# Online Efficiency Optimization and Speed Sensorless Control of Single-Phase Induction Motors
## 1 Introduction  
The increasing of electricity prices and the public awareness of environmental problems associated with conventional power plants motivate major attention in enhancing the efficiency of electrical devices. Numerous domestic, commercial and some light-duty industrial applications, near which there is no easy access to three-phase networks, utilize single-phase induction motors (SPIMs). SPIMs have the advantages of low cost and high reliability. However, they have a relatively low efficiency compared with alternative solutions such as permanent magnet synchronous motors. As such, the amount of power loss caused by SPIMs is significant. Therefore, enhancing the efficiency of SPIMs can have a major impact on the overall power losses of electrical systems. An SPIM is comprised of two stator windings (i.e., main and auxiliary winding) and a squirrel cage rotor. The two windings, which are in space quadrature, usually have different impedances. A phase shifting capacitor is placed in series with the auxiliary winding to produce an appropriate phase shift in the auxiliary winding current. The combination of auxiliary winding and the capacitor is paralleled with the main winding, which is connected to the motor terminals. The SPIM can be directly connected to the electrical grid. To control the speed and optimize its efficiency, a variable speed electric motor drive can be employed, in which a power electronic converter is connected between the SPIM and the grid. Figure 1 depicts the schematic of the SPIM drive, in which a Single-Phase Bridge Inverter is used to control a capacitor-run SPIM. The AC line voltage is rectifiedand filtered via a diode-bridge and a capacitor, and is fed to the Inverter. The amplitude and frequency of the sinusoidal reference is governed by the controller. A number of methods are proposed for speed control of induction motors, including rotor-flux-oriented control [1; 2], vector control [3], direct torque control [4; 5], sliding mode control [6], neural network based control methods [7; 8] and a model predictive control method [9]. Furthermore, various speed sensorless control methods have been proposed, among them are a method based on DC-link measurements [10], speed measurement based on rotor-slot-related harmonic detection in machine line current [11], field-orientation control with a speed estimation scheme [11] and speed estimation using adaptive nonlinear observers [12] and sliding mode observer [13]. The mentioned control schemes, which are focused on the speed control dynamics, do not take action for optimization of the efficiency. The efficiency of induction motors can be enhanced by adjusting the applied voltage and hence the flux at the optimum operating point [14]. In order to obtain the optimum operating point corresponding with each speed and load, one can use the loss model of the motor [15; 16]. In [17] the dynamic model of induction machine has been used to derive an algorithm for calculating the optimum flux. However, the aforementioned approaches require the exact value of the motor parameters through a parameter identification scheme [18]. A real time efficiency optimization method has been presented in [19]. In this method, the optimal flux is first calculated analytically based on the estimated motor parameters and then updated in real time using annealing method. The methods of [15; 16; 17; 18; 19] are focused on three-phase induction motors. Reference [20] proposes a control scheme for enabling symmetrical and balanced operation of SPIM during both starting and steady-state operating conditions. Although such symmetrical operation enhances the efficiency, the requirement of two full bridge converters for main and auxiliary windings increases the cost of the system. In a method presented in [21] the ratio of main to auxiliary stator winding currents (Stator Current Ratio) is used as the optimization parameter. It is shown that controlling this parameter can lead us to the optimum operating condition, regardless of the motor speed and torque.