IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN POWER ELECTRONICS, VOL. 1, NO. 4, DECEMBER 2013 247
Passivity-Based Control of a Grid-ConnectedSmall-Scale Windmill With Limited
Control AuthorityRafael Cisneros, Fernando Mancilla–David, Member, IEEE, and Romeo Ortega, Fellow, IEEE
Abstract— The problem of controlling small-scale wind tur-bines providing energy to the grid is addressed in this paper.The overall system consists of a wind turbine plus a permanentmagnet synchronous generator connected to a single-phase acgrid through a passive rectifier, a boost converter, and an inverter.The control problem is challenging for two reasons. First, thedynamics of the plant are described by a highly coupled setof nonlinear differential equations. Since the range of operatingpoints of the system is very wide, classical linear controllersmay yield below par performance. Second, due to the use ofa simple generator and power electronic interface, the controlauthority is quite restricted. In this paper we present a highperformance, nonlinear, passivity-based controller that ensuresasymptotic convergence to the maximum power extraction pointtogether with regulation of the dc link voltage and grid powerfactor to their desired values. The performance of the proposedcontroller is compared via computer simulations against theindustry standard partial linearizing and decoupling plus PIcontrollers.
Index Terms— Nonlinear control, passivity-based control,power control, renewable energy systems, wind speed, windmills.
I. INTRODUCTION
W IND power is becoming increasingly popular aroundthe world, with most of the effort focused on the
development of utility scale wind power. In recent years,small-scale wind turbines (1–100 kW) have been receivingattention as serious contributors for powering homes, farmsand small businesses as well as energy providers for thepower grid, which is the scenario considered in this paper.The American Wind Energy Association reported 198 MW(151 300 turbines) of installed capacity of small-scale windpower in the United States at the end of 2011 [1]. The reportalso states 91% of the sales in 2011 correspond to grid-connected units. Further efforts to increase the penetrationof distributed wind power are being made through renewableportfolio standards and market-based schemes [2].
Manuscript received June 17, 2013; revised October 1, 2013; acceptedOctober 4, 2013. Date of publication October 10, 2013; date of currentversion October 29, 2013. The work of F. Mancilla-David was supportedby the French National Program DIGITEO. Recommended for publication byAssociate Editor Wenzhong Gao.
R. Cisneros and R. Ortega are with the Laboratoire de Signaux et Sys-tèmes, Supélec, Plateau de Moulon, Gif-sur-Yvette 91192, France (e-mail:[email protected]; [email protected]).
F. Mancilla-David is with the Department of Electrical Engineering,University of Colorado Denver, CO 80217 USA (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JESTPE.2013.2285376
Because of their higher efficiency and power density, aclass of small-scale wind turbines very often use permanentmagnet synchronous generators (PMSGs) built either as radialor axial flux machines [3]–[7]. Moreover, because of costreasons, the connection to the grid is achieved via a simplepower electronic interface. In this paper, we consider a sys-tem consisting of a small-scale wind turbine plus a PMSGconnected to a single-phase ac grid through a passive rectifier,a dc–dc boost converter and a single-phase H-bridge inverter[8]–[10]. Because of this architecture, the controller designof the overall system becomes complicated for two reasons.On one hand, the generator dynamics cannot be neglected—asusually done for large wind turbines [11], [12]—leading to asystem behavior described by highly coupled set of nonlineardifferential equations. On the other hand, due to the use ofa simple generator and power electronic interface, the controlauthority is quite restricted.1 In this paper, we present a highperformance, nonlinear, passivity-based controller that ensuresasymptotic convergence to the maximum power extractionpoint together with regulation, to their desired values, of thedc link voltage and grid power factor.
Some MMPT strategies make use of algorithms based on theso-called hill-climbing search procedures. Many publications,including some variants of this approach, can be found in theliterature. See, for instance, [13]–[15] and references therein.The most widely used method, the perturb observe algorithm,presents some undesirable drawbacks. These include oscilla-tions around the maximum power point [16] and the failureto track fast-changing wind [17]. There are also papers, like[17] and [18], which consider the linearization of the systemdynamics. In [19] and [20], the problem is tackled using theextremum seeking control technique. Some fuzzy logic-basedschemes have been developed, for example, [21] and [22]. Inthis paper, a passivity-based strategy is derived to address theproblem. It is worth mentioning that, in contrast with the pre-vious approaches, our work is model-based, with all (relevant)nonlinearities of the dynamic system considered in the model.
In [23], we studied the problem of maximum powerextraction of a similar system but connected to a battery.Our focus on that paper was the development of a windspeed estimator [24], which is needed to determine the optimalgenerator speed, and was assumed to be unknown. Althoughthe wind speed estimator can be also used in this paper, our
1More precisely, the number of control signals (three) is smaller than theorder of the system (six) and equals the number of signals to be regulated.
248 IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN POWER ELECTRONICS, VOL. 1, NO. 4, DECEMBER 2013
Fig. 1. Circuit schematic of a grid-connected windmill system.
attention is concentrated on the replacement of the battery by agrid-connected single-phase inverter, which makes the controlproblem significantly more complicated. Indeed, besides theneed to include in the design model the dynamics of theinverter, in addition to regulation of the generator speed,there are two more control objectives to be satisfied, namely,regulation of the dc link voltage and power factor of the currentinjected into the ac grid.
To design the controller, the overall system is decomposedas a cascade connection of two subsystems. One consistingof the windmill and PMSG and the other containing theboost converter and the inverter. The PMSG rotor speed ofthe first subsystem is regulated around the maximum powerextraction point with a standard passivity-based controller(SPBC)—which is a nonlinear, dynamic, state feedback thatshapes the energy of the subsystem and adds damping andan integral action [25]. The dc link voltage and the injectionof reactive power to the grid is controlled in the secondsubsystem, via a tracking PI PBC whose reference is suitablytailored to compensate for the coupling term coming fromthe first subsystem. The development of this tracking PI is anontrivial extension of the regulation PI schemes (for systemswhose incremental model is passive) reported in [26] and [27].The main result of this paper is the proof that the overallcontrolled system has an asymptotically stable equilibriumpoint at the desired operating regime. Simulation results areshown comparing the controller’s performance with that of thetypical PI control utilized in power engineering.
This paper is organized as follows. Section II presents themathematical model considered in this paper, followed by thecontrol problem formulation in Section III. The aforemen-tioned system decomposition is given in Section IV. SectionV develops the SPBC of the first subsystem and Section VIcontains the tracking PI of the second subsystem. The mainresult of this paper is given in Section VII, followed by adiscussion of the control scheme implementation in SectionVIII. A benchmark system used for comparison is describedin Section IX, followed by computer simulations in Section X.Finally, the concluding remarks of Section XI close this paper.
Notation: Given a vector x = col(x1, . . . , xn) and twointegers 1 ≤ i < j ≤ n, we denote xi j = col(xi , xi+1, . . . , x j ).In addition, x ∈ R
n+ denotes xi > 0, i = 1, . . . , n.
II. MATHEMATICAL MODEL OF THE SYSTEM
In this section, we describe the components of the system,whose schematic diagram is shown in Fig. 1. The systemconsists of a wind turbine with a PMSG, a passive diode
bridge rectifier, a boost converter, a dc link, and an inverterconnected to the grid through a simple L filter. Althoughthe passive rectifier injects current harmonics into the PMSG,this topology is preferred due to its low cost and simplicityof implementation. On the other hand, it is clear that itsignificantly reduces the available control authority.
In [23], we considered a similar system, replacing the dclink by a battery and removing the inverter and the grid. Thereader is referred to [23] for additional details on the modelingof the first three elements that, in the interest of brevity, areonly briefly summarized in the following.
A. Dynamics of the PMSG
The electrical equations, which describe the behavior of thesurface-mounted PMSG in the rotor (dq) reference frame aregiven by
L�i d = −Rid + Liqωe − vd
L�i q = −Riq − Lidωe + φωe − vq
(1)
where id , iq , vd , and vq , are the currents and voltages in thed − q reference frame, L and R are the stator winding’sinductance and resistance, ωe is the electric frequency thatis related to the mechanical speed via
ωe = P
2ωm
φ is the permanent magnetic flux produced by the rotor mag-nets, and P is the number of poles. The magnetic flux φ is aconstant that depends on the material used for the realization ofthe magnets. A detailed derivation of this standard model maybe found in [28]. An important observation is that, in normaloperating conditions, i2
d + i2q > 0. See Remark 3.2 below.
B. Mechanical and Wind Turbine Dynamics
The mechanical dynamics is described by
J ωm = Tm − Te (2)
where J is the rotor inertia, ωm is the shaft’s rotational speed,and Te is the electrical torque defined as
Te = 3
2
P
2φiq
and Tm is the mechanical torque applied to the windmill shaft,that is given by
Tm = 1
2ρ Ar
Cp(λ)
λv2w
CISNEROS et al.: PASSIVITY-BASED CONTROL OF A GRID-CONNECTED SMALL-SCALE WINDMILL 249
Fig. 2. Power coefficient for a typical small-scale windmill.
where Cp(λ) is the power coefficient,2 λ, which is defined as
λ := rωm
vw(3)
is the blades’ tip speed, r the blades’ radius, ρ the air density,A is the area swept by the blades, and vw the wind speed,which is assumed constant and known.
The shape of the function Cp(λ) depends on the geometryof the windmill. Fig. 2 shows a typical curve that can beobtained from experimental measurements. In this paper, weare interested in operating the system at the point of maximumpower extraction. Namely
λ� := arg max Cp(λ)
which is assumed to be known. It is important to note that, ifvw is known the control task boils down, in view of (3), toregulation of the shaft’s speed ωm around the reference speed
ω�m = λ�vw
r.
C. Power Electronic Interface and Grid
As shown in Fig. 1, the PMSG is linked to the grid througha passive rectifier a dc–dc boost converter, a dc link, and aninverter. As discussed in [23], in this configuration, the PMSGvoltages may be expressed as
vd = id√i2d +i2
q
Mvdc D
vq = iq√i2d +i2
q
Mvdc D(4)
where D is the duty ratio of the dc–dc boost converter, M =(π/3
√3) is the gain of the passive diode rectifier, and vdc is
the voltage of the dc link.The well-known average model of the dc link and inverter
(in dq coordinates) is given by
C vdc = 3M
2
(i2d + i2
q
)u1 − 1
2Id u2 − 1
2Iq u3
Lg Id = −Rg Id + LgωIq − Vd + vdcu2
Lg Iq = −LgωId − Rg Iq + vdcu3 (5)
where C is the capacitance of the dc link, Id , Iq are the dand q components of the grid currents, respectively, Vd is
2The power coefficient is also a function of the blade pitch angle, which actsas an additional control input. In the paper we are interested in the operationregime where this angle is kept constant, consequently we have omitted thisadditional argument in the function C p .
the amplitude of the grid’s voltage, Lg, Rg are the inverter’sinductance and resistance. We defined in (5)
u1 := D√i2d + i2
q
and u2 and u3 are the d and q components, respectively, ofthe inverter modulating signal. Finally, ω is the (constant)frequency of the ac grid voltage. It is well known that theinverter is operational only if the voltage of the dc link iskept above a minimal (positive) value. See Remark 3.2 in thefollowing.
D. Overall Dynamic Model
Collecting (1), (2), (4), and (5), defining the state vector
x := col(id , iq , rωm , vdc, Id , Iq )
the constants
C2 := ρ A
3, L1 := P L
2r, φ1 := Pφ
2r, J1 := 2J
3r2 (6)
and the function
(x3) := C2v3w
x3Cp
(x3
vw
)(7)
the system state equations may be written as
Lx1 = −Rx1 + L1x2x3 − Mx1x4u1
Lx2 = −Rx2 − L1x1x3 + φ1x3 − Mx2x4u1
J1x3 = −φ1x2 + (x3)
Cx4 = 3M
2
(x2
1 + x22
)u1 − 1
2x5u2 − 1
2x6u3
Lgx5 = −Rgx5 + Lgωx6 − Vd + x4u2
Lgx6 = −Lgωx5 − Rg x6 + x4u3 (8)
where u := col(u1, u2, u3) is the control input vector.Remark 2.1: The total energy of the system (8) is given by
H (x) = 1
4x�diag{3L, 3L, 3J1, 2C, Lg, Lg}x
whose derivative, as expected, verifies the power balanceequation
H = −3R
2
(x2
1 + x22
) − Rg
2
(x2
5 + x26
)︸ ︷︷ ︸
dissipation
+ 3
2(x3)x3
︸ ︷︷ ︸mechanical power
− 1
2x5Vd .
︸ ︷︷ ︸electrical power
As indicated in Section I, the control scheme proposedherein can be rendered adaptive incorporating the wind speedestimator of [24].
III. ASSIGNABLE EQUILIBRIA AND
PROBLEM FORMULATION
The control objective is threefold: 1) to operate the systemof (8) in the point of maximum wind power extraction, whichtranslates into an optimal shaft speed x�
3; 2) to keep the dc linkvoltage at a desired constant value x�
4 > 0; and 3) to injectcurrent in to the grid at a given power factor. It is assumed in
250 IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN POWER ELECTRONICS, VOL. 1, NO. 4, DECEMBER 2013
this paper, the current is injected at unity power factor, that is,x�
6 = 0. These objectives should be achieved independently ofthe wind speed.
Under the assumptions of known λ� and vw , the controltask reduces to a standard problem of stabilization of an(assignable) equilibrium point x� of the system (8) with
x�3 := λ�vw > 0, x�
4 = v�dc > 0, x�
6 = 0. (9)
A. Assignable Equilibria
The proposition below characterizes the set of assignableequilibria compatible with the constraint (9).
Lemma 3.1: Consider the system (8). Fix x�6 =0 and x�
3 >0.Then, for any x�
4 > 0, the set of assignable equilibria is givenby
E :={
x ∈ R6 | 1(x1) = 0, x2 = �
φ1, 2(x1, x5) = 0
}(10)
where
1(x1) := x21 − φ1
L1x1 + 2
�
φ21
(11)
2(x1, x5) := Rg x25 + Vd x5 + 3Rφ1
L1x1 − 3x�
3�
(12)
with � := (x�3).
Proof: Fix x3 = x�3. From the third equation in (8), we
obtain
x�2 = �
φ1. (13)
Then, eliminating u1 from the first and second equations in(8) and using (13), we obtain
L1(x�1)2x�
3 − φ1x�1 x�
3 + L12
�
φ2 x�3 = 0 (14)
which is equivalent to 1(x�1) = 0.
Finally, at the equilibrium, the power balance equation ofRemark 2.1 is equal to zero. Then, substituting x�
2 from (13),the balance equation becomes
0 = Rg(x�5)
2 + Vd x�5 + 3R
[(x�
1)2 + 2�
φ2
]− 3�x�
3 . (15)
In addition, from (14)
(x�1)
2 = φ
L1x�
1 − 2�
φ21
which is substituted in (15) to complete the proof.Remark 3.1: Necessary and sufficient conditions for the
existence of equilibria are
φ21
2L1≥ �
V 2d ≥ 12Rg
(Rφ1
L1x�
1 − x�3�
)(16)
where x�1 is a solution of 1(x�
1) = 0.
Fig. 3. Cascade connection between subsystems (19) and (20).
B. Control Problem Formulation
Given the system (8) and an equilibrium x� ∈ E , verifying(9), find (if possible) a state-feedback controller that ensuresasymptotic stability of the closed-loop system.
Remark 3.2: As explained in Section II, the physical opera-tion of the system is restricted to a subset of R
6. In particular,it is necessary that
x21(t) + x2
2(t) ≥ κ1
x4(t) ≥ κ2 (17)
for some κ1, κ2 > 0, and all t ≥ 0. We will prove below thatthe closed-loop system is asymptotically stable. This, togetherwith the fact that x� ∈ R
6+, ensures that (17) holds if the initialconditions are sufficiently close to the equilibrium.
Remark 3.3: In reality the control signals, being dutycycles, live in a compact set. Unfortunately, the theoreticalresults presented later cannot take into account thisconsideration.
IV. CASCADE DECOMPOSITION OF THE SYSTEM
The following decomposition of the system allows us tosimplify the controller design task. Recalling (17), and defin-ing the new control signal
v1 := −Mx4u1 (18)
it is possible to write the overall system (8) as a cascadeconnection of the subsystem
Lx1 = −Rx1 + L1x2x3 + x1v1
Lx2 = −Rx2 − L1x1x3 + φ1x3 + x2v1
J1 x3 = −φ1x2 + (x3)
y1 = (x2
1 + x22
)v1 (19)
with input v1 and output y1, and the subsystem
Cx4 = −1
2x5u2 − 1
2x6u3 − 3
2x4y1
Lgx5 = −Rgx5 + Lgωx6 − Vd + x4u2
Lgx6 = −Lgωx5 − Rg x6 + x4u3 (20)
with external input y1 and controls u2 and u3. A cascadeconnection between the subsystems is shown in Fig. 3.
The cascade decomposition from Fig. 3 suggests the fol-lowing controller design procedure.
(S1) Design an SPBC to generate the control signal v1 thatrenders the desired equilibrium x�
13 of subsystem (19)asymptotically stable. This step is similar to the onedone in [23], but with the difference that, to improvethe transient performance of the closed-loop system, wehave followed the suggestion made in the concluding
CISNEROS et al.: PASSIVITY-BASED CONTROL OF A GRID-CONNECTED SMALL-SCALE WINDMILL 251
remarks of [23] and explored an alternative constructionof the SPBC. See Remark 5.1 below.
(S2) In the spirit of [27], design for the subsystem (20)a PI controller for an output with respect to whichthe incremental model is passive. Here, again, thereis a fundamental difference with respect to the designproposed in [27]. Indeed, due to the presence of theterm coupling the two subsystems, we are dealing nowwith a tracking and not a regulation problem like theone addressed in [27].
Since the controller design is based on the aforementioneddecomposition, to enhance readability, we first present theindividual controller designs in the next two sections. Themain result of this paper, given in Proposition 7.1, is theproof that the overall controller renders the equilibrium of thecomplete system (8) asymptotically stable.
V. STANDARD PBC OF SUBSYSTEM (19)
The control design for the subsystem (19) is based on theSPBC, which is a variation of PBC that is particularly suitedfor systems described by Euler–Lagrange equations. As shownin [25], it has been successful in a wide range of applicationsincluding mechanical, electromechanical and power electronicsystems.
For the sake of clarity, we present in this section, thethree steps that are followed to design an SPBC. First, theEuler–Lagrange representation of (19) is given in Section V-A.Second, since SPBC requires a stable invertibility condi-tion [25], the stability of the zero dynamics of (19)—forsome suitably defined output—is established in Section V-B.The design of the stabilizing SPBC is carried out inSection V-C. Finally, to improve performance, further dampingand an integral robustifying term are added in Section V-D.
A. Euler–Lagrange Model
To apply SPBC, the subsystem (19) are written in Euler–Lagrange form3
Dx13 + [C(x3) + R]x13 = G(x12)v1 + b(x3) (21)
where we defined the generalized inertia, damping, and inter-connection matrices
D := diag{L, L, J1},R := diag{R, R, 0}
C(x3) :=⎡⎣
0 −L1x3 0L1x3 0 −φ1
0 φ1 0
⎤⎦. (22)
The right-hand side terms in (21) are the external forces, where
b(x3) :=⎡⎣
00
(x3)
⎤⎦ , G(x12) :=
⎡⎣
x1x20
⎤⎦ . (23)
Notice that, consistent with their physical interpretation
D > 0, C(x3) = −C�(x3), R ≥ 0.
3See [25] and [29] for further details on Euler–Lagrange systems in controlapplications.
Hence, differentiating the systems energy function
H1(x13) = 1
2x�
13Dx13 (24)
yields the power-balance equation
H1 = −R|idq |2︸ ︷︷ ︸dissipation
− |vdq ||idq |︸ ︷︷ ︸elec. power
+ 2
3Tmωm,
︸ ︷︷ ︸mech. power
where | · | is the Euclidean norm.
B. A Stable Invertibility Property of the System (21)
As a Stable Invertibility explained in Section III-A of [25],SPBC performs a partial inversion of the system dynamics.Indeed, the controller is a copy of part of the system’sequations with the remaining states set equal to constants—plus some damping injection terms, which vanish at the equi-librium. Therefore, to ensure internal stability, it is necessarythat the zero dynamics of the system, with respect to theoutputs (the states that are fixed to constant) is asymptoticallystable.
The SPBC proposed here takes as output the state x1. Hence,the need of the lemma below.
Lemma 5.1: Given an assignable equilibrium x� ∈ E veri-fying (9). The zero dynamics of the system (19) with outputx1 − x�
1 has an asymptotically stable equilibrium at (x�2, x�
3).Proof: Setting x1 = x�
1 and x1 = 0, in the first equationof (21) yields the (zeroing output) control4
v1 = R − L1
x�1
x2x3
which replaced in x2 yields
x2 = − L1
Lx�
1x3 + φ1
Lx3 − L1
Lx�1
x22 x3
= − L1
Lx�1
x3
((x�
1)2 − φ1
L1x�
1 + x22
)
= L1
x�1
x3[(x�2)
2 − x22 ]
=: m1(x2, x3) (25)
where we have used (10) and (11) to obtain the second andthird equations. The zero dynamics is completed with the thirdequation of (21)
x3 = −φ1
Jx2 + 1
J(x3) =: m2(x2, x3). (26)
The Jacobian of the zero dynamics vector fieldcol(m1(x2, x3), m2(x2, x3)) is given by
[ − 2L1Lx�
1x2x3
L1Lx�
1[(x�
2)2 − x2
2 ]−φ1
J1
1J1
′(x3)
]
which evaluated at the equilibrium (x�2, x�
3) yields[ − 2L1
Lx�1
x�2x�
3 0
−φ1J1
1J1
′(x�3)
]. (27)
Now, from the definition of (x3) given in (7) and the factthat Cp(λ�) > 0 and C ′
p(λ�) = 0, we conclude that
′(x�3) < 0.
4To avoid cluttering, but with some obvious abuse of notation, we use thesame symbols for the system and its restricted dynamics.
252 IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN POWER ELECTRONICS, VOL. 1, NO. 4, DECEMBER 2013
This, together with the fact that (x�2 x�
3/x�1) > 0 ensures that
(27) is a Hurwitz matrix. The proof is completed invokingLyapunov’s first method.
C. Design of the SPBC
To enhance readability, the SPBC design is done in threesteps: 1) energy shaping; 2) damping injection; and 3) explicitdefinition of the controller.
The first step in the SPBC procedure is to modify the energyfunction (24) assigning to the closed-loop system the energyfunction
W1(e13) := 1
2e�
13De13 (28)
wheree13 := x13 − xd
13. (29)
is an error signal and the vector xd13, which is a signal that will
converge to x�13, is defined below. Toward this end, a copy of
the system dynamics is proposed
Dx d13+[C(x3)+R]xd
13 = b(x3)+G(x12)v1 +⎡⎣
00
R3ae3
⎤⎦ (30)
where R3ae3, with R3a > 0, is an additional damping injectionsignal. Subtracting (21) and (30) and using (29) yields the errorequation
De13 + [C(x3) + Rd ]e13 = 0 (31)
whereRd := R + diag{0, 0, R3a} > 0.
Taking the derivative of (28), along the trajectories of (31),yields
W1 = −e�13Rd e13
≤ −2min{R, R3a}max{L, J1} W1
establishing that e13(t) → 0, exponentially fast.The controller dynamics is obtained setting xd
1 = x�1 in (30),
which yields
Lxd2 = −L1x3x�
1 − Rxd2 + φ1xd
3 + x2v1
J1 x d3 = −φ1xd
2 + (x3) − R3a(x3 − xd
3
)
v1 = 1
x1
(Rx�
1 − L1xd2 x3
). (32)
Notice that the control signal v1 is obtained from the firstequation in (30), which becomes an algebraic equation becausex�
1 is a constant.The stability properties of this SPBC are summarized in the
following.Proposition 5.1: Consider the system (19) in closed-loop
with the controller (32). The equilibrium x�13 is asymptotically
stable.Proof: The derivations above established that x13 − xd
13is bounded, and |x13(t) − xd
13(t)| → 0, exponentially fast.Therefore, it only remains to prove that, for a suitable set ofinitial conditions, xd
13 is bounded and xd13(t) → x�
13. Towardthis end, replace the control v1 in the first equation of (32)
and write—with obvious definitions—the controller equationsin the compact form
x d2 = f1
(x1, x3, xd
2 , xd3
)x d
3 = f2(x3, xd
2 , xd3
).
Make now the key observation that these functions verify
f1(x�
1, xd3 , xd
2 , xd3
) = m1(xd
2 , xd3
)f2
(xd
3 , xd2 , xd
3
) = m2(xd
2 , xd3
)
where the functions mi (x2, x3), i = 1, 2 are defined in (25)and (26)—and correspond to the vector field of the asymptot-ically stable zero dynamics studied in Lemma 5.1. Adding andsubtracting the functions mi (xd
2 , xd3 ), the controller equations
can be written in the form
x d2 = m1
(xd
2 , xd3
) + �1(xd
2 , xd3 , e1, e3
)x d
3 = m2(xd
2 , xd3
) + �2(xd
2 , xd3 , e3
)
with the signals
�1(xd
2 , xd3 , e1, e3
) := f1(e1 + x�
1, e3 + xd3 , xd
2 , xd3
)
− f1(x�
1, xd3 , xd
2 , xd3
)
�2(xd
2 , xd3 , e3
) := f2(e3 + xd
3 , xd2 , xd
3
) − f2(xd
3 , xd2 , xd
3
)
viewed as perturbations to an asymptotically stable system.The proof is completed recalling that e13(t) → 0 exponentiallyfast, noting that
�1(xd
2 , xd3 , 0, 0
) = 0, �2(xd
2 , xd3 , 0
) = 0
and invoking standard results of (local) asymptotic stability ofcascaded systems.
The corollary below, is instrumental for the analysis ofthe overall closed-loop system. Its proof follows immediatelyfrom the definition of Lyapunov stability of an equilibrium,Lemma 3.1 and Proposition 5.1.
Corollary 5.1: Consider the system (19) in closed-loopwith the controller (32). For any ε > 0, there exists (suf-ficiently small) δ > 0 such that for all initial conditionsverifying
|col(x13(0), xd2 (0), xd
3 (0)) − col(x�13, x�
2, x�3)| ≤ δ
the corresponding trajectory satisfies the following.
(P1) x13(t) ∈ R3+ and xd
23(t) ∈ R2+ for all t ≥ 0.
(P2) The error signal y1 := y1 − y�1, with
y�1 := −1
3[Rg(x�
5)2 + Vd x�
5] < 0 (33)
verifies
|y1(t)| ≤ ε ∀t ≥ 0
limt→∞ y1(t) = 0 (exp). (34)
Remark 5.1: In the SPBC above we fixed xd1 = x�
1, thisshould be contrasted with the SPBC of [23], where we fixedxd
2 = x�2 instead. As discussed in [23], this was motivated by
the fact that the zero dynamics of the system with the outputx1 − x�
1 has a unique (asymptotically stable) equilibrium pointin the operating region, as shown in Lemma 5.1. On the otherhand, it was shown in Lemma 2 of [23], that the zero dynamicswith output x2 − x�
2 has two equilibria in the operating region,
CISNEROS et al.: PASSIVITY-BASED CONTROL OF A GRID-CONNECTED SMALL-SCALE WINDMILL 253
one of them unstable. Therefore, it is reasonable to expect thatthe domain of attraction of the new controller is larger thanthe one in [23].
D. Performance Improvement
As usual in PBC, the performance of the controller canbe improved injecting additional damping and incorporatingintegral actions [25]. In this case, the third right-hand sideterm in (30) can be replaced by col(R1ae1, R2ae2, R3ae3), withR1a, R2a > 0, to inject additional damping and improve theconvergence speed of the error e13.5
An integral term can easily be added to the SPBC replacingthe second equation in (32) by
J1x d3 = −φ1xd
2 + (x3) − R3a(x3 − xd
3
) + z (35)
where z is an integral term defined by
z = −Kiwe3 (36)
with Kiw = K �iw > 0 an integral gain. The error equation (31)
becomes
De13+[C(x3)+R+diag{R1a, R2a, R3a}]e13 =⎡⎣
001
⎤⎦z. (37)
Stability of the modified scheme is established with the totalenergy function
V1(e13, z) = 1
2e�
13De13 + 1
2Kiwz2
whose derivative verifies
V1 ≤ min{R + R1a, R + R2a, R3a}|e13|2.The interested reader is referred to [25] for additional details.
VI. TRACKING PI PBC FOR SUBSYSTEM (20)
In this section, we design the controller for the subsystem(20). We notice that the subsystem is perturbed by the couplingterm coming from the first subsystem, which we view as anadditive (measurable) disturbance y1(t). In the absence of thelatter, the PI PBC of [27] would solve the problem. To considerthis disturbance, we add to the regulation PI schemes of [26],[27] a suitably tailored reference, yielding a tracking PI PBC.
Proposition 6.1: Consider subsystem (20) with y1 andexternal signal verifying (34). Define the tracking PI controller
ξ = y2
u23 = ud23 − K p y2 − Kiξ (38)
with K p = K �p > 0, Ki = K �
i > 0 and
y2 = 1
2
[x�
4x5 − xd5 x4
x�4 x6
](39)
ud23 =
⎡⎣
− 3x�
4 xd5
y1
Lgωxd
5x�
4
⎤⎦ (40)
and xd5 the solution of the differential equation
Lg xd5 = −Rgxd
5 − Vd − 3
xd5
y1, xd5 (0) > 0. (41)
5This fact, together with the performance improvement discussed in Remark5.1, were verified by simulations, but are omitted from Section IX for brevity.
There exists εc > 0 such that for all ε ≤ εc the closed-loopsystem with state (x46, ξ, xd
5 ) has a globally asymptoticallystable equilibrium point at (x�
46, 0, x�5).
Proof: First, we show that under the conditions of theproposition the solution of (41) is well defined and verifiesxd
5 (t) > 0, for all t ≥ 0, and xd5 (t) → x�
5. Toward this end,we write (41) in the equivalent form
Lg xd5 = −Rgxd
5 − Vd − 3
xd5
y∗1 − 3
xd5
y1. (42)
Using (33), the equation above with y1 ≡ 0 becomes
Lg xd5 = −Rg xd
5 − Vd + 1
xd5
[Rg(x�
5)2 + Vd x�
5
]
which has an asymptotically stable equilibrium at x�5. More-
over, since xd5 (0) > 0 and x�
5 > 0, the set {xd5 > 0} is invariant.
The proof that these properties are preserved for the perturbedequation is completed by invoking (34), a continuity argumentand taking εc sufficiently small.
The subsystem (20) can be expressed in the following form:x46 = Ad(u23)x46 + E (43)
where
Ad(u23) := A + B2u2 + B3u3
E := −⎡⎢⎣
32C x4
y1VdLg
0
⎤⎥⎦ , A :=
⎡⎢⎣
0 0 0
0 − RgLg
ω
0 −ω − RgLg
⎤⎥⎦
B2 :=⎡⎣
0 − 12C 0
1Lg
0 0
0 0 0
⎤⎦ , B3 :=
⎡⎣
0 0 − 12C
0 0 01
Lg0 0
⎤⎦.
Similar to the construction in [27], the key observation is thatthe matrix
P :=⎡⎣
C 0 00 1
2 Lg 00 0 1
2 Lg
⎤⎦ > 0
verifies6
P Ad(u23) + A�d (u23)P = −1
2diag{0, Rg, Rg} ≤ 0. (44)
Now, define the vector xd46 := col(x�
4, xd5 , 0). With
(39)–(41), it is easy to prove that
x d46 = Ad
(ud
23
)xd
46 + E .
Hence, the error e46 := x46 − xd46 verifies
e46 = Ad(u23)e46 + (eu
2 B2 + eu3 B3
)xd
46 (45)
where we defined the input error
eu23 := u23 − ud
23.
Notice that substituting y1 of (19) and u1 of (32), the vectorud
23 in (40) becomes
ud23 =
⎡⎣− 3|x12|2
x1x�4 xd
5
(Rx�
1 − L1xd2 x3
)
Lgωxd
5x�
4
⎤⎦. (46)
6The interested reader is referred to [27] for further details on this construc-tion for general converter topologies.
254 IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN POWER ELECTRONICS, VOL. 1, NO. 4, DECEMBER 2013
Motivated by (44) define the function
W2(e46) := 1
2e�
46 Pe46
whose derivative along the trajectories of (45) satisfies
W2 = − Rg
2|e56|2 + y�
2 eu23.
This proves that (45) defines a passive map eu23 → y2, hence
it can be controlled with a PI.The proof is completed with the proper Lyapunov function
candidateV2(e46, ξ) = W2(e46) + 1
2ξ�Kiξ
whose derivative yields
V2 = − Rg
2|e56|2 + y�
2 eu23 + ξ�Ki y2
= − Rg
2|e56|2 − y�
2 K p y2
which establishes stability of the equilibrium. The proof ofattractivity follows doing some standard signal chasing.
Remark 6.1: The proof given above relies in a, far fromelegant, perturbation argument. This argument is avoided in theproof of the main result in Section VII, where a basic lemmaof cascades of asymptotically stable systems is invoked.
VII. MAIN RESULT
The proposition below shows that the combination of bothcontrollers ensures the control objective is asymptoticallyachieved. The proof relies on the following lemma establishedin [30].
Lemma 7.1: Consider the following cascaded system:
x = f (x, y, t)
y = g(y, t)
with |∂ f/∂x | and |∂y/∂y| bounded and
0 = f (x�, y�, t)
0 = g(y�, t).
The following statements are equivalent.
(C1) (x�, y�) is a uniformly asymptotically stable (UAS)equilibrium of the cascaded system.
(C2) y� is a UAS equilibrium of y = g(y, t) and x� is aUAS equilibrium of x = f (x, y�, t).
Proposition 7.1: Consider the system (8) and an equilib-rium x� ∈ E , verifying (9), in closed-loop with the dynamicstate-feedback controller.
1) Standard PBC
Lxd2 = −L1x3x�
1 − Rxd2 + φ1xd
3
+ x2
x1
(Rx�
1 − L1xd2 x3
)(47a)
J1x d3 = −φ1xd
2 + (x3) − R3a(x3 − xd
3
) + z (47b)
z = −Kiw(x3 − xd
3
)(47c)
u1 = − 1
Mx1x4
(Rx�
1 − L1xd2 x3
). (47d)
Fig. 4. Block diagram of Proposition 7.1.
2) PI PBC
Lgxd5 = −Rg xd
5 −Vd − 3
x1xd5
|x12|2(Rx�
1 −L1xd2 x3
)(48a)
ξ = 1
2
[x�
4 x5 − xd5 x4
x�4x6
](48b)
u23 =⎡⎣ − 3|x12|2
x1x�4 xd
5
(Rx�
1 − L1xd2 x3
)
Lgωxd
5x�
4
⎤⎦
−K p
[x�
4 x5 − xd5 x4
x�4x6
]− Kiξ (48c)
with xd5 (0) > 0, R3a > 0, Kiw = K �
iw > 0, K p = K �p >
0, Ki = K �i > 0. The equilibrium point of the closed-loop
system (x�, x�
23, 0, 0, x�5
) ∈ R11
is asymptotically stable.Proof: From the derivations of Proposition 5.1, we identify
the cascade of subsystems �1 and �2 given by
�1 : e13 = A(t)e13
�2 : x d23 = m12
(xd
23
) + �12(xd
23, e13)
whereA(t) := −D−1[C(x3(t)) + Rd ]
and m12(xd23),�12(xd
23, e13) are given in the proof. The cas-cade fits into the paradigm of Lemma 7.1, with �1 UGAS(actually, exponentially) and �2 AS, therefore the cascade isUAS, see Fig. 4.
Now, from Proposition 6.1, we identify the cascade of astatic map
y1 = H(xd
23, e13)
with the systems �3 and �4 given by
�3 : x d5 = m3
(xd
5
) + �3(xd
5 , y1)
�4 :[
e46
ξ
]= F
(e46, ξxd
5 (t), y1(t)).
See (42) and (45). �3 is AS and �4 is UAS (actually,globally). Invoking Lemma 7.1, we conclude AS of the overallsystem. A block diagram of Proposition 7.1 is shown in Fig. 4.
VIII. CONTROL IMPLEMENTATION
The implementation of the grid-connected windmill’s con-trol scheme of Section VII readily follows from the dynamicalsystems defined by (47) and (48). A block diagram of thecontrol scheme is shown in Fig. 5. As suggested in this figure,the control action is partitioned into two coupled blocks. TheSPBC block controls the windmill’s shaft speed (x3) actingupon the dc–dc converter’s duty cycle (D), while the PI PBC
CISNEROS et al.: PASSIVITY-BASED CONTROL OF A GRID-CONNECTED SMALL-SCALE WINDMILL 255
Fig. 5. Block diagram implementation of the proposed control scheme.
Fig. 6. Block diagram of the PI-based benchmark control scheme.
block regulates both the dc voltage and the ac grid currentacting upon the H-bridge inverter’s modulation index (m). Itis noteworthy the SPBC and PI PBC blocks are the directimplementation of (47) and (48), respectively.
The block diagram also shows the translation of physicalvariables into the state vector defined in Section II-D. It isassumed the full state vector is available for measurement.The block diagram’s inputs are the state vector (x16), windspeed (vw), ac grid voltage (V ), and references for the dc linkvoltage (x�
4 = 400 V) and reactive current (x�6 = 0). The
outputs of the block diagram are the control handles, namelyD and m. It is further observed in the figure that various blocksare linked to their corresponding equations within this paperthrough specifying their equation number. Moreover, blockstagged as “dynamical system” represent differential equationswhose state variables are highlighted on the right corner at thebottom of the block.
As customary in dq coordinates-based control schemes,the controller on the ac grid side relies on a suitable phase-locked loop scheme for the synthesis of the ac grid frequency(ω) and phase angle (θgrid) [31], [32]. Once ω and θgrid arecomputed, a consistent pair of matrices, namely T dq
a andT a
dq , map the single-phase and dq variables into one another.On the windmill side, the electrical frequency (ωe) is obtained
from the shaft speed (ωm ) utilizing the definition given inSection II-A. The value of ωe is in turn used todefine the transformation of three-phase quantities into dqvariables (T dq
abc).
IX. BENCHMARK SYSTEM
The performance of the controller introduced in this paperis compared against an industry standard PI control-basedarchitecture [33]. The block diagram of the control schemeused as a benchmark is shown in Fig. 6. Please refer toSection VIII for interpretation of various control parameters.
A. Parameters of the System (8)
As is customary, the power coefficient is assumed to begiven by the function
Cp(λ) = e− cp1λ
(cp2
λ− cp3
)+ cp4λ
where the coefficients cpi , i = 1, . . . , 4—that are windmillspecific, but independent of vw and ωm—are known. Thesecoefficients were taken from [10], [34], and have the followingvalues: cp1 = 21.0000, cp2 = 125.229, cp3 = 9.7803,and cp4 = 0.0068. This yields λ� = 8.1 and C�
p = 0.48.Parameters for the windmill system were taken from [35], and
256 IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN POWER ELECTRONICS, VOL. 1, NO. 4, DECEMBER 2013
TABLE I
GRID-CONNECTED WINDMILL SYSTEM PARAMETERS
adapted to fit the physical constraints of the dc–dc converter.Table I shows various numerical values. In this paper, to usea boost dc–dc converter, a larger dc link voltage of v�
dc =400 (V), is considered—this can be implemented with just adiode and a MOSFET as suggested in Fig. 1 [36]. The stiffnessconstraints required by the converter [37] are naturally givenby the inductor of the PMSG on the pole side of the converter,and by the dc link capacitor in the throw side.
B. Controllers
As stated, the performance of two different schemes ofcontrol is compared. For simplicity, we will refer as PBCcontrollers for the controllers whose methodology is proposedin this paper. The structure of this scheme has been definedin the previous sections. On the other hand, the PI-basedbenchmark control scheme of Fig. 6 will be labelled as PIcontrollers. Various gains of both controllers were tuned usingthe well-known pole placement method. Numerical values aregiven below.
1) PBC Controller Gains
R3a = 0.8, Kiw = 0.5
K p =[
0.007 00 0.009
], Ki =
[1 00 0.90
].
2) PI Controllers Gains
K p_x3 = 0.00967472, Ki_x3 = 0.10516
K p_x4 = −4, Ki_x4 = −5
K p_x5 = 0.05, Ki_x5 = 0.8
K p_x6 = 0.05, Ki_x6 = 0.8.
C. Wind Speed Profile
The wind speed profile shown in Fig. 7 is used for thesimulation studies. It was constructed using real measurements
Fig. 7. Real wind speed profile used in the simulation studies.
collected by the National Wind Technology Center in Boulder,Colorado, USA. The wind speed was measured at 100 Hz at36.6 m above the ground using a cup anemometer. As maybe observed in the figure, the profile is rich in turbulence andexhibits gusty behavior at times.
X. COMPUTER SIMULATIONS
In this section, the performance of both the PBC (Fig. 5) andPI (Fig. 6) controllers are tested via computer simulations con-sidering a detailed model for the system of Fig. 1, with para-meters specified in Section IX. All simulations are executedusing the MATLAB–Simulink mathematical analysis softwarepackage.
Recalling (9), the theoretical considerations for PBC designwere developed assuming x�
3 = λ�vw , that is, the referencefor the optimal shaft rotational speed shall be computed usingthe actual wind speed. However, in a real setting the windspeed will likely be filtered before being used to track x3.This is performed to provide smooth power to the turbineshaft. Alternatively, if the wind speed is unavailable formeasurement—which will very likely be the case for a small-scale wind turbine—one could estimate its value. Severalestimation algorithms featuring an acceptable performancehave been proposed in [38].
In the simulation study, we consider all three cases. Westudy the system’s response using to generate x�
3 the actualwind speed (x�
3 = λ�vw), a filtered wind speed (x�3 =
λ�vfw) and an estimated wind speed (x�
3 = λ�vw). Thefiltered wind speed is computed using a simple low-passfilter
v fw(s) = 1
1 + τ svw(s)
with τ = 0.7s. The estimation of wind speed is performedthrough the Immersion Invariance (I and I) estimator proposedin [24],
˙vw = γ
J1[φ1x2 − (x3, vw + γ x3)]
where γ = 0.2 is an adaptation gain.Fig. 8 shows the results. The left, middle, and right columns
presents the comparative evaluation using the actual, filtered,and estimated wind speed, respectively. PBC controllers relyon a copy of the system’s zero dynamics to generate refer-ences. Because of this, when using the actual wind speedthe PBC controller is unable to accurately track x�
3. Thus,the efficiency of the power extraction process is sacrificedas observed in the tip-speed ratio plot. On the other hand,
CISNEROS et al.: PASSIVITY-BASED CONTROL OF A GRID-CONNECTED SMALL-SCALE WINDMILL 257
Fig. 8. Simulation results. The left, middle, and right columns presents the comparative evaluation using the actual, filtered, and estimated wind speed,respectively. Recall x := col(id , iq , rωm , vdc, Id , Iq ), hence the units for the various plots are: x1 (A), x2 (A), x3 (m/s), x4 (V), x5 (A), and x6 (A). Controlsignals are unitless and because of their physical interpretation they must be contained within 0 and 1. (a) Actual wind speed. (b) Filtered wind speed.(c) Estimated wind speed. (d)–(f) Tracking states. Red lines: x�
i . Green lines: xi . (g)–(i) Remaining states. (j)–(l) Control signals. (m)–(o) Tip-speed ratio(λ� = 8.1).
PI controllers are able to quickly respond and properly per-form the tracking function. This comes at a cost of intro-ducing large noise levels on the states and control signals.
However, when using a filtered or estimated wind speedsignals to feed the control schemes, the PBC controller approx-imately matches the tracking performance of the PI controllers.
258 IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN POWER ELECTRONICS, VOL. 1, NO. 4, DECEMBER 2013
Fig. 9. Robustness test. Red line: x�3 . Green line: x3. A 30% step-like change
in Lg is applied. We show the response of x3. It may be observed that underthe PBC controllers, x3 is able to follow x�
3 indifferent to the variation ofLg . On the other hand, the PI controllers are unable to perform the trackingfunction, rendering the system unstable.
Both controllers track the tip-speed ratio to a similar degreeof accuracy, and the compromise between the speed and noisestill exists. It is worth mentioning that the system’s responsewas very sensitive to the gain’s tuning for the PI controller,while the PBC controller exhibited an acceptable performancethroughout a large range of gains’ selection. Furthermore, therobustness of both controllers was tested by introducing a30% disturbance in the value of the filter/grid inductance.Results are presented in Fig. 9. It may be observed that,unlike the PI controller, the PBC controller is able to remainstable.
XI. CONCLUSION
An asymptotically convergent PBC for a basic windmillsystem connected to the grid, which ensures maximum powerextraction and regulation of the dc link voltage and injection ofreactive power has been proposed. To design the controller, theoverall systems have been divided in two coupled subsystems:the windmill with the PMSG and the power converters with thegrid. For the first subsystem, a SPBC, similar to that of [23],was realized. The second subsystem was controlled by meansof a PI controller destined to track assignable trajectories.
Two modifications to a previous work were introducedto improve the transient performance of the PBC. First, asexplained in Remark 5.1, we design a new version of SPBCwhose domain of attraction is larger than that reported in [23].Second, endowing the PI controller with tracking capabilitiesallows for a faster response with respect to the standardregulation PI of [27]. This performance improvement, togetherwith a comparison with the industry standard scheme, wasverified through detailed computer simulations.
A first approach to quantify the performance of the PBC-regulated windmill in a more realistic distribution feeder wassuccessfully demonstrated through varying the grid equivalentinductance (Lg). The performance of the windmill embeddedin an actual distribution feeder is currently being investigatedby the authors. Furthermore, the authors are also validating theproposed controller through experimentation. Results on bothof the aforementioned topics will be reported in a dedicatedpaper in the future.
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Rafael Cisneros was born in Mexico in 1985. Hereceived the M.S. degree in control systems fromCentro de Investigacion y Estudios Avanzados delInstituto Politecnico Nacional, Guadalajara, Mexico,in 2011. He is currently pursuing the Ph.D. degreewith Laboratoire de Signaux et Systemes, Paris,France.
His current research interests include nonlinearsystems and port-Hamiltonian systems applied toelectrical systems.
Fernando Mancilla–David (S’05–M’07) receivedthe B.S. degree in electrical engineering from Uni-versidad Tecnica Federico Santa Mariia, Valparaiso,Chile, and the M.S. and Ph.D. degrees in electri-cal engineering from the University of Wisconsin-Madison, Madison, WI, USA, in 1999, 2002, and2007, respectively.
He is currently an Assistant Professor with theUniversity of Colorado Denver, Denver, CO, USA.He was a Visiting Scientist with ABB CorporateResearch, Vasteras, Sweden, in 2008, and has held
Visiting Professor positions with L’Ecole Superieure d’electricite, Supelec,Paris, France, in 2009 and 2010, Technische Universitat Berlin, Berlin,Germany, in 2011, and Universita Degli Studi Roma Tre, Rome, Italy, in 2012.His current research interests include power system analysis, energy systems,utility applications of power electronics, control systems, and optimizationproblems.
Romeo Ortega (F’99) was born in Mexico. Hereceived the B.S. degree in electrical and mechanicalengineering from the National University of Mexico,Mexico City, Mexico, the Master of Engineeringdegree from the Polytechnical Institute of Leningrad,Leningrad, Russia, and the Docteur D’Etat degreefrom the Politechnical Institute of Grenoble, Greno-ble, France, in 1974, 1978, and 1984, respectively.
He is currently with Laboratoire de Signaux etSystemes, Paris, France. He has published threebooks and more than 200 scientific papers in inter-
national journals, with an h-index of 56. He has supervised more than 30Ph.D. theses. His current research interests include nonlinear and adaptivecontrol with special emphasis on applications.
Dr. Ortega has been a member of the French National Researcher Councilsince June 1992. He has served as a Chairman in several IFAC and IEEEcommittees and editorial boards.
