PREDICTIVE CURRENT CONTROLLER WITH MTPA CONTROL STRATEGY – SIMULATION RESULTS

. Electric and hybrid vehicle drive systems are commonly equipped with Permanent Magnet Synchronous Motors, because they offer high efficiency in broad range of speed and torque, and also due to special current development techniques they can work with high speeds. The paper presents simulation results for predictive current controller used in PMSM drive system. The results were obtained for high speed operations with Maximum Torque Per Ampere (MTPA) control strategy, which allows to develop set torque value with the smallest possible currents values. Predictive current controller equations are based on IPMSM mathematical model which considers L d and L q inductances changes. Additionally MTPA strategy also considers inductances changes in d- and q-axis.


Introduction
PM excited machines are receiving increased attention in drive systems applications instead of induction motor because of high power density, efficiency, high torque-inertia ratio and wide speed operation capabilities which makes it particularly suitable for all automotive applications where weight and geometry dimensions are very important. In particular, an IPMSM with permanent magnets buried inside steel rotor core perform mechanically robust rotor structure which is suitable for high-speed operation applications. Very important aspect of electric vehicle drive |system is an energy efficient torque development, which can be achieved with MTPA control strategy and proper current controller or other adaptive or heuristics methods [4]. Additionally Field-Weakening (FW) strategy increases Constant Power Speed Range (CPSR). Wide speed range with constant power is required especially for the traction drive applications. CPSR depends on inductances and magnet flux values and power supply constraints. Unfortunately magnetic saturation may occur under different speed and load conditions resulting in the variation of the d-and q-axis inductances [1,5,6]. The variation of inductances is especially strong in IPMSMs and it is important to note that the performance of the drive is rather sensitive to machine parameters, especially for high torque regions.
In this paper there will be described the proposed algorithm starting from the mathematical model used to calculate the dand q-axis reference currents in the constant torque region. In the constant torque region the requirements are maximum efficiency of the drive or a good dynamic performance of the drive. The paper present equation set and achieved results. Fig. 1 shows a four pole IPM motor, which was used in the simulations. The following equations that describe work of a PM Synchronous motor are basic voltage and flux-current equations described in synchronous dq reference frame under several assumptions [8]:

IPMSM modeling
Torque developed by a PM excited motor can be expressed as:

IPMSM torque development
Torque developed by IPM motor can be divided into PM generated and reluctance torque components (Fig. 2).
The first term is PM generated torque, and the second term is reluctance torque which is proportional to the difference in stator inductances, L d and L q . In the analyzed IPMSM, L q is higher than L d (due to the lower q-axis path reluctance), because the magnetic flux flowing along the d-axis has to cross through the magnet cavities in addition to the rotor air gap, while the magnetic flux of the q-axis crosses only the air gap. The d-axis is also basically magnetized with PM [2].
Properly controlled currents in motor windings give a possibility to control synchronous machine in a similar manner to a separately excited DC machine, where flux and torque (with constant flux) can be controlled separately [1,4].

MTPA control strategy
Basic FOC (Field Oriented Control) structure consist of four PI controllers (Fig. 3 As shown in Fig. 2 the main torque component is developed with q-axis current. With MTPA control strategy application it is also possible to take advantage of reluctance torque according to (5,6). This way the smallest possible phase currents develop set torque value. The copper losses are minimized, and under assumption that core losses are negligible (which for low speeds and torques is generally true) the maximum efficiency of the IPMSM can be obtained.
It is needed to estimate and use L d , L q inductances values for different stator currents [7] according to equation (6). Calculated inductance values for different currents are shown in the Fig. 4.
The i di q planes with MTPA curve for different L d and L q values are presented in Fig. 5 and 6. The i d -i q palne with MTPA considering changing values of L d and L q (according to Fig. 4) is shown in Fig. 7.   The problem arises when the flux from magnets gives high electro-motive force, in the high speed regions, which results in exceeded supply voltage. Using d-axis negative current main flux is decreased and thus it is possible to stay in the voltage limit. This method is called field weakening method. The control strategy meets several limitations: Evaluating voltages and current constraints it is possible to identify two main regions [4]: 1) Constant Torque Region, 2) Constant Power Region.
The second one will be a subject of the next author paper.

Predictive current controller
The paper is focused on nonlinear current controllers. Each control system, which uses information of an object in order to develop optimal control signals can be included to the predictive controllers group. Fig. 8 presents the idea of a predictive current controller work.  where i d , i qstator currents, u d , u qstator voltages, Rstator phase resistance, L d , L qd-and q-axis inductances, PM  permanent magnet flux in the air gap. Considering very small time stepsmaller then electromechanical and electrical time constant it is possible to assume that the position and speed change is negligible. In this case equations (1, 2) can be written as: Finally, the equations that represent future current component vector values, that are developed for the selected voltage vector may be expressed as: where: is a input voltage vector for all seven inverter key configurations, DC U is a DC bus voltage and T is sample time. Such prediction is realized according to rules of linear extrapolation, that could be based on only one sample [3,4]. Details are presented in [2][3][4].
At start of a calculation period, the current and the voltage measurements are done in order to evaluate F, M and H matrices.
The future values of current vector are evaluated for each inverter key configuration. From this calculation it is possible to evaluate current change vector for each inverter key configuration (Fig. 8).
In the classic predictive current control structure the chosen key configuration is the one that fulfils the control quality indicator. Most popular criteria is the angle between the set current vector and the future current vector or the distance between ends of set and future current vectors. The chosen aim function is expressed as: On the basis of predicted current vectors three inverter key configurations (two active and one non-active) which cause current to achieve set value have to be chosen. Two non-active voltage vectors give the same current values but using the proper one it is possible to minimize number of inverter key switchings. Additionally on the basis of simple mathematical rules the times of application for each key configuration is calculated [2,4]. Nonlinear current controller acts in this way as a voltage space vector modulator. Linear predictive controllers also use equations presented in this paragraph but take advantage of a Pulse Width  id-iq plane current error area

Summary
Presented modified predictive current controller with MTPA control strategy and L d , L q changes consideration shows very good characteristics in comparison with common current controllers [2][3][4]. It shows excellent static characteristics and very good dynamic performance. The MTPA strategy with modified predictive current controller performs high dynamic, high efficiency and high torque capabilities which seems to fulfil the requirements for an electric vehicle drive system. A disadvantage of the proposed algorithm is number of calculations which demands powerful DSP to execute calculations in proper time. The current controller may be applied in F28335 Delfino Digital Signal Processor from T.I. thanks to its powerful calculation capabilities and many external interfaces. Not only prediction but also mathematical transformations needed for time recalculation cause high loadings for processor [2,3].