# A Quasi-Steady State Model of a Singularly Perturbed Wind Turbine with an Induction Generator and Nonlinear Frictions: A Bond Graph Approach

Authors: Aaron Padilla-Garcia; Gilberto Gonzalez-Avalos; Noe Barrera-Gallegos; Gerardo Ayala-Jaimes
Journal: Journal of Engineering
 (2020)

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## Abstract

The modelling in bond graph of a class of nonlinear systems with singular perturbations is presented. The class of nonlinear systems modelled by bond graphs is defined by the product of state variables and nonlinear dissipation elements. In order to obtain the mathematical model of the singularly perturbed nonlinear systems, a lemma based on the junction structure of the bond graph with a preferred integral causality assignment is proposed. The quasi-steady state model of the system is obtained by assigning a derivative causality to the storage elements for the fast dynamics and an integral causality to the storage elements for the slow dynamics. The proposed methodology to a wind turbine connected to an induction generator is applied. Simulation results of the exact and reduced models of this case study are shown.


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## Body

## 1. Introduction

A common simplification used in developing control and system analysis is the reduction of system order. Hence, the application of the singular perturbation methods for separation of different time scales can be applied [1, 2]. Other forms of model reduction in the literature use the concept of activity which is a measurement of energy in the elements of the system and excluding those with lower energetic activity [3]. Systems with singular perturbation are characterized by a small parameter multiplying the derivatives of the state variables called fast dynamics compared to the other variables called slow variables of the system. When the small parameter becomes zero the order of the system can be reduced [1, 2]. In the last years, the amount of electrical energy produced by wind energy has been increased. Hence, the wind turbines have great influence in the dynamics of electrical power systems. There is a considerable amount of papers about modelling, analysis, and control of wind turbines, and it is important to make a difference between the contribution of this paper and the state of the art of wind turbines models.In [4], the modelling and simulation of the wind turbine using two dynamic controllers are designed. In [5], the dynamic behavior of a typical wind turbine connected to the grid is described by using Matlab/Simulink with a modular structure. The use of energy and reliability of wind turbines depending on the applied control strategy are proved in [6].One important part of the wind turbine is the electrical machinery. The induction machine has been commonly used in wind turbines and its model is important to describe the dynamic of the energy produced. Most books and papers present the traditional mathematical models of the induction machine. Procedures for estimation and control of induction machines are provided in [7]. A relatively easy way to present the model of induction machines in electrical power systems and industrial electronics is shown in [8]. Different control strategies for the induction machine are given in [9].The need to have computational tools for the simulation of induction machines has been presented for many years. The computational methods for a representation of a symmetrical machine in the arbitrary reference frame are described in [10]. The electrical machines’ control and the determination of the steady state response using the transformation to the reference frame are described in [9]. It is well known that the wind turbines models have different energy domains. Hence, these models can be analyzed through the bond graph methodology which provides a direct and graphic alternative. The main characteristics of the bond graph theory are [11–15](1)
Bond graph theory provides a formal and systematic language for modelling dynamic systems(2)
A bond graph permits to represent systems with different energy domains(3)
Structural properties (stability, controllability, and observability) of a system can be obtainedSome papers determining bond graphs models of induction machines have been published. A bond graph of an induction machine considering mechanical components has been developed in [16]. The wind turbine models determine a set of high-order nonlinear differential equations. Also, an electrical power system with a wind turbine has electrical and mechanical components. Hence, the singular perturbation theory to this system can be applied. A singularly perturbed system is characterized by having slow and fast dynamics. Normally, the fast dynamics represent the electrical state variables and the mechanical state variables are the slow dynamics of an electromechanical system. Then, reduced systems applying the singularly perturbed theory can be obtained. Some papers using singular perturbations methods for bond graph models have been published. A useful bond graph methodology to simplify the analysis of linear systems of two time scales is proposed in [17]. In [18], it is introduced as a methodology to separate the slow and fast dynamics of a system and these dynamics can be estimated by causal loop gains. The direct determination of the quasi-steady state model of a system with singular perturbations in the physical domain is presented in [19]. A bond graph model for a closed-loop system with singular perturbations is proposed in [20]. A procedure to obtain approximate bond graph models for LTI MIMO systems with singular perturbations based on new junction structures is presented in [21].The quasi-steady state model for a class of nonlinear systems with singular perturbations modelled by bond graphs is proposed in [22]. Hence, this paper is based on the previous work developed in [22]. The contribution of this paper is to analyze another class of nonlinear systems with singular perturbations with respect to [22] in a bond graph approach. The class of nonlinear systems of this paper consists of the product of state variables with nonlinear dissipation elements. Then, the quasi-steady state model is obtained from a bond graph whose storage elements of the slow dynamics have an integral causality assignment and the storage elements that represent the fast dynamics, and a derivative causality is assigned.In this paper, an interesting case study described by the connection of a wind turbine to an induction generator is proposed. The complete system is modelled by bond graphs and the quasi-steady state model of the nonlinear system with singular perturbations applying the proposed methodology is determined. This paper has, in common, the use of the causality assignment to obtain reduced models, with respect to [20–22]. However, the papers [20, 21] are dedicated to open and closed-loop linear systems, respectively. Also, in this paper, the class of nonlinear systems has been extended with respect to [22]. The junction structure was originally presented by [14]. It was also used for different purposes in [15, 23–25]. In this paper, the junction structure is used to extend the singular perturbation methodology to a class of nonlinear systems. The organization of the paper is as follows. The singular perturbation theory in the traditional approach is described in Section 2. The class of nonlinear systems with singular perturbations in a bond graph approach is proposed in Section 3. The quasi-steady state model of the nonlinear system in the physical domain is presented in Section 4. The proposed methodology is applied to the connection of a wind turbine and an induction machine in Section 5. The conclusions are given in Section 6.


## 2. The Standard Singular Perturbation Model

Consider the singularly perturbed system model:(1)x˙1=fx1,x2,u,t,x1t0=x10,(2)εx˙2=gx1,x2,u,t,x2t0=x20,where x1 and x2 are n- and m-dimensional state vectors, respectively, u is an r-dimensional control vector, t is the time, and ɛ is a small positive parameter; f and g are continuously differentiable with respect to all arguments. The system defined by (1) and (2) is called the full system. The reduced system of the full system is defined by setting ɛ = 0 in (2):(3)x1¯•=fx1¯,x2¯,u,t,(4)0=gx1¯,x2¯,u,t.Assume that (4) can be solved uniquely, and this condition means the existence of the global solution of (4) is defined by h, for x2, and denote this solution as(5)x2¯=hx1¯,u,t.Then,x2 can be eliminated from (3) and the equation gives the reduced system:(6)x˙1=fx1¯,hx1¯,u,t,u,t.The reduced system (6) plays a key role in the analysis and the design problem of the singularly perturbed system (1) To obtain the fast components we rewrite (1) and (2) in the fast scale τ:(7)dx1dτ=εfx1,x2,u,t′+ετ,(8)dx2dτ=gx1,x2,u,t′+ετ,where(9)τ=t−t′ε.In the limit,ɛ ⟶ 0. Hence, dx1/dτ=0, x1 is a constant in the fast time scale, and the only fast variations are the deviations of x2 from its quasi-steady state x2¯. Considering x2f=x2−x2¯ and setting ɛ = 0 in (7) and (8), we determine(10)dx2fdτ=gx10,x20+x2fτ,u,t0,x2f0=x2−x2¯.The fixed instantt′ has been chosen to be t0, and hence the model constants are t0, x10, x20¯=x2¯t0, which is suitable for the fast phenomena occurring near t0.From (3) and (4) as the slow model and (10) as the fast model, one expects to approximate x1 and x2 by(11)x1t≃x1¯t,x2t≃x2¯t+x2ft−t0ε,where x2f is expressed in the t time scale.Finally, the eigenvalues of∂g/∂x2 evaluated for ɛ = 0 along x1¯t have real parts smaller than a fixed negative number defined by(12)Reλ∂g∂x2≤−c≤0.In Section3, the nonlinear systems with two time scales and nonlinear dissipation elements modelled by bond graphs are proposed.


## 3. Bond Graph Models of Nonlinear Systems with Singular Perturbations

Bond graph models are based on the exchange of power in a system, which is the product of an effort variableet and a flow variable ft whose graphical representation in BG is a power bond shown in Figure 1 [11].Figure 1
A bond.In BG, the different physical elements of a dynamic system can be built using the following elements:(i)
1-port active elements or sources denoted byMSe,MSf(ii)
1-port passive elements denoted byR,C,I which are resistances, capacitances, and inertias(iii)
2-port elements denoted byMTF,MGY representing modulated transformers and gyrators(iv)
Junction elements denoted by0,1 that determine the different connections of the elementsA diagram of a Bond Graph in an integral causality assignment [11, 12, 14] (NBGI) containing different elements of a nonlinear system is shown in Figure 2. The class of nonlinear systems of this paper is obtained by using MTF and MGY modulated by state variables which represents the product of state variables and nonlinear constitutive relations for the dissipation elements.Figure 2
Bond graph of a class of nonlinear systems with integral causality.The key vectors according to Figure2 are the following:(i)
x1t∈ℜn and x2t∈ℜm are the energy variables pt and qt related to I and C storage elements in integral causality for the slow and fast dynamics, respectively(ii)
z1t∈ℜn and z2t∈ℜm are the coenergy vectors of the storage elements for the slow and fast dynamics, respectively(iii)
Dinlt∈ℜr1 and Doutlt∈ℜr1 are the mixture of et and ft indicating the power exchanges of the linear dissipation field and the junction structure(iv)
Dinnlt∈ℜr2 and Doutnlt∈ℜr2 represent the power exchanges of the nonlinear dissipation field and the junction structure(v)
ut∈ℜp denotes the plant inputThe constitutive relationship of the storage field for the slow dynamics is(13)z1t=F1x1t,for the fast dynamics is(14)z2t=F2x2t,and for the linear dissipation field is(15)Doutlt=LDinlt.The constitutive relationship of the nonlinear dissipation field can be obtained when a resistor has variation with respect to any slow dynamics, for example, the friction where the speeds of rotation of the mechanical parts for a wind turbine by this phenomenon are affected. Hence, this dissipation element changes its constitutive relationship depending of the velocity which is a slow dynamic. Then, the constitutive relationship of the nonlinear dissipation field can be defined by(16)Doutnlt=ϕLx1Dinnlt,where ϕLx1 contains the nonlinear functions in terms of x1.Lemma 1.
Consider a bond graph model of a class of nonlinear systems defined by the product of the state variables and dissipation elements with a nonlinear constitutive relationship of the typeDoutnlt=ϕLx1Dinnlt, where the storage elements of both dynamics have an integral causality assignment (NBGI) whose junction structure is defined by(17)
Then, a state space representationx∈x1,x2 is defined by(18)x˙1t=A11xx1t+A12xx2t+B1xut,(19)εx˙2t=A21xx1t+A22xx2t+B2xut,where(20)A11x=S1111x+S1211xLTMxSL1x+S1212xϕLxTϕSxSϕ1xF1,(21)A12x=S1112x+S1211xLTMxSL2x+S1212xϕLxTϕSxSϕ2xF2,(22)A21x=F2−1S1121x+S1221xLTMxSL1x+S1222xϕLxTϕSxSϕ1xF1,(23)A22x=F2−1S1122x+S1221xLTMxSL2x+S1222xϕLxTϕSxSϕ2xF2,(24)B1x=S1311x+S1211xLTMxSBLx+S1212xϕLxTϕSxSBϕx,(25)B2x=F2−1S1321x+S1221xLTMxSBLx+S1222xϕLxTϕSxSBϕx,with(26)SL1x=S2111x+S2212xMϕxS2121x,(27)SL2x=S2112x+S2212xMϕxS2122x,(28)Sϕ1x=S2121x+S2221xMLxS2111x,(29)Sϕ2x=S2122x+S2221xMLxS2112x,(30)SBLx=S2311x+S2212xMϕxS2321x,(31)SBϕx=S2321x+S2221xMLxS2311x,(32)TMx=ML−1xL−S2212xMϕxS2221xL−1,(33)TϕSx=Mϕ−1xϕLx1−S2221xMLxS2112xϕLx1−1,(34)MLx=LI−S2211xL−1,(35)Mϕx=ϕLx1I−S2222xϕLx1−1.
The proof is presented in AppendixA.
It is important to know that the submatrices of the junction structureSijpqx with i, j, p, q = 1, 2, 3 and Mx can yield singularities problem due to the numerical values for the elements of the system.
Also, it is necessary to validate the assumption in (12) for the particular class of nonlinear systems given in (20) and (21); then, from (2) and (21), the nonlinear function of the fast dynamics is(36)g=A21xx1t+A22xx2t+B2xut.Then, (14) is defined by(37)Reλ∂A21xx1t∂x2+∂A22xx2t∂x2+∂B2xut∂x2≤0.


## 4. A Quasi-Steady State Model of a System with Nonlinear Dissipation Elements in the Physical Domain

When a system is characterized as a singularly perturbed system, reduced models of this system can be obtained. It is common that the slow dynamic model of a system gives interesting properties of the dynamic performance for the complete system. Working with reduced models is simpler for analysis and synthesis compared to effort required in the analysis of large complex systems. Thus, a quasi-steady state model of a nonlinear system with singular perturbations using bond graphs is presented.The key contribution of this paper is the causality assignment of the energy storage elements into the bond graph model. The quasi-steady state model of the system is a reduced model, considering slow and fast dynamics have reached the steady state response under their operation condition. Thus, a bond graph whose storage elements for the slow and fast dynamics have an integral and derivative causality assignment, respectively, [11,12] for a class of nonlinear systems with singular perturbations (SPNBG), determines the quasi-steady state model shown in Figure 3.Figure 3
Bond graph of a class of nonlinear systems with integral and derivative causality for slow and fast dynamics, respectively.The characteristics of the bond graph of Figure3 are(i)
The storage elementsx˙2,z2 for the fast dynamics have a derivative causality assignment(ii)
The storage elementsx˙1,z1 for the slow dynamics have an integral causality assignment(iii)
The dissipation elementsDinhl,DouthlandDinhnl,Douthnl will have a causality assignment in order to construct a correct bond graphThe following lemma presents the main contribution of this paper.Lemma 2.
Consider a bond graph model according to the scheme of Figure3, where the storage elements for the fast dynamics have a derivative causality assignment and the storage elements for the slow dynamics have an integral causality assignment. This bond graph represents a class of nonlinear systems with singular perturbations defined by the product of states variables and nonlinear constitutive relations of the dissipation field satisfying (16), and the junction structure is defined by(38)with the constitutive relations of the dissipation field:(39)Douthlt=LhDinhlt,(40)Douthnlt=ϕLhx1Dinhnlt.
Then, the quasi-steady state model for the fast dynamics is described by(41)x2¯t=A21hx1x1¯t+B2hx1ut,and for the slow dynamics is expressed by(42)x1¯•t=A11hx1x1¯t+B1hx1ut,where(43)A11hx1=H1111x1+H1211x1LhTQx1HQ1x1+H1212x1ϕLhx1TϕHx1Hϕ1x1F1,(44)A21hx1=F2−1H1121x1+H1221x1LhTQx1HQ1x1+H1222x1ϕLhx1TϕHx1Hϕ1x1F1,(45)B1hx1=H1311x1+H1211x1LhTQx1HBQx1+H1212x1ϕLhx1TϕHx1HBϕx1,(46)B2hx1=F2−1H1321x1+H1221x1LhTQx1HBQx1+H1222x1ϕLhx1TϕHx1HBϕx1,with(47)HQ1x1=H1211x1+H2212x1ϕQx1H2121x1,(48)Hϕ1x1=H2121x1+H2221x1Qx1H2111x1,(49)HBQx1=H2311x1+H2212x1ϕQx1H2321x1,(50)HBϕx1=H2321x1+H2221x1Qx1H2311x1,(51)TQx=QL−1xLh−H2212xQϕxH2221xLh−1,(52)TϕHx=Qϕ−1xϕLx1−H2221xQLxH2112xϕLhx1−1,(53)QLx=LhI−H2211xLh−1,(54)Qϕx=ϕLhx1I−H2222xϕLhx1−1.
The proof is presented in AppendixB.
From the second line of (38) and (41) with (14),(55)x2t=A21hxx1t+A22hxx˙2t+B2hxut,where x˙2t=0 in order to get the quasi-steady state for the fast dynamics.(56)x2¯t=A21hxx1¯t+B2hxut,if the terms A21hx and B2hx depend of both dynamics, and this algebraic system has to be solved and (41) is proved.
From (55),(57)x˙2t=−A22hx−1A21hxx1t+A22hx−1x2t−A22hx−1B2hxut,comparing (57) with (19), the following relations are determined:(58)A22hx=A22−1xε,A21hx=−A22−1xA21x,B2hx=−A22−1xB2x,from the first line of (38) and (42)(59)x˙1t=A11hxx1t+A12hxx˙2t+B1hxut.
By substituting (57) into (59),(60)x˙1t=A11hx−A12hxA22hx−1A21hxx1t+A12hxA22hx−1x2t+B1hx−A12hxA22hx−1B2hxut,comparing (18) with (60), the following relations are obtained:(61)A12hx=A12xA22−1xε,(62)A11hx=A11x−A12xA22−1xA21x,(63)B1hx=B1x−A12xA22−1xB2x,from (41) with (61)–(63), the quasi-steady state model for the slow dynamics given by (42) is proved.
The matricesQLx and Qϕx and submatrices of the junction structure Hijpqx with i, j, p, q = 1, 2, 3 can determine singularities’ problems due to the numerical values of the parameters of the system. This is important to detect in order to avoid stability and simulating problems. Another important condition is that A22x must be nonsingular.
The bond graph model and the quasi-steady state model of a wind turbine system are obtained in Section5.


## 5. An Induction Generator and Mechanical Models of a Wind Turbine Modelled by Bond Graphs

In this section, the proposed methodology to a nonlinear singularly perturbed system is applied. In the electromechanical systems, the mechanical variables represent the state variables for the slow dynamics and the electrical variables are the state variables for the fast dynamics. The air friction at high speeds, as a fluid with nonlinear resistance can be considered. This mechanical phenomenon occurs when there is a relative velocity between a fluid and a body; the body experiences a drag force that opposes the relative motion and points in the direction in which the fluid flows relative to the body. The drag force is sometimes called air resistance or friction [26].The magnitude of the drag force is related to the relative speedv and the drag coefficient C according to [26]:(64)D=12CρAv2,where ρ is the air density and A is the effective cross-sectional area of the body. Considering all constant parameters, the following relation is obtained:(65)c=12CρA.In general, the friction force is defined by(66)Fresist=−bv−cv2,if v≪1, drop the second term; if v≫1, ignore the first one.


### 5.1. Schematic Diagram of the Three-Mass Equivalent Model of a Wind Turbine

A three-mass equivalent model of a wind turbine in the mechanical system is proposed in Figure4. The first mass stands for the blades, the second mass stands for the hub, and the third mass stands for the generator including the gear box, high-speed shafts, and generator rotor [27], where Hb, Hh, and Hg are the constant inertia for the blades, hub, and generator, respectively; Dbh is the damping coefficient between the blades and hub; Dhg is the damping coefficient between the hub and the generator rotor; Db, Dh, and Dg are the damping coefficients for the blades, hub, and generator, respectively; Kbh and Khg are the stiffness of the blades and low speed shaft [28].Figure 4
Schematic diagram of the three-mass equivalent model of a wind turbine, considering resistance quadratic in the dissipation elements.Considering quadratic resistance in the dissipation elements, as shown in Figure4, the constant Dg is c in equation (65); this constant multiplies the square of the speed of this section of a wind turbine wg2=p13/Hg2, where p13 is the angular momentum.


### 5.2. Induction Machine Modelled by Bond Graphs

One of the most common electrical motor used in many applications is the induction motor. Figure5 shows a scheme of a 2-pole induction machine with symmetrical 3-phase windings in the stator and rotor [29]. The stator and rotor windings are identical and sinusoidally distributed and the s and r subscripts denote the variables and parameters associated to the stator and rotor circuits, respectively.Figure 5
Schematic of symmetrical 3-phase asynchronous or induction machine.The Park transformation gives an important simplification in the mathematical model of the induction machine. The property of the Park transformation is to change all stator variables and parameters from phasesa,b,c to new variables, the frame of reference of which moves with the rotor [29].The Park transformation for voltages is defined by(67)idq0=Piabc,where(68)P=23cosθcosθ−2π3cosθ+2π3−sinθsinθ−2π3sinθ+2π3121212.The angle between thed-axis and the rotor is(69)θ=wt+π2,where w is the rated angular frequency in rad/s. It is easy to find similar expressions for currents and flux linkages:(70)idq0=Piabc,λdq0=Pλabc.The equivalent circuits of the induction machine are shown in Figure6. It is convenient to refer all the rotor variables to the stator windings using the turns ratios which is indicated by vdr′,vqr′,v0r′, idr′,iqr′,i0r′, λdr′,λqr′,λ0r′, Llr′,Lls′,andRr′. Also, w is the synchronous velocity and wr is the rotor velocity.Figure 6
Arbitrary reference-frame equivalent circuits for a 3-phase, symmetrical induction machine.A bond graph model of an induction machine connected to a wind turbine on thed − q axes is shown in Figure 7. This bond graph contains(i)
I: Ld and I: Lq are the I fields of the magnetic coupling of the self and mutual inductances of the windings on the d-axis and q-axis, respectively(ii)
idr,λdr and iqr,λqr denote the electrical currents and flux linkages for the rotor circuits on the d-axis and q-axis, respectively(iii)
R: Rr and R: Rs are the resistances for the rotor and stator circuits, respectively(iv)
ids,λds,vds and idr,λdr,vdr denote the electrical currents, flux linkages, and voltages for the stator circuits on the d-axis and q-axis, respectively(v)
I: Hg, I: Hh, and I: Hb are the gear box, hub, and blade inertias, respectively(vi)
R:Dg/Hg, R:Dh/Hh, and R:Db/Hb are the gear box, hub, and blade friction coefficient, respectively(vii)
C: Khg and C: Kbh denote the stiffness for the shaft and blades, respectively(viii)
R:Dhgp29/Hh−p13/Hg and R:Dbhp36/Hb−p29/Hh are the nonlinear friction coefficients for the gear box and blades, respectively(ix)
MSe: Ta is the plant input representing the aerodynamic torqueFigure 7
Bond graph in an integral causality assignment of the wind turbine.The key vectors of the bond graph for the dissipation field and input are(71)Dinl=f2f8f19f20,Doutl=e2e8e19e20,Dinnl=f23f27f30f34f37,Doutnl=e23e27e30e34e37,u=e1e21e38f24,for the slow dynamics(72)x1=p13q26p29q33p36,x˙1=e13f26e29f33e36,z1=f13e26f29e33f36,and for the fast dynamics(73)x2=p4p5p6p7,x˙2=e4e5e6e7,z2=f4f5f6f7,where f24 = ωF is the synchronous velocity.The constitutive relations of the fields are(74)L=diagRs,Rr,Rs,Rr,(75)ϕL=diagDgHgp13,Dhg1Hhp29−1Hgp13,Dbh1Hbp36−1Hhp29,DbHbp36,(76)F1−1=diagHg,Khg,Hh,Kbh,Hb,(77)F2−1=LsLm00LmLr0000LsLm00LmLr.The junction structure of the bond graph is given by(78)x˙1x˙2DinlDinnl=S11xS12xS13xS21xS22xS23xz1z2DoutlDoutnlu.The developed junction structure is defined in AppendixC.The mathematical model of the complete system is determined. Firstly, for the slow dynamics, from (13) to (17) and (20) with (75) to (C.1), the matrix A11x is expressed by(79)A11x=σ111KhgDhgHhp13Hg−p29Hh00−1Hg01Hh00DhgHgp29Hh−p13Hg−1Khgσ33−1KbhDbhHbp36Hb−p29Hh00−1Hh01Hb00DbhHhp13Hb−p29Hh1Kbhσ55,where(80)σ11=1HgDhgp13Hg−p29Hh−DgHgp13,σ33=−1HhDbhp36Hb−p29Hh−Dhgp13Hg−p29Hh+DhHhp29,σ55=−1HbDbhp36Hb−p29Hh+DbHbp36,the relationships between the slow and fast dynamics denoted by A12x is obtained from (13) to (17) and (21) with (75) to (C.1):(81)A12x=1Δ−ηLmp7ηLsp7ηLmp5−ηLsp50000000000000000,where Δ=LrLs−Lm2.From (16), (24), (75), and (C.1), the matrix B1 is(82)B1=00000000000000000010.The matrices of the state variables representation for the fast dynamics are obtained. From (13) to (16) and (22) with (75) to (C.1), the matrix A21x is given by(83)A21x=−ηLmHgp70000−ηLrHgp70000ηLmHgp50000ηLrHgp50000.By substituting (75) to (C.1) into (23) with (13) to (16), the matrix A22x is(84)A22x=1ΔLm2Rr−LrLsRsLmLsRs−LmLsRr00LmLrRr−LmLrRsLm2Rs−LrLsRr0000Lm2Rr−LrLsRsLmLsRs−LmLsRr00LmLrRr−LmLrRsLm2Rs−LrLsRr.From (15), (16), and (25) with (75) to (C.1), the matrix B2x is(85)B2x=−Ls00Lsp6+Lmp7−Lm00Lsp6+Lmp70−Ls0−Lsp4+Lmp50−Lm0−Lsp4+Lrp5.It can be seen that the system order isn + m = 9, where the slow dynamics are n = 5 and the fast dynamics are m = 4.


## 5.1. Schematic Diagram of the Three-Mass Equivalent Model of a Wind Turbine

A three-mass equivalent model of a wind turbine in the mechanical system is proposed in Figure4. The first mass stands for the blades, the second mass stands for the hub, and the third mass stands for the generator including the gear box, high-speed shafts, and generator rotor [27], where Hb, Hh, and Hg are the constant inertia for the blades, hub, and generator, respectively; Dbh is the damping coefficient between the blades and hub; Dhg is the damping coefficient between the hub and the generator rotor; Db, Dh, and Dg are the damping coefficients for the blades, hub, and generator, respectively; Kbh and Khg are the stiffness of the blades and low speed shaft [28].Figure 4
Schematic diagram of the three-mass equivalent model of a wind turbine, considering resistance quadratic in the dissipation elements.Considering quadratic resistance in the dissipation elements, as shown in Figure4, the constant Dg is c in equation (65); this constant multiplies the square of the speed of this section of a wind turbine wg2=p13/Hg2, where p13 is the angular momentum.


## 5.2. Induction Machine Modelled by Bond Graphs

One of the most common electrical motor used in many applications is the induction motor. Figure5 shows a scheme of a 2-pole induction machine with symmetrical 3-phase windings in the stator and rotor [29]. The stator and rotor windings are identical and sinusoidally distributed and the s and r subscripts denote the variables and parameters associated to the stator and rotor circuits, respectively.Figure 5
Schematic of symmetrical 3-phase asynchronous or induction machine.The Park transformation gives an important simplification in the mathematical model of the induction machine. The property of the Park transformation is to change all stator variables and parameters from phasesa,b,c to new variables, the frame of reference of which moves with the rotor [29].The Park transformation for voltages is defined by(67)idq0=Piabc,where(68)P=23cosθcosθ−2π3cosθ+2π3−sinθsinθ−2π3sinθ+2π3121212.The angle between thed-axis and the rotor is(69)θ=wt+π2,where w is the rated angular frequency in rad/s. It is easy to find similar expressions for currents and flux linkages:(70)idq0=Piabc,λdq0=Pλabc.The equivalent circuits of the induction machine are shown in Figure6. It is convenient to refer all the rotor variables to the stator windings using the turns ratios which is indicated by vdr′,vqr′,v0r′, idr′,iqr′,i0r′, λdr′,λqr′,λ0r′, Llr′,Lls′,andRr′. Also, w is the synchronous velocity and wr is the rotor velocity.Figure 6
Arbitrary reference-frame equivalent circuits for a 3-phase, symmetrical induction machine.A bond graph model of an induction machine connected to a wind turbine on thed − q axes is shown in Figure 7. This bond graph contains(i)
I: Ld and I: Lq are the I fields of the magnetic coupling of the self and mutual inductances of the windings on the d-axis and q-axis, respectively(ii)
idr,λdr and iqr,λqr denote the electrical currents and flux linkages for the rotor circuits on the d-axis and q-axis, respectively(iii)
R: Rr and R: Rs are the resistances for the rotor and stator circuits, respectively(iv)
ids,λds,vds and idr,λdr,vdr denote the electrical currents, flux linkages, and voltages for the stator circuits on the d-axis and q-axis, respectively(v)
I: Hg, I: Hh, and I: Hb are the gear box, hub, and blade inertias, respectively(vi)
R:Dg/Hg, R:Dh/Hh, and R:Db/Hb are the gear box, hub, and blade friction coefficient, respectively(vii)
C: Khg and C: Kbh denote the stiffness for the shaft and blades, respectively(viii)
R:Dhgp29/Hh−p13/Hg and R:Dbhp36/Hb−p29/Hh are the nonlinear friction coefficients for the gear box and blades, respectively(ix)
MSe: Ta is the plant input representing the aerodynamic torqueFigure 7
Bond graph in an integral causality assignment of the wind turbine.The key vectors of the bond graph for the dissipation field and input are(71)Dinl=f2f8f19f20,Doutl=e2e8e19e20,Dinnl=f23f27f30f34f37,Doutnl=e23e27e30e34e37,u=e1e21e38f24,for the slow dynamics(72)x1=p13q26p29q33p36,x˙1=e13f26e29f33e36,z1=f13e26f29e33f36,and for the fast dynamics(73)x2=p4p5p6p7,x˙2=e4e5e6e7,z2=f4f5f6f7,where f24 = ωF is the synchronous velocity.The constitutive relations of the fields are(74)L=diagRs,Rr,Rs,Rr,(75)ϕL=diagDgHgp13,Dhg1Hhp29−1Hgp13,Dbh1Hbp36−1Hhp29,DbHbp36,(76)F1−1=diagHg,Khg,Hh,Kbh,Hb,(77)F2−1=LsLm00LmLr0000LsLm00LmLr.The junction structure of the bond graph is given by(78)x˙1x˙2DinlDinnl=S11xS12xS13xS21xS22xS23xz1z2DoutlDoutnlu.The developed junction structure is defined in AppendixC.The mathematical model of the complete system is determined. Firstly, for the slow dynamics, from (13) to (17) and (20) with (75) to (C.1), the matrix A11x is expressed by(79)A11x=σ111KhgDhgHhp13Hg−p29Hh00−1Hg01Hh00DhgHgp29Hh−p13Hg−1Khgσ33−1KbhDbhHbp36Hb−p29Hh00−1Hh01Hb00DbhHhp13Hb−p29Hh1Kbhσ55,where(80)σ11=1HgDhgp13Hg−p29Hh−DgHgp13,σ33=−1HhDbhp36Hb−p29Hh−Dhgp13Hg−p29Hh+DhHhp29,σ55=−1HbDbhp36Hb−p29Hh+DbHbp36,the relationships between the slow and fast dynamics denoted by A12x is obtained from (13) to (17) and (21) with (75) to (C.1):(81)A12x=1Δ−ηLmp7ηLsp7ηLmp5−ηLsp50000000000000000,where Δ=LrLs−Lm2.From (16), (24), (75), and (C.1), the matrix B1 is(82)B1=00000000000000000010.The matrices of the state variables representation for the fast dynamics are obtained. From (13) to (16) and (22) with (75) to (C.1), the matrix A21x is given by(83)A21x=−ηLmHgp70000−ηLrHgp70000ηLmHgp50000ηLrHgp50000.By substituting (75) to (C.1) into (23) with (13) to (16), the matrix A22x is(84)A22x=1ΔLm2Rr−LrLsRsLmLsRs−LmLsRr00LmLrRr−LmLrRsLm2Rs−LrLsRr0000Lm2Rr−LrLsRsLmLsRs−LmLsRr00LmLrRr−LmLrRsLm2Rs−LrLsRr.From (15), (16), and (25) with (75) to (C.1), the matrix B2x is(85)B2x=−Ls00Lsp6+Lmp7−Lm00Lsp6+Lmp70−Ls0−Lsp4+Lmp50−Lm0−Lsp4+Lrp5.It can be seen that the system order isn + m = 9, where the slow dynamics are n = 5 and the fast dynamics are m = 4.


## 6. A Quasi-Steady State Model of the Wind Turbine in a Bond Graph Approach

In this section, the main contribution of this paper is described. In order to get the quasi-steady state model of a nonlinear system with singular perturbations, the traditional approach given in Section2 requires many mathematical steps. Hence, the corresponding bond graph of Figure 7 describes the mathematical model of the case study of this paper. This bond graph contains all the storage elements of both dynamics in integral causality assignment. For the storage elements with fast dynamics, a derivative causality is assigned. For the storage elements with slow dynamics, an integral causality is maintained and a quasi-steady state model of a singularly perturbed nonlinear system can be obtained. The key vectors of the storage elements and inputs for this SPNBG are the same with respect to NBGI. Also, the new key vectors of the bond graph of Figure 8 for the linear dissipation elements are(86)Dinhl=e2e8e19e20,Douthl=f2f8f19f20,and the constitutive relation is(87)Lh=diag1Rs,1Rr,1Rs,1Rr.Figure 8
Bond graph with a derivative causality assignment for the fast dynamics of the wind turbine.From (38), the junction structure of the bond graph of Figure 8 is described in Appendix D.From (38)–(41) and (75) to (C.1) with (87) and (D.1), the real roots of the algebraic equation can be obtained by(88)p4¯p5¯p6¯p7¯=ηHg−Tmrp7¯0000−Trrp7¯0000Tmrp5¯0000Trrp5¯0000p13¯q26¯p29¯q33¯p36¯+−Tss00Tssp6¯+Tmrp7¯−Tms00Tmsp6¯+Trrp7¯0−Tss0−Tssp4¯−Tmrp5¯0−Tms0−Tmsp4¯−Trrp5¯e1e21e38f24,where(89)Tss=LsRs,Trr=LrRr,Tms=LmRs,Tmr=LmRr.By substituting (77)–(79) with (86) and (87) into (42) with (43) and (45), the quasi-steady state model is defined by(90)p13¯•q26¯•p29¯•q33¯•p36¯•=α11Khg−DhgHh1Hgp13¯−1Hhp29¯00−1Hg01Hh00−DhgHg1Hgp13¯−1Hhp29¯−1Khgα21KbhDbhHb1Hbp36¯−1Hhp29¯00−1Hh01Hb00DbhHh1Hbp36¯−1Hhp29¯−1Kbhα3p13¯q26¯p29¯q33¯p36¯+000ηRrp5¯2+p7¯20000000000000010e1e21e38f24,where α1, α2, and α3 are(91)α1=−1Hgη2Rrp5¯2+p7¯2−Dhg1Hgp13¯−1Hhp29¯+DgHgp13¯,α2=−1HhDbh1Hbp36¯−1Hhp29¯−Dhg1Hgp13¯−1Hhp29¯+DhHhp29¯,α3=−1HbDbh1Hbp36¯−1Hhp29¯+DbHbp36¯.By solving the first, second, and fourth lines of (88), we have(92)p5¯=K2K3+K5K4K5K6−K1K3=K7,(93)p4¯=K7K1+K2K5,(94)p7¯=ηTrrHgP5¯p13¯−Tmse21−TmsP4¯+TrrP5¯f24,where(95)K1=ηTmrHgp13¯−ηTrrHgp13¯+2Trr+Tssf24−TmrTss+Trrf242,K2=TmrTmse21ηHgp13¯−f24−Tsse1−Tss2f24e21,K3=TrrTmsf24ηHgp13¯−f24−TssTmsf242,K4=TrrTmse21ηHgp13¯−f24−Tmse1−TssTmsf24e21,K5=1+TmrTmsf24−ηHgp13¯+f24+Tss2f242,K6=1+ηHgp13¯ηTrr2Hgp13¯−2Trr2+TmrTmsf24+TmrTms+Trr2f242.By substituting (93) to (96) into (88) and (90), the quasi-steady state model is defined by(96)p13¯•=−1Hgη2Rrp5¯2+p7¯2−Dhg1Hgp13¯−1Hhp29¯+DgHgp13¯p13¯+1Khgq26¯−DhgHh1Hgp13¯−1Hhp29¯p29¯+ηRrp5¯2+p7¯2f24,(97)q26¯•=−1Hgp13¯+1Hhp29¯,(98)p29¯•=−DhgHg1Hgp13¯−1Hhp29¯p13¯−1Khgq26¯−−1HhDbh1Hbp36¯−1Hhp29¯−Dhg1Hgp13¯−1Hhp29¯+DhHhp29¯p29¯++1Kbhq33¯+DbhHb1Hbp36¯−1Hhp29¯p36¯,(99)q33¯•=−1Hhp29¯+1Hbp36¯,(100)p36¯•=DbhHh1Hbp36¯−1Hhp29¯p29¯−1Kbhq33¯−1HbDbh1Hbp36¯−1Hhp29¯+DbHbp36¯p36¯+e38.Section7 describes the performance of the state variables of the wind turbine connected to an induction machine.


## 7. Simulation Results

In order to verify the quasi-steady state model of the complete system, the simulations results are obtained. Hence, the parameters of the induction generator were obtained from the literature [30]. These parameters are Hg = 0.089 kgm2, Rs = 0.435 Ω, Rr = 0.816 Ω, Ls = 0.0713 H, Lr = 0.0713 H, Lm = 0.0693 H, η = 2 pairs of poles, Vnom = 220 V, and ωF = 377 rad/s. And, for the mechanical model, Hb = 0.05 kgm2, Hh = 0.03 kgm2, Dbh = 0.4 N-m-s, Dhg = 0.4 N-m-s, Db = 0.4 N-m-s, Dh = 0.3 N-m-s, Dg = 0.1 N-m-s, Kbh = 1e−3N-m/rad, Khg = 1e−3 Nm/rad, and Ta = 100 Nm.It is assumed that the machine is connected to a bus whose voltage and frequency remain constant, at the rated values. This is commonly known as an infinite bus. It is necessary to verify (12) with (36) and (37):(101)Reλ∂A21xx1t∂x2+∂A22xx2t∂x2+∂B2xut∂x2≤0.The first partial derivative is∂A21xx1/∂x2=0, according to (12), then (101) is reduced to(102)Reλ∂A22xx2t∂x2+∂B2xut∂x2≤0.By solving (102),(103)ReλLm2Rr−LrLsRsLrLs−Lm2LmLsRs−LmLsRrLrLs−Lm2Lsf24Lmf24LmLrRr−LmLrRsLrLs−Lm2Lm2Rs−LrLsRrLrLs−Lm2Lmf24Lrf24−Lsf24−Lmf24Lm2Rr−LrLsRsLrLs−Lm2LmLsRs−LmLsRrLrLs−Lm2−Lmf24−Lrf24LmLrRr−LmLrRsLrLs−Lm2Lm2Rs−LrLsRrLrLs−Lm2≤0,and substituting the numerical parameters of the system into (103),(104)Reλ6.0719−6.694726.880126.12616.6947−7.322926.126126.8801−26.8801−26.12616.07192−6.6947−26.1261−26.88016.6947−7.3229.Then, the eigenvalues are(105)λ1=−0.6255+j53.0055,λ2=−0.6255−j53.0055,λ3=−0.6255+j0.7547,λ4=−0.6255−j0.7547,and (12) is satisfied, and we can conclude that this singularly perturbed system has internal stability.It is instructive to observe the variables for the slow and fast dynamics considering the complete and reduced models. Figures9–11 show the angular momentum of the hub, blades, and gear box for the exact p29,p36,p13 and reduced p29¯,p36¯,p13¯ models, respectively.Figure 9
Performance of the slow variablep29 and the quasi-steady state model p29¯.Figure 10
Performance of the slow variablep36 and the quasi-steady state model p36¯.Figure 11
Performance of the slow variable and its respective quasi-steady responses:p13 and.p13¯.The oscillations of the angles for the exactq26,q33 and reduced q26¯,q33¯ models, respectively, are shown in Figures 12 and 13, where q26 is the angle between the generator and the hub and q33 the angle related to the hub and the blades of the wind turbine.Figure 12
Performance of the slow variableq26 and the quasi-steady state response q26¯.Figure 13
Performance of the slow variableq33 and its respective quasi-steady state response q33¯.In order to prove that this system has singular perturbations, Figure14 illustrates the electrical current on d-axis f5 that represents a fast dynamic, and the slow dynamic into this figure corresponds the angular velocity of the rotor f13.Figure 14
(a) Performance of the current ond-axis, id (fast variable). (b) The angular velocity of the rotor, ωg.
(a)
(b)Note that the proposed methodology allows to obtain the quasi-steady state model by assigning a derivative causality to the storage elements for the fast dynamics and an integral causality assignment to the storage elements for the slow dynamics of a nonlinear system with singular perturbations. The previous simulation results were obtained considering a constant torqueTa. Now, a variable torque is supplied to the bond graph model of Figure 7. This torque is obtained from a real wind speed data which is shown in Figure 13, taken from the measurement of a wind turbine installed in the University of Michoacan in Mexico.The torque Ta is determined by(106)Ta=KVwind2,where Vwind is the wind speed and K is the constant to transfer wind speed to a torque. The dynamic behavior of the exact and approximate variables of the system is shown in Figures 14–25.Figure 15
Real wind speed data.Figure 16
Full time simulation of the slow variablep13 and the quasi-steady state model p13¯.Figure 17
Close up of Figure16.Figure 18
Full-time simulation of the slow variablep29 and the quasi-steady state model p29¯.Figure 19
Close up of Figure18.Figure 20
Full-time simulation of the slow variablep36 and the quasi-steady state model p36¯.Figure 21
Close up of Figure20.Figure 22
Full-time simulation of the slow variableq26 and the quasi-steady state model p26¯.Figure 23
Close up of Figure22.Figure 24
Full-time simulation of the slow variableq33 and the quasi-steady state model q33¯.Figure 25
Close up of Figure24.


## 8. Conclusion

(i)
A class of singularly perturbed nonlinear systems modelled by bond graphs has been presented
(ii)
The product of state variables and nonlinear dissipation elements represent the class of nonlinear systems of this paper
(iii)
The state space representation of the complete nonlinear system of the bond graph with integral causality assignment for both dynamics is developed
(iv)
When a derivative causality to the storage elements for the fast dynamics is assigned and the storage elements for the slow dynamics an integral causality is maintained, then the quasi-steady state model can be determined
(v)
The nonlinear physical system formed by a wind turbine connected to an induction generator has been modelled by bond graphs
(vi)
The quasi-steady state model of this electromechanical system in the physical domain has been obtained
(vii)
The performance for both dynamics based on simulation results has been shown



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