A modulated model predictive control scheme for the brushless doubly-fed induction machine

This paper proposes a modulated model predictive control (MMPC) algorithm for a brushless double-fed induction machine. The Brushless Doubly-Fed Induction Machine has some important advantages over alternative solutions for brushless machine applications. The proposed modulation technique achieves a fixed switching frequency, which gives good system performance. The paper examines the design and implementation of the modulation technique and shows the comparison of finite control set-model predictive control (FCS-MPC) and MMPC. Simulation results verify the operation of the proposed modulation technique.


I. INTRODUCTION
ith the current concern about climate change, the world is paying more attention to the power sector. To reduce the emissions of carbon dioxide and minimize reliance on fossil fuels, wind power has turned into one of the best options for electrical grid power generation. In the report of 'the 2016 edition of the Global Wind Energy Outlook,' it was shown that wind power could reach 2,110 GW and supply up to 20% of global electricity by 2030 [1][2]. The generators used in these wind power applications, therefore, need to supply high-quality and efficient electrical power to the grid [3][4][5][6][7]. 1 The permanent magnet synchronous machine (PMSM) [3][4][5] and the doubly-fed induction machine (DFIM) [6][7] are the most common machines used in these applications. In a PMSM, Manuscript  the gearbox is not needed, which leads to high reliability and longevity, but the price is increased as the power converter must process all the electrical energy generated. The DFIM often has a high performance-price ratio when adopted in wind generation applications, partly due to the fact that the power converter only processes a fraction of the electrical energy generated. Other advantage of the DFIM include wide operational speed range, variable speed operation, and full power control capability. However, the gearbox and brushes of DFIM can lead to low reliability and higher maintenance costs [6].
To obtain improved performance, the dual stator winding induction machine (DSWIM) and brushless doubly-fed induction machine (BDFIM) have been used as alternatives to the DFIM. The DSWIM is suitable for electric aircraft applications [8][9][10][11]. In this paper, due to the absence of brushes, the maintenance costs of a BDFIM are lower than a traditional doubly-fed machine. The BDFIM is, therefore, suitable for drive systems in applications requiring minimal maintenance, such as Wind Energy Conversion Systems (WECS) [12][13]. The BDFIM, which is shown in Fig. 1, consists of two stator windings and one rotor winding [14][15][16]. The pair of stator windings includes a power winding (PW) and a control winding (CW). These windings have a different number of pole pairs so that they can produce the magnetic field and transfer energy through indirect interaction using the rotor winding.
To obtain high dynamic performance, vector control (VC) [17][18][19][20][21][22][23] and direct torque control (DTC) [24][25][26][27] have been proposed for this machine configuration. VC is a popular control method which uses a dynamic model within a unified reference frame. A VC usually includes two control loops, one fast inner current control loop, and one outer flux/speed control loop. In general, its linear nature does not consider the discrete operation of voltage source converter, and a vector conversion needed in the implementation, which may be complex and time-consuming. Furthermore, VC is sensitive to parameter detuning, so the machine parameters should be chosen and monitored carefully to satisfy the system stability and performance [22]. In applications where a DTC is applied, coordinate transformation and pulse-width modulation (PWM) are not required, the variables are controlled through torque and flux control in both steady and transient operating conditions. However, low-frequency current fluctuation, which is a feature of this technique, is often a serious disadvantage.
With the rapid development of digital signal processors and power devices, the finite control set-model predictive control (FCS-MPC) is now being considered as an alternative control method of power converters due to its advantages such as fast dynamic response, easy inclusion of nonlinearities and constraints of the system, and the flexibility to include other system requirements in the controller [28][29][30]. However, the switching frequency is variable, which decreases the system performance. To achieve a fixed switching frequency, modulated model predictive control (MMPC) has been proposed [31][32][33][34][35]. This paper presents the application of an MMPC strategy for the BDFIM. MMPC incorporates a modulation technique inside the FCS-MPC algorithm. The modulation uses a cost function to minimize the predictive algorithm by choosing two active vectors and one zero vector. Compared with FCS-MPC, MMPC achieves a fixed switching frequency, higher reactive power control accuracy, and lower torque ripple. Simulation and experimental results are presented to show the feasibility and effectiveness of the proposed approach as well as to allow an analysis of the performance. The presented MMPC strategy in this paper has been described in [36], more principle details, simulation and experimental results are added in this paper.
This paper is organized as follows. In Section II, the MMPC scheme for BDFIM is described, which includes the model of BDFIM and the theoretical comparison of MMPC and FCS-MPC. Then in section III and IV, simulation results of FCS-MPC and MMPC are presented to demonstrate the feasibility of the proposed methods, and experimental results verify the correctness of the simulation.

A. Model of the BDFIM
The BDFIM is composed of two three-phase stator windings and a special three-phase rotor winding. One of the stator winding is denoted as power winding (PW) and the other one is called control winding (CW).
Assume that the PW is directly connected to the grid in positive phase sequence, and the CW is connected to the grid via a voltage source inverter in reverse phase sequence connection. To simplify the model of BDFIM, the harmonic magnetic fields and their coupling effects are ignored. PW, CW, and rotor are described in dq reference frames which based on PW flux orientation with p1-type pole-pair. The voltages and currents vectors of PW, CW, and rotor, which are represented as , , and , should be transformed from their reference frames to the dq reference frame.  (3) where , are the angular frequencies of PW and CW; and are pole pairs of PW and CW; is the angular speed of the rotor. To simplify the expressions, the vector superscripts dq are removed.
The model of the BDFIM with the voltage and flux equations in dq reference frame can be shown as follows. 11 d Rj dt where , , , , , , and are the PW stator voltage vector, CW stator voltage vector, PW stator current vector, CW stator current vector, rotor current vector, PW stator flux vector, CW stator flux vector and rotor flux vector; , , , , and are the PW stator resistance, CW stator resistance, rotor resistance, PW stator inductance, CW stator inductance and rotor inductance; is the coupling inductance between the PW stator and rotor winding; is the coupling inductance between the CW stator and rotor winding; , and Q are the electromagnetic torque, load torque and reactive power; J and μ are the moments of inertia and friction coefficient; * is the conjugate symbol; Im is a symbol to obtain the imaginary part of vector.
Since the rotor resistance is much lower than its reactance, reference [37] ignores the rotor resistance and verifies the feasibility of the simplified model. When the BDFIM is running at steady state, , equation (6) can be simplified as: Since the rotor resistance has been ignored in this simplified model, the rotor flux The simplified model is derived as follows:   (15) by using a forward Euler approximation and the relation between the discrete-time variables can be described as: (18) where , , I is a 2*2 unit matrix, is the sample period.
B. Current control of the BDFIM a) Model of the converter Fig. 2 shows the topology of a back-to-back bidirectional power converter and its valid switching states. This converter prohibits the state that two switches in each leg are turned on at the same time, therefore the short-circuit switch states should be precluded. Six active vectors are represented by switching states 1-6 and the zero vectors are represented by switching states 7-8. Sj (j {1, 2, 3, 4, 5, 6}) equals to '1' when switch Sj is turned on and equals to '0' when switch Sj is turned off, respectively. Assume the value of dc-link voltage is . A constant dc-link voltage is achieved from the three-phase voltage source rectifier which adopts a Space Vector Pulse Width Modulation (SVPWM) strategy [38]. The output voltage in dq reference frame is expressed as:

b) Control strategies of FCS-MPC and MMPC
A superior control strategy could make the controlled variables catch up with the corresponding reference value respectively at the end of the sampling period. The operation of the BDFIM is required to satisfy two targets: (i), the desired reactive power * Q in PW should be obtained; (ii), the speed of the BDFIM should track the reference speed. According to equation (16), the target (i) can be obtained by setting: 1 11 To achieve the target (ii), a proportional-integral (PI) regulator shown in Fig. 3 is adopted. The output of this PI regulator is a reference electromagnetic torque, which can be satisfied with an appropriate according to equation (17). The reference value is: The predictive CW current in the next sampling period can be derived from the discrete BDFIM model. To obtain these two targets, the cost function g is defined as the deviation between the reference CW current and the predictive CW current. And it is expressed as:

1) FCS-MPC
The FCS-MPC is a well-known technique which uses the finite number of all possible switching states generated by the power converter to realize the optimal control. In this paper, the possible switching states are generated by the bidirectional power converter shown in Fig. 2. Eight predictive values are evaluated with the reference in the cost function g. Then, the first optimum vector switching state with the least value of the evaluation is chosen in the next sampling time. Fig. 4 shows the diagram of an FCS-MPC for the BDFIM.

2) MMPC
The proposed MMPC strategy shown in Fig. 3 evaluates an extra cost function based on the FCS-MPC. To obtain a fixed switching frequency, two adjacent active vectors and , and a zero vector will be chosen in each sampling period. It is assumed that the cost functions g of these vectors are gj, gk and g0 respectively. , and are the duty cycles of these vectors. is the sampling period. K is a positive constant. The duty cycles for the two active vectors are calculated by solving: Thus, the duty cycles for each vector are given as: Then the new cost function Cost can be defined as: At last, the two active vectors of the minimum cost function value are the optimal solutions and are used to the bidirectional power converter at the next sampling time. Thereafter, the appropriate ratio which is determined by the duty cycles ( , , ) is shown in Fig. 5 within each sampling period.  Fig. 3 shows the diagram of the proposed hybrid PI-MMPC scheme, which can be divided into two parts: outer speed loop and inner current loop.

C. Proposed PI-MMPC control scheme
The outer speed loop consists of a rotor speed regulator and a torque transfer function. The rotor speed regulator is used to adjust the rotor speed to the set value. PI-based controllers will be used for the outer loop. The inner current loop consists of a reactive power transfer function, an MMPC controller of CW current and a phase-locked loop (PLL). The reference current value is obtained according to the torque transfer function. The reference current value is obtained according to the reactive power transfer function. The MMPC controller generates two active vectors and a zero vector according to the cost function, which are used to the converter at the next sampling time. Fig. 6 shows the calculation of and the PLL is used to track the phase angle of the three-phase grid voltage accurately.

III. SIMULATION RESULTS
The operation and implementation of an MMPC for the BDFIM are verified using a simulation in MATLAB/Simulink. Table I shows simulation parameters of the system. According to equation (12), when PW and CW rated frequency are respectively 50 Hz and 0 Hz, the natural rotor speed is 750 r/min. The PW stator is directly connected to the grid. And the CW stator is fed by the back-to-back bidirectional power converter. Test 1: the expected reactive power is 0 Var, the reference rotor speed increases from 600 r/min to 800 r/min during the period 3-5 seconds when the load torque is 50 N· m.
Test 2: at t = 3 s, the load torque changes from 50 N· m to 25 N· m, and the reference of rotor speed is 600 r/min. Fig. 7 shows simulation results of the conventional FCS-MPC and Fig. 8 shows the results of the proposed MMPC. Both results show the waveforms of the rotor speed, the dq-axis CW stator current and the reactive power. It is obvious that the dq-axis CW stator current ripple of the MMPC scheme is lower than the conventional FCS-MPC scheme. And the MMPC scheme features higher control accuracy. Since the reactive power is indirectly controlled by the dq-axis CW stator current, results with a higher accuracy will be produced in power tracking due to lower CW stator current ripple. In both cases, satisfying dynamic responses are observed in the dq-axis CW stator currents.
The conventional FCS-MPC predicts the behavior of all possible switching states according to its mathematical model at each sampling period. The output voltage vector which is represented by the chosen switching states is different, which means the switching frequency is variable. For this reason, it may cause a high current ripple in some cases.  In the proposed MMPC, a modulation technique which is similar to the space vector modulation (SVM) is added in the conventional FCS-MPC. Two active vectors and one zero vector are selected for the next sampling period, and the duty cycles for the chosen vectors are calculated by the cost function during the sampling period. Then, a switching pattern procedure is adopted to choose two active vectors and one zero vector. A higher and fixed switching frequency is obtained by using this modulation. The current ripple does not have much impact on the mechanical robustness of the system, and speed dynamics are smooth for both FCS-MPC and MMPC. However, the reactive power under the MMPC scheme is closer to zero than the reactive power under FCS-MPC scheme, which will improve the power quality.
To obtain a good comparison and show the effectiveness of the proposed control strategy, the harmonic spectra for the CW current based on CW static 2 reference frame are shown in Fig. 9 and 10. The harmonic components for MMPC in Fig. 10 are shown to be lower throughout the frequency range compared to FCS-MPC in Fig. 9. These results show that superior system performance in terms of current harmonics can be obtained when using the proposed MMPC scheme in comparison to the FCS-MPC scheme.

IV. EXPERIMENTAL RESULTS
In order to further investigate the proposed MMPC strategy, a prototype of the BDFIM shown in Fig. 1 is implemented, and the experimental parameters are same with the simulation parameters shown in Table 1. The experimental system arrangement is shown in Fig. 11. To test the performance of the proposed control method with different loads, a 30 kW induction motor which is coupled with the BDFIM is worked as a load machine. The PW stator of BDFIM is directly connected to the power grid (380 V/50 Hz), the CW stator is fed by a back-to-back voltage source converter. The constant dc-link voltage which is achieved from the three-phase voltage source rectifier is 650V. The proposed control structure is implemented in a combination of a digital signal processor (DSP) TM320F28335 and a field-programmable gate array (FPGA) EP2C8T144C8N.
The maximum current limit is selected as 40A to sufficiently supply the CW current of 20 A. This is chosen within the actual maximum limits for control design verification. And a sampling frequency of 4 kHz (250 μs sampling period) is chosen for the MMPC scheme. Case 1: In Fig. 12 and Fig. 13, the expected reactive power is 0 Var, and the reference rotor speed is set to 600 r/min and 800 r/min which means that the BDFIM works in the sub-synchronous mode and super-synchronous mode, respectively.
It can be seen that the rotor speed tracks its reference value well, the error between them is controlled at about 2 r/min. The reactive power fluctuates round ±400 Var at 600 r/min and 800 r/min. If the BDFIM is operating at a rated power of 30 kW, the input power factor is still nearly unity even if the reactive power fluctuation is ±400 Var. The CW stator currents are also stable with small fluctuates. Decoupled from the model calculation in (20) and (21), corresponds to the reactive power of PW stator and corresponds to the electromagnetic torque . Although there are some oscillations in these four variables, the results are acceptable in practical applications.  Case 2: To synchronize the output frequency of the BDFIM with the grid, it is apparent that the rotor speed must track the rotor's speed reference value. In this case, the reference rotor speed is increased from 600 r/min to 800 r/min during the 3-5 seconds. And the load torque and its power value are set to 50 N· m and 3 kW. The conditions of the experiment in Case 2 are the same as those of Test 1 in the simulation. Fig. 14 shows the waveforms of the PW line voltage , the PW phase current , and the CW phase current in an oscilloscope. Within 5 seconds, Fig. 14(a) shows four states of and in Case 2 and Fig. 14(b) shows the experimental waveform of . These three variables change rapidly and stabilize quickly after the rotor speed has stabilized. In Fig.  15(a), speed tracking performance is presented when the rotor speed varies, it can be seen that the rotor speed tracks the reference value fast after the rotor speed is changed.
has a maximum change of about 10 A and the maximum reactive power variation is about 1500 Var when the rotor speed is changed. Although the CW current and the reactive power have some oscillations in Fig. 15(b)-(c), the results are acceptable in practical applications.  Like case 2, we recorded waveforms of , , and in the oscilloscope in Fig. 16. And in Fig. 17, as the load changes, a few adjustments take place in the rotor speed, CW current, and the reactive power. Moreover, the rotor speed increases because the electromagnetic torque is larger than the load torque at the change moment, then the rotor speed recovers to its original state when the electromagnetic torque decreases due to the decrease of . After about 0.2 seconds, these four variables tend to reach a steady state and decreases to a lower value in the steady state. This experiment confirms the designed MMPC-based controller is capable of maintaining stable operation of the drive system when the load changes and it can also achieve an excellent dynamic control performance. Compared with the simulation results, the tracking error of the CW current and the reactive power are larger in experiments, which are mainly caused by the non-ideal power sources, inaccurate motor parameters, and the converter losses.
However, the results are acceptable in practical applications. In sum, the correctness and merits of the MMPC used in a BDFIM are verified. Through the proposed method, excellent dynamic response and low ripple current are obtained.

V. CONCLUSION
An MMPC scheme with a fixed switching frequency for the BDFIM has been proposed in this paper. The modulation technique incorporates the SVM to the FCS-MPC. With a simple implementation structure, a satisfying tracking performance of the reactive power and rotor speed are achieved by a cost function. Compared with the results of the FCS-MPC, low current ripple and higher control accuracy are obtained. An outstanding system performance has been achieved and simulation results show the feasibility and effectiveness of the proposed method. The steady-state performance and fast dynamic performance of MMPC are verified by experimental results.
The MMPC retains the desirable advantage over the FCS-MPC, such as fast dynamic performance. Moreover, a fixed switching frequency is obtained by using a PWM scheme in this technique. The MMPC strategy is also dependent on the machine parameters, the evaluation on parameter sensitivity of the MMPC strategy is worthy of study in the future work. Rivera was awarded the Chilean Academy of Science Doctoral Thesis Award (Premio Tesis de Doctorado Academia Chilena de Ciencias), for the best PhD thesis published in 2011, selected from among all national and international students enrolled in any exact or natural sciences program in Chile and also he was awarded as an Outstanding Engineer in 2015. His research interests include matrix converters, predictive and digital controls for high-power drives, four-leg converters, renewable energies and development of high performance control platforms based on Field-Programmable Gate Arrays.