**1. Introduction**

Permanent magne<sup>t</sup> synchronous machines (PMSM) have a higher efficiency, torqueto-weight ratio, and power per volume value, which causes their popularity in motor drives, where efficiency and compactness are two of the main requirements. However, efficient control of PMSM requires full utilization of their potential and minimization of the consumed current at the commanded torque [1,2].

The feature of PMSMs is the presence of a reluctance torque component besides the main magnetic torque component. The reason for this feature is magnetic asymmetry along direct and quadrature axes, which is typically caused by the machine design. However, even-surface-mounted permanent magne<sup>t</sup> synchronous motors (SPMSM), which have equal direct and quadrature inductances at low-load conditions, typically demonstrate magnetic asymmetry under load. Therefore, efficient control systems have to consider this feature and utilize a reluctance torque component as well. Special control algorithms, which provide full utilization of motor torque, are called maximum torque per ampere (MTPA) techniques. These techniques differ by the type of operation (offline or online), usage of motor parameters (used or not required), etc., but an MTPA control became an

 Dianov, A.; Anuchin, Design of Constraints for Seeking Maximum Torque per Ampere Techniques in an Interior Permanent Magnet Synchronous Motor Control.*Mathematics* **2021**, *9*, 2785. https://doi.org/10.3390/ math9212785

**Citation:**

Academic Editor: Nicu Bizon

Received: 19 September 2021 Accepted: 29 October 2021 Published: 3 November 2021

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inevitable part of PMSM control systems, and it is hard to find a commercial motor drive without one of the MTPA algorithms.

The conventional approach to the calculation of the MTPA trajectory is differentiating the machine torque equation with respect to the amplitude of a stator current. For the purpose of simplification, the motor parameters are supposed to be stable and their dependencies on other variables are not considered [3,4]. However, motor parameters vary with the change of operating conditions: steel saturation impacts inductances, rotor temperature impacts flux-linkage, etc. Therefore, these variations have to be taken into account [5–7].

In order to adapt the MTPA control techniques to parameters variation, different online approaches were proposed. The authors of [8–15] proposed the usage of conventional MTPA equations together with the motor parameters online estimation techniques. The papers [8–10] considered motor inductance variation and proposed estimators for online inductance monitoring, whereas the papers [11,12] took into account only flux-linkage change and [13–15] considered simultaneous variation of these motor parameters. All these methods use the conventional MTPA equations obtained under the assumption that motor parameters are constant and do not vary; however, the motor parameters are updated using various estimation techniques. As a result, these approaches demonstrate the acceptable tolerance for motor drives where motor parameters vary slowly and their derivatives may be neglected, but they fail in the fast-dynamic systems, where motor parameter derivatives are significant. One more disadvantage of these methods is the necessity of motor parameter prior knowledge, which is used as a reference value for estimators and includes tuning of the adaptation mechanism.

In order to exclude these inconveniences, a group of techniques, which do not require motor parameters, have been proposed. These methods are called seeking algorithms, and they track minimum current consumption, which corresponds to the MTPA trajectory, using either injection of the additional signal [16–29] or perturbing the motor drive and analyzing response [30–38].

The algorithms proposed in [16–18] were designed for the direct torque control (DTC) and field-oriented control schemes, which use injections of high-frequency signals toward the motor, followed by an analysis of its response, which is used for extremum tracking. The main idea is to estimate the local torque derivative at a constant current and to find the position of a zero derivative, which corresponds to the MTPA condition. Similar methods adapted for open-loop v/f-constant control were studied in [21,22]. These techniques may operate without information of motor parameters. However, the knowledge of their approximate values may be used as a starting point for tuning and may significantly improve the dynamic.

The main disadvantage of injection-based techniques is noise and vibration caused by signal injection, which significantly restricts the application area of these methods; therefore, recently, a virtual signal injection control (VSIC) [23–29] was proposed, which eliminates the abovementioned drawback. The methods of this group inject a high-frequency signal into the system mathematically, without real modification of the signals applied to the motor. They use measured parameters: voltage current, speed, etc., and estimate the sign of the motor torque local derivative, which indicates the position of the operational point related to the torque extremum at the MTPA curve. At the next stage, this information is used to shift the operational point toward the torque extremum and is used to track that point. Since the stator resistance is neglected, when the authors developed VSIC, these algorithms worked better at higher speeds, but may fail in the low-speed range, where the applied voltage is comparable with the voltage drop across the stator. Furthermore, the sensitivity to stator resistance makes implementation of VSIC for motor drives impossible, which has a relatively high resistance (e.g., with aluminum windings) or which has to operate in a wide speed range.

Another approach for online tracking of MTPA conditions includes methods for real-time nonmodel-based optimization [39] and uses the perturb and observe (PandO) principle. The main idea of this approach is to perturb a system by modification of its input

signal in the "test" purpose and observe the response to the applied perturbation. For this purpose, the authors of [31–34] suggested modification of the stator current phase with the following analysis of changes in its magnitude, while the authors of [36] adopted a similar idea to the stator voltage vector. The PandO techniques do not use either system models or parameters, which makes them convenient for a wide range of applications having unknown parameters or variables in a wide range. However, the main drawback is poorer dynamic and stability issues. If the operation of a seeking algorithm is not limited, it may incorrectly detect the gradient of the observing signal and move the system far from extremum, significantly worsening the performance and sometimes making the system unstable.

Most of the seeking techniques were designed under the assumption that the torque is constant or changes slowly. Therefore, they may fail in transients, where speed and torque are modified by the external factors. As a result, the system response to the injected signal or applied perturbation is mostly defined by that external factor, so seeking algorithms fail and move the operational point far away from the MTPA trajectory. As soon as the transient ends, the system operates as expected and tracking algorithms can return the operational point at the MTPA trajectory. In order to limit deviation from the MTPA trajectory, the seeking algorithms have to use constraints, which limit uncontrollable modification of the stator current and its phase. Furthermore, proper design of these constraints significantly decreases deviation from the correct trajectory and notably decrease reaction time of the system. In turn, lower deviations from the desired trajectory increase system efficiency in the dynamic modes. Therefore, proper selection of constrains is extremely important for the seeking techniques, where the dynamic response is one of the weakest points.

Despite numerous publications dedicated to the online techniques, the problem of constraints design was not studied in detail, and it is hard to find recommendations on their selection. In [37] it was recommended to limit the maximum phase of the stator current with a predefined constant; however, recommendations on its selection were not provided. The authors of [34] use the adjustment of the stator current phase in order to track the MTPA trajectory. They suggested the limitations of this angle with theoretical minimum and maximum values plus some gaps. This approach provides system stability, however, it results in poorer dynamic stability, because this limitation does not depend on the current operational point in the MTPA curve. In [17] it was suggested to use a theoretical MTPA curve calculated for rated and unsaturated parameters, as a reference value and slightly modify it with a small angle calculated by the MTPA tuning block. However, recommendations on the selection of the maximum modification angle were not provided.

As is clearly seen, the suggestions on the selection of constraints are general and quite simple; therefore, online techniques demonstrate poorer dynamic behavior and efficiency. Furthermore, the previous researchers did not consider the effects of motor parameters variations and the corresponding change of MTPA trajectory. Therefore, after analysis of the published research, it was decided to develop an algorithm for the design of constraints for the MTPA trajectory.

The contribution of this paper is the development of a constraint design algorithm for MTPA seeking techniques. The proposed algorithm was developed for operation with control systems of commercial motor drives; however, it can be easily extended to self-commissioning systems. The developed algorithm is compatible with all MTPA seeking techniques such as [30–32] and improves their dynamic behavior and efficiency. The proposed method was implemented and compared with existing algorithms. The experimental results proved superiority of the proposed technique and improvement of dynamic behavior and efficiency of the test motor drive.

## **2. Conventional MTPA Approach**
