An Investigation of limit induced bifurcation in dynamic model of wind power system

Liu Yang, Liu Junyong, School of Electrical Engineering

and Information Sichuan University

Chengdu, China liuyanglee@163.com

Zhang Siming University of Liverpool

Liverpool L7 7AD Merseyside

United Kingdom sarah9-9@hotmail.com

Gareth Taylor, Liu Youbo Brunel University London UB8 3PH

Middlesex United Kingdom

gareth.taylor@brunel.ac.uk

Abstract—In this paper the dynamic model for Limit

Induced Bifurcation in wind power systems is constructed

and used to locate these bifurcation points of wind farms in

WSCC－9Bus system. For the sake of system stability, the

Limit Induced Bifurcation is investigated under different

wind speed conditions. Moreover, the initial condition of an

optimal model is calculated through the method of Quasi

Steady-State Analysis (QSS). It is safe to conclude that wind

farms operating with higher wind velocity will have larger

load margins, trading the probability of Limit Induced

Bifurcation. Moreover, systems with wide hard-limits for

dynamic devices and large fluctuations in wind speed will

suffer from a varying bifurcation, either Limit Induced

Bifurcation or Saddle Node Bifurcation, thus becoming

more vulnerable against system voltage collapse.

Keywords-Wind power system; Limit induced bifurcation;

Load margin; Quasi Steady-State; Voltage collapse；

I INTRODUCTION

According to the traditional bifurcation analysis of power systems, Ordinary Differential Equation (ODE)[1-2] representation has extended to Differential Algebraic Equations (DAE)[3-4]; the singularity of Jacobian Matrix is to determine the bifurcation type[5]. Unfortunately, for Limit induced bifurcation, which by definition is the bifurcation occurred when the system reaches its utmost value, its Jacobian Matrix is no longer singular. In the current study of Limit induced bifurcation, synchronous machines are generally treated as nodes with constant voltage[6-9]. This can no longer serve in wind power

systems, where node voltage of the wind power generator will drop significantly, because of the fact that asynchronous generators absorb reactive power from the utility grid. Therefore, the dynamic model of wind power systems should be constructed before analyzing its Limit induced bifurcation phenomenon.

This paper is based on a thoroughly considered dynamic model of the wind power system, and further presents the model for searching Limit induced bifurcation point in wind farms. To be more precise, QSS analysis gives the initial value for the model; tracing along the system's PV curve finds the limit induced bifurcation point; quantize the influence of wind speed and hard-limits on Limit induced bifurcation.

II WIND POWER SYSTEM MODEL

This paper takes constant speed constant frequency system wind power generating as the investigation subjects to form the wind power model [10-11].

A. Wind Turbine Model

The relation between the wind velocity and the torque is shown below[10]:

3 2

31 102

Nw p

N

UR vT cP

πρλ

−= × （1）

The gear box which gives the dynamic equation:

1 ( )dT m

d T Tdt Tτ

ω = − （2）

where ωd is the speed, Tτ is the inertia time

The National Natural Science Foundation of China (No.50977059)

978-1-4244-6255-1/11/$26.00 ©2011 IEEE

constant of the gear box, Tm is mechanical torque of the asynchronous machine. When wind turbines operation stable, its speed is essentially the same and can be regarded as equal T τ and Tm

[10].

1 ( )Tw T

h

dT T Tdt T

= − （3）

Th is the inertia time constant，TT is the torque at the input side of the gear box.

B. Model For The Asynchronous Motor

By neglecting the electromagnetic transient state of the stator winding in the asynchronous generator, the electromagnetic equation is following:

'

'

'' ' '

0 0'0

'' ' ' '

0 0'0

'

'

1 [ ( ') 2 ]

1 [ ( ) 2 ]

q s q d q

d s d q d

qq d d d

d

dd q d q

d

U r i x i E

U r i x i E

dEE x x i f T sE

dt T

dE E x x i f T sEdt T

π

π

⎧ = − − +⎪

= − + +⎪⎪⎪⎨ = − − − −⎪⎪⎪ = − + − +⎪⎩

（4）

where：

s mx x x= +

' r ms

r m

x xx xx x

= ++

'

02r m

dr

x xTf rπ

+=

Ed' ,Ud and id stand for transient EMF, stator voltage

and stator current in the direction of d axis, while Eq', Uq

and iq are the q-axis constituents. rs and xs represent the stator resistance and stator reactance respectively. rr and xr are the rotor resistance and stator leakage reactance. xm means magnetizing reactance, and Td

' is the time constant. s is defined as the slip of the asynchronous machine, which is less than zero when the machine acts as a generator. f0 is the rated system frequency.

The mechanical angular acceleration and the unbalanced torque endured by the rotor satisfy the following equation:

1 ( )T Ed T Tdt Tω = − （5）

Where T is the inertia time constant of the induction

generator, and TE is the electromagnetic torque. Synchronous generator is expressed by forth-order

equations, simply skipping the transient state. Induction generators, synchronous generators and exciters contribute to the ODE equations of the wind power system; stator constraints of the asynchronous generator, rotor constraints of the synchronous generator, network constraints and the final power output give the Algebraic equation set. The linearization at the equilibrium point of the DAE set will not be discussed here, and can be referred to Appendix [12].

III LOCAL BIFURCATION IN WIND POWER SYSTEM

Power electric systems can be represented as a set of dynamic algebraic equations (DAE):

.( , , , )

0 ( , , , )x f x y p

g x y pλλ

⎧⎪ =⎨

=⎪⎩ （6）

Where f(·)∈R represents the ODEs for the system; g(·)∈Rm consists of the Algebraic equations; x∈Rn

specifies state variables associated with generators, loads and system controllers; y∈Rm is the vector of algebra variables; λ∈R1 is the change factor of the load; p∈Rr is the parameters for every device in the system. Obviously, with regard to the linear property at the equilibrium point, Eq.6 can be transformed as:

.

0x y

x y

f f xxg g y

⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦⎣ ⎦⎣ ⎦ （7）

.1( )x y y x sysx f f g g x A x−Δ = − Δ = Δ （8）

In this paper, as a step forward from the Saddle

Node Bifurcation analysis model[13-14]

, an optimal model for studying Limit Induced Bifurcation is proposed.

λ

V

1λ 2λ

limQ Q= 'limQ Q=

A

B

C

D

E

3λ0λ

sV

'1λ

'A'E

'2λ

Figure 1. PV curves for generator

In Fig.1, the provided voltage remains constant at the generator node [6-9], whichever method is applied, continuous power flow method or optimization method. Therefore, by comparing the voltage when the load transmission reaches a maximum with the assigned voltage of the generator, the stable A and unstable E Limit induced bifurcation points can be determined. In the light of the definition of bifurcation given in Reference [6], A and E, altogether, are referred as the constraint exchange points, while only A belongs to the Limit induced bifurcation.

As mentioned previously, the induction generator demands reactive power from the gird, which will inevitably lower the voltage at the generator node. If the wind power generator node and the synchronous generator node are made identical, the Limit induced bifurcation will suffer inaccuracy. As illustrated in Fig.1, with the decrease of voltage across the wind generator, Limit induced bifurcation has already occurred prior to E'.

λ

V

1λ 2λ

limQ Q= 'limQ Q=

AB

C

D

E

3λ0λ Figure 2. Limit induced bifurcation diagram

However, when we take the limit into account, two possible situations are examined with details. As Fig.2 illustrates, DA and BE are PV curves for the generator node or the load node with system reactive power keeping its maximum. Note that D and E are the intersects with the original PV curves. When the operating point reaches D, the system hits reactive power limit. Then the operating point moves along the trace DA to point A, when system load margin is λ=λ1-λ0 and there is no Limit induced bifurcation; at point E, the moment generators encounter reactive power limit, instability

occurs with the load margin λ=λ2-λ0. E is the limit induced bifurcation point. In the following subsections, the numerical solution for limit induced bifurcation will be provided.

A. Saddle-node bifurcations of Keeping System Threshold

There is only one Saddle-node bifurcations point in every power system. Supposing that a certain dynamic device is running at its maximum capability, Eq.6 becomes:

.' lim

' lim

( , , , , )

0 ( , , , , )i

i

x f x x y p

g x x y p

λλ

⎧⎪ =⎨

=⎪⎩ （9）

' 'sys x xA w wλ=

Let '[ , ]y

T T Tx

w w w= ， and ' '1

y y x xw g g u−= − ，

where 'xw is the right eigenvector of Matrix sysA .

Where 'x ∈Rn-1，xi

lim is the limit of the state variable xi Maintaining the system parameters at maximum, saddle-node bifurcations can be calculated by:

' lim

' lim

' ''

'

' '

min | |

0 ( , , , , )

0 ( , , , , )

1

i

i

x xx y

y yx y

Tx x

f x x y p

g x x y p

w wf fw wg g

w w

αλλ

α

⎧ =⎪

=⎪⎪⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎨

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎪⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦⎪⎪ =⎩

（10）

This equation set is characterized by 2（n+m-1）+2 variables and consists of 2（n+m-1）+1 equations and an optimization objective. Also the node voltages at the saddle-node bifurcations are denoted as VSNi , and the load parameter is λSN.

B. Constraint Exchange Point on PV Curves

Besides the original system equation set for optimizing the PV curve, an equality constraint is added to find the critical point where system variables surpass their limits. The complete equation set is presented below:

min( )λ−

lim

0 ( , , , )0 ( , , , )0

0

0

0

1 0

0

0

x xx y yx xx xy

x xy y yy xy xx

x xx y yx

x xy y yy

T Txx xx xy xy

T Txx xy xy xx

i i

f x y pg x y pf w f w w w

f w f w w w

g w g w

g w g w

w w w w

w w w w

x x

λλ

α βα β

⎧ =⎪⎪ =⎪

= + − +⎪⎪ = + − −⎪⎪ = +⎨⎪ = +⎪⎪ − − =⎪⎪ + =⎪⎪ − =⎩

（11）

Where wx=wxx+jwxy，wy=wyx+jwyy. This equation set (11) is characterized by 3（n + m ）+3 variables with 3（n + m ）+3 equality constraints and an optimal object function. The threshold voltage is denoted as Vlimi , and loading factor is λlim

C. The Criterion for Limit induced bifurcation

Vlimi<VSNi，limit induced bifurcation occurs with load marginλ=λlim-λ0；

Vlimi>VSNi ， no limit induced bifurcation ， load marginλ=λSN-λ0；

Vlimi=VSNi，critical point for limit induced bifurcation, load margin λ=λSN-λ0=λlim-λ0.

IV BIFURCATION ANALYSIS EXAMPLE

WSCC－9Bus system[3] is used as a demonstration of bifurcation analysis in this paper.

Figure 3. Wind farms connect to 3-generator system

Wind farm model is connected with the transmission system at node 8, and the base MAV of the asynchronous generator is 600kW. rs=0.025117,

xs=0.30082, rr=0.011244, xr=0.32857, xm=10.686, Tτ=10, Th=10，xl=0.2 . The rated wind velocity of the system is

ν=13.5m/s, and ν*=1 is used in actual calculations in per-unit system.

Wind generators are placed in a Multimachine Power system, whose frequency is 50Hz, and its base Volts-Amps is 100MVA. The model for the system load obeys one principle that the power of the load remains constant for all times, so the mode of change is restricted. All in all, the load of the system (at node 5, 6, 8) increases and can be modeled as:

0

0

(1 )(1 )

li l i

li l i

P PQ Q

λλ

= +⎧⎨ = +⎩

（12）

In the simulation for the actual operation of the wind power system, with the growing load demand, synchronous generator 1, 2 and 3will output real and reactive power according to the same loading factor, in order to tune in equal proportions the power consumption of the static and the dynamic parts of the loads; asynchronous generator 4 output real power following the change in wind speed. As a result, when the wind speeds up, the excitation voltage for the synchronous generators will soar up accordingly. Consequently, it is quite common for synchronous generators to surpass the reactive power limit. Note that, considering the reactive power limit is the same as considering the limit on excitation voltage Efd. In

this example, the excitation voltage of generator 3 is selected as the limit induced bifurcation parameter in wind farms.

A. Constant wind velocity

Firstly, the case of constant wind speed is examined, while the per-unit value is set as ν*=1.25. In Table 1, the voltages across the generator 1 and 11 are listed the moment when bifurcation occurs. According to Table 1, when Efd3max=2, the loading factor λSN at the saddle node bifurcation point is

always larger than that at the critical change point of the hard-limit λlim on the PV curve. In the limit induced bifurcation model, it is possible for Vlimi> VSNi,

hence only saddle node bifurcation but limit induced bifurcation occurs, and the load margin is

λ=λSN=0.5885. When Efd3max=2.3, the corresponding load margin becomes λ=λSN=0.7238. When changing to a higher excitation voltage Efd3max=3.5, the

synchronous generators lose a small portion of their voltages across; meanwhile, wind power generator 11 suffers a dramatic voltage decrease. This proves that there will be comparatively big error if the bifurcation model treats the generator voltages as constant by only taking static power system model into consideration. The simulation result clearly indicates the situation when Vlim1<VSN1 and Vlim11<VSN11, limit induced

bifurcation occurs with the load margin λ=λlim=0.987<λSN. To briefly summarize, in the

dynamic model of the wind system with invariant wind speed, the higher system limit means more reactive power output by synchronous generators, as well as wider load margin, but more possible limit induced bifurcation. This will endanger the stability of the system, which is demonstrated in Fig.4.

λ

V

3max 2fdE = 3max 2.3fdE = 3max 3.5fdE =

Figure 4. Bifurcation diagram for Considering the limit of Efd3

TABLE I. Tab.1 Considering ν*=1.25 and different limit of Efd3

Efdmax Vlim1 VSN1 Vlim11 VSN11 λlim λSN

Efd3max=2 1.0397 1.0393 0.9801 0.9348 0.4347 0.5885

Efd3max=2.3 1.0395 1.039 0.9631 0.8914 0.6296 0.7238

Efd3max=3.5 1.0387 1.0389 0.6015 0.8296 0.987 1.1989

B. Changing Wind Speed

In real operation at wind farms, wind speed is changing every second, and the simulated wind speed is

between ν*=0.8 and ν*=1.8 shown in Table Ⅱ. When

Efd3max=2, the load margin only increases 0.0673 as the

speed changes from ν*=0.8 to ν*=1.8, which implies

that the wind speed can affect the load margin, but little increase load margin at the saddle node bifurcation point. When Efd3max=2.3, the rise of wind speed leads to

higher excitation voltage provided by synchronous generators, along with more reactive power output to balance the consumption from the wind power generator. When ν*=1.8, there is limit induced bifurcation, and the load margin is λ=λlim=0.7905,

which is smaller than when ν*=1.5. The reason is that limit induced bifurcation happens before saddle node bifurcation, and when the wind is growing stronger, the former makes a transaction to the latter, leaving a shrinking load margin. When Efd3max=3.5, in the range

from ν*=1.25 to ν*=1.8, system encounters limit induced bifurcation, and the expansion in load margin is more noticeable as wind speeds up, as illustrated in

Table Ⅱ.

Simulation reveals the fact that increasing wind speed can delay saddle node and limit induced bifurcation, especially for the latter case. Nevertheless, faster wind doesn’t necessary means wider load margin, on the contrary, under particular circumstances with certain device hard-limits, it could cause limit induce bifurcation and deduce the load margin. In extreme cases, if the system is heavily loaded, the voltage will fluctuate, resulting in system collapse.

TABLE II. The load margins for considering different wind speed and

limit of Efd3

Efd3max

ν*

2 2.3 3.5 4

0.8 0.5339（S） —— —— ——

1 0.5448（S） 0.7228（S） —— ——

1.25 0.5885（S） 0.7238（S） 0.9870（S） ——

1.5 0.5924（S） 0.8141（S） 1.2128（L） 1.2543（L）

1.8 0.6012（S） 0.7905（L） 1.3201（L） 1.3635（L）

C. Discussions

Every hard-limit comes from the physical properties of the device itself. So for a particular wind power system, the peak values of the excitation voltage are deterministic

in nature. In terms of heavy-load systems, when Efd3max=2,

if the load margin needs to be λ=0.5339, ν*=1.8, hence Δλ1=0.6012-0.5339=0.0673. As the acceptable variation of wind speed is Δν1*=1.8-0.8=1, no saddle node

bifurcation will happen. It explains that fluctuation in wind has little effect on system load margin when excitation voltage limit is rather low; even with heavy loading, the system can avoid saddle node bifurcation be restricting the wind speed change.

When Efd3max=3.5, λ=0.987 and ν*=1.8, the change in load margin Δλ2=1.3201-0.987=0.3331, in speed Δν2*=1.8-1.25=0.55. It can be seen that Δλ2>>Δλ1, but Δν2*<Δν1*, meaning when the excitation voltage limit is

higher, wind speed will have more say on load margin. In extreme cases, with heavily loaded system, a tiny disturbance in the wind will trigger large change in load margin and limit induced bifurcation. As the analysis goes, when the excitation limit is high, under no circumstances should the system be connected with large loads, otherwise the sensitiveness of the wind speed will drive the system unsteady.

When Efd3max=2.3, faster wind stimulate the

transformation from saddle node to limit induced bifurcation. In Table Ⅱ, ν*=1.8 is when limit induced bifurcation occurs. As wind keeps rising, load margin goes through minor changes as up and down and finally up. If the local wind tends to vary, the bifurcation will change between saddle node and limit induced bifurcation, which contributes to the random change in load margin. Consequently, wind power systems should never be over-loaded, in comparison to the traditional power systems.

V CONCLUSION

In this paper, it is confirmed that Limit induced bifurcation does exist in wind power systems by constructing a universal optimal model to search for bifurcation points in such systems. In terms of limit induced bifurcation, wind power systems have unique characteristics when compared with conventional power systems. For instance, when the wind blows faster, the reactive power consumption of the wind turbines will rise. As a result, the wind farm will be more likely to operate in

the condition where dynamic equipments are pushed towards their hard-limits, delaying the Saddle node bifurcation and causing limit induced bifurcation. In addition, in real wind farms, when the wind speed fluctuates a lot, the power system will switch between saddle node bifurcation and limit induced one back and forth; in other words, any change of wind velocity will narrow down the load margin. Obviously, due to constant unpredicted variation of the wind speed, it is more easily for bifurcation phenomenon to happen, and in a more complicated manner.

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