IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, … · B. G. Rawn is with the Department of Electrical Sustainable Energy, Delft University of Technology, Delft 2628 CD, The Netherlands

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 21, NO. 5, SEPTEMBER 2013 1795

Disturbance Margin for Quantifying Limits onPower Smoothing by Wind Turbines

Barry G. Rawn, Member, IEEE, Peter W. Lehn, Senior Member, IEEE, and Manfredi Maggiore, Member, IEEE

Abstract— Wind turbines can, in principle, be operated tosmooth wind power fluctuations by allowing wider variations inturbine speed and generator torque to store and release energy.This ability must be constrained by turbine speed and generatortorque limits. To present, work in the literature is conceptual anddoes not indicate what extent of smoothing is possible beforecomponent limits are reached, nor does it quantify sensitivityto variations in the input wind speed. This paper introducesa method for quantifying how much wind variation a windturbine can absorb in variable speed mode while still beingguaranteed to operate within its component limits. One can applythis method to obtain the dependence of maximum tolerablewind disturbance on the smoothing time constant, and thus makedesign decisions. This paper shows that the analysis of torquespeed intersections, as standardly applied in electric machinetheory, is of limited use for studying power smoothing. The newconclusions and design choices made available by the proposedmethod are illustrated with a series of computation examples. Themethod is shown to agree asymptotically with two limiting casesthat can be calculated based on torque-intersection analysis. Themethod is based on new theory for computing invariance kernelsfor nonlinear planar systems, and can be adapted to assess therobustness of other control laws.

Index Terms— Control theory, kinetic energy, mathematicalanalysis, nonlinear systems, state space methods, wind powergeneration.

I. INTRODUCTION

THE MECHANICAL behavior of electric machines hastraditionally been analyzed using torque-speed diagrams.

Though the system dynamics are typically nonlinear, operatingpoints are readily determined by the intersection of loadtorque-speed and machine torque-speed curves. Transients ofsuch systems have commonly been viewed merely as transi-tions between two steady state operating points on the torque-speed plane. This simple analysis approach provides usefulinsights into the electro-mechanical system, but it relies onthe assumption that: 1) the load (or source) torque takes ona constant steady state value and 2) electrical dynamics aremuch faster than the mechanical dynamics of the system.

An example of a widely-found system that violatesthese assumptions is the variable speed, converter-interfaced

Manuscript received September 1, 2011; accepted June 20, 2012. Manu-script received in final form July 19, 2012. Date of publication August 27,2012; date of current version August 12, 2013. Recommended by AssociateEditor G. E. Stewart.

B. G. Rawn is with the Department of Electrical Sustainable Energy,Delft University of Technology, Delft 2628 CD, The Netherlands (e-mail:[email protected]).

P. W. Lehn and M. Maggiore are with the Department of Electrical andComputer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada(e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TCST.2012.2210222

wind turbine. To achieve peak power tracking, generator torqueis typically set to be an algebraic function of generator speedto establish a static curve of desired operating points inthe torque-speed plane [1]. The shaft torque of the machinechanges with wind speed. Thus, the machine and converterare subject to a time varying torques, violating the firstassumption.

With wind energy systems providing an ever growingportion of total generation, expectation of these systems tosupport frequency regulation in the power grid is emerging.Proposals to exploit wind turbine kinetic energy for smoothingwind power [2]–[5], damping oscillations in power systemfrequency [6], [7], or counteracting momentary dips in fre-quency [8], [9] are the subject of growing interest in theliterature. These methods all introduce electrical dynamicsthat, through control action, are on a time scale comparable tothe mechanical dynamics. This violates the second assumptionof static torque-speed analysis. In sum, static torque-speedanalysis approaches are insufficient to assess this new classof system and alternative analysis techniques are required.

This paper introduces the notion of “wind disturbancemargin” and provides an analysis technique to compute it.The margin in question is the largest bound on the windinput signal variation for which the turbine is guaranteed toremain within a pre-defined safe operating region. For a givenmean wind speed, the margin characterizes the robustness ofthe energy conversion system to wind disturbances. Coupledwith advanced wind speed measurement techniques [10], themargin could be used to guarantee safe operation while pro-viding smoothing or other services that exploit turbine kineticenergy.

The computation of the margin employs a new techniquethat can be applied to planar nonlinear systems with a singleinput. The solution technique begins by first identifying theturbine’s safe operating region and mapping this region fromthe torque-speed plane into the state-space plane. The resultis a set of safe operating points in the state plane. This setis then tested for the existence of an invariance kernel- thelargest set of initial conditions leading to trajectories that cannever leave the set of safe operating points for a given classof wind disturbance signals. The class of wind disturbancesignals is characterized only by its mean wind speed, and abound on its range of variation. The bound is increased to findthe disturbance margin, which is the largest possible boundallowed before the invariance kernel becomes empty. Insteadof brute force simulation, computation of the invariance kernelemploys an analytical method that calls only for the integrationof a finite number of special trajectories. This paper works

1063-6536 © 2012 IEEE

1796 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 21, NO. 5, SEPTEMBER 2013

with the example of a wind turbine operating in variablespeed mode. The turbine is subjected to wind variations thatare bounded but otherwise unspecified. The generator is thussubject to a commensurate shaft torque, which is also boundedbut otherwise unspecified. The generator is regulated viatorque control, where the torque reference is derived by low-pass filtering the generator speed. This achieves smoothing ofthe wind turbine output power, allowing the kinetic energyof the rotor to absorb rapid input energy changes throughvariation of the rotor speed. Such a system is effective inreducing the impact of wind turbine power variations on thepower grid. However, too large a filtering time constant maylead to collapse [3] or overshoot of the rotor speed to the pointwhere pitch control intervenes and causes power variationsthat defeat the purpose of smoothing. The simplified two-state model and the analysis presented offer a new means ofassessing such power smoothing control schemes present inthe literature.

The structure of this paper is as follows. In Section II, theapplicability of a simplified wind turbine model is discussed.Section III introduces the disturbance margin concept andassociated definitions. Section IV demonstrates computationof the margin for special cases and highlights limitations ofstandard analysis techniques. Section V explains the planarinvariance kernel analysis technique and its application to thewind turbine problem. Section VI demonstrates computationof the margin for the general case, and Section VII providesconclusions.

II. WIND TURBINE MODELING AND CONTROL

The power conversion efficiency of a wind turbine dependson the tip-speed ratio

λ = Rω

vw(1)

where R is the radius of the rotor disc, ω is the rotationalspeed of the turbine rotor, and vw is the wind speed. Thereis also a dependence on the pitch angle of the blades whichis not considered in this paper. The dependence of conversionefficiency on λ is shown in Fig. 1. Efficiency has a maximumat the tip-speed ratio λopt. For any larger tip speed ratio λdel,conversion efficiency is lowered. A conversion efficiency of atleast Cp(λdel) is available over the range of tip-speed ratios[λ′

del, λdel]. Operation at λopt is referred to as optimal, andoperation away from λopt is referred to as de-loaded.

The aerodynamic torque exerted on the shaft of the windturbine is as follows:

Taero(ω, vw) =12ρπ R2Cp (λ) v3

w

ω(2)

where ρ is the density of air. The dependence of the aerody-namic torque on both rotor speed and wind speed is illustratedin Fig. 2, where curves corresponding to two values of windspeed (v low

w and vhighw ) are shown in the torque-speed plane.

The values chosen are associated with variable speed operationbetween the rotor speeds ωlow and ωhigh.

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

λdel λoptλdel

Cp(λ

)

λ

Fig. 1. Power conversion efficiency curve Cp(λ) for optimal constant pitchangle. Optimal operation occurs at λopt , while de-loaded operation occurs forvalues λ′

del and λdel.

A. Control Regimes and Component Limits

There are multiple ways in which the physical limitationson wind turbine components affect wind turbine controls [1].Wind turbines begin operating above a certain cut-in windspeed. For low wind speeds, the rotor is maintained closeto its minimum speed by a steep generator torque [SectionsA and B in Fig. 2]. Above a rotor speed ωlow (greater thanthe minimum allowed speed by some amount [11]), variablerotor speed operation occurs (Sections B and C) up to a rotorspeed ωhigh (less than maximum by some amount). Aboveωhigh, both generator torque and pitch feedback controls arealtered to keep rotor speed at or below the maximum value of1.0 per-unit (C and D) [12]. The maximum torque and rotorspeed (D) are reached at the rated wind speed. For wind speedsabove rated, electric power is limited to the rated value for thewind turbine using a constant power generator torque curve(D and E) and the adjustment of pitch to reduce aerodynamictorque.

In variable rotor speed operation (B and C), pitch is setto the optimal value and pitch control is not active [13].Rotor speed is free to vary between speeds ωlow and ωhigh,provided the maximum torque limit Trated is never exceeded.The generator torque Tgen in modern variable speed windturbines is established through control of power electronics[14]. Between the points B and C Tgen is set to

Tgen(ω) = kload(λ�)ω2 (3)

kload(λ�) = 1

2ρπCp(λ�)

R5

λ3�

(4)

where λ� is a desired tip-speed ratio. The generator torquecurve in the torque speed plane is often referred to as the loadcurve. The load curve is intended to allow the rotor speed tocontinually adjust to new setpoints as the wind speed changes.The choice of (3) and (4) has the effect that for a constantwind speed the efficiency Cp(λ�) is reached in the steady state.

RAWN et al.: DISTURBANCE MARGIN FOR QUANTIFYING LIMITS ON POWER SMOOTHING BY WIND TURBINES 1797

0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

0.2A

B

C

D

E

ω (p.u.)

vwlow

vwhigh

TratedT

orqu

e(p

.u.)

ωhighωlow

Taero(ω↪ vw)Tgen(ω)

Fig. 2. Aerodynamic (solid) and generator torque-speed curves (thick) forvariable speed wind turbine. Letters label operating points, where variouscontrol schemes become active. Safe region for variable speed operation isenclosed by a dashed box.

B. Simplified Dynamic Model for Power Smoothing

For control design related to power smoothing, a model ofwind turbine dynamics in the variable speed range (B and C)can neglect the pitch control, which is inactive, and can bevery simple [15], [13], consisting only of the single nonlineardifferential equation for rotor speed dynamics

Jdω

dt= Taero(ω, vw) − Tgen (5)

where J is the rotational inertia of the turbine hub andgenerator. For operation at and above rated wind speed (D andE), structural vibrational modes, such as the tower deflectionare usually also modeled because of their interaction with pitchcontrol [13] and relatively large excitation at high wind speeds[16]. However, in variable speed operation (B and C), theexcitation is much smaller and not enough to warrant extrapitch actuator wear [16]. Tower deflections also have littlediscernable effect on rotor speed [17]. For these reasons, towerand other structural modes can be safely excluded from thismodel, as is widely done for control design purposes [18].

The objective of power smoothing is to exploit variablespeed operation to absorb wind power fluctuations using therotational kinetic energy of the turbine. The concept has beenexplored with control schemes that set constant power ortorque [4], [5] or that introduce a low-pass filter into thesignal path used to generate these control references [2], [3],[19]. Most of these works acknowledge that depending on thechosen time constant of smoothing and amount of de-loading,instability may result from power smoothing. None of theseworks quantify how much smoothing is practically viable.

To study how power smoothing controls should safely bedesigned, this paper examines the controller introduced in [2].A low-pass filter is applied to a measurement of rotor speed,and a new variable ωfilt is used to generate a power reference.The power reference is extracted regardless of rotor speedby setting the torque command depicted in Fig. 3, whichequals the desired level of power divided by the unfiltered

11 + sτ

TgenPgenωfiltω ÷kload(λ )ω3filt

Fig. 3. Block diagram of power smoothing generator torque Tgen(ω,ωfilt)based on filtered rotor speed [2].

rotor speed ω

Tgen(ω, ωfilt) = kload (λ�) ωfilt3

ω. (6)

Dynamics are now described by two differential equations

Jdω

dt= Taero(ω, vw) − kload (λ�) ωfilt

3

ω

τdωfilt

dt= ω − ωfilt (7)

where τ is the time constant of rotor speed filtering. Thechoice of a power reference with a low-pass filter in the signalpath has two advantages. First, such control structures arealready in place (with small time constants) in some industryimplementions [7], [20]. Second, in the limit for large τ , (7)corresponds with constant power smoothing schemes studiedin the literature [4], [5].

C. Limiting Cases

Two limiting cases of speed filtering can be considered toprovide insight into system operation. For τ = 0, ωfilt =ω, and the generator torque (6) simplifies to the standardmaximium power tracking law (3), in which case the rotorspeed variations are dictated by

Jdω

dt= Taero(ω, vw) − kload(λ�)ω

2. (8)

For τ = ∞, ωfilt is a constant ωfilt0, and rotor speed

variations are instead dictated by

Jdω

dt= Taero(ω, vw) − kload(λ�)

(ωfilt

0)3

ω. (9)

Setting ωfilt0 to the steady state rotor speed associated with a

wind speed vw

ωfilt0 = λ�

Rvw (10)

simplifies (9) to

Jdω

dt=

12ρπ R2

ω

(Cp (λ) v3

w − Cp(λ�)v3w

)(11)

where vw sets the constant power level demanded and couldbe, for example, set to the mean or minimum value of windspeed expected in a certain period, as is suggested in otherwork [4], [5].

When the generator torque control law has no dynamics ofits own as in (8) or (11), movement in the torque speed-planeis confined to the load curve. When instead the control input(6) is employed, movement in the torque-speed plane dependson dynamics of both ω and ωfilt and can leave the load curve.


III. WIND DISTURBANCE MARGIN DEFINITION

In this paper, we consider a wind speed input

vw(t) = vw + δvw(t) (12)

where vw is a mean speed, and δvw(t) is a deviation.To define the wind disturbance margin, we use a number

of sets [21]. We begin by defining the collection of all winddeviations bounded from above by a constant .

Definition 1 (Class N () of Wind Deviation Signals):Denote by N () the class of all wind deviations boundedby

N () � {δvw(t) : |δvw(t)| ≤ for all t ∈ R} (13)

so thatvw(t) ∈ [vw − , vw + ] . (14)

Next, we define the set of all feasible states of the wind turbinemodel (7) corresponding to arbitrary constant wind speeds inthe interval [vw − , vw + ].

Definition 2 (Equilibrium Set E): The equilibrium setE(, vw, λ�) of (7) is the collection of all equilibria of (7)assuming that the wind is constant, and that its value rangesover the interval [vw − , vw + ]. In other words

E(, vw, λ�) �{(ω, ωfilt) : ω = λ�

Rv0w,ωfilt

= ω, v0w ∈ [vw − , vw + ]

}. (15)

The equilibrium set E(, vw, λ�) is depicted on the ω-ωfiltstate plane in Fig. 4 for a specific choice of , vw , and λ�.Also shown in Fig. 4 is a dashed line corresponding to therotor speed limits and torque limits of the machine. The rotorspeed limits are the same as those defined in Fig. 2. Imposinga limit on the absolute value of generator torque Tgen to beless or equal than the rated torque Trated requires in turn a limiton the filter state ωfilt, through the relation (6). The necessarylimits on ωfilt can be obtained by setting Tgen = ±Trated andsolving for ωfilt . Outside of these limits, safety controls willintervene to maintain rotor speed through pitch action or tolimit converter currents. Intervention by safety controls willdisrupt the behavior intended by the control law (6). The rotorspeed limits, combined with the machine torque limits, formbounds on a region in the ω-ωfilt state plane. The region offree operation whose borders are enforced by safety controlswill be called the safety set.

Definition 3 (Safety Set S):

S �{(ω, ωfilt) : ωlow ≤ ω ≤ ωhigh,− 3

√Tratedω

kload(λ�)

≤ ωfilt ≤ 3

√Tratedω

kload(λ�)

}. (16)

In this paper, we will also refer to this set’s boundary usingthe notation ∂S.

For a chosen control law, the wind turbines’ state-overtime will be determined by the dynamical equations and thewind signal input, which we have restricted for this paper to

0.6 0.8 1.0

−1.0

−0.5

0

0.5

1.0

ω (p.u.)

ωfil

t

S

∂S

S∂S

E

Fig. 4. Safety set S , equilibrium set E , and the invariance kernel S� thatdefine disturbance margin M for (7).

signals of the class defined in Definition 1. We will associatethe desired behavior of the wind turbine over time with theexistence of a positively invariant set. To this end, let x(t, x0)denote the solution of (7) with initial condition x0. Fix vw and, and consider (7) with a wind signal vw(t) = vw + δvw(t).

Definition 4 (Positively Invariant Set I): A set I ∈ R2 is

said to be positively invariant for (7) if ∀x0 ∈ I, and ∀δvw(t) ∈N (), x(t, x0) ∈ I, ∀t ≥ 0.

Initial conditions belonging to a positively invariant setproduce trajectories that can never leave that set, no matterwhat wind signal in the class vw(t) = vw + δvw(t) ∈ N ()affects the wind turbine. Definition of a positively invariant setcan analogously be made for the control law (3) and dynamics(8), or other control laws.

Of particular interest are positively invariant sets containedin the safety set. Indeed, if I is such a positively invariantset, initial conditions in I lead to solutions of (7) that satisfythe safety limits of the turbine for any wind signal in the classvw(t) = vw+δvw(t), with δvw(t) ∈ N (). In this context, it isnatural to look for the largest positively invariant set containedin the safe set.

Definition 5 (Safety Set Invariance Kernel S�): Given thesafety set S in Definition 3, a chosen bound , a mean windspeed vw , a desired tip-speed ratio λ�, and a filter time constantτ , the set S�(, vw, λ�, τ ) is the maximal set contained in thewind turbine safety set S that is positively invariant for (7).

Given a class of wind signals and values of λ� and τ , ifit happens that the invariance kernel S� is empty, then forall initial conditions in S, there exists a wind signal makingthe solution x(t) exit the safety set. In such a situation wewould conclude that the wind turbine cannot tolerate arbitrarywind signals in the stated class. A class of wind signalsis therefore feasible for the wind turbine if the associatedinvariance kernel is not empty. More precisely, given valuesof , vw, λ�, and τ , the least one must require in order toconclude that the turbine can tolerate arbitrary wind signalsvw(t) = vw +δvw(t) with δvw(t) ∈ N (), is that S� containsthe equilibrium set E associated with all constant winds inthe interval [vw − , vw + ]. This observation inspires ourdefinition of wind disturbance margin.

Definition 6 (Wind Disturbance Margin M): Given thesafety set S, a mean wind speed vw , a desired tip-speedratio λ�, and a filter time constant τ , the wind disturbance


-0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

+ω (p.u.)ω− ω+

vw = vw + Δ

vw = vw

ωhighωlow

Tor

que

(p.u

.) Taero(ω, vw)Tgen(ω)

− Δ

0.9

Fig. 5. Determination of S� for τ = 0 based on intervals of guaranteedacceleration and deceleration (shaded). S = [ωlow, ωhigh], E = [ω−, ω+],and S� = S .

margin M(vw, λ�, τ ) is the largest value of for which theinvariance kernel S�(, vw, λ�, τ ) is nonempty, and containsthe equilibrium set E(, vw, λ�).

IV. WIND DISTURBANCE MARGIN COMPUTATION:SPECIAL CASES

The boundary cases of power smoothing with τ = 0 orτ = ∞ are degenerate cases where the variable ωfilt is eitherequal ω, or is constant, with no dynamics. Analysis of thesecases can be conducted in the torque speed plane by studyingthe single differential equation (8) or (11). Because there isonly one state, the sets S, E, and S� are all intervals of thereal line. A simplification of the safety set results because thegenerator torque depends only on ω, and the torque limit Tratedis not reached for any rotor speed in [ωlow, ωhigh]. Therefore,ensuring that rotor speed is contained in [ωlow, ωhigh] will alsoguarantee torque limits are not exceeded. The safety set Sreduces to

S �{ω : ωlow ≤ ω ≤ ωhigh

}. (17)

The cases τ = 0 and τ = ∞ will be used to introduce theconcept of the disturbance margin and show the limitations oftorque-speed analysis.

A. Degenerate Cases τ = 0 and τ = ∞For the degenerate case τ = 0, the effect of all wind

deviations in N () can be examined using the pair of extremeaerodynamic torque curves shown in Fig. 5 (light solid), alongwith the generator torque given by (3) and also shown (heavysolid). For the practical range of tip-speed ratios encountered,Taero(ω, vw) depends monotonically on vw . Consequently theextreme values of Taero(ω, vw) occur at the wind speeds vw−and vw + . The equilibrium set E is defined for the steadystates of (8) instead of the steady states of (7) and coincideswith the interval [ω−, ω+].

The invariance kernel S� is an interval whose torque-speed points are associated with torque-speed intersections.As depicted in Fig. 5, for all time and for any deviation inN (), there are some rotor speeds that experience a consistent

acceleration (shaded +) or deceleration (shaded −). Theseshaded intervals are bounded at one end by one of the limitingspeeds ωlow or ωhigh, and at the other end by a rotor speedcorresponding to torque-speed intersections of the extremeaerodynamic torque curves. For the single pair of intersectionsoccurring for the case τ = 0, the intersection associated withthe lowest wind speed vw − is ω− and with the highest windspeed vw + is ω+. At the lower speed ωlow, accelerationis consistently experienced, and at the upper speed ωhigh,deceleration is consistently experienced. For the value of chosen in Fig. 5, the rotor speed cannot be driven out of theentire safe interval [ωlow, ωhigh]. Therefore, S� is [ωlow, ωhigh].

The wind disturbance margin M for the case τ = 0 is thelargest bound that still ensures

[ω−, ω+]

is contained withinthe safety set

[ωlow, ωhigh

]

M(vw, λ�, 0) =⎧⎨

⎩v

highw − vw, vw > R

λ�

(ωhigh−ωlow

)

2

vw − v loww , vw ≤ R

λ�

(ωhigh−ωlow

)

2

(18)

where v loww and v

highw are defined as follows:

v loww = Rωlow

λ�(19)

vhighw = Rωhigh

λ�. (20)

For the degenerate case τ = ∞, analogous argumentsapply using the aerodynamic torque curve and generator torqueindicated in (11). The equilibrium set E is again [ω−, ω+],but there are two complications. Fig. 6 shows an examplewhere de-loading has set conversion efficiency to 10% lowerthan optimal. The first complication is that two torque speedintersections exist for each wind speed. Thus, the two extremeaerodynamic torque curves produce two pairs of intersectionsω+,ω′+ and ω−, ω′−. The second complication is that for eachvw ∈ [vw − , vw + ] there is a corresponding generatorcurve. Fig. 6 depicts the case vw = vw.

When operating with de-loading, the constant powerextracted is less than or equal to the maximum available fora range of speeds below vw . For all possible wind deviationsin N () and the choice of vw depicted in Fig. 6, althoughthe wind deviations cause deceleration at ωlow, acceleration isexperienced at ω′−, and deceleration at ωhigh. Thus, S� is notempty for the case shown- it is equal [ω′−, ωhigh].

However, for a sufficiently large drop in wind speed,an intersection does not exist between Tgen(ω, vw) andTaero(ω, vw − ). When vw = vw + , the drop requiredis the smallest. In this case, the intersections at ω− and ω′−exist only deviations smaller than

(vw) � vw

(1 − 3

√C p(λ�)

C p(λopt)

)

(1 + 3

√C p(λ�)

C p(λopt)

) (21)

which is the deviation for which ω′− = ω−.It is also necessary to analyze the deviations in wind speed

for which ω− = ωlow or for which ω+ = ωhigh. Referring toFig. 6, the rotor speed ω+ is the intersection point between the


0.90.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

+ --ω (p.u.)

ω− ω+ω−ω+

vw = vw + Δ

vw = vw − Δ

vw = vw

ωhighωlow

Tor

que

(p.u

.)

Taero(ω, vw)Tgen(ω, vw)

Fig. 6. Determination of S� for τ = ∞ based on intervals of guaranteedacceleration and deceleration (shaded). S = [ωlow, ωhigh], E = [ω−, ω+],and S� = [ω′, ωhigh].

aerodynamic torque curve for vw = vw + and the generatortorque curve. The largest value of ω+ is obtained when vw =vw − . An implicit definition for high (vw), the deviationfor which ω+ = ωhigh in the steady state, can be identified.Substituting into (11) as follows: (1) for λ, ω = ωhigh, vw andvw as just mentioned, and = high (vw), yields

Cp

(Rωhigh

vw+high(vw)

)

(vw−high(vw)vw+high(vw)

)3 = Cp (λ�). (22)

Similarly, a deviation low(vw) is associated with the rotorspeed ω−, which is the intersection point between the aero-dynamic torque curve for vw = vw − and the generatortorque curve. The smallest value of ω− is obtained whenvw = vw + , leading to a second implicit definition

Cp

(Rωlow

vw−low(vw)

)

(vw+low(vw)vw−low(vw)

)3 = Cp (λ�). (23)

For a given mean wind speed, it is the smallest of the threedeviations low(vw),(vw), and high(vw) that determinesthe wind disturbance margin. The wind disturbance marginM for the case τ = ∞ is

M(vw, λ�,∞) =

⎧⎪⎨

⎪⎩

low(vw), vw < v loww

(vw) , v loww < vw < v

highw

high(vw), vhighw ≤ vw.

(24)

The threshold value v loww of mean wind speed where

low(vw) gives over to (vw) is where the deviations becomeequal. This threshold speed can be defined implicitly by settinglow(vw) = (vw) in (23) and substituting v low

w for vw

Cp

(Rωlow

2v loww

(

1 + 3

√C p(λopt)C p(λ�)

))

⎛

⎝1+ 3

√Cp(λopt)

Cp (λ�)

1+ 3√

Cp (λ�)

Cp(λopt)

⎞

⎠

3 = Cp (λ�). (25)

0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

Taero(ω, vw)Tgen(ω, vw)

M (vw, λ , 0) = 1.18

M (vw, λ , ∞) = 0. 06

ω (p. u.)

Tor

que

(p.u

.)

ωhighωlow

Fig. 7. Torque-speed curves used to compute disturbance marginM(vw, λ�, τ ) for systems (8) (τ = 0) and (11) (τ = ∞) at vw = 7 anda de-loading of 5%, λ� = 7.74.

Similarly the threshold value vhighw where high(vw) takes over

from (vw) is also defined implicitly

Cp

(Rωhigh

2vhighw

(1 + 3

√C p(λ�)

C p(λopt)

))

⎛

⎝1+ 3

√Cp (λ�)

Cp(λopt)

1+ 3√

Cp(λopt)Cp (λ�)

⎞

⎠

3 = Cp (λ�). (26)

B. Limitations of Torque-Speed Intersection Analysis

The computations of M available from analysis of thelimiting cases of τ are illustrated for an example in Fig. 7.At the chosen mean wind speed and de-loading, analysis ofthe case τ = 0 suggests that wind deviations in N () for ≤ 1.18 are acceptable (M = 1.18). Larger disturbancescause the rotor speed to exceed ωhigh. Analysis of the caseτ = ∞ suggests that only values of ≤ 0.06 are acceptable(M = 0.06). Larger disturbances allow collapse of the rotorspeed to occur, because the torque-speed intersection no longerexists for some wind speeds in the range [vw − , vw + ].The disturbance margin M quantifies robustness of a controlalgorithm to wind disturbances. For the degenerate cases ofτ = 0 or τ = ∞ shown in Fig. 7, torque-speed analysis issufficient to exactly compute the disturbance margin.

The cases shown in Fig. 7 are of limited help in assessingthe case of 0 < τ < ∞. Fig. 8 shows that the steady stateresponse of (7) with the same vw and λ� but with τ = 8 s(dashed) is stable for a periodic disturbance with amplitudesup to 0.6. Since the torque-speed variation reaches the upperrotor speed limit ωhigh, M (vw, λ�, 8) is surely no greaterthan 0.6, well below M (vw, λ�, 0). However, the marginmost likely far exceeds an amplitude of 0.06 as determinedfrom M (vw, λ�,∞). A computation of Ms dependence on τrequires the analysis technique presented in the next section.

V. PLANAR INVARIANCE KERNEL ANALYSIS

A wind turbine subjected to a generator torque based ona filtered rotor speed has dynamics described by two statevariables, as introduced in (7). This case involves a 2-D vector


0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

vw + 0.6

vw − 0.6

ωhighωlow

Tor

que

(p.u

.)

ω (p.u.)

Fig. 8. Response of (7) with τ = 8 s (dashed) compared with responseof (8) (thick solid) to periodic wind deviation around a mean wind speedvw = 7 m/s with an amplitude of 0.6 m/s and a period of 40 s, and thesame mean wind speed and de-loading as in Fig. 7. Thin solid lines showaerodynamic torque-speed curves for vw ± 0.6.

field in the (ω, ωfilt) state plane. It will be helpful to discussthis vector field in abstract form

x = f (x) + g(x) h(x, u) (27)

where x = [x1, x2] with x1 = ω, x2 = ωfilt, u = vw , and

f (x) =[

− k1 x23

x1x1−x2

k2

]

(28)

g(x) =[ k3

x10

](29)

h(x, u) = Cp

(k4x1

u

)u3 (30)

with constants k1 = kload(λ�)/J , k2 = τ , k3 = 1/2ρπ R2/J ,k4 = R, and function Cp(λ) as shown in Fig. 3. This choiceof g(x) and h(x, u) highlights the slight difference between(27) and a system affine in the control u.

A. Concepts and Definitions

If the input u is bounded [as per (14)] then at each point inthe state plane, there is a cone of possible directions for thevector field (27). This is depicted in Fig. 9. Since f (x) andg(x) are fixed for a given x , the spread of the cone depends onthe known vector g(x) scaled by h(x, u), which varies betweenhmin and hmax. For the wind turbine system, u corresponds tothe wind speed. For the Cp curve and domain of wind androtor speeds studied in this paper, h(x, u) has a monotonicdependence on u. This fact is established in the Appendix,where a condition to check monotonicity for a wind turbinemodel is derived from the function h(x, u) and numericallychecked. Thus, extreme values hmin and hmax and thereforealso the edges of the cone are obtained for the extreme valuesof the input u.

Instead of working with the functions f (x) and g(x),it is more convenient to work with the vectors that arethe edges of the cone in Fig. 9. These edge vectors aregiven by

f1(x) = f (x) + g(x) hmin(x) (31)

f2(x) = f (x) + g(x) hmax(x). (32)

xa

f1f2

f

g · hming · hmax

Fig. 9. Cone of possible variations due to bounded wind variation at a pointxa . Representation using original vectors f and g (dashed) is equivalent toconvex combination of vectors f1 and f2 (solid).

The right-hand side of (27) can be written as a convexcombination of f1(x) and f2(x)

x = α f1(x) + (1 − α) f2(x) (33)

with α ∈ [0, 1]. Varying α ∈ [0, 1] in (33) corresponds tovarying u ∈ [vw − , vw + ] in (27).

We are in search of the invariance kernel, S�, shown inFig. 4. It has been shown [22] that the invariance kernel is aclosed set with a boundary ∂S� that is the union of parts ofthe boundary ∂S of the safety set S, and of trajectories of thefields f1 and f2. On ∂S�, we require both edges of the coneto either point to the interior of S� or to be tangent to ∂S�.Since S� is the largest possible positively invariant subset of Sat least one of the edge fields will be tangent to the boundary∂S� for those pieces not coinciding with ∂S [22].

We proceed with identification of S� by first discerningbetween the left and the right edges of the cone. Fig. 10illustrates the need for further definition. At one point xa ,the cone is found to the left of the field f1(x). But fora nearby point xb, the two fields could be parallel, andat another point xc, the cone might now be found to theright of f1(x). The plane can be partitioned into three setsbased on the possible angle relation between f1(x) and f2(x).Points such as xa where f2(x) points to the left-hand sideof f1(x) (det[ f1, f2] > 0) comprise a set R+. Points suchas xc where f2(x) points instead to the right-hand side of f1(det[ f1, f2] < 0) comprise a set R−. Points where the vectorsf1(x) and f2(x) are parallel (xb) or anti-parallel (xd ) comprisethe collinearity set.

Definition 7: We define the collinearity set L and its subsetsusing the determinant and dot product as follows:

L = {x ∈ R2 : det[ f1(x) f2(x)] = 0}

with subsets L+ = {x ∈ L : 〈 f1(x), f2(x)〉 > 0}, L− = {x ∈L : 〈 f1(x), f2(x)〉 < 0} and the sets R+ and R− similarly

R+ = {x ∈ R2 : det[ f1(x) f2(x)] > 0}

R− = {x ∈ R2 : det[ f1(x) f2(x)] < 0}.

By assessing whether a point is in R+ or R−, the fieldvector giving a chosen edge of the cone can always beidentified. The following definition for a new vector field thenbecomes possible.

Definition 8: The extremal vector fields fR(x) and fL (x)are defined as

fL(x) ={

f1(x), x ∈ R+

f2(x), x ∈ R− , fR(x) ={

f2(x), x ∈ R+

f1(x), x ∈ R−.


R+

R−

L+

L+

L−f1

f1

f1

f1

f2

f2

f2

f2

xa

xb

xc

xd

Fig. 10. Example of extremal trajectory of fL (x) (thin solid) and relationbetween f1 and f2 on the sets R+, R−, and L (dash-dot line).

The subscript L and R indicates to which side of the vectorthe rest of the cone lies.

Solutions of the extremal fields exist everywhere on theplane, and are unique almost everywhere.1 The trajectories ofextremal solutions on the plane are called extremal arcs, withtrajectories of fL(x) being called L-arcs and trajectories offR(x) R-arcs.

The boundary of the invariance kernel S� is composedof concatenations of extremal arcs that pass through specialpoints. Theory has been advanced to prove this assertion,which can be rigorously justified given some generic assump-tions on S and the vector fields [22]. Further, it has beenshown that these special points and arcs can be computed viaan algorithm having a finite number of steps [21], [22]. Inorder to apply the algorithm, three additional definitions arenecessary.

Definition 9: A connected subset of ∂S along which bothf1(x) and f2(x) point inside of S or are tangent to ∂S is saidto be an invariant arc of ∂ S. Each endpoint of an invariantarc of ∂S is called a t∂ point. An invariant arc contains theseendpoints and is thus closed.

This definition specifies the character of any endpoint t∂

of an invariant arc. An endpoint is a point in the curve ∂Sthat is the boundary between two connected subsets of thecurve. In one subset, both f1(x) and f2(x) point inside of S,while in the other at least one of the fields points out. If ∂S isdifferentiable (i.e., is a class C1 curve) in a neighborhood ofa t∂ point, then at least one of the vector fields f1(x), f2(x)must be there tangent to ∂S.

The orientation of the extremal arcs is already given by thetime parametrization of the corresponding extremal solutions,so that the orientation indicates the direction of increasingtime. We give ∂S a positive orientation so that a point movingalong ∂S finds the interior of S to its left-hand side.

1Two solutions may converge or diverge from a point x0 ∈ L−, wheref1(x) and f2(x) are anti-parallel. Also the extremal vector fields fL , f Rare discontinuous on L. However, the existence and uniqueness of extremalsolutions has been discussed in detail in [22], and is supported by the body ofwork on differential equations with discontinuous right-hand side pioneeredby Filippov [23].

0.6 0.7 0.8

−1.5

−1

−0.5

0

0.5

1

1.5

ω (p.u.)

ωfil

t(p

.u.)

t∂

t∂

t∂

t∂

0.9

Fig. 11. Identification of special points (open circles) and closed extremalarc (thin solid). Invariant arcs of ∂S (thick solid) are bounded by t∂ pointswithin ∂S (dashed).

At an equilibrium point x of f1(x) or f2(x), we define twospecial types of extremal arcs. Recall that by Definition 8, anextremal arc will coincide with the trajectory on the plane ofa solution of either f1(x) or f2(x).

Definition 10: Suppose that x is an equilibrium of f1(x)[respectively, f2(x)]. An extremal arc through x is said to bean equilibrium extremal arc through x if on a neighborhoodof x it coincides with a trajectory of f1(x) (respectively, atrajectory of f2(x)). If, instead, the extremal arc coincideswith a trajectory of f2(x) (respectively, a trajectory of f1(x))in a neighborhood of x , then it is said to be a nonequilibriumextremal arc through x .

Definition 11: A point p in L− is called a t− point if thetrajectories of f1(x) and f2(x) through p remain in the closureof R+ or the closure of R− for some time interval containingt = 0 (i.e., 〈 f1, f2〉 has constant sign along the trajectories off1(x) and f2(x) through p for small time).

B. Computing S� for Wind Turbine Providing Smoothing

Based on the definitions of the previous section, an algo-rithm has been devised to compute the invariance kernel in afinite number of steps. The algorithm is stated in the Appendixand has been rigorously justified in an earlier publication [22].It is assumed that any closed extremal arcs are known. Thevalidity of the algorithm rests on eight generic assumptions onS and the vector fields, all of which are met by the simplifiedwind turbine model.

The algorithm is applied in this section to find the invariancekernel S� of the safety set S as defined in (16), under thedynamics of the wind turbine model (7) with parameters givenin [11]. For this example, vw = 6.4 m/s, λ = 7.43, = 0.700,and τ = 4 s.

1) Initialization (Fig. 11): For the operating point and filterparameter chosen for this example, t∂ points are the onlyspecial points found; the equilibria are foci, and there areno t− points. There is one closed extremal arc of fL(x).Fig. 11 shows the objects identified in the initializationstep of the algorithm.


0.6 0.7 0.8 0.9

−1.5

−1

−0.5

0

0.5

1

1.5

ω (p.u.)

ωfil

t(p

.u.) L−

Fig. 12. Integration from special points (open circles) until stoppingconditions reached (filled circles).

1 2

4

3

5

8

7

9

6γ1γ2

γ3

γ4

γ5

γ6γ7

Fig. 13. Point numbering, arc partitioning, and pruning (axes omitted forclarity). Arcs γ1, . . . , γ7 will be pruned.

1

5L

5R

Fig. 14. Vertices and edges of graph G constructed after pruning. One cycleexists (gray vertices).

2) Integration (Fig. 12): Of the four t∂ points, only tworequire integration. Both of them are at the tail of aninvariant arc (see Fig. 11), with fL(t∂ ) tangent to theboundary. This corresponds to the first entry of Table I.Therefore, the L-arc through each point is integrated inreverse time (solid arcs) and the R-arc is integrated inforward time. As shown in Fig. 12, the L-arcs (thin solid)hit invariant arcs of ∂S (thick solid), while the R-arcs(thin dashed) hit L−.

3) Pruning (Fig. 13): Fig. 13 shows points and ori-ented arcs resulting from carrying out the labeling and

0.5 0.6 0.7 0.8 0.9

0

0.2

0.4

0.6

0.8

ω (p.u.)

ωfil

t(p

.u.)

Fig. 15. Invariance kernel S� (shaded region) identified by the algorithmcontains the closed extremal arc, which in turn contains the equilibrium set E(solid black line). Cone of possible directions is depicted at multiple points(arrows).

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

ω(p.u.)

ωfil

t(p

.u.)

E(0.807)Δ = 0.807

Δ = 0.800

Δ = 0.750

Δ = 0.700

Fig. 16. Computation of M for a single value of τ by incrementing andfinding invariance kernels (shaded). vw = 6.4 m/s, λ� = 7.43, τ = 4 s.

partitioning of Step 3. There are no arcs sharing thesame two endpoints, so there is no action for Step 3.2.Pruning proceeds in several executions as follows:

EXEC. 1: STEP 3.3: γ1 , γ2 STEP 3.4: γ3 , γ4EXEC. 2: STEP 3.3: γ5 STEP 3.4: γ6EXEC. 3: STEP 3.3: γ7 STEP 3.4: NO ACTION

No action is required for Step 3.6, and in Step 3.7 thepoints 2, 3, 4, 6, 7, 8, and 9 are removed.

4) Graph Construction (Fig. 14): Fig. 14 shows the resultsof applying the rules of Step 4. A single vertex has beencreated for t∂ point 1 (by 4.1), while two are createdfor the integration endpoint 5 (by 4.2). The only arcsremaining after pruning are either L-arcs or invariantarcs, so Step 4.3 is applied. Graph edges are added fromv1 to vL

5 and from vL5 to v1, forming the only cycle in

the graph. Step 4.5 is applied to the significant point 5so that an edge is added from v R

5 to vL5 .

5) Cycle Analysis (Fig. 15): From the graph, there isone cycle that corresponds to a closed curve in theplane composed of an invariant arc of ∂S and anL-arc, which we will refer to as a concatenation. Fromthe initialization step, there was also a closed extremalarc. Both of these closed curves are plotted in Fig. 15.


100

101

102

1030

0.5

1

1.5

M

τ

M (vw λ , 0)

M (vw,λ ,∞)10

010

110

210

30

0.5

1

1.5

M

τ

M (vw ,λ , 0)

M (vw,λ , ∞)

(a) (b)

Fig. 17. M(vw, λ�, τ ) over a range of τ ∈ [0,∞] (solid with dots), compared with special cases obtained from analysis of torque-speed intersections (dashed)for degenerate cases τ = 0 [using (18)] and τ = ∞ [using (24)]. (a) Optimal (λ� = 6.91), vw = 7.5 m/s. (b) 20% de-rating (λ� = 8.62), vw = 6 m/s.

TABLE I

RULES OF INTEGRATION THROUGH SPECIAL POINTS

Initial extremal integrationcondition arc direction

fL is L rev.t∂ point, tangent R fwd.

tail of inv. arc fR is do nothingtangent

fL is do nothingt∂ point, tangent

head of inv. arc fR is L fwd.tangent R rev.

L fwd.t− point L rev.

R fwd.R rev.

non-eq fwd.node rev.

stable or (unstable) eq., fast rev. (fwd.)manifold rev. (fwd.)

non-eq fwd.rev.

saddle eq., stable rev.manifold rev.

eq., unstable fwd.manifold fwd.

The union of the regions enclosed by these two closedcurves is the invariance kernel, shaded in Fig. 15. As averification, Fig. 15 also depicts the cone of directions atseveral points along ∂S and ∂S�, showing that, indeed,S� is positively invariant while S is not.

VI. WIND DISTURBANCE MARGIN COMPUTATION:GENERAL CASE

To practically determine M, one can set the disturbancebound to a small value and then increase it incrementally,finding the planar invariance kernel S� at each step. As statedin Definition 5, the largest value of for which S� is both notempty and also contains E is the wind disturbance margin M.

A. Computation for Single Value of τ

Fig. 16 demonstrates the incremental process of computingM for a single value of τ . The invariance kernel S� is

plotted for increasing values of ∈ [0.700, 0.807]. The closedextremal curve produced in Step 1 of the algorithm expands,while the concatenation produced by Steps 2–4 contracts. Theconcatenation and the closed extremal curve coincide for thevalue of = 0.807 m/s. Beyond this critical value, S� isempty. The wind disturbance margin M(vw,) is thereforeequal 0.807 m/s.

The important arc of the concatenation is the trajectorypassing through points 1, 9, and 5 in Fig. 13. That trajectoryintersects itself when = M. This observation could leadto a more direct method of computing M based on analyzingthe sensitivity of the trajectory in question. However, differentcritical trajectories will determine M for other parametervalues of vw, λ�, and τ . The incremental process is a flexibleapproach that can be generally applied.

B. Results for Range of τ

The disturbance margin as determined through the incre-mental process over a range of τ is plotted in Fig. 17 forλ� = λopt and in Fig. 17 for λ� = 8.62 (a de-loading of 20%).The margins for the extreme cases τ = 0 and τ = ∞ havebeen computed from (18) and (24) and plotted as horizontallines for comparison.

For small values of τ , the margin approaches the valuecalculated for the limiting case of τ = 0 for both choicesof de-loading. For large values of τ , the margin approacheszero for the selection λ� = λopt, and approaches the quan-tity (21) for the de-loaded selection. The analysis of planarinvariance kernels is required to describe anything other thanthe asymptotic behavior of the disturbance margin. Withoutsuch analysis, one could not be sure that smoothing operationwould be stable, particularly in the case of λ = λ�. The abilityto compute the margin makes it possible to find the largestallowable τ for a given range of expected wind disturbance,or vice-versa.

VII. CONCLUSION

This paper has introduced an efficient method of rigorouslystudying the behavior of wind turbines that implement powersmoothing by exploiting the kinetic energy of the rotor. Itwas argued that traditional approaches to analyzing machinedynamics are not adequate for studying the problem. It wasshown that by performing a number of integrations of the


dynamical equations of a simplified model from certain specialpoints, it is possible to determine whether a given controllaw will produce acceptably bounded behavior in responseto a class of bounded wind variations. By applying thisnew method to determine the largest safe bound on suchwind variations, the concept of a disturbance margin for thecontrolled wind turbine has been quantified.

The disturbance margin enables a systematic design of τ toobtain maximal smoothing within available freedom. In thispaper, it has been used to prove that smoothing operation canbe stable without de-loading. These two results could not beobtained using the traditional technique of speed-torque inter-section analysis. Using the new techniques and disturbancemargin concept introduced in this paper, a model of the windturbines’ maximal smoothing capability over its operatingrange could be created. Such models could be combined withinformation about expected turbulence intensity to evaluate theavailability of the smoothing capability over a chosen timeperiod. A practical evaluation would likely not be based onthe absolute range of the signal, but on a chosen fraction ofits probability mass, so as to indicate highly probable ratherthan guaranteed smoothing capability.

The benefit of the disturbance margin introduced in thispaper is a quantification of the robustness of a given controllaw. The method presented here for finding the invariancekernel allows computation of the disturbance margin for anysystem that can be formulated as a planar nonlinear affinecontrol system. This can now be used to compare differentcontrol laws or different wind turbine technologies on acommon basis of equal robustness, and in future to informon-line adjustments of control parameters in response to windconditions to safely maximize exploitation of wind turbinekinetic energy.

APPENDIX

A. Sufficient Conditions on h(x, u)

The theory presented and applied in this paper is provenin [22], using the properties of the linearization at equilibriaof f1(x) and f2(x). For the linearization to exist, the vectorfields (31) and (32) must be differentiable in x at equilibria.Ensuring this requires analysis of the function h(x, u), whosemaximum and minimum values hmin and hmax are obtained agiven point x with the inputs

ξmin(x) = arg minu∈[umin,umax]

h(x, u) (34)

ξmax(x) = arg maxu∈[umin,umax]

h(x, u). (35)

By the chain rule, we also require differentiability at equilibriafrom the minimizing and maximizing functions ξmin(x) andξmax(x). For (7), these requirements are sufficiently satisfiedby the strong property that h(x, u) is monotonically dependenton u in the domain of interest. For, where ever

∂

∂uh(x, u)

∣∣∣∣x,u

≥ 0 (36)

then functions ξmin(x) and ξmax(x) are constant and equal tothe minimum and maximum values of the input u

ξmin(x) = umin (37)

ξmax(x) = umax. (38)

Considering the function h(x, u) for (7), as given by (30)

∂

∂uCp

(k4x1

u

)u3

∣∣∣∣x�

1,u�

=[−dCp

dλ

k4x�1

u�2 u�3+3Cp

(k4x�

1

u�

)u�2

]

(39)

the substitution

λ� = k4 x�1

u�(40)

allows a simplification showing the derivative (39) is equal tozero when

λ� = 3Cp(λ�)

dC pdλ

∣∣∣λ�

(41)

which for Cp(λ) can happen at two values of λ. For theaerodynamic model [11] studied here, (36) holds for all λ� ∈[4, 16.9]. The domain of wind speeds vw ∈ [4, 11] and rotorspeeds ω ∈ [0.92, 1.5] studied here imply, by the tip-speedratio (1), a maximum possible range of λ� ∈ [3.5, 15.8], andthus the condition (41) is never violated.

B. Algorithm to Find Planar Invariance Kernel S�

1. InitializationDetermine:

1.1. t∂ points in S;1.2. t− points in S;1.3. nodes and saddles of f1 or f2 in S;1.4. closed extremal arcs in S.

2. IntegrationUsing the integration rules in Table I, generate extremalarcs from all points computed in Part 1.The stopping criteria for the integration are:

2.1. The solution hits L− at a point which is not a t−point.

2.2. The solution hits ∂S at a point which does not lieon an invariant arc of ∂S.

2.3. The solution hits an invariant arc of ∂S comingfrom int S.

2.4. The solution is detected to reach (in finite orinfinite time) an equilibrium of f1 or f2 or to spiral(in positive or negative time) around a limit set.

3. PruningLabel all points identified in Part 1 (steps 1.1–1.4) asspecial points. Label as significant all special points, allthe integration endpoints, and all points of intersectionbetween extremal arcs generated in Part 2 or betweenextremal and invariant arcs of ∂S. Thus, special pointsare significant, but not vice versa.

3.1. Partition each extremal arc resulting from an inte-gration performed in Part 2 and invariant arcs of∂S into sub-arcs whose heads and tails are the


significant points. The sub-arcs inherit the orienta-tion of the parent arc. In the rest of the algorithmbelow, these sub-arcs will be simply referred to asextremal arcs.

3.2. Prune one L-arc γ and one R-arc η if γ and ηhave the same endpoints, and if neither endpointis special.

3.3. Prune any L-arc (respectively, R-arc) with head ata point p which is not special if there is no L-arc(respectively, R-arc) with tail at p.

3.4. Prune any extremal arc whose head or tail is at apoint where no other arc is connected.

3.5. Repeat Steps 3.3–3.4 until there is not more arc toprune.

3.6. Prune extremal arcs that spiral around limit sets inpositive or negative time.

3.7. Eliminate from the list of significant points allpoints with no arcs attached, and points connectingonly two arcs of the same type (L or R).

4. Graph ConstructionConstruct a graph G = (V , E), with V the set of verticesof G and E the set of edges of G as follows.Vertices of G. Let P denote the set of significant pointsin S that remain after the pruning in Part 3.

4.1. For every point p ∈ P which is special, create avertex v p .

4.2. For every p ∈ P which is not special, create twovertices, denoted vL

p and v Rp .

Edges of G. Create directed edges between verticesassociated with extremal arcs and invariant arcs of ∂Sas follows.

4.3. If p is the tail of an L-arc or an invariant arc of∂K with head at q , create a directed edge from v p ,or vL

p , to vq , or vLq .

4.4. If p is the tail of an R-arc with head at q , createa directed edge from vq , or v R

q , to v p , or v Rp .

4.5. For every (vLp , v R

p ) pair, create a directed edge fromv R

p to vLp .

5. Cycle Analysis

5.1. Find all simple cycles (i.e., closed paths thatdo not visit any vertex more than once) in thegraph G.

5.2. Discard any cycles containing two vertices v Rp , vL

pthat are not consecutive (when travelling in thedirection of the edges of the graph).

5.3. For each remaining cycle in G, check whether theregion in the plane delimited by the path associatedto the cycle is positively or negatively invariant. Ifit is negatively invariant, discard the cycle.

5.4. S� is the union of all regions enclosed by closedpaths associated to graph cycles and by closedextremal trajectories in S.

Remark 1: The test in Step 5.3 can be done simply by pick-ing any nonspecial point p in the closed path and discardingthe cycle if f1(p) points outside the region delimited by thepath.

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Barry G. Rawn (M’10) received the B.A.Sc. degreein engineering science, and the M.A.Sc. and Ph.D.degrees in electrical engineering, from the Universityof Toronto, Toronto, ON, Canada in 2010.

He is currently a Post-Doctoral Researcher withthe Electrical Power Systems Group, Delft Uni-versity of Technology, Delft, The Netherlands. Hiscurrent research interests include nonlinear dynamicsand sustainable energy infrastructure.

Peter W. Lehn (SM’05) received the B.Sc. andM.Sc. degrees in electrical engineering from theUniversity of Manitoba, Winnipeg, MB, Canada,in 1990 and 1992, respectively, and the Ph.D.degree from the University of Toronto, Toronto, ON,Canada, in 1999.

He was with the Network Planning Group,Siemens AG, Erlangen, Germany, from 1992 to1994. He is currently a Professor with the Universityof Toronto. His current research interests includemodeling and control of converters, and integration

of renewable energy sources into the power grid.

Manfredi Maggiore (M’99) was born in Genoa,Italy. He received the Laurea degree in electronicengineering from the University of Genoa, Genoa,Italy, in 1996, and the Ph.D. degree in electricalengineering from Ohio State University, Columbus,in 2000.

He has been with the Edward S. Rogers Sr.Department of Electrical and Computer Engineering,University of Toronto, Toronto, ON, Canada, since2000, where he is currently an Associate Professor.He was a Visiting Professor with the University of

Bologna, Bologna, Italy, from 2007 to 2008. His current research interestsinclude mathematical nonlinear control, particularly methods from dynamicalsystems theory and differential geometry.

Documents

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, … · B. G. Rawn is with the Department of Electrical Sustainable Energy, Delft University of Technology, Delft 2628 CD, The Netherlands