An Overmodulation Algorithm With Neutral-Point Voltage Balancing for Three-Level Converters in High-Speed Aerospace Drives

In this article, a virtual space vector based overmodulation algorithm is presented for three-level neutral-point (NP) clamped converters in high-speed aerospace motor drives. With the proposed inscribed polygonal-boundary compression technique, the output voltage capability is enhanced under a lower crossover angle and compression coefficient. As a result, it brings an opportunity for the operation of the studied aircraft electric starter/generator (ESG) systems easily extending from the linear modulation range into the overmodulation region. Furthermore, an active capacitor voltage balancing control method is investigated to recover NP potential imbalance in the case of high modulation index and low power factor operating conditions. To simplify the digital implementation of the algorithm, a fast calculation approach is adopted in this work. The modulation performance of the proposed strategy is verified by both simulation and experimental results with a 45 kW, 32 k r/min ESG prototype system.

is the electric starter/generator (ESG) system. It can deal with several disadvantages that are common for state of the art starter/generator (S/G) systems [7]. Nowadays, the active rectification technology [8] opens opportunities for the integrated aircraft ESG systems, which employ new machine types that are free from existing limitations and allow for the novel design of high-performance S/G systems in the MEA. The key component of the ESG system is a high-speed machine and a bidirectional converter, as shown in Fig. 1. A permanent magnet synchronous machine (PMSM) is chosen since high torque-to-current ratio, high power density, and high robustness can be achieved. Despite the fact that the efficiency of the machine can sometimes be improved by increasing pole number, the low carrier-pulse ratio issue becomes harsh, particularly for the converter-fed high-speed motor drives [9]. For this reason, a 6-pole 36-slot surface-mounted machine is chosen as the optimal candidate that also has the lowest eddy-current losses [8]. Compared with a typical two-level converter, the three-level neutral-point-clamped (3L-NPC) topology, as shown in Fig. 2, is selected due to the utilization of lower blocking voltage switches, better output harmonic performance, and reduced EMI emissions. 0885-8993 © 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
In the ESG system, the PMSM is mechanically coupled to an engine shaft via a gearbox and electrically connected with a 3L-NPC converter, which interfaces the S/G system to the onboard electric power system. In the startup process, the converter serves as an inverter that drives the PMSM to crank and accelerate engine shaft. Once the engine is up to a self-sustained speed, the engine ignition activates. After that, the ESG system goes into standby mode. When the PMSM shaft speed approaches 20 k r/min, the generation mode starts. In this mode, the engine drives the machine to generate electrical power, and the converter works as a rectifier to convert the variable frequency ac into dc that supplies onboard loads.
Due to the high-speed operation of the machine, the converter uses deep flux-weakening control to ensure that the back electromotive force is within converter limits. However, not only large flux-weakening currents cause more machine copper losses but also the output electromagnetic torque capability in the constant power region is weakened as the speed of PMSM increases, and is constrained by stator voltage. In addition, the 3L-NPC topology has an inherent NP potential deviation because of the split dc-link capacitor bank, which results in extra overstress across switching devices and output distortion, particularly for the ESG system operating in stringent conditions, i.e., high modulation index and low power factor. Consequently, these significant challenges are required to be settled for high-performance three-level converter-fed high-speed aerospace motor drives in MEA applications.
In recent years, numerous advanced pulsewidth-modulation (PWM) strategies have been studied to tackle the unbalanced capacitor voltage problem in 3L-NPC converters [10]- [12]. Besides, the model-predictive control (MPC) based NP voltage balancing methods [13], [14] are incorporated in the space-vector-PWM (SVM) modulator to obtain exceptional dynamic performance of systems. Compared with the aforesaid approaches, the nearest-three virtual space vector (NTV 2 ) modulation concept presented in [15] is more attractive as it achieves inherent NP voltage balance for all loads. Although the advantage is at the expense of additional switching losses, more degrees of freedom by this technique can bring other merits, such as common-mode voltage reduction [16].
To boost the dc-link voltage utilization, a number of overmodulation algorithms have been developed by researchers for two-level [17]- [20], flying capacitor [21], [22], cascaded [23], [24], and diode-clamped [25]- [32] multilevel inverters. The existing literature about the overmodulation strategy for the NPC topology is discussed in the following. In [25], the authors propose an overmodulation method by calculating three-phase modulation waves, which is beneficial for simple implementation in digital controllers. With the help of the neural network, a classification SVM technique [26] in overmodulation region has been studied for 3L-NPC converters, which ensures a smooth linear transition and lower complexity. It is noted that the authors employ the nearest-three space vector (NTV) scheme and predefine a series of switching sequences to maintain two capacitor voltages. For the high-power applications with a low pulse-ratio, a synchronized SVM strategy [27] is proposed to operate induction motor drives in the overmodulation region. In order to avoid trigonometric computations, a duty cycle-based overmodulation strategy is presented in [28], together with a dedicated NP voltage balancing control method. In [29], a timing-correction-based algorithm is investigated to gain the output voltage under overmodulation conditions. Also, the approach is implemented in a field-programmable-gate-array (FPGA) based real-time control platform to attain the minimum computational resources. With respect to low power factor loads, however, the aforementioned methods pose a challenge for capacitor voltage balance, as well as elimination of low-frequency NP potential oscillation. To address these deficiencies, a hybrid SVM strategy is introduced in [30] to operate a 3L-NPC inverter in the overmodulation region. This approach abandons medium vectors to mitigate voltage fluctuation in the NP, but harmonics performance of phase current may suffer from nonnearest-three-vectors in a higher output voltage region. In [31], a carrier-based PWM (CBPWM) scheme for 3L-NPC inverter-fed induction motor drive systems with dynamic loads has been studied to extend the operating point into the overmodulation region. Besides, the technique is capable of keeping two capacitor voltage in a balanced state when NP potential deviation occurs. It is worth noting that calculating an appropriate zero-sequence voltage components is required. The work reported in [32] discusses a novel overmodulation scheme that achieves the balanced capacitor voltages and alleviated low-frequency NP voltage fluctuation over the entire range of modulation indices and power factor angles, even for n-level NPC topologies. While the performance is desirable by applying virtual space vectors (VSVs), maintaining capacitor voltage balanced is not discussed when NP voltage shifts. Additionally, the presented hexagonal compression limit may impair overmodulation capability to some extent.
In this article, a new variant of the VSV-based PWM algorithm is proposed for the operation of aircraft ESG systems extending into the overmodulation region. Compared with the prior method [32], the presented trajectory of the reference voltage vector contributes to a better overmodulation capability for converterfed high-speed aerospace motor drives. With the introduced NP potential balancing control strategy, two capacitor voltages are effectively kept at a balanced state if NP voltage drifts away. Notably, the technique is easily implemented with a fast calculation approach, which aims to reduce the computational complexity for the digital controller.
The rest of this article is organized as follows. The principles of the ESG system are briefly introduced in Section II. The proposed capacitor voltage balancing control-based overmodulation algorithm is studied in Section III. Simulation and experimental results are presented in Section IV to evaluate the modulation performance. Finally, Section V concludes this article.

A. PMSM Model and Operating Constrains
The PMSM mathematical model equations in the dq coordinates are expressed as follows: where T em represents the electromagnetic torque, n p refers to the number of pole pairs, ϕ f is the permanent magnet flux linkage, V d , V q and i d , i q indicate the direct and quadrature components of motor voltage and current, respectively, L d and L q are the dq-axis inductances, ω e denotes electrical angular frequency, and R is the stator winding resistance. For the applied surface-mounted PMSM in the ESG system, L d equals L q . Hence, the electromagnetic torque of machine is only provided by permanent magnet torque, which is the first term in (1). Since the maximum machine stator voltage, indicated by V lim , is determined by dc-link voltage and PWM strategy, the corresponding limitation can be written as Considering the capacity of designed machine/converter, the maximum current value, manifested by I max , should fulfill The solution of (4) and (5) is an overlapped area of voltage and current limit circle, which, therefore, restrict the allowable operating points of the ESG system.

B. Control Design of ESG System
The whole system control block includes flux-weakening, machine speed, current loop, and droop control, as illustrated in Fig. 3. Speed control is activated in the startup process. DC-link current is manipulated by droop characteristics in the generation mode. The vector control technique is used for machine decoupling purpose, which allows controlling flux and torque independently. Therefore, machine torque represented as active power is controlled by q-axis current. As the maximum phase voltage is limited by a definite dc-link voltage supply of 270VDC, a large negative current needs to be injected into d-axis during the high-speed operation, which results in a lower power factor angle.

C. Three-Level NPC Converter, SVM Strategies, and NP Voltage Imbalance Issue
As shown in Fig. 2, each phase of the converter consists of four semiconductor devices (V T x1 -V T x4 ) and two clamping diodes (DZ x1 and DZ x2 ), where x∈{A, B, C}. Two equal capacitors (C 1 and C 2 ) are series-connected to form the dc-link. V T x1 is complimentarily switched with V T x3 . V T x2 and V T x4 follow the same manner. The switching state [P] means gating signal (S x1 , S x2 ) is (1,1), which represents output voltage of V dc /2. The switching state [N] manifests gating signal (S x1 , S x2 ) is (0,0), which denotes output voltage of −V dc /2, where V dc refers to the full dc-link voltage. Otherwise, gating signal (S x1 , S x2 ) is chosen as (0,1), the output voltage is zero and switching state is expressed as [O]. The switching principle of the 3L-NPC converter is detailed in Table I.
With regard to the magnitude of space vectors listed in Table II, 27 switching states generated by the converter can be grouped into four categories: zero, small, medium, and large vectors, wherein 12 are small ones while the number of large and medium ones are both 6, and the rest are zero vectors.
It is assumed that the reference voltage vector is located in the subsector 3 of Sector-I. The NTV scheme uses V 1 , V 3 , and V 4 to synthesize V ref , as shown in Fig. 4(a). According to the volts-second balance rule, (6) can be obtained where d 1 , d 2 , and d 3 indicate the duty cycle of V 1 , V 3 , and V 4 , respectively. And then, the solution of abovementioned equations can be yielded where θ represents reference voltage vector rotating angle, m refers to the modulation index defined by Here, the four-state synthesis approach, i.e., reference voltage vector is composed of four switching states from the selected NTVs, is adopted to discuss the NP voltage imbalance issue. For example, if V ref falls into subsector 3 of Sector-I, the timeaveraged NP current is given by where σ∈[0, 1] denotes a distribution factor of redundant small vector pairs. Thus, the capacitor voltage deviation for each switching period can be calculated as where C and T s represent the dc-link capacitance and a switching period, respectively. By regulating σ in (9), NP currents produced by small vectors can be used to compensate for the accumulated NP potential rendered by medium vectors. However, it should be pointed out that the strength of capacitor voltage balancing becomes weak when the modulation index arises to unity and power factor angle towards ±90 • [12].
In contrast, the NTV 2 scheme employs a series of VSVs that is modified by basic space vectors with a specific proportion. The definition of zero and large vectors, denoted as V 0 , V L1 , and V L2 , are the same as the NTV scheme, whereas V S 1 , V S 2 , and V M 1 are virtual small and medium vectors defined by When V ref falls into subsector 4 of Sector-I, as shown in Fig. 4(b), the corresponding duty cycle is calculated as As the polarity of NP currents associated with redundant small vector pairs are opposite and the sum of balanced three-phase currents is zero, VSVs have no impact on NP potential deviation. Therefore, the NTV 2 scheme achieves two balanced capacitor voltages for all load conditions.

III. PROPOSED VSV-BASED OVERMODULATION ALGORITHM
In accordance with the modulation index defined by (8), the operating regions of a converter can be classified into the following three parts: linear modulation (LM), overmodulation mode-I (OM-I), and overmodulation mode-II (OM-II).
In the LM region, the reference voltage vector traces a circular trajectory inside the hexagon. Accordingly, modulation index (0 ≤ m ≤ 1) arises with increasing magnitude of output voltage until the trajectory is inscribed in the hexagon side. Within linear range, the duty cycle for each space vector can be calculated by (6). The OM-I denotes the radius of desired reference voltage vector is between the inscribed circle and circumscribed circle of the hexagon, which indicates a higher achievable modulation index (1 < m ≤ 1.05). In this mode, the tip of modified reference voltage vector transits from a circular trajectory and moves along the hexagon side when it exceeds the boundary of hexagon. Thus, a crossover angle (θ c ) that keeps the trajectory inside half of the sextant needs to be controlled for obtaining adequate modulation depth, as shown in Fig. 5. To further increase modulation index, the reference voltage vector is fixed at the vertex for a period of time, and then jumps to the straight line of the hexagon by a holding angle (θ h ). When θ h equals 30 • , the discrete six-step mode operates, which constrains m max of OM-II at 1.1 [17].
Since the output variables of a converter steeply deteriorate in the OM-II [18], [20], the acoustic noise and torque ripple issues are much severe, which is not suitable for the applications of high-speed aerospace motor drives. As a result, the scope of this study primarily focuses on the operation of OM-I, which provides a chance to achieve higher dc-link voltage utilization, enhance output torque capability, and reduce large-required fluxweakening current for the aircraft ESG system. For the purpose of simplifying PWM calculation process, the proposed VSVbased overmodulation algorithm is carried out in a transformed reference frame. The detailed descriptions of each subsection are given in the following.

A. Fast Calculation Approach
With (13), the traditional orthogonal plane can be shifted into a sextant one for simplifying the duty cycle expression using trigonometric functions By projecting (6) into this frame, V g and V h are yielded as where  Fig. 4(b).

B. Inscribed Polygonal-Boundary Compression (IPBC) Technique
As mentioned above, before tracking circular trajectory in linear range again, the reference voltage vector command is programmed to coincide with the hexagon side for the purpose of a higher fundamental component in output voltage. As seen from Fig. 5, with the geometric relationships of VSV-based SVM, the following equations are given: To approximate the red straight-line in Fig. 5, the duty cycle of each VSV can be easily calculated by Table III and (15). As shown in the Appendix, however, the dwell time of virtual medium vector (d 3 T s ) constantly equals zero at that time. It means that only large vectors are used, which, therefore, violates the three-level operation of the converter. This phenomenon not only lowers the efficiency of the ESG system but also incurs a steady-state capacitor voltage error if NP potential imbalance initially exists. To address the demerit, the hexagon side was scaled down by a compression coefficient λ [32]. Thus, it makes full use of virtual medium vector and takes its dwell time to 3(1−λ), as derived in the Appendix. Fig. 6 shows the space vector diagram of polygonal-boundary limit, in which the red straight-line denotes the prior hexagonal-boundary compression (HBC) limit [32] that is formed by A(λ HBC ,0) and C(0,λ HBC ) in the first sextant.
In order to fully utilize the margin of the hexagon, not only the tip of V ref passes through A (λ IPBC ,0) and C (0, λ IPBC ), but also the middle point of hexagon side is involved, as shown in Fig. 6, which is an IPBC limit represented by a blue dodecagon and forms the basis of the proposed solution. For example, B(0.5,0.5) is a turning point for Sector-I. It is worth noting that both two boundaries gradually coincide with the hexagon side when the compression coefficient converges to 1, and output voltage eventually reaches the upper limit of OM-I.
For a given λ, the presented trajectory of normalized |V ref | in a quarter cycle is given by and θ = ωt (ω is the angular fundamental frequency). Given that the selected compression coefficient is 0.9, the αβ-axis components of the proposed technique and its counterpart under different θ c are shown in Fig. 7, in which red and blue waveforms refer to V α and V β generated by the HBC and IPBC scheme. As shown, the proposed approach makes these two components more sinusoidal. Hence, it would be expected to attain less distortion in output voltage and current.
The introduced track of V ref with its output phase voltage in β-axis is shown in Fig. 8. The amplitude of its fundamental component can be, thus, calculated by Fourier series expansion   With (17), the desired modulation indices with different compression coefficients and crossover angles can be attained and described by Figs. 9 and 10. As shown, the dc-link voltage utilization of the IPBC scheme is higher than that of the HBC scheme under the same conditions. In other words, λ and θ c of the proposed technique are allowed us to be set at a smaller value to achieve the same overmodulation capability as the prior method. Besides, it can be seen that m max of the proposed approach is the same as that of the counterpart.

C. NP Voltage Balancing Control
Although the applied VSVs can maintain NP voltage naturally balanced irrespective of modulation index and power factor angle, the voltage balance between two capacitors cannot be facilitated by itself if the initial NP potential imbalance exists under overmodulation conditions. Therefore, active capacitor voltage balancing control must be involved to handle the divergence. When extra electric charges are accumulated in the last control period, the capacitor voltage shifts. With (10), the required NP electric charge for the convergence of capacitor voltage within a sampling time can be expressed as where ΔQ is defined as the compensated charge.
To meet the abovementioned requirement, a straightforward solution is to manipulate the dwell time of space vectors that can provide NP current with appropriate polarity. Moreover, it is known that when the converter runs for rectifying, NP potential tends to arise up with more participation of N-type small vectors and vice versa. Hence, according to the operational mode of the ESG system and dc-bias in the NP, the dwell time of these vectors for composing VSVs can be regulated by an allocation coefficient k, as given by (19), where V dc1 and V dc2 indicate the upper and lower side capacitor voltage, respectively When V ref falls into subsector 4 of Sector-I, the modified switching intervals under overmodulation conditions are shown in Fig. 11, in which (g, h) is the real-time coordinate of the blue line in Fig. 6. To evaluate the ability of the introduced NP voltage balancing control method, the fundamental component of output voltage should be set at the same value. For example, the compression coefficient of HBC and IPBC limit are 0.98 and 0.95, respectively, and the crossover angle is set to be 12.5 • . By doing so, the desired modulation indices are both at 1.01, and the reference voltage vector is synthesized in subsectors 3-5. Here, Sector-I is chosen as a representative example to investigate. The dwell time of active VSVs and the compensated charge ΔQ based on (10) and (18) are illustrated in Fig. 12(a) and (b), respectively, in which "+" and "−" represent the maximum and minimum value by corresponding approaches. As can be seen, although the compensation capability of the presented IPBC  technique is a bit weak as compared to that of the prior method, it can still resolve the NP voltage imbalance issue for all loads.
Noteworthy, the output line-to-line voltage might be distorted by the modified duty cycles of small vectors to some extent. Nevertheless, it facilitates a tradeoff between capacitor voltage balance and short-term output harmonic performance. In addition, the distortion caused by the onsite closed-loop control will be quickly dismissed when the NP potential is maintained at a balanced state again.

IV. PERFORMANCE ANALYSIS
In this section, the performance of the proposed overmodulation algorithm is validated by simulation and experimentation. The 3L-NPC converter-fed PMSM drive systems are simulated in the Simulink/PLECS environment. The ESG system initially runs in the startup process. The flux-weakening control is triggered at 0.5 s. The power generating mode starts by connecting a 10 kW resistive load to the dc-link at 1 s. The presented approach is activated at 1.1 s. The experimental validation is tested on a 45 kW, 32 k r/min laboratory-built ESG prototype system, as shown in Fig. 13. A 150 kW prime mover that emulates aircraft engine shaft and an oil-cooled high-speed machine are located in an isolated room for the safety viewpoint. The controller platform uses a TMS320C6713 chip and an Actel FPGA-ProAsic3 A3P400 kit. The torque of the PMSM is measured by ET1303 Torquemeter. dq-axis currents are displayed by the graphic user interface software. AC-side parameters are monitored by PPA5530 power analyzer. Since the direction of phase current changes rapidly in light-load conditions, it weakens NP voltage balance in the OM-I. Thus, the experimental verifications mainly focus on the motor drives operating in the overmodulation region  during the generation mode, particularly in a light-load condition, whereas the results in the starter mode are not presented as it accounts for a short time compared with the whole operation time. The system parameters are detailed in Table IV.

A. Improvement of DC-Link Voltage Utilization
Without loss of generality, θ c of the proposed strategy is set at 12.5 • and λ is chosen as 0.95. The simulation results of dq-axis currents illustrated in Fig. 14(a) describe the required torque and flux-weakening currents for the motor drives in entire speed range operation. As shown, i d is significantly reduced after 1.1 s. It indicates that the IPBC scheme offers a higher dc-link voltage utilization, thereby extending the operation of the ESG system into the overmodulation region, while the ripple in i q is unavoidably increased, which trades an enlarged converter output voltage range against output harmonics. The experimental comparison with the same θ c is performed at a machine speed of 20 k r/min, while in order to validate the dc-link voltage utilization improvement by the IPBC scheme and the correctness of m = f (θ c , λ) shown in Fig. 10, the λ of the prior scheme and the presented one refer to 0.97 and 0.95, respectively. Besides, the test can demonstrate that a more reduced flux-weakening current is achieved with a lower compression factor for the IPBC scheme owing to a higher output modulation index. As evident in the results from Fig. 14(b), both methods can lower i d , whereas the provided algorithm, contributing to further deepen 1.5% of modulation depth, brings this value from −115 to −107.5 A, as opposed to i d of −110 A with the HBC scheme at the same operating conditions. Moreover, it can be observed that the ripple of i q by the IPBC scheme is smaller than the HBC counterpart.

B. Operation of High-Speed Machine
The whole operating trajectory and relevant limits given by (4) and (5) are shown in Fig. 15(a). In the startup process, denoted as the constant torque region, the maximum torque per amp line is followed by point A. The flux-weakening control activates at point B in the constant power region. Then, the trajectory tracks the voltage limits based on the PMSM speed and current limits set by the converter. The ESG system is in standby mode after a machine speed of 10 k r/min, which is denoted by point C. i q maintains at zero and operation status is at point D until the generation mode starts. At that time, the S/G systems run in the power generating region and the modulation index is ramping up to 0.95. As the flux-weakening current is dropped by a boosted dc-link voltage utilization, the operating point subsequently moves from point E to F.
To be consistent with the experimentation presented in subsection-A of Section-IV, the maximum torque by the HBC (λ = 0.97) and the IPBC (λ = 0.95) scheme is measured at different machine speeds, which is helpful to demonstrate the output torque capability improvement. It can be seen from Fig. 15(b) that the proposed strategy gains T em = 14.3 N · m at a shaft speed of 20 k r/min, which means the output torque capability is roughly increased by 3.6% as compared to the HBC limit. In comparison with the LM operation, the proposed modulation algorithm enhances 10% output torque capability for the studied drives.

C. Steady-State Performance
When the PMSM operates at a speed of 20 k r/min, the steadystate performance under the LM (m = 0.95) and the OM-I with the proposed IPBC method (λ = 0.95 and θ c = 12.5 • ) are tested. The results from Fig. 16(a) show that the line-to-line voltage by the proposed overmodulation technique features five levels and constant clamping instants in some periods. Apart from that, two capacitor voltages are well balanced.

D. Dynamic-State Performance
To validate the dynamic-state performance of the proposed overmodulation strategy, rotation speed, and load are both changed, where λ and θ c are set at 0.95 and 12.5 • , respectively. Initially, the output active power of the ESG system is 3 kW under a machine speed of 20 k r/min, in which the q-axis reference current command is −6 A. After that, the PMSM speed descends to 18 k r/min while the load is increased to 10 kW, where the q-axis reference current demand is −25 A. The results from Fig. 16(b) show a smooth transient performance regarding NP voltage, phase current, and line-to-line voltage.

E. Capacitor Voltage Balancing Performance
The active capacitor voltage balancing performance is verified under a dc-link voltage utilization ratio of 1.02. That is, λ for the HBC and IPBC scheme is 0.98 and 0.95, while θ c equals 12.5 • . As shown in Fig. 17, the initial NP voltage deviation is 200 V in the LM region, with an average modulation index of 0.95 and power factor of 0.12. When switching to the proposed overmodulation strategy, the two capacitor voltages can be kept at a balanced state again. Compared with the HBC scheme, an additional time of 50 ms is required to correct the unbalanced NP voltage. Fig. 18 presents the fast Fourier transform (FFT) analysis of the stator current when the compression coefficient is set at 0.95 and the crossover angle is controlled at 12.5 • . It can be seen that the total harmonic distortion (THD) by the proposed strategy is less than that by the HBC scheme, which is consistent with moretical analysis in Section III. The weighted THD (WTHD) comparison of the line-to-line voltage under different λ and desired m is illustrated in Fig. 19. From the results, it can conclude that the presented inscribed polygonal-boundary limit obtains a lower WTHD as compared to the hexagonal compression limit.

G. Power Losses Analysis
Power losses analysis of the semiconductors is carried out by our applied Infineon F3L400R07ME4_B22/B23 insulatedgate bipolar transistor module that has the maximum collectoremitter voltage of 650 V and continuous collector current of 400 A. According to the manufacturer's datasheet, the losses can be obtained through the PLECS tools. Given that θ c is set at 10 • , while λ for HBC and IPBC limits is selected as 0.97 and 0.94, respectively. Switching frequency is set to be 16 kHz. Before extending into the overmodulation region, the modulation index maintains at 0.95in the linear range. Under these operating conditions, with 35 kW of active power generation at 20 k r/min,  the switching device losses distribution map is elaborated in Fig. 20. As can be seen, the proposed overmodulation algorithm can lower conduction losses compared with the linear range operation. Besides, due to increased dc-bus terminal clamping instants, the switching losses are reduced as compared to that by the LM. In order to experimentally test the converter's total losses under the above overmodulation conditions, a power analyzer PPA5330 is used to monitor the input and output parameters when the target system delivers an active power of 25 kW at a rotation speed of 20 k r/min. A resultant of 1.24 kW power loss is measured under the presented approach, while that value in the case of the benchmark method is about 1.30 kW.
By importing the data of stator currents into the finite element analysis model of the employed PMSM, the machine copper loss can be calculated in a steady state. Compared with typical operation in the LM, that value is significantly decreased from 189.96 to 160.18W with the upper limit of the OM-I, which means a nearly 15.7% reduction.
As the overmodulation conditions lead to a lower required flux-weakening current, the loss issue can be thus mitigated. Nevertheless, the more boosted dc-link voltage utilization, the less output harmonic performance will be for the aviation electric power system. Consequently, it comes as a tradeoff between efficiency and stability to some extent.

H. Computational Time Performance
The execution time of the proposed overmodulation algorithm is tested by the applied TMS320C6713 DSP Starter Kit with a 225 MHz system clock. The results present that it costs 57.2 μs to implement the presented technique in the gh reference frame, while 73.8 μs is the time to implement the same algorithm in the conventional αβ reference frame, which indicates nearly 30% computational burden reduction.

I. Comparison With Other NP Control-Based Overmodulation Strategies
Compared with existing NP control-based overmodulation strategies [25]- [31] for 3L-NPC converters, the proposed strategy embraces the following advantages.
1) As the configured switching patterns generate extra NP currents to realize net-zero NP current, the proposed capacitor voltage balancing control is effective for high modulation index and low power factor operating conditions. However, the methods in [25]- [29] can hardly achieve this target.
2) Due to the adopted nearest-three VSVs, less output distortion would be expected as compared to the work of [30] that omits space vector numbers in high output voltage region.
3) The complexity caused by massive trigonometric functions is reduced by algebraic operations in the gh plane. While the CBPWM scheme reported in [31] gains overmodulation conditions, the appropriate ZSVs need to be calculated. 4) With the proposed IPBC technique, the output voltage converges to the upper limit of LM, which is beneficial for smooth transition among various compression coefficients, particularly for traction and servo drives, where loading conditions and speed change frequently [31]. On the other hand, due to the nonmonotonous relationship between θ c and m shown in Fig. 10, the continuity of boosted dc-link voltage utilization would be affected to some extent in comparison with the HBC scheme. Nonetheless, the minimum modulation depth by the provided strategy gaining near unity might make this issue marginal. The dedicated compensation method will be explored in our future study.

V. CONCLUSION
In this article, a VSV-based PWM algorithm has been presented for the 3L-NPC converter-fed high-speed aerospace drives in the MEA, which aims at the operation of aircraft ESG systems easily extending into the overmodulation region. With the provided IPBC technique, the major contributions of this work lie in the following.
1) DC-link voltage utilization is improved under a lower crossover angle and compression coefficient, which offers further benefits of flux-weakening current and machine copper loss reduction, together with an enhanced maximum output torque for the applied PMSM. 2) In order to remain a balanced NP potential under stringent operating conditions, an active capacitor voltage control method is developed to supplement the algorithm.
3) A fast-executed approach is adopted as an alternative to overcome the computational burden caused by massive trigonometric functions in the orthogonal plane, which is helpful for semiconductors with a short switching period. The conclusions are applicable to all 3L-NPC topologies in electric motor drive systems. Simulation and experimental results confirmed the good overall performance of the proposed strategy in terms of overmodulation capability, NP voltage balance, losses, phase current, and ease of use.

APPENDIX
The duty cycle of virtual medium vector is calculated as follows: 1) without HBC limit = 0 2) with HBC limit