A CONTINUOUS ENERGY-EFFICIENCY OPTIMIZATION CONTROLLER FOR FIELD-ORIENTATION INDUCTION MOTOR DRIVES

The current paper discusses the adaptive choice of a filter time constant for filtering the steady-state flux reference in the energy-efficient control problem of field-orientation induction machines in transient behavior when load and speed conditions are changing taking into account the effect of the main induction saturation. It is shown that by appropriately managing the flux linkage rate of change the energy losses per full operation cycle under torque changes can be significantly reduced compared to the conventional cases. The analysis for the appropriate choice of the filter time constant as a fraction of the rotor time constant is based on a numerical study and simulation results for three different induction machines with different rated powers.


Statement of the problem and analysis of the recent research and
publications. The induction machine is widely used in industrial applications due to its robustness and its low cost compared to permanent magnet synchronous machines. However, in part-load operation, the efficiency of the induction machine dramatically decreases when the flux is kept at the nominal level. To address this issue different strategies have been developed in the past to increase the efficiency of the induction machine in a large operation range [1], [2]. However, when the induction machine is operated under changing loads these methods will not yield maximum efficiency. A solution to this problem in the framework of vehicle applications is given in [3]. However, the motor for these applications will often operate in dynamics with changing torques and speed up to voltage and current limits. The development of parametrized prototypes is discussed in [4]. Unfortunately, without voltage and current limits as well. These limits require the knowledge of the behavior of the process quantities, which can be obtained from offline optimization [5]. Purpose of the study. The current paper discusses a different approach. The motivation is to give a simple and easy implementable solution to the problem of energy consumption minimization, which is characterized by a continuity of action, converges to the optimal flux level in the steadystate and expresses a compromise approach to the problem with changing loads in dynamics as well as provide an alternative solution to model predictive control [6].
Statement of main research material. State-space model. An induction machine operated with high dynamics is controlled as a rule via a field-oriented scheme. The approach in the sequel implies the orientation along the rotor flux linkage Ψ2, i.e. the flux linkage phasor is aligned with the d-axis of the rotating frame. The full model thus has four state variables: field-producing current i1d, torque-producing current i1q, rotor flux Ψ2, and motor speed ω2; and two controls: the stator voltage phasor components u1d and u1q. The continuous state-space model of an induction machine is given as follows (Г-inverse equivalent circuit parameters are used): the characteristic matrix of the system, given by the following expression with stator and rotor resistances R1 and R2 respectively; L σ is the stray inductance and Lm denotes the main inductance; and ω1 as the synchronous speed in electrical radians per second, obtained from where ωslip is the slip speed.
(4x2) Β ∈ ℝ is the input matrix given by C∈ ℝ is given by The electromagnetic torque is given by Main inductance saturation. The modelling of an induction machine to be adequate must consider the nonlinear nature of the machine magnetic characteristics. The measured main inductance data points are shown in Fig. 1. The minimum allowable set consists of 5-6 points. In addition, with a small amount of data, "abnormal" results cannot be included in the sampling sequence. Hence, the bigger the number of points, the better. The experimentally measured data of nineteen points is further approximated using the method of least squares. An obvious "applicant" immediately appears here in a form of a high-degree polynomial whose curve passes through all measured points. But this option is, more often than not, simply incorrect and reflects the main trend poorly. Thus, the desired function should be relatively simple and at the same time reflect the dependence adequately. We will restrict the degree of resulting polynomial to fifth degree. In this case, the objective consists of adjusting the parameters of a model function m 1d ( ) : To fit the model to a data it is required to find the optimal parameter values by minimizing the sum of squared residuals given in the following form  4  3  2  1  2  3  4  5  6  k  1 k  2 k  3 k  4 k  5 k  6  1   k   min , , , , , n k k n k r F P P P P P P Y P P P P P P where P1-P6 are the function parameter values to be found; k λ and k Y are measured field-producing current and main inductance values, respectively.
In essence, it is necessary to solve the problem of finding the minimum of the function of six variables. First things first, let us find all partial derivatives of the first order. According to the rule of linearity, it is allowed to differentiate under the sum sign: Now compose a typical system equating each expression from (8) The obtained system (9)   2. The rotor flux linkage is set to its new optimum steady-state value.
In the first case, the torque-producing current rapidly increases to its steady-state value, but under the new value of the torque on the shaft, the power consumption will not be optimal. In the second case, if we consider the peak value of the power losses during the transient period in Fig� 2 its value will be much greater than in case 1�� It was noticed that the field regulator attempts to establish a new steady-state optimal value for the rotor flux linkage as quickly as possible and as a result uses a high magnitude of the field-producing current and reaches its output almost in no time. This is the main contribution to short-term high losses according to the stepwise approach denoted in Fig. �a. This fact means that it is not profitable to use solely the conventional flux controller in dynamic mode�� due to high instantaneous power loss overshoots under changing torques and, in addition, it leads to the increase of the total energy consumed per duty cycle. That is, the peak power loss is much lower in case 1. This statement is also based on the fact that by condition there is no change in the rotor flux linkage before the change in load occurs. Thus, at the very first instance of time, the speed controller sets up the torque-producing current for the value of the rotor flux, which was optimal until any changes in the torque. Numerical study. Let us consider the impact of first-order filtering. To simplify the calculations, we will assume in the sequel that the flux regulator is fast enough such that the flux linkage follows its reference closely. In addition, assume that the speed and current controllers of field-oriented control have high enough performance to ensure the control characteristic close to perfectly rigid, that is, the dynamics of the stator phasor components is significantly higher than the dynamics of the magnetic flux and speed. In this case, we can assume for the flux linkage dynamics the first-order differential equation: Thus, under zero initial conditions, the output of the object is calculated as the product of its transfer function by the image representation of the input ss ss 2,opt 2,opt ss 2,opt Now, using the principle of superposition for images, we calculate the original output signal: The solution of this differential equation for non-zero initial conditions is given by   Adaptive line search. In contrast to [7], where filter time constant with factor k had fixed value over wide operation range, it is suggested in the i1q,ref . It is assumed that the function is being positively defined ( ) J k + ∈ ℝ . One of the most common search methods is the gradient descent method, which is formulated in a continuous time frame as follows: x with 0 k > as a constant value. result of a numerical study shows that the optimal value of the multiplier k is in a range between 0.5 and 1. This solution is simple to implement and can be easily integrated into existing inverters, and not less importantly, the same algorithm is used both to minimize power loss in statics and dynamics when load and speed conditions are changing.