Cogging Torque Suppression of Modular Permanent Magnet Machines Using a Semi-Analytical Approach and Artificial Intelligence

The cogging torque of permanent magnet machines with a modular stator is affected by additional harmonic components due to the segmentation of the stator lamination. This paper proposes a novel approach based on the shaping of the stator tooth tips with sinusoidal profiles to minimize the cogging torque of such machines. A theoretical study and a design formula are proposed to determine the spatial frequency of the sinusoidal profiles, while an optimization procedure based on genetic algorithm and artificial neural networks is adopted to determine their amplitudes and phase shifts. The proposed method is validated through finite element analysis considering two different case studies. Also, a comparison with other approaches from the literature is presented to highlight the effectiveness of the proposed technique. Finally, an additional analysis is reported to demonstrate the effectiveness of the proposed method against manufacturing and assembling tolerances.


I. INTRODUCTION
Permanent magnet machines (PMMs) with a modular, i.e. segmented, stator core are gaining increasing interest. In the last years, different studies have highlighted many advantages of such machines if compared to conventional PMMs with a one-piece stator. In fact, a modular stator core allows to enhance the slot fill factor, to ease the manufacturing process and coil winding and to reduce the end winding and iron wastage [1], [2], [3]. These advantages are noticeable both The associate editor coordinating the review of this manuscript and approving it for publication was Wei Wei .
in large machines, such as wind and tidal generators [4], [5], as well as in low power and small size PMMs [6], [7], [8]. Also, many works studied the features of PMMs with a segmented stator core. For instance, [2], [7], and [9], show how the additional stator gaps between the stator modules influence the electromagnetic performances. The influence of the stator segmentation on mechanical vibrations and acoustic noise is analyzed in [3], while [10] proves the resistance capabilities of modular surface mounted PMMs against irreversible demagnetizations.
However, the additional stator gaps in the flux path of modular stator PMMs generate additional harmonic components (AHCs) of the cogging torque [11]. That is, an undesired torque pulsation also at no-load condition, which increases vibrations, acoustic noise and speed pulsations. Compared to the native harmonic components (NHCs) of the cogging torque in conventional PMMs with a one-piece slotted stator, the AHCs have lower harmonic orders and higher amplitudes [11], [12].
The minimization of the NHCs of the cogging torque in one-piece stator PMMs has been addressed using rotor or stator skewing [13], [14], dummy slots or notches on the tooth tips [15], [16], slot openings design [17], [18] and teeth pairing [19], [20]. Both analytical and meta-heuristic approaches are adopted to determine the optimal design solutions. For instance, in [21], an analytical approach using conformal mapping method is employed to determine the optimal shifting angle of the slot openings to minimize the cogging torque. In [22], the optimum arrangement of PMs to reduce cogging torque is found by means of analytical studies in which the PMs magnetization is described using the Fourier series and the effect of the slots on the airgap flux density is taken into account with the equivalent magnetizing current method. In [23], a genetic algorithm coupled with finite-element analysis (FEA) is proposed to determine the position of PMs which minimizes the cogging torque. In [23], the cogging torque of a brushless DC motor is minimized using a genetic algorithm and Kriging surrogate models.
However, such methods cannot be applied for the minimization of the AHCs in modular stator PMMs. In fact, the rotor skewing has no influence on low order AHCs [25], while the design of a unique shape for all the stator tooth tips has a limited effectiveness in AHCs minimization as shown in [26] and [27]. Thus, placing equally spaced dummy slots in the stator is ineffective as well.
The explicit minimization of the cogging torque of PMMs with modular stators has received poor attention from the research community. In [11], an optimal number of uniform stator modules or an optimal combination of non-uniform stator modules are proposed. However, although strict limitations are imposed in the machine design phase, the results showed a non-negligible residual cogging torque. In [4], the stator slot openings shifting is proposed to mitigate the cogging torque of a modular stator PMM with E-shaped stator modules. Nevertheless, the effectiveness of such method fails in case of some specific topologies due to the limited impact of the slot openings on the cogging torque. In [26] and [27], a method based on the design of multiple shapes of the tooth tips with a topological optimization is presented. The optimal geometry of the tooth tips is found by means of a genetic algorithm (GA) and a finite element analysis (FEA) in [26] or surrogate models trained with FEA results in [27]. Compared to [4], this approach handles arbitrary topologies with uniform stator segments. However, this method requires a complex design procedure to discretize the tooth tips in the finite element model. Moreover, the number of design variables depends on the number of stator modules, stator slots, machine poles and on the number of binary elements in which each tooth tip is discretized. Therefore, the computational efforts required by [26] and [27] significantly increase in case of specific topologies where the number of design variables is very high. This paper proposes an alternative design method aimed at minimizing the AHCs of the cogging torque of modular PMMs. This method uses sinusoidal profiles to shape the tooth tips, which is an approach mainly used to contour the permanent magnets in order to increase the generated torque while reducing the NHCs of the cogging term in one-piece stator superficial PMMs (SPMMs) [28], [29]. As in [26] and [27], the proposed method handles arbitrary topologies with uniform stator segments. However, compared to the topological optimization in [26] and [27], the design procedure is simplified. In fact, sinusoidal profiles are easily reproducible in the finite element model and their number depends only on the number of AHCs to minimize, ensuring a computationally-efficient approach. A theoretical study supports the choice of the spatial frequency of the sinusoidal profiles, while an optimization procedure is set up to find their amplitudes and positions.
The main contributions and features of this work are as follows: 1) A novel method to reduce the cogging torque of modular PMMs is proposed. 2) Compared to [4], this method has a wider range of application since it handles different modular topologies. 3) Compared to [11], this method does not impose limitations in the machine design phase. 4) Compared to [26] and [27], this method offers a more simple and computational-efficient solution.

II. COGGING TORQUE OF PMMs WITH MODULAR STATOR CORES
The cogging torque of slotted PMMs with modular stators can be expressed as in [11] by: where ϑ r is the rotor angular position, T NHC expresses the NHCs caused by the stator slots and T AHC expresses the AHCs caused by the modular stator core. The angular frequency of the NHCs is an integer multiple of the least common multiple (LCM) between the number of stator slots N s and the number of poles 2p, while the frequency of the AHCs is an integer multiple of the LCM between 2p and the number of stator core modules m. Thus, it results that: where T NHCi , T AHCi , ϕ NHCi and ϕ AHCi are the amplitudes and the phase shifts of the i-th harmonic components. 39406 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
Since the number of stator slots is an integer multiple of the number of stator core modules, then LCM (2p, m) ≤ LCM (2p, N s ) and the harmonic orders of the AHCs can be lower than those of the NHCs. Moreover, once the number of poles is fixed, the amplitudes of the AHCs increase when LCM (2p, m) decreases [11].

III. TOOTH TIPS HARMONIC SHAPING
This section describes the method to minimize the cogging torque based on the harmonic shaping of the tooth tips. Subsection III-A presents an analytical study aimed at determining the number and the spatial frequency of the sinusoidal profiles required to suppress the cogging torque harmonics. Subsection III-B deals with determining the optimal amplitudes and phase shifts of the introduced sinusoidal profiles. Finally, Subsection III-C describes the proposed approach to solve the optimization problem formulated in the previous section.

A. THEORETICAL STUDY AND DESIGN FORMULA
Following the approach proposed in [27], the present study investigates the introduction of additional harmonics in the airgap permeance function to suppress the AHCs caused by the modular stator core. The cogging torque of a PMM featuring tooth tips with a modified shape can be expressed as: where T IHC are the cogging torque harmonics introduced by the modified shapes of tooth tips, namely the introduced harmonic components (IHC). In [27], the IHCs are investigated considering the coenergy method and the superposition principle. The latter allows to study the IHCs considering a non-modular stator core, i.e., these torque harmonics can be expressed as: where k g is a constant which depends on the geometrical parameters of the PMM, ′ tts is a component of the squared airgap permeance function introduced by the modified shapes of the tooth tips and F m is the rotor magneto-motive force (MMF); α is the angular displacement along the stator circumference. In [27] the authors proved how the frequencies of the harmonic components of ′ tts (α) are equal to the spatial frequencies of the harmonic components of the function expressing the additional airgap flux-path length due to the modified shape of the tooth tips. In this case, the additional airgap flux-path length depends on the sinusoidal profiles shaping the tooth tips. Therefore, it can be stated that a unique harmonic of ′ tts is introduced for each sinusoidal profile and that these share the same spatial frequencies.
Considering (5), for the orthogonality of the trigonometric functions, only the isofrequential harmonic components of ′ tts (α) and F 2 m (α, ϑ r ) contribute to the cogging torque.
Let N sp be the number of introduced sinusoidal profiles and f IHC k be the spatial frequency, expressed in rad −1 , of the k-th sinusoidal profile introduced to shape the tooth tips. Hence, the Fourier series of the IHCs produced by N sp sinusoidal profiles used to shape the tooth tips is as follows where T IHC k and ϕ IHC k are the amplitude and the phase shift of the k-th harmonic component, respectively.
To suppress the AHCs, the IHCs should have the same frequencies. Let now N AHC ⊂ N be the set of the harmonic orders of the AHCs to minimize. Considering (3) and (6), the following relations must be satisfied Equation (8) represents the design formula which allows to choose the frequency of the sinusoidal profiles employed to shape the tooth tips of the PMM with a modular stator core.
Note that the same methodology can be also applied to reduce the NHCs. Let be N NHC ⊂ N the set of the NHCs to minimize. According to (2) and (6), the IHCs required to suppress the NHCs should have the following frequencies Note how (9) can be also employed for the suppression of the cogging torque of conventional PMMs with a one-piece stator core. Note also that from (8) and (9), a sinusoidal profile should be introduced for each cogging torque harmonic component to be suppressed. Therefore, N sp is equal to the number of harmonic components of the cogging torque to be minimized.

B. FORMULATION OF THE OPTIMIZATION PROBLEM
Equations (8) and (9) allow the designer to choose the number and frequency of the sinusoidal profiles which introduce cogging torque harmonic components with the same frequencies of the AHCs and NHCs to be minimized. However, to suppress both the AHCs and NHCs, it is also necessary to properly set the amplitude and position of the sinusoidal profiles. Fig. 1 shows a linearized representation of two tooth tips shaped through the composition of two sinusoidal profiles, where: in which A k and ϕ k are the amplitude (mm) and phase shift of the introduced sinusoidal profiles, respectively. To determine the optimal amplitudes and phase shifts of the introduced VOLUME 11, 2023 sinusoidal profiles, the following optimization problem is defined: (15) in which x is the vector of the design variables and A lim is the upper bound for the amplitude of the sinsuodial profiles. Moreover, the following constraint is introduced: The manufacturing feasibility of the tooth tips is ensured by setting A lim less or equal to half of the tooth tips height.

C. HEURISTIC OPTIMIZATION USING GA AND ANN-BASED SURROGATE MODELS
The optimization problem in the previous section is solved using a GA which employs ANN-based surrogate models to compute the objective function in (14). In fact, an analytical approach to compute the cogging torque of the PMM, even though computationally fast, would result poorly accurate and difficult to set-up due to the complex geometries of the machines and to the non-linearities of the magnetic materials. FEA is commonly employed to overcome the limitations of analytical models [30] and meta-heuristic algorithms are adopted to solve this class of optimization problems [23], [23], [31]. However, this approach requires a huge computational burden due to the number of iterations needed by the meta-heuristic algorithms to find an acceptable solution. Therefore, the use of surrogate-models trained by a small amount of FEA simulations chosen with a proper design of experiment (DoE) can result an effective alternative since it guarantees a good accuracy while mitigating, at the same time, the computational effort required by the evaluation and minimization of the objective function [32], [33]. The optimization procedure proposed in this paper is divided into three main stages:

2) SURROGATE MODELS DESIGN AND TRAINING
A multi-layer feed-forward (FF) ANN, with the structure depicted in Fig. 2 is chosen as surrogate model. This FF ANN consists of N HL hidden layers with N HN neurons per layer, where each neuron features a hyperbolic tangent sigmoid as activating function. The input layer is the vector of the design variables x. The output layer consists of N pos linear neurons whose output is T cog,NN (x, ϑ r ), i.e. the vector containing the N pos values of the cogging torque over a two-poles pitch angle. ANNs adopted as surrogate models for optimization purposes ensure high approximation performances with low computation burden [34]. However, to optimize the accuracy and the computational cost of an ANN, a careful design of its topology is required, in particular with respect to the number of neurons per layer and the number of hidden layers [35]. In this paper, to choose the optimal topology of the ANN-based surrogate models, the following procedure is developed: • Step 1: random splitting of the dataset obtained with the DoE in training data (80%), validation data (15%), and test data (5%).
• Step 2: training instances of the ANNs with all the possible topologies with N HL ∈ N HL and N HN ∈ N HN , where N HL and N HN are the set of the allowed values of the number of hidden layers and number of neurons per layer, respectively. To avoid the overfitting of the ANN, a training stop criterion based on the maximum validation failures is considered. The training of an ANN is stopped if the estimation error on the validation data fails to improve for ten epochs in a row. At the end of the training, the estimation error on the test data is stored.
• Step 3: steps 1 and 2 are repeated N tr times. • Step 4: the ANN topology with the lowest average estimation error on the test data is chosen as surrogate model. • Step 5: final training of the chosen surrogate model with a random splitting of the dataset in training data (80%) and validation data (20%). In this case, the training of the surrogate model is stopped if the estimation error on the validation data fails to improve for ten epochs in a row.

3) GA OPTIMIZATION AND FEA VALIDATION
A GA is finally implemented to solve the optimization problem (14)-(16) considering the following surrogate objective function: This objective function expresses the optimization problem (14) considering the use of the surrogate model to compute the cogging torque of the modular PMM. The GA is a stochastic evolutionary algorithm that modifies a population of individual solutions according to rules that mimic biological evolution. At each iteration, the GA selects individuals from the current population to be parents producing the individuals for the next generation.
A drawback of the GA is that it can easily converge to suboptimal solutions if the number of individuals and generations is not carefully chosen [36], [37]. The main limitation to the use of high numbers of individuals and generations is the computational effort required for the objective function computation. However, due to the to the high computational efficiency of the surrogate models, a GA with a high number of individuals and generations can be used to perform the optimization (17) thus reducing the risk to converge to suboptimal solutions.
Due to the approximation error of the surrogate models, a validation using a TWM FEA is performed to rigorously evaluate the solution x found by the GA. Therefore, the objective function (18) is finally evaluated using a TWM FEA:

IV. RESULTS
The proposed method is validated on two different topologies of a modular annular PM generator (APMG) designed for a low power ducted wind turbine. This generator has a large diameter and a small axial length which justifies the adoption of a modular structure to simplify the manufacturing and assembling process. The main design aspects of the APMG can be found in [38], while its parameters are summarized in The results obtained with sinusoidal profiles designed in agreement with the design formula (8) are reported in Subsection IV-A, while Subsection IV-B reports the results obtained with sinusoidal profiles designed in disagreement with the design formula. Subsection IV-C shows the cogging torque obtained with the methods proposed in [4] and [27] and Subsection IV-D reports a detailed comparison of the optimized and basic designs of the APMG at rated current. Finally, Subsection IV-E analyzes the influence of manufacturing and assembling tolerances on the performance of the proposed method.
A. RESULTS USING THE DESIGN FORMULA Fig.3 shows the basic design of the APMGs with 20 and 60 stator core modules. According to (3), for these two topologies, the following harmonic orders of the AHCs are expected VOLUME 11, 2023  in an electrical period, respectively: Instead, according to (2), the following harmonic orders of the NHCs are expected in an electrical period: Fig. 4 shows the cogging torque waveforms and harmonic spectra of the two AMPG topologies obtained by means of a TWM FEA (N pos = 150), which requires about 90 minutes to be performed. Note that the orders of the AHCs agree with (19) and (20). Moreover, the amplitude of the NHCs is negligible if compared with the amplitude of the AHCs. The peak-to-peak values of the cogging torque are 16.68 Nm and 1.39 Nm, respectively. The dominant harmonic components of the APMGs with 20 and 60 stator core modules are respectively the 2 nd and the 6 th ones; moreover, most of the cogging torque of these machines can be reduced by suppressing these harmonic components. Therefore, only one sinusoidal profile is introduced to suppress the 2 nd harmonic component of APMG with 20 stator core modules. According to the design formula (8), the spatial frequency of this profile must be 100/2π rad −1 . Instead, to suppress the 6 th harmonic component of the APMG with 60 modules, a sinusoidal profile with a spatial frequency of 300/2πrad −1 is adopted.
To determine the amplitudes and phase shifts of the introduced sinusoidal profiles, the approach described in Subsection III-C is employed . TABLE 3 and TABLE 4 report the results of five optimizations performed on the APMG with 20 stator core modules with N DOE = 100 and N DOE = 250, respectively. These tables report the optimized surrogate model topology, the mean squared error (MSE) on the test data, the solution x of the GA and the corresponding objective function evaluation f NN (x), and the result of the FEA validation, f FEA (x), for each one of the five optimizations. Each sample has been obtained by using a TWM FEA with N pos = 24. Note that, to better evaluate the performance of the proposed method, each of the five optimizations is performed by repeating the whole procedure described in Subsection III-C. An average reduction of the peak-to-peak cogging torque of 92.5% and 97.6% is obtained in the two cases, respectively. The accuracy of the surrogate models increases with the number of samples: in particular, the mean absolute percentage error (MAPE) between the surrogate objective function evaluation f NN (x) and the TWM FEA f FEA (x), is equal to 73.4% and 7.4% in the two cases, respectively. It can be also noticed that the optimization of the ANNs topology converges to similar solutions in the two cases. This result highlights how the ANNs accuracy depends on the topology adopted and confirms the importance of optimizing the ANNs topology .  TABLE 5 and TABLE 6 summarize the results on the APMG with 60 stator core modules with N DOE = 100 and N DOE = 250, respectively. Each sample has been obtained by using a TWM FEA with N pos = 36. Also in this case, a significant average reduction of the peak-to-peak cogging torque (by 83.5% and 89.2% in the two cases) is achieved.
The MAPE between f NN (x) and f FEA (x) is equal to 20.8% and 11.1% in the two cases, respectively. Considerations about the accuracy and topology optimization of the surrogate models made for the APMG with 20 stator core modules can be extended to the 60 modules machine. Fig. 5 shows the evolution of the best individuals among the GA generations obtained for the best designs of the APMG with 20 and 60 stator core modules, i.e. the ones obtained considering the parameters reported in the second row of TABLE 4 and  TABLE 6, respectively. In both cases, the GA met the stop criterion before reaching the maximum number of generations. Approximately 4 × 10 4 evaluations of the objective function f NN (x) with the ANNs have been performed during the GA optimization. The optimized shapes of the tooth tips of these two designs are shown in Fig. 6 and Fig. 7. It can be demonstrated that considering a profile with a     spatial frequency equal to 100/2πrad −1 , 6 different shapes are obtained. Instead, when the spatial frequency is equal to 300/2πrad −1 , only two different shapes are obtained.      profiles. This is particularly noticeable comparing the 12 th and 18 th harmonic components of the basic and optimized designs of the APMG with 60 stator core modules in Fig. 9(b). Fig. 8 also reports the cogging torque profile at reverse rotation direction, demonstrating how the cogging torque is not affected by the rotation direction, according to the theoretical study. Finally, TABLE 7 reports the computational times of the main steps of the proposed optimization procedure performed on an Intel Xeon CPU E5-1620 v3 @ 3.50 GHz. The computational effort required by the ANN to calculate the cogging torque corresponding to a given solution x is very limited: this means that most of the time spent by the proposed optimization procedure is required by the evaluation of the DoE.

B. RESULTS WITHOUT USING THE DESIGN FORMULA
To further validate the theoretical study presented in Subsection III-A, this subsection provides the results achieved by introducing sinusoidal profiles with spatial frequencies which do not respect (8). In particular, Fig. 10 shows the results of a TWM FEA (N pos = 150) performed on the three different designs of the APMG with 20 stator core modules with the following sinusoidal profiles: • First case: spatial frequency, amplitude and phase f IHC 1 = 150/2πrad −1 , A 1 = 0.3 mm and ϕ 1 = 0.
• Third case: spatial frequency, amplitude and phase f IHC 1 = 150/2πrad −1 , A 1 = 0.3 mm and ϕ 1 = 0 and f IHC 2 = 200/2πrad −1 , A 2 = 0.3 mm and ϕ 2 = 0. In the first case, the introduced sinusoidal profile does not affect the cogging torque as predicted by the theoretical study, i.e., since the harmonics of F 2 m have a spatial frequency which is an integer multiple of 100 2πrad −1 , no harmonics of F 2 m are isofrequential with the airgap permeance function harmonic related to the introduced sinusoidal profile; thus, according to (5), the contribution to the cogging torque is none. Compared to the basic design, the amplitude of the harmonics is slightly reduced due to the average length increase of the airgap caused by the introduction of the sinusoidal profile.
In the second case, as expected, the introduced sinusoidal profile affects only the 4 th harmonic component. In particular, compared to the basic design, the amplitude of the 4 th harmonic component is increased since the amplitude and phase shift of the introduced sinusoidal profile have not been optimized to reduce the cogging torque. Note that, in the last case, the resulting torque is similar to the torque of the APMG with the sinusoidal profile with a spatial frequency of 200 2πrad −1 . The amplitudes of the harmonics are slightly reduced since the average airgap length is greater than in the previous case. This result confirms the proposed theoretical study and the validity of the superposition principle adopted to study the effect of the airgap permeance function harmonics on the cogging torque.

C. COMPARISON WITH OTHER METHODS
In this subsection, a comparison with two other existing methods is presented. The first method is the topological optimization (TO) of the tooth tips proposed in [27], while the second approach is based on the slot openings shifting proposed in [4]. Both methods have been implemented on the APMG with 60 modules. In particular, according to the design formula (20) of [27], two different shapes of the tooth tips must be independently optimized by means of the topological optimization for the considered machine. Similarly, according to [4], in order to shift the slot openings, two different widths of the tooth tips are employed.
The topological optimization has been performed considering the same approach based on surrogate models described in Section III-C and the tooth tips have been discretized with 7 elements, called subteeth, as shown in Fig. 11. The vector of the design variables includes a binary quantity (0:air 1:iron) for each subteeth of the two tooth tip shapes plus a final variable representing the depth of the subteeth. The results obtained with N DOE = 250 and N DOE = 400 are shown in TABLE 8 and TABLE 9, respectively. An average reduction of the peak-to-peak cogging torque by 58.1% and 50.1% is obtained, while the MAE between f NN (x) and f FEA (x) is equal to 73.3% Nm and 78.4%, respectively. Note that, also in this case the optimization of the ANNs topology converges to similar solutions, but the accuracy of the surrogate models is lower compared to the ones obtained with the harmonic shaping of the tooth tips. This is the reason why the performances of the TO are lower if compared with the proposed method, even assigning a remarkable advantage to the former, i.e., a higher quantity of samples available to train the surrogate models. The high number of design variables required by the TO makes the prediction of the cogging torque more difficult. Therefore, compared to the method proposed in this paper, the   TO requires more sophisticated approaches to suppress the cogging torque of modular machines [27]. The TO has also been performed with 5 and 9 subteeth with N DOE = 400. An average reduction of the peak-to-peak cogging torque of 72.0% and 49.9% has been achieved, with a MAPE between f NN (x) and f FEA (x) equal to 59.8% and 78.2%, respectively. Note that the improvement of the performances obtained with 5 subteeth further highlights the relationship bewteen the accuracy of the surrogate models and the number of the design variables.
The slot opening shifting is based on a unique design variable, i.e. the shift angle γ , as shown in Fig. 12. Due to the simplicity of the problem, the method can be investigated by evaluating a set of solutions obtained by the uniform sampling   of γ in its permissible range [0 γ lim ], where 0 corresponds to the basic slot openings position and γ lim is the position corresponding to the minimum width of the stator tooth tip (i.e. the width of the stator teeth). Fig. 13 shows the results of 240 solutions in this interval obtained directly using the FEA with N pos = 36, while Fig. 14 shows the cogging torque profile and harmonic spectrum with γ = 0.0052rad. As expected, the method affects the 6 th harmonic component of the cogging torque. However, only few solutions slightly reduce the cogging torque of the considered machine. In fact, the lowest value of the peak-to-peak cogging torque obtained is 1.20 Nm, with a reduction of 12.4% with respect to the basic machine. This can be explained by considering that the cogging torque harmonic introduced by the slot opening shifting method may not match the amplitude of the 6 th AHC. Indeed, since the width of the slot openings is fixed, the amplitude of the introduced harmonic component cannot be regulated to suppress the AHC. Instead, with the harmonic shaping and the TO methods, the amplitude of the introduced VOLUME 11, 2023  harmonic components can be adjusted to compensate the undesired AHCs.

D. ANALYSIS AT RATED CURRENT
This subsection reports the analysis of the basic and optimized designs of the APMG with 20 stator core modules at rated current, with the phase angle selected in agreement with the maximum torque per Ampere strategy. A sketch of the optimal machine with the modified shape of the tooth tips together with the flux density distribution under rated conditions is reported in Fig. 15. Fig. 16 shows the torque profiles and harmonic spectra of the two machines, highlighting how the 2 nd harmonic component of the torque of the basic design has been perfectly suppressed at the expense of a slight reduction of the average torque (2.6%). Therefore, the impact of the proposed method on the rated torque is negligible. Also, the phase flux linkages before and after the optimization (Fig. 17) do not present appreciable differences. In fact, the average THD of the phases flux linkage is 2.24% for the basic design and 1.90% for the optimized design. Moreover, the average amplitudes are 0.166 Wb and 0.161 Wb for the basic and optimized design. Table 10 reports a comparison of the loss components of the two machines at rated speed and current. The optimized  design exhibits a slight reduction of the PM and iron losses and the impact on the overall efficiency is negligible. This analysis clearly proves that the proposed method ensures a significant improvement of the torque ripple of the APMG with negligible effects on other performance indexes of the machine.

E. ANALYSIS CONSIDERING MANUFACTURING AND ASSEMBLING TOLERANCES
This subsection analyzes the performance of the proposed method considering assembling and manufacturing tolerances of the modular PMM in the finite element model. The parameters affected by tolerances are the positions of magnets and stator modules, the shape of the tooth tips and the phase shift of the introduced sinusoidal profiles, as shown in Fig. 18 and Fig. 19. In these figures the manufacturing and assembling errors are accentuated to be more perceptible. Note that these errors affect both AHCs and IHCs of the cogging torque and may potentially compromise the effectiveness of the optimization. It is also worth noticing that manufacturing and assembling tolerances are sources of other AHCs of the cogging torque, as illustrated in [39] and [40].
According to typical manufacturing techniques, the considered positioning and phase shift tolerances are ±0.05 • [39]. Instead, regarding the tooth tips shape, typical tolerances of the wire Electric Discharge Machining (EDM) technique are considered, i.e., ±0.0063 mm. To analyze the impact of the assembling and manufacturing tolerances,  the reference basic and optimized models of the APMG with 20 and 60 modules have been modified by introducing parameter errors using the non-uniform uncertainties method (non-UUM). According to the non-UUM, each magnet and stator module of the APMG has its own position error, while each point of the sinusoidal shape of the tooth tips has its own amplitude error. The errors are generated assuming a normal distribution with zero-mean and standard deviation σ = (USL − LSL)/6, where USL and LSL are the upper and lower boundaries of the tolerance range, respectively [39]. Fig. 20 reports the analysis of the basic and optimized APMG with 60 and 20 stator core modules. The average  peak-to-peak cogging torque values of the basic APMGs considering the assembling and manufacturing tolerances are 2.34 Nm and 17.32 Nm. Therefore, the cogging torque of the non-optimized APMGs on average is increased by 0.98 Nm and 0.65 Nm, respectively (gaps between blue and cyan lines). Instead, the average peak-to-peak cogging torque values of the optimized APMGs considering the assembling and manufacturing tolerances are 1.56 Nm and 1.42 Nm, with a reduction of 0.78 Nm and 15.90 Nm compared to the average cogging torque of the basic machines with manufacturing and assembling uncertainties (gaps between blue and red lines). The difference between the cogging torque of the reference basic and optimized designs of the two APMGs are instead 1.25 Nm and 16.28 Nm, respectively (gaps between cyan and magenta lines). By comparing the gaps between cyan and magenta lines with gaps between blue and red lines, the differences are 0.47 Nm and 0.38 Nm. Therefore, in both cases, the performances of the methods are marginally affected by the manufacturing and assembling tolerances. This is more evident when comparing the harmonic spectra of two designs of the basic and optimized APMGs affected by tolerances, as in Fig. 21 and Fig. 22. In particular, the designs no. 25 have been analyzed for both APMGs, which have peak-to-peak cogging torque value near the average values shown in Fig. 18. The dominant harmonic components of both APMGs (i.e. 2 nd and 6 th harmonic component for the APMGs with 20 and 60 modules, respectively) have been almost suppressed. However, the cogging torque is affected by other AHCs introduced by the tolerances.

V. CONCLUSION
In this paper, an effective design methodology to suppress the cogging torque of modular permanent magnet machines has been presented. Sinusoidal profiles have been used to shape the stator tooth tips of the machine. The quantity and the frequency of the sinusoidal profiles to be introduced have been determined through an analytical study. In particular, a simple design formula has been derived which allows a fast computation of the spatial frequency of the sinusoidal profiles. Moroever, the optimal amplitude and phase shift of the sinusoidal profiles have been determined by using ANN-based surrogate models and a GA.
The effectiveness of the proposed method has been investigated considering two PMMs with different number of stator core modules. In both cases, a reduction of the cogging torque higher than 80% has been obtained, with a perfect suppression of the dominant cogging torque harmonics. Also, the comparison with other existing methods further higlighted the strengths of the proposed approach. Finally, an additional analysis demonstrates the effectiveness of the proposed method against manufacturing and assembling tolerances.
FRANCESCO CUPERTINO (Senior Member, IEEE) received the Laurea and Ph.D. degrees in electrical engineering from the Polytechnic University of Bari, Bari, Italy, in 1997 and 2001, respectively. Since 2001, he has been with the Department of Electrical and Information Engineering, Polytechnic University of Bari, where he is currently a Full Professor of converters, electrical machines, and drives. He is the Scientific Director of four public/private laboratories with the Polytechnic University of Bari, which enroll more than 50 researchers; the Laboratory Energy Factory Bari, with GE AVIO, aimed at developing research projects in the fields of aerospace and energy; the More Electric Transportation Laboratory, with CVIT SpA (BOSCH Group), aimed at developing technologies for sustainable mobility; Cyber Physical Systems AROL Bari, with AROL SpA, focused on closure systems for food and beverage; and Innovation for Mills, with Casillo Group and Idea75, focused in the Industry 4.0 applications for wheat processing. He has authored or coauthored more than 130 scientific articles in his research fields. His research interests include the design of synchronous electrical machines, the motion control of high-performance electrical machines, the applications of computational intelligence to control, and the sensorless control of ac electric drives. He was a recipient of two Best Paper Awards from the Electrical Machines Committee of the IEEE Industry Application Society and the Homonymous Committee of the IEEE Industrial Electronics Society, in 2015. He is currently the rector of the Polytechnic University of Bari. Open Access funding provided by 'Politecnico di Bari' within the CRUI CARE Agreement