Pitch Control of Wind Turbine Blades Using Fractional Particle Swarm Optimization

Citation: Karami-Mollaee, A.; Barambones, O. Pitch Control of Wind Turbine Blades Using Fractional Particle Swarm Optimization. Axioms 2023,12, 25. https://doi.org/10.3390/ axioms12010025 Academic Editor: Fevrier Valdez Received: 20 November 2022 Revised: 20 December 2022 Accepted: 22 December 2022 Published: 26 December 2022 Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). axioms Article Pitch Control of Wind Turbine Blades Using Fractional Particle Swarm Optimization Ali Karami-Mollaee 1and Oscar Barambones 2,* 1Faculty of Electrical and Computer Engineering, Hakim Sabzevari University, Sabzevar 9617976487, Iran 2Automatic Control and System Engineering Department, University of the Basque Country, UPV/EHU, Nieves Cano 12, 48940 Vitoria, Spain *Correspondence: oscar[email protected]; Tel.: +34-945013235; Fax: +34-945013270 Abstract: To achieve the maximum power from wind in variable-speed regions of wind turbines (WTs), a suitable control signal should be applied to the pitch angle of the blades. However, the available uncertainty in the modeling of WTs complicates calculations of these signals. To cope with this problem, an optimal controller is suitable, such as particle swarm optimization (PSO). To improve the performance of the controller, fractional order PSO (FPSO) is proposed and implemented. In order to construct this approach for a two-mass WT, we propose a new state feedback, which was first applied to the turbine. The idea behind this state feedback was based on the Taylor series. Then, a linear model with uncertainty was obtained with a new input control signal. Thereafter, the conventional PSO (CPSO) and FPSO were used as optimal controllers for the resulting linear model. Finally, a comparison was performed between CPSO and FPSO and the fuzzy Takagi–Sugeno–Kang (TSK) inference system. The provided comparison demonstrates the advantages of the Taylor series with combination to these controllers. Notably, without the state feedback, CPSO, FPSO, and TSK fuzzy systems cannot stabilize WTs in tracking the desired trajectory. Keywords: wind turbine; pitch angle control; fractional particle swarm optimization; fuzzy inference system; Taylor series MSC: 93D15 1. Introduction Solar or wind, as clean renewable viable energies, are accessible worldwide and are clean. However, due to economic reasons, the use of wind energy and wind turbines (WTs) is popular. There are two kinds of WT, fixed speed WTs (FWTs) [ 1 , 2 ] and variable-speed WTs (VWTs) [ 3 ]. It is not capable for FWTs to work such that the maximum power of wind can be harnessed [ 4 ]. Therefore, VWTs have recently been developed and constructed. To capture the maximum power of wind in VWTs, its operation regions are divided into four important sections using cut-out, rated, and cut-in boundaries [ 5 ]. Below the cut-out wind speed, VWTs will be shut down, to balance economic performance between the cut-out and rated wind speeds by controlling the generator torque [ 5 ]. Moreover, between the rated and cut-in boundaries of wind speeds, the pitch angle of turbine blades is used as the input control [ 6 ]. Finally, above the cut-in boundary, the VWT will be shut down again to protect it from fatigue damage [7]. On the other hand, mechanical stresses are another important challenge, which require powerful optimal or adaptive approaches to protect WTs [ 7 ]. Therefore, some pitch angle controllers have been proposed for WT blades between cut-out and rated wind speeds [6–14] . In [ 6 ], a digital controller was designed; classical controllers such as PID (proportional–integral–derivative) are proposed in [ 7 , 8 ]; a PID controller with an adaptive self-tuning regulator (STR) was constructed in [ 9 ]; a gain-scheduled PID controller was designed in [ 10 ]; a PI controller scheme is shown in [ 11 ]; and a combination of adaptive Axioms 2023,12, 25. https://doi.org/10.3390/axioms12010025 https://www.mdpi.com/journal/axioms Axioms 2023,12, 25 2 of 16 and PI controllers is presented in [ 12 ]. Some simple nonlinear feedback controllers are proposed in [ 13 , 14 ]. Finally, to improve the performance of WTs, variable frequency converter controls to regulate the rotor speed were also used in [3,15]. Among these approaches, fractional controllers can have better performance [ 16 , 17 ] because they can precisely describe the behavior of many dynamical systems in physical, mathematical, and engineering fields [ 18 – 20 ]. Hence, many studies have focused on fractional subjects to develop their theories [ 21 , 22 ]. Therefore, the fractional calculations have progressed in various phenomena due to their applications in dynamic systems [ 17 ]. In the other hand, particle swarm optimization (PSO) is a power tool for optimization [ 23 ] and controllers [ 24 ]. Therefore, based on the advanced properties of fractional calculus and PSO, we improved the performance of conventional PSO (CPSO) using a combination of fractional and PSO. The proposed approach is fractional PSO (FPSO), and was applied to WTs for pitch angle control. Initially, state feedback was applied to the WT model; then, FPSO forced the WT rotor angular velocity to track its reference while the pitch angle of the blades was regulated. To demonstrate the advanced performance of FPSO, comparison was performed with CPSO and the Takagi–Sugeno–Kang (TSK) fuzzy system with similar parameters [25]. Hence, the proposed controller is demonstrated in five sections. First, the WT model and their subsystems are explained in Section 2. Then, the controller details, consisting of state feedback, with reference to rotor angular velocity, PSO, and TSK system, are provided in Section 3. The simulation results and comparisons of FPSO, CPSO, and TSK systems are presented in Section 4. Section 5presents the conclusion. 2. Wind Turbine (WT) Model The generator and drivetrain are two WT subsystems, their electrical and mechanical sections, respectively [ 16 ]. Other important subsystem of a WT is the aerodynamic section [16]. These subsystems are depicted in Figure 1. Axioms 2023, 12, x FOR PEER REVIEW 2 of 18 tive self-tuning regulator (STR) was constructed in [9]; a gain-scheduled PID controller was designed in [10]; a PI controller scheme is shown in [11]; and a combination of adaptive and PI controllers is presented in [12]. Some simple nonlinear feedback controllers are proposed in [13,14]. Finally, to improve the performance of WTs, variable frequency converter controls to regulate the rotor speed were also used in [3,15]. Among these approaches, fractional controllers can have better performance [16,17] because they can precisely describe the behavior of many dynamical systems in physical, mathematical, and engineering fields [18–20]. Hence, many studies have focused on fractional subjects to develop their theories [21,22]. Therefore, the fractional calculations have progressed in various phenomena due to their applications in dynamic systems [17]. In the other hand, particle swarm optimization (PSO) is a power tool for optimization [23] and controllers [24]. Therefore, based on the advanced properties of fractional calculus and PSO, we improved the performance of conventional PSO (CPSO) using a combination of fractional and PSO. The proposed approach is fractional PSO (FPSO), and was applied to WTs for pitch angle control. Initially, state feedback was applied to the WT model; then, FPSO forced the WT rotor angular velocity to track its reference while the pitch angle of the blades was regulated. To demonstrate the advanced performance of FPSO, comparison was performed with CPSO and the Takagi–Sugeno–Kang (TSK) fuzzy system with similar parameters [25]. Hence, the proposed controller is demonstrated in five sections. First, the WT model and their subsystems are explained in Section 2. Then, the controller details, consisting of state feedback, with reference to rotor angular velocity, PSO, and TSK system, are provided in Section 3. The simulation results and comparisons of FPSO, CPSO, and TSK systems are presented in Section 4. Section 5 presents the conclusion. 2. Wind Turbine (WT) Model The generator and drivetrain are two WT subsystems, their electrical and mechanical sections, respectively [16]. Other important subsystem of a WT is the aerodynamic section [16]. These subsystems are depicted in Figure 1. Figure 1. The WT subsystems. 2.1. The Aerodynamic Subsystem Considering a WT with blade length r , power coefficient p C, the captured power can calculated using the following equation: ),(C 2 vr Pp 22 aλβ ρπ = (1) where ρ is the air density and )t(v is the wind speed, which are dependent on environment conditions. The power coefficient is dependent on the tip speed ratio, λ , and the pitch blades, β [5], which are defined as follows: v rr ω =λ (2) Figure 1. The WT subsystems. 2.1. The Aerodynamic Subsystem Considering a WT with blade length r, power coefficient Cp, the captured power can calculated using the following equation: Pa=ρπ r2v2 2Cp(β,λ)(1) where ρ is the air density and v(t) is the wind speed, which are dependent on environment conditions. The power coefficient is dependent on the tip speed ratio, λ , and the pitch blades, β[5], which are defined as follows: λ=rωr v(2) Cp(β,λ) = d1d2 λi−d1d3β−d1d4e−d5 λi+d6λ 1 λi=1 λ+0.08β−0.035 β3+1 (3) Axioms 2023,12, 25 3 of 16 where ωris the rotor side angular velocity of the turbine blades, and: d1=0.5176, d2=116, d3=0.4, d4=5, d5=21, d6=0.0068 (4) Then, the generated rotor torque is described by: Ta=Pa ωr =ρπ r3v2 2λCp(β,λ)(5) 2.2. The Drivetrain Subsystem The two-mass mechanical drivetrain, which shows the transient response and steadystate response in the presence of the controller, is described by the following equations and is depicted in Figure 2[26]. Jr . ωr=−Krωr+Ta−Tls Jg . ωg=−Kgωg−Tg+Ths (6) Axioms 2023, 12, x FOR PEER REVIEW 3 of 18 1 035.0 08.0 11 dedddd dd ),(C 3 i 6 d 4131 i 21 p i 5 +β − β+λ = λ λ+        −β− λ =λβ λ − (3) where r ω is the rotor side angular velocity of the turbine blades, and: 0068.0d,21d,5d,4.0d,116d,5176.0d 654321 ====== (4) Then, the generated rotor torque is described by: ),(C 2 vrP Tp 23 r a aλβ λ ρπ = ω = (5) 2.2. The Drivetrain Subsystem The two-mass mechanical drivetrain, which shows the transient response and steady-state response in the presence of the controller, is described by the following equations and is depicted in Figure 2 [26]. hsggggg lsarrrr TTKJ TTKJ +−ω−=ω −+ ω −= ω   (6) Figure 2. The drivetrain structure. In Figure 2, g ω and r ω are angular velocity, g J and r J are inertia, g K and r K are the external dapping, g T and a T are the torque output, on the generator and rotor side, respectively, and finally, hs T and ls T are the braking torque in the high-speed and low-speed shaft. The ratio of gearbox is defined as: r g g nω ω = (7) Using the second part of Equation (6) results in: () ()         +−ω−=ω g ls grggrgg n T TnKnJ  (8) or: Figure 2. The drivetrain structure. In Figure 2, ωg and ωr are angular velocity, Jg and Jr are inertia, Kg and Kr are the external dapping, Tg and Ta are the torque output, on the generator and rotor side, respectively, and finally, Ths and Tls are the braking torque in the high-speed and low-speed shaft. The ratio of gearbox is defined as: ng=ωg ωr(7) Using the second part of Equation (6) results in: Jgng . ωr=−Kgngωr−Tg+Tls ng(8) or: ng2Jg . ωr=−ng2Kgωr−ngTg+Tls (9) Finally, adding this equation to the first part of Equation (6) results in [5]: Jt . ωr=−Ktωr+Ta−ngTg(10) Such that Jt=Jr+ng2Jg and Kt=Kr+ng2Kg . In fact, all the parameters are transferred to a low-speed shaft [5]. Axioms 2023,12, 25 4 of 16 2.3. The Generator Subsystem The Tg , i.e., the output torque of the generator, can be modeled using the first-order dynamic, where Tref is the reference torque and τg=15 s is the generator time constant. . Tg=Tref −Tg τg(11) We focused on pitch control; thus, the generator torque reference was set as Tref =Treted . Moreover, the produced output power delivered to the grid can be written as Pg=ηgωgTg, where the efficiency of the generator is ηg. 3. The Optimal Controller Design In this section, we first used state feedback and then calculated the desired rotor angular velocity. The CPSO and FPSO optimal controllers are also described. 3.1. State Feedback Initially, we calculated the derivative of the power coefficient of Equation (3) with respect to the pitch angle of the blades. dCp dβ=∂Cp ∂λi ∂λi ∂β+∂Cp ∂β =−c1c2 λi2+c1c2c5 λi3−c1c3c5β λi2−c1c4c5 λi2−0.08 (λ+0.08β)2+0.035 (β3+1)2e−c5 λi−c1c3e−c5 λi (12) Therefore, the Taylor series of Equation (5) around its optimal operating points βopt and λopt would be: Ta=ρπ r3v2 2λopt dCp dββ=βopt λ=λopt (β−βopt) + HOT (13) where HOT is used to denote higher-order terms; thus: . ωr=−Kt Jt ωr−ng Jt Tg+Ta Jt =−Kt Jt ωr−ng Jt Tg+ρπ r3v2 2Jtλopt dCp dββ=βopt λ=λopt (β−βopt) + ∆(14) Due to the convergence of the Taylor series around the operating points, the unknown uncertainty ∆=HOT Jt is bounded, i.e., |∆|≤η . Then, the following state feedback with the new input signal, u, and the arbitrary parameter, a, can be used. β=ng JtTg+Kt Jt−aωr+u ρπ r3v2 2Jtλopt dCp dββ=βopt λ=λopt +βopt (15) Then, system Equation (14) can be rewritten as follows: . ωr=−aωr+u+∆(16) In which ∆ is an unknown uncertain function. We aimed to design an optimal approach such that in this linear system, the rotor angular velocity, ωr , tracked the desired signal, ωrd . 3.2. Reference of Rotor Angular Velocity As mentioned in the Introduction and based on Figure 3, the VWT operation modes were divided into four regions using wind speed boundaries of cut-in, rated, and cut-out. The critical point is the rated wind speed, such that below this point, the pitch of turbine Axioms 2023,12, 25 5 of 16 blades is fixed and generator torque is controlled; hence, the rotor speed is increased to have the maximum of power coefficient. Moreover, above the rated wind speed, the generator reference torque is fixed and is set to its rated value. In this region, the pitch angle would be increased to reduce the rotor speed. Finally, out of the cut-in and cut-out wind speeds, the turbine would be shut down due to the economic criterion and fatigue damages, respectively [5]. Axioms 2023, 12, x FOR PEER REVIEW 5 of 18 opt p optt 23 r t t g t g opt opt d dC J2 vr ua J K T J n β+ βλ ρπ +ω        −+         =β λ=λ β=β (15) Then, system Equation (14) can be rewritten as follows: Δ++ ω −= ω ua rr  (16) In which Δ is an unknown uncertain function. We aimed to design an optimal approach such that in this linear system, the rotor angular velocity, r ω , tracked the desired signal, rd ω . 3.2. Reference of Rotor Angular Velocity As mentioned in the Introduction and based on Figure 3, the VWT operation modes were divided into four regions using wind speed boundaries of cut-in, rated, and cut-out. The critical point is the rated wind speed, such that below this point, the pitch of turbine blades is fixed and generator torque is controlled; hence, the rotor speed is increased to have the maximum of power coefficient. Moreover, above the rated wind speed, the generator reference torque is fixed and is set to its rated value. In this region, the pitch angle would be increased to reduce the rotor speed. Finally, out of the cut-in and cut-out wind speeds, the turbine would be shut down due to the economic criterion and fatigue damages, respectively [5]. Figure 3. Operation regions of the VWT. In this study, we focused on the pitch angle control in region three, whereas the rotor angular velocity should be reduced with the increased of wind speed. Therefore, the reference of rotor angular velocity is as follows: ratedoutcut rated ratedratedrd vv vv − − ω−ω=ω − (17) Based on Equation (2), one can conclude that: Figure 3. Operation regions of the VWT. In this study, we focused on the pitch angle control in region three, whereas the rotor angular velocity should be reduced with the increased of wind speed. Therefore, the reference of rotor angular velocity is as follows: ωrd =ωrated −ωrated v−vrated vcut−out −vrated (17) Based on Equation (2), one can conclude that: ωrated =λoptvrated r(18) 3.3. Particle Swarm Optimization (PSO) and Controller Structure According to the previous sections, the aim was to determine the angular velocity of the rotor, i.e., ωr tracks the desired trajectory, ωrd . To this end, the error, e= (ωr−ωrd)2 , is applied to the PSO. PSO is applied to calculate the input control signal, u , in Equation (16), while the error signal, e , converges to zero. In CPSO, the velocity of each particle is updated as follows [23]: . vi(t) = c1φ1(pb−xi(t))+c2φ2pg−xi(t): i =1, 2, . . . , n (19) where n is the number of particles, vi(t) is the velocity of each particle, φ1 and φ2 are uniformly random functions between 0 and 1, pb is the best position of each particle, pg is the global best position between all of the particles, xi(t) is the current position of the each particle, and coefficients c1 and c2 are constant numbers. Then, the position of any particle is updated as follows [23]: . xi=vi(t)(20) There are several definitions for fractional differentiations and integrations, such as Grünwald–Letnikov, Riemann–Liouville, and Caputo formulae [ 27 ]. Among them, the Axioms 2023,12, 25 6 of 16 Caputo method is popular because initial conditions are considered [ 17 , 27 ]; thus, the Caputo definition was used in this study. Definition 1. Caputo q-order integration and the differentiation of variable v(t) with respect to time, t, is defined as follows [27]: t0Iq tv(t) = 1 Γ(q)Zt t0 (t−τ)q−1v(τ)dτ(21) t0Dq tv(t) = 1 Γ(1−q)Zt t0 v0(τ) (t−τ)qdτ(22) Here, t>t0 and t0 is the initial time; 0 <q< 1and Γ(q) = R∞ 0τq−1e−τdτ is the Gamma function . Remark 1. In this study, we considered the zero initial condition, i.e., t0= 0. Moreover, for simplicity, subscript t was also eliminated; hence, we use Dqv(t) instead of t0Dq tv(t) and Iqv(t) instead of t0Iq tv(t). To improve the performance of the CPSO, we propose FPSO, as follows: Dqvi(t) = c1φ1(pb−xi(t))+c2φ2pg−xi(t): i =1, 2, . . . , n (23) For a valid comparison, both CPSO and FPSO were implemented. Therefore, the implemented diagram of the proposed approach is illustrated in Figure 4. From this figure, one can see the combination of two feedbacks for nonlinear systems of Figures 1and 2. The first state feedback of Equation (15) is based on the theory of the Taylor series with a new input control signal, u(t) , to obtain a linear system, as in Equation (16). Then, in the second feedback, FPSO or CPSO were applied to this linear system in order to minimize the error signal. Moreover, we used the fuzzy TSK systems. Axioms 2023, 12, x FOR PEER REVIEW 7 of 18 input control signal, )t(u , to obtain a linear system, as in Equation (16). Then, in the second feedback, FPSO or CPSO were applied to this linear system in order to minimize the error signal. Moreover, we used the fuzzy TSK systems. Figure 4. The implemented structure of the proposed controller. 3.4. Takagi–Sugeno–Kang (TSK) Controller Structure The structure of proposed TSK controller is shown in Figure 4, with two inputs )(e rdr ω − ω = and its derivative and one output )t(u . For each input and output, five triangular membership functions are defined as negative-large (NL), negative-small (NS), zero (Z), positive-small (PS), and positive-large (PL). The range of inputs was set between −2 and +2, and the range of output was also set to between −200 and 200. Therefore, 25 rules with the aggregation defuzzification were used. 4. Simulations Results We used the two-mass 5 MW, VWT in National Renewable Energy Laboratory (NREL) located at Colorado, with the aerodynamic parameters in Table 1 and drivetrain parameters in Table 2 [28]. Table 1. The aerodynamic parameters of VWT. Parameter Value Unit rated T 4 103094.4 × mN × incut v− 3 s/m rated v 5.10 s/m outcut v− 25 s/m opt β 0 deg opt λ Scalar 55.7 Figure 4. The implemented structure of the proposed controller. 3.4. Takagi-Sugeno-Kang (TSK) Controller Structure The structure of proposed TSK controller is shown in Figure 4, with two inputs e= (ωr−ωrd) and its derivative and one output u(t) . For each input and output, five triangular membership functions are defined as negative-large (NL), negative-small (NS), zero (Z), positive-small (PS), and positive-large (PL). The range of inputs was set between Axioms 2023,12, 25 7 of 16 − 2 and +2, and the range of output was also set to between − 200 and 200. Therefore, 25 rules with the aggregation defuzzification were used. 4. Simulations Results We used the two-mass 5 MW, VWT in National Renewable Energy Laboratory (NREL) located at Colorado, with the aerodynamic parameters in Table 1and drivetrain parameters in Table 2[28]. Table 1. The aerodynamic parameters of VWT. Parameter Value Unit Trated 4.3094 ×104N×m vcut−in 3 m/s vrated 10.5 m/s vcut−out 25 m/s βopt 0 deg λopt Scalar 7.55 Table 2. The mechanical drivetrain parameters of VWTs. Notation Value Unit r 21.62 m ρ1.308 kg/m3 Jr3.25 ×105kg ×m2 Jg34.4 kg ×m2 Kr27.36 (N×m)/(rad/s) Kg0.2 (N×m)/(rad/s) Kls 9.5 ×103(N×m)/rad Bls 2.691 ×105(N×m)/(rad/s) ng43.165 Scalar For a reliable comparison, all of the simulations were performed using MATLAB software with a sample time of 0.01. The wind speed is shown in Figure 5, with a mean value 16 and maximum disturbance of 5. Notably, this is between 11.4 and 25, i.e., in region 3: vrated = 11.4 <v(t)<vcut−out = 25. In addition, the initial value for the rotor angular velocity is set to 3, i.e., ωr( 0 ) = 3, and the feedback parameter is set as a= 2. Moreover, Figure 6shows the reference of rotor speed in region 3 denoted by Equation (17). Axioms 2023, 12, x FOR PEER REVIEW 8 of 18 Table 2. The mechanical drivetrain parameters of VWTs. Notation Value Unit r 62.21 m ρ 308.1 3 m/kg r J 5 1025.3 × 2 mkg × g J 4.34 2 mkg× r K 36.27 )s/rad/()mN( × g K 2.0 )s/rad/()mN( × ls K 3 105.9 × rad/)mN( × ls B 5 10691.2 × )s/rad/()mN( × g n 165.43 Scalar For a reliable comparison, all of the simulations were performed using MATLAB software with a sample time of 0.01. The wind speed is shown in Figure 5, with a mean value 16 and maximum disturbance of 5. Notably, this is between 11.4 and 25, i.e., in region 3: 25v)t(v4.11v outcutrated =<<= − . In addition, the initial value for the rotor angular velocity is set to 3, i.e., 3)0( r= ω , and the feedback parameter is set as 2a =. Moreover, Figure 6 shows the reference of rotor speed in region 3 denoted by Equation (17). Figure 5. The profile of time series wind speed. Figure 5. The profile of time series wind speed. Axioms 2023,12, 25 8 of 16 Axioms 2023, 12, x FOR PEER REVIEW 9 of 18 Figure 6. Reference of the rotor speed in region 3 of VWTs. Example 1. The CPSO approach. As the first result, simulations of the CPSO in Equation (19) are shown in Figure 7– 10. The parameters of the PSO are 5.1cc 21 == , with 20 particles and 4000 iterations. Figure 7. Convergence of the CPSO. Figure 6. Reference of the rotor speed in region 3 of VWTs. Example 1. The CPSO approach. As the first result, simulations of the CPSO in Equation (19) are shown in Figures 7–10. The parameters of the PSO are c1=c2=1.5, with 20 particles and 4000 iterations. Axioms 2023, 12, x FOR PEER REVIEW 9 of 18 Figure 6. Reference of the rotor speed in region 3 of VWTs. Example 1. The CPSO approach. As the first result, simulations of the CPSO in Equation (19) are shown in Figure 7– 10. The parameters of the PSO are 5.1cc 21 == , with 20 particles and 4000 iterations. Figure 7. Convergence of the CPSO. Figure 7. Convergence of the CPSO. Axioms 2023,12, 25 9 of 16 Axioms 2023, 12, x FOR PEER REVIEW 10 of 18 Figure 8. Angular velocity of the rotor in CPSO. Figure 9. The input control signal of state feedback in CPSO. Figure 8. Angular velocity of the rotor in CPSO. Axioms 2023, 12, x FOR PEER REVIEW 10 of 18 Figure 8. Angular velocity of the rotor in CPSO. Figure 9. The input control signal of state feedback in CPSO. Figure 9. The input control signal of state feedback in CPSO. Axioms 2023,12, 25 16 of 16 18. Kilbsa, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: New York, NY, USA, 2006. 19. Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications; Academic Press: San Diego, CA, USA, 1998. 20. Qiu, F.; Liu, Z.; Liu, R.; Quan, X.; Tao, C.; Wang, Y. Fluid flow signals processing based on fractional Fourier transform in a stirred tank reactor. ISA Trans. 2019,90, 268–277. [CrossRef] 21. Li, Y.; Chen, Y.; Podlubny, I. 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