Predictive Sensorless Control of Induction Motor Drives - IEEE Xplore

Predictive Sensorless Control of Induction Motor Drives Y. Zbede, S. M. Gadoue, D. J. Atkinson and M. A. Elgendy School of Electrical and Electronic Engineering, Newcastle University, Newcastle upon Tyne, NE1 7RU, U.K. E-mail: [email protected]

Abstract— This paper presents a control system of an Induction Machine (IM) with predictive controller-estimator. A novel speed estimator is proposed, and the estimator operation is examined at different operational conditions. The new estimator shows an improvement in the speed estimation during the transient operation compared to the conventional estimators, and it also shows a satisfactory operation at different operating conditions. Simulations results are presented and the proposed experimental rig is described. Index Terms; Induction Machines; Vector Control; Speed Estimation; Predictive Controller; Predictive Estimator.

I.

INTRODUCTION

Vector Control is often applied when fast dynamic response is required. Like any other cascaded control system, vector control performance is limited by its inner loop speed of response, which means a better current controller in the inner loop can lead to a better overall system behavior. In the literature, different current controllers have been used for vector control of induction machines. These controllers can be classified into nonlinear, such as the hysteresis controllers, and linear like the PI controllers, for example [1]. Also predictive current controllers, artificial intelligence-based controllers and sliding mode controllers have been applied recently [1]. Traditionally, hysteresis controllers were applied very often because of their simplicity of design, simpler implementation and their fast dynamic response. However, these controllers suffer from the problem of variable PWM frequency in their output which is largely dependent on the hysteresis band and the controlled system parameters [2]. This has limited the industrial application of these controllers. In industry, controllers which operate at fixed frequency are preferred because their output has a well-defined harmonic content [2]. However the tracking error is the main drawback of these controllers [3-5]. PI controllers are a good example of fixed-frequency regulators. In general, by using PI controllers, constant switching frequency operation can be achieved with fairly negligible tracking error. Unfortunately the PI dynamic response is inferior to that of the hysteresis controller, and if

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the PWM maximum modulation index is exceeded in the output of PI controllers, the system may lose its stability and this can lead to excessively high currents [6]. The sliding mode controller (SM) has been described as having a robust and good dynamic response for the control of electric drives. However, in general, SM controllers require smaller sampling period in comparison with other controllers in order to reduce the toque ripple [7] and therefore fast microprocessors should be used. Recently, a group of modem controllers with real-time optimization have been employed. These controllers are classified as predictive controllers and they precalculte the behaviour of the controlled system to find a proper control signal before the error between the reference and real value really occurs [8]. This unique feature, introduces an important advantage of these controllers over the rest of the controllers, which only take the present and past tracking errors into consideration. In addition, these controllers allow the integration of system constraints into the controller design, and they ensure a near optimum performance of the system by minimizing the cost function in the control law [9]. However the high computational requirement of these controllers is a drawback that limits their use [9]. In many induction machine vector control applications, sensorless speed estimation is applied to reduce the system’s cost, size and maintenance requirements, and to increases its mechanical robustness and hence its reliability [10]. In the literature, many sensorless strategies have been introduced, and most of them can be categorized in two main categories, model-based sensorless strategies [11-14], which use the machine basic model to estimate the speed of the rotor, and spectral analysis strategies which use the rotor positiondependence feature which exists in many AC motors to determine the rotor speed or position [12, 14-16]. Among model-based sensorless strategies, Model Reference Adaptive System (MRAS) estimators have gained widespread popularity, due to their simplicity and low computational effort. Generally Rotor flux-MRAS is the most often applied MRAS strategy and many attempts have been made to improve the scheme performance especially at low speeds. In this paper, a novel predictive rotor flux MRAS has been introduced for induction motor sensorless vector control drives.

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A predictive current controller is also used in the current loop to control the motor current. To evaluate the new system performance, a rotor-flux MRAS with PI current controllers has been built, and the estimated speeds of both systems have been compared using Matlab/Simulink. II.

consider the past states to optimize the future to a specified control horizon. This class of controllers is described in more detail in the next Section [18].

PREDICTIVE CONTROLLERS

Traditionally, linear PID controllers were used in electric drives which do not require any knowledge about the system except during the controller design. With the development of microprocessors and digital control techniques it is becoming possible to use the plant mathematical model to pre-calculate the system behavior and hence to choose the optimum actuating variables. This represents the main principle of the predictive controllers [17]. The typical structure of predictive controllers is shown in figure 1, where a position control of electric drive system is chosen as an example [8]. The system measured states in this case are the machine current, speed and position. These states are fed to the machine and power electronic model which predict the future state values depending on both the current states and the system mathematical model. The future states are then compared with the desired behavior within the prediction/calculation block and an optimum actuating variable is chosen to minimize the error between the actual and the desired values. Finding the optimum value for the actuating variable depends mainly on the desired optimum condition, [8] for example minimizing the current distortion or minimizing the current error...etc.

B. Model-based predictive controllers The functional principle of the model-based predictive controllers (MPCs) is illustrated in figure 2. The core part is the model which predicts the behavior of the system up to the prediction horizon. This prediction consists of two parts [18]: The free response: which shows the system output behavior when the future actuating values are equal to zero. The forced response: forms the added component when applying nonzero actuating values. The sum of the free and forced response forms the total response. Many techniques can be used to optimize the actuating variables in order to reduce the error between the desired values and the predictive ones.

Figure 2. The functional principle of the model-based predictive controllers

C. Predictive Current Controllers (PCC) The design of the predictive current controller is based on the simplified IM model. In this model, the IM is modelled as an inductance with an EMF connected in series. The small motor resistance can be neglected in the design [19]. Following the previous assumption the stator current can be calculated as [19]:

Figure 1. The functional principle of the model-based predictive controllers

This principle can be considered common to all predictive controllers, except the prediction-calculation block where it differs between one controller and another, and according to this, predictive controllers can be classified into three different categories [8].

𝑑𝑖𝑠 (𝑢𝑠𝑐𝑜𝑚 − 𝑒) = 𝑑𝜏 𝜎𝐿𝑠

A. Predictive controller classification Based on the operational principle, predictive controllers can be classified into three main categories: hysteresis-based predictive controllers, trajectory-based predictive controllers and model-based predictive controllers [18]. In the hysteresis-based controller the main objective is to keep the controlled variable value within a predefined tolerance area or tolerance band. The trajectory-based predictive controllers force the system states to follow predefined trajectories and remain there until an outside change is enforced. Finally the model-based predictive controllers

(1)

If the sampling period is 𝑇𝑠 , equation (1) can be written in discrete time domain as: 𝑖𝑠 (𝑘) − 𝑖𝑠 (𝑘 − 1) 𝑢𝑠𝑐𝑜𝑚 (𝑘 − 1) − 𝑒(𝑘 − 1) = 𝑇𝑠 𝜎𝐿𝑠

(2)

In this equation, the known elements are the commanded voltage 𝑢𝑠𝑐𝑜𝑚 (𝑘 − 1) and the current 𝑖𝑠 (𝑘 − 1), while 𝑖𝑠 (𝑘) and 𝑒(𝑘 − 1) need to be predicted. According to [20], 𝑒(𝑘 − 1) can be simply predicted based on the value of 𝑒(𝑘 − 2) and 𝑒(𝑘 − 3):

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Figure 3. predecive current controllers structure

𝑒̂ (𝑘 − 2) =

𝜎𝐿𝑠 (𝑖𝑠 (𝑘 − 2) − 𝑖𝑠 (𝑘 − 1)) + 𝑢𝑠𝑐𝑜𝑚 (𝑘 − 2) (3) 𝑇𝑠

(k) is calculated from: Finally the voltage ucom s pred

𝜎𝐿𝑠 (𝑖𝑠 (𝑘 − 3) − 𝑖𝑠 (𝑘 − 2)) 𝑒̂ (𝑘 − 3) = + 𝑢𝑠𝑐𝑜𝑚 (𝑘 − 3) (4) 𝑇𝑠 To find 𝑒̂ (𝑘 − 1), the EMF position change ∆𝜑𝑒 can be used. ∆𝜑𝑒 is calculated from: ∆𝜑𝑒 [(𝑘 − 2), (𝑘 − 1)] = ∆𝜑𝑒 = 𝜑𝑒 (𝑘 − 2) − 𝜑𝑒 (𝑘 − 3) (5) The problem associated with (5) is that two arctangent are required. Alternatively (6) can be used: 𝑒̂𝛼 (𝑘 − 2)𝑒̂𝛽 (𝑘 − 3) − 𝑒̂𝛼 (𝑘 − 3)𝑒̂𝛽 (𝑘 − 2) ∆𝜑𝑒 = 𝑎𝑟𝑐𝑡𝑎𝑛 𝑒̂𝛼 (𝑘 − 2)𝑒̂𝛼 (𝑘 − 3) + 𝑒̂𝛽 (𝑘 − 2)𝑒̂𝛽 (𝑘 − 3)

(6)

If the flux is kept constant the back EMF e will be a function of speed and due to the large mechanical time constant, the EMF changes are limited. With this assumption 𝑒̂ (𝑘 − 1) can be predicted by rotating 𝑒̂ (𝑘 − 2) vector with the angle ∆𝜑𝑒 as in (7): 𝑒 𝑝𝑟𝑒𝑑 (𝑘 − 1) = 𝐶𝐸𝑀𝐹 𝑒̂ (𝑘 − 2)

(7)

(k) = ucom s

𝐶𝐸𝑀𝐹 = [

− sin(∆𝜑𝑒 )

] cos(∆𝜑𝑒 )

(8)

After back EMF prediction, stator current prediction can be carried out using the following equation: 𝑝𝑟𝑒𝑑

𝑖𝑠

(𝑘) = 𝑖𝑠 (𝑘 − 1) + [𝑢𝑠𝑐𝑜𝑚 (𝑘 − 1) − 𝑒 𝑝𝑟𝑒𝑑 (𝑘 − 1)] ×

𝑇𝑠 𝜎𝐿𝑠

(9)

∆is (k) =

(k) icom s

− is

pred

(k)

+ epred (k)

epred (k) = C2EMF ê(k − 2)

(12)

(13)

and C2EMF = [

cos(2∆φe ) − sin(2∆φe )

sin(2∆φe ) ] cos(2∆φe )

(14)

(k) which is chosen to minimize The voltage vector ucom s ∆is (k + 1) the stator current regulation error at (k+1) is then applied. Equation (12) is derived from (2) and DIs is a correction part which can be calculated from: DIs = −Wcurr CEMF ∆is (k) + Wcurr C2EMF ∆is (k − 1) (15) The controller structure is shown in figure 3, where trial and error method is used to tune 𝑊𝑐𝑢𝑟𝑟 in order to minimize the current regulation error. ROTOR FLUX MRAS ESTIMATOR

Figure 4 shows the rotor flux MRAS estimator structure. The estimator consists mainly of two mathematical models that calculate the rotor flux components; the reference model is independent of the rotor speed, while the adaptive model is speed dependant. The induction machine stator voltage equation in stationary reference frame (DQ) can be written as:

The minimization of the current regulation is chosen as a cost function to optimize the PCC. This error is expressed as: (k − 1) − is (k − 1) ∆is (k − 1) = icom s

Ts

III.

sin(∆𝜑𝑒 )

(k) + DIs ]

where

where cos(∆𝜑𝑒 )

(k + 1) − is σLs [icom s

𝑣𝑠𝐷 = 𝑅𝑠 𝑖𝑠𝐷 + 𝜎𝐿𝑠

𝑑𝑖𝑠𝐷 𝐿𝑚 𝑑𝜓𝑟𝑑 + 𝑑𝑡 𝐿𝑟 𝑑𝑡

(16)

𝑣𝑠𝑄 = 𝑅𝑠 𝑖𝑠𝑄 + 𝜎𝐿𝑠

𝑑𝑖𝑠𝑄 𝐿𝑚 𝑑𝜓𝑟𝑞 + 𝑑𝑡 𝐿𝑟 𝑑𝑡

(17)

(10) (11)

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Figure 5. Adaptive model with rotor frame implementation

Figure 4. Rotor Flux MRAS structure

In the same reference frame, the rotor voltage equation of the induction machine is similarly: 1 𝐿𝑚 𝑑𝜓𝑟𝑑 0 = 𝜓𝑟𝑑 − 𝑖𝑠𝐷 + + 𝜔𝑟 𝜓𝑟𝑞 𝑇𝑟 𝑇𝑟 𝑑𝑡 0=

𝑑𝜓𝑟𝑞 1 𝐿𝑚 𝜓𝑟𝑞 − 𝑖𝑠𝑄 + + 𝜔𝑟 𝜓𝑟𝑑 𝑇𝑟 𝑇𝑟 𝑑𝑡

(18) (19)

Equations (16) to (19) have the rotor flux components in stationary reference frame as a common output. Equations (16) and (17) are speed independent and form the reference model, whereas (18) and (19) equations depend on a flux calculation on the speed and therefore can be used as an adaptive model. The cross coupling presence of the speed dependant components in the adaptive model can lead to instability issue [21], therefore in the literature the rotor flux equations represented in rotor frame are used: 𝜓̅𝑟𝑟 = 𝐿𝑚 𝑖̅𝑟𝑠 −𝑇𝑟 where 𝑖̅𝑟𝑠 = 𝑖̅𝑠𝑠 𝑒 −𝑗𝜃𝑟

𝑑𝜓̅𝑟𝑟 𝑑𝑡

(20) (21)

and the flux components in the stationary reference frame can be found from: 𝜓̅𝑟𝑠 = 𝜓̅𝑟𝑟 𝑒 𝑗𝜃𝑟

Figure 6. Rotor flux MRAS adaption mechanisim

(22)

IV.

THE NOVELPREDICTIVE ESTIMATOR

As mentioned previously, predictive current controllers use the past, present and future current values to choose a proper voltage vector that can reduce the current error between the demand and actual currents, not only at the present time instant but also all over the prediction horizon. The same principle is used in designing the proposed predictive speed estimator. In other words, if the error used to estimate the speed in the rotor flux MRAS estimator, equation (23), is also predicted then a better estimation can be carried out by choosing a speed that not only minimizes the current speed estimation error but also the future errors over the prediction horizon. Consequently, a faster estimation dynamic can be obtained and the estimated speed will converge to the real speed in a shorter time that depends on the length of the prediction horizon. In order to estimate the speed estimation error, equation (23), both future currents and fluxes are required. The future currents can be obtained from the output of the predictive current controller while the future fluxes can be estimated depending on the machine model. Figure 7 illustrates the model of the proposed estimator.

The implementation of the rotor frame based model is clarified in figure 5. The design of the adaption mechanism is based mainly on the hyperstability theory [22], and as a result of applying this theory, the speed tuning error signal can be written as: 𝜀𝜔= 𝜓𝑟𝑞 𝜓̂𝑟𝑑 −𝜓𝑟𝑑 𝜓̂𝑟𝑞

(23)

A PI controller can then be used to minimize this error, which in turn generates the estimated speed at its output as shown in equation (24). 𝜔 ̂𝑟 = (𝐾𝑝 +

𝐾𝑖 )𝜀 𝑠 𝜔

(24)

Figure 7. the model for the proposed estimator

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V.

RESULTS AND DISCUSSION

A model of 7.5 kW, 415 V, 50 Hz, 4-pole star-connected, 3-phase, squirrel-cage induction motor which has the following parameters: R s = 0.776 Ω , R r = 0.703 Ω , Lls = 4.51 mH, Llr = 4.51 mH, Lm = 103.22 mH, has been used as a test machine. Two different sensorless vector control models have been built the first scheme uses predictive current controllers alongside the proposed predictive rotor flux MRAS estimator, whereas the second one uses PI current controllers with PI- based rotor flux MRAS estimator. The simulation of both systems has been carried out using Matlab/Simulink environment. In figure 8, the speed reference signal has been changed from 0 to rated speed 157.07 rad/s at the time instant 0s. The red signal represents the estimated speed when the conventional rotor flux MRAS and PI current controller are applied, while the green signal is obtained by applying the novel predictive estimator and predictive current controller. It can be noticed that the proposed scheme shows better response during the transient stage with smaller overshoot and less oscillations. In steady state, it also shows superiority with negligible oscillation around the speed demand. Both estimators reach the steady state at the same instant of time giving no advantage in terms of settling time for any of the estimators.

during the transient operation and negligible oscillation after reaching the steady state. Figure 11 shows the operation in the regenerative region. The reference speed has been reduced gradually from 157.07 red/s to -157.07 red/s, and the full load has been applied at t = 5 s where the speed reference crosses the zero point. The proposed scheme shows a better performance with faster dynamics and less over/undershoots during the transient and smoother estimation during the steady state. The proposed scheme is currently being validated experimentally by appling the proposed algorithm on an experimental test rig as shown in figure 12.

Figure 9. full load application at rated speed

Figure 8. transient and full speed comparison

In figure 9, the rated torque has been applied at 5 s, and the performance of both estimators has been tested. It is obvious that the proposed scheme shows a considerable improvement, in terms of ripple reduction in the estimated speed, in comparison with the conventional MRAS-PI scheme. Additionally, the motor speed recovers faster after applying the load with smaller under-shoot value, which means the proposed scheme has a better transient performance with faster dynamic response. Zero speed operation of both schemes has also been tested. The reference speed has been reduced gradually from rated speed to zero, and after reaching steady-state the full load has been applied. Figure 10 illustrates the results and it is obvious that the predictive estimator shows superior performance at zero speed with less undershoot and faster dynamic response,

Figure 10. full load application at zero speed

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Figure 11. full load application in the regenerating region

Figure 12. The experiment circuit

IV.

CONCLUSION

In sensorless electric drives it is desirable to maximise the performance in all operational regions. In this paper, a predictive rotor flux MRAS estimator is presented for a predictive current controller-based vector controlled drive. This proposed scheme is shown to have superior results, compared to a conventional PI current controller-based MRAS sensorless drive, in terms of reducing speed estimation ripple during the steady state and faster dynamic performance during the transient operation. This means both transient and steady state operation are shown to be improved. REFERENCES

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