Impact of Large-scale Wind Power Integration on Small Signal Stability Based on Stability Region Boundary

Up until now, study results on the impact of large-scale wind power integration on small signal stability have often been in conflict. Sometimes, the conclusions are even completely opposite, making people unable to agree on which is right. The reason behind this phenomenon is that most of these studies are based on a certain grid and typical working conditions, so conclusions are reached by comparing changes in oscillation mode, one by one. This study method lacks a broader perspective, and often reflects only a part of the grid conditions. However, the small signal stability region boundary describes the critical operating range of power system small signal stability as a whole, making possible an overall evaluation of the system from a more macro perspective. Thus it is more suitable for analysis of the impact of large-scale wind power integration on small signal stability. Based on the above, using the model of wind farm integration to the single-machine infinite bus power system, this paper studies the impact of wind power integration scale and the coupling strength with synchronous generator on small signal stability through the comparison of the stability region boundaries, thus providing a new method and support for analyzing the impact of wind power integration on small signal stability.


Introduction
With the aggravation of the energy crisis and increasingly prominent environmental problems, renewable energy has become the most effective solution [1].Among the various methods of renewable energy generation, wind power generation, seen as the most technologically sophisticated and economically beneficial [2], has been developing rapidly [3,4].However, there are great differences in structure and control between wind turbine and conventional units [5].Thus, great attention [6,7] has been paid to large-scale wind power integration.Specifically, the impact of large-scale wind power integration on power system small signal stability is a high research priority.
To date, many scholars have researched this subject.Shi et al. [8] adopted the Monte Carlo simulation technique to figure out the relationship between small signal stability and the wind generation intermittence.Bu et al. [9] studied the impact of stochastic uncertainty of grid-connected wind generation on probabilistic small signal stability of the power system based on the Gram-Charlier expansion and the probabilistic density function of critical eigenvalues.Soleimanpour and Mohammadi [10] presented a new method based on two-point estimation for probabilistic eigenvalues analysis in power systems containing wind energy, obtaining information on probabilistic system stability with less computation.Slootweg et al. [11] took a system with two generators and a two-area test system with four generators as their example and studied the impact of wind power integration on power system small signal stability by observing the movement of the eigenvalues, investigated by gradually replacing the power generated by the synchronous generators with wind power.It was concluded from the results that due to the reduction of the number of synchronous generators participating in the oscillation, wind power integration will improve the damping of the power system in which constant speed wind turbines have better effects than variable speed turbines.Anaya-Lara et al. [12] tested a three-generator system and came to similar conclusions.However, not all researchers agreed.Mendonca and Peas Lopes [13] tested a three-area system with six generators to study the influence of wind power integration, using the eigenvalue method and multi-scenario technology.Their results indicated that when the change of wind power output counteracts that of the conventional units in the same area, the damping of oscillation mode does not change significantly.Knuppel et al. [14] analyzed the impact of full-load converter interfaced wind turbines on small signal stability of power system with wind energy integration based on the eigenvalue method and sensitivity method.Their results also indicate that wind energy does not participate in the power system oscillation mode and the characteristics of the system oscillation mode are virtually unchanged by wind energy penetration.Meanwhile, some other studies indicated that wind power integration will reduce the power system small signal stability in certain circumstances.Gautam et al. [15] replaced the DFIG (Doubly Fed Induction Machine) machines with conventional generators and then studied the impact of wind power penetration on the power system small signal stability by evaluating the sensitivity of its eigenvalues.Their results showed that wind power exerts a positive influence on part modes but adversely affected the other modes.Hagstrom et al. [16] proposed that integration of wind power generated by DFIG and DDSG (Direct Drive Synchronous Generator) will reduce the damping of oscillation mode by time-domain simulation analysis of Nordic grid.Sanchez-Gasea et al. [17] pointed out that wind power integration had different effects on small signal stability under different operation modes.Compared with the synchronous machines, large-scale wind power integration enhances the damping of oscillation mode.However, when the scale of wind power is small, the damping of oscillation mode is reduced.Meanwhile, wind power taking the place of the synchronous machines will lead to the disappearance of part oscillation modes.
Generally speaking, the above studies are divided into two kinds of approaches.One is based on probability, which considers the uncertainty of wind energy and determines the probability distribution of a power system's critical eigenvalues to find the impact of wind power's stochastic uncertainty factors on power system small signal stability.Another approach is a deterministic method, which applies to a certain grid and typical working conditions; conclusions are reached by comparing changes in oscillation mode one by one with the eigenvalue method or time-domain simulation.These study results are often in conflict, making people unable to agree on which is right.The reason is that most of these research methods are based on a certain grid and typical working conditions, reflecting only part conditions of power system small signal stability but lacking a more macro perspective.
However, the small signal stability region boundary [18][19][20] describes the critical operating range of small signal stability of the system as a whole, making overall evaluation of the system state possible from a more macro perspective.Thus it is more suitable for analysis on the impact of large-scale wind power integration on small signal stability.At present, the small signal stability region boundary theory is mainly adopted in the field of conventional power systems and has no application in the study of the impact of wind power on small signal stability yet.
Based on this, the theory of the small signal stability region boundary was studied first.By using the model of wind farm integration to the single-machine infinite bus power system, this paper studies the impact of wind power integration scale and the coupling strength with synchronous generator on small signal stability through comparison of the stability region boundaries, searching for a new way of thinking about and analyzing the impact of wind power integration on small signal stability.

Basic Theory
The dynamics model [21] of power systems can be described by a set of differential-algebraic equations: ( , , ) 0 ( , , ) In Equation ( 1), x∈R m is the stated variable of the system, y∈R n is the algebraic variable, and k∈R p is the system parameter.
Holding the variable k unchanged, Equation (2) satisfies the formula for the equilibrium point of the system.
Then Equation (2) can be linearized as Equation (3): in which: When the gy is nonsingular, the algebraic variables can be eliminated in Equation (3) to create Equation ( 5).
( , , ) where 1 ( , , ) In Equation ( 6), J is the characteristic matrix of the system.According to Lyapunov's first theorem, the small signal stability of the system is decided by the eigenvalues of the characteristic matrix J.

Composition of Small Signal Stability Region Boundary
Under normal circumstances, the eigenvalues are normally less than zero and the system works steadily.However, with the change of parameter K, three circumstances may appear in which the eigenvalue makes the system achieve the small signal stability region boundary: (1) A pair of real eigenvalues cross the imaginary axis, causing saddle node bifurcation (SNB); (2) A pair of conjugate complex eigenvalues cross the imaginary axis, causing Hopf bifurcation (HB); and (3) Matrix gy is singular, causing singularity induced bifurcation (SIB).Thus the small signal stability region boundary ( sssr  ) consists of these three types of point set.
In Equation (7), SNBs is the point set of saddle node bifurcation, HBs is the point set of Hopf bifurcation, and SIBs is the point set of singularity induced bifurcation.Among these, Hopf [22,23] bifurcation relates to the oscillatory instability of the power system.This paper mainly focuses on the small signal stability region boundary formed by Hopf bifurcation in the active power injection space.

Method and Steps
Supposing power sources 1 and 2 are two arbitrary power supplies in the power system associated with the dominant oscillation mode, the balancing machine chosen is one large unit that is unrelated to the dominant oscillation mode and far away from the oscillatory areas.Keeping the other nodal injection power constant except for the balancing machine, the injected active power P1 of power source 1 and P2 of source 2 will form a two-dimensional small signal stability region boundary.
The system small signal stability region boundary is determined by way of system simulation, point by point.The specific method and steps are as follows: (1) Select one small signal stable operating point in the power system as the initial searching point.
(2) Keep the other power injections constant, change the active power output P1, and get the system a new balance point.Then, according to the basic theory in Section 2.1, recalculate the system's eigenvalue of the dominant mode accordingly.
(3) Determine whether the Hopf bifurcation shown in Section 2.2 occurs in the system according to the eigenvalues.If Hopf bifurcation occurs, record the parameters; if Hopf bifurcation does not occur, adjust the active power output P1 according to the real parts of the eigenvalues until Hopf bifurcation occurs in the system.(4) Change the active power output P2 with a fixed step size.For each P2, repeat steps ( 2) and (3) to obtain the new Hopf bifurcation point.(5) When the power output reaches the upper or lower limits or the power flow has no solution in step (4), stop searching.
Connect the Hopf bifurcation points with a smooth curve to obtain the Hopf bifurcation point set of the power system.Considering the existence of upper and lower limits of actual power output, the upper and lower limits of actual power output and the Hopf bifurcation point set together make the power system small signal stability region boundary.The whole process is shown in Figure 1.

Model of DFIG Wind Turbine
This paper adopts the current mainstream wind turbine, the doubly fed induction generator (DFIG) [24][25][26], to study the impact of wind power integration on small signal stability.The 2MW wind turbine built in the software DIgSILENT/PowerFactory was used to do simulations.Its mathematical models are as follows.

The Aerodynamic Model
The aerodynamic model of wind turbines is: In Equation ( 8), PM is the mechanical power of the wind turbine, ρ is the air density, R is the rotor radius, λ is the tip speed ratio; β is the blade pitch angle, and Cp is the power coefficient as a function of λ, β.Cp changes along with λ and β, and Vw is the wind speed.

The Model of the Drive-Train System
The drive-train system of DFIG adopts the so-called two-mass model.In this, a mass block includes the low speed shaft's flexibility to reflect the wind turbine's inertia, and another mass block reflects the generator's inertia.The model of the drive-train system is given by a system of differential equations, as follows: In Equation set (9), θs is the twist in the shaft system, ωt is the speed of the wind turbine, ωr is the speed of the generator rotor, Ht is the inertia constant of the wind wheel, Hg is the inertia constant of the generator's rotor, Tm is the wind turbine's mechanical torque, Te is the generator's electromagnetic torque, and D and Ks respectively represent the drive-train system's equivalent damping coefficient and stiffness coefficient.

Model of the Induction Generator
In the reference d-q coordinate system, the stator adopts the generator convention, while the rotor adopts motor convention.Thus the voltage equations of the doubly fed induction generator are given as follows: In Equation set (10), all the quantities are normalized values.Here, u, I, R, and ψ respectively represent voltage, current, resistance, and flux; the subscripts d and q respectively represent the d axis and q axis components; and the subscripts r and s respectively represent the rotor and stator.The synchronous rotational speed is represented by ω; ωs = ω -ωr = sω (the slip speed), where s is the slip ratio.
The flux equations are: in which Lss = Ls + Lm, Lrr = Lr + Lm, Ls, Lr, Lm are the self-inductance and mutual inductance of stator and rotor, respectively.Arrange Equations ( 10) and ( 11) and set the stator voltage vector of DFIG as the q axis.In the d-q coordinate system, select the stator's flux (ψds, ψds) and the rotor's current (idr, iqr) as state variables.The state equations for DFIG are: where

The Overall Control Model of DFIG
The overall control system of a doubly fed variable speed wind turbine is shown in Figure 2.
The control system mainly consists of two sub-systems: wind turbine control and DFIG control.Wind turbine control includes a speed controller and a pitch angle controller.The pitch angle controller is mainly used to limit the active power of a wind turbine when wind speed exceeds the rated wind speed.The pitch angle is kept at 0° under the normal working state.The speed controller provides a reference value of active power, ref grid P , for the rotor-side converter.DFIG control includes rotor-side converter controller and grid-side converter controller, which are mainly used to control its active and reactive power.For the reactive power control of the converter, set a fixed value for the rotor-side converter's ref grid Q , according to the requirement for the reactive power exchange degree of the interconnection node at steady operation state; set the value of rotor-side converter's ref conv Q to 0 to ensure no reactive power change between the rotor and the grid at normal operation state.Through the converter control, the DFIG can realize the decoupling of active power and reactive power.More details of the DFIG control can be found in document [26] (Hansen et al., 2003).

Wind Farm Model
As the capacity of a single wind turbine is very small, the wind farm is often composed of large numbers of wind turbines.In this paper, m doubly fed induction generators are equivalent to one according to the single machine equivalent method [27,28] (Fernández et al., 2008;Ali et al., 2008).The equivalent model is given in Equation ( 13).
m eq i eq i eq i i i i m s r m eq s eq r eq s r s eq r eq m m m eq i eq i eq i i i i In which the subscript i is the serial number of each wind turbine, eq is the equivalent parameters, m is the total number of equivalent units, and H, K, and D respectively represent the inertial time constant, the shafting stiffness coefficient, and the shafting damping coefficient of the wind turbine.

The Simulation System
At present, the equivalent wind farm is often used to represent the large-scale wind power integration.We can adjust the output of the wind farm to reflect different integration scales of wind power.In order to make this paper's research universal, the wind farm is integrated into the single-machine infinite-bus system through lines, as shown in Figure 3.In Figure 3, bus C is the infinite bus, bus A is a PV node which connects to the conventional generator G1 with excitation system, and bus B is a PQ node which connects to the DFIG wind farm with the capacity of 210MVA.In the analysis, the wind farm is equivalent to one machine.The parameters of one signal wind turbine are shown in Table 1.The excitation system of generator unit G1 adopts the A-type exciter in Appendix D of reference [29].Parameters of power generation unit G1 and the excitation system are shown in Tables 2 and 3.In Table 2, the voltage and power reference values are this machine's rated voltage and capacity.Line type is LGJ-400, L1 of 100 km in length, L2 of 200 km in length, and L3 of 15 km in length.
The model was built using DIgSILENT/PowerFactory simulation software.

Impact of Wind Power Integration Scale
This paper substitutes the constant power source, synchronous generator and synchronous generator with an excitation system of the same capacity for the wind farm in Figure 3, studying the impact of the scale of the wind power integration on small signal stability through the comparison of small signal stability region boundaries.The formation of a small signal stability region boundary is obtained by the method in Section 2. In this study, we mark the active power output of generator G1 as P1, the DFIG and its corresponding substitute power source's active power output as P2, and synchronous generator, which replaced the power supply, as G2, whose parameters are shown in Table 4.The Hopf bifurcation sampling points are shown in Appendix A. According to Tables A1 and A2 in the Appendix A, the small signal stability region boundary contrast between a wind farm and a constant power source when they connect to the power system is shown in Figure 4.As can be seen from Figure 4, the stability region boundaries cross over when the wind farm and constant power source connect to the power system.Below the crossover point, namely when the wind power integration scale is relatively small, the stability region boundary of the wind farm is higher than that of the constant power source, suggesting that the small signal stability of wind farm integration is better than that of constant power source integration.However, above the crossover point, namely when the wind power integration scale is relatively large, the stability region boundary of the wind farm is lower than that of the constant power source, suggesting that the small signal stability of wind farm integration is worse than that of constant power source integration.
In order to further validate the argument, we selected two operating points of the power system: point 1 (220, 180) and point 2 (276.7,30), which are both located in the region surrounded by the two boundaries.Point 1 is above the crossover point, while point 2 is below the crossover point.The method of time-domain simulation was adopted to compare the small signal stability between wind power integration and constant power source integration.The small disturbance was set as the mechanical torque's increasing 0.01 p.u of generator G1 at 1 s, then restored to the original level at 2 s.The simulation time was set at 20 s.This resulted in the power angle swing curves shown in Figures 5 and 6.
As can be seen from Figure 5, when the power system operates at point 1, generator G1's power angle oscillation amplitude will become bigger and bigger with wind farm integration under small disturbance, making the system eventually lose stability.On the other hand, when the constant power source is integrated into the power system, generator G1's power angle oscillation amplitude will become smaller and smaller and ultimately tends to be stable.However, Figure 6 shows that when the power system operates at point 2, generator G1's power angle oscillation amplitude will decrease slowly with wind farm integration under small disturbance; when the constant power source is integrated into the power system, generator G1's power angle is almost in constant amplitude oscillation.The contrast is not very obvious because the region below the crossover point is small and the operating point 2 is close to both instability boundaries.Therefore, the difference between the two stable levels is not obvious.The simulation results of Figures 5 and 6 verify this section's conclusion.

Contrast between Wind Farm and Conventional Generator
In this Section, the synchronous generator G2 with the same capacity was used to substitute for the wind farm.In the range of power flow equations that have solutions, the system is always stable, with small disturbance.In the meantime, the small signal stability region boundary consists of the boundary of power flow equations that have solutions and the upper output limit of the power source.The detailed boundary sampling points refer to Table A3 in the Appendix.
According to Tables A1 and A3 in the Appendix A, the small signal stability region boundary contrast between wind farms and synchronous generators when they connect to the power system is shown in Figure 7.As can be seen from Figure 7, when the wind farm replaces the conventional generators, the small signal stability region boundary is greatly narrowed and the power system small signal stability is reduced.
Operating point 3 (260, 120) is located in the region surrounded by the two boundaries.The same method of time-domain simulation was adopted to validate the argument.The set value of the small disturbance remained unchanged.This resulted in the power angle swing curves shown in Figure 8.
As can be seen from Figure 8, when the power system operates at point 3, generator G1's power angle oscillation amplitude will become bigger and bigger with wind farm integration under small disturbance, making the system eventually lose stability.By contrast, when the constant power source is integrated into the power system, generator G1's power angle oscillation amplitude will become smaller and smaller, ultimately tending to be stable.The simulation results verify this section's conclusion.According to Tables A1 and A4 in the Appendix A, the stability region boundary contrast between a wind farm and a synchronous generator with excitation system when they connect to the power system is shown in Figure 9.
As can be seen from Figure 9, the stability region boundaries also cross over each other when the wind farm and the conventional generator with excitation system connect to the power system.However, these circumstances are contrary to Figure 4. Below the crossover point, namely when the wind power integration scale is relatively small, the stability region boundary of the wind farm is lower than that of the synchronous generator with excitation system, suggesting that the small signal stability of wind farm integration is worse than that of a synchronous generator with excitation system.Above the crossover point, namely when the wind power integration scale is relatively large, the stability region boundary of the wind farm is higher than that of the synchronous generator with excitation system, suggesting that the small signal stability of wind farm integration is better than that of a synchronous generator with excitation system.We selected two operating points, point 4 (180, 180) and point 5 (278, 60), and adopted the same method of time-domain simulation to validate the argument.Point 4 is above the crossover point and Point 5 is below the crossover point; both are located in the region surrounded by the two boundaries.The set value of the small disturbance remains unchanged.This resulted in the power angle swing curves shown in Figures 10 and 11.
As can be seen from Figure 10, when the power system operates at point 4 with wind power integration under small disturbance, generator G1's power angle oscillation amplitude will become smaller and smaller, ultimately tending to be stable.By contrast, when the synchronous generator with excitation system is integrated into the power system, generator G1's power angle oscillation amplitude will become bigger and bigger under small signal, making the system eventually lose stability.The results in Figure 11 are just the opposite and thus verify the correctness of this section's conclusion.The above analyses show that, in terms of small signal stability, compared with a constant power source, synchronous generator, and synchronous generator with excitation system, a wind farm shows significant differences.For a fixed power grid structure, wind power integration does not always increase or decrease the small signal stability of the power system, depending on the scale of wind power integration and the contrast power source type.

Impact of the Coupling Strength of Wind Power and Synchronous Generators on Small Signal Stability
At present, especially in China, large-scale wind power is often transmitted by way of wind-thermal-bundled power transmission [30] (Gao. et al., 2013).The coupling strength of wind power and conventional units has become an important factor affecting power system small signal stability.This section mainly analyzes this subject.As shown in Figure 3, when line L1 is longer, this indicates that the coupling strength between wind power and conventional units is weaker.Thus the length of L1 is adjusted to describe different coupling strengths; in this case, the total length of lines L1 and L2 remains unchanged.We set the length of line L1 to be 50 km, 150 km, and 200 km, respectively, and obtained the sampling points of small signal stability region boundary as shown in Tables A5-A7 in Appendix B.
A comparison of these stability region boundaries is given in Figure 12.The number after the legend represents the length of line L1 in km.As can be seen from Figure 12, the smaller the distance between the wind power and synchronous generator, the stronger the coupling effect between them.Also, the further to the lower left the border line drifts, the smaller the corresponding stable region and the worse the small signal stability.
In order to further validate the argument in this section, we set L1 at 50 km and 200 km and chose operating point 6 (210, 150), which is located in the region surrounded by the two boundaries.The same method of time-domain simulation was adopted to validate the argument.The set value of the small disturbance remained unchanged.This resulted in the power angle swing curves shown in Figure 13.


As can be seen from Figure 13, when the power system operates at point 6, generator G1's power angle oscillation amplitude will become bigger and bigger when L1 = 50 km, making the system eventually lose stability.On the other hand, when L1 = 200 km, generator G1's power angle oscillation amplitude will become smaller and smaller, ultimately tending to be stable.The simulation results in time-domain verify the correctness of this section's conclusion.

Discussion with Respect to Existing Research Results
Based on the conclusions above, this paper attempts to discuss the results of some of the published literature through comparison.As noted above, the results of previous studies are often in conflict.The reason is that most of these studies are based on a certain grid and typical working conditions, which study the impact of different integration scales of wind power on small signal stability by replacing the conventional units with wind turbines.
The published papers often use wind turbines as substitutes for the synchronous machine with excitation system to study the power system small signal stability with wind power integration, only changing the power output of the wind turbine and its substitute conventional units, i.e., keeping P1 fixed and only changing P2 in Figure 9.We labeled the intersection point of the small signal stability boundary lines as point A, whose P1 is P1a.We then redrew it to create Figure 14.Different working conditions correspond to different sizes of P1, which means the line P1 = const is at different positions on the horizontal axis.Based on previous analyses, we got that when P1 < P1a, wind power integration would improve power system small signal stability, which corresponds to the conclusions of reference [11,12]; and that when P1 > P1a, wind power would reduce power system small signal stability, which corresponds to the conclusions of reference [15][16][17].When the size of P1 and P1a is close, the impact of wind power on small signal stability would be very small, which corresponds to the conclusions of reference [13,14].
Therefore, although the conclusions of the current literature seem to be contradictory, they in fact only reflect a part of the grid.Analysis based on small signal stability boundary offered a good interpretation from a qualitative point of view.

Figure 1 .
Figure 1.Flow chart of small signal stability boundaries calculation.

Figure 2 .
Figure 2. Overall control system scheme of DFIG.

4 .
Research on the Impact of Large-Scale Wind Power Integration on Small Signal Stability

Figure 4 .
Figure 4.The stability region boundary contrast between wind farm and constant power source.

Figure 5 .
Figure 5.The angle swing curves when the system is at operation point 1.

Figure 6 .
Figure 6.The angle swing curves when the system is at operation point 2. (deg) 

Figure 7 .
Figure 7.The stability region boundary contrast between wind farm and synchronous machine.

Figure 8 .
Figure 8.The angle swing curves when the system is at operation point 3.

3 .
Contrast between Wind Farm and Synchronous Generator with Excitation System

Figure 9 .
Figure 9.The stability region boundary contrast between wind farm and synchronous generator with excitation system.

Figure 10 .Figure 11 .
Figure 10.The angle swing curves when the system is at operation point 4.

Figure 12 .
Figure 12.The stability region boundary of small-signal stability under different lengths of L1.

Figure 13 .
Figure 13.The angle swing curves when the system is at operation point 6.

Figure 14 .
Figure 14.The research contrast between this article and the published papers.