{array}{l}-k_{\mathrm{p}}\mathrm{p}\mathrm{\Delta }{V}_{\mathrm{dc}}+\mathrm{p}\mathrm{\Delta }{I}_{\mathrm{g}d}^{*}=k_{\mathrm{i}}\mathrm{\Delta }{V}_{\mathrm{dc}},\\ k_{\mathrm{p}}\mathrm{p}\mathrm{\Delta }(V_{\mathrm{s}}{I}_{\mathrm{g}q})+\mathrm{p}\mathrm{\Delta }{I}_{\mathrm{g}q}^{*}=-k_{\mathrm{i}}\mathrm{\Delta }(V_{\mathrm{s}}{I}_{\mathrm{g}q}),\end{array}

(9)\{\begin{array}{l}{k}_{\mathrm{p}}\mathrm{p}\mathrm{\Delta }{I}_{\mathrm{g}d}+X_{\mathrm{g}}\mathrm{p}\mathrm{\Delta }{I}_{\mathrm{g}q}-k_{\mathrm{p}}\mathrm{p}\mathrm{\Delta }{I}_{\mathrm{g}d}^{*}+\mathrm{p}\mathrm{\Delta }{V}_{\mathrm{g}d}-\mathrm{p}\mathrm{\Delta }{V}_s\\ =-k_{\mathrm{i}}\mathrm{\Delta }{I}_{\mathrm{g}d}+k_{\mathrm{i}}\mathrm{\Delta }{I}_{\mathrm{g}d}^{*},\\ -X_{\mathrm{g}}\mathrm{p}\mathrm{\Delta }{I}_{\mathrm{g}d}+k_{\mathrm{p}}\mathrm{p}\mathrm{\Delta }{I}_{\mathrm{g}q}-k_{\mathrm{p}}\mathrm{p}\mathrm{\Delta }{I}_{\mathrm{g}q}^{*}+\mathrm{p}\mathrm{\Delta }{V}_{\mathrm{g}q}\\ =-k_{\mathrm{i}}\mathrm{\Delta }{I}_{\mathrm{g}q}+k_{\mathrm{i}}\mathrm{\Delta }{I}_{\mathrm{g}q}^{*},\end{array}

where the subscript g denotes the circuit, usually a filter or a transformer, between the GSC and the stator.

For the small-disturbance analysis, the transmission system is reduced with the SGs and DFIGs only,

(10)I_{xy}=Y{V}_{xy}.

The relation of the dq and xy frames is given by

(11)\left[\begin{array}{l}{V}_x\\ V_y\end{array}\right]=\left[\begin{array}{ll}\sin \delta & \cos \delta \\ -\cos \delta & \sin \delta \end{array}\right]\left[\begin{array}{l}{V}_d\\ V_q\end{array}\right].

Finally, the state equation of the SGs and the DFIGs are derived

(12){\mathit{F}}_{\mathrm{DFIG}}p\mathrm{\Delta }{x}_{\mathrm{DFIG}}+{\mathit{G}}_{\mathrm{DFIG}}p\mathrm{\Delta }{z}_{\mathrm{DFIG}}={\mathit{A}}_{\mathrm{DFIG}}\mathrm{\Delta }{x}_{\mathrm{DFIG}}+{\mathit{B}}_{\mathrm{DFIG}}\mathrm{\Delta }{z}_{\mathrm{DFIG}},

(13){\mathit{F}}_{\mathrm{SG}}p\mathrm{\Delta }{x}_{\mathrm{SG}}={\mathit{A}}_{\mathrm{SG}}\mathrm{\Delta }{x}_{\mathrm{SG}}+{\mathit{B}}_{\mathrm{SG}}\mathrm{\Delta }{z}_{\mathrm{SG}},

where F_SG, A_SG, and B_SG are the coefficient matrices of the SG, F_DFIG, G_DFIG, A_DFIG, and B_DFIG are the coefficient matrices of the DFIGs, p is the derivative operator.

Combining the SGs and the DFIGs, the state equation of the wind power system is given by Eq. (14), where A and C are the state matrix and the observation matrix respectively.

(14)\{\begin{array}{l}p\mathrm{\Delta }x=\mathit{A}\mathrm{\Delta }x,\\ \mathrm{\Delta }y=\mathit{C}\mathrm{\Delta }x.\end{array}

The eigenvalues of A show the critical modes, e.g., the LFOs caused by the SGs or the governors, or the DFIGs, and the right-most modes, which threaten the security and stability of the power system, thus are to be observed. The LFO modes of the SGs are defined by the frequency and electromechanical loop participation ratio r. The latter is extended to define the LFOs caused by the DFIGs,

(15){\rho }_{\mathrm{DFIG,}i}=\left|{\displaystyle \frac{{\displaystyle \sum _{\Delta x_k\in \Delta x_{\mathrm{DFIG}(\mathrm{MT})}}{p}_{ki}}}{\mathrm{1}-{\displaystyle \sum _{\Delta x_k\in \Delta x_{\mathrm{DFIG}(\mathrm{MT})}}{p}_{ki}}}}\right|,

where Dx_DFIG(MT) of the mechanical transient corresponds to the transfer shaft dynamics and the pitch control with a slower response than the electromagnetic transient, and p_ki is the participation factor of mode i to Dx_k.

2.2 Outputs of SG and DFIG to observe LFO modes

The outputs y of the SG and DFIG, i.e., the active output (P), the reactive output (Q), and the voltage magnitude (V) are chosen to observe the LFO modes. The incremental forms are given by

(16)\{\begin{array}{l}\mathrm{\Delta }{P}_{\mathrm{SG}}=\mathrm{\Delta }(V_{\mathrm{SG}d}{I}_{\mathrm{SG}d})+\mathrm{\Delta }(V_{\mathrm{SG}q}{I}_{\mathrm{SG}q}),\\ \mathrm{\Delta }{Q}_{\mathrm{SG}}=\mathrm{\Delta }(V_{\mathrm{SG}q}{I}_{\mathrm{SG}d})-\mathrm{\Delta }(V_{\mathrm{SG}d}{I}_{\mathrm{SG}q}),\\ \mathrm{\Delta }{V}_{\mathrm{SG}}=\mathrm{\Delta }(\sqrt{V_{\mathrm{SG}d}^{\mathrm{2}}+V_{\mathrm{SG}q}^{\mathrm{2}}}),\end{array}

(17)\{\begin{array}{l}\mathrm{\Delta }{P}_{\mathrm{DFIG}}=\mathrm{\Delta }[V_{\mathrm{s},d}(I_{\mathrm{g},d}-I_{\mathrm{s},d})],\\ \mathrm{\Delta }{Q}_{\mathrm{DFIG}}=-\mathrm{\Delta }[V_{\mathrm{s},d}(I_{\mathrm{g},q}-I_{\mathrm{s},q})],\\ \mathrm{\Delta }{V}_{\mathrm{DFIG}}=\mathrm{\Delta }{V}_{\mathrm{s},d},\end{array}

(18)\{\begin{array}{l}\mathrm{\Delta }{P}_{\mathrm{PMSG}}=\mathrm{\Delta }(V_{\mathrm{PCC,}d}{I}_{\mathrm{g},d}),\\ \mathrm{\Delta }{Q}_{\mathrm{PMSG}}=-\mathrm{\Delta }(V_{\mathrm{PCC,},d}{I}_{\mathrm{g},q}),\\ \mathrm{\Delta }{V}_{\mathrm{PMSG}}=\mathrm{\Delta }{V}_{\mathrm{PCC,},d},\end{array}

where the subscript s means the stator, and g means the grid or the GSC. The power angle d may be applied as the output, if it is measurable.

It is clear that Dy are composed of the state variables Dx, i.e., I_s,d, I_s,q, I_g,d, and I_g,q, and the algebraic variables Dz, i.e., V_SG,d, V_SG,q, I_SG,d, I_SG,q, V_s,d, and V_PCC,d. Since Dx may be given by the participation factors of the modes, and Dy is given by Dx, the modes are observable from Dy, after Dz is eliminated.

3 Improvement to mode observability measure

3.1 Existing definition to observability measure

The mode observability may be decided by the binary criterions, e.g., a system is observable if the canonical forms are satisfied,

(19)\mathrm{rank}\left[\begin{array}{l}\mathit{C}\\ \mathit{C}\mathit{A}\\ ⋮\\ \mathit{C}{\mathit{A}}^{n-\mathrm{1}}\end{array}\right]=n,

(20)\mathrm{rank}\left[\begin{array}{l}{\lambda }_i\mathit{I}-\mathit{A}\\ \mathit{C}\end{array}\right]=n,i=\mathrm{1},\mathrm{2},\mathrm{...},n,

where I is the identity matrix of the dimension n, l_i = a_i + w_i is the ith eigenvalue. The decaying coefficient a_i quantifies the distance to the stability boundary, and w_i is the angular frequency. The binary criterion is straightforward, but may yield error or no conclusion for large systems due to the finite accuracy of the computers [34].

With Dx(0) known, Dx(t) is given by

(21)\Delta \mathit{x}(t)={\displaystyle \sum _{i=\mathrm{1}}^n[{\mathit{\Phi }}_i{\mathit{\psi }}_i\mathrm{\Delta }x(\mathrm{0})e^{{\lambda }_{i}t}]},

where Y_i is the transpose of the ith left eigenvector, and F_i is the ith right eigenvector of A. By substituting Eq. (21), the output is further given by

(22)\mathrm{\Delta }y(t)={\displaystyle \sum _{k=\mathrm{1}}^n\left[c_k{\displaystyle \sum _{i=\mathrm{1}}^n{\varphi }_{ki}\left({\displaystyle \sum _{j=\mathrm{1}}^n{\psi }_{ij}{x}_j(\mathrm{0})}\right)e^{{\lambda }_{i}t}}\right]}.

The continuous observability measure M_ji^a to the ith mode from the jth output is defined in Eq. (23), with which the observability from different outputs is comparable.

(23)M_{ji}^a=\left|{\mathit{c}}_{\mathit{j}}{\mathit{\Phi }}_i\right|.

3.2 Extended definitions to observability measure

By adding the contribution of the modes to y(0), M_ji^a is an absolute index. If Dy is small compared with y(0), the mode may be not obvious even with the signal processing technique, e.g., fast Fourier transformation (FFT).

Therefore, a relative measure is defined as

(24)M_{ji*}^a={\displaystyle \frac{\left|c_j{\mathit{\Phi }}_i\right|}{y_j(\mathrm{0})}},

where M_ji^a and M_ji*^a are valid at t = 0₊. It is possible that the modes with large measure are not notable due to attenuation. Therefore, the observability measure is further defined as

(25)M_{ji}^b=M_{ji}^a{e}^{-{\alpha }_i},

(26)M_{ji*}^b=M_{ji*}^a{e}^{-{\alpha }_i},

which are useful for the modes with different attenuation speeds. It is possible to differentiate the modes from the same output.

The proposed measures consider different features of power system, e.g., the ratio of the output to the incremental mode and attenuation of the mode. There are other features important but difficult to include, e.g., the decaying speed of y(t), hence V and Q are not often applied to observe the LFOs.

3.3 Improvement to mode observability

If the observability measure, e.g., M_ji^b, is lower than expected, some parameters, e.g., t, may be adjusted. A sensitivity model of the observability measure to t is derived,

(27){\displaystyle \frac{\partial M_{ji}^b}{\partial \tau }}=\left[\mathrm{Re(}{\mathit{C}}_j{\mathit{\Phi }}_i){\displaystyle \frac{\partial \mathrm{Re(}{\mathit{C}}_j{\mathit{\Phi }}_i)}{\partial \tau }}+\mathrm{Im}({\mathit{C}}_j{\mathit{\Phi }}_i){\displaystyle \frac{\partial \mathrm{Im(}{\mathit{C}}_j{\mathit{\Phi }}_i)}{\partial \tau }}\right]{\displaystyle \frac{e^{{\alpha }_i}}{M_{\mathrm{o}ji}}}+M_{ji}^b{\displaystyle \frac{\partial {\alpha }_i}{\partial \tau }},

where with the partial derivative to AF_i = l_iF_i, the sensitivity of F_i to t is given by

(28){\displaystyle \frac{\partial {\mathit{\Phi }}_i}{\partial \tau }}=-{({\lambda }_i\mathit{I}-\mathit{A})}^{-\mathrm{1}}\left({\displaystyle \frac{\partial {\lambda }_i}{\partial \tau }}\mathit{I}-{\displaystyle \frac{\partial \mathit{A}}{\partial \tau }}\right){\mathit{\Phi }}_i.

If (l_iI–A)^–1 is difficult to find, as an alternative scheme, the relation of the observability measure with t is shown in Fig. 1. With Dt small enough, the sensitivity at the operation point O, i.e., the slop of OB, is estimated by the slope of OA (29). The error is small, since the permissible range of t is very limited.

(29){\displaystyle \frac{\partial M}{\partial \tau }}\approx {\displaystyle \frac{\Delta M}{\Delta \tau }}.

Adjustment to the parameters should not reduce the stability of the system Eq. (30), and the damping ratio x_i of the LFO modes Eq. (31). The difficulty lies in the derivative of A to t, as given by the multi-step derivative (32).

(30){\displaystyle \frac{\partial {\lambda }_i}{\partial \tau }}={\mathit{\Psi }}_i{}^T{\displaystyle \frac{\partial \mathit{A}}{\partial \tau }}{\mathit{\Phi }}_i={\displaystyle \frac{\partial {\alpha }_i}{\partial \tau }}+j{\displaystyle \frac{\partial {\omega }_i}{\partial \tau }},

(31){\displaystyle \frac{\partial {\xi }_i}{\partial \tau }}={\displaystyle \frac{{\omega }_i}{{({\alpha }_i^{\mathrm{2}}+{\omega }_i^{\mathrm{2}})}^{\frac{\mathrm{3}}{\mathrm{2}}}}}\left(-{\omega }_i{\displaystyle \frac{\partial {\alpha }_i}{\partial \tau }}+{\alpha }_i{\displaystyle \frac{\partial {\omega }_i}{\partial \tau }}\right),

(32){\displaystyle \frac{\partial \mathit{A}}{\partial \tau }}={(\mathit{F}+\mathit{G}\mathit{\kappa })}^{-\mathrm{1}}\left[-{\displaystyle \frac{\partial (\mathit{F}+\mathit{G}\mathit{\kappa })}{\partial \tau }}{(\mathit{F}+\mathit{G}\mathit{\kappa }\kappa )}^{-\mathrm{1}}(\mathit{A}\text{'}+\mathit{B}\text{'}\mathit{\kappa })+{\displaystyle \frac{\partial (\mathit{A}\text{'}+\mathit{B}\text{'}\mathit{\kappa })}{\partial \tau }}\right].

4 Numerical analysis

The modified RTS system is studied (Fig. 2), which has 9 thermal plants, a hydro plant, and a SC [35]. A wind farm with 100 DFIGs at bus 3 is aggregated to a DFIG. The SGs and DFIGs are integrated through the transformers. The basic parameters of the DFIGs may be referred to in Ref. [17]. The control parameters of the DFIGs are given in Table 1. A DFIG have a PI controller for the pitch angle control, 4 controllers for the RSC, and 4 controllers for the GSC. The numbers of the control loops are given in the bracket.

For the small disturbance analysis, a SG has 15 state variables, i.e., DE_q', ΔE_dq", Dw, Dd, EE1-EE4 of the excitation system, Gov1-Gov3 of the governor, and PSS1-PSS3 of the PSS. The SC does not have the governor and PSS. A DFIG has 20 state variables.

The following analysis is based on the MATLAB program written by the author.

4.1 Observability measure of SMIB system

The SMIB is based on the scenarios of the RTS system with the terminal voltage fixed. For the SG 1 at bus 25, the eigenvalues and the related state variables (with the participation factor larger than 0.02 p.u.) are listed in Table 2. There are 3 oscillation modes, but no LFO mode is found.

For the DFIG, there are 4 oscillation modes, as can be found in Table 3, where 17/18 is the LFO mode with w = 0.1331 Hz, x = 0.1021, and r>1000.

The observability measures of the modes are tabulated in Table 4. Judged by M_ji^a, the observation from P is more obvious than that from Q. However, judged by M_ji^b or M_ji^c, the observation from P is not necessarily more obvious. For the fast decaying mode 6/7 of the SG, M_ji^c is much smaller than M_ji^a and M_ji^b.

Although the capacity of the DFIGs is less than that of the SGs, the observability measure of the DFIGs is not negligible, i.e., it is possible to observe them from their outputs.

4.2 Observability measure of RTS system

The system has 10 × 15+ 9+ 20+ 17= 196 state variables. The critical modes are given in Table 5. There are 10 LFO modes, where 143/144 is related to the DFIG.

To the LFO mode 62/63 related to the SG, the observability measures are demonstrated in Table 6. According to M_ji^a, the observation from PSG, VSG, and QSG is effective. For M_ji^b, the conclusion is still valid. Due to little var output of the SG 1 (0.03 p.u.), the relative measures M_ii*^a and M_ji*^b from QSG are quite obvious. However, since the oscillation of Q is damped more quickly than those of the LFO modes, the observability from Q is less applied. It is found that the outputs of the DFIG, i.e., d and P, are also suitable to observe the LFO of the SGs.

To the mode 143/144 introduced by the DFIG, the observability measures are illustrated in Table 7. Based on M_ji^a, the observation from P_DFIG, d_DFIG, or P_SG, is effective to observe the mode. Compared with mode 62/63, 143/144 damps slower, the difference between the relative and the absolute measures are less obvious. Therefore, M_ji^b and M_ji*^b are more suitable to mode 62/63 than to mode 143/144.

4.3 Validation of observability of critical modes

To validate the observability measures, the step-by-step simulation is applied. A 3-pahse short fault occurs on line 4–9 at 0.02 s, and is tripped at 0.12 s. The fault is cleared after 0.5 s to restore the line. The simulation lasts for 100 s, with the step of 0.005 s. The active outputs of the SGs and the DFIGs are depicted in Fig. 3.

The band-pass Butterworth filters are designed to find the component consistent with the mode frequency from the eigen-analysis, as exhibited in Fig. 4, where the parameters are flexibly set to consider the frequency of the mode and sampling frequency.

To observe the mode 62/63 caused by the SG, the filtered active outputs are displayed in Fig. 5. The oscillation of the output of SG is more intensive that that of the DFIG. The frequency magnitude from the FFT analysis is decided by the oscillation amplitude and attenuation. The mode is also observable from the DFIGs, which is consistent with the observability measures.

To observe the LFO mode 143/144 caused by the DFIG, the filtered active outputs are presented in Fig. 6. Due to the smaller decaying coefficient, the oscillation is much longer than that of mode 62/63. The oscillation of the output of SG is intensive, i.e., this mode is also observable from the SGs, consistent with the observability measures.

4.4 Improvement to mode observability

The sensitivities of mode 143/144 to the control parameters of the DFIG are given in Table 8, where the subscripts 1–8 of t are the control loops shown in the Appendix. Larger k_p1, k_i2, k_i3, k_p4, k_p6, k_i6, and k_i8, or smaller k_i1, k_p2, k_i4, and k_i7, will increase the stability and the damping ratios of the mode. The sensitivities of k_p1, k_i1, and k_i6 are larger than those with other parameters.

The sensitivities of the observability measures are summarized in Table 9. The increase in k_p1 or k_i6 enhances M_ji^a, while the increase in k_i1 weakens it. The control effects are opposite to M_ji^b. For the relative measures, the increase of k_p1, k_i1, or k_i6 will decrease M_ji*^a, but increase M_ji*^b.

The change of k_i6 of the active power control loop of the GSC is the most influential to the LFO mode caused by the DFIG and its observability measure, which is useful to the design and coordination of the control parameters of the DFIGs.

4.5 Impacts of shunt compensation on LFO modes and observability measures

The observability measures are dependent on the configuration parameters and the operational parameters of the power systems, such as the reactive power (var) compensation at the step-up transformer, which is helpful to the voltage control and the fault ride-through of the WTGs, especially for the fixed-speed WTGs, i.e., the induction generators (IGs). For the variable-speed WTGs, i.e., DFIGs and PMSGs, if the capacity of the GSC is larger than that needed to transit the active power, the remaining capacity is used to produce the reactive power; otherwise, the var compensation is needed to maintain the reactive power balance.

To validate the contribution of reactive power compensation on the LFO modes and the observability measures, a shunt capacitor of 0.2 p.u. is added to bus 3, the high-voltage side of the step-up transformer of the wind farm. The LFO modes are shown in Table 10. Compared with Table 5, it is found that the stability margin is increased with the shunt compensation, and most of the oscillation frequencies of the LFO modes related to the SGs increase, i.e., the LFOs are improved by the shunt compensation. However, the oscillation frequency related to the DFIG has little change. This mode is more closely dependent on the DFIG than other equipment in the power system.

With the shunt compensation, the observability measures of the LFO mode 143/144 related to the DFIG are shown in Table 11. Compared with Table 7, the observability measures seen from the voltage magnitude, the voltage angles, or the active outputs may increase, but those seen from the reactive output may decrease.

5 Conclusions

The pre-disturbance output and the mode attenuation speed are considered to improve the existing definitions of the observability measure to the LFO modes in the wind power systems with the DFIGs. The sensitivities of the observability measures to the control parameters are proposed. With the FFT analysis after the band-pass filter to the large-disturbance, the observability of the LFO modes caused by and observed from the SGs and the DFIGs are validated. Some conclusions are given as following,

The LFO modes caused by the DFIGs are not negligible. They are observable from both the SGs and the WTGs. Besides, the initial outputs and attenuation speeds of the LFOs have influence on the observability measures. Moreover, the sensitivities of the observability measure to the control parameters help to design or adjust the latter, and improve the stability and the damping characteristics of the power systems, and the observability of the critical modes. Furthermore, system parameters, e.g., the shunt compensation, may change the LFO modes related to the SGs, but they have little impact on the LFO mode related to DFIGs.

Based on the LFO modes and their observability, future research will be focused on the further improvement of the observability, thus damp of the LFO modes.

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