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Design of observer of flux linkage of synchronous drive of rolling mill

2016, vol. 16, no. 3, pp. 84–92 84 Introduction Variable frequency synchronous motors (SM) with the vector control of electromagnetic torque are used at main electric drives of the rolling mills [1–3]. It prevents motor dropout of synchronism and allows motor thrust operation as at DC electric drive at overloads in the torque-limiting mode. As vector control implies operation in coordinates oriented according to one of the magnetic flux linkage, the SM control system may be designed on the orthogonal coordinate axes of the stator (α, β) or rotor (d, q). As a rule, control is implemented for the simplification purpose as a system rotating together with the controlled vector, namely, the d, q system. Consequently, this approach enables vector control due to regulation of scalar values of its projection of the coordinate axes. In this case, the torque control may be represented with the formula: M = 3 2 Zp(ψ − ψ ). (1) A significant disadvantage of this method is the need for controlling all four changing values, which complicates the control system. As a solution of this issue, a rotating coordinate system m, l is proposed in [4], which axis m is oriented in the direction of the flux linkage vector of the stator ψ , while the stator current projection is put on axis l. Switching to a new coordinate system helps preventing consumption of reactive power in the synchronous machine by means of excitation field control. Against this background, the stator current ̅ being orthogonal to the flux linkage ψ coincides with the stator voltage vector. Therefore, the torque control is reduced to scalar regulation of the stator flux linkage ψ and projection of the stator current , which form is similar to control of the DC motor current: M = 3 2 Zpψsil. (2) The equation (2) provides for separate SM torque control – by the flux linkage and stator current. Relying on the required torque, the control system generates stator current setting, which specifies the stator voltage vector with current controller and corresponding coordinate transformations. Then, the inverter of the frequency converter generates this vector by means of pulse-width modulation. The main problem at implementation of this approach is complexity of measuring stator flux linkage ψ required for calculation of the SM internal angle. Strict dynamic requirements to electric drives of the rolling mills cannot be met without quick-response feedbacks according to coordinates unavailable for direct measuring. Under these conditions, the use of state observer appears to be rather promising, because it provides estimate of unknown value according to the known motor coordinates (current, voltage and rotation rate). The methods of observer selection and setting for asynchronous motors are thoroughly studied and proposed by foreign and domestic researchers [5–7]. The issues of observer computing for the synchronous motor with a salient-pole rotor and damper windings have not been practically covered.


Introduction
Variable frequency synchronous motors (SM) with the vector control of electromagnetic torque are used at main electric drives of the rolling mills [1][2][3]. It prevents motor dropout of synchronism and allows motor thrust operation as at DC electric drive at overloads in the torque-limiting mode.
As vector control implies operation in coordinates oriented according to one of the magnetic flux linkage, the SM control system may be designed on the orthogonal coordinate axes of the stator (α, β) or rotor (d, q). As a rule, control is implemented for the simplification purpose as a system rotating together with the controlled vector, namely, the d, q system. Consequently, this approach enables vector control due to regulation of scalar values of its projection of the coordinate axes. In this case, the torque control may be represented with the formula: (1) A significant disadvantage of this method is the need for controlling all four changing values, which complicates the control system. As a solution of this issue, a rotating coordinate system m, l is proposed in [4], which axis m is oriented in the direction of the flux linkage vector of the stator ψ , while the stator current projection is put on axis l. Switching to a new coordinate system helps preventing consumption of reactive power in the synchronous machine by means of excitation field control. Against this background, the stator current ̅ being orthogonal to the flux linkage ψ coincides with the stator voltage vector. Therefore, the torque control is reduced to scalar regulation of the stator flux linkage ψ and projection of the stator current , which form is similar to control of the DC motor current: The equation (2) provides for separate SM torque control -by the flux linkage and stator current.
Relying on the required torque, the control system generates stator current setting, which specifies the stator voltage vector with current controller and corresponding coordinate transformations. Then, the inverter of the frequency converter generates this vector by means of pulse-width modulation.
The main problem at implementation of this approach is complexity of measuring stator flux linkage ψ required for calculation of the SM internal angle. Strict dynamic requirements to electric drives of the rolling mills cannot be met without quick-response feedbacks according to coordinates unavailable for direct measuring. Under these conditions, the use of state observer appears to be rather promising, because it provides estimate of unknown value according to the known motor coordinates (current, voltage and rotation rate). The methods of observer selection and setting for asynchronous motors are thoroughly studied and proposed by foreign and domestic researchers [5][6][7]. The issues of observer computing for the synchronous motor with a salient-pole rotor and damper windings have not been practically covered.

Observer of magnetic field state of synchronous motor stator
The authors of this paper offer description of state observer of magnetic flux linkage of the stator of synchronous motor based on the Luenberger observer with error-feedback.
Two main principles are generally applied at implementation of the state observers for asynchronous motors (AM): observer with sensors (current meter and angular position encoder of the rotor) and sensorless observer (measuring current and voltage) [8,9]. In the first case, when designing observer, a mathe- Here, observer parameters are computed with due regard to active resistance of the rotor assumed to be constant. However, during operation under dynamic overloads, the rotor winding is greatly heated; its active resistance rises. Consequently, observer functioning under severe motor operation conditions cannot be appropriately because of changed initial design parameters. A solution of this matter is designing complex adaptive state observers.
When designing a sensorless observer, the value of angular rate of the rotor is calculated with mathematical method; the equation of dynamic AM equilibrium based on the stator voltage is used together with (3): A shortage of this method is the need for integration of differences (U sα − I sα R s ) and (U sβ −I sβ R s ) at determining projections of the stator flux linkage. As the value of active resistance of high power machine stator is extremely negligible, voltage drop at the stator is also low; hence, integration of variables upon availability of DC components may ultimately cause the system integral saturation.
To eliminate the above drawbacks of the observer of powerful SMs of the rolling mill, the proposed approach combines positive properties of the aforementioned principles. Accordingly, stator flux linkage is calculated with mathematical models of a synchronous machine reflecting its current and voltage and without notice of a non-linear nature of the magnetic system. The rotor rotation rate is supposed to be a known measurable value; the state observer is believed to be tolerant to its change.

Luenberger observer
The mathematical model of continuous linearcontrolled stationary entity with many inputs and outputs is presented in a standard (5) and transformed (6) view: where u -multidimensional vector of control actions (r×1); y -measurable m-dimensional output (m×n); x -variables setting the system within the state space (n×1); where x 1 -n-dimension vector ((n-m)×1), for which an observer is designed; x 2 -measurable m-dimensional vector (m×1); I -identity matrix (m×m). Expression (6) may be written as an equation system: According to Luenberger [10], this equation system is reduced to the observer of the following type: Equation (8) enables deriving a formula for error calculation: The use of the Luenberger observer to deduce flux linkage of the synchronous machine stator to be considered below shows that error calculation does not involve a critical difficulty, because matrix elements А 11 and А 21 varydepending on the rotor rate ω r having a real value. Coefficients K shall be selected in such a manner as to have eigenvalues of the matrix (А 11 -KА 21 ) in the left portion of the complex plane to provide a stable nature of the transient process.
Then, the observer of vector x 1 with error forecasting according to the feedback may take the following form [10]: This observer contains a derivative of output values, which is undesirable because it results in intensification of distortions present in initial signals. There under, the derivative is replaced with a dummy variable: When combining (10) and (11), a final observer formula may be written as the following equation system: The structural diagram of the reduced-order Luenberger observer with feedback is shown in Fig. 1.

Observer of current and flux linkage of SM stator
Equations of SM voltage equation shall be written as follows: u q = pψ q − ω r ψ d + R a i q ; 0 = pψ yd + R yd i yd ; 0 = pψ yq + R yq i yq ; ψ yq =L aq i q + L yq i yq ; where L d = L ad + L σa ; L q = L aq + L σa ; L yd = L ad + L σyd ; L yq = L aq + L σyq ; L f = L ad + L σf . Due to this replacement, the equation (13) will take the following form:  Expression (15) shall be presented in a matrix form: Equation (16) is reduced to a standard type of n-dimension mathematical model of synchronous motor (6), where

Observer parameter calculation
This calculation was performed for a real synchronous motor of the cold rolling mill, which parameters are provided in Table 1. As an observer being robust to variations of the rotation rate within the operating speed range is needed, coefficients k 12 , k 13 , k 21 , k 23 are taken to be equal to zero [11]. It enables weakening influence of the rotor rotation rate ω r on observer functioning. As a result, we obtain a characteristic equation for determination of design coefficients k 11   .
The observer has a rather high immunity at speed changes within a wide range both at real and complex conjugate roots of the characteristic equation. A stabilized speed control loop is an essential pre-requisite for powerful electric drives of the rolling mills [11].
Based on the belief that response speed of the observer shall exceed that of the system, real roots of the characteristic equation for case ω r = 0 are assumed to be located in the left-hand semi-plane at point -1,000. This root distribution corresponds to the following coefficient matrix: With rising |ω |, roots obtain an imaginary component; they are arranged at points -707.1 ± j707.1 on a complex plane. Variations of parameters k 11 , k 12 of the coefficient depending on the rotor rotation rate are offered in Table 2 and Fig. 2; here, the rated rotation rate ω r.rated = 120.95 rad/s is taken as a basic value.

Modeling results
At the investigations performed, the proposed observer of flux linkage rate was integrated into a mathematical model of the cold rolling mill SM. The motor is represented with complete SM equations in transient processes and provides for change of main engineering coordinates -the static torque. At the model output, main motor coordinates are computed, namely -currents (i d , i q , i f ), voltages (u d , u q , u f ) and rotor rotation rate (ω r ). The observer is based on the above mathematical provisions and structural diagram in Fig. 1. Controller parameters comply with data in Table 2. The model was implemented with the MATLAB Simulink software.
Modeling results are shown in Fig. 3 at acceleration and load rise (changes of torque and rotation rate  are shown in Fig. 3, а, b), where ψ d.s and ψ q.s -projections of flux linkages calculated with the known formulas (Fig. 3, c), ψ d.e and ψ q.e -projections of flux linkages supplied by the state observer (Fig. 3, d). Fig. 4 shows error of the observer: Δψ d = ψ d.e -ψ d.s and Δψ q = ψ q.e -ψ q.s , Wb, its relative value does not exceed 0.15 %. Such a high accuracy may be explained by the fact that parameters in the state observer complied with those of the SM model during modeling.

Conclusions
The paper highlights main principles of designing up-to-date systems based on the vector method of generating electromagnetic torque of the variable frequency synchronous electric drive of the rolling mill.
The authors offer the design of observer of magnetic flux linkage state of the stator of synchronous motor based on the Luenberger observer with errorfeedback.