Speed controller design for three-phase induction motor based on dynamic adjustment grasshopper optimization algorithm

Received Jun 19, 2020 Revised Jul 26, 2020 Accepted Sep 22, 2020 Three-phase induction motor (TIM) is widely used in industrial application like paper mills, water treatment and sewage plants in the urban area. In these applications, the speed of TIM is very important that should be not varying with applied load torque. In this study, direct on line (DOL) motor starting without controller is modelled to evaluate the motor response when connected directly to main supply. Conventional PI controller for stator direct current and stator quadrature current of induction motor are designed as an inner loop controller as well as a second conventional PI controller is designed in the outer loop for controlling the TIM speed. Proposed combined PI-lead (CPIL) controllers for inner and outer loops are designed to improve the overall performance of the TIM as compared with the conventional controller. In this paper, dynamic adjustment grasshopper optimization algorithm (DAGOA) is proposed for tuning the proposed controller of the system. Numerical results based on well-selected test function demonstrate that DAGOA has a better performance in terms of speed of convergence, solution accuracy and reliability than SGOA. The study results revealed that the currents and speed of TIM system using CPIL-DAGOA are faster than system using conventional PI and CPIL controllers tuned by SGOA. Moreover, the speed controller of TIM system with CPIL controlling scheme based on DAGOA reached the steady state faster than others when applied load torque.


INTRODUCTION
Some applications like paper mills, water treatment plants and several other industrial factories are used three-phase induction motor (TIM) because it is simple, rugged and low maintenance [1,2]. Controlling method of induction motor is very difficult due to complexity and nonlinearity of the system [3][4][5]. Different load torque may apply to the system, and that would effect on speed performance. Therefore, speed of induction motor has to be controlled in order to let the system work with high performance at different load torque [3].
PI controller has been widely used in industrial applications due to its structural simplicity and the ability to solve practical control problems [6]. There are several existing works revealing that the use of PI controller will lead to poor performances as well as causing instability for the controlled system. Several researches have been done on the speed controller of TIM. In [1] particle swarm optimization (PSO) was used for tuning PI speed controller of TIM. Universal developed drive software of a speed controller of single

SYSTEM DYNAMIC MODEL AND DESCRIPTION
Induction motor consists from two parts constant part, which is stator, and moving part, which is rotor. Matlab/simulink environment is used to implement of three-phase induction motor. The model is built using the stator and rotor equations in alpha and beta axis [18], in order to designing a controlling method for the current and speed of the system in simple way. Three phase induction motor in alpha ( ) and beta ( ) equations are shown below: Where , , , and are stator voltage and current in -axis respectively; and are stator flux in -axis; is stator resistance; and are rotor flux in -axis; , , and are magnetising inductance, stator inductance and rotor inductance respectively; is leakage coefficient; and are rotor current in -axis. Three-phase induction motor equations from (1) to (10) are used to build the system in Matlab/Simulink as shown in Figure 1. In this paper, the system parameters are illustrated in Table 1. From Table 1, it can be seen that the starting current of induction motor is about 250 A, speed is 1475 rpm and torque is about 160 N.M. The purpose of making speed controller for the TIM is to let the system work with same speed 1500 rpm under various load torques. Three-phase induction motor is modelled using Matlab/Simulink block in -axis as shown in Figure 1. TIM is connected as direct on line (DOL) to three-phase main supply 415 v to check system behaviour as depicted in Figure 2. In DOL connection the three-phase supply (A, B, C) is converted to alphabeta axis using Clark transformation as illustrated in Figure 3.  The loop from mechanical torque of the system and add it with load torque to produce speed in RPM using following equation.
The mechanical (11) of the system is represented in Simulink block as depicted in Figure 4. The stator current, motor torque and motor speed responses for TIM without controller and without load torque are demonstrated in Figure 5. Additionally, the responses for the same variables of TIM but with applying load torque are depicted in Figure 6. In Figure 5, it shows that starting current is above 200 A at transient with oscillation and motor starting torque is above 500 N.M during transient, while motor speed is 1500 RPM but takes 0.45 s to reach the steady state with high oscillation during transient. Figure 6, it shows that three-phase induction motor speed is reduced to 1452 RPM with oscillation when applied load torque to a system at 0.5 s.

CONTROLLING TECHNIQUES 3.1. Conventional PI controller
The proportional plus integral controllers are the dominating control action for a wide range of industrial processes. The mathematical representation of PI control signal is as follows: where and are the proportional gain and the integral time constant, respectively. The design objective is to obtain optimal values for and such that an objective function with desired specifications is minimized. For this target, the system is excited by a reference signal, and the system response is obtained in terms of a number of parameters. In this paper, the excitation signal is a unit step with the transient response being characterized by the settling time [19].
Conventional PI controller is designed for and currents. The plant that used to design such controller is derived from (14) and (15) below. Characteristic equation is used to determine the PI controller parameters, which are and . This design depend on natural frequency ( ) and damping ratio ( ) of the characteristic equation formulated in (13). Current controller of the system should have very fast response because the system plant have resistance and reactance, which identify the electrical circuit time constant. In addition, the parameters and are chosen to let the system work with a powerful response. Moreover, the controller depends on resistance and reactance Therefore; the current controller should be faster than speed controller, which depends on moment of inertia ( ) and friction. The value of and for and currents controller are chosen 200 HZ and 0.7 respectively.
In (14) and (15) are used to find the transfer function of the plant. It can be seen from equations have same resistance , reactance and leakage coefficients . Therefore, the system plant are same for both and . Therefore, PI controllers have same parameters for and currents. PI controller parameters are = 75 and =0.0746. The for the PI controller is time constant which is equal to = . Figure 7 illustrate the desigof three-phase induction motor with current controller using PI controller. The value of is 12.3 A and is 20.55 A. It shows that * and * references are subtracted from and feedback from system after converting and to and using park transformation in block Simulink illustrated in Figure 8. The response of TIM in terms of currents with PI controller is demonstrated in Figure 9. The stator quadrature current of the system reached steady state at 0.006 s after designing controller using characteristic equation with natural frequency 200 Hz and ζ =0.7. The speed controller for the system is designed in the same manner followed in the design of the stator direct and quadrature currents controllers. Motor speed depends on the mechanical components, which are motor inertia and friction . For simplicity, the friction is chosen to be zero in this research work. Moreover, the speed loop frequency is chosen to be much lower the current loop frequency, which is because the motor speed needs more time to change from one speed to another, i.e. it is not practical to change motor speed within milliseconds. Therefore, (13) can be used to find PI controller parameters with natural frequency = 2 Hz and = 0.7. From Figure 10, it can be seen that speed of induction motor with conventional PI controller arrived steady state at 0.85 s, this is because of determination PI controller parameters and values of natural frequency and damping ratio. These values has highly effect on current controller response and would effect on speed controller. It can be seen that from Figure 11, speed reference is subtracted from feedback speed of the system and the error is enter to PI controller. The output of controller is current reference to the system. Therefore, it is very interesting to modified current response by using proposed combined PI-lead (CPIL) compensator. In addition, it is very important to achieve current with proposed controller faster than current with PI controller in order to get very fast speed response.

Proposed controller
The drawbacks of PI controlling scheme are slow response speed and poor robust performance compared with the exogenous disturbances. In this research work, the phase-lead controller is used due to its superiority against conventional PI control action in terms of improve the dynamic performance as well as the transient response of the system. In case of a phase-lead compensator, the zero ( ) of the control law is located nearer to the origin of the S-plane as compared to the compensator's pole ( ) [20].
The design of the proposed combined PI-lead (CPIL) compensator is developed to provide a better steady-state tracking, faster recovery from disturbances and prefect stability of the closed loop system. The inner loop controller has impact over the outer one; therefore, the behavior of the inner control law must influence the essential process variables in a predictable way. Finally, the response of the inner loop is quicker than the outer one. Finally, this permits the secondary action adequate time to compensate for inner loop feedback and thereafter they can improve the outer loop performance [21]. The transfer function formulation of the proposed CPIL controller is as follows: The integral of time-multiplied square error (ITSE) is used as an objective function in this research work to design the proposed controller's parameters, and it's as follows [22]: where, is the simulation time period, the error function for the inner loop represents the and currents deviation and for the outer loop expressed by the speed deviation.

OPTIMIZATION ALGORITHMS 4.1. Standard grasshopper optimization algorithm
The SGO algorithm is a swarm intelligence algorithm that tries to mimic the swarm behaviour of the grasshopper insects in nature. In spite of the fact that grasshoppers are usually seen individually in nature, they join in one of the biggest swarms of all creatures. In their route, they eat practically all vegetation. After this reaction, when they become an adult, they establish a swarm in the air to preparing for migrate over large distances. It was confirmed by the research work based on SGOA that it is able of outperforming several well-known nature-inspired algorithms such as genetic algorithm (GA), particle swarm optimization (PSO), differential evolution (DE), ant colony optimization (ACO), firefly algorithm (FA), bat algorithm (BA) [17]. The main characteristics of the grasshoppers are foraging, target seeking and team behaviour in either larval or adulthood phases. In the larval stage, they reveal slow motion with small steps. In adulthood stage, they do long-range and abrupt movement in the way to find the farming areas. The swarm-intelligence algorithms divide the search routine into two directions: exploration and exploitation. In exploitation, the grasshoppers are tended to move locally, while they encouraged to move suddenly during exploration. Therefore, grasshoppers carry out these two properties, as well as target pursuing naturally [23].
GOA simulates the forces between the grasshoppers, which are the attraction and repulsion. The attraction forces exploit promising space while the repulsion forces enable the grasshoppers to explore the search area. The zone at which the two attraction and repulsion forces are equal is called comfort zone. The position of grasshoppers with the optimum objective function is considered as the closest one to the destination target; accordingly, the rests try to relocation toward that place through swarm interaction during the algorithm procedure; the comfort zone is changed until the finest solution is attained [24]. The position updating for the swarming behaviour of the grasshopper can be modelled mathematically as follows [25]: where, , is the current position in iteration for d dimension, c is the comfort zone coefficient, N is the swarm size, and are the upper and lower limits of the design variables, is the distance between , and , grasshoppers, ̂ is the target position in the d th dimension that represent the best solution determined so far, S(r) is a function, which calculates either repulsion or attraction forces, f represents the strength of the attraction, l s is the length of attraction and r is a distance that has been considered in the interval [0, 15].
The comfort zone coefficient is necessary to be decreased linearly w.r.t. the number of iterations to enhance the exploration and exploitation properties and is represented as follows: where, c max is the maximum limit, c min is the minimum limit, and T is the maximum number of iterations.

Dynamic adjustment grasshopper optimization algorithm
The SGOA has some drawbacks, such as the slow convergence to the optimal solution and the problem of falling in the local optimum [26]. In order to overcome these two deficiencies, a dynamic adjustment GOA scheme is proposed that combines the random initialization-based Gaussian distribution and the nonlinear decreasing comfort zone coefficient to obtain an appropriate exploitation and exploration balancing. Firstly, the enhancement for GOA is based on random candidate solution to increase population diversity, which can be initiated according to the following equation: where, ( ) is the Gaussian density function that utilized to take smaller step vector inside the search space bounded by lower and upper limits to precisely explore and achieve a faster convergence. The Gaussian density function is as follows [26]: where, 2 is the variance for each individual of the population and is the Gaussian random number between [0, 1].
The second improvement is based on the parameter that responsible for moderating the grasshoppers' movement towards the solution, and it are used twice in the position equation. The outer one acted to balances the exploration and exploitation of the search agents around the target. The inner coefficient minimizes the repulsion or attraction forces between grasshoppers relative to the number of iterations [27]. Therefore, in order to improve the comfort zone, attraction zone and repulsion zone between grasshoppers the following adjustment is proposed that leads to exploit the global optima in the search space. where, = 1.4, = 1.12 and = 0.3. The search procedure of the proposed DAGOA is outlined as follows: Step 1: Initialize the input parameters of the algorithm; swarm size, no. of iterations, no. of tuned parameters (dimension size), maximum and minimum bound of the tuned parameters, and .
Step 2: Assign the position value for the first iteration for all the grasshoppers in the search space.
Step 3: Determine the system design parameters.
Step 4: Calculate the objective function value for each individual in the swarm.
Step 5: Sort the objective function values in the descending order and thereafter nominate the minimum value as ̂.
Step 6: Update the improved nonlinear decreasing comfort zone coefficient.
Step 7: Normalize the distance between grasshoppers to find out .
Step 8: Determine the social forces strength.
Step 10: Examine the termination condition; if current iteration number is equal to the maximum no. of iterations, collect the optimal solution for each dimension ̂ and save the global best design parameters. Otherwise, increase the iteration counter and go to Step 3.

Numerical results for performance evaluation of algorithms
The purpose of this subsection is to assess the performance of the proposed optimization algorithm quantitatively. The ordinary procedure used is to utilize chosen benchmark test functions with known optimal values to perform a capabilities' comparison between the SGOA and DHGOA algorithms on the basis of solution accuracy.
The numerical results contain three sets of various test functions; Unimodal set (F1-F5), Multimodal set (F6-F8) and Composite set (F9 & F10) [17,[28][29][30]. Each test was repeated 30 times in order to obtain reliable statistical results for 10 mathematical benchmark functions with the average best (AB), the median best (MB) and the standard deviation (SD) that are tabulated as in Table 2.
The obtained results of the selected three sets of test functions show that the proposed DAGOA scheme provides competitive results throughout touching the global optimum solution for all the 10 test problems with an average executing time of about 5 seconds. Therefore, the modifications performed on the SGOA enhance the abilities of the proposed algorithms in terms of exploitation and exploration balancing and maximizing the search capability and stability.

Simulation results of TIM
The system current controller using CPIL is implemented in Matlab/Simulink as illustrated in Figure 12. Where, the CPIL is connected to the system as current controller. SGOA and DAGOA algorithms are used for tuning CPIL controller parameters. From Figure 13, speed controller of TIM using CPIL is implemented in Matlab/Simulink, which is tuned by SGOA and DAGOA algorithms.
From Figure 14, it can be seen that the stator quadrature current arrived steady state at 1.2× 10 −7 s when CPIL-DAGOA is used for tuning the controller parameters. In addition, with CPIL-SGOA is faster than using conventional PI controller. From Figure 15, it can be seen that current arrived steady state at 0.1× 10 −7 s when CPIL-DAGOA is used for tuning controller. Furthermore, with CPIL-DAGOA is faster than using CPIL-SGOA and using conventional PI controller. Therefore, when current settling time reach very fast value that is mean, the system speed response is faster because the output of the speed controller is considered * reference to the system.  Figures 16 and 17, it can be seen that starting current is about 200 A which is with in limit and with the system requirements. In addition, from Figure 16, speed of the system with CPIL-SGOA for tuning controller parameters arrived to steady state at 0.53 s while speed of the system with CPIL-DAGOA for tuning controller parameters reached to settle at 0.28 s. Hence, that is mean system with proposed controller using DAGOA is faster than the system with CPIL-SGOA by two times.

Performance comparison of controlling techniques
In this section, three-phase induction motor current controller using CPIL-DAGOA is compared with current controller using CPIL-SGOA and conventional PI current controller for tuning parameters. In addition, speed controller of the system with CPIL-DAGOA is compared with system speed using CPIL-SGOA for tuning parameters and system speed using PI controller. It can be seen that the system with CPIL-DAGOA scheme for tuning parameters has higher performance and faster than the system using the CPIL-SGOA and system with PI using SGOA as depicted in Figure 18. Figure 18 shows that current of the TIM with CPIL using DAGOA reached settling time of 0.1× 10 −7 s, while current of the system with CPIL-SGOA reached settling time of 1.4× 10 −7 s and system with PI controller arrived steady state at 0.006 s. Finally, that is mean; system with CPIL using DAGOA is faster than system with SGOA and Conventional PI controller as illustrated in Table 3. Figure 18. Stator quadrature current of the system using the three controlling methods Table 3. Performance comparison between the three types of controllers

Controlling techniques
Settling time ITSE Stator quadrature current of the system using PI controller 0.006 s 0.3796 Speed of the system using PI controller 0.85 s 3373 Stator quadrature current of the system using CPIL controller tuned by SGOA 1.4× 10 −7 s 6.52× 10 −6 Speed of the system using CPIL controller tuned by SGOA 0.55 s 1500 Stator quadrature current of the system using CPIL controller tuned by DAGOA 0.1× 10 −7 s 4.42× 10 −9 Speed of the system using CPIL controller tuned by DAGOA 0.28 s 290 In Figure 19, the speed response of the three-phase induction motor with CPIL-DAGOA scheme reached settling time at 0.3 s, while, system speed with CPIL using SGOA reached settling time at 0.55 s and system with PI controller arrived steady state at 0.85 s. Therefore, that is mean; the system with CPIL using DAGOA is faster than system with CPIL-SGOA and conventional PI controller as demonstrated in Table 3. In Figure 20, it can be seen that the speed of the TIM with CPIL using DAGOA method reached settling time faster than system with CPIL using SGOA and Conventional PI controller when load torque is applied at 1 s. Additionally, the disturbance due to load torque has less impact on system with CPIL using DAGOA for tuning parameters. While, the disturbance from load torque has high impact on system speed with PI controller, where speed dropped to below 1000 rpm and system has higher under shoot due to load torque when applied at 1 s. Figure 20. Motor speed of the system using three controlling methods when applied a load torque at 1 s

CONCLUSION
The mathematical modelling of the three-phase induction motor has been implemented in Matlab/ Simulink in α and β coordinates. Design of conventional PI controller for the current and speed of the system has been performed. Proposed CPIL controller for the current and speed have been connected with the system to improve the performance. The SGOA algorithm has been used for tuning the proposed controller parameters of current and speed of the system. Additionally, the SGOA has been modified and a DAGOA for tuning CPIL controller parameters for currents and speed of TIM is proposed. Furthermore, the effect of applying load torque on TIM after 1 s has been carried out to examine the proposed controller and optimization algorithm robustness. The computational efficiency and convergence rate of the developed DAGOA algorithm has been validated by ten benchmark functions. The simulation results indicate and confirm the effectiveness of the proposed CPIL tuned by DAGOA for the current and speed of the system. Moreover, the current and speed controlling of TIM with CPIL using DAGOA are faster than current and speed of TIM with proposed CPIL using SGOA and PI controller. Furthermore, load torque has less impact on speed of the system with proposed CPIL-DAGOA when load torque is applied at 1 s, whereas load torque has a high impact on system speed with conventional PI controller, where the system speed dropped to below 1000 rpm when load torque is applied. Finally, it is concluded that the proposed compensator and optimization technique is robust and satisfying the design requirements.