An enhanced whale optimized fractional order pi λ controller based power quality improvement for sensorless brushless dc motor under dynamic operating conditions

Abstract

This paper presents an Enhanced Whale Optimization Algorithm (EWO) approach for tuning to perfection of Fractional Order Proportional Integral and integral order Controller (FOPI^λ) is used to sensorless speed control of permanent magnet Brushless DC (PMBLDC) motor under the operating dynamic condition such as (i) speed change by set speed command signal (ii) varying load conditions, (iii) integrated conditions and (iv) controller parameters uncertainty. On the other hand, it deals with a reduced THD (Total Harmonic Distortion) under dynamic operating conditions to improve the power quality for the above control system. Here present are three optimization techniques, namely (i) Enhanced Whale Optimization (EWO), (ii) Invasive Weed Optimization (IWO), and (iii) Social Spider Optimization (SSO) for fine-tuning of the FOPI^λ controller parameters with reduction of THD. The proposed optimization algorithm optimized FOPI^λ controller are compared under various BLDC motor operating conditions. Based on the results of MATLAB/Simulink models, the proposed algorithms are evaluated. Here, both the simulation and the results of the experiments are validated for the proposed controller technique. It demonstrates that the effectiveness of the proposed controllers is completely validated by comparing the three intelligent optimization techniques mentioned above. The EWO optimized FOPI^λ controller for speed control of sensorless PMBLDC motor clearly outperforms the other two intelligent controllers by minimizing the time domain parameters, THD, performance Indices error, convergence time, control efforts, cost function, mean and standard deviation.

Keywords

BLDC motor drives whale optimization algorithm sensorless speed control techniques

1 Introduction

Brushless direct current (BLDC) motors have become popular in recent years for an industrial purpose because the nominal characteristics are lightweight, high torque, compact structure, high performance, controllability, low maintenance, and better efficiency [1]. The BLDC motors are permanent magnet motors with trapezoidal/rectangular EMF waves, and position sensors are essential for proper operation. The broadcast of the BLDC is activated electronically using an inverter and a counter-counter position sensor i.e. hall sensors. The rotor position information is still very important and can usually be measured with a Hall sensor. To reliably measure the rotational speed of a BLDC motor using a signal transmitted from the built-in hall sensor, the rotor pole of the BLDC motor should be aligned with the Hall Sensor [2].

In sensorless BLDC motor, the information of rotor position is highly needed which is used for the current communication towards the stator windings [3]. Many researchers have been focused on sensorless communication for BLDC motor drives which optimally gives better performance with proper control strategies [4].

Furthermore, the FOPI^λ controller is a development of the FOPID (Integer Order Proportional Integral Derivative) controller [5]. Proportional gain (K_p), Integral gain (K_i), and Integral order (λ) are the three functional DOF parameters of a standard FOPI^λ controller [6]. The FOPI^λ controller provides greater flexibility and better transient and steady-state characteristics. The controller control parameters to be fine-tuned in order to obtain optimal results using optimization techniques [7]. However, Conventional optimization techniques produce more computation time, tot solving non-linearity problems, longer time for settling, steady-state error fluctuation, and poor robust speed control to compute optimal results which limits system performance.

In literature, [8] presented the work related to FOPI^λ controller speed control of induction motors can be fine-tuned offline. From the offline tuning results, steady-state error and settling time are maximum values due to sudden external disturbance of the system. MGA, ABC, and BA are used to tune the FOPID controller to perfection [9, 10]. MATLAB simulations based on sensorless BLDC motor drive speed step response features, 50%load, and 100%load conditions achieve MGA, ABC, and specific BA strong scheme. This analysis does not focus on changing set speed conditions and integrated conditions.

The PSO optimized PI controller for BLDC motor speed regulation is created [11]. The controller adjusts the gain parameters and the PSO algorithm’s performance optimized PI controller parameters to measure time-domain parameters. The proposed [12] PSO–GSA optimised fuzzy PI controller design for BLDC motor. The proposed algorithm PSO–GSA is used to tune controller parameters can obtain improved time-domain performance. From the results, it is noted that poor performance and more settling time. In this study [13], the WOA optimized FO fuzzy PI controller for BLDC motor speed control. From the results, the performance of the control system has been investigated at various speeds to assess its robustness.

For BLDC motors, a firefly algorithm based FOPID controller is employed to provide desired torque and speed control [14]. The reference torque is compared to the estimated torque, and the error is sent to the FOPID controller in a direct instantaneous torque control system. It controls the motor torque with reduced ripple over a wide range of speed.

Optimal coronavirus Optimization algorithm (CVOA) is used to choose the best PID controller for a BLDC motor [15]. To alter the time domain parameters, a multi objective function can be employed. The coronavirus Optimization algorithm is a well-known metaheuristic optimization techniques used to improve the dynamic performance (minimum rise time, minimum settling time and overshoot) of the BLDC motor [16].

This paper examines optimization strategies for BLDC motor speed control based on optimal tuning of the FOPI^λ controller in terms of time-domain characteristics minimization. The proposed intelligent strategies improved FOPI controller is simulated in MATLAB-R2019a, M-file on a PC Intel Core i5-4200U CPU, 2.30 GHz speed system running Windows 10®.

This paper’s major contributions include:

Using Enhanced Whale Optimization (EWO), Invasive Weed Optimization (IWO), and Social Spider Optimization, create a MATLAB/Simulink model of a FOPI controller for sensorless speed control of a BLDC motor (SSO).

Various operating conditions for BLDC motors to be considered under (i) Changes in speed based on step response (ii) change the loading conditions, (iii) integrate the conditions, and (iv) keep the load constant.

The simulation results are compared to the proposed intelligent technique EWO, IWO and SSO optimized FOPI^λ controllers for Sensorless BLDC motor speed control. The optimum controller for the above-mentioned system is validated and recommended based on the results.

The FOPI^λ controller is used to generate the pulse width modulated signal used in the inverter and investigated the THD. In addition, to improve the power quality by reduction of Total Harmonics distortion for above dynamic operating conditions.

Under the above operating conditions, we measured the control efforts, cost function, mean deviation, standard deviation, convergence time, performance Indices (ITAE, ISE, IAE, and RMSE), and time-domain parameters (t_r, t_p, t_s, e_ss) of an EWO optimised FOPI controller for BLDC motor.

In order to demonstrate the utility of the suggested controller design, an experimental model is created, validated for specific operating conditions, and the results compared to simulation results.

The article’s content organization.is shown in Fig. 1.

Fig. 1

Content organization of the article.

2 Formulation of objective functions and system description

This section emphasizes the necessity of formulation of the objective function and system model of proposed FO-PI^λ controller in the present scenario of speed control systems. A briefing of sensorless speed control relevant to the proposed work is presented.

2.1 Formulation of objective functions

The cost function is a design constraint assigned to determine optimum parameters for controllers [17]. A typical mathematical cost function optimization formulation can be expressed as:

(i) Minimization of the cost function (f_cost)

A cost function (f _cost ) is utilised to find the best tuning settings for the FOPI^λ controller under variable load and set speed (Ns) conditions of the BLDC motor.

The set speed is N(t)_set, the estimated speed is N(t)_est. of the BLDC motor for each sample simulation time (T) for the optimization of the FOPI^λ controller $f_{cont 1} = \sqrt{\sum_{i = 0}^{T} \frac{{(N (t)_{{set}_{i}} - N (t)_{{est}_{i}})}^{2}}{T}}$ (1)

(ii) Performance indices are minimised. $f_{cost 2} = IAE = \int | e (s) | dt = \int_{0}^{T} | N (t)_{{set}_{i}} - N (t)_{{est}_{i}} | dt$ (2) $f_{cost 3} = ITAE = \int t | e (s) | dt = \int_{0}^{T} t * (| N (t)_{{set}_{i}} - N (t)_{{set}_{i}} |) dt$ (3) $f_{cost 4} = ISE = \int e^{2} dt = \int_{0}^{T} (| N (t)_{{set}_{i}} - N (t)_{{set}_{i}} |) dt$ (4) $f_{cost 5} = RMSE = \frac{\int e^{2}}{T} dt = \int_{0}^{T} \frac{{(| N (t)_{{set}_{i}} - N (t)_{{set}_{i}} |)}^{2}}{T}$ (5)

(iii) The scope of the FOPI^λ controller’s optimal tuning parameter is given in Equations (6) to (8) as,, $K_{p - \min} \leq K_{p} \leq K_{p - \max}; K_{p - \min} = 0 and K_{p - \max} = 3$ (6) $K_{i - \min} \leq K_{i} \leq K_{i - \max}; K_{i - \min} = 0 and K_{i - \max} = 2$ (7) $λ_{\min} \leq λ \leq λ_{\max}; λ_{\min} = 0.02 and λ_{\max} = 0.98$ (8)

2.2 A system model of proposed FOPI^λ controller

A FOPI^λ controller creates an output signal which is a combination of three controller parameters (K_p, K_i,λ) gain values. It produces a zero-control error and is insignificant for measurement channel interference [18]. To configure the PI^λ controller, the integration time must be set to zero, and the maximum aspect ratio must be specified. The FOPI^λ allows for greater flexibility in the design of the BLDC motor’s speed controller. The model of fractional order PI^λ controller is shown in Fig. 2.

Fig. 2

System model of Fractional Order PI^λ controller.

$F_{2} (s) = D_{2} (s) Q (s)$ (9)

The response can be acquired from fractional-order PI^λ controller transfer function $D_{2 (i ω)} = Z_{q} (1 + Z_{j} (i ω)^{- λ})$ (10) $= Z_{q} (1 + Z_{j} ω^{- λ} Cos (λ \frac{π}{2}) - {iZ}_{j} ω^{- λ} Sin (λ \frac{π}{2}))$ (11)

The phase and gain are as follows, $arg [D_{2} (i ω)] = - Arc tan \frac{Z_{j} ω^{- 1} Sin (\frac{λ π}{2})}{1 + Z_{j} ω^{- λ} Cos (\frac{λ π}{2})}$ (12) $| D_{2} (i ω) | = Z_{Q} \sqrt{(1 + Z_{j} ω^{- λ} Cos (λ \frac{π}{2}))^{2} + (Z_{j} ω^{- λ} Sin (λ \frac{π}{2}))^{2}}$ (13)

The open-loop response F₂ (iω) is $F_{2} (i ω) = D_{2} (i ω) Q (i ω)$ (14) $= - Arc tan \frac{Z_{j} ω^{- 1} Sin (\frac{λ π}{2})}{1 + Z_{j} ω^{- λ} Cos (\frac{λ π}{2})} - Arc tan (ω t) - l ω$ (15)

The phase of F₂ (iω) is $arg [F_{2} (i ω_{d})] = - Arc tan \frac{Z_{j} ω_{d}^{- λ} Sin (\frac{λ π}{2})}{1 + Z_{j} ω_{d}^{- λ} Cos (\frac{λ π}{2})} - Arc tan (ω_{j} t) - l ω_{d}$ (16) $arg [F_{2} (i ω_{d})] = π + φ_{n}$ (17)

The following equation represents the relationship between Z_j and λ $Z_{j} = \frac{- tan [Arc tan (ω_{d} t) + φ_{n} + l ω_{d}}{N}$ (18)

where $N = ω_{d}^{λ} Sin (\frac{λ π}{2}) + ω_{d}^{λ} Cos (\frac{λ π}{2}) tan [Arc tan (ω_{d} t) + φ_{n} + l ω_{d}]$ (19)

The gain is $= {[Arc tan \frac{Z_{j} ω^{λ} Sin (\frac{λ π}{2})}{1 + Z_{j} ω^{λ} Cos (\frac{λ π}{2})} Arc tan (ω t) l ω]}_{ω = ω_{d}}^{'}$ (20)

The equation shows about Z_j and λ $B_{2} Z_{j}^{2} + [2 ω_{d}^{λ} Cos (λ \frac{π}{2}) B_{2} - λ ω_{d}^{λ - 1} Sin (λ \frac{π}{2})] Z_{j} + B_{2} ω_{d}^{2 λ} = 0$ (21) $B_{2} ω_{d}^{2 λ} Z_{j}^{2} + A_{2} Z_{j} + B_{2} = 0$ (22)

where $B_{2} = \frac{t}{1 + (ω_{d} t)^{2}} + l; A_{2} = 2 B ω_{d}^{λ} Cos (λ \frac{π}{2}) λ ω_{d}^{λ 1} Sin (λ \frac{π}{2})$

Then $Z_{j} = \frac{A_{2} \pm \sqrt{A_{2}^{2} 4 B_{2}^{2} ω_{d}^{2 λ}}}{2 B_{2} ω_{d}^{2 λ}}$ (23)

Z_Q can be established by $| F_{2} (i ω) | = | D_{2} (i ω_{d}) Q (i ω_{d})$ (24)

Then $Z_{Q} = \sqrt{\frac{1 + (ω_{d} t)^{2}}{1 + {[1 + Z_{j} ω_{d}^{λ} Cos (\frac{λ π}{2})]}^{2} + {[Z_{j} ω_{d}^{λ} Sin (\frac{λ π}{2})]}^{2}}}$ (25)

3 Generation of pulse width modulated signal using optimized FOPI^λ controller

The FO-PI^λ is an exceptional instance of a FOPI^λ controller that is less sensitive to controller parameter variables. The fractional order PI^λ controller offers more flexibility to configure the controller using two additional configuration options. $μ D_{T}^{γ} = {\begin{matrix} \frac{d^{γ}}{dT} Re (γ) > 0 \\ 1 Re (γ) = 0 \\ \int {dT t (d τ)}^{- γ} Re (γ) < 0 \end{matrix}$ (26) $μ D_{t}^{γ} F (t) = lim_{p \to 0} p^{- γ} \sum_{b = 0}^{[(t - μ / b)]} U_{s}^{b} F (t - sb)$ (27)

Where $U_{s}^{γ} = \frac{{(- 1)}^{b} Γ (γ + 1)}{b! Γ (γ - b + 1)}$ and Γ is gamma function. $\int_{0}^{\infty} e^{- jt} 0_{} D^{γ} F (t) dt = j^{γ} f (j) - \sum_{v = 0}^{x - 1} j^{v}_{0} D_{t}^{γ - v - 1} F (t) |_{t = 0}$ (28)

Where x is the integer such that x - 1 < γ ≤ x.

If initial value $_{0} D_{t}^{γ - v - 1} F (t) |_{t = 0} = 0$ , then Laplace transform of $_{0} D_{t}^{γ} F (t)$ can be represented as (29). $G {_{0} D_{t}^{γ} F (t)} - j^{γ} f (j)$ (29)

At the z^th sampling time, calculate error E (z): $E (z) = i - p (z)$ (30)

Where i the set is value and p (z) is the system output. Calculate PI^λ controller output: $k (u) = U_{z} E (u) + U_{R} b^{γ} \sum_{s = 0}^{v} u_{s}^{- γ} E (u - s) + U_{D} b^{- γ} \sum_{s = 0}^{v} u_{s}^{α} E (u - s)$ (31) where k (u) is the controller output, U_Z, U_R, U_D are proportional gain, integral gain and differential gain, b is sampling period, and E (u - s) is error from previous sampling time. The coefficient u can be calculated more simply by the following recurrence. $u_{0}^{- γ} = 1$ (32) $u_{s}^{- γ} = [1 - \frac{(- γ) + 1}{s}] u_{s - 1}^{- γ}$ (33) $u_{0}^{α} = 1$ (34) $u_{s}^{α} = [1 + \frac{α + 1}{s}] u_{s - 1}^{α}$ (35)

Generation of pulse width modulated signal using optimized FOPI^λ controller

Input: Limits of operators, fractional order
Output: Fractional generation
Step 1 Define the basic operator of PID $μ D_{T}^{γ}$
Step 2 Evaluate the continuously differentiable function F (T)
Step 3 Define the Laplace transform of the fractional-order using
Equation (28)
Step 4 Analyze the transfer function of FOPI^λ controller in Equation (29)
Step 5 Calculate error E (z) at sampling time
Step 6 Calculate PI^γD^α controller output in Equation (31)
Step 7 Obtain the value of fractional calculus using Equation (32)
Step 8 BLDC motor speed can be controlled using Equation (38)

The open loop transfer function of BLDC motor is given by $L_{c} = \frac{15}{1.08 j^{2} + 6.1 j + 1.63}$ (36)

Therefore, FO-PI controller has been used, which is given by $L_{P} = V_{c} + \frac{V_{r}}{j^{γ}} + V_{D} j^{\infty}$ (37)

The objective function is given by $Of = \int_{0}^{\infty} t | E (t) dt |$ (38)

By using this, reduce the objective function to control the BLDC motor’s speed to the set point state. This will help us to manage the speed of the BLDC motors. The BLDC motors’ speed can be controlled by the PID controller and thus it is efficient for the generation of width pulse modulated signal used in the inverter and the control signal used in the capacitor. Also, reduce the torque and speed control using the motor. The working function of the proposed PWM signal generation is given in Algorithm 1.

Algorithm 1- Drive control using SSO

Input: weight of spiders, the distance between the spiders
Output: a fittest spider is evaluated
Step 1 Define the vibration of the spider Rxui_x using Equation (44)
Step 2 Update the female position using Equation (45)
Step 3 Evaluate the female’s moment I_x (T + 1)
Step 4 Movement of ND males is represented using Equation (46)
Step 5 Movement of D males is represented using Eq. (47)
Step 6 Use mating operator for modifying the search agents
Step 7 Calculate the fitness of each spider
Step 8 Evaluate the fittest spider

3.1 Fractional Order PI controller (FOPI^λ):

The FOPI^λ controller is a closed loop feedback signal system which is used for process control and motor speed control systems [19]. It has three parameters degrees such as proportional gain (K_P), Integral gain(K_I) & Integral order (λ). Figure 3 depicts the FOPI^λ controller’s basic construction. The error detector e(t) is compared with the set point and measured variable Y(s). The purpose of FOPI controller has been optimal tuned with the help of EWO, IWO and SSO optimization algorithms in order to minimize the cost function, control effort, steady state error, settling time and performance indices.

Fig. 3

FOPI^λ controller’s basic construction.

The transfer function of FOPI is $G_{c} (s) = \frac{U (S)}{E (S)} = K_{p} + K_{I} S^{- λ}$ (39)

The equation for the FOPI controller in the time domain is $U (t) = K_{p} e (t) + K_{I} D^{- λ} e (t)$ (40)

The closed loop control system’s unit step response using a FOPI controller is given by $\sum_{k = 0}^{n} a_{k} D^{β k} y (t) + K_{p} y (t) + K_{I} D^{- λ} y (t)$ (41) $= K_{p} ω (t) + K_{I} D^{- λ} ω (t)$ (42)

From Equation (43), the transfer function of a closed loop system is expressed as, $G_{closed} (s) = \frac{K_{p} S^{λ} + K_{I}}{\sum_{K = 0}^{n} a_{K} S^{β} K + K_{p} S^{λ} + K_{I}}$ (43)

Laplace inversion of Equation (43) yields the unit step response G_closed (t).

3.2 Sensorless speed control

The permanent magnet BLDC motors are controlled electronically, it requires the rotor position information to achieve suitable commutation of current in the stator winding. For rotor position information, hall sensors are being used [20]. But the use of hall sensors is not desirable because it increases the cost, complexity in structure and also the sensor fails at high-temperature applications. Sensorless BLDC motor control is also known as sensorless trapezoidal BLDC motor control. Back EMF refers to the voltage that is proportional to the BLDC motor speed. Back EMF can be used to calculate flux, torque, and the rotor speed and position of a motor. The sensorless BLDC motor speed control is shown in Fig. 4.

Fig. 4

Sensorless BLDC motor speed control.

3.3 Proposed meta-heuristic optimization algorithms

3.3.1 Social spider optimization (SSO)

Cuevas et al. proposed the social spider optimization (SSO) algorithm [21]. This is a swarm-based technique that takes into account the social intelligence of the colony’s spiders. [22]. The main motivation for this method is the social interaction of spiders using web vibrations. In the same way, search optimization problems are treated as a spider web and a search optimal colony [23]. The Equation (44) shows the mathematic equation of vibration of spider ${Rxui}_{x} = a_{y} m^{l_{x . y}^{2}}$ (44)

It is the status of the search agents that determines the operator for gender. Female updates the position as follows: $\begin{matrix} I_{x} (T + 1) = I_{x} (T) \\ + (φ . {Rxuk}_{x} . (N_{k} - I_{x} (T)) + φ . {Rxuu}_{x} . (N_{u} - I_{x} (T)) + γ . (v - 0.5)) \end{matrix}$ (45)

For ND-type spiders:

$\begin{matrix} I_{x} (T + 1) = I_{x} (T) \\ + (φ . {Rxuz}_{x} . (N_{z} - I_{x} (T)) + γ . (v - 0.5)) \end{matrix}$ (46)

For D-type spiders:

$I_{x} (T + 1) = I_{x} (T) + (ϕ . (\frac{\sum_{y = 1}^{S_{e}} e_{y}^{c} . A S_{z} + y}{\sum_{y = 1}^{S_{e}} A S_{z}})) - I_{x} (T)$ (47)

Except for female movements, these Equations (46) reveal that ND-type spiders can only move females and cannot go backward. The mating distance can be more than one male and one female. Therefore, the roulette wheel algorithm selects parents according to their fitness values. The new spider has a combination of male genes (variables). If a new spider is more suitable than another spider, the unsuitable spider will be removed and added to the population.

3.3.2 Invasive Weed Optimization (IWO)

Invasive weed optimization (IWO) is a one-of-a-kind numerical stochastic optimization algorithm that was inspired by current weed invasion [24]. Weeds are a major threat to healthy plants and actively growing fruit crops. It is highly resistant and adaptable to environmental changes [25]. This algorithm reflects the resistance, adaptability, and incompatibility of the model community. This method is extremely simple, but it is useful for combining relevant solutions with key features such as weed colonization, growth, and competition. [26]. $α_{i} = {(\frac{I - i}{I})}^{x} (α_{INITIAL} - α_{FINAL}) + α_{FINAL}$ (48)

The following is the definition of this map: $\begin{matrix} n_{b + 1} = {in}_{b} (1 - n_{b}) \\ for 0 < i \leq 4, b = 0, 1, 2, \dots n_{b} \in [0, 1] \end{matrix}$ (49)

$\begin{matrix} α (i) = {(\frac{I - i}{I})}^{x} (α_{INITAL} - α_{FINAL}) \\ + α_{FINAL} + α_{FINAL} \times w (i) \end{matrix}$ (50)

Algorithm 2 - Drive control system using invasive weed optimization

Input: Positions of the weed
Output: Best position of the weed
Step 1 Define the number of grains of grasses
Step 2 Evaluate the seeds produced in the normal distribution
Step 3 Define the map using Eq. (49)
Step 4 Update standard deviation formula using chaotic mapping in Eq. (50)
Step 5 Define the best position using Eq. (51)

$n_{1}^{i + 1} = α (i) \times n_{1}^{i} + (N_{BEST} - N_{t}^{i})$ (51)

In which i^th iteration is the new weed position $n_{1}^{i + 1}$ is the best weed found in the entire population. With variations between previous and best position levels, as well as constant variables, the new position in (51) is updated. Subsequently, the algorithm evaluates these new conditions as well. Because the standard deviation balances local and global searches, this equation enhances the local and global search algorithm [27].

3.3.3 Enhanced Whale Optimization (EWO)

For sensorless speed control of BLDC motors, the Enhanced Whale Optimization [28] is used to optimise the FOPI controller parameter to achieve design constraints (i) settling time≤1 sec (ii) steady-state error (e_ss)≤1 1%. The conventional Whale Optimization Algorithm (WOA) is affected search space, convergence problems and updated its position issues. To overcome the above problems of the WOA, the enhanced method Lévy flight strategy [29] is used in WOA with the enhanced mutation operator based on ranking are presented into the WOA. The improved whale optimization method can identify the global best solution by balancing exploration and exploitation. $Q = | P . A^{*} (i) - A (i) |$ (52) $X = 2 x . t - x,$ (53) $P = 2 t$ (54) $A (i + 1) = {\begin{matrix} A^{*} (i) - X . Q f < 0.5 \\ Q^{'} . E^{nz} . cos (2 π z) + A^{*} (i) f \geq 0.5 \end{matrix}$ (55) $Q = | P . A_{tan d} (r) - A (r),$ (56) $A (i + 1) = A_{tan d} (r) - X . Q$ (57)

Algorithm 3- Drive control system using advanced whale optimization

Input: Current position of the whale
Output: Updated location of the whale
Step 1 Evaluate the model of encircling prey using Equations (52) to (54)
Step 2 Calculate the auxiliary coefficients X and P
Step 3 Define the spiral evaluating a position A (i + 1)
Step 4 The location of whales has been updated by using Equation (55)
Step 5 Define the behavior of the whale Q
Step 6 Update the location of the whale according to the value of c
Step 7 Obtain the highest score and largest weight of the individual using Equation (57)

4 Proposed EWO optimized sensorless BLDC motor speed control with the FOPI^λ controller

This sector focused proposed EWO optimized Sensorless BLDC motor speed control with the FOPI^λ controller. An inverter is a static equipment that produces AC voltage from DC-DC boost converter voltage by triggering the power devices. The switching timing is determined by the switching logic. The voltage and current values are measured in the sensorless technique, and the rotor speed is approximated by using the Kalman filter algorithm. The Kalman Filter (KF) method is used to estimate the motor speed [30]. The following factors must be considered when estimating the motor speed are (i) A linear machine’s dynamic state model (ii) Model selection for the machine and (iii) The system gradient and measurement matrices are calculated. The voltage and current on the stator winding line are used to do the calculation. The linear system’s state-space model in the digital domain. Figure 5 shows the WOA- FOPI^λ controller functional blocks for BLDC motor.

Fig. 5

WOA- FOPI^λ controller functional blocks for BLDC motor.

The speed estimation is achieved with the help of voltage and current. The error detector receives the estimated speed in rpm (N_e) and manually sets the reference speed i.e. set speed to 1500 rpm (N_s). The error detector generates a difference value that is send to the FOPI^λ controller. The controller produces the required tuning parameters K_p, K_i, λ with the help of proposed intelligent techniques namely WOA. The switching logic circuits receive the control signal u(t) from the FOPI^λ controller. The switching logic circuits generate the switching sequence to energize the stator winding. The PWM signal is generated by a pulse generator for controlling the voltage with the aid of a varying firing angle. The motor speed is controlled by varying the voltage, which activates the semiconductor switches in the PWM inverter.

4.1 Flowchart of proposed EWO- FOPI λ controller

The steps for obtaining the FOPI^λ controller parameters (Kp, Ki, and λ) can be depicted in a flow chart. Recognize the suggested EWO-FOPI controller for BLDC motor’s design constraint as Equations (58 & 59) and use the proposed EWO algorithm to acquire the initial FOPI^λ controller parameters Kp, Ki, Kd, λ, Figure 6 depicts the flowchart of an upgraded genetic algorithm-based FO PI^λ controller for sensorless BLDC motor drive.

Fig. 6

Flowchart of proposed EWO-FO PI^λ controller.

The following design constraint is considered

Reduction of settling time (t_s) $Settling time \leq 1 s$ (58)

Reduction of steady-state error (e_ss)

A minimum steady-state error is required since the system’s ultimate value should approach the reference input. $\begin{matrix} Fractional Order {PI}^{λ} Conroller = \min | e_{ss} | \\ Steady - state error {(e}_{ss}) \leq 1 % \end{matrix}$ (59)

4.2 Optimal FOPI^λ controller parameters

The key parameters are the cost function’s minimal value and the control effort to decide the EWO optimized FOPI^λ controller to be better than SSO and IWO. The optimal FOPI^λ controller parameters, cost function, and control effort for sensorless BLDC motor are shown in Table 1.

Table 1
Controller parameters of varying load conditions

Varying load Conditions Controller Name FOPI^λ Controller Parameters Cost function (J) Control Effort

K_p K_i Λ

No load (T_L = 0 Nm) SSO - FOPI^λ 2.612 1.603 0.492 3.632 32.52

IWO - FOPI^λ 2.123 1.472 0.402 2.952 29.09

EWO - FOPI^λ 1.925 1.254 0.359 2.105 28.74

20 %load (T_L = 0.5 Nm) SSO - FOPI^λ 2.636 1.593 0.492 3.524 28.35

IWO - FOPI^λ 2.147 1.427 0.358 3.142 27.43

EWO - FOPI^λ 1.984 1.221 0.304 2.854 25.41

40 %load (T_L = 1 Nm) SSO - FOPI^λ 2.536 1.456 0.568 3.526 26.59

IWO - FOPI^λ 2.124 1.254 0.456 3.524 24.01

EWO - FOPI^λ 1.984 1.112 0.325 2.564 23.08

60 %load (T_L = 1.5 Nm) SSO - FOPI^λ 2.436 1.362 0.642 3.362 24.35

IWO - FOPI^λ 2.145 1.242 0.542 2.652 22.08

EWO - FOPI^λ 1.987 1.114 0.464 2.118 21.27

80 %load (T_L = 2 Nm) SSO - FOPI^λ 2.407 1.315 0.862 3.243 20.78

IWO - FOPI^λ 2.235 1.352 0.684 2.865 18.80

EWO - FOPI^λ 1.984 1.112 0.532 2.104 17.75

Full load (T_L = 2.5 Nm) SSO - FOPI^λ 2.317 1.305 0.999 3.020 18.52

IWO - FOPI^λ 2.182 1.241 0.845 2.451 16.99

EWO - FOPI^λ 1.954 1.110 0.624 2.012 15.74

Varying load Conditions	Controller Name	FOPI^λ Controller Parameters	Cost function (J)	Control Effort
No load (T_L = 0 Nm)	SSO - FOPI^λ	2.612	1.603	0.492	3.632	32.52
	IWO - FOPI^λ	2.123	1.472	0.402	2.952	29.09
	EWO - FOPI^λ	1.925	1.254	0.359	2.105	28.74
20 %load (T_L = 0.5 Nm)	SSO - FOPI^λ	2.636	1.593	0.492	3.524	28.35
	IWO - FOPI^λ	2.147	1.427	0.358	3.142	27.43
	EWO - FOPI^λ	1.984	1.221	0.304	2.854	25.41
40 %load (T_L = 1 Nm)	SSO - FOPI^λ	2.536	1.456	0.568	3.526	26.59
	IWO - FOPI^λ	2.124	1.254	0.456	3.524	24.01
	EWO - FOPI^λ	1.984	1.112	0.325	2.564	23.08
60 %load (T_L = 1.5 Nm)	SSO - FOPI^λ	2.436	1.362	0.642	3.362	24.35
	IWO - FOPI^λ	2.145	1.242	0.542	2.652	22.08
	EWO - FOPI^λ	1.987	1.114	0.464	2.118	21.27
80 %load (T_L = 2 Nm)	SSO - FOPI^λ	2.407	1.315	0.862	3.243	20.78
	IWO - FOPI^λ	2.235	1.352	0.684	2.865	18.80
	EWO - FOPI^λ	1.984	1.112	0.532	2.104	17.75
Full load (T_L = 2.5 Nm)	SSO - FOPI^λ	2.317	1.305	0.999	3.020	18.52
	IWO - FOPI^λ	2.182	1.241	0.845	2.451	16.99
	EWO - FOPI^λ	1.954	1.110	0.624	2.012	15.74

Table 2 shows the controller parameters, cost function, and control effort values for varying set speed conditions. To validate the competence of the SSO, IWO and EWO optimized the Sensorless BLDC motor speed control with the FOPI controller by various tests, the controller parameters value, Cost function (J), and control effort are measured. Table 3 shows the values for integrated conditions controller settings, cost function, and control effort. Here varying set speed condition are analyzed for different tests are described below:

Table 2

Variable set speed conditions with controller parameters

Changing set speed conditions	Controller Name	FOPI^λ Controller Parameters			Cost function (J)	Control Effort
		K_p	K_i	Λ
Test 1	SSO - FOPI^λ	1.959	1.202	0.369	2.724	26.07
	IWO - FOPI^λ	1.592	1.104	0.302	2.214	24.42
	EWO - FOPI^λ	1.444	0.941	0.269	1.578	20.03
Test 2	SSO - FOPI^λ	1.977	1.195	0.369	2.643	24.46
	IWO - FOPI^λ	1.610	1.070	0.269	2.356	21.54
	EWO - FOPI^λ	1.488	0.916	0.228	2.140	18.06
Test 3	SSO - FOPI^λ	1.902	1.092	0.426	2.644	23.90
	IWO - FOPI^λ	1.593	0.941	0.342	2.643	19.94
	EWO - FOPI^λ	1.488	0.834	0.244	1.923	16.25
Test 4	SSO - FOPI^λ	1.827	1.022	0.482	2.521	22.26
	IWO - FOPI^λ	1.609	0.932	0.407	1.989	19.56
	EWO - FOPI^λ	1.490	0.836	0.348	1.588	16.87
Test 5	SSO - FOPI^λ	1.805	0.986	0.647	2.432	20.64
	IWO - FOPI^λ	1.676	1.014	0.513	2.148	18.15
	EWO - FOPI^λ	1.488	0.834	0.399	1.578	15.52

Table 3

Controller parameters for integrated conditions

Integrated conditions	Controller Name	FOPI^λ Controller Parameters			Cost function (J)	Control Effort
		K_p	K_i	Λ
Test 1 + 60%	SSO - FOPI^λ	3.918	2.404	0.738	5.448	32.59
Load	IWO - FOPI^λ	3.184	2.208	0.603	4.428	30.53
	EWO - FOPI^λ	2.887	1.881	0.538	3.157	25.04
Test 2 + 60%	SSO - FOPI^λ	3.954	2.389	0.738	5.286	30.58
Load	IWO - FOPI^λ	3.220	2.140	0.537	4.713	26.93
	EWO - FOPI^λ	2.976	1.831	0.456	4.281	22.58
Test 3 + 60%	SSO - FOPI^λ	3.804	2.184	0.852	5.289	29.88
Load	IWO - FOPI^λ	3.186	1.881	0.684	5.286	24.93
	EWO - FOPI^λ	2.976	1.668	0.487	3.846	20.31
Test 4 + 60%	SSO - FOPI^λ	3.654	2.043	0.963	5.043	27.83
Load	IWO - FOPI^λ	3.217	1.863	0.813	3.978	24.45
	EWO - FOPI^λ	2.980	1.671	0.696	3.177	21.09
Test 5 + 60%	SSO - FOPI^λ	3.610	1.972	0.293	4.864	25.80
Load	IWO - FOPI^λ	3.352	2.028	0.026	4.297	22.69
	EWO - FOPI^λ	2.976	1.668	0.798	3.156	19.40

Test 1 - Change the speed - 500 rpm to 1500 rpm.

Test 2 - Change the speed - 1000 rpm to 1500 rpm.

Test 3 - Change the speed - 1500 rpm to 1000 rpm.

Test 4 - Change the speed - 1500 rpm to 500 rpm.

Test 5 - Change the speed - 1000 rpm to 500 rpm.

5 Discussions and results

The MATLAB simulation results are presented to demonstrate the performance of the EWO optimised FOPI^λ controller for sensorless BLDC motor speed response under various operating conditions.

5.1 Simulation setup and results

This section evaluates the proposed performance of the sensorless BLDC motor using SSO, IWO and EWO optimized FOPI^λ controller by simulation platform. The optimization techniques are utilised to fine-tune the FOPI^λ controller for BLDC motor speed control under various working conditions. The proposed optimization techniques based on FOPI^λ controllers are implemented in Windows 10® using a MATLAB 2020a M file. The suggested EWO-FOPI^λ controller design is compared to the IWO- FOPI^λ and SSO- FOPI^λ controller methods for performance. The parameters of optimization algorithms are shown in Table 4.

Table 4
Parameters of Optimization Algorithms

Social Spider Optimization Invasive Weed Optimization Enhanced Whale Optimization

Population size (Xi) 10 Population 10 Population size (Xi) 10

No of dimension 3 S. diffusion 2 Random vector 0; 1

Number of iteration 10 Number of iterations 10 Number of Iterations 10

Loudness (A) 90 Lower bounds [0.001 0.001 0.001] Random number (p) –1; + 1

Pulse rate (R) [1,10, 1,10] Upper bounds [20 20 20] Reference whale > 1 or < 1

Changing Frequency 1 –90 kHz Random walk 0.75 Probability percentage 50 %

Echolocation 8–10 ms – – – –

Wavelength 2–14 mm – – – –

Social Spider Optimization	Invasive Weed Optimization	Enhanced Whale Optimization
Population size (Xi)	10	Population	10	Population size (Xi)	10
No of dimension	3	S. diffusion	2	Random vector	0; 1
Number of iteration	10	Number of iterations	10	Number of Iterations	10
Loudness (A)	90	Lower bounds	[0.001 0.001 0.001]	Random number (p)	–1; + 1
Pulse rate (R)	[1,10, 1,10]	Upper bounds	[20 20 20]	Reference whale	> 1 or < 1
Changing Frequency	1 –90 kHz	Random walk	0.75	Probability percentage	50 %
Echolocation	8–10 ms	–	–	–	–
Wavelength	2–14 mm	–	–	–	–

The EWO controller, as compared to the other two intelligent controllers, reduces the magnitude of stator current. Here, magnitude of stator current is 13 amps at the time of starting of BLDC motor. The EWO optimized FOPI^λ controller can decrease the energized time of stator winding. It is observed that the stator back EMF is better time-domain characteristics by EWO optimized FOPI^λ controller. For a proposed controller, the stator current and back EMF of a BLDC motor is shown in Figs. 7 8.

Fig. 7

Stator current of BLDC motor for proposed controllers.

Fig. 8

Back EMF of BLDC motor for a proposed controller.

It has been demonstrated that different load situations, such as 10%load, 30%load, 50%load, 70%load, and 90%load condition. The EWO optimised FOPI^λ controller for sensorless BLDC motor speed control is more effective, according to the results. It depicts the settling time of the proposed EWO- FOPI^λ controller is very low in terms of 3%, 8%, 7%for SSO and IWO techniques respectively. The proposed EWO- FOPI^λ controller has a very low steady-state error (rpm) in terms of 25%, 3%, and 10%for SSO, and IWO techniques respectively. Figure 9 shows the optimization-based FOPI^λ controller for varying load conditions.

Fig. 9

BLDC motor speed response under conditions of varying load.

The proposed FOPI^λ controller’s rise time is very low compared to FOPI^λ controller in terms of 9%, 28%, 33%for SSO, IWO, and IWO techniques respectively. The peak time of the proposed FOPI^λ controller is very low compared to FOPI^λ controller in terms of 7%, 19%, and 16%for SSO, IWO, and EWO techniques respectively. It depicts the Settling time of the proposed FO-PI controller is very low compared to the FO-PI controller in terms of 9.3%, 12.5%, and 9.3%for SSO, IWO, and IWO respectively. In comparison to the FO-PI controller, the proposed FO-PI controller has a very low steady-state error in terms of the 7%, 13.3%, and 20%for SSO, IWO, and EWO respectively. The performance in time domain under varying load conditions is shown in Table 5.

Table 5

Performance in time domain under varying load conditions

AlgorithmTime Response	Rise time t_r (s)	Peak time t_p (s)	Settling time t_s (s)	Steady-state error e_ss (%)	Performance Indices
					IAE	ITAE	ISE
Load 0%(T_L = 0 Nm)
SSO - FOPI^λ	0.522	0.585	0.735	0.801	0.014	0.006	0.012
IWO - FOPI^λ	0.465	0.510	0.685	0.675	0.007	0.003	0.008
EWO - FOPI^λ	0.122	0.310	0.362	0.341	0.006	0.003	0.007
Load 20%(T_L = 0.5 Nm)
SSO - FOPI^λ	0.575	0.735	0.842	0.735	0.012	0.006	0.010
IWO - FOPI^λ	0.436	0.675	0.735	0.612	0.007	0.004	0.007
EWO - FOPI^λ	0.210	0.392	0.455	0.311	0.007	0.002	0.007
Load 40%(T_L = 1 Nm)
SSO - FOPI^λ	0.613	0.789	0.885	0.800	0.008	0.005	0.011
IWO - FOPI^λ	0.566	0.661	0.785	0.603	0.004	0.004	0.009
EWO - FOPI^λ	0.240	0.565	0.596	0.332	0.003	0.002	0.006
Load 60 %(T_L = 1.5 Nm)
SSO - FOPI^λ	0.622	0.740	0.902	0.726	0.012	0.006	0.007
IWO - FOPI^λ	0.590	0.659	0.812	0.525	0.008	0.004	0.006
EWO - FOPI^λ	0.389	0.570	0.698	0.472	0.0087	0.003	0.006
Load 80%(T_L = 2 Nm)
SSO - FOPI^λ	0.796	0.865	0.941	0.801	0.012	0.004	0.009
IWO - FOPI^λ	0.597	0.711	0.765	0.526	0.008	0.003	0.008
EWO - FOPI^λ	0.452	0.672	0.663	0.332	0.007	0.002	0.007
Load 100 %(T_L = 2.5 Nm)
SSO - FOPI^λ	0.809	0.888	0.961	0.795	0.008	0.003	0.010
IWO - FOPI^λ	0.741	0.830	0.910	0.461	0.007	0.002	0.006
EWO - FOPI^λ	0.676	0.760	0.864	0.322	0.006	0.001	0.005

The optimization techniques based on BLDC Motor speed response under Test 1 conditions are indicated in Fig. 10. In terms of peak time, the suggested FO-PI controller has a much shorter peak time than the FOPI^λ controller in terms of the 8%, 5%, and 4%for SSO, IWO, and EWO respectively. The proposed FOPI^λ controller has a very low steady-state error (rpm) when compared to the FOPI^λ controller in terms of the 22.5%, 20%, and 17.5%for SSO, IWO, and EWO respectively. The suggested FOPI^λ controller has a much lower steady-state error (rpm) than the FOPI^λ controller in terms of the 8%, 5%, and 3%for SSO, IWO, and EWO respectively. Performance in time domain under varying set speed conditions is shown in Table 6.

Fig. 10

Optimization techniques based BLDC Motor speed response under Test 1 conditions.

Table 6

Performance in time domain under varying set speed conditions

AlgorithmTime Response	Rise time t_r (s)	Peak time t_p (s)	Settling time t_s (s)	Steady-state error e_ss (%)	Performance Indices
					IAE	ITAE	ISE
Test 1
SSO - FOPI^λ	0.551	0.741	0.840	0.821	0.012	0.014	0.010
IWO - FOPI^λ	0.470	0.620	0.731	0.765	0.009	0.008	0.006
EWO - FOPI^λ	0.251	0.552	0.662	0.514	0.008	0.008	0.005
Test 2
SSO - FOPI^λ	0.551	0.641	0.840	0.868	0.011	0.012	0.006
IWO - FOPI^λ	0.470	0.520	0.631	0.424	0.006	0.011	0.005
EWO - FOPI^λ	0.331	0.492	0.609	0.303	0.005	0.010	0.004
Test 3
SSO - FOPI^λ	0.682	0.786	0.892	0.904	0.012	0.022	0.010
IWO - FOPI^λ	0.511	0.701	0.814	0.536	0.004	0.012	0.009
EWO - FOPI^λ	0.434	0.638	0.795	0.338	0.004	0.010	0.008
Test 4
SSO - FOPI^λ	0.642	0.749	0.865	0.871	0.012	0.011	0.009
IWO - FOPI^λ	0.504	0.612	0.641	0.407	0.009	0.007	0.008
EWO - FOPI^λ	0.411	0.516	0.596	0.270	0.008	0.006	0.007
Test 5
SSO - FOPI^λ	0.682	0.726	0.892	0.968	0.036	0.012	0.006
IWO - FOPI^λ	0.506	0.591	0.654	0.805	0.020	0.010	0.006
EWO - FOPI^λ	0.305	0.443	0.595	0.504	0.012	0.006	0.005

The optimization techniques-based BLDC Motor speed response under Test 4 conditions is shown in Fig. 11. Reduce the proposed FOPI controller’s settling time value is higher compared to FOPI^λ controller in terms of the 0.8%, 0.5%, and 0.3%for SSO, IWO, and EWO respectively. When compared to the FOPI^λ controller, the proposed FOPI controller has a higher peak time value in terms of 4.5%, 3%, and 2 %for SSO, IWO and EWO respectively. The steady-state error of the proposed FOPI^λ controller is very low compared to the FOPI^λ controller in terms of the 5%, 12%, and 25%for SSO, IWO, and EWO respectively. The performance in time domain under integrated conditions (Case A and varying load) is shown in Table 7.

Fig. 11

optimization techniques based BLDC Motor speed response under Test 4 conditions.

Table 7

Performance in Time domain under integrated conditions (Case A and varying load)

Operating	AlgorithmTime Response	Rise	Peak	Settling	Steady-state	Performance Indices
Conditions		time t_r (s)	time t_p (s)	time t_s (s)	error e_ss (%)	IAE	ITAE	ISE
Case A and	SSO - FOPI^λ	0.712	0.908	0.989	0.802	0.014	0.016	0.012
No Load	IWO - FOPI^λ	0.611	0.806	0.920	0.632	0.012	0.010	0.008
	EWO - FOPI^λ	0.326	0.718	0.861	0.269	0.010	0.010	0.007
Case A and	SSO - FOPI^λ	0.716	0.833	0.994	0.967	0.014	0.016	0.008
20%Load	IWO - FOPI^λ	0.611	0.696	0.820	0.490	0.008	0.014	0.007
	EWO - FOPI^λ	0.430	0.640	0.792	0.367	0.007	0.013	0.005
Case A and	SSO - FOPI^λ	0.807	0.900	0.993	0.801	0.016	0.029	0.013
40 %Load	IWO - FOPI^λ	0.664	0.811	0.908	0.667	0.005	0.016	0.012
	EWO - FOPI^λ	0.564	0.809	0.894	0.467	0.005	0.013	0.010
Case A and	SSO - FOPI^λ	0.835	0.914	0.996	0.826	0.016	0.014	0.012
60%Load	IWO - FOPI^λ	0.655	0.796	0.833	0.533	0.012	0.009	0.010
	EWO - FOPI^λ	0.534	0.671	0.775	0.426	0.010	0.008	0.009
Case A and	SSO - FOPI^λ	0.787	0.844	0.996	0.962	0.047	0.016	0.008
80 %Load	IWO - FOPI^λ	0.658	0.768	0.850	0.733	0.026	0.013	0.008
	EWO - FOPI^λ	0.397	0.576	0.774	0.467	0.016	0.008	0.007
Case A and	SSO - FOPI^λ	0.802	0.912	0.994	0.801	0.047	0.016	0.008
100 %Load	IWO - FOPI^λ	0.736	0.876	0.950	0.667	0.026	0.013	0.008
	EWO - FOPI^λ	0.466	0.642	0.790	0.467	0.016	0.008	0.007

From these above results, the EWO optimized FOPI^λ controller has a minimal output characteristic value, indicating superiority over SSO and IWO optimized FOPI^λ controller for the BLDC motor’s various operating conditions. The comparison of different optimization methods for the FOPI^λ controller for sensorless speed regulation of BLDC motor drives for fine-tuning FOPI^λ controller under various optimization techniques observed that the time-domain characteristics, i.e. peak time, rise time, settling time, peak overshoot and steady-state error are minimised.

5.2 Hardware experimental setup

The experimental setup shows the sensorless speed control of BLDC motor by EWO optimized FOPI^λ controller. It is made up of an autotransformer, TMS 320–DSP kit, power converter, BLDC motor with mechanical loading arrangements, electrical parameters measuring instruments, and power quality analyzer for THD measurements. The DSP kit interface between BLDC motor and power quality analyzer. Figure 12 shows the Measurement of Total Harmonics Distortion (THD) for BLDC motor load with the EWO-FOPI^λ controller.

Fig. 12

A snapshot of the hardware setup in real time.

The proposed controller’s major impact is to minimise the settling time, steady-state error of the BLDC motor, and harmonics of the inverter. The intelligent controller’s performance is evaluated and compared to the simulation results. Figures 13 (a&b) shows the measurement of phasor unbalance, voltage waveform, voltage fluctuation, and Total Harmonics Distortion (THD) for the EWO-FOPI^λ controller-based BLDC motor.

Fig. 13(a)

Phasor for EWO-FOPI^λ controller.

Fig. 13(b)

Voltage waveform for EWO-FOPI^λ controller.

Fig. 14(a)

THD for EWO-FOPI^λ controller.

Fig. 14(b)

Voltage fluctuations for EWO-FOPI^λ controller.

The comparison of experiment and simulation results for the BLDC motor is shown in Table 8. From this result, the proposed EWO optimized FOPI^λ controller gives the control signal tracks the speed accurately with less settling time and very small steady-state error. The comparison of simulation and experimental results of changing speed conditions is shown in Table 9. The Experimental speed response performance parameters such as rise time, peak time, settling time, and steady-state error are close to the performance parameters of simulation results.

Table 8

Comparison of Simulation and Experimental results for BLDC motor

Name of the Intelligent controller	Simulation		Experiment
	%V_THD	%I_THD	%V_THD	%I_THD
SSO - FOPI^λ	16.12	15.36	18.23	17.5
IWO - FOPI^λ	13.15	12.31	14.52	13.55
EWO - FOPI^λ	7.52	4.28	9.52	5.52

Table 9

Comparison of Simulation and Experimental results of varying speed conditions

	Performance Parameters
Operating	Rise time		Peak time		Settling time		Steady state error
Conditions	Simulation	Experiment	Simulation	Experiment	Simulation	Experiment	Simulation	Experiment
Case A	0.251	0.821	0.552	1.290	0.662	1.458	5	6
Case B	0.331	0.846	0.492	1.185	0.609	1.598	3	5
Case C	0.434	0.735	0.638	1.176	0.795	1.568	3	5
Case D	0.411	0.876	0.516	1.365	0.596	1.495	2	4
Case E	0.305	0.869	0.443	1.278	0.595	1.477	5	7

6 Conclusion

The sensorless speed control of FOPI^λ controller using the proposed optimization technique in a BLDC motor is presented. The proposed three optimization techniques such as Social Spider Optimization (SSO), Invasive Weed Optimization (IWO), and Enhanced Whale Optimization (EWO) are used to create the signal with a pulse width modulation used in the inverter and the control signal. The simulation results show that the proposed EWO optimised FOPI controller is effectiveover the SSO-FOPI^λ and IWO-FOPI^λ controller.

Under varying load conditions, the settling time of the proposed EWO-FOPI^λ controller is very low when compared to other FOPI^λ controllers in terms of 8%, 7%, and 3%for SSO, IWO, and EWO techniques respectively. The steady-state error of the proposed EWO- FOPI^λ controller is very low compared to the existing FO-PI^λ controller in terms of 12%, 8%, and 5%for SSO, IWO, and EWO techniques respectively.

Under varying set speed conditions, the settling time of the proposed FO-PI controller is very low compared to the existing FO–PI^λ controller in terms of 12.3%, 9.5%, and 6.3%for SSO, IWO, and EWO respectively. The steady-state error of the proposed EWO-FOPI^λ controller is very low compared to other FOPI^λ controllers in terms of the 7%, 13.3%, and 20%for SSO, IWO, and IWO respectively.

Under integrated conditions, the settling time of the proposed EWO-FOPI^λ controller is very low compared to other FO–PI^λ controllers in terms of 14%, 10.2%, and 7.5%for SSO, IWO, and EWO respectively. The steady-state error of the proposed FO-PI^λ controller is very low compared to the FO-PI^λ controller in terms of the 11%, 7%, and 5.2%for SSO, IWO, and EWO respectively.

According to the simulation results, the EWO-based FOPI^λ controller outperforms the IWO and SSO. Under the above operating conditions, the sensorless speed control of a BLDC motor was optimised in terms of cost function minimization (J), controller energy, mean, standard deviation, performance indices (ITAE, ISE, IAE, RMSE), and improved time domain parameters (tr, tp, ts).

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