Complexity

Complexity / 2021 / Article
Special Issue

Cyber-Enabled Intelligence Control and Security Optimization for Complex Microgrid Networks

View this Special Issue

Research Article | Open Access

Volume 2021 |Article ID 6697792 | https://doi.org/10.1155/2021/6697792

Minghao Zhou, Siwei Cheng, Long Xu, Likun Wang, Qingbo Guo, William Cai, "Robust Switching Gain-Based Fractional-Order Sliding Mode Control for Wind-Powered Microgrids", Complexity, vol. 2021, Article ID 6697792, 12 pages, 2021. https://doi.org/10.1155/2021/6697792

Robust Switching Gain-Based Fractional-Order Sliding Mode Control for Wind-Powered Microgrids

Academic Editor: Ruoli Tang
Received22 Oct 2020
Revised16 Feb 2021
Accepted24 Feb 2021
Published24 Mar 2021

Abstract

This study proposes a novel fractional-order sliding-mode control strategy with robust switching gain to achieve reliable and high quality of wind-powered microgrid systems. Three fractional-order sliding mode controllers are designed to generate continuous control signals and regulate the outer DC-link voltage loop and inner current loop in the grid-side inverters. High robustness and stability of the grid-side inverter can be guaranteed even in the presence of parameter variations and external disturbances. Owing to the fractional-order sliding manifold and fractional-order integral control law, the chattering is attenuated. The fractional-order robust adaptive switching gain is designed to avoid overestimating the upper bound of matched/unmatched uncertainties, save the control energy, and guarantee the rapidity and robustness of the convergence. Simulations validate the proposed method.

1. Introduction

Distributed generation (DG) has paid more and more attention because of its advantages, such as low investment, clean environment protection, high reliability of power supply, and flexible power generation [1]. Beside the aforementioned advantages, DG unit may bring some problems to the grid, such as voltage fluctuations, voltage deviations, and bidirectional power flow. To give full play to the efficiency of the distributed generation systems, the most effective way is to combine the distributed generation and loads to form a microgrid connecting to the main grid. The active power and reactive power between the microgrid and the main grid can be flexibly exchanged. If the main power grid fails, the microgrid will be instantly separated from the main grid and run in the isolated island mode to guarantee the power supply of important loads [2].

In the wind-powered microgrid system, the voltage source inverters (VSIs) are widely used as grid-side inverters. However, this type of converter has a characteristic that the high-frequency harmonics exist in the voltage waveform. Therefore, filters must be used to filter the harmonics out. LCL filters, as commonly used filters, have superior performance in harmonic attenuation and lower current ripple if compared with the L filter, another commonly used filter. Due to the structures and resonances of LCL filters, the LCL filters are rather complicated, which means the control schemes for LCL filters are difficult to implement. The traditional control methods cannot have good performance when applied in the control of LCL filters [3, 4], which may have poor reference tracking and stability problems. In [5], a proportional resonant (PR) controller with a PI controller is designed to get a better dynamic response in terms of disturbance compensation. However, the high gain was introduced by PR, which can cause a larger steady-state error [6].

There are many uncertainties existed in the wind powered microgrid system, such as parameter variations of LCL filters, grid voltage sag, and frequency fluctuation. Sliding mode control (SMC) is extensively used to deal with aforementioned problems due to its high robustness, fast dynamics responses, and simplicity in the control algorithm. It is suitable to use SMC for the control of the grid-side inverters (GSI), on account of its attractive advantages [712]. In [9], an integral-type terminal sliding mode control is proposed to get a high dynamics response in the wind energy conversion system with LCL filters. In [13], a sliding mode controller with Kalman filter (KF) is proposed in a three-phase unity power factory rectifier. An SMC method by modifying reaching law for the variable-speed direct-driven wind energy conversion systems is proposed in [14]. In [15, 16], PI-type sliding surfaces are presented to remove the steady-state error. In [17], a super-twisting sliding mode (STSM) for a gearless wind turbine is developed, so that the system has distinguished robustness against disturbances. In [18], a novel direct active and reactive power control of grid-connected DFIG-based wind turbine systems is presented, which employed a novel SMC to calculate the required rotor control voltage. In [19], an advanced SMC scheme is designed for wind energy conversion systems with a nonlinear disturbance observer.

The aforementioned SMC strategies are all based on integer order calculus which restricts the development of the control theory [20]. With the development of fractional-order calculus, the fractional-order SMC (FOSMC) has been a topic of intense research. Compared with traditional integral order calculus, fractional calculus increases the variability of differential and integral degrees of freedom, which brings new flexibility to the design of control systems [21, 22], so that it has both advantages of fractional calculus and the SMC. The memorial and genetic traits are the typical characteristics of FOSMC, which can predict the control behavior of the system to a certain extent, reduce the switching probability of the control behavior in the reaching process, and thus improve the continuity of the control [23, 24]. A new time-delay estimation-based fractional-order NTSM controller and an intelligent robust fractional-order LSM controller are proposed in [25, 26], but the boundary layer leads to steady-state errors. An adaptive fractional-order TSM is developed in [27]; however, the singularity exists.

To achieve higher performances of wind-powered microgrid systems, this study proposes a fractional-order SMC method with robust adaptive switching gain for GSI. Fractional-order sliding mode controllers are designed for the outer DC-link voltage loop and inner current loop in GSI. The DC-link voltage and current errors can be controlled to converge to zero and chattering can be avoided. High performances of GSI can be guaranteed even in the presence of external disturbances and parameter variations. The remainder of this study is organized as follows: Section 2 briefly introduces the fractional calculus and the wind-powered microgrid systems with a squirrel cage induction generator (SCIG) and the model of GSI with LCL. Section 3 presents the fractional-order SMC design for GSI. Section 4 shows the simulation results. Finally, Section 5 gives a conclusion.

2. Preliminaries

Composed of a wind turbine, an SCIG, a converter, and an LCL-type filter, a SCIG wind turbine system is shown in Figure 1. Figure 2 shows the detailed circuit of the wind-powered microgrid control system, and the definition of symbols in Figure 2 is given in Table 1. The PWM converter consists of a machine-side rectifier (MSR), a DC-link capacitor C with its voltage uDC, and a grid-side inverter. The wind turbine delivers the energy PGe to MSR through the SCIG. The MSR and GSI are controlled separately using fractional-order SMC strategies proposed in the study.


SymbolDefinition

Generator voltages and currents
Inductors/resistances of the L-type filter of the generator
iLOutput current of the MSR
iIInput current of GSI
iDCCurrent through the capacitor
ejGrid voltages
ij, iGjCurrents flowing through the LCL filters LIj, LGj
uCjVoltages in LCL filter capacitors Cj

Due to the existence of the uncertainties in the GSI system, which may be caused by aging, temperature, saturation effects, and some parameter variations have to be considered as follows when modeling the system:where Δ stands for the parameter variations.

With the harmonic of grid voltages and the loss of switches ignored, the model of the GSI systems can be expressed as follows:where Sj, j = a, b, c are the inputs of the PWM modulator, and iI = Saia + Sbib + Scic.

In view of the uncertainties of parameters, the model of the GSI systems (2) can be rewritten in the vector form aswhere S = [Sa, Sb, Sc]T, iG = [iGa, iGb, iGc]T, uC = [uCa, uCb, uCc]T, i = [ia, ib, ic]T, and e= [ea, eb, ec]T, and the parameters LI = [LIa, LIb, LIc]T, Rk = diag, k = I, G, C; Ll = diag, l = I, G; and C = diag. It can be assumed that the uncertainties σG, ΔdG, σC, and ρC are upper-bounded.

For the controller design, the model of the GSI system (3) can be transformed to the form in the αβ-stationary coordinate by the Clarke transformation; then, equation (3) can be rearranged as follows:where S2 = [Sα, Sβ]T, iG2 = [i, i]T, i2 = [iα, iβ]T, uC2= [u, u]T, and e2= [eα, eβ]T.

3. FOSM Controller Design for GSI

The GSI control system is presented in Figure 3. There are three feedback control loops in the system: the DC-link voltage loop, the current loop, and the LCL filter capacitor voltage loop, where ∗ stands for the reference. To depress the effect of unbalanced gird-voltages and deal with the parameter variations, three FOSM controllers with robust switching gains are designed for the three loops. The fractional-order calculus used in three controllers is defined by αth-order Caputo.

To ensure the stability of the controller, the following are the reasonable assumptions:where UDC, UG, and UC are the unknown upper bounds of the uncertainty vectors.

Figure 4 shows the flow chart of the whole algorithms of the fractional-order controllers for the grid-side converters. These three controllers are designed to guarantee that the errors of the DC voltage, current voltage, and the LCL capacitor voltage can converge to zero.

Lemma 1. (see [28]). The nonautonomous fractional-order system is Mittag–Leffler stable at the equilibrium point x(t) = 0, if a continuous and differentiable Lyapunov function V(t, x(t)): R+ ⟶ R and class-K function ϕi (i = 1, 2, 3) satisfies

3.1. DC-Link Voltage Controller

Define the DC-link voltage error as . Then, the error dynamics of the DC-link voltage can be obtained from equation (4) as follows:

A fractional-order sliding manifold for the DC-link voltage error dynamic (7) is chosen as follows:where 0 < α < 1 is the fractional order, and λDC > 0 is a constant.

Theorem 1. Along the fractional-order sliding manifold (8), the DC-link voltage error dynamic (7) can be guaranteed to converge to its equilibrium point, if a fractional-order integral virtual control law with its robust adaptive switching gain is designed as follows:where kDC > 0 is the robust adaptive switching gain, 0 < β < 1 is the fractional order of the fractional integrator, and μ is a positive constant.

Proof. Substituting the error dynamics of the DC voltage (7) into the fractional-order sliding manifold (8) yieldsConsidering the fractional-order integral virtual control laws (9a) and (9b) givesConsider a Lyapunov function. Differentiating VDC givesSubstituting the fractional-order integral virtual control law (9c) with its robust adaptive switching gain (9d) into the above yieldswhich indicates that the DC-link voltage error dynamic (7) will reach to the equilibrium point. On sDC = 0, from (8), it can be seen that the system will behave in the identical fashion asBased on Lemma 1, it can be obtained that when sDC = 0, the DC-link voltage error dynamic (7) will converge to its equilibrium points along the fractional-order sliding manifold. This completes the proof. And the block diagram of the control algorithm of DC-link voltage is shown in Figure 5.
Based on equation (4), the following power equation set can be obtained:Due to the complex power , and the references for the instantaneous active and reactive powers satisfy and q = 0, respectively; the references for the output currents can be obtained as follows:

3.2. Current Controller

Define the current error as . Then, current error dynamics can be obtained according to (4) as follows:

A fractional-order sliding manifold for current error dynamic (17) is chosen as follows:where 0 < α < 1 is the fractional order, and λG2 = diag (λG2α, λG2β), λG2α, λG2β> 0 are all constants.

Theorem 2. Along the fractional-order sliding manifold (18), the current error dynamic (17) can be guaranteed to converge to its equilibrium point, if a fractional-order integral virtual control law with its robust adaptive switching gain is designed as follows:where kC2n > 0 is the robust adaptive switching gain, 0 < β < 1 is the fractional order of the fractional integrator, and μ> 0 is a constant.

Proof. Substituting the current error dynamic (17) into the fractional-order sliding manifold (18) yieldsConsidering the fractional-order integral virtual control laws (19a) and (19b) givesDifferentiating the above with respect to time t givesConsider a Lyapunov function . Differentiating VG2 givesSubstituting the fractional-order integral virtual control law (19c) with its robust adaptive switching gain (19d) into the above yieldswhich means that the error dynamics of the current (17) will reach to zero. On sG2 = 0, it can be seen from (18) that the system will behave according to the following equation:Based on Lemma 1, it can be obtained that when sG2 = 0, the current error dynamic (26) will converge to its equilibrium points along the fractional-order sliding manifold. This completes the proof.

3.3. Filter Capacitor Voltage Controller

The switching control signal S2 in the system (4) is designed to force the actual voltage to track its virtual control reference. Define the current error as . Then, filter capacitor voltage error dynamics are as follows:

A fractional-order sliding manifold for filter capacitor voltage error dynamic (17) is chosen aswhere 0 < α < 1 is the fractional order, and λC2 = diag (λC2α, λC2β), λC2α, λC2β> 0 are all constants.

Theorem 3. Along the fractional-order sliding manifold (27), the LCL capacitor voltage error dynamics (26) can be guaranteed to converge to its equilibrium point, if a fractional-order integral actual control law with its robust adaptive switching gain is designed as follows:

Proof. Substituting the LCL capacitor voltage error dynamic (26) into the fractional-order sliding manifold (27) yieldsConsidering the fractional-order integral virtual control laws (28a) and (28b) givesDifferentiating the above with respect to time t givesConsider Lyapunov function . According to the fractional-order integral virtual control law (24c) with its robust adaptive switching gain (24d) into the above yieldswhich indicates that the error dynamics of filter capacitor (26) will reach to zero. On sC2 = 0, it can be seen from (27) that the system will behave in the following identical fashion:Based on Lemma 1, it can be obtained that when sC2 = 0, the current error dynamics (25) will converge to its equilibrium points along the fractional-order sliding manifold. This completes the proof.

4. Simulations

In order to verify the effectiveness of the proposed method, some simulations are carried out using Matlab. Table 2 provides the parameters of GSI. The parameters of the controllers are as follows: k = 60, kd = 100, η = 100, C = diag (80, 80), μ = diag (3/5, 3/5), and T = diag (100, 100).


ParameterValue (kW)

Rated power2
DC-link voltage700
DC-link capacitor6800
Filter inductor1.5
Phase resistor0.4
Phase voltage of the grid311

The simulations of PI and FOSMC are shown in Figure 6. It can be seen that the DC-link voltage is quickly converged to 700 V and stays around at 700 V under both control strategies. Compared with PI, the proposed FOSMC method can effectively improve the dynamic response of DC-link voltage and reduce the peak value and settling time of DC-link voltage, which enhance the disturbance rejection. From Figures 6(b) and 6(c), the instantaneous active power and reactive power are controlled to converge to their reference value. Therefore, it can be concluded that the control objective has been achieved.

The simulations of the current controller are shown in Figure 7. In Figure 7(a), it can be seen that the current can track the reference quickly and precisely. In Figure 7(b), and have a high degree of sinusoidal, which means the harmonics barely exist in the voltage. Meanwhile, total harmonic distortion (THD) of the output currents of GSI are shown in Figure 7(c) which is a measurement of the harmonic distortion present. It can be known from Figure 7(c) that THD is extremely closed to zero, which means that the harmonic in the current is eliminated and the quality of grid energy can be guaranteed.

Figures 810 show the simulation results when the filter inductances and resistors are changed. Figure 8 depicts the case that values of filter inductances are changed and the resistors keep unchanged. The filter inductances are changed from 1.5 mH to 2 mH. Figure 9 depicts the case that values of the resistors are changed from 0.4 Ω to 0.6 Ω, and three filter inductances keep unchanged. Figure 10 depicts the case for varying resistors and varying filter inductances. The values of resistors and the filter inductances are changed to 1.5 Ω and 2 mH. It can be seen that the changes are not obvious under FOSMC in the view of parameter change, which means that the proposed method has a strong robustness against the parameter variation. From the simulation results in Figures 8(c)10(c), it can be concluded that the voltage and current have the same phase angle, which means that the unit power factor can be guaranteed. The basics of fractional calculus are shown in appendix.

5. Conclusions

In this study, a novel fractional-order sliding mode control strategy with robust switching gain is proposed to achieve reliable and high quality of wind-powered microgrid systems. Fractional-order sliding mode controllers generate continuous control signals and regulate the outer DC-link voltage loop and current loop in GSI. Considering the parameters variation and external disturbance, the outer DC-link voltage and inner current can track their references.

Appendix

A. Basics of Fractional Calculus

The theory of fractional-order calculus is closely related to the traditional integral order calculus. Grünwald–Letnikov (GL), Riemann–Liouville (RL), and Caputo definitions, shown as follows, are three most definitions of the fractional-order differentiation used in literature and practical applications.

Definition 1. (see [29]). The αth-order GL fractional derivative for a function f(t): R+ ⟶ R is defined aswhere α ∈ R is the fractional order, t0 and t are the limits of the operation, [⋅] denotes the integer value function, and h is the time increment, andwhere Γ (⋅) is the well-known Euler gamma function.

Definition 2. (see [29]). The αth-order RL fractional derivative for a function f(t): R+ ⟶ R is defined aswhere u − 1 < α ≤ u, u ∈ N.

Definition 3. (see [29]). The αth-order Caputo fractional derivative for a continuous function f(t): R+ ⟶ R is defined aswhere u is defined in (2). Meanwhile, the αth-order Caputo fractional integration for a function f(t): R+ ⟶ R is defined as

Data Availability

The data used in the simulations are carried out using Matlab-Simulink and are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the Postdoctoral Science Foundation of Heilongjiang Province of China (2901051432), the National Natural Science Foundation of China under Grant (51907042 and 62003086), the Fundamental Research Funds for the Central Universities (2242020R20015), and in part by China Postdoctoral Science Foundation funded project (2020M671294).

References

  1. X. Lu, J. Lai, and G. P. Liu, “Master-slave cooperation for multi-DC-MGs via variable cyber networks,” IEEE Transactions on Cybernetics, Early Access, 2020. View at: Google Scholar
  2. J. Lai, X. Lu, Z. Dong, and S. Cheng, “Resilient distributed multiagent control for AC microgrid networks subject to disturbances,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, pp. 1–11, 2021. View at: Publisher Site | Google Scholar
  3. J. Dannehl, C. Wessels, and F. W. Fuchs, “Limitations of voltage-oriented PI current control of grid-connected PWM rectifiers with $LCL$ filters,” IEEE Transactions on Industrial Electronics, vol. 56, no. 2, pp. 380–388, 2009. View at: Publisher Site | Google Scholar
  4. F. Blaabjerg, R. Teodorescu, M. Liserre, and A. V. Timbus, “Overview of control and grid synchronization for distributed power generation systems,” IEEE Transactions on Industrial Electronics, vol. 53, no. 5, pp. 1398–1409, 2006. View at: Publisher Site | Google Scholar
  5. M. Liserre, R. Teodorescu, and F. Blaabjerg, “Multiple harmonics control for three-phase grid converter systems with the use of PI-RES current controller in a rotating frame,” IEEE Transactions on Power Electronics, vol. 21, no. 3, pp. 836–841, 2006. View at: Publisher Site | Google Scholar
  6. G. Shen, X. Zhu, J. Zhang, and D. Xu, “A new feedback method for PR current control of LCL-filter-based grid-connected inverter,” IEEE Transactions on Industrial Electronics, vol. 57, no. 6, pp. 2033–2041, 2010. View at: Google Scholar
  7. Y. Feng, F. Han, and X. Yu, “Chattering free full-order sliding-mode control,” Automatica, vol. 50, no. 4, pp. 1310–1314, 2014. View at: Publisher Site | Google Scholar
  8. Y. Feng, X. Yu, and Z. Man, “Non-singular terminal sliding mode control of rigid manipulators,” Automatica, vol. 38, no. 12, pp. 2159–2167, 2002. View at: Publisher Site | Google Scholar
  9. X. Zheng, Y. Feng, F. Han, and X. Yu, “Integral-type terminal sliding-mode control for grid-side converter in wind energy conversion systems,” IEEE Transactions on Industrial Electronics, vol. 66, no. 5, pp. 3702–3711, 2019. View at: Publisher Site | Google Scholar
  10. M. Zhou, Y. Feng, and C. Xue, “Deep convolutional neural network based fractional-order terminal sliding-mode control for robotic manipulators,” Neurocomputing, vol. 416, pp. 143–151, 2020. View at: Publisher Site | Google Scholar
  11. B. Sun, Y. Cao, Z. Guo et al., “Sliding-mode stabilization of memristive neural networks with leakage delays and control disturbance,” IEEE Transactions on Neural Networks and Learning Systems, 2019. View at: Google Scholar
  12. Y. Wang, Y. Cao, Z. Guo, T. Huang, and S. Wen, “Event-based sliding-mode synchronization of delayed memristive neural networks via continuous/periodic sampling algorithm,” Applied Mathematics and Computation, vol. 383, Article ID 125379, 2020. View at: Publisher Site | Google Scholar
  13. R. Guzman, L. G. de Vicuna, J. Morales, M. Castilla, and J. Matas, “Sliding-mode control for a three-phase unity power factor rectifier operating at fixed switching frequency,” IEEE Transactions on Power Electronics, vol. 31, no. 1, pp. 758–769, 2016. View at: Publisher Site | Google Scholar
  14. S. M. Mozayan, M. Saad, H. Vahedi, H. Fortin-Blanchette, and M. Soltani, “Sliding mode control of PMSG wind turbine based on enhanced exponential reaching law,” IEEE Transactions on Industrial Electronics, vol. 63, no. 10, pp. 6148–6159, 2016. View at: Publisher Site | Google Scholar
  15. J. Liu, F. Zhou, and C. Zhao, “A PI-type sliding mode controller design for PMSG-based wind turbine,” Complexity, vol. 2019, Article ID 2538206, 12 pages, 2019. View at: Publisher Site | Google Scholar
  16. R. Errouissi and A. Al-Durra, “A novel PI-type sliding surface for PMSG-based wind turbine with improved transient performance,” IEEE Transactions on Energy Conversion, vol. 33, no. 2, pp. 834–844, 2018. View at: Publisher Site | Google Scholar
  17. M. Nasiri, S. Mobayen, and Q. Zhu, “Super-twisting sliding mode control for gearless PMSG-based wind turbine,” Complexity, vol. 2019, Article ID 6141607, 15 pages, 2019. View at: Publisher Site | Google Scholar
  18. J. Hu, H. Nian, B. Hu, Y. He, and Z. Q. Zhu, “Direct active and reactive power regulation of DFIG using sliding-mode control approach,” IEEE Transactions on Energy Conversion, vol. 25, no. 4, pp. 1028–1039, 2010. View at: Publisher Site | Google Scholar
  19. T. D. Do, “Disturbance observer-based fuzzy SMC of WECSs without wind speed measurement,” IEEE Access, vol. 5, pp. 147–155, 2017. View at: Publisher Site | Google Scholar
  20. Y. Wang, L. Gu, Y. Xu, and X. Cao, “Practical tracking control of robot manipulators with continuous fractional-order nonsingular terminal sliding mode,” IEEE Transactions on Industrial Electronics, vol. 63, no. 10, pp. 6194–6204, 2016. View at: Publisher Site | Google Scholar
  21. Y. Tang, X. Zhang, D. Zhang, G. Zhao, and X. Guan, “Fractional order sliding mode controller design for antilock braking systems,” Neurocomputing, vol. 111, pp. 122–130, 2013. View at: Publisher Site | Google Scholar
  22. H. Delavari, P. Lanusse, and J. Sabatier, “Fractional order controller design for a flexible link manipulator robot,” Asian Journal of Control, vol. 15, no. 3, pp. 783–795, 2013. View at: Publisher Site | Google Scholar
  23. L. Gao, Z. Wang, K. Zhou, W. Zhu, Z. Wu, and T. Ma, “Modified sliding mode synchronization of typical three-dimensional fractional-order chaotic systems,” Neurocomputing, vol. 166, pp. 53–58, 2015. View at: Publisher Site | Google Scholar
  24. M. P. Aghababa, “Design of a chatter-free terminal sliding mode controller for nonlinear fractional-order dynamical systems,” International Journal of Control, vol. 86, no. 10, pp. 1744–1756, 2013. View at: Publisher Site | Google Scholar
  25. Y. Wang, G. Luo, L. Gu, and X. Li, “Fractional-order nonsingular terminal sliding mode control of hydraulic manipulators using time delay estimation,” Journal of Vibration and Control, vol. 22, no. 19, pp. 3998–4011, 2016. View at: Publisher Site | Google Scholar
  26. H. Delavari, R. Ghaderi, A. Ranjbar, and S. Momani, “Fuzzy fractional order sliding mode controller for nonlinear systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 4, pp. 963–978, 2010. View at: Publisher Site | Google Scholar
  27. D. Nojavanzadeh and M. Badamchizadeh, “Adaptive fractional‐order non‐singular fast terminal sliding mode control for robot manipulators,” IET Control Theory & Applications, vol. 10, no. 13, pp. 1565–1572, 2016. View at: Publisher Site | Google Scholar
  28. Y. Li, Y. Chen, and I. Podlubny, “Mittag-Leffler stability of fractional order nonlinear dynamic systems,” Automatica, vol. 45, no. 8, pp. 1965–1969, 2009. View at: Publisher Site | Google Scholar
  29. I. Podlubny, Fractional Differential Equations, Academic Press, Cambridge, MA, USA, 1999.

Copyright © 2021 Minghao Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views107
Downloads216
Citations

Related articles

Article of the Year Award: Outstanding research contributions of 2020, as selected by our Chief Editors. Read the winning articles.