Co-Design Optimization of Direct Drive Pmsgs for Offshore Wind Turbines Based on Wind Speed Profile

energies

Article Co-Design Optimization of Direct Drive PMSGs for Offshore Wind Turbines Based on Wind Speed Proﬁle

Linh Dang, Serigne Ousmane Samb, Ryad Sadou and Nicolas Bernard *

IREENA, Nantes University, CRTT 37 Bd de l’Université, 44600 Saint-Nazaire, France; [email protected] (L.D.); [email protected] (S.O.S.); [email protected] (R.S.) * Correspondence: [email protected]

Abstract: This paper presents a new method to optimize, from a working cycle defined by torque and speed profiles, both the design and the control strategy of permanent magnet synchronous generators (PMSGs). The case of a 10 MW direct-drive permanent magnet generator for an Offshore wind turbine was chosen to illustrate this method, which is based on the d–q axis equivalent circuit model. It allows to optimize, with a reduced computation time, the design, considering either a flux weakening control strategy (FW) or a maximum torque per Ampere control (MTPA) strategy, while respecting all the constraints—particularly the thermal constraint, which is characterized by a transient regime. The considered objective is to minimize the mass and the average electric losses over all working points. Thermal and magnetic analytical models are validated by a 2D finite element analysis (FEA).

Keywords: PMSG; co-design optimization; flux weakening control; MTPA control; wind speed profile; offshore wind energy Citation: Dang, L.; Ousmane Samb, S.; Sadou, R.; Bernard, N. Co-Design Optimization of Direct Drive PMSGs for Offshore Wind Turbines Based on 1. Introduction Wind Speed Profile. Energies 2021, 14, Offshore wind generation has taken an increasingly important place in the European 4486. https://doi.org/10.3390/ wind power development, in recent years. It presents high availability, stable wind speed en14154486 and less environmental constraints. In order to reduce the costs, increasing the turbine power is a strong trend. However, it leads to increase in active and structural masses, Academic Editor: Davide Astolfi which are limited by technology, transport and installation. Therefore, maximizing the power density is a crucial criterion in the design process. Received: 18 June 2021 In that case, variable-speed wind turbines with pitch control are used to optimize the Accepted: 21 July 2021 Published: 24 July 2021 turbine output power [1,2]. Generally, the working cycle is not taken into account in the design process. In most cases, the generator is only designed for the rated power [3–6].

Publisher’s Note: MDPI stays neutral Such a method can lead to oversize the generator, particularly when it works in a variable with regard to jurisdictional claims in thermal regime. On the other hand, the maximization of the energy efficiency, that can be published maps and institutional affil- achieved by a flux weakening mode, for example [7], implies that all working points have iations. to be taken into account [8]. One of the most important issues in a design process which considers several thou- sands of working points is that, in addition to optimizing the geometric parameters, it must also optimize the time-dependent control parameters id(t) and iq(t) while respecting all the constraints in each point, leading to a huge computation time. To overcome this Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. problem, the solutions currently proposed in the electrical engineering literature limit the This article is an open access article optimization problem to the most representative working points [9–11], which makes the distributed under the terms and result approximate, because the control strategy, as well as the thermal transient, is not conditions of the Creative Commons managed. Attribution (CC BY) license (https:// The aim of this paper is to present an optimal design methodology to solve this creativecommons.org/licenses/by/ problem. The proposed method allows to optimize simultaneously the geometry as well as 4.0/). the control parameters (id(t) and iq(t)) of each working point for the following two cases:

Energies 2021, 14, 4486. https://doi.org/10.3390/en14154486 https://www.mdpi.com/journal/energies Energies 2021, 14, x FOR PEER REVIEW 2 of 17

Energies 2021, 14, 4486 The aim of this paper is to present an optimal design methodology to solve this prob-2 of 17 lem. The proposed method allows to optimize simultaneously the geometry as well as the control parameters (𝑖푑(푡) and 𝑖푞(푡)) of each working point for the following two cases: a maximum torque per Ampere (MTPA) control with 𝑖푑(푡) = 0 and a flux weakening (FW) a maximum torque per Ampere (MTPA) control with i (t) = 0 and a flux weakening (FW) control with 𝑖 (푡) = 𝑖 (푡) ≠ 0. The case of a 푞 −phasesd 10 MW direct-drive surface control with 푑i (t) =푑i 표푝푡 (t) 6= 0. The case of a q—phases 10 MW direct-drive surface mounted PMSGd was chosend opt to illustrate our study, with an offshore wind speed profile mounted PMSG was chosen to illustrate our study, with an offshore wind speed profile measured in the North Sea and the two following objective functions: mass minimization measured in the North Sea and the two following objective functions: mass minimization and energy loss minimization. and energy loss minimization. The paper is organized as follows. In Section 2, the principle of the design methodol- The paper is organized as follows. In Section2, the principle of the design methodology is presented and, in Section 3, the sizing model and the constraints used are given. In ogy is presented and, in Section3, the sizing model and the constraints used are given. In Section 4, the results are presented and discussed. Finally, the selected optimal machine Section4, the results are presented and discussed. Finally, the selected optimal machine is isvalidated validated by by the the use use of of a a magnetic magnetic and and thermal thermal 2D 2D finite finite element element analysis analysis (FEA). (FEA). AtAt last, last, let let u uss note thatthat aa first first presentation presentation of of the the methodology methodology was was partially partially presented pre- sentedat the at International the International Conference Conference on Electrical on Electrical Machines Machines ICEM ICEM 2020 2020 [12 ],[12] where, where only only the theFW FW control control was was considered. considered. The The articlearticle proposesproposes a more complete complete version version,, where where the the twotwo controls controls (FW (FW and and MTP MTPA)A) are arestudied studied and andcompared. compared. The mecha The mechanicalnical constraints constraints con- sideredconsidered are also are more also more realistic. realistic.

2.2. Optimal Optimal Design Design Methodology Methodology TheThe presented presented methodology methodology is is based based on on the the d d–q–q axis axis equivalent equivalent circuit circuit model model taking taking ironiron losses losses into into account account via via the the iron iron loss loss resistance resistance ℛR휇µ(푡()t) (see(see Figure Figure 1)[) [1313,,1414]]..

(a) (b)

FigureFigure 1. 1. (a()a d)- d-axisaxis equivalent equivalent circuit circuit;; (b ()b q)- q-axisaxis equivalent equivalent circuit circuit..

TheThe optimization optimization parameters parameters can can be be categorized categorized into into three three groups groups as as follows follows::

• • TheThe time time-dependent-dependent control control variables variables are the d–qd–q axis currentscurrents ioq𝑖표푞(t()푡)and andiod 𝑖(표푑t)(.푡 In). ourIn ourmethod, method, they they will will be be analytically analytically expressed expressed for for the the two two control control strategies strategies considered. consid- • eredThe. rotor variable Bf m, the magnitude of the air-gap ﬂux density created by the ( ) • Themagnets rotor variable represented 퐵푓푚 in, the the magnitude circuit via the of the electromotive air-gap flux force densitye0 t created. It will by be optimizedthe mag- analytically to minimize the energy losses for the considered working cycle. Note that nets represented in the circuit via the electromotive force 푒0(푡). It will be optimized analyticallythe magnets to are minimize sized (shape the energy and remanence) losses for afterwards,the considered from workingBf m opt. cycle. Note • thatThe the stator magnets geometry are sized variables (shape are andR remanence), rs, rw, wmag afterwards,, ns, τLR pfromand 퐵q푓푚(see 표푝푡. Figure 2), which are in the expressions of coefﬁcient kφ, armature reactance (X(t)) and resistance Energies 2021, 14, x FOR PEER REVIEW• The stator geometry variables are 푅, 푟푠, 푟푤, 푤푚𝑎푔, 푛푠, 휏퐿푅 푝 and 푞 (see Figure3 of2) 17, (R , R (t)) (see Figure1). These parameters will be optimized by the use of a genetic whichc areµ in the expressions of coefficient 푘휙 , armature reactance (푋(푡)) and re- algorithm to minimize both the mass and the energy losses. sistance (ℛ푐, ℛ휇(푡)) (see Figure 1). These parameters will be optimized by the use of a genetic algorithm to minimize both the mass and the energy losses.

Figure 2. Design and geometric parameters of the PMSG. Figure 2. Design and geometric parameters of the PMSG. The proposed optimization method is performed in three steps. The proposed optimization method is performed in three steps. In step 1, for the FW control, the optimal current 𝑖표푑 표푝푡(푡) that minimizes the total electric losses 푃푡표푡 for each working point is expressed analytically. For the MTPA control, this current is zero. For both of these controls, the 푞-axis current is directly imposed by the torque. The analytical expressions of the currents allow to express 푃푡표푡 as a function 표 of the other optimization parameters, so that 푃푡표푡 = 푓 (푅, 푟푠, 푟푤, 푤푚𝑎푔, 휏퐿푅, 푛푠, 푝, 푞, 퐵푓푚). In step 2, from the previous expression of 푃푡표푡, the optimal flux density 퐵푓푚 표푝푡 that minimizes the energy losses 푊푡표푡 is analytically expressed, allowing to express 푊푡표푡 as a function of the remaining optimization parameters as 푊푡표푡 = 표 푓 (푅, 푟푠, 푟푤, 푤푚𝑎푔, 휏퐿푅, 푛푠, 푝, 푞). In step 3, a genetic algorithm is used to minimize both the energy losses 푊푡표푡 = 표 푓 (푅, 푟푠, 푟푤, 푤푚𝑎푔, 휏퐿푅, 푝, 푞) obtained in step 2 and the mass of the generator. Let us note that, with the elimination of the time-dependent optimization parameters 𝑖표푞(푡) and 𝑖표푑(푡) in the objective function 푊푡표푡, the computation time is significantly reduced to an acceptable value (a few minutes), while it would have been of several months otherwise.

2.1. Basics Equations Due to the high inertia of the turbine, speed and torque variations are very slow. Thus, it is possible to neglect the terms in d/dt, which means that, from a sizing point of view, the machine operation can be seen as a succession of static points. Then, the main equations, such as the d–q axis stator voltages (푣표푑(푡), 푣표푞(푡)) and the electromagnetic power (푃푒푚(푡)), can be expressed as

푣표푑(푡) = 푋(푡)𝑖표푞(푡) (1)

푣표푞(푡) = −푋(푡)𝑖표푑(푡) + 푘휙Ω(푡) (2)

푃푒푚(푡) = 𝑖표푞(푡) 푘휙Ω(푡) (3)

The expression of the copper losses 푃푐(푡) is given by 2 2 푃푐(푡) = ℛ푐(𝑖푑(푡) + 𝑖푞(푡)) (4) with

푣표푑(푡) 𝑖푑(푡) = 𝑖표푑(푡) − 𝑖휇푑(푡) = 𝑖표푑(푡) − (5) ℛ휇(푡)

푣표푞(푡) 𝑖푞(푡) = 𝑖표푞(푡) − 𝑖휇푞(푡) = 𝑖표푞(푡) − (6) ℛ휇(푡)

From (1)–(6), the copper losses, for a given power 푃푒푚 and a given speed 훺, can be expressed as follows:

Energies 2021, 14, 4486 3 of 17

In step 1, for the FW control, the optimal current iod opt(t) that minimizes the total electric losses Ptot for each working point is expressed analytically. For the MTPA control, this current is zero. For both of these controls, the q-axis current is directly imposed by the torque. The analytical expressions of the currents allow to express Ptot as a function of the o other optimization parameters, so that Ptot = f R, rs, rw, wmag, τLR, ns, p, q, Bf m . In step 2, from the previous expression of Ptot, the optimal ﬂux density Bf m opt that minimizes the energy losses Wtot is analytically expressed, allowing to express Wtot as a function o of the remaining optimization parameters as Wtot = f R, rs, rw, wmag, τLR, ns, p, q . In step 3, a genetic algorithm is used to minimize both the energy losses Wtot = o f R, rs, rw, wmag, τLR, p, q obtained in step 2 and the mass of the generator. Let us note that, with the elimination of the time-dependent optimization parameters ioq(t) and iod(t) in the objective function Wtot, the computation time is signiﬁcantly reduced to an acceptable value (a few minutes), while it would have been of several months otherwise.

2.1. Basics Equations Due to the high inertia of the turbine, speed and torque variations are very slow. Thus, it is possible to neglect the terms in d/dt, which means that, from a sizing point of view, the machine operation can be seen as a succession of static points. Then, the main equations, such as the d–q axis stator voltages (vod(t), voq(t)) and the electromagnetic power (Pem(t)), can be expressed as vod(t) = X(t)ioq(t) (1)

voq(t) = −X(t)iod(t) + kφΩ(t) (2)

Pem(t) = ioq(t) kφΩ(t) (3)

The expression of the copper losses Pc(t) is given by

2 2 Pc(t) = Rc id(t) + iq(t) (4)

with vod(t) id(t) = iod(t) − iµd(t) = iod(t) − (5) Rµ(t)

voq(t) iq(t) = ioq(t) − iµq(t) = ioq(t) − (6) Rµ(t)

From (1)–(6), the copper losses, for a given power Pem and a given speed Ω, can be expressed as follows:

2 X (t) 2 2Rc X(t)kφΩ(t) Pc(t) = Rc 1 + 2 i (t) − 2 iod(t) + ... Rµ(t) od Rµ(t) (7) 2 2 2 X(t)Pem(t) Pem(t) kφΩ(t) 2Pem(t) +Rc + + − kφΩ(t)Rµ(t) kφΩ(t) Rµ(t) Rµ(t) For the iron losses, we have:

2 2 vod(t) + voq(t) Pmg(t) = (8) Rµ(t)

From (1), (2) and (8) it is possible to write

2 2 2 2 2 ! X (t) 2X(t)kφΩ(t) kφΩ (t) X (t)P (t) P (t) = i2 (t) − i (t) + + em (9) mg R (t) od R (t) od R (t) 2 2 µ µ µ kφΩ (t) Energies 2021, 14, 4486 4 of 17

2.2. Analytical Expressions of d- and q-Axis Currents

The optimal currents allow the generator to satisfy the requested power Pem(t) and speed Ω(t). For a surface mounted PMSG, the q-axis current ioq opt(t) is directly imposed by the electromagnetic power. According to (3), whatever the control strategy (MTPA or ﬂux weakening controls), this current is expressed as follows:

Pem(t) ioq opt (t) = (10) kφΩ(t)

In an MTPA control, the d-axis current is zero:

iod MPTA (t) = 0 (11)

In an FW control, the d-axis current, that minimizes both the copper losses and the iron losses, can be analytically expressed. It’s possible to show that

kφΩ(t) X(t) B(t) i (t) = (12) od FW A(t)

where the terms A(t) and B(t) depend on the resistances and the reactance as follows:

X(t) 2 X2(t) A(t) = Rc + Rc + (13) Rµ(t) Rµ(t)

R + R (t) ( ) = c µ B t 2 (14) Rµ(t)

2.3. Analytical Expression of Bf mopt

The magnitude of the ﬂux density Bf m produced by the magnets in the air-gap is constant during the cycle. This parameter is then optimized by the minimization of the energy losses over the cycle. With kφ = kBf m (see (21)), it is possible to express optimal expression of the magnet ﬂux density. In an MTPA control, since (7), (9), (10) and (11), the lost energy can be written as

Z T B(t) 1 Z T A(t)P2 (t) Z T P (t) W = k2 B2 Ω2(t)dt + em dt − 2R em dt (15) tot MPTA φ f m R ( ) 2 2 2 c R ( ) 0 µ t kφBf m 0 Ω (t) 0 µ t Then, the ﬂux density that minimizes the energy losses for the MTPA control is

1  ( ) 2 ( )  4 R T A t Pem t dt 1 0 Ω2(t) B =   (16) f m MPTA opt  T ( )  k R B t Ω2(t)dt 0 Rµ(t)

In an FW current control, since (7), (9), (10) and (12), the lost energy can be written as

! Z T B(t) B2(t)X2(t) 1 Z T A(t)P2 (t) Z T P (t) W = k2 B2 − Ω2(t)dt + em dt − 2R em dt (17) tot MPTA φ f m R ( ) ( )R2 ( ) 2 2 2 c R ( ) 0 µ t A t µ t kφBf m 0 Ω (t) 0 µ t

Then, the ﬂux density that minimizes the energy losses for the MTPA control is:

  1 ( ) 2 ( ) 4 R T A t Pem t dt 1  0 Ω2(t)  Bf m FW opt =   (18) k  R T B(t) B2(t)X2(t) 2  − 2 Ω (t)dt 0 Rµ(t) A(t)Rµ(t) Energies 2021, 14, 4486 5 of 17

2.4. Analytical Expression of Energy Losses Finally, in an MTPA control, given (15) and (16), the expression of the lost energy is

s s Z T B(t) Z T A(t)P2 (t) Z T P (t) = 2( ) em − R em Wtot MPTA 2 Ω t dt 2 dt 2 c dt (19) 0 Rµ(t) 0 Ω (t) 0 Rµ(t)

and in an FW control, given (17) and (18), the expression of the lost energy is v u ! s uZ T B(t) B2(t)X2(t) Z T A(t)P2 (t) Z T P (t) = − 2( ) em − R em Wtot FW 2t 2 Ω t dt 2 dt 2 c dt (20) 0 Rµ(t) A(t)Rµ(t) 0 Ω (t) 0 Rµ(t)

In Equations (19) and (20), the remaining optimization variables are the geometrical ones (R, rs, rw, wmag, τLR, ns, p, q). Such an expression can be thereby minimized by the use of a genetic algorithm without an excessive computation time.

3. Modeling We consider a q-phase surface mounted permanent magnet synchronous generator. A 1D magnetic model is used, with steel parts assumed inﬁnitely permeable. The winding is concentrated with one slot/pole/phase and the slots are assumed skewed by one slot pitch to reduce the torque ripple. The slot width to slot pitch ratio is 0.6. We also neglect losses in permanent magnets, assuming it is possible to reduce signiﬁcantly their impact by the use of segmented magnets. The design, with only one pole pair represented, is shown in Figure2.

3.1. Electromagnetic Model

The back-emf eO(t) is proportional to kφ which can be written as √ √ 2 kφ = kBf m = 2 2 q nsτLRrsR pBf m (21)

The iron loss resistance Rµ(t) can be deduced from (8); for q phases:

2 ( ) + 2 ( ) 2 vod t voq t qVm Rµ(t) = = (22) Pmg(t) 2Pmg(t)

where Vm is the magnitude of the voltage, which can be expressed from the magnitude of the resulting ﬂux density in the air-gap Brm as follows:

Vm = 4pnsΩRLBrm (23)

The iron losses (eddy currents + hysteresis) can be also written as a function of Brm, as described in [15], such as 2 Pmg(t) = Brmγ(t) (24) with ! (1 + r )r2 r2 − r2 r ( ) = 2 2( ) + ( ) w s + w s s 2 γ t πkad kec p Ω t kh pΩ t 2 LR (25) p (1 − rw) ktrw Thus, from (20)–(23), it follows that

8 2 1 pΩR Rµ(t) = qns τLR (26) π k (k pΩ(t) + k ) 2 2 2 ad ec h 1 rw−rs + 1−rw r r 2 2 kt w s p (1−rw) Energies 2021, 14, 4486 6 of 17

The winding resistance (Rc) of a q-phase machine can be written as ! 16 k p2 1 R = L 2 c q ns τLR 2 2 (27) π (1 − kt)k f σc rw − rs R

where kL is the coefﬁcient that corrects the active length due to the end windings and k f the slot ﬁll factor. For the synchronous reactance X(t) of a q-phase PMSG it is possible to show that 2 8 2 rsR X(t) = qτLRµ0ns pΩ(t) (28) π wag + wPM

3.2. Mass Calulcation

Only the mass of the active parts will be considered here. Mc, MFe and MPM are, respectively, the mass of the copper, the mass of the iron and the mass of the magnets. They are calculated as follows: 2 2 3 Mc = πτLR rw − rs kr R ρc (29)

2 2 2 3 MFe = πτLR rw − rs kt + 1 − rw R ρFe + ··· 2 (30) + (2rs + rw − 1)R − 2wag − 2wPM (1 − rw)πτLR R ρFe. W M = 2πR r R − W − PM W β τ ρ (31) PM s ag 2 PM PM LR PM 6 ◦ In the proposed design, the electric magnet pole arc βPM is set to 7 . 180 and the magnet thickness to airgap thickness ratio WPM/Wag is set to 7/3.

3.3. Thermal Constraint During operation, the hottest point in the machine must remain smaller than the maximum permissible temperature in the winding θmax, such as:

θw(t) ≤ θmax (32)

For each evaluated machine, the dynamic behavior of the temperature in the winding θw(t) is calculated from the lumped parameter thermal model represented in Figure3b [16,17] . In this study, the heat ﬂow is assumed unidirectional in the radial direction and each cylindrical element can be modeled by an equivalent circuit, as shown in Figure3a. The thermal resistance, as well as the thermal capacity, is calculated from the geometry and the thermal properties of the materials via (33)–(35). At the internal radius Ro and at the external radius R, the heat is extracted by convection with, respectively, hint = 10 (for a 2 natural convection) and hext = 100 W/m K (for air cooled convection). The time-dependent temperature at node i is evaluated with (36).

 2  Rext Rext 1 2 R ln R R =  int int − 1 (33) x1 2λ β L  2  mat mat Rext − 1 Rint   Rext 1 2ln R R = 1 − int  (34) x2 2λ β L  2  mat mat Rext − 1 Rint 1 C = C ρ β R2 − R2 L (35) x p 2π mat mat ext int

dθi θi − θi−1 θi − θi+1 Cx + + = P (36) dt Rx2 Rx1 where P is heat generated inside the element. EnergiesEnergies 20212021, 14, 14, x, FOR 4486 PEER REVIEW 7 of7 17of 17

휃 Material with conductivity 휆푚𝑎푡, 𝑎푚 material density 휌푚𝑎푡, specific heat 퐶푝 ℛ푐푦

훽 ℛ푦2 푚𝑎푡 1 푅 𝑖푛푡 ℛ푦1 2 퐶푦 푅푒푥푡 퐿 ℛ푤2 ℛ푡2

휃𝑖−1 휃푤 4 ℛ푥2 ℛ푤1 ℛ푡1 퐶푤 퐶푡 휃𝑖 ℛ푥1 ℛ푐푤 ℛ푐푡 퐶푥 휃𝑎푚 휃 휃𝑖+1 𝑎푚 (a)(a) (b)(b)

FigureFigure 3. 3. (a(a) )Thermal Thermal equivalent equivalent circuit circuit of of a a cylindrical cylindrical element; element; (b ) (b lumped) lumped parameter parameter thermal thermal modelmodel of of PMSG PMSG..

3.4. Saturation Constraint 푅 2 푅 The maximum ﬂux densities in the yoke 2 and( 푒푥푡 teeth) 푙푛 must( 푒푥푡 be) limited at the saturation 1 푅𝑖푛푡 푅𝑖푛푡 level, such as 푅푥1 = − 1 (33) 2휆 훽 퐿 푅 2 푚𝑎푡 푚𝑎푡 1 ( 푒푥푡) − 1 Btm(t) = [Brm ≤푅 Bsat ] (37) kt 𝑖푛푡 푅 rs 푒푥푡 Bym(t) = 1 Brm ≤2푙푛Bsat( ) (38) p(1 − rw) 푅𝑖푛푡 푅푥2 = 1 − (34) 2휆 훽 퐿 푅 2 푚𝑎푡 푚𝑎푡 ( 푒푥푡) − 1 3.5. Electrical Constraint [ 푅𝑖푛푡 ] We consider a voltage limit Vlimit 1imposed by the power electronics converter. This 퐶 = 퐶 휌 훽 (푅2 − 푅2 )퐿 (35) voltage depends on the topology푥 of the푝 2 power휋 푚𝑎푡 converter푚𝑎푡 푒푥푡 and the𝑖푛푡 voltage rating of the power semiconductor devices. The two-level, back-to-back voltage source converter (BTB 2L-VSC) 푑휃 휃 − 휃 휃 − 휃 is mostly used in the wind turbines𝑖 for powers𝑖 𝑖−1 up to𝑖 a few𝑖+1 megawatts [18]. However, in 퐶푥 + + = 푃 (36) the 10 MW range and above, the increase푑푡 ℛ in푥 voltage,2 currentℛ푥1 and losses (switching losses) whererequires 푃 anis heat increase generated in the number inside ofthe components element. and the number of levels. Among all the proposed converter topologies, the three-level active neutral-point diode clamped converter 3.4.(3L-NPC) Saturation is one Constraint of the most popular [19]. Without going into the optimization of the power converter, which will be the subject of a future work, we impose a maximum phase voltage of 3000The V,maximum which would flux bedensities authorized in the by yoke the use and of mediumteeth must voltage be limited IGBT transistorsat the saturation (up levelto 6.5, such kV) [as20 ]. For q-phases machines, the voltage limitation is formulated as 1 q퐵 (푡) = r퐵q ≤ 퐵 (37) V푡푚2 + V2 ≤푘 푟푚V 푠𝑎푡 (39) d q 푡 2 limit 푟푠 퐵푦푚(푡) = 퐵푟푚 ≤ 퐵푠𝑎푡 (38) 3.6. Mechanical Constraints 푝(1 − 푟푤) Due to severe mechanical stresses, the obtained designs must respect minimum yoke 3.5.and Electrical tooth thicknesses, Constraint or the stresses will be transferred to the structural components [21]. Today, these constraints are the main limitation in the scale-up in off-shore wind power and the manufacturersWe consider a dovoltage not communicate limit 푉푙𝑖푚𝑖푡 theseimposed data, by which the power are strategic. electronics As a consequence,converter. This voltagethe academic depends literature on the presents topology a lot of of the dispersion power converter in the proposed and the constraints voltage valuesrating andof the powerdesigns. semiconductor For example, devices. for a power The oftw 10o-level MW,, theback minimum-to-back voltage thickness source of the converter stator yoke (BTB 2LW-yVSC) min observed is mostly in used papers in the varies wind between turbines 14 mmfor powers and 109 up mm to and a few the megawatts minimum width[18]. How- of ever,the teeth in theW 10t min MWvaries range between and above, 15.3 mm the and increase 50 mm. in On voltage, the basis current of these and observations, losses (switching we losses)set these requires values atan40 increase mm and in 20 the mm, number respectively of components [22,23]. The andslot depth the number to tooth of width levels. Amongratio is all also the limited proposed to 8 [ 24converter]. Concerning topologies, the outer the radius three-Rlevel, setting active a maximum neutral-point value diode is clampedmore difﬁcult, converter especially (3L-NPC) because is one the of smallest the most radii popular do not [19] necessarily. Without lead going to theinto lowest the optimization of the power converter, which will be the subject of a future work, we impose a maximum phase voltage of 3000 V, which would be authorized by the use of medium

Energies 2021, 14, x FOR PEER REVIEW 8 of 17

voltage IGBT transistors (up to 6.5 kV) [20]. For 푞-phases machines, the voltage limitation is formulated as 푞 √푉2 + 푉2 ≤ √ 푉 (39) 푑 푞 2 푙𝑖푚𝑖푡

3.6. Mechanical Constraints Due to severe mechanical stresses, the obtained designs must respect minimum yoke Energies 2021, 14, 4486 and tooth thicknesses, or the stresses will be transferred to the structural components8 of 17 [21]. Today, these constraints are the main limitation in the scale-up in off-shore wind power and the manufacturers do not communicate these data, which are strategic. As a consequence, the academic literature presents a lot of dispersion in the proposed constraints masses, here [25]. According to [21], if the external diameter is too large, the stress on the valuesmechanical and structuredesigns. For becomes examp toole great., for aConsequently, power of 10 MW according, the minimum to the observed thickness values, of the stator yoke 푊 observed in papers varies between 14 mm and 109 mm and the mini- we will limit the푦 푚𝑖푛 space requirement by limiting the outer radius Rmax to 5 m. mum width of the teeth 푊푡 푚𝑖푛 varies between 15.3 mm and 50 mm. On the basis of these observation4. Design Optimizations, we set these values at 40 mm and 20 mm, respectively [22,23]. The slot depth to toothIn this width section, ratio theis also methodology limited to previously8 [24]. Concerning presented the is outer applied radius to design 푅, setting a direct- a max- imumdrive PMSGvalue is for more a 10 difficult, MW wind especially turbine because (see specifications the smallest given radii in do Table not 1necessarily). A wind lead tspeedo the lowest profile ofmasses 4500, points here [25] (one. According point every to 10 [21] min),, if the measured external in diameter the North is Sea too during large, the stressone month, on the will mechanical be considered structure (Figure becomes4)[ 26]. too The great. case ofConsequently, a three-bladed according pitch-regulated to the ob- servedvariable values, speed windwe will turbine limit is the considered space requirement in this study. by It limiting operates the at the outer maximum radius power푅푚𝑎푥 to 5 m.point between a cut-in wind speed of 2.5 m/s and a rated wind speed of 12.4 m/s. Above this speed, the maximum power is limited and kept constant. 4. Design Optimization Table 1. Specifications of the considered wind turbine. In this section, the methodology previously presented is applied to design a direct- drive PMSG for aParameters 10 MW wind turbine (see specifications given Values in Table 1). A wind speed profile of 4500 pointsBlade radius (one point every 10 min), measured in 82 the m North Sea during one month, will be Maximalconsidered power (Figure 4) [26]. The case of a three 10-bladed MW pitch-regulated variable speed windCut-in turbine speed is considered in this study. It operat 2.5es m/s at the maximum power Rated speed 12 m/s point betweenCut-out a cut- windin wind speed speed of 2.5 m/s and a rated wind 25 m/s speed of 12.4 m/s. Above this speed, the maximum power is limited and kept constant.

Energies 2021, 14, x FOR PEER REVIEW 9 of 17

FigureFigure 4. Wind speedspeed profile profile measured measured at at the the North North Sea Sea in January.in January. The speed and power profiles of the PMSG can be deduced from the wind speed TableThe 1. Specification speed ands power of the considered profiles of wind the PMSG turbine can. be deduced from the wind speed profileprofile and and the the specifications specifications of theof the wind wind turbine, turbine, considering considering the four the different four diff operationerent operation modes [[27]27] of of the the wind wind turbine,Parameters turbine as, representedas represented in Figure in Figure5. 5. Values Blade radius 82 m Maximal power 10 MW Cut-in speed 2.5 m/s Rated speed 12 m/s Cut-out wind speed 25 m/s

Figure 5.5.Typical Typical power power curve curve of aof pitch-controlled a pitch-controlled wind wind turbine. turbine.

In the second region, between v and the rated speed v , the maximum power In the second region, betweencut−in 푣 and the ratedrated speed 푣 , the maximum point tracking approach (MPPT) is adopted푐푢 in푡− order𝑖푛 to maximize the captured푟𝑎푡푒푑 power [28]. power point tracking approach (MPPT) is adopted in order to maximize the captured power [28]. In the third region, the pitch angle is regulated to limit the turbine output power. The rotor speed and the output power in this region are constants and equal to their rated values. Finally, the speed and torque profiles of the generator can be obtained; these are presented in Figure 6.

Figure 6. Speed-torque profiles for the generator.

The main constant parameters used for the optimization are summarized in Table 2. The optimization parameters are listed in Table3.

Table 2. Constant parameters.

Parameters Values

퐵푠𝑎푡 1.6 T 푉푙𝑖푚𝑖푡 2.5 kV 푊푡 푚𝑖푛 20 mm 푊푦 푚𝑖푛 40 mm

푘𝑎푑 2 푘푒푐 0.035 푘ℎ 30 2 ℎ𝑖푛푡 10 W/m k 2 ℎ푒푥푡 100 W/m k 휃푚𝑎푥 140 °C

Energies 2021, 14, x FOR PEER REVIEW 9 of 17

The speed and power profiles of the PMSG can be deduced from the wind speed profile and the specifications of the wind turbine, considering the four different operation modes [27] of the wind turbine, as represented in Figure 5.

Figure 5. Typical power curve of a pitch-controlled wind turbine. Energies 2021, 14, 4486 9 of 17

In the second region, between 푣푐푢푡−𝑖푛 and the rated speed 푣푟𝑎푡푒푑 , the maximum power point tracking approach (MPPT) is adopted in order to maximize the captured Inpower the third [28] region,. In the the third pitch region, angleis the regulated pitch angle to limit is theregulated turbine outputto limit power. the turbine The rotor output speedpower. and The the rotor output speed power and in th thise output region power are constants in this andregion equal are to constants their rated and values. equal to Finally,their rated the speedvalues. and Finally, torque the proﬁles speed of and the generatortorque profiles can be of obtained; the generator these are can presented be obtained; inthese Figure are6 .presented in Figure 6.

FigureFigure 6. 6.Speed-torque Speed-torque proﬁles profiles for for the the generator. generator.

TheThe main main constant constant parameters parameters used used for for the the optimization optimization are are summarized summarized in Table in Table2. 2. TheThe optimizationoptimization parameters parameters are are listed listed in Tablein Table3.3.

TableTable 2. 2.Constant Constant parameters. parameters.

ParametersParameters Values Values

퐵푠𝑎푡 Bsat 1.61.6 T T Vlimit 2.5 kV 푉푙𝑖푚𝑖푡 2.5 kV Wt min 20 mm 푊푡 푚𝑖푛 20 mm Wy min 40 mm 푊푦 푚𝑖푛k ad 40 mm2 k 푘𝑎푑 ec 20.035 kh 30 푘푒푐 0.035 2 hint 10 W/m k 푘 2 ℎ hext 10030 W/m k ◦2 ℎ𝑖푛푡 θmax 10 W/m140 Ck ◦ ℎ θamb 100 W/m20 C2k 푒푥푡 3 ρc 8960 Kg/m 휃푚𝑎푥 140 °C 3 ρFe 7800 Kg/m 3 ρPM 7600 Kg/m Cpc 390 J/Kg/K

Table 3. Optimization parameters.

Parameters Min Max p 20 200 rs 0 1 rw 0 1 R 2 5 τLR (L/2R) 0.2 0.6 Wmag 10 mm 100 mm ns 1/2 10 Energies 2021, 14, x FOR PEER REVIEW 10 of 17

휃𝑎푚 20 °C 3 휌푐 8960 Kg/m 3 휌퐹푒 7800 Kg/m 3 휌푃푀 7600 Kg/m 퐶푝푐 390 J/Kg/K

Table 3. Optimization parameters.

Parameters Min Max 푝 20 200

푟푠 0 1 푟푤 0 1 푅 2 5

휏퐿푅 (퐿/2푅) 0.2 0.6 Energies 2021, 14, 4486 푊푚𝑎푔 10 mm 100 mm10 of 17 푛푠 1/2 10

4.1.4.1. Results Results TheThe NGSAII NGSAII algorithm algorithm is is used toto solvesolve the the two two objective objective functions: functions:

( 푊푡표푡 퐹푊 표푟 푊푡표푡 푀푃푇퐴 푀𝑖푛W{ tot FW or Wtot MPTA (40) Min 푀푔 = 푀푐 + 푀퐹푒 + 푀푃푀 (40) Mg = Mc + MFe + MPM Note that the method proposed in this paper would also minimize the cost of the generator,Note either that theby replacing method proposed the second in this objective paper wouldfunction also or minimize by adding the a costthird of objective the generator, either by replacing the second objective function or by adding a third objective function. In this paper, we have only chosen to minimize the mass of the generator, with- function. In this paper, we have only chosen to minimize the mass of the generator, without outminimizing minimizing its cost.its cost. According According to [21 ],to this [21] criterion, this criterion is indeed is essential, indeed today,essential in a, context today, in a contextof increasing of increasing wind turbine wind turbine power. However,power. However, we will present we will the present detailed the costs detailed of the two costs of thelightest two lightest generators generators obtained obtai for thened two for considered the two considered control strategies. control strategies. In Inorder order to to analyze analyze the the effect effect of thethe number number of of phases, phases, Figure Figure7 presents 7 presents the Pareto- the Pareto- optimaloptimal fronts fronts obtained obtained when when the the number ofof phases phasesq was푞 was ﬁxed fixed at three at three and ﬁve.and five For . For bothboth cases, cases, the the two two current current modemode controls control (FWs (FW and and MTPA) MTPA were) were considered. considered. The NSGA The IINSGA II algorithmalgorithm developed by by [29 [29]] and and available available in a in Matlab a Matlab code code [30] was [30] used was with used a numberwith a num- of generations and a population size of, respectively, 3000 and 500. ber of generations and a population size of, respectively, 3000 and 500.

(a) (b)

FigureFigure 7. Pareto 7. Pareto-optimal-optimal front front of ofoptimal optimal machines machines for q푞==3 (a )(a and) andq = 푞5= (b).5 (b).

AccordingAccording to to the the results, results, for a givengiven number number of of phases, phases, the the optimal optimal Pareto Pareto fronts fronts are are overlaid.overlaid. However, However, it can can be be seen seen that that the minimum the minimum mass is mass always is obtained always for obtained the MTPA for the MTPAcurrent current mode mode control. control. The result The shows result how shows the current how the mode current control mode can impact control the can result impact (performance and design) of the machine when it is taken into account in the optimization the result (performance and design) of the machine when it is taken into account in the process. For a three-phase machine, the variation is closed to 15%. Regarding the number of phases, the optimum is obtained for q = 3. As represented in Figure7, the mass of the machine increase with q. This result is mainly due to the reduction of the pole pair number p with q because of the limitation of maximum number of possible slots at a given slot width. However, the increase of q allows to reduce the phase current, which is necessary to reduce the constraints and the losses in the power converter when more powerful machines are investigated. Figure8 represents the proﬁle of the currents ido and iqo during the cycle for the lightest three-phase machine only. In the case of the FW current mode control, the current ido is adjusted at each working point to minimize the electrical power losses according to (12). Energies 2021, 14, x FOR PEER REVIEW 11 of 17

optimization process. For a three-phase machine, the variation is closed to 15%. Regarding the number of phases, the optimum is obtained for 푞 = 3. As represented in Figure 7, the mass of the machine increase with 푞. This result is mainly due to the reduction of the pole pair number 푝 with 푞 because of the limitation of maximum number of possible slots at a given slot width. However, the increase of 푞 allows to reduce the phase current, which is necessary to reduce the constraints and the losses in the power converter when more powerful machines are investigated. Figure 8 represents the profile of the currents 𝑖푑표 and 𝑖푞표 during the cycle for the Energies 2021, 14, 4486 lightest three-phase machine only. In the case of the FW current mode control,11 the of 17 current 𝑖푑표 is adjusted at each working point to minimize the electrical power losses according to (12).

(a) (b)

FigureFigure 8. Evolution 8. Evolution of the of theoptimal optimal d– d–qq axis axis currents currents forforq =푞 =3: FW: FW (a) and(a) and MTPA MTPA (b). (b).

In TableIn Table 4, 4the, the costs costs of of the fourfour lightest lightest machines machines presented presented in Figure in Figure7 are given 7 are for given for comparison. The following material costs were considered [25]: 50 EUR/kg for the magnets, comparison.3 EUR/kg The for ironfollowing and 15 material EUR/kg forcosts copper. were As considered it can be seen, [25] machines: 50 EUR optimized/kg for the magnets,considering 3 EUR/kg for a MTPA iron and control 15 areEUR less/kg expensive for copper. compared As it can to thebe seen, machines machines optimized optimized considconsideringering a MTPA an FW controlcontrol (28% are andless 18% expensive for q = 3 comparedand q = 5, respectively). to the machines Such aoptimized result is con- sideringmainly an due FW to control the volume (28% of and magnets, 18% lowerfor 푞 for= the and MTPA, 푞 = which5, respectively). represents a signiﬁcant Such a result is mainlypart due of the to costthe ofvolume the machine. of magnets, This difference lower for is inthe agreement MTPA, which with (16) represents and (18), where a significant B is lower than B . part off m the MPT cost opt of the machine.f m FW This opt difference is in agreement with (16) and (18), where 퐵 퐵 푓푚 푀푃푇Table 표푝푡 4. Costis lower of the optimalthan solutions.푓푚 퐹푊 표푝푡.

Table 4. Cost of the optimal solutions. q = 3 q = 5 FW MTPA FW MTPA 풒 = 3 풒 = 5 Cost of magnets (kEUR) 371 218 458 311 Cost of iron (kEUR) 135FW 126MTPA 141FW 138 MTPA Cost ofCost magnets of copper (k (kEUR)EUR) 296371 228218 315458 299 311 Total material cost (kEUR) 802 572 914 748 Cost of iron (kEUR) 135 126 141 138 Cost4.2. Optimal of copper Machine (kEUR) 296 228 315 299 Total materialFor high powercost (k offshoreEUR) wind turbines,802 mechanical572 constraints strongly914 affect the748 design of the machines. To satisfy the safety of the structure, it is important to limit the mass 4.2. Optimalof the nacelle Machine to be as low as possible. Therefore, here, we consider the lower mass machine, q i.e.,For the high lower power masse offshore machine withwind =turbines 3 optimized, mechanical for an MTPA constraints current mode strongly control. affect In the this particular case, and only because the steady state thermal regime is reached, we ﬁnd designthe of result the obtained machines. by classical To satisfy methods the onlysafety when of the the massstructure, is minimized it is important considering to the limit the massrated of the power nacelle with anto MTPAbe as low control. as possible. Beside the Therefore, fact that it is here, an element we consider of validation the of lower the mass machine,proposed i.e.,method, the lower it is masse important machine to note with that an푞 optimization= 3 optimized for thefor ratedan MTPA power current with a mode control.steady-state In this thermal particular regime case, not reached and only (which because was not the obvious steady in advance) state thermal would lead regime is reached,to oversizing we find the the machine. result obtained by classical methods only when the mass is mini- Table5 summarizes the optimal geometry of the optimal generator. mized considering the rated power with an MTPA control. Beside the fact that it is an

Energies 2021, 14, x FOR PEER REVIEW 12 of 17

element of validation of the proposed method, it is important to note that an optimization for the rated power with a steady-state thermal regime not reached (which was not obvious in advance) would lead to oversizing the machine. Table 5 summarizes the optimal geometry of the optimal generator.

Table 5. Optimal machine parameters.

Parameter Value Energies 2021, 14, 4486 12 of 17 푞 3 푝 156 푅 5 m Table 5. Optimal machine parameters. 퐿 (휏퐿푅) 1.15 m (0.23) Parameter Value 푟푠 0.968 q 푟푤 3 0.992 p 퐵 156 R 푟 5 m 1.2 T L (τLR) 퐵푓푚 1.15 m (0.23) 1 T r 0.968 s 퐵푡푚 1.59 T rw 0.992 퐵푦푚 0.79 T Br 1.2 T Bf m 푊𝑎푔 1 T 8 mm B 1.59 T tm 푊푃푀 18.7 mm B 0.79 T Total activeym material weight 61.75 tons Wag 8 mm WPMIron weight 18.7 mm 42.2 tons Total activeC materialopper weightweight 61.75 tons 15.2 tons Iron weight 42.2 tons CopperMagnet weight weight 15.2 tons 4.36 tons MagnetAverage weight losses 4.36 tons 253 kW AverageNominal losses voltage 푉 253 kW 2990 V Nominal voltage V 푠 2990 V Nominal currents Nominal current Is 퐼푠 1030 A 1030 A NominalNominalcosϕ 푐표푠휑 0.98 0.98

FiguresFigure9 sand 9 and 10 show 10 show the ﬂuxthe densitiesflux densities and the and temperature the temperature in the winding in the winding during during thethe cycle cycle for for the the optimal optimal machine. machine. Magnetic Magnetic and thermal and thermal constraints constraints are always are fulﬁlledalways fulfilled withwith the the control control of theof the thermal thermal transient transient regime. regime. Here, Here due to, due long to operating long operating times, the times, the permanentpermanent thermal thermal regime regime is reached. is reached. It should It should be noted be noted that, for that some, for applications some applications (tidal (tidal turbine for example), where the steady state thermal regime is not reached, the optimization turbine for example), where the steady state thermal regime is not reached, the optimiza- method presented in this article would avoid an oversizing of the generator. tion method presented in this article would avoid an oversizing of the generator.

FigureFigure 9. 9.Evolution Evolution of theof the maximum maximum ﬂux densityflux density in tooth in andtooth yoke and of yoke the optimal of the generator.optimal generator.

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Energies 2021, 14, 4486 13 of 17 Energies 2021, 14, x FOR PEER REVIEW 13 of 17

Figure 10. Evolution of the temperature in the winding. FigureFigure 10. 10.Evolution Evolution of the of temperaturethe temperature in the in winding. the winding. 4.3. FEA Validation 4.3. FEA Validation 4.3.In FEA this Validation part, the results for the optimum generator are validated by a 2D finite element In this part, the results for the optimum generator are validated by a 2D finite element analysisanalysisIn (FEA). (FEA).this part, Figure Figure the 11 results 11 shows shows for the thethe flux optimflux lines linesum and generatorand flux densityflux density are in validated the in optimal the byoptimal machine a 2D finite machine at element at thetheanalysis rated torquetorque (FEA). with with Figure the the current 11 current showsIs = 퐼the푠1.03= flux1kA.0 . lines ThekA. The average and a fluxverage torque density torque obtained in obtained the validates optimal validate the machines the at analytical model with a variation lower than 10% (see Table 6). The magnitude of the flux analyticalthe rated model torque with with a variation the current lower 퐼푠 than= 1 10%.0 kA (see. The Table average6). The magnitudetorque obtained of the flux validates the densitiesdensities,analytical,measured measured model inwith thein thea middle variation middle of the lowerof most the saturated thanmost 10% saturated teeth (see andTable teeth in the6) .and middleThe in magnitude the of the middle yoke of theof the flux yoke(seedensities Figure (see Figure ,11 measured), also 11 validates), also in thevalidate the middle analyticals the of analytical the model. most modelsaturated. teeth and in the middle of the yoke (see Figure 11), also validates the analytical model.

FigureFigure 11. FluxFlux lines lines and and ﬂux flux density density at full at full load. load. Figure 11. Flux lines and flux density at full load. TableTable 6.6. MaximumMaximum torque torque and and EMF EMF at 10 at MW 10 MW and 11- 11 rpm. rpm. Table 6. Maximum torque and EMF at 10 MW - 11 rpm. QuantityQuantity Analytical ModelAnalytical Model FEA FEA Variation Variation 퐵 Btm Quantity푡푚 1.59Analytical1.59 1.6Model 1.6FEA 0.7% Variation0.7% Bym 퐵 0.79 0.84 6.3 푦푚퐵푡푚 0.791.59 0.841.6 6.30.7% TorqueTorque (MNm) 퐵(MNm) 8.68.6 8.138.13 5.47% 5.47% Magnitude of the EMF푦푚 (1st 0.79 0.84 6.3 Magnitude of the EMF (1st harmonic)3.02 2.96 2% harmonic)Torque (kV) (MNm) 3.028.6 2.968.13 2%5.47% Magnitude of the(kV) EMF (1st harmonic) 3.02 2.96 2% (kV) In order to validate the thermal model and its transient regime, the evolution of the temperatureIn order in tothe validate winding the was thermal calculated model considering and its transient a step regime of power, the (see evolution Figure of 12 the) withtemperature copper and in the iron winding losses inwas the calculated yoke and considering tooth (see Tablea step 7of) atpower full load,(see Figure for 퐼푠 =12) with copper and iron losses in the yoke and tooth (see Table 7) at full load, for 퐼푠 =

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1.0 kA. The result shows that both the transient and the final temperature in the winding are in good agreement. Energies 2021, 14, 4486 14 of 17 Table 7. Losses of the optimal generator at the maximal power (10 MW, 1.03 kA).

Analytical Model FEA Variation InIron order losses to validate in the yoke the thermal model and20 kW itstransient regime,13 kW the evolution−35% of the temperatureIron losses in the in winding the teeth was calculated139 considering kW a step of173 power kW (see Figure25% 12) with copperTotal andiron ironof the losses stator in the yoke and tooth159 kW (see Table7) at full163kW load, for Is =3%1.03 kA. The result showsCopper that losses both the transient and the ﬁnal temperature173 kW in the winding are in good agreement.

FigureFigure 12. 12.Evolution Evolution of of the the temperature temperature in in the the winding winding at at the the maximal maximal power. power.

Table5. Conclusions 7. Losses of the optimal generator at the maximal power (10 MW, 1.03 kA). In this paper, we showed how to take into account all the operating points of a work- Analytical Model FEA Variation ing cycle with the control strategy in the optimization process of a PMSG. It should be notedIron lossesthat, by in theits yokeformulation, this 20 kWmethod is also applicable 13 kW to other kinds− of35% machines, eitherIron lossessynchronou in the teeths (with or without 139 kW magnets, with or without 173 kW salience) or variable 25% reluc- Total iron of the stator 159 kW 163kW 3% tance Coppermachine losses. Contrary to “classical” methods, which 173 reduce kW the problem to a few significant points of the working cycle in order to reduce the computation time, the method presented in this paper allows to consider all the points, which makes it possible to control 5. Conclusions the constraints at any point of the cycle, in particular the thermal one, characterized by a transientIn this regime. paper, we The showed 1D model how used to take was into validated account allby thea 2D operating finite element points ofanalysi a workings. The cycledynamic with thermal the control behavior strategy is inalso the controlled, optimization avoiding process an of oversizing a PMSG. It of should the machine be noted in that,case bythe its permanent formulation, thermal this method regime is alsonot reached. applicable Finally, to other this kinds approach of machines,, with its eitherquick- synchronousness and simplicity, (with or constitutes without magnets, a first step with toward or without a design salience) optimization or variable of the reluctance complete machine.turbine system, Contrary including to “classical” the power methods, electronics which reducecomponents the problem, which towill a fewbe discussed significant in pointsfuture of works. the working cycle in order to reduce the computation time, the method presented in this paper allows to consider all the points, which makes it possible to control the constraints6. Patents at any point of the cycle, in particular the thermal one, characterized by a transient regime. The 1D model used was validated by a 2D finite element analysis. The dynamicAuthor Contributions: thermal behavior “Conceptualization, is also controlled, L.D avoiding., S.O.S., R.S an. oversizingand N.B.; methodology of the machine, L.D in., S.O.S case ., theR.S permanent. and N.B.; software, thermal L.D regime. and N.B is not.; validation reached., L.D Finally,., S.O.S this., R.S approach,. and N.B.; with formal its analysis, quickness L.D. andand simplicity, N.B.; investigation, constitutes L.D a. and first N.B step.; resources, toward a N.B design.; data optimization curation, L.D of. and the N.B complete.; writing turbine—orig- system,inal draft including preparation, the L.D power. and electronicsN.B.; writing components,—review and editing which, N.B will.; bevisualizat discussedion, N.B in future.; super- works.vision, N.B.; project administration, N.B.; funding acquisition, N.B. All authors have read and agreed to the published version of the manuscript 6. Patents

Author Contributions: Conceptualization, L.D., S.O.S., R.S. and N.B.; methodology, L.D., S.O.S., R.S. and N.B.; software, L.D. and N.B.; validation, L.D., S.O.S., R.S. and N.B.; formal analysis, L.D. and N.B.; investigation, L.D. and N.B.; resources, N.B.; data curation, L.D. and N.B.; writing—original draft preparation, L.D. and N.B.; writing—review and editing, N.B.; visualization, N.B.; supervision, N.B.; project administration, N.B.; funding acquisition, N.B. All authors have read and agreed to the published version of the manuscript. Energies 2021, 14, 4486 15 of 17

Funding: This work was carried out within the framework of the WEAMEC, West Atlantic Marine Energy Community, and with funding from the Pays de la Loire Region of France, under the OCEOS project. https://www.weamec.fr/en/projects/oceos/. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Abbreviations

vd, vq d- and q- axis terminal voltages (V) id, iq d- and q- axis currents (A) e0 back electromotive force (V) Rc armature resistance (Ω) Rµ iron loss resistance (Ω) X synchronous reactance Pc copper losses (W) Pmg iron losses (W) kad additional iron loss coefficient kec eddy currents specific loss coefficient kh hysteresis specific loss coefficient k f slot fill factor kt tooth opening to the slot pitch ratio kL coefficient for correcting the active length L active length τLR length to outer stator radius ratio R outer stator radius Ro inner rotor radius (m) Rs inner stator radius Rr outer rotor radius rs reduced inner stator radius Rw outer winding radius rw reduced outer winding radius wag air-gap thickness wmag magnetic airgap (magnet + mechanical airgap) (m) wPM permanent magnet height (m) wt slot width (m) wy armature yoke thickness (m) bPM permanent magnet width (m) ns number of turns per phase per pole p number of pole pairs q number of phases βPM electrical magnet pole arc (rad) σc electric conductivity Ωm machine mechanical angular velocity (rad/s) ◦ θmax maximal permissible temperature ( C) ◦ θc temperature in the copper ( C) ◦ θamb ambient temperature ( C) 2 heq heat transfer coefficient (W/m K) 3 ρc copper density (kg/m ) 3 ρFe steel density (kg/m ) 3 ρPM permanent magnet density (kg/m ) Cpc specific heat capacity of copper (J/Kg/K) CpFe specific heat capacity of steel (J/Kg/K) Energies 2021, 14, 4486 16 of 17

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