Analysis of grid-connected voltage stability of FSCWT based on bifurcation theory

HE Ai-huan, ZHANG Rui-ping, DONG Hai-ying,2

(1. School of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China;2. School of Energy and Power Engineering, Lanzhou Jiaotong University, Wuwei 733000, China)

0 Introduction

If a great deal of wind power are integrated into the power grid, it will have influence on the voltage stability of power grid, and in some circumstances, it will even become the main factor for restricting the installed capacity of wind farm[1-3]. Due to the fluctuation and randomness of wind turbines’ angular velocity, the fluctuation of voltage at the grid-connected point will be caused, which can also be caused by reactive power from turbines. On the other hand, the load fluctuation is usually one of the factors causing the change of voltage at grid-connected point. The fluctuation of active power at load bus will cause the change of voltage at the grid-connected point, and its size will impact voltage stability at load bus[4]and even lead to the bifurcation of voltage at the grid-connected point. Many integrated wind farms make the system voltage hold the dynamic uncertainty.

On the subject of voltage stability of wind turbines, a large number of scholars have researched it. Ref.[5] introduced several bifurcation phenomena and their relationship with voltage stability in the analysis of voltage stability in power system. Taking the active power of asynchronous wind turbines at the injecting node and reactive power’s compensation capacities as the bifurcation control parameters, Refs.[6-7] analyzed and determined the types of bifurcation and its boundary, which were different from STATCOM and static var compensation (SVC). Ref.[8] gave the review about the current situation of voltage stability in power system and the influence of load on voltage stability. And it focused on that recovery characteristics and characteristics of losing stability of load have influence on the stability of system voltage. Under the guidance of IEEE-1system model, Ref.[9] made the analysis of the influence of the time constant of the excitation system regulator in several losing stability models on the voltage stability. Taking double fed induction generator (DFIG) or permanent magnetic synchronous generator (PMSG) as research object, the above all researches used the equivalent model to quantitatively analyze the bifurcation of voltage stability in wind power system. Under the guidance of IEEE-3system model, Ref.[10-12] made the analysis of the influence of wind speed fluctuation and excitation regulation in several losing stability models on the voltage stability. However, in the equivalent differential algebraic equations (DAE) model, angular velocity of wind turbines and the influence of load or feedback controller on voltage stability are not clearly reflected.

Taking the front-end speed controlled wind turbines (FSCWT) as the research object[13-15]that can achieve variable-speed input and constant-speed output[16-18], from the perspective of bifurcation theory, this paper analyzes the stability of static and dynamic bifurcation structure in power system. On the basis of this, a power system model containing FSCWT will be established after taking the angular velocity of wind turbines, the dynamic effect of load and feedback controller into account. In the node 3 system, this paper makes a simulation analysis of the influence of angular velocity of wind turbines, load and feedback controller on voltage stability and voltage bifurcation of FSCWT grid-connected integrating into system. Finally, through increasing the value of gainksof feedback controller, the voltage bifurcation at the grid-connected point can be delayed.

1 Mathematical foundation of bifurcation analysis and its basic concept

The instability of power system can be regarded as a bifurcation node occurring in space, and the parameter space composed by every bifurcation node forms the area with stable structure whose bifurcation surface is stable boundary[18]. The nonlinear dynamic characteristics of power system can be described as follows

(1)

wherex∈Rn, is state variable in power system, usually stands for the internal state variables related to the generator and load dynamic;y∈Rm, is state variable of algebraic in power system, usually stands for phase angle or voltage of node;λ∈Rk(k>1,k∈Z), as control parameter, is the bifurcation parameter in the stability analysis;n，m，kall stand for corresponding dimensions of variables;fstands for the differential equation describing systematically dynamic component;gdescribes the balance equation of power at every node[12].

If there is an equilibrium point or a set of equilibrium points (x0，y0，λ0) which satisfies Eq.(1), so exist

Whenλis equal toλ0, non-hyperbolic equilibrium point will occurs in system. And making a linear processing for Eq.(1) at the equilibrium point, we can get

(2)

(3)

whereAis the decreased Jacobin matrix of system.

If in Eq.(1), the bifurcation point of set system occurs around the equilibrium points (x0，y0，μ0),Ahas a zero eigenvalue, and the equilibrium point is a hyperbolic one at this moment, so that the system will lose structural stability. If system is disturbed slightly, locus topological structure surrounded by equilibrium points (x0，y0，λ0) will be changed, and saddle node bifurcation (SNB) will emerge. The point of SNB can be calculated according to[19]

(4)

whereυis the right characteristic vector which corresponds to the zero eigenvalue of matrixA.

The method of Newton-Raphson is applied to Eq.(4), and the bifurcation nodes which are under the effect of various control parameters can be obtained.

2 Power system model with FSCWT

2.1 System model of FSCWT

FSCWT consists of wind wheel, main speedup gearbox, fluid torque converter WinDrive and electrically excited synchronous generator (EESG), as shown in Fig.1[20-22]. The main speedup gearbox is composed by two-stage planetary gears whose transmission ratio is fixed, and it can accelerate in drive system. FSCWT employs EESG to regulate wind speedVpassing the main speedup gearbox and the dynamic fluid torque converter WinDrive connected with spindle as the constant inputωGof EESG[23].

Fig.1 Structure of FSCWT

Compared with wind turbines, such as DFIG and PMSG, FSCWT omits the transducer, the speed regulation of turbines can be realized by the main speedup gearbox and the fronted WinDrive, which guarantees its constant frequency output and improves the stability of turbines at the same time[22,24].

2.1.1 FSCWT model of transmission chain

For FSCWT, in order to analyze the dynamic characteristics of the transmission system, it is usually equivalent to a wind wheel rotor and low speed shaft. The speed torque characteristic can be described as

JRωR=MR-Mj-DωR,

(5)

whereDis damping coefficient of low speed shaft;ωRis the speed of wind turbine;JRis the inertia of turbine rotor;Mjis the torque of planetary carrier;MRis the torque of turbine and its size is shown as follows[21]

whereP1is the active power absorbed in by wind turbine;CPis utilization coefficient of wind power;ARis the area wind of turbines;ρis air density.

FSCWT’s torque of turbine and torque of planetary carrier must satisfy

(6)

whereiRjis the speed increasing ratio of main speedup gearbox;Tjis the speed of the planetary carrier.

In WinDrive, the dynamic equilibrium equation of pump impeller shaft and the turbine shaft is[13]

(7)

whereTPis the torque of pump impeller;Ttis the torque of sun gear;TTis the torque of turbine;Tqis the torque of gear’s outer ring;TGis the input mechanical torque of EESG.

There exists a relationship held by the structure of fluid torque converter

(8)

whereλPis torque ratio of fluid torque converter;ρoilis its oil density;Dis circle diameter;μis the torque coefficient of pump impeller.

The relationship among the WinDrive pump impeller, speed of wind wheel and speed of turbine must satisfies

(9)

whereitqis the speed ratio from sun gear to its outer ring;iTqis the speed ratio from turbine to its outer ring.

If the rotational inertia of the high speed shaft (EESG inputting shaft) is neglected, the rotor of synchronous generator and dynamic characteristics of speed increasing shaft can be expressed as

(10)

whereωGandδrespectively refer to EESG’s angular velocity and power angle;Meis its electromagnetic torque;DGis the damping coefficient of high speed shaft;JGis rotational inertia of EESG rotor.

2.1.2 EESG excitation system model[20]

In terms of electric excitation synchronous wind generator applied by front-end variable speed wind turbines, the brushless excitation system used in the electric excitation synchronous wind generator which is studied in present study can be described by Park equation as

(11)

whereidandiqrepresent the current of direct and quadrature-axis respectively;ifis the excitation current;RDis the damp resistance;Rfis the exciting resistance;ψDandψQstand for direct and quadrature-axis flux link age respectively,ψDandψQare the flux linkage of damping direct and quadrature-axis respectively;ψfis the flux linkage of exciting resistance;ωeis the time derivative of stator electrical degree corresponding to linkage revolving speed.

A simplified Park equation is obtained, after neglecting the influence of the damping winding, the transient state of the stator winding and rotational speed is

(12)

Collating Eqs.(11) and (12), the model of third-order synchronous generator can be obtained by

(13)

whereUcis the output of controller;cis the equivalent parameter as

Taking the influence of the exciting winding into consideration, the equation of motion of generator rotor and the equation of stator voltage can be described as follows

whereDis damping coefficient;δis the rotor angle;ωs=2πf0, is the reference frequency;Mis the inertial time constant;ωis the per-unit value of relative angular velocity;PmandPeare the per-unit values of mechanical power of generator and electromagnetic power respectively;Uis the voltage of stator;Ufis the exciting winding voltage.

Thus, the equation of output power of EESG can be obtained by

whereQeis the output reactive power;EqandXqstand for the induction electromotance and reactance of quadrature-axis respectively.

2.2 FSCWT integrating into system model

In this paper, the wind power plants containing FSCWT is processed by equivalent power source. So the dynamic load applies the form connecting walve composite load with statically constant power load, and the equivalent grid-connected structure adopts IEEE 3 node system. The equivalent structure of FSWCT is shown in Fig.2.

Fig.2 Equivalent structure of FSCWT integrating into power system

In Fig.2, node 1 is the one integrating into the equivalent wind farm, node 2 is load bus, node 3 is slack bus. The wind farm is composed by 20 front-end speed controlled wind turbines.

The dynamic characteristics of equivalent power source in FSCWT wind farm can be described by the EESG model given in Eqs.(10) and (12). The output power of wind farm, namely, the active powerPand reactive powerQat the node 1 respectively are

(14)

The power and system dynamic load model provided by network for load all apply the results offered by Ref.[23], in which the system load model applies the form connecting walve composite load with statically constant power load. From the Eqs.(10)-(14), the group of state equations describing the dynamic characteristics of the system can be obtained as

(15)

whereQξ=Q0-Q+Q1；Pξ=P0-P+P1；K1=Kqω·Kpu；K0=KpωKqu-K1；JG0is the rotational inertia of equivalent generator;TG0is the per-unit value of input torque of equivalent generator and its size can be derived from Eqs.(5)-(10).

3 Bifurcation analysis under effect of different control parameters

In order to test the stability of the FSCWT grid-connected voltage under the effect of different bifurcation control parameters, the grid-connected stability model containing the FSCWT wind farm is calculated and analyzed under the simulation environment of MatCont. The parameters of equivalent system, load model and CA2.0MW-WD are shown in Tables 1 and 2, and all the parameters use unitary value.

Table 1 Parameters of equivalent system and load model

Table 2 Wind turbine parameters of CA2.0MW-WD

3.1 Bifurcation analysis of voltage stability after disturbance of FSCWT parameter

In the process of operation, the fluctuation of voltage of load bus is analyzed according to the fluctuation ofωG. The systematically differential state variables selected in this paper areδ，ωG，θ2,u2. Taking load reactive power as the control parameter, this paper tests the influence ofGon the FSCWT stability of grid-connected voltage without considering the effect of excitation regulation. WhenGchanges from 0 to 2.5, andQ1is equal to 10.76, method of Newton-Raphson is applied to Eq.(15) to get the initial condition of system, which is: (δ，ωG，θ2，u2)=(19.011 3， 0， 6.438 5， 2.170 4).

Fig.3 shows the curve of the two-dimensional voltage with the fluctuation of wind turbines.

Fig.3 Variation of voltage of load bus with fluctuation of wind turbines’ angular velocity

From Fig.3, we can see that, after a period of fluctuation, the voltage of the load bus can reach a stable value, which indicates that the voltage of the load bus can also fluctuate with theωG.

WhenQ1is equal to 10, method of Newton-Raphson is applied to Eq.(15) to get the initial condition of system, which is: (δ，ωG，θ2，u2)=(19.236 1， 0， 2.431 6， 6.570 6， 0.483 6). And it can be learned from Fig.4 that the increase in reactive power of the wind turbines can delay the occurrence of SNB bifurcation node (LP) and raise the voltage of the load bus.

Fig.4 Q-u curve at node 2 when increasing reactive power

With the increase in reactive power of wind turbines, the capacity of holding the load by system will increase.

3.2 Bifurcation analysis of voltage stability under load fluctuation

Apply the Newton-Raphson to Eq.(15) to get the initial condition of system, which is: (δ，ωG，u2，p)=(0.553 572， 0， 0.587 784， 0.618 035). From Fig.5, it can be concluded that with the active power of consumption from load side increasing, the node voltage at the grid-connected point will decrease. Bifurcation node appears before that in SNB (LP).

Fig.5 p-u curve at node 2 when increasing active power

In order to delay the bifurcation phenomenon caused by the increase in active power and increase the FSCWT’s voltage stability integrating into power system, the feedback control of linear state has been applied to control LP. When the control variableeis added to the right side of the second equation in Eq.(15), a controlled closed-loop system can be obtained as

(16)

Because the power angle of generatorδ, angular velocity of rotorωGand the node voltageu2are measured easily, the power angle of generator is used as the feedback variable.

e=-ks(δ-δref),

(17)

whereksstands for controller gain;δrefis the reference angle ( 0≤δref≤π/2). Taking the valuesks=1,δref=0,ks=1 and substituting Eq.(17) into Eq.(16), we can obtain the value of parameter, which is (δ，ωG，u2，p)=(0.546 514，0，0.570 538，0.647 682) at the node LP of solid line by the Newton-Raphson method. The value of parameter at the node LP of dotted line is the same as that in Fig.5.

In order to analyze the influence of the active powerpat load bus on the node voltage stability, taking thepas the control parameter, this paper makes the single-parameter bifurcation analysis of the FSCWT before and after integrating into power system. The simulation results are shown in Fig.6. The dotted line in Fig.6 is thep-ucurve before the control; the solid line is thep-ucurve after the control.

Fig.6 p-u curve of node 2 before and after using feedback control of linear state

In order to analyze the influence of the control gain of FSCWTksand its active powerpon the voltage stability of node, takingksandpas the control parameters, this paper makes the bifurcation analysis of the controlled system and gets the boundary curve of LP system. The results are shown in Fig.7.

From Fig.7(a), in a certain range, the higher control gainksis, the larger active powerpis at the node LP, namely, the later the bifurcation occurs. Ifksbecome larger, influence of its value on controlling LP is no longer obvious. From Fig.7(b), if the value ofksbecomes smaller, increasingkscan reduce the value of voltage at the node LP. Ifksreaches a certain value (such as 21), increase inkswill not reduce voltage obviously at bifurcation nodes. Increasingksin a certain range, the bifurcation at the node LP will occur when voltage becomes low to improve the stability of the power system (voltage of load bus will decrease when occurring fault in power system).

Fig.7 Double-parameter bifurcation analysis of p and ks

4 Conclusion

According to the voltage stability of FSCWT grid-connection, taking the IEEE 3 node as an example, this paper researches the influence of FSCWT’s angular velocity, reactive source output and active power of load bus on the stability of grid-connected voltage of turbines, and establishes model of stability of grid-connected voltage of turbines. The research results show that, after operation of FSCWT grid-connection, the grid-connection voltage fluctuates with the fluctuation of the turbines’ angular velocity. The voltage at the grid-connected point can be increased by increasing reactive power of turbines. The increase in active power at the load bus can lead to the emergence of the LP bifurcation of voltage at the grid-connected point and the emergence of the LP bifurcation can be delayed by increasing gainksof feedback controller of linear state. But ifksbecomes too high, the effect of delaying LP bifurcation will not be obvious. On the other hand, increasing the feedback controller gainksin linear state can increase the value of active power at the load bus. Applying FSCWT grid-connection with feedback controller in linear state can improve the voltage stability of power system.

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