Single-Phase Mains Fed Three-Phase Induction Motor Drive Using Improved Power Quality Direct AC–AC Converter

This article presents an alternative ac–ac converter topology and its control for low-speed three-phase induction motor drives fed from single-phase ac mains. The proposed matrix converter based drive system eliminates the dc link capacitor, thus facilitating high power density and system reliability. The primary challenge in controlling a <inline-formula><tex-math notation="LaTeX">$1\phi$</tex-math></inline-formula> to <inline-formula><tex-math notation="LaTeX">$3\phi$</tex-math></inline-formula> matrix converter is to navigate the discrepancy of instantaneous power across the <inline-formula><tex-math notation="LaTeX">$1\phi$</tex-math></inline-formula> grid and <inline-formula><tex-math notation="LaTeX">$3\phi$</tex-math></inline-formula> load via a direct ac–ac converter. In order to address this concern, it is proposed to use mathematical modeling of the motor and input <italic>LC</italic> filter for state selection, in order to deliver improved current waveforms. Enhanced performance is demonstrated through simulations, and further validated using experimental results. Particularly improved power quality performance is achieved at low motor speeds, where precise low-speed sensorless operation is ensured through suitable speed observer.

1φ grid current and voltage. i in , v in Current and voltage at converter input. i abc , v ab 3φ motor currents and stator voltage (line-line). φ m , φ r , φ s Magnetizing, rotor and stator magnetizing flux. T s Sample time for predictions. ω b Base frequency (= 2π × 50) rad/s. ω r , ω m Electrical and mechanical rotor speeds. θ r Rotor flux position. R s , R r Stator and rotor resistances. X sl , X rl Stator and rotor leakage reactances. C F , L F , r F Input filter capacitance, inductance, and internal resistance of L F . I r , I s Root mean square values of rotor and stator currents. λ Cost function weight for 1φ mains current.
x opt Predicted variable "x" for the most optimum switching state in the previous sample. x Estimation of any variable "x.". x * Reference value for any variable "x.".

I. INTRODUCTION
Electric motors are the most predominant loads on power networks worldwide, with three-phase (3φ) induction motors often being the most preferred choice due to their ruggedness, power density, and cost benefits. However, in a large number of consumer applications where 3φ power supply is not available, less efficient single-phase (1φ) motors are used. Alternatively, 3φ motors may be used on 1φ ac mains by using 1φ-3φ static power converters. Typically, multistage dc link converters are used to drive 3φ motors from 1φ ac sources [1]. Nonetheless, such converters often have a poor lifetime and increased footprint due to the use of dc link electrolytic capacitor. These capacitors are one of the leading causes of unreliability in motor drive systems [2]. Besides being a common point of failure, dc link capacitors hamper high power density. Especially in 1φ-3φ power converters, dc link capacitors are the most sizeable components in the drive due to the pulsating nature of 1φ power supplies. Although naturally commutated ac-ac converters have been previously proposed as a compact alternative [3], they exhibit significantly adverse power quality performance. Some topologies for 1φ-3φ static power converters are discussed in [4], [5], [6], [7], however, these have sizeable interdispersed energy storage elements and/or higher device count when compared to the topology considered here. This article proposes the use of 1φ-3φ ac-ac matrix converter (MC) for driving a three-phase induction motor from a single-phase supply [8]. MCs are particularly popular for providing a nearly pure semiconductor solution for power conversion, which makes them much smaller than ac-dc-ac converter based drives [9]. Such motor drives are, therefore, ideal for applications where space and weight constraints exist. Additionally, in ac-dc-ac drives, the motor braking energy is fed back to the dc link capacitor, which needs to be sized enough to accommodate that energy, and be accompanied by a bleeder resistor to dissipate it. The MC-based drive besides being capacitorless is inherently bidirectional and feeds the braking energy from the motor back into the grid. Furthermore, in the coming years, monolithic bidirectional devices are expected to be commercialized and this could pave way for reducing the device count in MC fed drives. However, unlike 3φ-3φ MC, the converter is not quite popular in 1φ-3φ applications, due to an inevitable discrepancy in instantaneous power across this converter, which leads to distorted current waveforms on either the 3φ motor side [10] or the 1φ source side [11] of the converter. This article proposes a niche application and strategic control to overcome this limitation. A brief comparison of the proposed topology with previously documented converters for 1φ-3φ power conversion is outlined in Table 1.
The 1φ-3φ MC topology as shown in Fig. 1 has no significant energy storage devices. However, upon integration with a motor load and an input LC filter, distributed energy storage components are established in the circuit, which have potential to be strategically used to improve power quality metrics of the converter, especially with dedicated use in applications with low output frequency [8]. Thus, a low speed drive is an ideal candidate for such drives, so that the fluctuating 1φ ac power (pulsating at two times the frequency of 1φ ac mains), manifests itself as a high-order frequency corresponding to the low-frequency motor currents. Therefore, this fluctuating component can be smoothened out by the machine inductances. In this article, speed sensorless control of a 1φ to 3φ matrix converter fed induction motor drive is presented for low speed applications. The MC is controlled using two objective model predictive control with finite control set [12], [13], which employs discrete models of the motor load and input LC filter, for concurrent control of 3φ motor currents and 1φ source current. Further, the article also proposes an improved motor speed observer for enhanced precision of the low speed estimate.
It is well known that flux estimation and encoderless performance over the low speed range are challenging due to factors such as converter voltage drops, integration offset, and parameter variation being significant at low stator frequency. For precise low speed estimates, various observers are reported in the available literature [14]. Among these, the full order Luenberger observer is a promising technique [15]. However, full order adaptive observers often exhibit stability issues at low speeds, particularly during regeneration as the motor traverses through zero speed. Available literature proposes a number of methods for overcoming stability problems near zero stator frequency. One of the widely accepted methods is based on gain scheduling for the observer gains [16] to achieve stable operation near zero frequency. This article uses the discretized model of the induction motor without any observer gains to predict the flux and stator current variables for each possible converter switch state. The challenge of precision in low speed estimation in the presence of 1φ grid power pulsation is tackled by means of refinements in the speed estimation algorithm. First, the rotor speed is determined from stator current errors in the synchronously rotating (dq) frame instead of the stationary frame of reference (αβ). This facilitates the use of digital filters to enhance the quality of estimated signals. An additional compensation term correlated with the error in d-component of motor current is used in generating a more precise rotor speed estimate. The proposed drive system is a power-dense and reliable alternative for 3φ low-speed motor drives fed from 1φ ac mains in numerous consumer applications, such as automatic door closers, lifts and elevators, low speed centrifuges, treadmills, exercisers, conveyor belts, fuel filling machines at gas stations, and low-speed high volume fans.
The rest of this article is organized as follows. Twoobjective finite set predictive control for 1φ-3φ matrix converter is described in Section II. The control of 3φ motor currents is detailed in Section II-A and that of 1φ ac mains current is outlined in Section II-B. Section III discusses state estimation and proposed modifications to attenuate 1φ power pulsations from the motor speed estimate. The proposed control and estimation algorithm is validated through simulation and experimental results reported in Section IV. Finally, conclusions regarding the suitability of 1φ-3φ MC fed drive are drawn in Section V.

II. TWO OBJECTIVE PREDICTIVE CONTROL FOR 1φ-3φ MATRIX CONVERTER
The main objective in 3φ-3φ matrix converter is often the concurrent control of grid currents and load voltages (or load currents). However, in the less explored 1φ-3φ MC topology, literature reports either sinusoidal 1φ source current or sinusoidal 3φ load waveforms [8]. Thus, one objective is achieved at the cost of compromise on the other control objective. This is because of the ineptitude of the ac-ac converter to handle instantaneous power imbalance between its two sides, in the absence of an intervening dc link [8]. Despite the fact, that lack of energy storage exacerbates power quality deterioration in this converter, it is crucial to retain this feature in order to impart compactness, high power density, and reliability in comparison to other ac-dc-ac converter counterparts.
This article addresses simultaneous power quality improvement on either side of the 1φ-3φ, in a targeted low output frequency application for 1φ mains fed motor drive systems. This is achieved by means of a two objective finite set model predictive control (TO-FS-MPC) technique, which mathematically models the grid side filter as well as the load side induction motor, for concurrent control on both sides of the converter. A sinusoidal source current with no reactive power flow inevitably generates an ac component in the source power, which pulsates at twice the grid frequency. To attenuate this 1φ pulsating component on the 3φ stator side, the drive is proposed for use in low speed applications. This implies that stator current harmonics that are produced due to pulsating component of 1φ power, present as harmonics of higher order in comparison to the fundamental stator currents at low frequency. Therefore, the algorithm uses the motor as an energy storage device to compensate for the lack of dc link energy storage, while the 1φ mains current quality is regulated by the input LC filter.
To undertake the control of 3φ motor currents, estimates of 3φ converter output voltages (v j abc ) for all converter states required for 1φ mains current control. For this purpose, the converter switching possibilities, and representative estimates of 3φ output voltages and 1φ mains current are given in Table  2.

A. CONTROL OF 3φ MOTOR CURRENTS
To control the motor currents, the commanded 3φ currents (i * abc ) are generated through speed and flux controllers, using the standard vector control algorithm. The commanded dq components of motor currents in the synchronously rotating frame of reference are Once the rotor flux position (θ r ) is determined, as discussed later in (11), the reference 3φ motor currents (i * abc ) are further deduced from their dq components using inverse Park's transformations.
A discretized model of the induction motor is employed to compute flux as well as currents corresponding to each state of the 1φ-3φ MC. The discretized flux equations for the jth state are where X eq is defined as Then, predictions of motor currents are made from the flux variables given in (2) It must be noted that all reactance values are calculated with a base value of supply frequency, irrespective of the speed of the motor. As such these reactance values are constants, and immune to any changes in motor speed.
The 3φ current references are individually compared with predictions of motor currents for each converter state (i j abc ). The resulting error (e j 3φ ) constitutes the cost function that is eventually minimized

B. CONTROL OF 1φ AC MAINS CURRENT
For the control of 1φ mains current, a suitable reference signal (i * g ) needs to be generated. This article proposes to use the reference torque component ω m k ω p + k ω i s ω * m −ω m to generate the source current reference (i * g ). In fact, this implementation is an adapted version of active power balancing between both sides of the converter. Moreover, the motor copper losses are also incorporated in 1φ reference current generation. The amplitude of 1φ mains current command is therefore This adapted version of the input-output power balance offers better tracking performance, by continuing to adjust the source current reference until the motor attains the speed command, thereby overcoming any inadvertent performance deviation. Intuitively, i * g (t ) is allocated the same template as that of the 1φ mains voltage to warrant sinusoidal operation and IPF correction.
Ideally, 1φ mains current and 3φ load currents are sinusoidal signals. However, any attempted enhancement in 3φ motor current waveforms has detrimental effects on the 1φ mains current quality [11]. It is proposed to deal with this problem by modeling the input LC filter and the induction motor, and incorporating both models in the switching decision making process. The input filter model is given as The error between the reference 1φ current (i * g ) and the predicted 1φ currents of all states, constitutes the other weighted objective of the cumulative cost function TO-FS-MPC evaluates the cumulative cost function (F j ob j ) in each sample time, and makes the switching decision in favor of the least error state As the switching state is found using TO-FS-MPS, predictions of flux variables (φ opt ), stator currents (i opt abc ), 1φ grid current (i opt g ), and voltage (v opt in ) at input of the matrix converter for the least error state are carried forward. These optimum variables are recursively used in the control algorithm as feedback terms for the motor model, the filter model, and also for subsequent estimation of speed. Since the computation algorithm uses predictions of the winning state, it is possible to eliminate voltage sensor from input filter and current sensor from 1φ ac mains.

III. STATE ESTIMATION
To use vector control of the motor, it is essential to determine rotor flux (φ r ) and corresponding position (θ r ). Rotor speed measurement or estimation is also required. When a switch state is realized by TO-FS-MPC, rotor flux predictions of the best state are carried ahead to derive an estimate of the rotor fluxφ Furthermore, the rotor flux position in electrical radians can be foundθ In line with (6), the generation of the 1φ grid current reference requires estimates of the 3φ stator current (Î s ), rotor current (Î r ), and rotor speed (ω m ). The stator current estimate corresponds to the prediction for the most optimum switching stateÎ Similarly, the rotor current estimate can be calculated from the optimum flux variables Speed sensorless drive systems are well known to be more reliable and economically feasible as compared to measurement-based systems [14]. However, speed observation over the low speed range is challenging on account of parametric variation and significant resistive drops at low motor speeds. Generally, full order adaptive observers are most commonly preferred when a wide range of speed estimation is desired. Speed estimate in the synchronous reference frame for conventional full order estimators, is thereby given aŝ Considering the constraints of the converter and its proposed application, two modifications in the observer algorithm are made to enhance precision in speed estimation. These are, 1) As well as the quadrature axis stator current error, the speed update uses an additional compensation term with a factor (η) for the d-component of stator current error. 2) Digital low pass filtering is used for the dq-components of stator current errors. This enhances precision of the speed estimate by weakening the effects of 1φ grid power pulsations, as well as switching noise that may otherwise appear on the rotor speed estimate. Hence, the modified speed estimate iŝ (15) where f denotes the low pass digital filter given by Since motor drives are often susceptible to stator resistance drift, a stator resistance update algorithm is used to enhance the performance of the drive system during parametric inconsistency. This helps to overcome any variations or drift in stator resistance, thereby enhancing the speed estimation performance. The stator resistance can also be estimated using the filtered error terms described in (15) s is the initial assumption of stator resistance.

IV. PERFORMANCE ANALYSIS
To validate the performance of the 1φ-3φ matrix converter fed motor drive, detailed simulation analysis and experimentation are undertaken. The proposed algorithm maintains a tradeoff between the 1φ source current and the 3φ stator current quality. The estimation technique is also validated. System parameters are given in Table 3. Simulations have been conducted using the same parameters of the filter and motor (Table 3) as used in experimentation. Experimentation has been carried out with a larger sampling time than the simulations, due to limitations of the controller dSPACE 1202 used in the tests. Test conditions, speed range, and load torque are similar for both validation environments.

A. SIMULATION RESULTS
The performance of the motor drive during low speed start up at rated torque, as shown in Fig. 2, which shows the 1φ ac voltage (v g ), 1φ mains current (i g ), the converter output voltage (v ab ), the 3φ motor currents (i abc ), the stationary rotor flux signals (φ αβr ), the electromagnetic torque (T e ), and the rotor speed estimate (N r ). It can be seen that the source current waveform is sinusoidal and that the pulsating 1φ power is lower in the motor currents (at low stator frequency). Moreover, the flux variables (φ αβr ) are completely sinusoidal, thus demonstrating the strategic use of motor load to compensate for the lack of energy storage in the converter. Fig. 3 shows the comparative harmonic performance of single-objective FS-MPC for motor current control versus two-objective FS-MPC for the control of both grid and motor currents. Although, single-objective FS-MPC helps achieve optimum motor current waveforms [see Fig. 3(b)], the source current quality [see Fig. 3(a)] is poor with significant presence of lower order harmonics. It can be seen that grid current with TO-FS-MPC is sinusoidal with negligible reactive power flow, as observed in Fig. 3(c). It is a significant improvement over work reported in the past [11]. With sinusoidal mains current, the 1φ mains power pulsates at two times the grid frequency, which can have adverse impact on the load side due to lack of energy storage. It is possible to alleviate these fluctuations from appearing on the load side by modeling the distributed load and filter energy storage components, and subsequently incorporating these models into the control algorithm. The 3φ motor currents and corresponding harmonic profile for the proposed algorithm are shown in Fig. 3(d). The best balance of grid and motor current quality is achieved through twoobjective FS-MPC.
The weighting factor is a critical parameter in multiobjective FS-MPC, as it determines the relative attainment of objectives. For cost functions involving more than a few weighting factors, optimal tuning of weighting factors is needed [17]. However, since this work is based on twoobjective FS-MPC with only one weighting factor, i.e., λ for grid current control, this has been heuristically tuned for simplicity. Table 4 shows the choice of weighting factor λ and its effect on different power quality parameters such as total harmonic distortion (THD) in motor current, THD in grid An important aspect of the low speed drive is the estimation of motor speed. To show the efficacy of proposed revisions in the speed update law, the reference motor speed (N * r ) and the estimated speed (N r ) are given for different scenarios in Fig. 4. The classical speed estimator using traditional full order adaptive observer, that only uses q-component stator current error, is shown in Fig. 4(a). Fig. 4(b) shows the estimator performance with an additional factored (η) d-component of current error added in the speed update, besides the conventional qcomponent. Fig. 4(c) depicts the estimator performance with a digital low-pass filter ( f ) used over the q-component of error alone, and finally Fig. 4(d) shows the cumulative effect of digital low-pass filter ( f ) on both dq components of current errors. It is evident that the combination of both revisions in speed estimation law promises a more accurate speed estimate.
In Fig. 5, the performance of the stator resistance estimator (17) is shown for the drive during parametric inconsistency. Fig. 5(a) shows the speed estimation performance when the stator resistance is exactly known. Fig. 5(b) shows the speed estimation performance assuming a 15% error in stator resistance without any corrective estimator, and Fig. 5(c) shows the estimation performance with the online stator resistance estimator in place assuming a 15% error in stator resistance. The online stator resistance estimator takes necessary corrective action to rectify the value of stator resistance and eventually improves the speed estimation performance.

B. EXPERIMENTAL RESULTS
To validate the proposed control algorithm for the 1φ-3φ MC fed motor drive, a broad range of experimental results are presented. The steady-state performance of the drive is shown in Fig. 6. Fig. 6(a) shows the load voltage and 3φ load currents, Fig. 6(b) shows the αβ plot of rotor flux, and Fig. 6(c) shows the steady-state reference and estimated rotor speed, and electromagnetic torque.
The steady-state grid side performance of the drive can be seen in Fig. 7. The 1φ grid voltage and grid current at unity input power factor are shown in Fig. 7(a), while the harmonic spectrum of source current is shown in Fig. 7(b), which has a total harmonic distortion (THD) of merely 3.3%. It must be noted that it is possible to further improve the quality of stator currents, by setting the weighting factor "λ" to zero in (9), however, this is achieved at the expense of severe lower order harmonics and poor THD in the grid current, which is undesirable.
The quality of the control and estimation algorithm is demonstrated through exhaustive speed control and torque perturbation in the low speed range, as shown in Fig. 8. The rotor speed is first decreased in steps of 50 r/min, and  then further increased back to 100 r/min. The corresponding speed and torque dynamics are shown in Fig. 8(a). Fig. 8(b) shows the speed and torque dynamics as the motor is suddenly subjected to a torque perturbation from full load to no load.
Furthermore, the versatility of the motor drive is demonstrated by means of speed reversal. Fig. 9(a) shows   the dynamics in speed and torque as the command speed is reversed from +100 to −100 r/min, while corresponding reversal of phase in 3φ stator currents is shown in Fig. 9(b). It can be seen from Fig. 9(a) that the electromagnetic torque reverses once the rotor speed reverses. However, during the small interval when the motor is decelerating, the electromagnetic torque is still positive, and the motor operates under regenerative braking, feeding energy back to the grid.

V. CONCLUSION
In this article, it is concluded that high power density of 1φ-3φ MCs can be favorably exploited in consumer applications involving low-speed 3φ motor drives fed from 1φ ac mains. Through results and analysis it is demonstrated that despite the lack of dc energy storage, the mismatch in instantaneous power between the two sides of the converter can be mitigated, by accounting for the energy storage capabilities of the motor and input LC filter. This is achieved using mathematical models of the input filter and motor being incorporated in the switching decision process through TO-FS-MPC. The inherently low voltage conversion ratio of 1φ-3φ MC coupled with the possibility of improved power quality at low stator frequency make this converter an ideal candidate for low speed drives. To achieve precision in low speed estimation, a full order predictive observer with revised speed update is presented. Digital low-pass filters are used on stator error terms to attenuate noise and power pulsations from the speed estimate. Additionally, the factored d component of stator current error can be employed while updating the speed estimate for increased overall precision. Impact of the revised speed update and efficacy of the control/switching algorithm in a low-speed motor drive are validated via simulation and experimental analysis. Thus, the 1φ-3φ MC fed low speed induction motor drive is advocated as a reliable and power-dense alternative, offering improved current quality for consumer applications like elevators, low speed centrifuges, automatic door closers, treadmills, exercisers, conveyor belts, fuel filling machines at gas stations, and low speed-high volume fans, operated from 1φ ac mains. Although limited by its operational range, this work can be further extrapolated to explore the suitability of 1φ-3φ MCs for low power-factor motors like synchronous reluctance machines and bespoke motor designs, where the performance metrics and speed range can be improved. The control concepts documented in this article can be used interchangeably for the control of single phase to three MCs in grid-tied applications with high-frequency ac link and grid frequency three phase output.