Investigating the Impact of Wind Power Integration on Damping Characteristics of Low Frequency Oscillations in Power Systems

Abstract

This paper investigates the impact of doubly-fed induction generator (DFIG) wind farms on system stability in multi-generator power systems with low-frequency oscillations (LFOs). To this end, this paper establishes the interconnection model of the equivalent generators and derives the system state equation. On this basis, an updated system state equation of the new power system with integrated wind power is further derived. Then, according to the updated system state equation, the impact factors that cause changes in the system damping characteristics are presented. The IEEE two-area four-machine power system is used as a simulation model in which the LFOs occur. The simulation results demonstrate that the connection of wind power to the power feeding area (PFA) increases the damping ratio of the dominant mode of inter-area oscillation from −0.0263 to −0.0107, which obviously improves the system stability. Furthermore, the wind power integration into PFA, as the connection distance of the wind power increases, gradually decreases the damping ratio of the dominant mode of the inter-area oscillation to −0.0236, approaching that of no wind power in the system. Meanwhile, with the increase in the wind power output capacity, the damping ratio of the dominant mode of the intra-area and inter-area oscillation increases, and the maximum damping ratios reach 0.1337 and 0.0233, respectively.

1. Introduction

The expansion of the integration of clean and efficient renewable energy sources such as wind and solar energy has exacerbated the gradual expansion of the modern power grid interconnection and caused an unprecedented complexity of the energy system [1,2,3]. Hence, the security and stability of the power system should be taken seriously [4]. A typical hazard in a power system is the interaction among generators or regional power grids, which results in the damping characteristic changes of the power systems and thus may produce low-frequency oscillations (LFOs) [5]. The synchronous generator angles start oscillating with each other in low frequencies of 0.1–2.5 Hz following grid disturbances. These oscillations are usually poorly damped and can lead to system collapse [6]. A famous example of a grid failure is the blackout that occurred at Western Systems Council Coordinated (WSCC) in August 1996 [7]. A series of system failures occurred in eastern North America, which eventually led to system instability and resulted in the blackout disaster on August 14, 2003 [8]. There was a major grid disturbance in the northern region at 02:33 on 30 July 2012. The northern regional grid load was about 36,000 MW at the time of the disturbance. Subsequently, there was another grid disturbance at 13:00 on 31 July 2012, resulting in the collapse of the northern, eastern, and north-eastern regional grids. A total load of about 48,000 MW was affected in this blackout [9]. The power oscillations at different stages and degrees were observed during these big blackouts. LFO seriously affects the power transmission capacity and threatens the stable operation of the power grid and even causes power failure accidents due to grid instability [10].

Due to the increase in power demand and to reduce global warming, the power grid is moving more and more to renewable energy integration, in which wind power is the most important renewable source of energy all over the world. The impact of integrated wind power on the power system stability is also a long-term problem which in turn arouses the interest of many researchers around the world [11]. The authors in [12] proposed the stability problems of the power systems early, including large-scale wind power in which the impact of intra-area and inter-area oscillations is studied under the condition of wind power integrated into the power systems. It is found that the damping effect of the constant speed wind turbine (WT) on the power systems is more effective than that of the variable speed WT because of the effects of the induction generator. In [13], the authors have analyzed the power system oscillation modes of the two-area four-machine system with small disturbance and studied the changes of the oscillation modes under the situation of a wind farm replacing a synchronous generator in the system. Aiming at the integration of the wind power into Norwegian power grids, E. Hagstrøm et al. [14] studied the impact of the integrated wind power on the interval oscillation damping of northern European power grids and tested the impact of different types of generators on the inter-area oscillation damping of Norway and Sweden. In [15], the generic dynamic model of WT with a doubly-fed induction generator (DFIG) is presented, and the small-signal stability mathematical model is derived. G. Tsourakis et al. [16] studied the response modes of DFIG in wind power plants and their impact on the electromechanical oscillation of interconnected power systems. L. Yang et al. [17] carried out a mathematical model for DFIG. Through eigenvalue sensitivity analysis, the most sensitive parameters of the Hopf bifurcation of DFIG power generation system can be determined. Thus, the Hopf bifurcation boundaries of the key parameters are analyzed to facilitate the parameterization of DFIG WT system and ensure system stability. H. Li et al. [18] proposed a fuzzy control strategy to suppress the LFOs of the power system after the doubly-fed induction generators (DFIGs) are integrated into the power grid. M. Singh et al. [19] studied the potential damping interval oscillation mode of the wind farm. The result shows that the intra-area oscillation mode is a phenomenon of oscillation in a single or a group of generators with another set of generators on a weak transmission link. R. Effatnejad et al. [20] studied the LFO suppression strategy of power systems in the case of the integrated wind power. The angular velocity difference signal of the two generators is used as the input signal of the damping controller. The conventional damping controller is replaced by the fuzzy controller, and the controller parameters are optimized by the intelligent optimization algorithm. In [21], a method was proposed for the separate examination of the two impact factors, the change of load flow and dynamic interaction brought about by the DFIG on the electromechanical oscillation modes of the power systems. J. G. Slootweg and W. L. Kling [22] established a 7-order WT mathematical model for doubly-fed wind turbines and revealed the impact of shafting and the wind speed on the system’s forced LFOs in the condition of the wind power integrated into the power systems through the small-signal linear analysis method. In order to solve the dynamic stability problem of the large wind farms, X. Zhang et al. [23] proposed a power grid operation method based on variable inertia control. A variable inertia control strategy for WT was proposed to improve the damping and frequency stability of power oscillation between the areas. In [24], a line modal potential energy (LMPE) method was proposed for the inter-area oscillation analysis and damping control of DFIGs. The study attempts to design the damping control loop using the LMPE approach from the network viewpoint. L. Simon et al. [25] described an optimization enabled wide-area damping control (WADC) for DFIG to mitigate both the intra-area and the inter-area oscillations. The main purpose in [26] was to improve the synchrony generator’s transient and dynamic performance using local DFIGs. The impact of wind power integrated into the existing power system is multi-dimensional. There are still many uncertain results that cannot be ignored in the coordinated development of wind power and the large power grid to maintain system stability in the future power grid. Based on the above investigation and explanations, the main contributions of this research are summarized as follows:

• The two-area interconnection system and its mathematical model are established and analyzed, and the system state equation is deduced before and after the wind power integration.
• Three impact factors of the integrated wind power on the LFOs of the power system are proposed and the eigenvalues of the intra-area and inter-area oscillation of the power system are identified by the total least squares-estimation of signal parameters via rotational invariance techniques (TLS-ESPRIT) algorithm.
• By comparing the changes of the damping characteristics of the system before and after wind power integration, the impact on the LFOs of the system is obtained, which provides a reference for the design of the wind power plants, as well as the stability of the power system with small disturbance.

The remaining sections of this paper are organized as follows: Section 2 establishes the model of the two-area interconnection power system and derives the system state equation. The damping characteristics of the system integrated with the wind power consisting of the DFIGs are analyzed. The impact factors of wind power connection on the LFOs are mainly related to the area option of the integrated wind power, including a power feeding area (PFA) or a power receiving area (PRA), the connection distance, and the output capacity of the integrated wind power. Section 3 shows the example simulation and result analysis based on the simulation of the IEEE two-area four-machine power system. Firstly, the eigenvalues of the intra-area and inter-area oscillation modes of the system are calculated. Secondly, considering the variation of each impact factor, the changes in the damping characteristics of the simulation model are reflected by the eigenvalues of the modes of the intra-area and inter-area oscillation. Finally, through the quantitative analysis of the modal eigenvalues and the changes of the damping ratios, the effect of the different impact factors of the integrated wind power on the LFOs of the power system is obtained and discussed. Section 4 addresses the conclusion of the paper.

2. Analysis of Damping Characteristics

The LFOs of large-scale power systems usually include intra-area and inter-area oscillation. The intra-area oscillation is mainly the relative sway between different generators in the area. The inter-area oscillation is the relative swing between multiple generator groups in one area and multiple generator groups in other areas. According to the equivalent principle of the center of inertia (COI) [27], multiple generator groups in the area are taken to be COI equivalent, which can simplify the damping analysis of the system. Hence, the interconnection between two groups of generators can be studied by using a simplified system consisting of two equivalent generators.

2.1. Analysis of Damping Characteristics of the Two-Area Interconnection Power Systems

The equivalent model of the two-area interconnection system is shown in Figure 1. The inter-area oscillation is equivalent to the oscillation between the synchronous generators G_eq1 and G_eq2. Taking the active power transmitted from area A1 to area A2 as an example, it can be said that area A1 is PFA, while area A2 is PRA. Then, the equivalent rotor angular velocity increment of G_eq1 and G_eq2 will show the same frequency reverse characteristic, as well as the equivalent power angle increment.

Figure 1. Equivalent model of the two-area interconnection system.

The classical second-order equations are adopted as the rotor motion mathematical model of G_eq1 and G_eq2. The mechanical power of the prime movers is constant, and the excitation system dynamics are ignored while the load is treated as constant impedance. The rotor motion equation of the COI equivalent synchronous generator is represented as follows:

$$\{\begin{array}{l}\mathsf{\Delta }\stackrel{·}{{\delta }_j}=\mathsf{\Delta }{\omega }_j\\ \mathsf{\Delta }{\stackrel{·}{\omega }}_j=\frac{1}{T_{\mathrm{J}j}}(-\mathsf{\Delta }{P}_{je}-D_j\mathsf{\Delta }{\omega }_j)\end{array}$$

(1)

Using the Heffron–Phillips model, rewriting (1) in the matrix form, it is represented as follows:

$$\left[\begin{array}{c}\mathsf{\Delta }\stackrel{·}{{\delta }_1}\\ \mathsf{\Delta }\stackrel{·}{{\delta }_2}\\ \mathsf{\Delta }{\stackrel{·}{\omega }}_1\\ \mathsf{\Delta }{\stackrel{·}{\omega }}_2\end{array}\right]=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ -\frac{K_{11}}{T_{\mathrm{J}1}}& -\frac{K_{11}}{T_{\mathrm{J}1}}& -\frac{D_1}{T_{\mathrm{J}1}}& 0\\ -\frac{K_{21}}{T_{\mathrm{J}2}}& -\frac{K_{21}}{T_{\mathrm{J}2}}& 0& -\frac{D_2}{T_{\mathrm{J}2}}\end{array}\right]\left[\begin{array}{c}\mathsf{\Delta }{\delta }_1\\ \mathsf{\Delta }{\delta }_2\\ \mathsf{\Delta }{\omega }_1\\ \mathsf{\Delta }{\omega }_2\end{array}\right]={\mathit{A}}_{eq}\left[\begin{array}{c}\mathsf{\Delta }{\delta }_1\\ \mathsf{\Delta }{\delta }_2\\ \mathsf{\Delta }{\omega }_1\\ \mathsf{\Delta }{\omega }_2\end{array}\right]$$

(2)

According to the system’s circuit analysis, based on the superposition theorem of the model shown in Figure 1, the output electromagnetic power $P_{1e}$ and $P_{2e}$ are calculated as represented in (3).

$$\{\begin{array}{c}{P}_{1e}=\frac{E_1^2}{\left|Z_{11}\right|}\sin {\alpha }_{11}+\frac{E_1{E}_2}{\left|Z_{12}\right|}\sin ({\delta }_{12}-{\alpha }_{12})\\ P_{2e}=\frac{E_2^2}{\left|Z_{22}\right|}\sin {\alpha }_{22}+\frac{E_1{E}_2}{\left|Z_{21}\right|}\sin ({\delta }_{21}-{\alpha }_{12})\end{array}$$

(3)

where

$$\begin{gathered} Z_{12}=Z_{21},Z_{11}=\left|Z_{11}\right|\angle {\phi }_{11},Z_{22}=\left|Z_{22}\right|\angle {\phi }_{22},Z_{12}=\left|Z_{12}\right|\angle {\phi }_{12},{\alpha }_{11}=90°-{\phi }_{11},{\alpha }_{22}=90°-{\phi }_{22}, \\ {\alpha }_{12}=90°-{\phi }_{12}. \end{gathered}$$

At a certain operating point, the rotor angle difference of G_eq1 and G_eq2 has the relation ${\delta }_{120}=-{\delta }_{210}$. According to (3), the synchronous torque coefficients $K_{jk}$ can be calculated as follows:

$$\{\begin{array}{c}{K}_{11}=-K_{12}=\frac{E_1{E}_2}{\left|Z_{12}\right|}\cos ({\delta }_{120}-{\alpha }_{12})\\ K_{22}=-K_{21}=\frac{E_1{E}_2}{\left|Z_{21}\right|}\cos ({\delta }_{210}-{\alpha }_{12})\end{array}$$

(4)

The state matrix ${\mathit{A}}_{eq}$ can be extracted from (2), and its characteristic equation is represented by:

$$\det \left({\mathit{A}}_{eq}-\lambda \mathit{I}\right)=0$$

(5)

Through matrix transformation and the Schur theorem, Equation (5) can be transformed into a fourth-degree equation with one variable with respect to $\lambda$, as represented in (6).

$${\lambda }^4+(\frac{D_1}{T_{\mathrm{J}1}}+\frac{D_2}{T_{\mathrm{J}2}}){\lambda }^3+(\frac{K_{11}}{T_{\mathrm{J}1}}+\frac{K_{22}}{T_{\mathrm{J}2}}+\frac{D_1}{T_{\mathrm{J}1}}\frac{D_2}{T_{\mathrm{J}2}}){\lambda }^2+\frac{1}{T_{\mathrm{J}1}{T}_{\mathrm{J}2}}(D_1{K}_{22}+D_2{K}_{11})\lambda =0$$

(6)

Idealizing the damping moment coefficient and the same inertia time constant of the equivalent generator results in $D_1=D_2,T_{\mathrm{J}1}=T_{\mathrm{J}2}$, or $\frac{D_1}{T_{\mathrm{J}1}}=\frac{D_2}{T_{\mathrm{J}2}}$. Therefore, (6) can be simplified as follows:

$$\lambda (\lambda +\gamma )({\lambda }^2+\gamma \lambda +\frac{K_{11}}{T_{\mathrm{J}1}}+\frac{K_{22}}{T_{\mathrm{J}2}})=0$$

(7)

The eigenvalues are obtained by solving the equation as follows:

$$\{\begin{array}{l}{\lambda }_1=0\\ \\ {\lambda }_1=-\gamma =-\frac{D_1}{T_{\mathrm{J}1}}\\ {\lambda }_{3,4}=\sigma \pm j\omega =-\raisebox{1ex}{$D_1$}\!\left/ \!\raisebox{-1ex}{$2T_{\mathrm{J}1}$}\right.\pm \frac{j\sqrt{{\left(-\raisebox{1ex}{$D_1$}\!\left/ \!\raisebox{-1ex}{$T_{\mathrm{J}1}$}\right.\right)}^2+4\left(\raisebox{1ex}{$K_{11}$}\!\left/ \!\raisebox{-1ex}{$T_{\mathrm{J}1}$}\right.+\raisebox{1ex}{$K_{22}$}\!\left/ \!\raisebox{-1ex}{$T_{\mathrm{J}2}$}\right.\right)}}{2}\end{array}$$

(8)

In general, $K_{11}$, $K_{22}$, $D_1$, $D_2$ are greater than zero in normal operation. According to the Routh system stability criterion, Equation (8) shows that the real parts of the three non-zero eigenvalues are all negative, and the conjugate eigenvalues ${\lambda }_3$ and ${\lambda }_4$ represent the electromechanical oscillation modes of the interconnected system. If the system is abnormal, such as $D_1\le 0$ or $D_2\le 0$, that is $-D_1/T_{\mathrm{J}1}\ge 0$ or $-D_2/T_{\mathrm{J}2}\ge 0$, the real part of eigenvalue will be located on the imaginary axis or the right half-space of the complex plane. The corresponding consequences are that the system damping is zero or negative, which results in continuous or growth oscillation of the power systems. If it is not intervened in, the power systems under the conditions of instability will be out-of-step, splitting finally.

2.2. Analysis of Damping Characteristics of the Two-Area Interconnection Power Systems Integrated with Wind Power

The wind farm marked by ellipse dotted lines in Figure 2 is incorporated into the two-area interconnection system. The wind farm can be represented by a single machine of a large-capacity DFIG [25]. The schematic diagram of a DFIG integrating the power grid is shown in Figure 3.

Figure 2. Two-area interconnection system integrated with wind farm.

Figure 3. Schematic diagram of DFIG integrated into power grid.

Neglecting the system line loss, the electromagnetic power increments $\mathsf{\Delta }{P}_{1e}$ and $\mathsf{\Delta }{P}_{2e}$ of G_eq1 and G_eq2 mentioned in (1) are rewritten as (9).

$$\{\begin{array}{c}\mathsf{\Delta }{P}_{1e}=\frac{E_1{U}_{\mathrm{w}}}{X_{13}}\cos ({\delta }_{10}+{\delta }_{\mathrm{w}0})(\mathsf{\Delta }{\delta }_1+\mathsf{\Delta }{\delta }_{\mathrm{w}})\\ \mathsf{\Delta }{P}_{2e}=\frac{E_2{U}_{\mathrm{w}}}{X_{23}}\cos ({\delta }_{20}+{\delta }_{\mathrm{w}0})(\mathsf{\Delta }{\delta }_2+\mathsf{\Delta }{\delta }_{\mathrm{w}})\end{array}$$

(9)

Labeling $k_1=\frac{E_1{U}_{\mathrm{w}}}{X_{13}}\cos ({\delta }_{10}+{\delta }_{\mathrm{w}0}),k_2=\frac{E_2{U}_{\mathrm{w}}}{X_{23}}\cos ({\delta }_{20}+{\delta }_{\mathrm{w}0})$, Equation (9) can be simplified as follows:

$$\{\begin{array}{c}\mathsf{\Delta }{P}_{1e}=k_1(\mathsf{\Delta }{\delta }_1+\mathsf{\Delta }{\delta }_{\mathrm{w}})\\ \mathsf{\Delta }{P}_{2e}=k_2(\mathsf{\Delta }{\delta }_2+\mathsf{\Delta }{\delta }_{\mathrm{w}})\end{array}$$

(10)

As the system structure is stable and the load remains unchanged, the active power of the PFA and PRA is increased or decreased by the same amount. Hence, the active power increment of the interconnected two-area systems in Figure 2 remains in the balance after the integration of the wind farm to the grid, and the mathematical relationship is represented as follows:

$$\mathsf{\Delta }{P}_{1\mathrm{e}}+\mathsf{\Delta }{P}_{\mathrm{w}}=\mathsf{\Delta }{P}_{2\mathrm{e}}$$

(11)

$\mathsf{\Delta }{P}_{\mathrm{w}}$ is not a state variable. When LFOs occur in the system, the changing trend of the power increment at any node is consistent with that of the frequency increment at the same point of the system. The power and frequency for any point have the same oscillation mode. Therefore, it can be considered that their increments have a proportional relationship, which can be simplified as follows:

$$\mathsf{\Delta }{P}_{\mathrm{w}}=g_1\cdot \mathsf{\Delta }{f}_{\mathrm{B}}$$

(12)

Similarly, there is a linear relationship between the frequency increment of the system and the rotor angular velocity of the system generator. In the case that the regional structure is not completely symmetrical, the proportional relation can be represented as follows:

$$\mathsf{\Delta }{f}_{\mathrm{B}}=g_2\cdot \mathsf{\Delta }{\omega }_1=g_3\cdot \mathsf{\Delta }{\omega }_2$$

(13)

The linear relationships between (12) and (13) with regard to the state variables $\mathsf{\Delta }{\omega }_1$ and $\mathsf{\Delta }{\omega }_2$ are represented in (14).

$$\{\begin{array}{c}\mathsf{\Delta }{P}_{\mathrm{w}}=k_3\mathsf{\Delta }{\omega }_1=k_4\mathsf{\Delta }{\omega }_2\\ k_3=g_1{g}_2\\ k_4=g_1{g}_3\end{array}$$

(14)

Uniting (10)–(14), the state variables can represent $\mathsf{\Delta }{P}_{1e}$ and $\mathsf{\Delta }{P}_{2e}$. Combining with rotor motion (1) derived from the above second-order classical model, the wind farm is incorporated into the PFA corresponding to area A1, and the state equation is represented in (15). When the wind power is incorporated into the PRA corresponding to area A2, the state equation is represented in (16).

$$\left[\begin{array}{c}\mathsf{\Delta }\stackrel{·}{{\delta }_1}\\ \mathsf{\Delta }\stackrel{·}{{\delta }_2}\\ \mathsf{\Delta }{\stackrel{·}{\omega }}_1\\ \mathsf{\Delta }{\stackrel{·}{\omega }}_2\end{array}\right]=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ -\frac{k_1{k}_2}{T_{\mathrm{J}1}(k_1+k_2)}& \frac{k_1{k}_2}{T_{\mathrm{J}1}(k_1+k_2)}& -\frac{D_1}{T_{\mathrm{J}1}}+\frac{k_1{k}_3}{T_{\mathrm{J}1}(k_1+k_2)}& 0\\ -\frac{k_1{k}_2}{T_{\mathrm{J}2}(k_1+k_2)}& \frac{k_1{k}_2}{T_{\mathrm{J}2}(k_1+k_2)}& \frac{k_1{k}_3}{T_{\mathrm{J}2}(k_1+k_2)}& -\frac{D_2}{T_{\mathrm{J}2}}\end{array}\right]\left[\begin{array}{c}\mathsf{\Delta }{\delta }_1\\ \mathsf{\Delta }{\delta }_2\\ \mathsf{\Delta }{\omega }_1\\ \mathsf{\Delta }{\omega }_2\end{array}\right]$$

(15)

$$\left[\begin{array}{c}\mathsf{\Delta }\stackrel{·}{{\delta }_1}\\ \mathsf{\Delta }\stackrel{·}{{\delta }_2}\\ \mathsf{\Delta }{\stackrel{·}{\omega }}_1\\ \mathsf{\Delta }{\stackrel{·}{\omega }}_2\end{array}\right]=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ -\frac{k_1{k}_2}{T_{\mathrm{J}1}(k_1+k_2)}& \frac{k_1{k}_2}{T_{\mathrm{J}1}(k_1+k_2)}& -\frac{D_1}{T_{\mathrm{J}1}}& \frac{k_1{k}_4}{T_{\mathrm{J}2}(k_1+k_2)}\\ -\frac{k_1{k}_2}{T_{\mathrm{J}2}(k_1+k_2)}& \frac{k_1{k}_2}{T_{\mathrm{J}2}(k_1+k_2)}& 0& -\frac{D_2}{T_{\mathrm{J}2}}+\frac{k_1{k}_4}{T_{\mathrm{J}2}(k_1+k_2)}\end{array}\right]\left[\begin{array}{c}\mathsf{\Delta }{\delta }_1\\ \mathsf{\Delta }{\delta }_2\\ \mathsf{\Delta }{\omega }_1\\ \mathsf{\Delta }{\omega }_2\end{array}\right]$$

(16)

The impact of integrated wind power on the power system damping is mainly investigated through the quantitative changes of the system damping characteristics. Before and after the integration, the change of damping characteristics can be discussed according to (2), (15), and (16).

Keeping the system structure unchanged, the damping sum of all the modes of the system is constant and equals the sum of all the characteristic roots of the system [28]. Furthermore, the sum of diagonal elements ${\mathit{A}}_{eq}$ in (2) is always equal. When wind power is incorporated into the PFA, it can be seen from (15) that the damping sum of the system is $-\frac{D_1}{T_{\mathrm{J}1}}-\frac{D_2}{T_{\mathrm{J}2}}+\frac{k_1{k}_3}{T_{\mathrm{J}1}(k_1+k_2)}$. The damping increment is $\frac{k_1{k}_3}{T_{\mathrm{J}1}(k_1+k_2)}$, which is the relation of $k_1$, $k_2$, $k_3$. $k_1=\frac{E_1{U}_{\mathrm{w}}}{X_{13}}\cos ({\delta }_{10}+{\delta }_{\mathrm{w}0})$, $k_2=\frac{E_2{U}_{\mathrm{w}}}{X_{23}}\cos ({\delta }_{20}+{\delta }_{\mathrm{w}0})$, and $k_3=g_1{g}_2$ are a relation of $E_1$, $E_2$, $U_{\mathrm{w}}$, $X_{13}$, and $X_{23}$, which are fixed. However, ${\delta }_{20}$, ${\delta }_{\mathrm{w}0}$, $g_1$, and $g_2$ may result in changes of $k_1$, $k_2$, and $k_3$, which cause the change of $\frac{k_1{k}_3}{T_{\mathrm{J}1}(k_1+k_2)}$. When the wind power is incorporated into the PRA, it can be observed from (16) that the damping sum of the system is $-\frac{D_1}{T_{\mathrm{J}1}}-\frac{D_2}{T_{\mathrm{J}2}}+\frac{k_2{k}_4}{T_{\mathrm{J}2}(k_1+k_2)}$. The damping increment is $\frac{k_2{k}_4}{T_{\mathrm{J}2}(k_1+k_2)}$, in which there is $k_4=g_2{g}_3$. Similarly, ${\delta }_{20}$, ${\delta }_{\mathrm{w}0}$, $g_2$, and $g_3$ result in the changes of $k_1$, $k_2$, and $k_4$, which cause the change of $\frac{k_1{k}_4}{T_{\mathrm{J}2}(k_1+k_2)}$.

Based on the mathematical derivation and analysis above, the system damping increment may be positive or negative in the impact of the integrated wind power on the power systems. The effect depends on the various parameters of the integrated wind power, such as how far it is connected to the grid, how large-scale the capacity is, and in which area the integrated location is, PFA or PRA. The wind power impact is related to the actual changes of system damping, according to the practical structure of the power systems containing the wind farm.

3. Simulation Example and Result Analysis

3.1. The Impact of the Different Areas of Wind Power on the System LFOs

In this paper, the wind farm is connected to the double loop line of the IEEE two-area four-machine power systems [27]. Before the connection of the wind farm, area A1 is mainly composed of generators G1 and G2, and area A2 is mainly composed of generators G3 and G4. Area A1 is set to transmit 413 MW of active power to area A2 through the tie line. Hence, area A1 is the PFA, and area A2 is the PRA. According to the literature [25], the wind farm consisting of DFIGs can be simulated by an equivalent DFIG denoted with G_w, which is connected to node 7 in the PFA and node 9 in the PRA, as shown in Figure 4 and Figure 5, respectively. The green ellipse dotted line frame in Figure 4 and Figure 5 is the integrated wind farm. The wind farm is equivalent, practically, to the WT of 25 DFIGs, which has the capacity of 2 MW for each one. Hence, the capacity of G_W equals 50 MW.

Figure 4. The wind power integrated into the PFA.

Figure 5. The wind power integrated into the PRA.

Before the DFIG reaches a stable operating state, it needs a period of 10–15 s from the start to the stable state. Therefore, the small disturbance is set at t = 20 s after the DFIG starts. The square wave pulse signals, with 5% of the stable transmission voltage, which is set as the small disturbance with a duration of 0.1 s, are added to G1 and G3 at the same time. Under the perturbation excitation, the intra-area oscillation of the rotor angular velocity of the synchronous generator and the inter-area oscillation of the active power of the tie line between node 7 and node 9 are observed.

In the paper, three operating conditions are studied under the condition of the no-power system stabilizer (PSS) in the system. First, the IEEE two-area four-machine power systems suffer small disturbances without integrated wind power. Second, the power system suffers small disturbance with the wind power integrated into the PFA, as shown in Figure 4. Third, the power system suffers small disturbance with the wind power integrated into the PRA, as shown in Figure 5. According to the data of the oscillation modes in Table 1, Table 2 and Table 3, the mode distributions of the intra-area oscillation in G1–G4 are drawn out in Figure 6a–d and that of the intra-area oscillation in the tie line are illustrated in Figure 6e in three operating conditions. The variables $\sigma$ and $\omega$ denote the real part and imaginary part of the eigenvalue, respectively. The eigenvalues of the oscillation modes are identified by the TLS-ESPRIT algorithm [29] in the paper. The variable $\xi$ is the damping ratio of the corresponding mode, which is equal to $\frac{-\sigma }{\sqrt{{\sigma }^2+{\omega }^2}}$. The intra-area oscillation modes are characterized with the oscillation mode parameters of the rotor angular velocity of the four generators (G1–G4). The inter-area oscillation modes are characterized with the oscillation mode parameters of the active power in the tie line [30]. The conjugate eigenvalues are symmetrically distributed about the real axis in the complex plane; so, the imaginary part is not depicted in Figure 6.

Table 1. Oscillation modes of the system without wind power.

Object	Mode	$\mathit{\sigma }$	$\mathit{\omega }$ (rad/s)	$\mathit{\xi }$
G1	Mode #1	−0.5809	±7.0282	0.0824
	Mode #2	0.0871	±4.0228	−0.0217
	Mode #3	−1.0705	±1.1010	0.6971
G2	Mode #1	−0.6319	±7.0404	0.0894
	Mode #2	0.0620	±4.0274	−0.0154
	Mode #3	−0.1028	±0.8767	0.1165
G3	Mode #1	−0.5637	±7.2388	0.0776
	Mode #2	0.0927	±4.0292	−0.0230
	Mode #3	−0.8445	±1.0654	0.6212
G4	Mode #1	−0.6301	±7.2762	0.0863
	Mode #2	0.1092	±4.0185	−0.0272
	Mode #3	−0.0681	±0.8490	0.0800
PL	Mode #1	−0.2247	±6.9636	0.0323
	Mode #2	0.1072	±4.0257	−0.0266
	Mode #3	−0.0695	±0.7260	0.0953

Table 2. Oscillation modes of the system with wind power integrated into PFA.

Object	Mode	$\mathit{\sigma }$	$\mathit{\omega }$ (rad/s)	$\mathit{\xi }$
G1	Mode#1	−0.5868	±7.0125	0.0834
	Mode #2	0.0463	±4.0702	−0.0114
	Mode #3	−1.3740	±1.7076	0.6269
G2	Mode #1	−0.5559	±6.9174	0.0801
	Mode #2	0.0438	±4.0686	−0.0108
	Mode #3	−0.0484	±0.6640	0.0727
G3	Mode #1	−0.5835	±7.2156	0.0806
	Mode #2	0.0429	±4.0650	−0.0106
	Mode #3	−0.1245	±0.7685	0.1599
G4	Mode #1	−0.6608	±7.2553	0.0907
	Mode #2	0.0451	±4.0676	−0.0111
	Mode #3	−0.0244	±0.7069	0.0345
PL	Mode #1	0.0436	±4.0634	−0.0107
	Mode #2	0.0303	±1.1072	−0.0274

Table 3. Oscillation modes of the system with wind power integrated into PRA.

Object	Mode	$\mathit{\sigma }$	$\mathit{\omega }$ (rad/s)	$\mathit{\xi }$
G1	Mode #1	−0.5953	±7.0269	0.0844
	Mode #2	0.1081	±4.0544	−0.0267
	Mode #3	−0.9478	±1.0427	0.6726
G2	Mode #1	−0.5597	±6.9155	0.0807
	Mode #2	0.1108	±4.0523	−0.0273
	Mode #3	−0.0468	±0.5066	0.0920
G3	Mode #1	−0.5480	±7.2324	0.0756
	Mode #2	0.1084	±4.0557	−0.0267
	Mode #3	0.1571	±0.6128	−0.2483
G4	Mode #1	−0.5347	±7.1724	0.0743
	Mode #2	0.1089	±4.0558	−0.0268
	Mode #3	−0.0682	±0.4766	0.1417
PL	Mode #1	0.1092	±4.0525	−0.0269
	Mode #2	0.2013	±0.7349	−0.2642

Figure 6. The modes distributions of intra-area oscillation and inter-area oscillation. (a) G1; (b) G2; (c) G3; (d) G4; and (e) tie line.

The following are the discussions on Figure 6:

(1) The green squares corresponding to the eigenvalues of each mode under the three conditions are mainly located on the left of the red angle. The eigenvalues on the corresponding complex plane generally shift to the left under the condition of wind power integrated into the PFA, compared with no wind power in the system. The damping characteristics are improved and the LFOs are suppressed.

(2) The blue circles corresponding to the eigenvalues of each mode in the three operating conditions are mainly located on the right of the red angle. The eigenvalues on the corresponding complex plane generally shift to the right when the wind power is integrated into the PRA, compared with the condition of no wind power in the system. When the wind power is integrated into the PRA, the damping characteristics become worse and the LFOs are aggravated slightly.

(3) The wind power integration has an impact on the dominant mode of the rotor angular velocity oscillation of the four generators G1~G4. The variation range of the eigenvalues is small, and the impact is not significant. However, the intra-area oscillation of the active power of the tie line is obviously suppressed, which can be observed from the existence of the red angle but no blue circle or green square nearby in Figure 6e. In addition, the corresponding eigenvalue of the blue circle in the lower right corner of Figure 6e indicates that wind power integrated into the PRA increases the oscillation modes and weakens the damping characteristics of the system.

Figure 7 represents the fact that the damping ratio of Mode#1 of the dominant mode of the intra-area oscillation fluctuates in the range of 0.07 to 0.09, which shows that the wind power has a slight impact. However, with the wind power integration into the PFA, the damping ratio of the dominant mode of the inter-area oscillation changes from −0.0266 to −0.0107, which in turn improves the damping characteristics of the system. In contrast, the damping ratio of the wind power integration into PRA is −0.0269, which is a slight decrease. The minimum damping ratio of the intra-area oscillation is reduced to −0.2483, corresponding to Mode#3 on G3, and that of the inter-area oscillation in Mode#2 is reduced to −0.2642.

Figure 7. Changes of damping ratio before and after wind power integration.

3.2. The Impact of Connection Distance of the Wind Power on the System LFOs

The schematic diagram of a single line of the IEEE two-area four-machine power systems, including the integrated wind power whose connection distance can be adjusted, is shown in Figure 8. Area A1 consists of the generators G1 and G2 and the wind farm. The capacity of the wind farm is set at 50 MW, which is equivalently provided by 25 DFIGs of 2 MW in the simulation. The thick red line marked with $l_{\mathrm{w}}$ represents the connection distance of the wind power and is used as an adjustable object. The area A2 mainly consists of G3 and G4. As area A1 is set to transmit 413 MW of active power to area A2 through the tie line, A1 and A2 represent the PFA and the PRA, respectively. The small disturbance, including the disturbance amplitude and the disturbance period, is set the same as Section 3.1. $l_{\mathrm{w}}$ is set as 10 km, 50 km, 100 km, 200 km, and 300 km, respectively. With the different connection distances of the integrated wind power, the oscillation modes of the rotor angular velocity of G1–G4 and the active power of the tie line are tested and recorded in Table 4, Table 5, Table 6, Table 7 and Table 8.

Figure 8. Schematic diagram of adjustable connection distance of integrated wind power.

Table 4. $l_{\mathrm{w}}$ = 10 km.

Object	Mode	$\mathit{\sigma }$	$\mathit{\omega }$ (rad/s)	$\mathit{\xi }$
G1	Mode #1	−0.5869	±7.0114	0.0834
	Mode #2	0.0514	±4.1136	−0.0125
	Mode #3	−1.5415	±2.0295	0.6049
G2	Mode #1	−0.5544	±6.9159	0.0799
	Mode #2	0.0471	±4.1054	−0.0115
	Mode #3	−0.0512	±0.6694	0.0763
G3	Mode #1	−0.5832	±7.2138	0.0806
	Mode #2	0.0429	±4.1124	−0.0104
	Mode #3	−0.1772	±0.7630	0.2262
G4	Mode #1	−0.6592	±7.2562	0.0905
	Mode #2	−4.9058	±3.2432	0.8342
	Mode #3	0.0481	±4.1112	−0.0117
	Mode #4	−0.0398	±0.7003	0.0567
PL	Mode #1	0.0468	±4.1099	−0.0114
	Mode #2	−0.2904	±6.2922	0.0461
	Mode #3	0.0964	±0.6716	−0.1421

Table 5.  $l_{\mathrm{w}}$ = 50 km.

Object	Mode	$\mathit{\sigma }$	$\mathit{\omega }$ (rad/s)	$\mathit{\xi }$
G1	Mode #1	−0.5897	±7.0044	0.0839
	Mode #2	0.0721	±3.9967	−0.0180
	Mode #3	−0.2326	±0.7583	0.2933
G2	Mode #1	−0.5646	±6.9093	0.0814
	Mode #2	0.0699	±3.9986	−0.0175
	Mode #3	−0.0710	±0.6669	0.1059
G3	Mode #1	−0.5054	±7.2594	0.0695
	Mode #2	0.0683	±3.9903	−0.0171
	Mode #3	−0.0708	±0.9064	0.0779
G4	Mode #1	−0.5606	±7.1206	0.0785
	Mode #2	0.0732	±3.9895	−0.0183
	Mode #3	−0.0461	±0.6661	0.0690
PL	Mode #1	0.0696	±3.9922	−0.0174
	Mode #2	0.2988	±0.6986	−0.3933

Table 6. $l_{\mathrm{w}}$ = 100 km.

Object	Mode	$\mathit{\sigma }$	$\mathit{\omega }$ (rad/s)	$\mathit{\xi }$
G1	Mode #1	−0.5768	±7.0187	0.0819
	Mode #2	0.0801	±3.9932	−0.0201
	Mode #3	−0.8717	±0.8444	0.7183
G2	Mode #1	−0.5540	±6.8997	0.0800
	Mode #2	0.0788	±3.9940	−0.0197
	Mode #3	−0.1318	±0.6801	0.1903
G3	Mode #1	−0.5027	±7.2622	0.0691
	Mode #2	0.0766	±3.9886	−0.0192
	Mode #3	0.0419	±0.8395	−0.0498
G4	Mode #1	−0.5194	±7.1183	0.0728
	Mode #2	0.0800	±3.9868	−0.0201
	Mode #3	−0.1182	±0.7211	0.1618
PL	Mode #1	0.0775	±3.9902	−0.0194
	Mode #2	0.3224	±0.6805	−0.4281

Table 7. $l_{\mathrm{w}}$ = 200 km.

Object	Mode	$\mathit{\sigma }$	$\mathit{\omega }$ (rad/s)	$\mathit{\xi }$
G1	Mode #1	0.0891	±3.9848	−0.0224
	Mode #2	−0.4344	±7.0341	0.0616
	Mode #3	0.0302	±1.0444	−0.0289
G2	Mode #1	−0.5224	±6.8893	0.0756
	Mode #2	0.0840	±3.9819	−0.0211
	Mode #3	−0.1453	±0.9354	0.1535
G3	Mode #1	0.0837	±3.9824	−0.0210
	Mode #2	−0.5783	±7.2186	0.0799
	Mode #3	0.1729	±1.0368	−0.1645
G4	Mode #1	0.0868	±3.9804	−0.0218
	Mode #2	−0.3566	±7.0398	0.0506
	Mode #3	0.0629	±1.0646	−0.0590
PL	Mode #1	0.0857	±3.9860	−0.0215
	Mode #2	0.3259	±0.6785	−0.4330

Table 8. $l_{\mathrm{w}}$ = 300 km.

Object	Mode	$\mathit{\sigma }$	$\mathit{\omega }$ (rad/s)	$\mathit{\xi }$
G1	Mode #1	−0.3745	±7.0774	0.0528
	Mode #2	0.0915	±3.9652	−0.0231
	Mode #3	0.1439	±1.3245	−0.1080
G2	Mode #1	0.2916	±7.5212	−0.0387
	Mode #2	−0.4981	±6.8604	0.0724
	Mode #3	0.0912	±3.9643	−0.0230
	Mode #4	0.3169	±1.3077	−0.2355
G3	Mode #1	−0.6302	±7.0251	0.0893
	Mode #2	0.0913	±3.9695	−0.0230
	Mode #3	0.3995	±1.3254	−0.2886
G4	Mode #1	−0.2189	±6.9959	0.0313
	Mode #2	0.0937	±3.9657	−0.0236
	Mode #3	0.3436	±1.3607	−0.2448
PL	Mode #1	0.1813	±7.2515	−0.0250
	Mode #2	0.0935	±3.9681	−0.0236
	Mode #3	0.2956	±0.7389	−0.3714

The following are discussions on Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13:

Figure 9. Rotor angular velocity oscillation of G1: (a) Oscillation output; (b) eigenvalues distribution.

Figure 10. Rotor angular velocity oscillation of G2: (a) Oscillation output; (b) eigenvalues distribution.

Figure 11. Rotor angular velocity oscillation of G3: (a) Oscillation output; (b) eigenvalues distribution.

Figure 12. Rotor angular velocity oscillation of G4: (a) Oscillation output; (b) eigenvalues distribution.

Figure 13. Active power oscillation of the tie line: (a) Oscillation output; (b) eigenvalues distribution.

According to Figure 9a, Figure 10a, Figure 11a, Figure 12a and Figure 13a, as the connection distance of the wind power increases from 10 km to 300 km; the vibration amplitudes of both intra-area and inter-area oscillations present an increasing trend. In the period of 20 s after the small disturbance occurs, the overall damping ratio of the system decreases. Therefore, the LFO problem worsens, and the system stability weakens.

Figure 9b, Figure 10b, Figure 11b, and Figure 12b describe the eigenvalue distributions of the oscillation modes of the rotor angular velocity of G1–G4 related to the different connection distances of the wind power. Figure 13b describes the eigenvalue distributions of the active power oscillation modes of the tie line with the connection distance changes. It is illustrated that the eigenvalues of each mode generally move to the right in the complex plane with the connection distance increased, which makes the overall system stability worse. Although mode#1′s eigenvalue of G3 moves to the left in the complex plane, the weak and negative damping characteristics of the other two dominant modes determine the increasing amplitude of the intra-area LFOs. On the other hand, it is shown that the inter-area LFOs increase significantly with the connection distance increased in Figure 13.

In addition, it can be seen from Figure 14 that the damping ratio of the dominant mode of the intra-area oscillation is little changed with the increase in distance. With the change of distance from 10 km to 300 km, the damping ratios of the active power of the tie line are −0.0114, −0.0174, −0.0194, −0.0215, and −0.0236, successively, showing an obvious decrease. The damping characteristic of the inter-area oscillation obviously deteriorates, and the system stability becomes worse.

Figure 14. Changes of damping ratio for a different connection distance.

3.3. The Impact of the Capacity of the Wind Power on the System LFOs

The wind farm with adjusted capacity is integrated into the IEEE two-area four-machine power systems, as shown in Figure 15. The wind farm is equivalent to a DFIG whose capacity can be set at a different value. The equivalent DFIG is connected to bus node 7 in area A1 of the power systems. The connection distance $l_{\mathrm{w}}$ is fixed at 15 km, which benefits the wind power transmission. In order to study the impact of the capacity of the integrated wind power on the LFOs of the power systems, the capacity of the wind power is set at different values. The wind power output $P_{\mathrm{w}}$ is adjusted by the number of DFIGs of 2 MW, and the other parameters of the system were the same as those in Figure 6 in Section 3.2.

Figure 15. Schematic diagram of the adjustable capacity of integrated wind power.

The capacity of the integrated wind power is set as 10 MW, 50 MW, 100 MW, 300 MW, and 500 MW, respectively. The impact of different capacities on the oscillation modes of the intra-area and inter-area LFOs is tested. The oscillation modes of the rotor angular velocity of G1–G4 and the active power of the tie line are tested and shown in Table 9, Table 10, Table 11, Table 12 and Table 13. Figure 16a, Figure 17a, Figure 18a, Figure 19a and Figure 20a show that the amplitude of both the intra-area oscillations and the inter-area oscillations presents a decreasing trend with the capacity of the wind power increased from 10 MW to 500 MW. Figure 16b, Figure 17b, Figure 18b, Figure 19b and Figure 20b describe the eigenvalues of the oscillation modes of the rotor angular velocity of G1–G4 and the active power of the tie line with the changes of the capacity of the integrated wind power.

Table 9. $P_{\mathrm{w}}$ = 10 MW.

Object	Mode	$\mathit{\sigma }$	$\mathit{\omega }$ (rad/s)	$\mathit{\xi }$
G1	Mode #1	−0.5686	±7.0010	0.0810
	Mode #2	0.0968	±4.1159	−0.0235
	Mode #3	0.0681	±0.6361	−0.1065
G2	Mode #1	−0.5362	±6.9257	0.0772
	Mode #2	0.0978	±4.1147	−0.0238
	Mode #3	−0.0478	±0.5757	0.0827
G3	Mode #1	−0.5566	±7.2391	0.0767
	Mode #2	0.0991	±4.1140	−0.0241
	Mode #3	−0.0799	±0.5212	0.1515
G4	Mode #1	−0.5426	±7.1518	0.0757
	Mode #2	0.1018	±4.1155	−0.0247
	Mode #3	−0.0372	±0.4784	0.0775
PL	Mode #1	0.0988	±4.1140	−0.0240
	Mode #2	0.1435	±0.6649	−0.2110

Table 10. $P_{\mathrm{w}}$ = 50 MW.

Object	Mode	$\mathit{\sigma }$	$\mathit{\omega }$ (rad/s)	$\mathit{\xi }$
G1	Mode #1	−0.5867	±7.0125	0.0834
	Mode #2	0.0464	±4.0702	−0.0114
	Mode #3	−0.0257	±0.7032	0.0365
G2	Mode #1	−0.5559	±6.9174	0.0801
	Mode #2	0.0438	±4.0686	−0.0108
	Mode #3	−0.0484	±0.6640	0.0727
G3	Mode #1	−0.5834	±7.2156	0.0806
	Mode #2	0.0430	±4.0650	−0.0106
	Mode #3	−0.1244	±0.7685	0.1598
G4	Mode #1	−0.6607	±7.2553	0.0907
	Mode #2	0.0451	±4.0676	−0.0111
	Mode #3	−0.0244	±0.7069	0.0345
PL	Mode #1	0.0436	±4.0635	−0.0107
	Mode #2	0.0303	±1.1071	−0.0274

Table 11. $P_{\mathrm{w}}$ = 100 MW.

Object	Mode	$\mathit{\sigma }$	$\mathit{\omega }$ (rad/s)	$\mathit{\xi }$
G1	Mode #1	−0.6124	±6.9808	0.0874
	Mode #2	0.0349	±3.9510	−0.0088
	Mode #3	−0.0466	±0.7909	0.0588
G2	Mode #1	−0.5914	±6.8905	0.0855
	Mode #2	0.0320	±3.9483	−0.0081
	Mode #3	−0.0318	±0.6593	0.0482
G3	Mode #1	−0.6149	±7.1886	0.0852
	Mode #2	0.0341	±3.9480	−0.0086
	Mode #3	−0.0616	±0.6992	0.0878
G4	Mode #1	−0.5889	±7.0858	0.0828
	Mode #2	0.0399	±3.9481	−0.0101
	Mode #3	−0.0265	±0.6084	0.0435
PL	Mode #1	0.0339	±3.9422	−0.0086
	Mode #2	0.0674	±1.2951	−0.0520

Table 12. $P_{\mathrm{w}}$ = 300 MW.

Object	Mode	$\mathit{\sigma }$	$\mathit{\omega }$ (rad/s)	$\mathit{\xi }$
G1	Mode #1	−0.7655	±6.8725	0.1107
	Mode #2	−0.0346	±3.6981	0.0094
	Mode #3	−0.0061	±0.6995	0.0087
G2	Mode #1	−0.7323	±6.7197	0.1083
	Mode #2	−0.0253	±3.6977	0.0068
	Mode #3	−0.0282	±0.6665	0.0423
G3	Mode #1	−0.7612	±7.0388	0.1075
	Mode #2	−0.0295	±3.7020	0.0080
	Mode #3	−0.1315	±0.7283	0.1777
G4	Mode #1	−0.7511	±6.8930	0.1083
	Mode #2	−0.0321	±3.7044	0.0087
	Mode #3	−0.0316	±0.7136	0.0442
PL	Mode #1	−0.0250	±3.6985	0.0068
	Mode #2	−0.0864	±1.5045	0.0573

Table 13. $P_{\mathrm{w}}$ = 500 MW.

Object	Mode	$\mathit{\sigma }$	$\mathit{\omega }$ (rad/s)	$\mathit{\xi }$
G1	Mode #1	−0.9017	±6.6822	0.1337
	Mode #2	−0.0750	±3.3351	0.0225
	Mode #3	−0.0097	±0.6907	0.0140
G2	Mode #1	−0.8103	±6.4869	0.1240
	Mode #2	−0.0950	±3.3221	0.0286
	Mode #3	−0.0361	±0.6752	0.0534
G3	Mode #1	−0.7154	±6.9459	0.1025
	Mode #2	−0.0820	±3.3208	0.0247
	Mode #3	−0.0320	±0.7344	0.0435
G4	Mode #1	−0.8395	±6.6771	0.1247
	Mode #2	−0.0685	±3.3253	0.0206
	Mode #3	−0.0496	±0.6883	0.0719
PL	Mode #1	−0.0776	±3.3239	0.0233
	Mode #2	−0.1143	±0.7413	0.1524

Figure 16. Rotor angular velocity oscillation of G1: (a) Oscillation output; (b) eigenvalues distribution.

Figure 17. Rotor angular velocity oscillation of G2: (a) Oscillation output; (b) eigenvalues distribution.

Figure 18. Rotor angular velocity oscillation of G3: (a) Oscillation output; (b) eigenvalues distribution.

Figure 19. Rotor angular velocity oscillation of G4: (a) Oscillation output; (b) eigenvalues distribution.

Figure 20. Active power oscillation of the tie line: (a) Oscillation output; (b) eigenvalues distribution.

The following are the discussions on Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20:

According to Figure 16a, Figure 17a, Figure 18a, Figure 19a and Figure 20a, in the period of 20 s after the small disturbance occurs, the overall damping ratio of the system increases. Therefore, the LFOs have been alleviated, and the power system stability is enhanced. In addition, from the point of view of the eigenvalue change shown in Figure 16b, Figure 17b, Figure 18b, Figure 19b and Figure 20b, the eigenvalue of each mode generally moves to the left in the complex plane. It means that the system’s stability is enhanced. The imaginary part of the modal eigenvalues shows a downward trend as a whole. Therefore, the dominant mode frequency of the local oscillation and the interval oscillation decreases with the increase in the wind power output. It is worthy of attention that the damping of the dominant modes of the active power of the tie line increases significantly with the increase in the wind power output capacity. The larger capacity of the integrated wind power suppresses the LFOs better; the damping characteristics of the system are improved, and the system stability is obviously enhanced.

As shown in Figure 21, the damping ratio of the dominant mode of the intra-area oscillation (Mode#1) shows an increasing trend, and the damping ratio of the dominant mode of the regional oscillation also shows an increasing trend. The maximum damping ratio will reach 0.1337 and 0.0233, respectively, and the stability of the system will be improved.

Figure 21. Changes of damping ratio for different output capacities.

4. Conclusions

This paper conducts a detailed theoretical derivation and analysis of the damping characteristics of the two-area interconnected power system. The mathematical relations are used to analyze the changes in the damping characteristics caused by wind power integration into the two-area interconnected system. The LFO model of the IEEE two-area four-machine power system is selected as an example to analyze the impact of wind power integration on the damping characteristics of the power system. The results are summarized as follows:

(1) The wind power consisted of DFIGs which are integrated into the PFA to suppress the system‘s LFOs. In contrast, the integration to the PRA increases the LFOs slightly.

(2) With the increase in the connection distance of the wind power, the amplitudes of both the intra-area and the inter-area oscillations tend to increase. The system’s LFOs increase, and its stability weakens.

(3) As the capacity of the integrated wind power increases, the amplitudes of the intra-area and inter-area oscillations tend to decrease, and the damping ratio of the system increases as a whole. The stability of the system is enhanced, and the LFO problem is alleviated to some extent.

As is known from the analysis of the above results, wind power integration not only changes the structure of the power systems, but also alters the damping characteristics of the large-scale interconnected systems. Not only should the design and construction of the wind farms be optimized, but the appropriate grid parameters should also be constructed to improve the stability of the power system.