Efficient operation of anisotropic synchronous machines for wind

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Efficient operation of anisotropic synchronous machines for wind energy systems Hisham Eldeeb† , Christoph M. Hackl†,? and Julian Kullick†

†

?

All authors (in alphabetical order) contributed equally to this paper; Corresponding author Research group “Control of Renewable Energy Systems (CRES)” Munich School of Engineering, Technical University of Munich, 85748 Garching, Germany ?

E-mail: [email protected], [email protected] , [email protected]

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Abstract. This paper presents an analytical solution for the Maximum-Torque-per-Ampere (MTPA) operation of synchronous machines (SM) with anisotropy and magnetic cross-coupling for the application in wind turbine systems and airborne wind energy systems. For a given reference torque, the analytical MTPA solution provides the optimal stator current references which produce the desired torque while minimizing the stator copper losses. From an implementation point of view, the proposed analytical method is appealing in terms of its fast online computation (compared to classical numerical methods) and its efficiency enhancement of the electrical drive system. The efficiency of the analytical MTPA operation, with and without consideration of cross-coupling, is compared to the conventional method with zero direct current.

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Nomenclature N, R, C are the sets of natural, real, and complex numbers. x ∈ R or x ∈ C is a real or complex scalar. x ∈ Rn (bold) is a real valued vector with n ∈ N. X ∈ Rn×m (capital bold) is a real valued matrix with n ∈ N rows and m ∈ N columns. xks = (xds , xqs )> ∈ Rn×m is a stator space vector expressed in the synchronously rotating k-coordinate system (with orthogonal axes known as the direct (d) and quadrature (q) axes) and may represent voltage, current or flux linkage.

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1. Introduction For wind turbine systems (WTS), the optimal choice for the generator (electrical machine) adopted is still controversially discussed [1]. Doubly-fed induction machines (DFIMs) were considered one of the favourable choices for WTS, owing to their accompanied cost effective power converters and simple control of active/reactive power [2]. However, the depletion of DFIMs with respect to synchronous machines (SMs) became more pronounced due to three reasons: (i) partially rated power converters of DFIMs offer a rather poor voltage ride through capability [3, 4], (ii) SMs do not have slip rings (i.e. lower maintenance), eliminate the need for a gearbox (or allow for a drastic reduction of the gear ratio), have a simpler terminal wiring, and offer better grid support capabilities, and (iii) the increased calls for energy-efficient drives discriminates between SMs and DFIMs [5]. Particular interest has been set to permanent magnet SMs (PMSMs) and PM-excited reluctance SMs (PME-RSMs) [6], since they possess lower electrical losses than DFIMs due to the absence of the copper losses in the rotor. In general, for all WTS topologies such as large-scale or small-scale WTSs or airborne wind energy systems (AWESs), the selected SMs must be robust, cheap, and, most importantly, efficient. For pumping-mode AWESs [7, Ch. 2 & 3], the SM will operate in generator and also motor

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mode. Albeit PM machines can be controlled to operate at high power factor [5], this does not necessarily imply that the electrical losses within the adopted SM are minimised [8]. Thus, an optimal control strategy for the SMs is vital to fully realize its intrinsic efficiency. For PMSMs or PME-RSMs, the quadrature-axis stator current iqs is proportional to the produced electromagnetic torque mm if the direct-axis stator current ids is set to zero. However, in most cases, the electromagnetic torque is a nonlinear function of both stator currents [6, 9–11]. Therefore, for a given reference torque mm,ref , ids,ref optimal stator current references ids,ref and iqs,ref mm,ref current reference q is,ref must be computed (see Fig. 1). These optimal generation (MTPA) current references allow for reduced stator copper Figure 1: Generation of the optimal current refer- losses and higher efficiencies than conventional methods with ids,ref = 0 [10, 11]. Typically, (i) ences with analytical solution (1). magnetic cross-coupling between the d- and q-axes is neglected (even though it has a direct effect on the machine torque; see (4) or [9, 10]) and (ii) numerical solutions are employed to compute the optimal references [12]. In [12], optimal feedforward torque control—known as Maximum-Torque-per-Ampere (MTPA)—was suggested by formulating an optimisation problem where a reference current vector with minimal magnitude was computed numerically, which in turn due to the anisotropy gives a noticeable efficiency increment compared to the conventional ids,ref = 0 approach [13]. For high speeds, further efficiency enhancements can be achieved by considering the iron losses in the machine as well [9, 14]. However, modelling of iron losses is not straight forward (e.g. requires finite element analyses [9]) and, due to the rather low mechanical speeds in WTS, iron losses can be neglected [10]. Nevertheless, solving the MTPA optimisation problem requires nonlinear solvers such as the bisection method (known for its numerical stability but its poor convergence rate [12]) or the Newton-Raphson method (known for its fast convergence but its rather weak numerical stability depending on the initial condition [12]). Moreover, those solvers impose a rather high computational burden on the fixed-point processors adopted for such drives. Only very few results exist which compute an analytical solution to the MTPA optimization problem. In [8] and [15], an analytical expressions for the optimal d-axis reference current (the q-axis reference current comes from the outer speed control loop) has been presented, but magnetic cross-coupling and its impact on the electromagnetic torque were neglected. In this paper, an analytical solution for the MTPA optimization problem considering magnetic cross-coupling (i.e. a non-zero mutual inductance Lm 6= 0) is presented which allows to compute the optimal d- and q-axis current references

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iks,ref = (ids,ref , iqs,ref )> = MTPA(mm,ref , Lm , . . . , other machine parameters)

(1)

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online based on the machine parameters and a given (feasible1 ) reference torque mm,ref . The presented method, derived from Lagrangian optimization, is applicable to PMSMs, PME-RSMs or electrically-excited SMs (EESMs) with non-negligible anisotropy and magnetic cross-coupling. Due to space limitations, in the remainder of the paper, only the derivation for permanent-magnet synchronous machines (PMSMs) is shown and discussed. 2. MTPA with analytical solution for PMSMs with magnetic cross-coupling In this section, the main result of the paper is derived based on the dynamical model of anisotropic PMSMs with magnetic cross-coupling. The result emanates from a Lagrangian formalization of the problem and invoking Ferrari’s method to solve quartic polynomials [16]. 1

It is assumed that the currents can actually produce the desired reference torque within the electrical drive system; e.g. voltage constraints/maximum-torque-per-volt (MPTV) are not considered.

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2.1. Dynamical model of anisotropic PMSMs with magnetic cross-coupling The generic model of an anisotropic PMSM in the k = (d, q)-reference frame is given by [17]  k k k =:ψ s (t) =:is (t) =:us (t)  =:J   z }| { z }| { z }| { z }| {    d d d   0 −1 ψs (t) us (t) is (t) k k k 2 d = Rs q +np ωm (t) + dt ψ s (t), ψ s (0) = ψpm ∈ R  q q 1 0 us (t) i (t) ψ (t) (2) s  s 
mt (t)  0 k d 1  ωm (0) = ωm ∈ R   dt ωm (t)= Θ mm is (t) + gr − νf ωm (t) ,    0 d φ (t)= ω (t), φ (0) = φ ∈ R m m m dt m with stator voltages uks = (uds , uqs )> (in V2 ; e.g. applied by a voltage source inverter), stator currents iks = (ids , iqs )> (in A2 ), flux linkages ψ ks = (ψsd , ψsq )> (in Wb2 ), stator resistance Rs (in Ω), number of pole pairs np , mechanical angle φm (in rad) and speed ωm (in rad s ), inertia Θ (in kg m2 ), machine torque mm and turbine torque mt (both in N m), gear ratio gr and viscous ms friction coefficient νf (in Nrad ). The stator flux linkage vector  d    ψpm Ls Lm k k k (3) ψ s (is ) = is + 0 Lm Lqs | {z } | {z } k

k

k >

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=:Ls =(Ls ) >0

=:ψpm

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is assumed to be an affine2 function of the stator current vector iks and the permanent-magnet k flux linkage vector ψpm (in Wb2 ). The inductance matrix Lks with stator inductances Lds > 0, q Ls > 0 and mutual (cross-coupling) inductance Lm 6= 0 is symmetric and positive-definite (i.e. Lds Lqs − L2m > 0 [17]). If the anti-diagonal elements Lm of the inductance matrix Lks are non-zero, a change in the d-axis current ids imposes a change in the q-axis flux linkage ψsq and vice versa. This effect is referred to as magnetic cross-coupling. Moreover, if the main-diagonal elements Lds and Lqs have different values, i.e. Lds − Lqs 6= 0 H, the (three-phase) stator flux linkage depends on the actual rotor position/angle, and the machine is said to be anisotropic. Fig. 2 illustrates the third quadrant (generator mode) of the flux linkage (3) of the simulated PMSM (see Sec. 3). Due to magnetic cross-coupling, the flux maps are slightly tilted. The electro-mechanical torque for such an anisotropic machine is given by3
mm (iks ) = 23 np (iks )> J ψ ks (iks ) = 32 np ψpm iqs + 32 np (Lds − Lqs )ids iqs + 23 np Lm (iqs )2 − (ids )2 , | {z } | {z } | {z } (i)

(ii)

(4)

(iii)

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and can be split into three components: (i) permanent magnet torque, (ii) reluctance torque, and (iii) torque due to magnetic cross coupling (which is usually neglected [10]). Clearly, there exist infinitely many current vectors iks to produce the same torque (see (4) and/or Fig. 3). 2.2. Problem formulation The MTPA optimization problem can be formulated as: Find the minimum current magnitude kiks k2 = (iks )> iks for a given reference torque mm,ref (see e.g. mm,ref as in (17) in regime II), i.e. iks,ref = MTPA(mm,ref , Lds , Lqs , Lm , ψpm , . . . ) := arg minik kiks k2 such that mm (iks ) = mm,ref . s

2

k Ls

d

q

Note that a constant matrix is considered. Most publications deal with constant inductances Ls and Ls while the cross-coupling inductance Lm is neglected [9, 10, 12]. Future work will consider nonlinear flux linkages. 3 For the sake of readability, the time-dependency will be omitted in the following. The factor 3/2 is due to an amplitude-correct Clarke transformation [18].

ψsq / Wb

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ψsd / Wb

0,2 0,1

0 0

0 −0,1 −0,2 0

0

−20

−40 −40

iqs / A

ids

0

−20

−20

−20

−40 −40 iqs / A

/A

ids / A q

d

(b) q-component ψs of the stator flux linkage.

(a) q-component ψs of the stator flux linkage.

Figure 2: Flux linkage maps (3) of an anisotropic PMSM with parameters as in (16).

Due to the equality constraint, the optimization problem can be rewritten as Lagrangian   max(ik ,κ) L(iks , κ) where L(iks , κ) := −(iks )> iks + κ mm (iks ) − mm,ref

(5)

s

(3),(2)

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with Lagrangian multiplier κ ∈ C \ {0}. Clearly, (iks )> J Lks iks =   k k > > k and (iks )> J ψpm = (ψpm ) J is . Then, by defining I 2 := 10 01 , "

T :=

k 3 4 np (J Ls

+

Lks J > )

=

3 2 np

d

q

−Lm

Ls −Ls 2

Ls −Ls 2

Lm

d

q

(4)

=T

= (6)

=

>

(4),(6)

and t :=

k 3 4 np J ψpm

 + Lks J > ) iks

 =

0



3np ψpm 4

, (6)

(iks )> T iks + 2t> iks and the Lagrangian

=

  −(iks )> I 2 iks + κ 32 np (iks )> J ψ ks iks − mm,ref h i k > > k ) J is − κmm,ref (iks )> −I 2 + κ 34 np (J Lks + Lks J > ) iks + κ 32 np (ψpm

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=

k 2 (J Ls

1

#

the machine torque can be expressed as mm (iks ) in (5) can be rewritten as quadric as follows L(iks , κ)

(iks )>

  (iks )> − I 2 + κ T iks + κ 2t> iks − κ mm,ref .

(7)

g L (iks , κ)

:=

dL(iks , κ)

!>

d(iks , κ)

=

d k dis d dκ

L(iks , κ)

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2.3. Analytical solution for the current references To find the maximum of (5), it is sufficient to (i) set the gradient of (7) to zero and find its ? (optimal) solutions in ik,? s and κ and (ii) check negative definiteness of the Hessian of (7). The gradient and the Hessian of (7) are given by !   2 − I 2 + κ T iks + κ 2t

!

L(iks , κ)

=

(iks )> T iks

and H L (iks , κ) respectively.

:=

d2 L(iks , κ) d(iks , κ)2

!>

" = 2

 − I2 + κ T ,

(iks )> T >

>

+t ,

+

T iks + t 0

2t> iks

− mm,ref

∈ R3

(8)

#

= H L (iks , κ)> ∈ R3×3 , (9)

(i) Setting the gradient (8) to zero, i.e. g L (ik,? s , κ) = 03 , and solving the first two rows for k ik,? = i yields s s #−1  "   −1 0 −1 − κ 23 Lm np κ 43 np (Lds − Lqs ) . (10) = − − I2 + κ T κt = − κ 3 κ 34 np (Lds − Lqs ) −1 + κ 32 Lm np 4 np ψpm

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ik,? s (κ)

Inserting ik,? s (κ) into the last row of the gradient (8) gives a fourth-order polynomial a4 κ4 + a3 κ3 + a2 κ2 + a1 κ + a0 = 0

(11)

in κ with five coefficients a0 , a1 , . . . , a4 ∈ R. The coefficients of (11) for the considered anisotropic PMSM model (2) depend on the machine parameters and are given by (details are omitted) a0 = −16mm,ref , a1 = −18n2p (ψpm )2 ,     2 81 2 2 d q 3 2 + Lm np (ψpm ) , a2 = mm,ref 36Lm np + 3np Ls − Ls a3 = 0, and 2   2   81 4 2 d q 2 2 2 d q 2 a4 = − np 2mm,ref 4Lm + (Ls − Ls ) + 3np (ψpm ) 4Lm + (Ls − Ls ) . 32

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2

p

4

s

s

m

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Note that the resulting quartic is a depressed quartic (since a3 = 0), which simplifies the calculation of its four roots κ?1 , . . . , κ?4 . The four roots depend on the coefficients ai , i = 0, . . . , 4 only (and, hence, on machine parameters and reference torque only) and can be computed analytically e.g. by invoking Ferrari’s method [16]. ? ? (ii) To obtain the optimal solution (ik,? s (κ ), κ ) of the optimization (maximization) problem (5), the Hessian matrix (9) must be negative definite. The root κ? ∈ {κ?1 , . . . , κ?4 }, which ? ? renders the Hessian H L (ik,? s (κ ), κ ) negative definite, is the optimal Lagrangian multiplier. Therefore, the Hessian matrix must be evaluated for all κ?i , i = 1, . . . , 4 and checked for negative definiteness. This can be done by employing Sylvester’s criterion [19, Prop. 8.2.8] and [19, Prop. 2.7.1]4 : First and third leading principle minor of (9) must be negative whereas the second leading principle minor must be positive, which is the case if (for details see the Appendix)  =⇒ −1 − 32 np Lm κ?i < 0  (a) κ?i > − 3 1  2 np Lm   ? ? 1 (12) r  (b) |κi | < =⇒ det − I 2 + κi T > 0.   3 d q 2 2  1 n L −L +L Both conditions (a) and (b) must be satisfied simultaneously, hence the following must hold −

1 3 2 np Lm

< κ? := κ?i
T iks + 2 t> iks = 0

where T and t are as in (6). Solving this quadric for iqs yields the explicit formula  d q d (Ls −Ls )is +ψpm   + −  2Lm  r   2mm,ref d q 2 2 d 2 d q d 2  [(Ls −Ls ) +4Lm ](is ) +2ψpm (Ls −Ls )is +4Lm 3p +ψpm d TRQ(is , mm,ref ) = + , Lm = 6 0 2Lm     mm,ref   , , Lm = 0 3  q d d

(14)

2 np ψpm +(Ls −Ls )is

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as function of ids which holds for all ids 6= ψpm /(Lds − Lqs ). Exemplary torque hyperbolas are plotted in Fig. 3 (see green lines). Note that, for Lm = 0 and Lds = Lqs (isotropic case), the torque hyperbola becomes a horizontal line (see Fig. 3a). To derive an expression for the MTPA hyperbola, in the first two rows of the gradient (8)   (i.e. 2 − I 2 + κ T iks + κ 2t = 02 ), the Lagrangian multiplier κ can be eliminated. Solving this for κ and rearranging gives the implicit expression of the MTPA hyperbola as follows ( " ) # d q 3 3
n (L − L ) n L s 2 p m MTPA := iks ∈ R2 (iks )> 4 p3 s iks + 34 np ψpm , 0 iks = 0 . d q 3 n L − n (L − L ) s s 2 p m 4 p Solving this quadric for iqs yields the explicit expression of the MTPA hyperbola d

q

Ls −Ls

 d (is = 0)

, Lds 6= Lqs

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MTPA(ids ) =

q   2Lm ids ± [(Lds−Lqs )2 +4L2m ](ids )2 +(Lds−Lqs )ψpm ids

(15)

, Lm = 0 ∧ Lds = Lqs .

In Fig. 3b, it can be clearly seen that, for a machine with Lm 6= 0, neglecting the magnetic cross-coupling will have a significant effect on the shape of the MTPA hyperbola and, hence, on the current reference computation. The computed reference will not have a minimal magnitude.

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3. Simulation results For the upcoming simulations, a small-scale wind turbine system (SS-WTS) with a 17.7 kW PMSM is considered. The PMSM has rated torque mm,rated = 49.3 N m, rated speed ωm,rated = 360 rad s and the following electrical parameters Lds = 3.5 mH, Lqs = 1.5Lds , Lm = 0.15Lds , ψpm = 0.2 Wb, np = 3 and Rs = 0.12 Ω. Three different MTPA strategies are implemented:

(16)

• the proposed MTPA considering anisotropy and cross-coupling (i.e. Lds 6= Lqs and Lm 6= 0, see solid blue line ( ) in Fig. 4 and 5), • the standard MTPA(Lm = 0) considering anisotropy but neglecting cross-coupling (i.e. Lds 6= Lqs and Lm = 0, see dashed blue line ( ) in Fig. 4 and 5) [10], and

-0.75

ids / p.u. -0.5

-0.25

0

-1 0

mm = −0.25 mm,rated

ids / p.u. -0.75 -0.5 mm = −0.25 mm,rated

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0

-0.5 mm = −1.0 mm,rated

-0.75

mm = −1.0 mm,rated

MTPA(Lm = 0)

kiks k = imax

kiks k = imax

-1

k

-0.25

MTPA mm = −0.75 mm,rated

iqs / p.u.

-0.5

mm = −0.75 mm,rated

0

mm = −0.5 mm,rated

-0.25

mm = −0.5 mm,rated

-0.25

d

iqs / p.u.

-1

-0.75

-1 k

(a) Isotropic machine with mm (is ) = 23 ψpm is .

(b) Anisotropic machine with mm (is ) as in (4). d

q

Figure 3: Generator mode current loci of an isotropic machine (with Ls = Ls and Lm = 0) and an anisotropic machine with parameters as in (16). The axes are normalized with respect to the maximal admissible current imax (in A).

• the simple MTPA(ids = 0) usually used for isotropic machines with ids,ref = ids = 0 neglecting ) in anisotropy and cross-coupling (i.e. Lds = Lqs and Lm = 0, see dotted red line ( Fig. 4 and 5) [13].

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Simulations have been conducted in Matlab/Simulink (R2016a) to compare the performance and the effects of the MTPA strategies on the efficiency of the electrical drive system and the overall small-scale wind turbine system. Moreover, the proposed MTPA considering anisotropy and cross-coupling was implemented by using the analytical solution and a conventional numerical solution (Newton-Raphson). Both solutions match with high accuracy (deviation < 10−26 ), but the analytical solution was obtained 20 times faster than the numerical one. In the remainder, only the analytical solutions for the three MTPA strategies are implemented and discussed. Remark 2: Note that the proposed algorithm is applicable to any PMSM, PME-RSM or EESM with non-negligible anisotropy and magnetic cross-coupling independently of the machine’s power rating. Small-scale machines tend to have a higher ratio of reluctance torque to permanent magnet torque (represented by (i) and (ii) in (4), respectively) and, therefore, the application to small-scale wind turbine systems is discussed, since it illustrates better the efficiency gain of the proposed analytical MTPA strategy.

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3.1. Current loci and efficiency improvement of the electrical drive system Fig. 3 shows the current loci (third quadrant/generator mode) of an isotropic PMSM (see Fig. 3a) and an anisotropic PMSM with parameters as in (16) (see Fig. 3b). Anisotropy and magnetic cross-coupling are not negligible and should be considered in the MTPA strategy. Neglecting Lm leads to an incorrect computation of the reference currents (see Fig. 3b) and, hence, higher 2 and losses (see Fig. 4b). For copper losses PCu = 32 Rs kiks k2 , friction losses Pfric = νf ωm k 5 mechanical power Pmech = mm (is )ωm , the efficiencies η = (Pmech − PCu − Pfric )/Pmech = 1 − (PCu + Pfric )/Pmech are compared in generator mode for different speeds ωm using the simple MTPA(ids = 0), the conventional MTPA(Lm = 0) and the proposed MTPA (considering cross-coupling). As illustrated in Fig. 4, considering anisotropy and cross-coupling allows for a significant efficiency improvement differing for different machine speeds. The efficiency increase compared to the simple MTPA(ids = 0) is measured by the relative efficiency gain (see Fig. 4c) ∆η := η(X) − η(MTPA(ids = 0)) = 5

d

PCu (MTPA(is =0))−PCu (X) Pmech

with X ∈ {MTPA, MTPA(Lm = 0)}.

Iron losses are not considered. For WTS, the angular speed will (in most cases) not exceed the rated speed.

0.98

8

0.96

0.96

MTPA MTPA(Lm = 0) MTPA(ids = 0)

0.94

0.92

0.92

MTPA MTPA(Lm = 0) MTPA(ids = 0)

0.88

0.25

0.5 mm / p.u.

0.75

1

(a) Current magnitude.

0

4

= 25% = 50% = 75% = 100%

2

0.84

0

ωm ωm,rated ωm ωm,rated ωm ωm,rated ωm ωm,rated

6 ∆η / %

PCu / PCu (MTPA(ids = 0))

1

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kiks k / kiks (MTPA(ids = 0))k

1

0

0.25

0.5 mm / p.u.

0.75

1

(b) Copper losses.

0

0.25

0.5 mm / p.u.

0.75

1

(c) Efficiency improvement.

d MTPA(is

Figure 4: Comparison of MTPA, MTPA(Lm = 0) and = 0) for parameters as in (16) (generator mode). The horizontal axes are normalized with respect to the rated machine torque mm,rated (in N m).

2 mm,ref = −kp? ωm + νf ωm −

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3.2. Overall small-scale wind turbine system (SS-WTS) with anisotropic PMSM Now, three identical SS-WTSs with anisotropic PMSM (2) parametrized as in (16), gear ratio ms are implemented gr = 14.7, inertia Θ = 0.199 kg2 and viscous friction coefficient νf = 5 · 10−3 Nrad m in Matlab/Simulink. The SS-WTS has turbine power Pt = cp (λ)Pw (in W) and turbine torque 3 (in W), % = 1.293 kg3 , rt = 3.38 m and vw (in ms ) are mt = Pt ωgr [18] where Pw = 12 %πrt2 vw m m wind power, air density, turbine radius and wind speed, respectively. The power coefficient cp (·) from [20, Example III.2] with maximum c?p := cp (λ? ) = maxλ cp (λ) = 0.441 and optimal tip speed ratio λ? = 6.91 is used. For maximum power point tracking (MPPT) in region II (i.e. vw ∈ [5, 12] ms for the simulations), the nonlinear turbine speed controller [18, 21] Θv −Θ Tv ξ

where kp? :=

?

5 %rt π cp (λ ) 3 ? 3 2gr (λ )

 and ξ(s) = 1 −

1 1+s Tv

 ωm (s) (17)

?

∆Pext :=

(cp (λ)Pw −PCu −Pfric )−cp Pw ? cp P w

and

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with inertia compensation6 [21, p. 482] and (viscous) friction feedforward compensation is implemented for all three SS-WTS. The underlying current control loops [17] are identical for all three SS-WTS. The only difference between the three SS-WTS is that the reference torque mm,ref (controller output) is fed to one of the three introduced MTPA strategies (see above). The simulation results are depicted in Fig. 5 (line colors are as introduced above). The wind speed vw (·) varies within the interval [7, 12] ms . Only the SS-WTS with the proposed MTPA (considering cross-coupling) achieves maximum power point tracking (MPPT) at steady state, since λ → λ? for all wind speeds (see second subplot in Fig. 5). For the SS-WTS with MTPA(ids = 0) and MTPA(Lm = 0), the speed controller (17) does not guarantee MPPT; not even at steady state. Due to the incorrect current reference computation, the produced machine torque mm 6= mm,ref differs from the reference torque (17) (see also [22]) and λ → λ? can not be established. Hence, the maximum turbine power c?p Pw can not be extracted, i.e. Pt = cp (λ)Pw < c?p Pw . The relative extracted power ∆Pext (normalized with respect to the maximum turbine power c?p Pw ) and the relative torque deviation ∆mm (normalized with respect to the reference torque (17)) of each SS-WTS can be computed by ∆mm :=

mm −mm,ref , mm,ref

respectively. ∆mm and ∆Pext are shown in the third and fourth subplot of Fig. 5, respectively. For example, for the SS-WTS with MTPA(ids = 0) at vw = 12 ms , the relative torque deviation can reach more than 15% with a relative loss in extracted power of almost 8.5% . 6

2

d Note that ξ ≈ Tv dt ωm approximates the time derivative of the machine speed and Θv < Θ (in kg m ) is the desired reduced (virtual) inertia of the WTS. For the simulations, Θv = Θ/4 and Tv = 0.2 s were chosen.

vw /

m s

12 10 8

λ/1

rk Wo 7.5 7.25 7 6.75 6.5 6.25 6

λ∗

∆mm / %

0

-5

-10

∆Pext / %

-15

0

5

MTPA

10

MTPA (ids = 0)

MTPA (Lm = 0)

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-6 -7 -8 -9 -10 -11

15

20

25

time t / s

Figure 5: Simulation results for a SS-WTS with anisotropic PMSM (parametrized as in (16)) operated in region II (maximum power point tracking) under (step-like) varying wind speeds vw (·) ∈ [7, 12] ms .

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4. Conclusion In this paper, an analytical solution for the Maximum-Torque-per-Ampere (MTPA) optimization problem has been derived for permanent-magnet synchronous machines with anisotropy and magnetic cross-coupling. The proposed MTPA strategy has been applied to a small-scale wind turbine system. The main contributions of this paper have been (i) the derivation of the analytical solution considering magnetic cross-coupling (which has not been done in literature before) and (ii) performance and efficiency comparisons with both, the conventional method (with zero direct current) and the state of the art MTPA strategy (neglecting magnetic cross-coupling). It has been shown that including the magnetic cross-coupling into the optimization problem can significantly increase the efficiency of the system (copper losses are reduced). Moreover, the analytical solution allows for an almost instantaneous computation of the optimal reference currents. Compared to the numerical Newton-Raphson method, the analytical solution converged to the correct solution 20 times faster. The proposed analytical MTPA strategy and its reference current computation can be adopted within already existing small-scale or large-scale wind turbine systems (WTSs) without any further modifications. Independently of the machine’s power rating, the proposed solution can be applied to any permanent-magnet synchronous machine (PMSM), permanent-magnet-excited reluctance synchronous machine (PME-RSM) or electrically-excited synchronous machine (EESM) with non-negligible anisotropy and magnetic cross-coupling (see also Remark 2). Future work will focus on the extension of the analytical solution to nonlinear flux maps (i.e., when the inductances are functions of the stator currents). Acknowledgement This work is supported by the EU ITN AWESCO project funded by the European Union Commission under Marie Sklodowska-Curie ITN H2020 Ref.: 642682.

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Appendix Note that, by defining " #  −1 k I , − I2 + κ T (T is + t) M (iks ) := >2 ∈ R3×3 02 , 1

with

det M (iks ) = 1

and by invoking [19, Fact 2.16.2], the Hessian matrix    − I2 + κ T , 02 k k >
 −1
M (iks ) H L (is , κ) = 2M (is ) 0> − (iks )> T > + t> − I 2 + κ T T iks + t 2, can be written as product of three matrices. Hence,    
 −1
det H L (iks , κ) = −2 (iks )> T > + t> − I 2 + κ T T iks + t · det − I 2 + κ T , {z } |

(18)

=:α∈R

  (12) which, with = = 1 and det − I 2 + κ T > 0, implies that α =  −1 kT iks + tk2Q > 0 (a weighted norm with positive definite Q := − I 2 + κ T > 0) is positive k k k,? k,? for all non-zero vectors T is + t 6= 02 . Inserting is = is (κ) with is (κ) as in (10) yields det M (iks )

(10)

ing

T ik,? s (κ) + t

det M (iks )>

=

 −1  −1
−κT − I 2 + κ T t + t = − − I 2 + κT − I 2 + κ T t   −1  −1
− − − I2 + κ T − I2 + κ T + κT − I 2 + κ T t  
  −1  − I2  −κ T + κ T − I2 + κ T t

=

 −1 (12) − − I2 + κ T t 6= 02 for all κ = κ? with κ? as in (12).

=

=

Concluding, the third leading principle minor of the Hessian (9) is negative7 , i.e.,     det H L (iks , κ) < 0, for a positive second leading principle minor, i.e., det − I 2 + κ T > 0.

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Interesting to note that the third principle minor is negative in any case (due to (18)); even if the second leading    −1 principle minor is negative, i.e. − I 2 + κ T < 0 ⇐⇒ − I 2 + κ T < 0.

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