Optimal Design for Torque Ripple Reduction in a Traction Motor for Electric Propulsion Vessels

Abstract

Recently, as carbon emission regulations enforced by the International Maritime Organization (IMO) have become stricter and pressure from the World Trade Organization (WTO) to abolish tax-free fuel subsidies has increased, the demand for electric propulsion systems in the marine sector has grown. Most small domestic fishing vessels rely on tax-free fuel and have limited cruising ranges and constant-speed operation, which makes them well-suited for electric propulsion. This paper proposes replacing the internal combustion engine system of such vessels with an electric propulsion system. Based on real operating conditions, an Interior Permanent Magnet Synchronous Motor (IPMSM) was designed and optimized. The Savitsky method was used to calculate total resistance at a typical cruising speed, from which the required torque and output were determined. To reduce torque ripple, an asymmetric dummy slot structure was proposed, with two dummy slots of different widths and depths placed in each stator slot. These dimensions, along with the magnet angle, were set as optimization parameters, and a metamodel-based optimal design was carried out. As a result, while meeting the design constraints, torque ripple decreased by 2.91% and the total harmonic distortion (THD) of the back-EMF was lowered by 1.32%.

1. Introduction

The mobility industry is rapidly transitioning from fossil-fuel-based engines to eco-friendly electric propulsion systems as a result of increasingly stringent environmental regulations, the depletion of energy resources, and rising public concern about environmental problems. The International Maritime Organization (IMO), through a resolution of the Marine Environment Protection Committee (MEPC), announced a strategy to reduce greenhouse gas (GHG) emissions from ships by 50% by 2050 compared with 2008 levels [1]. In July 2023, this goal was further strengthened to target net-zero emissions by 2050 [2]. In line with these developments, regulatory measures have been gradually reinforced, and the World Trade Organization (WTO) has urged the termination of tax-exempt marine fuels provided as fuel subsidies [3]. As a result, developing electric propulsion systems to proactively address international environmental regulations has become increasingly urgent.

Small fishing vessels, which typically operate with simple navigation patterns and maintain steady cruising speeds, are regarded as suitable platforms to implement electric propulsion systems [4]. In this paper, resistance was evaluated based on the cruising speed most frequently observed in real-world operations using the Savitsky method [5,6]. Based on these results, the rated torque and speed of the electric motor were determined.

Electric propulsion vessels require motors with both high output and efficiency to compensate for the added weight due to onboard batteries and to ensure an adequate cruising range. Among the various motor types, the Interior Permanent Magnet Synchronous Motor (IPMSM) has been actively studied because of its ability to meet these requirements [7,8]. In particular, the hairpin winding method (using rectangular wire instead of traditional round wires) was introduced to support high power density. These conductors improve slot-space utilization and contribute to minimizing the end-turn length and motor diameter. As a result, overall system performance could be improved [9,10]. However, their larger cross-sectional area leads to a noticeable increase in AC copper losses. Therefore, designing IPMSMs using hairpin windings requires careful consideration of these losses and their effects [11].

In addition, maintaining the overall energy efficiency and operational stability of electric propulsion systems requires not only optimizing the drive motor itself but also coordinating the optimization of higher-level system components, including battery systems, thermal management structures, and energy management strategies. Recently, active research has been conducted on real-time global energy management strategies for PHEVs [12], cooling architectures that address temperature distribution in battery modules [13], and thermal control designs based on liquid-cooling plate structures [14].

Torque ripple is one of the key factors that significantly affect the operational performance of IPMSMs. It is primarily caused by cogging torque, which originates from the interaction between stator slots and rotor magnets and results in output torque irregularities. This phenomenon becomes particularly problematic in electric propulsion systems and outboard motor configurations where thrust is transmitted directly from the stern. These conditions can lead to vibration and unstable propulsion. Under cruising conditions where a constant speed must be maintained, such irregularities in output torque can lead to periodic fluctuations in thrust. These fluctuations in thrust cause speed deviations and the control system must repeatedly take corrective actions to correct them. This process not only reduces control responsiveness but also results in energy loss and decreased system efficiency. Therefore, effectively suppressing torque ripple enables stable speed maintenance and improved control performance. Although mechanical problems like cogging torque contribute to this, electrical factors—particularly the total harmonic distortion (THD) of the back electromotive force (back-EMF)—also play a significant role. Harmonics distort the current waveform by introducing non-sinusoidal components that cause torque to fluctuate and result in uneven output. Additionally, non-uniform variations in magnetic reluctance during rotor rotation (when coupled with the stator slot structure) induce harmonic components in the airgap flux density, which also act as a significant cause of torque ripple. Therefore, effective torque ripple reduction requires a combined approach that involves both mechanical structural improvements and electromagnetic design strategies aimed at suppressing harmonic components.

In previous studies, various approaches have been proposed to reduce torque ripple and cogging torque by implementing dummy slots. Some studies attempted to mitigate torque ripple by adjusting the position, number, or spacing of these slots [15,16,17,18,19]. However, most of these studies employed structurally symmetric dummy slot configurations and limited their analysis and design efforts to reducing torque ripple or cogging torque. Comprehensive evaluations that consider performance indicators such as the THD of the back-EMF, output performance, electromagnetic losses, and efficiency remain insufficient. Furthermore, the geometry of dummy slots has typically been determined empirically or through manual tuning without employing optimal design processes or conducting sensitivity analyses to evaluate design variables. Symmetric dummy slot structures, which have an identical width and depth, offer only limited effectiveness in reducing torque ripple and suppressing specific harmonic orders. Introducing asymmetry into the dummy slot structure alters the spatial distribution of the stator slot openings, which breaks the geometric symmetry conditions that allow harmonic components to accumulate. This modification changes the periodicity of the airgap permeance and disrupts the regularity of the slot harmonics. As a result, the magnetic circuit becomes non-uniform and the alignment between rotor flux harmonics and stator slot harmonics is weakened. This mechanism causes the partial cancellation of the torque ripple components generated by harmonics. These structures act as spatial harmonic filters and effectively attenuate low-order harmonic components that primarily amplify cogging torque and torque ripple under load conditions. Therefore, an asymmetric slot configuration that includes dummy slots can effectively suppress harmonics and help achieve a smoother torque output [20].

To overcome the limitations of previous studies and enhance torque ripple reduction performance, this study proposes an asymmetric dummy slot structure with a non-uniformly distributed width and depth, as opposed to conventional symmetric designs. This structure offers greater design flexibility compared with conventional symmetric dummy slot layouts. It improves magnetic flux distribution and helps reduce torque ripple and cogging effects caused by unwanted harmonic distortions in the electromagnetic field. Performance improvements were confirmed in key metrics such as average torque, the THD of the back-EMF, and overall efficiency. In this paper, a sensitivity analysis was conducted by applying the V-shaped angle of permanent magnets as a design variable to further reduce torque ripple. Based on the analysis results, metamodel-based optimization was carried out and the optimal design parameters were identified through a finite element analysis (FEA).

2. Design of an IPMSM for Electric Propulsion Outboard Systems

2.1. Calculation of the Required Output Based on the Cruising Speed

In this paper, an electric propulsion motor for a small fishing vessel is designed by calculating the hull resistance at a specific cruising speed and determining the required output and torque of the propulsion system accordingly. For the resistance calculation, the Savitsky method was applied, and it was validated for both practicality and computational efficiency. This method has been widely adopted in various studies involving small high-speed vessels and is commonly used in real-world analysis and design processes [5,6]. In addition, this study aims to replace the outboard motor (which is mounted on the exterior aft section of the hull and directly drives the propeller) with an eco-friendly and efficient IPMSM instead of a conventional internal combustion engine. The specifications of the vessel used in this paper are listed in

Table 1. Principal specifications of the target vessel.

Item	Unit	Specification
Overall length	m	10.10
Length between perpendiculars	m	7.78
Breadth	m	2.42
Depth	m	0.86
Full-load draft	m	0.32
Gross tonnage (GT)	-	1.78

[21].

In this paper, the main components of ship resistance are categorized into the following four types: viscous resistance ($R_V$), caused by the viscous friction between the hull surface and water; pressure resistance ($R_P$), which occurs due to the pressure distribution under the hull when the vessel runs at a specific trim angle on the water surface; spray resistance ($R_S$), which arises from the water spray generated at the bottom of the hull during movement; and air resistance ($R_A$), which is the aerodynamic drag generated as the superstructure cuts through the air.

First, the lift coefficient $C_{L\beta }$ required for the hull to maintain a planing condition is calculated using Equation (1). Here, $M$ is the vessel mass, $g$ denotes the gravitational acceleration, $B$ is the beam of the vessel, $U$ represents the vessel speed, and $p$ is the water density. The initial trim angle ${\tau }_0$, defined as the inclination between the vessel and the water surface, and the wetted surface ratio $\lambda$, representing the proportion of the vessel’s surface in contact with water, are initially assumed. Based on these assumptions, the trim angle is iteratively corrected using Equations (2) and (3), and the optimal actual trim angle ${\tau }_e$ is derived.

$$C_{L0}=C_{L\beta }-0.0065\cdot {\tau }_0\cdot C_{L0}^{0.6}$$

(1)

$$C_{L\beta }={\displaystyle \frac{Mg}{\frac{1}{2}\rho B^2{U}^2}}$$

(2)

$${\tau }_e={\tau }_0+\Delta$$

(3)

Changes in the trim angle affect the positions of the center of lift and the center of pressure, which directly influence the calculation of total resistance and thrust. Therefore, the trim angle was iteratively calculated based on the horizontal moment equilibrium condition, and a trim state satisfying a stable planing condition was derived. The pressure center coefficient $C_P$ was calculated using Equation (4) and used as a key parameter in the trim estimation process to account for the pressure distribution and moment balance.

$$C_p=0.75-{\displaystyle \frac{1}{5.21\cdot {\left(\frac{U^2}{gB}\right)}^2+2.39}}$$

(4)

$$R_V={\displaystyle \frac{1}{2}}\cdot \rho \cdot S_W\cdot U^2\cdot C_f$$

(5)

$$R_S={\displaystyle \frac{1}{2}}\cdot \rho \cdot U^2\cdot {\displaystyle \frac{B^2}{4\cdot \sin \left(2\alpha \right)\cdot \cos \left(\beta \right)}}\cdot C_f$$

(6)

$$R_P=M\cdot g\cdot \tan \left({\tau }_e\right)$$

(7)

$$R_A={\displaystyle \frac{1}{2}}\cdot {\rho }_{air}\cdot U^2\cdot A_T\cdot C_D$$

(8)

$$Torque=R_T\cdot r$$

(9)

Finally, the viscous resistance $R_V$, pressure resistance $R_P$, spray resistance $R_S$, and air resistance $R_A$ were calculated using Equations (5)–(8), and the total resistance can be obtained as their sum. By multiplying the total resistance calculated from Equation (9) by the propeller radius, the required torque for the propulsion system can be determined. Lastly, effective horsepower (EHP) refers to the actual propulsive power generated as the propeller pushes water to move the vessel. Delivered horsepower (DHP) represents the power transmitted to the propeller after accounting for mechanical losses, and brake horsepower (BHP) is the raw output of the motor without considering mechanical losses. These key relationships are illustrated in Figure 1. Assuming a mechanical transmission efficiency of 70% from the motor to the propeller, the required motor-side output (BHP) is 115 HP, which corresponds with 85.7 kW when converted to electric output. The propeller’s rotational speed can be calculated by dividing the vessel speed by the propeller pitch, and the motor’s rotational speed can be obtained by multiplying this value by the reduction ratio. At a cruising speed of 25 knots, the motor speed was calculated to be 3202 rpm. To achieve the required output of 85.7 kW, a torque of 256 Nm is needed. The vessel’s maximum speed was set to 35 knots. The required output conditions for the IPMSM used in the outboard propulsion system—based on both cruising and maximum speeds—are summarized in

Table 2. Required specifications of the IPMSM for an outboard motor.

Contents		Unit	Required Specification
Input	DC voltage	${\mathrm{V}}_{\mathrm{D}\mathrm{C}}$	800
	Rated line current	${\mathrm{V}}_{\mathrm{r}\mathrm{m}\mathrm{s}}$	135
Output	Continuous power	kW	85
	Maximum speed	rpm	4500
	Base speed	rpm	3200
	Continuous torque	Nm	256
	Maximum efficiency	%	above 92

.

2.2. IPMSM Model Design

Based on the previously estimated hull resistance and calculated output, the specifications of the IPMSM were determined to be a rated output of 85 kW, a rated torque of 256 Nm, and a rated speed of 3200 rpm. In addition, the maximum operating speed of the motor during navigation was set to 4500 rpm.

Because the electrical and mechanical specifications of the IPMSM are directly related to the propulsion performance of a vessel, selecting accurate design parameters is essential. Accordingly, the DC voltage of the electric propulsion system was set to 800 V_dc, corresponding with the battery supply voltage. The number of poles and slots selected were 8 poles and 48 slots, considering both electromagnetic performance and mechanical characteristics. Moreover, to achieve a high output and high efficiency, rectangular copper wire and a hairpin winding method were applied. A water-cooling system was assumed for the cooling method, and the allowable current density was set to 15 A/mm² for the design process [22]. The basic electromagnetic and structural specifications set during the IPMSM design are listed in

Table 3. Design parameters and materials of the proposed IPMSM.

Parameter	Unit	Value
Core	-	35JN440
Permanent magnet	-	N50SH
Continuous current	${\mathrm{V}}_{\mathrm{r}\mathrm{m}\mathrm{s}}$	135
Current density	$\mathrm{A}/\mathrm{m}{\mathrm{m}}^2$	15
Winding type	-	Hairpin
Wire dimension	mm	3.6 × 2.5
Outer diameter of stator	mm	230
Outer diameter of rotor	mm	153.6
Airgap length	mm	0.7

, and the motor geometry is illustrated in Figure 2.

An FEA was conducted using ANSYS Maxwell 2D (Release 2024 R1) commercial software to evaluate the electromagnetic performance of the designed model. At the rated speed of 3200 rpm and a current of 135 A (RMS), the average torque was calculated to be 259.6 Nm, meeting the design requirement. Torque ripple was 4.16%, and the corresponding torque waveform is shown in Figure 3.

3. Concept and Design of the Asymmetric Dummy Slot

Design of the Asymmetric Dummy Slot

The previously designed IPMSM base model satisfied the required output and speed conditions. However, a certain level of torque ripple was observed in the output torque, which could cause instantaneous fluctuations in torque and lead to uneven propulsion and vibration problems. This becomes more critical in electric propulsion vessels with outboard-type structures where thrust is directly transmitted from the stern, which significantly affects both navigational stability and comfort. Generally, torque ripple is caused not only by mechanical factors such as cogging torque but also by electrical effects, particularly harmonic distortions in the back-EMF that introduce unwanted torque fluctuations.

The conventional symmetric dummy slot structure has been used to reduce cogging torque by increasing the least common multiple of the pole–slot combination through the insertion of grooves with symmetric widths and depths into the stator slots, which was utilized to reduce torque ripple [15,16,17,18,19]. However, this symmetric structure, due to the repeated application of identical shapes, shows only limited effectiveness in reducing torque ripple and lacks the design flexibility needed to respond to different operating conditions. To address this limitation, this paper proposes an asymmetric dummy slot structure where two dummy slots with different widths and depths are placed within each slot instead of using the conventional symmetric approach. This geometric asymmetry increases the flexibility of design combinations compared with symmetric structures and provides structural characteristics that can help to reduce torque ripple. Specifically, the asymmetric configuration enables fine adjustments to the dummy slot geometry, which allows certain harmonic components to be suppressed and is, therefore, more effective at reducing torque ripple. To verify the effect of this structure, an FEA-based performance analysis was conducted by comparing it with the symmetric dummy slot model.

In addition, most previous studies focused on reducing the THD of the back-EMF, cogging torque, and torque ripple through the application of dummy slots. However, few researchers have analyzed and compared the losses associated with the application of dummy slots. Therefore, in this paper, the design process included efficiency calculations that considered not only the THD of the back-EMF but also the AC copper losses, which are particularly significant in hairpin winding structures that use rectangular conductors.

In the asymmetric dummy slot structure shown in Figure 4, two dummy slots are placed within each stator slot, and they are designed with different widths and depths. Accordingly, four design parameters (X1, X2, X3, and X4) are defined for each slot. The parameter ranges are set to ensure a minimum thickness of at least 0.6 mm between the tooth tip and the slot shoe during dummy slot placement. Four cases were defined based on combinations of the design parameters to examine the effect of the asymmetric structure on the reduction in output torque ripple. All cases use the same reference values (X1 = X3 = 1.35 mm; X2 = X4 = 0.5 mm), and only one parameter on either the left or right side of the slot is varied in each case for comparison. Case 1 varies the depth of the right dummy slot (X4) while keeping the left dummy slot fixed; Case 2 varies the depth of the left dummy slot (X2) while keeping the right slot fixed; Case 3 varies the width of the right dummy slot (X3) while keeping the left fixed; and Case 4 varies the width of the left dummy slot (X1) while the right is fixed.

A comparison of torque ripple characteristics across four asymmetric dummy slot configurations analyzed using the FEA showed that the asymmetric design consistently reduced torque ripple compared with the symmetric structure (see Figure 5 and Figure 6). In Cases 1 and 2, which involved variations in slot depth, a tendency for increased torque ripple was observed under certain conditions. However, overall, the asymmetric structure was confirmed to contribute to torque ripple control. A more noticeable reduction in torque ripple was observed in Cases 3 and 4, which involved variations in slot width. Among these, Case 3 reached a minimum torque ripple of 4.13%, approximately 1.04% lower than the 5.17% observed in the symmetric dummy slot structure. These results demonstrate that asymmetrically adjusting the width and depth of dummy slots can effectively reduce torque ripple, and the structure proposed in this paper indicated meaningful improvements in torque ripple reduction compared with the symmetric reference structure.

Among the analyzed cases, Case 3 showed the most effective torque ripple reduction compared with the symmetric dummy slot structure. It exhibited a torque ripple of 4.13% when the width of the left dummy slot was set to 0.2 mm. As shown in Figure 7, the total harmonic content also decreased from 1.2139 Nm to 1.1832 Nm, which resulted in a reduction of approximately 0.0307 Nm compared with the symmetric structure. These findings suggest that the asymmetric structure of dummy slots causes non-uniformity in the magnetic flux distribution, which leads to differences in harmonic characteristics due to changes in the phase and amplitude of harmonic components. In a symmetric structure, slots arranged with identical shapes and spacing maintain the spatial periodicity of the flux distribution, which makes specific harmonic components likely to consistently repeat in terms of phase and amplitude. In contrast, an asymmetric structure induces different flux modulation effects at each slot, resulting in phase shifts or amplitude imbalances among specific harmonic components, which can potentially weaken or cancel specific harmonics. In the symmetric configuration shown in Figure 7, the 12th, 24th, and 36th harmonics maintained relatively high amplitudes of 0.5862 Nm, 0.3872 Nm, and 0.2405 Nm, respectively, which directly contributed to torque ripple. In the asymmetric structure, the 12th and 36th harmonics were reduced to 0.4514 Nm and 0.126 Nm, which represent decreases of 23% and 47.6%, respectively. However, the 24th harmonic increased to 0.6562 Nm, i.e., a 56.5% increase, which suggests that not all harmonics are uniformly suppressed and harmonic responses can vary by order depending on the geometry. Thus, asymmetry in dummy slot geometry has a significant impact on specific harmonic orders, effectively reducing torque ripple. Such an asymmetric structure can, therefore, serve not only as a simple geometric alteration but also as an effective design variable for torque ripple reduction. The optimal geometric combination for reducing torque ripple should be derived through design optimization.

4. Optimal Design Using Asymmetric Dummy Slots and Metamodeling

4.1. Formulation of the Optimal Design Problem

In this paper, a metamodel-based optimal design was conducted based on a total of five design variables. The process is summarized in Figure 8. In the initial design stage, minimizing torque ripple was defined as the primary objective function, while meeting minimum performance requirements for average torque and flux density were set as the constraint conditions. The five design variables included left and right widths, depths, and spacing of the dummy slots, as well as the angle between the magnets. Subsequently, experimental points were generated using Design of Experiments (DOE) sampling, and a 2D FEA was performed for each point. Based on these data, a sensitivity analysis was conducted to evaluate the correlation between the design variables and the objective function and constraints, thereby identifying the design variables that are significant for optimization. Using these selected variables, metamodels were created by applying 11 different metamodel techniques. Subsequently, the prediction performance of each metamodel was compared based on the RMSE test to determine the optimal metamodels for the objective function and constraints.

Finally, optimal conditions were derived by combining the optimization algorithm with these selected metamodels, and the resulting optimal conditions were then verified through the FEA. If the results were unsatisfactory, the process of partially adjusting variables and rebuilding the model was repeated, ultimately deriving the final optimal conditions. “Minimizing torque ripple” was set as the objective function, and efficiency, “THD of the back-EMF”, and torque at the rated speed were set as constraints, as shown in Equations (10)–(13). Here, torque ripple is the value at the rated speed, torque refers to the average torque at the rated speed, and efficiency represents the performance at rated conditions. Moreover, “THD of the back-EMF” was evaluated under no-load conditions. The torque constraint was determined based on the required motor output. The THD constraint was established in accordance with acceptable design standards for IPMSMs, while the efficiency constraint was derived from the performance of the base model. As shown in Figure 9, five design variables were used. X1 to X4 correspond with the dummy slot parameters described in the previous section, and the parameter ranges were set to ensure a minimum thickness of at least 0.6 mm between the tip of the tooth and the slot shoe during dummy slot placement taking manufacturability into account. X5 represents the angle between the permanent magnets, which affects the magnetic flux distribution and airgap flux, thereby significantly impacting both torque ripple and back-EMF waveform quality. Therefore, X5 was selected as an additional parameter to improve overall electromagnetic performance. The design range was also defined to ensure a minimum thickness of 0.6 mm between the bridge and the magnet for manufacturability and to prevent magnetic flux saturation by keeping the flux density in the stator teeth below 2 T. The minimum and maximum values for each design variable are summarized in

Table 4. Range of the design variables.

Design Variable	Unit	Lower	Initial	Upper
Slot width of left dummy (X1)	mm	0	0	2.7
Slot depth of left dummy (X2)	mm	0	0	1
Slot width of right dummy (X3)	mm	0	0	2.7
Slot depth of right dummy (X4)	mm	0	0	1
Inner angle between PMs (X5)	°	140	149.5	160

.

To minimize Torque ripple

(10)

Subject to Torque ≥ 256 Nm

(11)

THD of the back-EMF ≤ 8%

(12)

Efficiency ≥ 96.91%

(13)

4.2. Sensitivity Analysis

A sensitivity analysis was conducted to identify the design variables that impacted the objective function and constraints the most. The number of experimental points was reduced using the OA (Orthogonal Array) method, which is designed to lower the number of trials required by the FFD (Full Factorial Design) technique. The sensitivity of each design variable was evaluated using an ANOVA (Analysis of Variance), calculated based on the SS (Sum of Squares) value, which represents the contribution of each variable to the overall output deviation. The results are shown in Figure 10. The variable that had the highest impact on torque ripple was X5, which accounted for 50% sensitivity. Among the dummy-slot-related variables, X1, X3, and X4 showed relatively high contributions of 14%, 19%, and 14%, respectively. Although X2 had a relatively low sensitivity of 4%, it could still affect the overall performance when combined with X1 because it represented a parameter of the left-side dummy slot. Moreover, X1 and X2 had the highest influence on rated torque and efficiency, respectively, while X4 contributed most to the THD of the back-EMF, with a sensitivity of 56%. Therefore, considering both the objective function and the constraints, all five design variables were included in the optimization process.

4.3. Metamodeling

Because the optimal metamodeling technique varies depending on the FEA results of the experimental points, 11 metamodeling methods were compared using the RMSE (Root Mean Square Error) test value to select the best-suited model for each objective and constraint function. Representative metamodeling techniques include EDT (Ensemble of Decision Trees) [23], which combines decision-tree-based models to enhance prediction performance; KRG (Kriging) [24], which utilizes spatial correlations; MLP (Multi-Layer Perceptron) [25], a multi-layer artificial neural network with one or more hidden layers; and PRG (Polynomial Regression) [26], which approximates the relationship between dependent and independent variables using polynomial functions. PRG-based models include the Backward Step (BS), Forward Step (FS), Full Quadratic (FQ), Linear Regression (LR), Simple Cubic (SC), and Simple Quadratic (SQ) models. In addition, RBF-based techniques—which transform input data into a high-dimensional feature space for prediction—include RBF Interpolation (Int) and RBF Regression (Reg) models [27].

The prediction performance of each metamodel was evaluated using the RMSE test, where values closer to zero indicate a smaller error relative to actual FEA results [28]. In this study, the number of experimental points was determined using Equations (14) and (15), and model training was performed accordingly. The number of experimental points used was 54, and the number of test points was seven. The RMSE test results using the test points are summarized in

Table 5. RMSE test results.

Method	Torque	Efficiency	Torque Ripple	THD of the Back-EMF
EDT	2.687532413	0.017803635	0.626197272	1.407876543
KRG	1.888395072	0.013437225	0.578776084	0.893450905
MLP	0.917006968	0.034439428	0.364933765	0.464566997
PRG (BS)	1.042229463	0.02034773	0.609251074	1.671807566
PRG (FS)	1.045633839	0.018921095	0.541693674	1.876204119
PRG (FQ)	1.501535645	0.02295552	0.601260253	1.679955251
PRG (LR)	1.045633839	0.018921095	0.541693674	1.876204119
PRG (SC)	3.184395327	0.032698192	1.011832783	2.098816844
PRG (SQ)	1.045633839	0.018921095	0.541693674	1.876204119
RBF (Int)	0.92366497	0.026957972	0.779280382	2.22014301
RBF (Reg)	3.06963340	0.078997461	0.804976201	0.84233359

, and the best-performing metamodels are indicated with shading in the table. MLP was selected as the optimal metamodeling technique for torque, torque ripple, and the THD of the back-EMF, while KRG was selected as the optimal model for efficiency.

$$\mathrm{n}\mathrm{E}\mathrm{X}\mathrm{P}>\mathrm{Min} \left[\right(\mathrm{nDV}+1) (\mathrm{nDV}+2)/2, 10 \times \mathrm{nDV}]+5\times \mathrm{nDV}$$

(14)

$$\mathrm{nTS}=\min [0.1\times \mathrm{nEXP}, 10\times \mathrm{nDV}]$$

(15)

4.4. Optimal Design Results

The optimal metamodels selected for the objective function and constraints were combined with an optimization algorithm to derive the optimal solution. The optimization algorithm used was PORSM (Progressive Optimization with Response Surface Method) [29]. PORSM performs a global search across the initial design space, followed by a local fine search near the optimum. This stepwise approach not only enables the faster and more accurate identification of the optimal solution but also improves overall computational efficiency by progressively narrowing the search range. The results of the optimal design are summarized in

Table 6. Optimal design results using the PORSM algorithm.

Item		Unit	Initial	Optimal (Metamodel)	Optimal (FEA)	Improvement Rate
Design variables	X1	mm	0	1.55	-	-
	X2	mm	0	0.3	-	-
	X3	mm	0	0.41	-	-
	X4	mm	0	0.3	-	-
	X5	°	149.4	160	-	-
Objective function	Torque ripple	%	4.61	1.51	1.7	+2.91%
Constraints	Torque	Nm	259.71	257.58	257.42	−0.88%
	THD of the back-EMF	%	7.98	6.79	6.66	+1.32%
	Efficiency	%	96.91	96.95	96.95	+0.04%

.

As a result of the optimization, as shown in Figure 11, asymmetric dummy slots with varying widths and depths were added to the stator slots in the optimal model (b), compared with the initial model (a), and the angle between the permanent magnets was also increased.

5. Discussion

Torque ripple, which was set as the objective function, was 4.61% in the baseline model. Under the optimal conditions derived using the metamodel and optimization algorithm, the predicted torque ripple was reduced to 1.51%. To verify this result, an FEA was conducted, yielding a torque ripple of 1.7%, which represented an approximate 2.91% reduction compared with the baseline. As shown in the torque waveform comparison in Figure 12, the optimal model demonstrated a more stable waveform across the full electrical angle range, with reduced amplitude fluctuation relative to the baseline model. The rated torque (which was considered to be a constraint) was reduced from 259.71 Nm to 257.85 Nm (a decrease of −0.88%) but it still satisfied the motor’s required output. The THD of the back-EMF decreased from 7.98% to 6.66%, which indicates a reduction of 1.32%. In addition, the efficiency slightly increased from the initial value of 96.91% to 96.95%, which confirms that the optimization of the asymmetric dummy slots and the magnet angle did not introduce additional losses while maintaining the original performance. Figure 13 compares the magnetic flux lines of the baseline with the optimal models. It confirms that in the optimal model (b), the concentration and path of the magnetic flux lines were more uniformly distributed. This also indicates that the optimal design of the asymmetric dummy slots and the adjusted angle between the permanent magnets contributed to improved magnetic flux flow and helped to suppress torque ripple caused by imbalances in flux density. Furthermore, as shown in Figure 14 and Figure 15, the peak-to-peak value of the cogging torque in the baseline model was 3.20 Nm, while that of the optimal model was 2.02 Nm, which indicates a reduction of 1.18 Nm. In terms of harmonic components, the 12th and 24th orders showed values that were lower by approximately 0.19 Nm and 0.13 Nm, respectively. These results indicate that not only was the amplitude of the cogging torque reduced but also that the suppression of specific harmonic components contributed to the reduction in torque ripple. As a result, the optimal model successfully reduced torque ripple while preserving both electromagnetic performance and overall efficiency.

6. Conclusions

This paper presents the design and optimization of an IPMSM for use in the electric propulsion system of small fishing vessels. Based on actual operating conditions, the required output and ship resistance were calculated using the Savitsky method. To reduce torque ripple, a stator structure with two asymmetric dummy slots per stator slot was introduced. Through an FEA and a metamodel-based optimization approach, the electromagnetic performance and efficiency were evaluated and the optimal design parameters were identified. The main findings and contributions of this study are as follows:

• A stator structure with two asymmetric dummy slots per stator slot (differing in width and depth) was proposed. This design demonstrated more effective torque ripple reduction compared with the conventional symmetric dummy slot configuration.
• A metamodel-based optimal design process was carried out, including the estimation of rated torque based on actual cruising speed conditions, taking into account efficiency, THD, and torque.
• The analysis confirmed that the optimized asymmetric dummy slot structure is not only a geometric modification but also a critical design element that significantly enhances harmonic suppression and reduces torque ripple.

Therefore, the proposed asymmetric dummy slot structure is expected to be applicable not only to small fishing vessels but also to various types of marine mobility platforms with outboard-type propulsion systems, including passenger boats, yachts, and small cargo ships. It is anticipated to practically contribute to the design of high-power, high-efficiency motors required for eco-friendly marine applications.