OPTIMAL GAIN OF PI SPEED CONTROLLER IN RELUCTANCE SYNCHRONOUS MOTOR USING PARTICLE SWARM OPTIMIZATION

This paper presents a new direct torque control (DTC) strategy for Synchronous Reluctance Motor using the particle swarm optimization (PSO) algorithm. In conventional direct torque controlled (DTC) Synchronous Reluctance Motor (SRM), there is usually undesired torque and flux ripple. So Tuning parameter of the PI-Controller (Kp, Ki) are essential to DTC system to improve the performance of the system. In this work, particle swarm optimization (PSO) is proposed to adjust the parameters (Kp, Ki) of the speed controller in order to improve the performance of the system, and run the machine at reference speed.


INTRODUCTION
The SRM has attracted significant interest of industry due to their main advantages are [1] [2]:  Simplicity and robustness  High torque overloads capacity  High efficiency over wide speed-range  Low machine inertia  Decreased maintenance requirements The absence of windings and magnets on the rotor enables SRM to run high speed and temperature. An SRM can produce large torque in a wide speed range.
All this reinforces the idea of the optimal design of a system of tracking in the aim to push system solar efficiency to an interval more incentive for investment.
The most modern technique is direct torque control method (DTC). The DTC offers many advantages like fast torque response, no need of coordinate transformation and less dependence on the rotor parameters [3]. The conventional PI (proportional, integral) control method is widely used in motor control system due to the simple control structure and easiness of design. However tuning the parameters of PI controller is a difficult task. To enhance the capabilities of traditional PI parameter tuning techniques, several intelligent approaches have been suggested such as the particle swarm optimization (PSO).
Particle Swarm Optimization (PSO) is one of the modern algorithms used to solve global optimization problems. Thus, to solve an optimization problem, PSO applies a simplified social model [4]. Compared to other methods [5], the advantages of PSO are that PSO possesses the capability to escape from local optima, it is easy to be implemented and has fewer parameters to be adjusted [6]. The PSO method is an excellent optimization methodology and a promising approach for solving the optimal PI controller parameters problem.

MATHEMATICAL MODELING OF SRMOTOR
The model adopted for the SRM suitable for DTC control is as follows [1] [2].
Where ( 1 ) The electromagnetic torque is expressed in the same frame by: The motor mechanical equation is written as follows: ( 3 )

DIRECT TORQUE CONTROL OF SRM
The DTC control is based on the direct determination of the command sequence used to switch a voltage inverter.
This choice is usually based on the use of hysteresis comparators whose function is to control the system state, namely the amplitude of stator flux and electromagnetic torque. A two levels classical voltage inverter can achieve seven separate positions in the phase corresponding to the eight sequences of the voltage inverter [1][2][3].
These positions are illustrated in Fig. 1. In addition Table 1 shows the sequences for each position, such as: S i =1, ..., 6, are the areas of localization of stator flux vector, on the other hand, the error Δφ, between the reference flux and the flux estimated, is introduced into a hysteresis comparator for two levels, which delivers '1' if the error is positive and '0' if it is negative as well, the error ΔT e , between the reference torque and estimated torque is introduced into a hysteresis comparator for three levels that delivers '1' if positive, '0' if zero and '-1' if negative.
The use of three levels to adjust the torque has been proposed to minimize the average switching frequency, because its dynamics is generally faster than the flux [1][2][3].   The synthetic sequence can be illustrated through the following example: Assuming that the flux vector is located in sector 1 ( Fig. 1), then if the error between the reference flux and the stator flux is positive, we must increase the flux this is only possible by applying a voltage vector in the same direction, according to (4) or V1(100), V2(110) or V6(101). However, applying voltages of opposite direction V3(010), V4(011) or V5(001) decreases the variation of the flux [1][2][3].
On the other hand, if the error between the reference torque and the electromagnetic torque is positive we must increase the electromagnetic torque by applying the voltage vectors in the half plane of positive angles, according to (5), i.e. V2(110), V3(010) or V4(011).
Combining these states we can decide which sequence should be applied [1][2][3]. The following diagram describes the process of DTC controlling an SRM associated with a two-level inverter supplied by controlled battery.

PARTICLE SWARM OPTIMIZATION
Particle swarm optimization is a heuristic global optimization method put forward originally by Doctor Kennedy and Eberhart in 1995. It is developed from swarm intelligence and is based on the research of bird and fish flock movement behaviour [4,[7][8][9][10].
PSO has two primary operators; velocity and position update. In this paper the main objective of PSO is minimization of speed error. Fig. 4 shows the block diagram for PI controller and the corresponding objective function is as shown in equation (7) and (8).

Step 1: Initialization
Each element of the swarm is initialized randomly within the effective operating limits [4] [5]. P initial The particles are initialized as follows as given in eq. (6) and v initial the velocity of particles initialized as given in eq. (7) Where, rand is a random positive number between 0-1. [4] ( 8 )  Where: p best =(p best , p best , ...., p best ) is the best previous position yielding the best fitness value for the i th particle; g best = (g best , g best , ...., g best ) is the best position discovered by the whole population [4] [5]. S k is the current position of individual. c 1 and c 2 are the acceleration constants reflecting the weighting of stochastic acceleration terms that pull each particle toward p best and g best positions, respectively. rand 1 and rand 2 are two random numbers in the range [0, 1].W max is the initial weight, W min is the final weight, iter max is the maximum iteration number and iter is the current iteration position [4][5].

Step 3: Inertia Weight Improved PSO (IWIPSO)
In this section, for getting the better global solution, the traditional PSO algorithm is improved by adjusting the weight parameter, cognitive and social factors. Where c 1min , c 1max : initial and final cognitive factors, c 2min , c 2max : initial and final social factors.

PROPOSED METHOD
In this work we used PI controller for optimal regulation of rotor speed at the desire speed. The general block diagram of the PI speed controller is shown in Fig.  3. The output of the speed controller (torque command) at nth instant is expressed as follows [4][5][7][8][9][10][11]: Input can be define as ( 1 9 ) That K i and K p are proportional and integral coefficient in PI controller.
Proportional integral (PI) controller can be used to control the speed of SRM. The PI controller is normally avoided because differentiation can be problematic when input command is a step. Generally, the speed error, which is the difference of reference speed (ω ref (n)) and actual speed (ω r (n)), is given as input to the controllers. These speed controllers process the speed error and give torque value as an input. Then the torque value is fed to the limiter, which gives the final value of reference torque. The speed error and change in speed error at n th instant of time are given as ( 2 0 ) ( 2 1 ) In PI controller design methods, the most common performance criteria are integrated absolute error (IAE), the integrated of time weight square error (ITSE), integrated of squared error (ISE) and integrated of time weight absolute error (ITAE) that can be evaluated analytically in the frequency domain [12,13]. These four integral performance criteria in the frequency domain have their own advantage and disadvantages. For example, disadvantage of the IAE and ISE criteria is that its minimization can result in a response with relatively small over shoot but a long settling time because the ISE performance criterion weights all errors equally independent of time. Although the ITSE performance criterion can overcome the disadvantage of the ISE criterion, the derivation processes of the analytical formula are complex and time-consuming [13]. The IAE, ISE, ITAE and ITSE performance criterion formulas are as follows [4][5][7][8][9][10][11]:

RESULTS AND ANALYSIS
In this section, simulation results related to the proposed controller PI-PSO for controlling speed of a Synchronous Reluctance Machine (SRM) will be presented and compared with those obtained by using the controller conventional PI. The rated values and parameters used in the simulation program are as follows:

Parameters
Symbols Values The main objective of this application is to provide as input a reference speed that must enslave Synchronous Reluctance Machine. For this, two case examples are studied.
In the first case, the reference speed is defined by a echelon which varies between (100 rpm/s and 200 rpm/s) to demonstrate the performance and efficiency of the proposed model (PI and PI-PSO) in an extreme case (Fig.  4), the mechanical load torque varies between (0N.m) and (3N.m) (Fig. 5). For the second case, the reference speed is represented by repetitive sequence of trapezoids (Fig. 6), the torque mechanical load being kept constant during the simulation time (T m = 3N.m) (Fig. 7).

First case: Control by echelon
In this case, the reference speed and mechanical load torque are defined by steps (Fig. 4). Fig. 4 show the time response of the machine to the reference speed using the two control strategies (controllers): the conventional PI and PI-PSO. Fig. 4 shows the reference speed used as the response time is not achieved in the case of a conventional PI controller. But the time response on using PI-PSO is obtained at time (t = 0.05s).
Moreover, to illustrate the performance and the efficiency of the proposed model, Fig. 5 show the electromagnetic torque response provided by these two controllers.
The response presented by electromagnetic torque Fig.  5 (blue colour) concerning the conventional PI controller, shows that the oscillations are not attenuated during the time of simulation. In Fig. 5 (red colour), the oscillations are reduced moderately by about (5%) by contribution in case of a PI controller.

Second case: Control by a trapezoidal sequence
In this case, the reference speed is defined by a repetitive sequence trapezoids, the torque is fixed to (3N.m). Fig. 6 shows the speed responses for the two strategies command used. As illustrated in Fig. 6, the control strategy by particle swarms PI-PSO is more suitable than the other strategy PI-Conventional in the different phase control of the SRM in terms of stability and response time required. Fig. 7 shows the electromagnetic torque response. This figure also confirms the results concluded previously.

CONCLUSION
In conclusion, due to the nonlinear behaviour of the system, disturbances of the variation of parameters and load torque, the conventional control strategy is inadequate for controlling the SRM. In effect, using the conventional PI controller, convergence is obtained occasionally and generally depends on a correct adjustment of PIparameters.
Therefore the controller based on the particle swarm is proposed and compared with that based on conventional PI controller. According to the simulation results, it is clear that the PI-PSO strategy provided better answers speed and accurately than the other strategy.