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Minghao Zhou, Siwei Cheng, Long Xu, Likun Wang, Qingbo Guo, William Cai, "Robust Switching GainBased FractionalOrder Sliding Mode Control for WindPowered Microgrids", Complexity, vol. 2021, Article ID 6697792, 12 pages, 2021. https://doi.org/10.1155/2021/6697792
Robust Switching GainBased FractionalOrder Sliding Mode Control for WindPowered Microgrids
Abstract
This study proposes a novel fractionalorder slidingmode control strategy with robust switching gain to achieve reliable and high quality of windpowered microgrid systems. Three fractionalorder sliding mode controllers are designed to generate continuous control signals and regulate the outer DClink voltage loop and inner current loop in the gridside inverters. High robustness and stability of the gridside inverter can be guaranteed even in the presence of parameter variations and external disturbances. Owing to the fractionalorder sliding manifold and fractionalorder integral control law, the chattering is attenuated. The fractionalorder 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 windpowered microgrid system, the voltage source inverters (VSIs) are widely used as gridside inverters. However, this type of converter has a characteristic that the highfrequency 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 steadystate 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 gridside inverters (GSI), on account of its attractive advantages [7–12]. In [9], an integraltype 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 threephase unity power factory rectifier. An SMC method by modifying reaching law for the variablespeed directdriven wind energy conversion systems is proposed in [14]. In [15, 16], PItype sliding surfaces are presented to remove the steadystate error. In [17], a supertwisting 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 gridconnected DFIGbased 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 fractionalorder calculus, the fractionalorder 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 timedelay estimationbased fractionalorder NTSM controller and an intelligent robust fractionalorder LSM controller are proposed in [25, 26], but the boundary layer leads to steadystate errors. An adaptive fractionalorder TSM is developed in [27]; however, the singularity exists.
To achieve higher performances of windpowered microgrid systems, this study proposes a fractionalorder SMC method with robust adaptive switching gain for GSI. Fractionalorder sliding mode controllers are designed for the outer DClink voltage loop and inner current loop in GSI. The DClink 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 windpowered microgrid systems with a squirrel cage induction generator (SCIG) and the model of GSI with LCL. Section 3 presents the fractionalorder 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 LCLtype filter, a SCIG wind turbine system is shown in Figure 1. Figure 2 shows the detailed circuit of the windpowered microgrid control system, and the definition of symbols in Figure 2 is given in Table 1. The PWM converter consists of a machineside rectifier (MSR), a DClink capacitor C with its voltage u_{DC}, and a gridside inverter. The wind turbine delivers the energy P_{Ge} to MSR through the SCIG. The MSR and GSI are controlled separately using fractionalorder SMC strategies proposed in the study.

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 S_{j}, j = a, b, c are the inputs of the PWM modulator, and i_{I} = S_{a}i_{a} + S_{b}i_{b} + S_{c}i_{c}.
In view of the uncertainties of parameters, the model of the GSI systems (2) can be rewritten in the vector form aswhere S = [S_{a}, S_{b}, S_{c}]^{T}, i_{G} = [i_{Ga}, i_{Gb}, i_{Gc}]^{T}, u_{C} = [u_{Ca}, u_{Cb}, u_{Cc}]^{T}, i = [i_{a}, i_{b}, i_{c}]^{T}, and e = [e_{a}, e_{b}, e_{c}]^{T}, and the parameters L_{I} = [L_{Ia}, L_{Ib}, L_{Ic}]^{T}, R_{k} = diag, k = I, G, C; L_{l} = diag, l = I, G; and C = diag. It can be assumed that the uncertainties σ_{G}, Δd_{G}, σ_{C}, and ρ_{C} are upperbounded.
For the controller design, the model of the GSI system (3) can be transformed to the form in the αβstationary coordinate by the Clarke transformation; then, equation (3) can be rearranged as follows:where S_{2} = [S_{α}, S_{β}]^{T}, i_{G2} = [i_{Gα}, i_{Gβ}]^{T}, i_{2} = [i_{α}, i_{β}]^{T}, u_{C2} = [u_{Cα}, u_{Cβ}]^{T}, and e_{2} = [e_{α}, e_{β}]^{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 DClink voltage loop, the current loop, and the LCL filter capacitor voltage loop, where ∗ stands for the reference. To depress the effect of unbalanced girdvoltages and deal with the parameter variations, three FOSM controllers with robust switching gains are designed for the three loops. The fractionalorder calculus used in three controllers is defined by α^{th}order Caputo.
To ensure the stability of the controller, the following are the reasonable assumptions:where U_{DC}, U_{G}, and U_{C} are the unknown upper bounds of the uncertainty vectors.
Figure 4 shows the flow chart of the whole algorithms of the fractionalorder controllers for the gridside 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 fractionalorder 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 classK function ϕ_{i} (i = 1, 2, 3) satisfies
3.1. DCLink Voltage Controller
Define the DClink voltage error as . Then, the error dynamics of the DClink voltage can be obtained from equation (4) as follows:
A fractionalorder sliding manifold for the DClink voltage error dynamic (7) is chosen as follows:where 0 < α < 1 is the fractional order, and λ_{DC} > 0 is a constant.
Theorem 1. Along the fractionalorder sliding manifold (8), the DClink voltage error dynamic (7) can be guaranteed to converge to its equilibrium point, if a fractionalorder integral virtual control law with its robust adaptive switching gain is designed as follows:where k_{DC} > 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 fractionalorder sliding manifold (8) yieldsConsidering the fractionalorder integral virtual control laws (9a) and (9b) givesConsider a Lyapunov function. Differentiating V_{DC} givesSubstituting the fractionalorder integral virtual control law (9c) with its robust adaptive switching gain (9d) into the above yieldswhich indicates that the DClink voltage error dynamic (7) will reach to the equilibrium point. On s_{DC} = 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 s_{DC} = 0, the DClink voltage error dynamic (7) will converge to its equilibrium points along the fractionalorder sliding manifold. This completes the proof. And the block diagram of the control algorithm of DClink 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 fractionalorder sliding manifold for current error dynamic (17) is chosen as follows:where 0 < α < 1 is the fractional order, and λ_{G2} = diag (λ_{G2α}, λ_{G2β}), λ_{G2α}, λ_{G2β} > 0 are all constants.
Theorem 2. Along the fractionalorder sliding manifold (18), the current error dynamic (17) can be guaranteed to converge to its equilibrium point, if a fractionalorder integral virtual control law with its robust adaptive switching gain is designed as follows:where k_{C2n} > 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 fractionalorder sliding manifold (18) yieldsConsidering the fractionalorder integral virtual control laws (19a) and (19b) givesDifferentiating the above with respect to time t givesConsider a Lyapunov function . Differentiating V_{G2} givesSubstituting the fractionalorder 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 s_{G2} = 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 s_{G2} = 0, the current error dynamic (26) will converge to its equilibrium points along the fractionalorder sliding manifold. This completes the proof.
3.3. Filter Capacitor Voltage Controller
The switching control signal S_{2} 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 fractionalorder sliding manifold for filter capacitor voltage error dynamic (17) is chosen aswhere 0 < α < 1 is the fractional order, and λ_{C2} = diag (λ_{C2α}, λ_{C2β}), λ_{C2α}, λ_{C2β} > 0 are all constants.
Theorem 3. Along the fractionalorder sliding manifold (27), the LCL capacitor voltage error dynamics (26) can be guaranteed to converge to its equilibrium point, if a fractionalorder 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 fractionalorder sliding manifold (27) yieldsConsidering the fractionalorder integral virtual control laws (28a) and (28b) givesDifferentiating the above with respect to time t givesConsider Lyapunov function . According to the fractionalorder 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 s_{C2} = 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 s_{C2} = 0, the current error dynamics (25) will converge to its equilibrium points along the fractionalorder 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, k_{d} = 100, η = 100, C = diag (80, 80), μ = diag (3/5, 3/5), and T = diag (100, 100).

The simulations of PI and FOSMC are shown in Figure 6. It can be seen that the DClink 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 DClink voltage and reduce the peak value and settling time of DClink 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.
(a)
(b)
(c)
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.
(a)
(b)
(c)
Figures 8–10 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.
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(b)
(c)
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(b)
(c)
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(c)
5. Conclusions
In this study, a novel fractionalorder sliding mode control strategy with robust switching gain is proposed to achieve reliable and high quality of windpowered microgrid systems. Fractionalorder sliding mode controllers generate continuous control signals and regulate the outer DClink voltage loop and current loop in GSI. Considering the parameters variation and external disturbance, the outer DClink voltage and inner current can track their references.
Appendix
A. Basics of Fractional Calculus
The theory of fractionalorder 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 fractionalorder differentiation used in literature and practical applications.
Definition 1. (see [29]). The α^{th}order GL fractional derivative for a function f(t): R^{+} ⟶ R is defined aswhere α ∈ R is the fractional order, t_{0} and t are the limits of the operation, [⋅] denotes the integer value function, and h is the time increment, andwhere Γ (⋅) is the wellknown Euler gamma function.
Definition 2. (see [29]). The α^{th}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 α^{th}order Caputo fractional derivative for a continuous function f(t): R^{+} ⟶ R is defined aswhere u is defined in (2). Meanwhile, the α^{th}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 MatlabSimulink 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).
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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.