# Voltage Stability Analysis of Front-End Speed Controlled Wind Turbine Integrated into Regional Power Grid Based on Bifurcation Theory

Authors: Kaide Ren; Hongwei Li; Shuaibing Li; Haiying Dong
Journal: Complexity
 (2020)

---

## Abstract

Since wind power has characteristics such as intermittent and fluctuation, the integration of large-scale wind turbines into the power grid will bring a great impact on the voltage stability of the system. In this paper, the influence of the front-end speed controlled wind turbine (FSCWT) on the system voltage stability is studied. An actual model of the wind turbines, including the FSCWTs, connected to a regional power grid in Zhangye, Gansu Province, is established. Firstly, differential-algebraic equations (DAEs) describing the dynamic characteristics of the wind turbine are given and the mathematical model of the system includes FSCWT is established. The continuation method is used to track the balance solution of the DEAs within given parameter intervals. Based on that, the influence of the reactive power variation and wind speed fluctuation on the stability of system voltage is analyzed through both the bifurcation theory and the time-domain simulation. Results show that the Hopf bifurcation (HB) and the saddle-node bifurcation (SNB) are inherited for the system, indicating that such bifurcations are the essence of nonlinear dynamics that lead to voltage instability. The greater the disturbance of the bifurcation parameterQ1, the shorter the time of voltage collapse and the smaller the stable operation area of the system. With the increase of wind speed, the amplitude of system voltage will increase slightly, but the HB point will appear in advance, which is more likely to lead to voltage instability and further reduce the stable operation area of system voltage.


---

## Body

## 1. Introduction

With the rapid development of wind power generation technology, wind energy has become increasingly utilized. As an important part of the power system, the new installed capacity and the total installed capacity of the wind turbine are increasing year by year [1, 2]. By the end of 2019, the installed capacity of wind power in China accounts for about 10.5% of the total capacity of the system [3]; among which, the installed capacity of wind power accounts for almost 25% of the total installed capacity in typical areas such as Gansu [4]. To make better use of wind power resources and improve economic benefits, wind power systems at home and abroad are increasingly developing in the direction of large capacity, ultra-high voltage, and long-distance transmission, which greatly increases the difficulty of maintaining the system voltage stability. The integration of wind turbines introduces the fluctuation and indirectness of wind speed into the power system, which will inevitably impact on the voltage stability of the integrated system [5, 6]. Besides, the voltage of the system will be disturbed by the reactive power fluctuation caused by wind speed variation. Research shows that in the modern power system, voltage stability accident is more sudden than frequency stability and synchronous operation stability [7].The new energy power system with wind turbines is essentially a nonlinear system [8, 9], in which the grid-connected voltage stability of wind turbines will be affected by many factors, such as the interference of natural wind speed and reactive power changes on wind turbines. The change of voltage stability state of wind turbines connected to the regional power grid is essentially the process of the whole system from a stable state to the bifurcation. The set of different bifurcation points forms a bifurcation hypersurface. Based on this, G. Aronovich and other scholars introduced the bifurcation theory into the field of power system voltage stability analysis, which has become a powerful tool for studying the structural stability of dynamic systems [10].Many scholars at home and abroad have done a lot of research on the voltage stability of conventional wind turbines connected to the regional power grid, and have achieved certain results. In [11], through the dynamic modeling of doubly fed wind turbines and the simulation calculation of the power system including wind farms, the results show that the appropriate wind power access is conducive to the improvement of the voltage level of the regional power grid. When the wind power access capacity is too large, it will lead to the phenomenon of line transmission power crossing and voltage instability. In [12], the stability of the grid-connected voltage of the wind turbine with doubly fed induction generator (DFIG) is studied based on the bifurcation theory, and with the increase of reactive power at load nodes, various bifurcation phenomena will occur in the system, which will lead to voltage collapse. With the increase of wind speed, the system voltage stability margin will be smaller and smaller. In [13], the bifurcation theory is used to analyze the direct-drive permanent magnet synchronous generator (PMSG) incorporated into a three-node simple power system, and the injected active power and reactive power compensation capacity are taken as the bifurcation parameters of the wind power system. The results show that in the absence of reactive power compensation, the system operates at a lower voltage level when the system is effectively compensated by capacitance, and the voltage of each node of the system will be significantly increased. Under the condition of high-power wind power injection, the system voltage will be unstable.The above research mainly focuses on the analysis of grid-connected voltage stability of the conventional wind turbines connected to the grid with converters. One typical feature of such turbines is that they are difficult to provide reactive power and inertia support in case of reactive power shortage or instability of the system [14], and there are many deficiencies in dealing with grid faults. This is mainly because the converter capacity of the DFIG is too small to protect against grid voltage sag. For the wind turbine with PMSG [15], its ability of dynamic response and reactive power regulation are limited owing to the lack of the gearbox. To sum up, although the bifurcation theory has been applied to the grid-connected voltage stability of conventional wind turbines (i.e., DFIG and PMSG), it has not been involved in the front-end speed controlled wind turbine (FSCWT). Based on this, it is of practical significance to analyze the grid-connected voltage stability of the FSCWT by using the bifurcation theory.In contrast, the FSCWT has many advantages, and which solves the above problems. The essential difference of the FSCWT is that it uses a hydraulic torque converter named WinDrive to regulate the speed of the generator at the front-end of the turbine [16], greatly improving the low voltage ride-through capability of the wind turbine and increasing the advantage of reactive power output [17]. At the same time, the EESG is directly coupled with the power grid, which can provide certain transient support for the system in case of failure or instability of the access system. Therefore, this paper proposes to use the bifurcation theory to study the voltage stability of the regional power grid integrated with FSCWTs.The rest of this paper is organized as follows: Section2 listed the structural equation of the power system and introduces different bifurcations, the dynamic differential equations of the system integrated with FSCWTs are modeled in Section 3, operation characteristics and dynamic equivalent modeling of the wind farm are given in Section 4, and voltage stability analysis of FSCWT connected to the regional power grid is simulated in Section 5. Conclusions are finally drawn in Section 6.


## 2. Differential-Algebraic Equation of the Power System

Generally, the stability of the power system can be described by the differential-algebraic equation (DAE), which consists ofn-dimensional differential equations and m-dimensional algebraic equations. Assumed that the two sets of equations are smooth throughout the analysis, the power system can be described using the following equations [18]:(1)x˙=fx,y,μ,0=gx,y,μ,where x is an n-dimensional differential state variable, x ∈ Rn; y is an m-dimensional algebraic state variable, y ∈ Rm; μ is a k-dimensional variable and also a bifurcation parameter in structural stability analysis, μ ∈ Rk (k > 1 and k ∈ Z), f is a differential equation describing the dynamic components of the system, and g is the balanced equation of each node.All solutions belong to (x0, y0, μ0) satisfying that the following equations can be taken as the equilibrium points of the system:(2)fx0,y0,μ0=0,gx0,y0,μ0=0.Consequently, equilibrium solutions can be expressed asM = {(x, y, μ)/f(x, y, μ) = 0, g(x, y, μ) = 0}.It can be seen that the greatest advantage of a general power system is to describe the dynamic behavior of the system with DAEs. To study the dynamic stability of the power system shown in (1), the differential transformation of the system at the equilibrium point (x0, y0, μ0) can be obtained as(3)dxdt=Dxfx0,y0⋅dx+Dyfx0,y0⋅dy,0=Dxfx0,y0⋅dx+Dyfx0,y0⋅dy.The differential-algebraic equation describing the dynamic characteristics of the system can be replaced by equation (3):(4)dxdt=Dxfx0,y0−Dyfx0,y0Dygx0,y0−1×Dxgx0,y0⋅dx,where A = Dxf(x0, y0), B = Dyf(x0, y0), C = Dxf(x0, y0), and D = Dyf(x0, y0).Which can be simplified as(5)x˙=A−BD−1Cx.According to the dynamic theory, the dynamic stability of the system can be determined by the reduced Jacobian matrixJ:(6)J=A−BD−1C.For a given power system, when the system parameters change continuously and pass a certain critical value, the system balance point or the number of periodic orbits will inevitably change. At this time, the system will have a bifurcation phenomenon. The corresponding parameter value is called bifurcation value, and the corresponding equilibrium point is called a bifurcation point [19]. Therefore, the instability of the power system is usually attributed to the occurrence of bifurcation points in the system parameter space, which include dynamic and static bifurcation points. The two bifurcation points can be distinguished by the eigenvalues of the Jacobian matrix shown in equation (6).


### 2.1. Solution of the Static Bifurcation Point

The static bifurcation mainly focuses on the behavior change of the number and stability of the equilibrium points of the system along with the parameter variation. Let the eigenvalue ofJ(μ0) be λa(μ) = αa(μ) + jβa(μ), where αa(μ) and βa(μ) are real functions of the system control parameter (a = 1, 2, …, l). According to the dynamic theory, when J(μ0) has zero eigenvalues, the left and the right eigenvectors of the zero eigenvalues can be defined, i.e., v ∈ Rn and w ∈ Rn. Therefore, the problem of solving the equilibrium solution for the system can be turned into solving the following equations:(7)Fz,μ=0,Fzz,μv=0,FμTz,μv=1,orwTFzz,μ=0,wTz,μ=1,where F(z, μ) = [fT(x, y, μ), gT(x, y, μ)]T is the equilibrium equation, while z = [xTyT]T stands for the state variable.Therefore, the saddle-node bifurcation (SNB, static bifurcation point) of the system can be obtained by using the Newton–Raphson method to solve equation (5) [20].


### 2.2. Solution of the Dynamic Bifurcation Point

During system operation, once the disturbance occurs, the equilibrium state of the system and the number of solutions of the equilibrium equation will change with the variation of the control parameters. For a given control vectorμ, suppose a nonhyperbolic equilibrium point x0 appears in the equilibrium solutions (x, y, μ) with μ = μ0 and x0 satisfies (1), and it is located on the surface of the equilibrium solution M. If a pair of conjugate complex roots with zero real parts appears on the reduced Jacobian matrix J in equation (6), it is considered that the structural stability of the system changes, resulting in dynamic bifurcation, in which the main dynamic bifurcation point is the Hopf bifurcation [21].In conclusion, the change of eigenvalues corresponding to the dynamic and static bifurcation points is clearly described in Figure1.Figure 1
Variation of eigenvalues corresponding to the dynamic and static bifurcation points: (a) static bifurcation; (b) dynamic bifurcation.
(a)(b)


## 2.1. Solution of the Static Bifurcation Point

The static bifurcation mainly focuses on the behavior change of the number and stability of the equilibrium points of the system along with the parameter variation. Let the eigenvalue ofJ(μ0) be λa(μ) = αa(μ) + jβa(μ), where αa(μ) and βa(μ) are real functions of the system control parameter (a = 1, 2, …, l). According to the dynamic theory, when J(μ0) has zero eigenvalues, the left and the right eigenvectors of the zero eigenvalues can be defined, i.e., v ∈ Rn and w ∈ Rn. Therefore, the problem of solving the equilibrium solution for the system can be turned into solving the following equations:(7)Fz,μ=0,Fzz,μv=0,FμTz,μv=1,orwTFzz,μ=0,wTz,μ=1,where F(z, μ) = [fT(x, y, μ), gT(x, y, μ)]T is the equilibrium equation, while z = [xTyT]T stands for the state variable.Therefore, the saddle-node bifurcation (SNB, static bifurcation point) of the system can be obtained by using the Newton–Raphson method to solve equation (5) [20].


## 2.2. Solution of the Dynamic Bifurcation Point

During system operation, once the disturbance occurs, the equilibrium state of the system and the number of solutions of the equilibrium equation will change with the variation of the control parameters. For a given control vectorμ, suppose a nonhyperbolic equilibrium point x0 appears in the equilibrium solutions (x, y, μ) with μ = μ0 and x0 satisfies (1), and it is located on the surface of the equilibrium solution M. If a pair of conjugate complex roots with zero real parts appears on the reduced Jacobian matrix J in equation (6), it is considered that the structural stability of the system changes, resulting in dynamic bifurcation, in which the main dynamic bifurcation point is the Hopf bifurcation [21].In conclusion, the change of eigenvalues corresponding to the dynamic and static bifurcation points is clearly described in Figure1.Figure 1
Variation of eigenvalues corresponding to the dynamic and static bifurcation points: (a) static bifurcation; (b) dynamic bifurcation.
(a)(b)


## 3. Dynamic Modeling of the FSCWT

### 3.1. Operation Principle of the FSCWT

The FSCWT adopts the transmission form of low-speed ratio gearbox + hydraulic torque converter speed control + brushless electric excitation synchronous generator, and its structure is shown in Figure2. The biggest advantage of the wind turbine is that the hydraulic torque converter WinDrive is used to realize the stepless speed regulation. In case of a sudden change of wind speed, the hydraulic torque converter WinDrive can withstand such shock and can convert the varied wind speed into the constant input speed of EESG, thereby, to realize the purpose of variable speed input and constant speed output.Figure 2
Structure of the FSCWT.To analyze the dynamic characteristics of the wind turbine drive chain system, it is usually equivalent to a multimass model composed of spring and damping. This equivalent method is also applied for the FSCWT. In the structure diagram of the drive chain provided in Figure3, the characteristics of the rotor speed and the low-speed shaft can be described in the following form [22]:(8)JRdωRdt=TR−Tj−D1ωR,while the rotor torque TR can be described as [23](9)TR=P1ωR=Cpλ,βSV3ρ2ωR.Figure 3
Structural chart of drive chain of FSCWT.In Figure3, the torque of the wind wheel and the planetary frame satisfies(10)TRω˙R=Tjω˙j,ωj=ωRiRj.For the torque converter, the dynamic balance equation between the pump wheel and the turbine wheel is(11)Jtω˙B=Tt−TB−TG,Jqω˙T=Tq−TT,TT=αTB,TB=λBρoilωB2D5.From the relationship between the turbine speed and the pump wheel speed of the hydraulic torque converter, it can be concluded that [24](12)ω˙T=1+iTq1+itqω˙R−iRj⋅itq⋅ω˙R.If the moment of inertia of the high-speed shaft is ignored, the dynamic characteristics of the high-speed shaft and the synchronous generator rotor can be expressed as [25](13)JGdωGdt=Te−TG−DGωG.


### 3.2. Modeling of EESG

For the EESG, the excitation system can be represented as [26].In this paper, the fifth-order synchronous generator model is selected for FSCWT for stability analysis [27], which is given as follows:(14)dEq′dt=EfdTd0′−Eq′Td0′−Xd−Xd′Td0Id,dEq′dt=aEfdTd0′+1Td0′−aTd0′Eq′−Eq″Td0′−kId,dEd″dt=−Ed″Tdo″+Xq−Xd″Tdo″Id,E˙fd=−Efdτm+3Ts+KPWMXadτm+3TsRfUC,where UC is the controller output, and K and c are equivalent parameters, and details for each parameter can be found in [28]:(15)K=Xd′−Xd″Tdo″+cXd′−cXd″Tdo′,c=Xd″−X1Xd′−X1.


## 3.1. Operation Principle of the FSCWT

The FSCWT adopts the transmission form of low-speed ratio gearbox + hydraulic torque converter speed control + brushless electric excitation synchronous generator, and its structure is shown in Figure2. The biggest advantage of the wind turbine is that the hydraulic torque converter WinDrive is used to realize the stepless speed regulation. In case of a sudden change of wind speed, the hydraulic torque converter WinDrive can withstand such shock and can convert the varied wind speed into the constant input speed of EESG, thereby, to realize the purpose of variable speed input and constant speed output.Figure 2
Structure of the FSCWT.To analyze the dynamic characteristics of the wind turbine drive chain system, it is usually equivalent to a multimass model composed of spring and damping. This equivalent method is also applied for the FSCWT. In the structure diagram of the drive chain provided in Figure3, the characteristics of the rotor speed and the low-speed shaft can be described in the following form [22]:(8)JRdωRdt=TR−Tj−D1ωR,while the rotor torque TR can be described as [23](9)TR=P1ωR=Cpλ,βSV3ρ2ωR.Figure 3
Structural chart of drive chain of FSCWT.In Figure3, the torque of the wind wheel and the planetary frame satisfies(10)TRω˙R=Tjω˙j,ωj=ωRiRj.For the torque converter, the dynamic balance equation between the pump wheel and the turbine wheel is(11)Jtω˙B=Tt−TB−TG,Jqω˙T=Tq−TT,TT=αTB,TB=λBρoilωB2D5.From the relationship between the turbine speed and the pump wheel speed of the hydraulic torque converter, it can be concluded that [24](12)ω˙T=1+iTq1+itqω˙R−iRj⋅itq⋅ω˙R.If the moment of inertia of the high-speed shaft is ignored, the dynamic characteristics of the high-speed shaft and the synchronous generator rotor can be expressed as [25](13)JGdωGdt=Te−TG−DGωG.


## 3.2. Modeling of EESG

For the EESG, the excitation system can be represented as [26].In this paper, the fifth-order synchronous generator model is selected for FSCWT for stability analysis [27], which is given as follows:(14)dEq′dt=EfdTd0′−Eq′Td0′−Xd−Xd′Td0Id,dEq′dt=aEfdTd0′+1Td0′−aTd0′Eq′−Eq″Td0′−kId,dEd″dt=−Ed″Tdo″+Xq−Xd″Tdo″Id,E˙fd=−Efdτm+3Ts+KPWMXadτm+3TsRfUC,where UC is the controller output, and K and c are equivalent parameters, and details for each parameter can be found in [28]:(15)K=Xd′−Xd″Tdo″+cXd′−cXd″Tdo′,c=Xd″−X1Xd′−X1.


## 4. Operation Characteristics and Dynamic Equivalent Modeling of the Wind Farm

### 4.1. Site Location of the Wind Farm for Case Study

In this paper, the Longyuan Pingshan Lake wind farm is selected for the case study. This wind farm is located near Pingshan Lake in Zhangye. The total capacity of this wind farm is 50 MW and includes 25 FSCWTs designed to withstand the sand blow and low-temperature environment in Northwest China [29].The wind farm is connected to the railway station through a 110 kV transmission line with a line length up to 80 km and then connected to 330 kV Zhangye substation The topological structure of the regional power system is shown in Figure4.Figure 4
The topological structure of the regional power grid in Zhangye.


### 4.2. Grid-Connected System Modeling

Figure5 presents the structure diagram of the FSCWTs integrated into the regional power grid. Each wind turbine is equipped with a step-up transformer (35 kV/2350 kVA) to boost the terminal voltage of the wind turbine from 0.69 kV to 35 kV and then transmitted to the 110 kV station through 35 kV power cables.Figure 5
Structure diagram of the FSCWT grid-connected system.As given in Figure4, the grid structure of the regional power grid is relatively complex and is also composed of many wind turbines. Owing to the large scale of the wind farm, the amount of simulation calculation is too complex, which greatly reduces the efficiency of simulation calculation. This is an important condition for the equivalent treatment of the FSCWT. For such consideration, the region includes the Longyuan Pingshan wind farm, and Zhangye thermal power plant integrated into the 330 kV Zhangye substation is taken as the case. Firstly, the system is simplified through equivalent modeling, the Zhangye thermal power plant is treated as the equivalent power source G1, and the wind farm with FSCWTs is treated as the equivalent power source G2. The equivalent circuit diagram is shown in Figure 6.Figure 6
Equivalent circuit diagram.Figure6 is the circuit diagram obtained by equivalent transformation, where node 1 and node 2 stand for the grid-connection point of the thermal power plant and the wind turbine, respectively. Node 3 represents the 330 kV Zhangye substation, while node 4 is the grid-connection point of the wind farm with FSCWT. E1′ and S1′ represent the equivalent potential and capacity of the power source G1, respectively. Accordingly, E2′ and S2′ represent the equivalent potential and capacity of the wind farm, while PL and QL stand for the load active power and reactive power of the equivalent system. The voltage and the corresponding phase angle of the wind turbine connected to the system are denoted as u and θ, respectively. Similarly, the voltage and phase angle of the equivalent power source G1 are treated as u1 and θ1. The equivalent admittance between nodes 1 and node 3 is denoted as y0′, while that between node 3 and node 4 is taken as ym′.To approach the actual situation, the IEEE first benchmark model for the grid-connected equivalent generator G2 composed of the FSCWT [30]:(16)δ·m=ω,Mω·=Tm+uEd′cosθ−δm+Eq′sinθ−δmxd′−Dmω,Td0′Ed′·=−xdEq′xd′+xd−xd′ucosθ−δmxd′+Efd,Tq0′Ed′·=−xqEd′xd′+xq−xd′usinθ−δmxd′,TAE·fd=−KE+Efd+KAVref−u.The network algebra equation including the FSCWT is listed as follows:(17)uEq′xd′sinδm−θ=−u1uymsinθ1−θ−u2ymsinθ1−θ,uEq′xd′cosδm−θ=u2xd′−u1uymcosθ1−θ+u2ymcosθ1−θ.The power provided by the network to the load side of the grid-connected system is(18)P=−E0′uy0′sinθ+θ1−Emuymsinθ−δm+θ1+y0′sinθ1+ymsinθu2,Q=E0′uy0′cosθ+θ1−Emuymcosθ−δm+θ1−y0′cosθ1+ymsinθu2.Here, the dynamic load model is composed of asynchronous motor and constant PQ power. Once denote the active load as the function of node voltage change rate with time and frequencyPli = Pli(ui, dui/dt, dδ/dt), while the reactive load is expressed as a function of node voltage and frequency Qli = Qli(ui, dδ/dt), it can be concluded that(19)P=P0+P1+K1θ˙+K2u2−Tu˙,Q=Q0+Q1+K3θ˙+K4u+K5u2.Therefore, according to equations (16)～(19), the state equation describing the dynamic performance of the system can be summarized as(20)δ·m=ω,Mω·=Tm+uEd′cosθ−δm+Eq′sinθ−δmxd′−Dmω,Td0Eq′·=−xdEq′xd′+xd−xd′ucosθ−δmxd′+Efd,Tq0Ed′·=−xqEd′xdc+xq−xd′usinθ−δmxd′,TAE·fd=−KE+Efd+KAVref−u,K3θ˙=−K4u−K5u2+Q−Q0−Q1,TK3K2u˙=K1K5u2+K1K4−K3K2u+K1Q0−Q+Q1−K3P0−P+P1.By solving the state differential equations of the system and selecting the control parameters that affect the voltage stability of the system, the stability of the grid-connected voltage is simulated in detail based on the bifurcation method.The values of the parameters for the equivalent system and load model are provided in Table1.Table 1
Parameter of the equivalent system and load model.
ParameterValueTm1.0M0.0637y020.0xq0.85Tq0′0.25xq′0.1TA0.2K10.4K20.3K52.1P00.6Dm0.05ym5.0xd0.90Td0′8xd′0.1uref1KE1.0K3−0.03K4−2.8T8.5Q01.3


## 4.1. Site Location of the Wind Farm for Case Study

In this paper, the Longyuan Pingshan Lake wind farm is selected for the case study. This wind farm is located near Pingshan Lake in Zhangye. The total capacity of this wind farm is 50 MW and includes 25 FSCWTs designed to withstand the sand blow and low-temperature environment in Northwest China [29].The wind farm is connected to the railway station through a 110 kV transmission line with a line length up to 80 km and then connected to 330 kV Zhangye substation The topological structure of the regional power system is shown in Figure4.Figure 4
The topological structure of the regional power grid in Zhangye.


## 4.2. Grid-Connected System Modeling

Figure5 presents the structure diagram of the FSCWTs integrated into the regional power grid. Each wind turbine is equipped with a step-up transformer (35 kV/2350 kVA) to boost the terminal voltage of the wind turbine from 0.69 kV to 35 kV and then transmitted to the 110 kV station through 35 kV power cables.Figure 5
Structure diagram of the FSCWT grid-connected system.As given in Figure4, the grid structure of the regional power grid is relatively complex and is also composed of many wind turbines. Owing to the large scale of the wind farm, the amount of simulation calculation is too complex, which greatly reduces the efficiency of simulation calculation. This is an important condition for the equivalent treatment of the FSCWT. For such consideration, the region includes the Longyuan Pingshan wind farm, and Zhangye thermal power plant integrated into the 330 kV Zhangye substation is taken as the case. Firstly, the system is simplified through equivalent modeling, the Zhangye thermal power plant is treated as the equivalent power source G1, and the wind farm with FSCWTs is treated as the equivalent power source G2. The equivalent circuit diagram is shown in Figure 6.Figure 6
Equivalent circuit diagram.Figure6 is the circuit diagram obtained by equivalent transformation, where node 1 and node 2 stand for the grid-connection point of the thermal power plant and the wind turbine, respectively. Node 3 represents the 330 kV Zhangye substation, while node 4 is the grid-connection point of the wind farm with FSCWT. E1′ and S1′ represent the equivalent potential and capacity of the power source G1, respectively. Accordingly, E2′ and S2′ represent the equivalent potential and capacity of the wind farm, while PL and QL stand for the load active power and reactive power of the equivalent system. The voltage and the corresponding phase angle of the wind turbine connected to the system are denoted as u and θ, respectively. Similarly, the voltage and phase angle of the equivalent power source G1 are treated as u1 and θ1. The equivalent admittance between nodes 1 and node 3 is denoted as y0′, while that between node 3 and node 4 is taken as ym′.To approach the actual situation, the IEEE first benchmark model for the grid-connected equivalent generator G2 composed of the FSCWT [30]:(16)δ·m=ω,Mω·=Tm+uEd′cosθ−δm+Eq′sinθ−δmxd′−Dmω,Td0′Ed′·=−xdEq′xd′+xd−xd′ucosθ−δmxd′+Efd,Tq0′Ed′·=−xqEd′xd′+xq−xd′usinθ−δmxd′,TAE·fd=−KE+Efd+KAVref−u.The network algebra equation including the FSCWT is listed as follows:(17)uEq′xd′sinδm−θ=−u1uymsinθ1−θ−u2ymsinθ1−θ,uEq′xd′cosδm−θ=u2xd′−u1uymcosθ1−θ+u2ymcosθ1−θ.The power provided by the network to the load side of the grid-connected system is(18)P=−E0′uy0′sinθ+θ1−Emuymsinθ−δm+θ1+y0′sinθ1+ymsinθu2,Q=E0′uy0′cosθ+θ1−Emuymcosθ−δm+θ1−y0′cosθ1+ymsinθu2.Here, the dynamic load model is composed of asynchronous motor and constant PQ power. Once denote the active load as the function of node voltage change rate with time and frequencyPli = Pli(ui, dui/dt, dδ/dt), while the reactive load is expressed as a function of node voltage and frequency Qli = Qli(ui, dδ/dt), it can be concluded that(19)P=P0+P1+K1θ˙+K2u2−Tu˙,Q=Q0+Q1+K3θ˙+K4u+K5u2.Therefore, according to equations (16)～(19), the state equation describing the dynamic performance of the system can be summarized as(20)δ·m=ω,Mω·=Tm+uEd′cosθ−δm+Eq′sinθ−δmxd′−Dmω,Td0Eq′·=−xdEq′xd′+xd−xd′ucosθ−δmxd′+Efd,Tq0Ed′·=−xqEd′xdc+xq−xd′usinθ−δmxd′,TAE·fd=−KE+Efd+KAVref−u,K3θ˙=−K4u−K5u2+Q−Q0−Q1,TK3K2u˙=K1K5u2+K1K4−K3K2u+K1Q0−Q+Q1−K3P0−P+P1.By solving the state differential equations of the system and selecting the control parameters that affect the voltage stability of the system, the stability of the grid-connected voltage is simulated in detail based on the bifurcation method.The values of the parameters for the equivalent system and load model are provided in Table1.Table 1
Parameter of the equivalent system and load model.
ParameterValueTm1.0M0.0637y020.0xq0.85Tq0′0.25xq′0.1TA0.2K10.4K20.3K52.1P00.6Dm0.05ym5.0xd0.90Td0′8xd′0.1uref1KE1.0K3−0.03K4−2.8T8.5Q01.3


## 5. Bifurcation Analysis of FSCWT Connected to the Regional Power Grid

### 5.1. Influence of Reactive Load Variation on Voltage Stability

The new energy power system with FSCWTs is a typical dynamic nonlinear system. The change of its voltage stability is essentially a process from a stable state to a bifurcation or collapse.For the grid-connected system, the reactive load powerQ1 is selected as the bifurcation parameter of the grid side, assuming that the bifurcation parameter Q1 = 2.1408, while the wind speed V = 10 m/s, and the active power of the load P1 = 0. Therefore, the initial equilibrium point of the system can be obtained using the Newton–Raphson method by solving (20), which is x = (δm, ω, Ed′, Eq′, Efd, θ, u) = (2.947, 0, −0.463, 0.26, 3.11, 0.118, 0.97). According to the simulation, the curve of voltage u varying with the control parameter Q1 at the connection point is shown in Figure 7.Figure 7
Q1−u curve simulation diagram.As given in Figure7, two bifurcation points can be found on the system’s equilibrium solution. One is a Hopf bifurcation point, marked as H1, and another marked as LP is a saddle-node bifurcation point. The solid blue line indicates that the system is in a stable state, while the dotted red line means the system is in an unstable state. Therefore, this article mainly focuses on the upper half of the Q1−u curve, namely, the area with yellow-shadow in Figure 7.Further numerical analysis indicates that the first Lyapunov coefficient is negative atH1, which means the system is operating in a stable state. The state parameters corresponding to the bifurcation points H1 and LP are shown in Table 2.Table 2
The parameter value of each bifurcation point (in p.u.).
ParameterBifurcation pointH1LPδm3.2343233.221205ω00Ed′−0.891554−0.887648Eq′0.0498830.049893Efd0.0030100.000133θ0.0364030.023048u1.0042151.000187Q11.2948551.296925Once a disturbance occurs and acts onH1 on the Q1−u curve, the changes of the δ−u phase locus plot and the u−t voltage curve can be depicted in Figures 8 and 9. It can be seen from Figure 8 that a limit cycle appears on the system phase locus plot, and the corresponding phenomenon is that the voltage u oscillates with the increase of time t, as is shown in Figure 9. Besides, it is also clear from Figure 8 that there is uncertainty phenomenon, namely, the chaos. Based on the above analysis, it can be concluded that the occurrence of a limit cycle will inevitably lead to the Hopf bifurcation.Figure 8
δ−u2 phase locus plot.Figure 9
u−t curve.According to Figure7 and Table 2, when the system operates at the point H1, the corresponding reactive power Q1 is 1.294855 p.u., and before this, the system is stable. However, once the system has interfered with a small interference, i.e., Q1 suddenly increases from 1.294855 p.u. to 1.39 p.u., or 1.294855 p.u. to 1.49 p.u., or from 1.294855 p.u. to 1.59 p.u., the corresponding simulation results in time-domain are given in Figure 10.Figure 10
u−t curve of the system after small interference.It can be seen from Figure10 that when the system is subject to small interference (Q1 = 1.39), after a short period of stable operation, the bus voltage at the load side will lose stability and eventually lead to voltage collapse at t = 0.57 s. If the value of Q1 is increased to 1.49 p.u. continually, it is clear that the stable operation time of the system will be shortened; at t = 0.24 s, the system voltage will become unstable and step into collapse. In conclusion, the stable operation time will become shorter and shorter as the disturbance of the reactive power (Q1) increases, or namely, the stable operation area of the system voltage will be gradually reduced as the reactive power increases.


### 5.2. Voltage Bifurcation Analysis with Variable Wind Speed

During operation, the stability of the regional power system includes FSCWTs is not only related to the reactive power but also affected by the wind speed fluctuation. Assuming the wind speedV = 10 m/s, the initial equilibrium point of the system is x = (δm, ω, Ed′, Eq′, Efd, θ, u) = (3.403, 0, −0.467, 0.392, 0.208, 0.416, 0.708). By implementing this case on MATCONT [31], the curve of voltage u changing with wind speed V can be obtained, which is shown in Figure 11.Figure 11
u−V simulation diagram.In Figure11, there are two Hopf bifurcation points (H1 and H2) and two saddle-node bifurcation points (LP1 and LP2) on the u−V curve. The corresponding state parameters of each bifurcation point are given in Table 3.Table 3
The parameter value of each bifurcation point.
ParameterBifurcation pointH1H2LP1LP2δm4.5866954.6630544.6228973.372529ω0000Ed′−0.136046−0.186939−0.570637−0.467516Eq′1.1534631.1626570.6290280.392306Efd−0.223735−0.2357360.023990.208029θ0.0302990.0870540.646247−0.416301u1.3132291.3300300.9616420.708759V15.26768118.12981726.32953223.031749In actuality, the cut-out wind speed of the FSCWT is designed as 25 m/s. According to Figure11 and Table 3, when the system operates to the point LP1, the wind speed has exceeded 25 m/s and beyond the limit wind speed of the wind turbine. The operation area beyond the rated wind speed of the wind turbine is marked in red, which is also an unstable operation area. While the remaining area on the left side is a stable operation area although three bifurcation points are available. This is because the occurrence time of both LP1 and LP2 is later than that of the dynamic bifurcation points H1 and H2.The above analysis is only limited to the single parameter bifurcation of system voltage. To better reflect the dynamic characteristics of the FSCWT when connected to the grid, the voltage at the connection point (V) and the load reactive power (Q1) are selected for the two-parameter bifurcation analysis. The two-dimensional Q1−u curve of the system under different wind speed is shown in Figure 12.Figure 12
Q1−u simulation curve under different wind speeds.From Figure12, it is clear that the number of bifurcation points on Q1−u curves is different when the wind speed is different.When the wind speedV reaches to 15 m/s, there is only one saddle-node bifurcation point LP3 on the Q1−u curve. As the wind speed decreases to 10 m/s, in addition to a saddle-node bifurcation point LP2, one Hopf bifurcation point H2 will occur on the Q1−u curve. As the wind speed continues to decrease to 5 m/s, no additional bifurcation points will occur except one Hopf bifurcation point H1 and one saddle-node bifurcation point LP1, but the location of the bifurcation points will move to the right side of the curve. The results show that the number of equilibrium solutions on different curves is not only related to the wind speed but also changes with the change of reactive power. With the increase of wind speed, the amplitude of system voltage will rise slightly, which has been marked with a black ellipse in Figure 12. However, the Hopf bifurcation point of the system will appear in advance, which is more likely to cause the system voltage instability and further reduce the stable operation area of the system. At the same time, as the wind speed increases gradually, the corresponding reactive power value of the saddle-node bifurcation point in each curve also decreases, which will lead to the reduction of the reactive power capacity of the grid-connected system.


## 5.1. Influence of Reactive Load Variation on Voltage Stability

The new energy power system with FSCWTs is a typical dynamic nonlinear system. The change of its voltage stability is essentially a process from a stable state to a bifurcation or collapse.For the grid-connected system, the reactive load powerQ1 is selected as the bifurcation parameter of the grid side, assuming that the bifurcation parameter Q1 = 2.1408, while the wind speed V = 10 m/s, and the active power of the load P1 = 0. Therefore, the initial equilibrium point of the system can be obtained using the Newton–Raphson method by solving (20), which is x = (δm, ω, Ed′, Eq′, Efd, θ, u) = (2.947, 0, −0.463, 0.26, 3.11, 0.118, 0.97). According to the simulation, the curve of voltage u varying with the control parameter Q1 at the connection point is shown in Figure 7.Figure 7
Q1−u curve simulation diagram.As given in Figure7, two bifurcation points can be found on the system’s equilibrium solution. One is a Hopf bifurcation point, marked as H1, and another marked as LP is a saddle-node bifurcation point. The solid blue line indicates that the system is in a stable state, while the dotted red line means the system is in an unstable state. Therefore, this article mainly focuses on the upper half of the Q1−u curve, namely, the area with yellow-shadow in Figure 7.Further numerical analysis indicates that the first Lyapunov coefficient is negative atH1, which means the system is operating in a stable state. The state parameters corresponding to the bifurcation points H1 and LP are shown in Table 2.Table 2
The parameter value of each bifurcation point (in p.u.).
ParameterBifurcation pointH1LPδm3.2343233.221205ω00Ed′−0.891554−0.887648Eq′0.0498830.049893Efd0.0030100.000133θ0.0364030.023048u1.0042151.000187Q11.2948551.296925Once a disturbance occurs and acts onH1 on the Q1−u curve, the changes of the δ−u phase locus plot and the u−t voltage curve can be depicted in Figures 8 and 9. It can be seen from Figure 8 that a limit cycle appears on the system phase locus plot, and the corresponding phenomenon is that the voltage u oscillates with the increase of time t, as is shown in Figure 9. Besides, it is also clear from Figure 8 that there is uncertainty phenomenon, namely, the chaos. Based on the above analysis, it can be concluded that the occurrence of a limit cycle will inevitably lead to the Hopf bifurcation.Figure 8
δ−u2 phase locus plot.Figure 9
u−t curve.According to Figure7 and Table 2, when the system operates at the point H1, the corresponding reactive power Q1 is 1.294855 p.u., and before this, the system is stable. However, once the system has interfered with a small interference, i.e., Q1 suddenly increases from 1.294855 p.u. to 1.39 p.u., or 1.294855 p.u. to 1.49 p.u., or from 1.294855 p.u. to 1.59 p.u., the corresponding simulation results in time-domain are given in Figure 10.Figure 10
u−t curve of the system after small interference.It can be seen from Figure10 that when the system is subject to small interference (Q1 = 1.39), after a short period of stable operation, the bus voltage at the load side will lose stability and eventually lead to voltage collapse at t = 0.57 s. If the value of Q1 is increased to 1.49 p.u. continually, it is clear that the stable operation time of the system will be shortened; at t = 0.24 s, the system voltage will become unstable and step into collapse. In conclusion, the stable operation time will become shorter and shorter as the disturbance of the reactive power (Q1) increases, or namely, the stable operation area of the system voltage will be gradually reduced as the reactive power increases.


## 5.2. Voltage Bifurcation Analysis with Variable Wind Speed

During operation, the stability of the regional power system includes FSCWTs is not only related to the reactive power but also affected by the wind speed fluctuation. Assuming the wind speedV = 10 m/s, the initial equilibrium point of the system is x = (δm, ω, Ed′, Eq′, Efd, θ, u) = (3.403, 0, −0.467, 0.392, 0.208, 0.416, 0.708). By implementing this case on MATCONT [31], the curve of voltage u changing with wind speed V can be obtained, which is shown in Figure 11.Figure 11
u−V simulation diagram.In Figure11, there are two Hopf bifurcation points (H1 and H2) and two saddle-node bifurcation points (LP1 and LP2) on the u−V curve. The corresponding state parameters of each bifurcation point are given in Table 3.Table 3
The parameter value of each bifurcation point.
ParameterBifurcation pointH1H2LP1LP2δm4.5866954.6630544.6228973.372529ω0000Ed′−0.136046−0.186939−0.570637−0.467516Eq′1.1534631.1626570.6290280.392306Efd−0.223735−0.2357360.023990.208029θ0.0302990.0870540.646247−0.416301u1.3132291.3300300.9616420.708759V15.26768118.12981726.32953223.031749In actuality, the cut-out wind speed of the FSCWT is designed as 25 m/s. According to Figure11 and Table 3, when the system operates to the point LP1, the wind speed has exceeded 25 m/s and beyond the limit wind speed of the wind turbine. The operation area beyond the rated wind speed of the wind turbine is marked in red, which is also an unstable operation area. While the remaining area on the left side is a stable operation area although three bifurcation points are available. This is because the occurrence time of both LP1 and LP2 is later than that of the dynamic bifurcation points H1 and H2.The above analysis is only limited to the single parameter bifurcation of system voltage. To better reflect the dynamic characteristics of the FSCWT when connected to the grid, the voltage at the connection point (V) and the load reactive power (Q1) are selected for the two-parameter bifurcation analysis. The two-dimensional Q1−u curve of the system under different wind speed is shown in Figure 12.Figure 12
Q1−u simulation curve under different wind speeds.From Figure12, it is clear that the number of bifurcation points on Q1−u curves is different when the wind speed is different.When the wind speedV reaches to 15 m/s, there is only one saddle-node bifurcation point LP3 on the Q1−u curve. As the wind speed decreases to 10 m/s, in addition to a saddle-node bifurcation point LP2, one Hopf bifurcation point H2 will occur on the Q1−u curve. As the wind speed continues to decrease to 5 m/s, no additional bifurcation points will occur except one Hopf bifurcation point H1 and one saddle-node bifurcation point LP1, but the location of the bifurcation points will move to the right side of the curve. The results show that the number of equilibrium solutions on different curves is not only related to the wind speed but also changes with the change of reactive power. With the increase of wind speed, the amplitude of system voltage will rise slightly, which has been marked with a black ellipse in Figure 12. However, the Hopf bifurcation point of the system will appear in advance, which is more likely to cause the system voltage instability and further reduce the stable operation area of the system. At the same time, as the wind speed increases gradually, the corresponding reactive power value of the saddle-node bifurcation point in each curve also decreases, which will lead to the reduction of the reactive power capacity of the grid-connected system.


## 6. Conclusions

In this article, the voltage stability of the FSCWTs connected to the regional power grid is studied. The whole process of the system voltage from instability to collapse is revealed by using bifurcation theory. The critical points of the system bifurcation and voltage collapse are determined under different parameter changes. The analysis and verification are carried out on the MATCONT simulation platform by combining the time-domain simulation method. The following conclusions can be drawn:(1)
The number of bifurcation points on the manifold of the system equilibrium solution is not only related to the fluctuation of wind speed but also related to the disturbance of reactive power of the system.(2)
When the system is subject to small interference, the voltage will fluctuate. The greater the disturbance of the reactive power injected to the grid, the shorter the time of voltage collapse and the smaller the stable operation area of the system. On the other hand, the smaller the operation area of the system, the shorter the stable operation time of the system, and the earlier the system will occur voltage collapse. In short, the shorter the time of system collapse.(3)
From voltage stability analysis using the two-parameter model, it is found that the value of reactive power corresponding to saddle-node bifurcation point LP in each curve will decrease with the gradual increase of wind speed, which will lead to the reduction of the reactive power capacity of the grid-connected system with FSCWTs.



---

*Source: 8816334-2020-10-07.xml*