Fractional-Order Modeling and Control of HBCS-MG in Off-Grid State

Abstract

Half-bridge converter series microgrid (HBCS-MG) is susceptible to a variety of uncertainties and disturbances during operation, and therefore, the use of the traditional integer-order models cannot accurately reflect the effects of environmental variations on internal components of the off-grid system, such as converters, filters, and loads, including factors like time delays, memory effects, and multi-scale coupling. The fractional-order control method is better equipped to deal with these disturbances, thereby enhancing the robustness and stability of the system. In the off-grid state, a fractional-order PI (FOPI) controller is employed for double-closed-loop control, and the load voltage feedforward control is utilized to offset the impact of load voltage fluctuations on the system. A new simplified equivalent circuit calculation method for the fractional-order inductor is proposed, and a complete fractional mathematical model of the system in the dq rotating coordinate system is established to obtain the transfer function between the load voltage and the input voltage. Furthermore, the impact of the fractional-order variation of the FOPI controllers and the fractional elements on system performance in the frequency domain and time domain is described in detail. The simulation results are compared with the theoretical analysis to demonstrate the accuracy of the mathematical model. The overshoot of the load voltage at the switching instant of 0.7 s is reduced by 4.2% compared with the integer-order PI controller, which proves that the fractional-order controller can improve the system control accuracy.

1. Introduction

The modernization of the global economy has benefited from the extensive utilization of fossil fuels, but the surge in energy consumption has led to an intensifying primary energy crisis. In contrast, renewable energy sources, including wind and solar, are abundant, and can satisfy the requirements of sustainable development of human society. Consequently, the accelerated advancement of renewable energy is imminent [1,2,3].

The majority of clean energy sources are distributed and are subject to volatility and uncertainty. As an emerging method of energy distribution and management, microgrids can flexibly integrate and optimize various energy sources. In light of the global objective of carbon neutrality, microgrids are expected to become a pivotal component of the prospective energy internet [4,5].

As a novel microgrid structure, the half-bridge converter series microgrid (HBCS-MG) addresses the inherent issues associated with the traditional parallel microgrid, including the presence of high harmonic content and the difficulty in suppressing circulating current. In comparison to the other two series-structured microgrid configurations, namely the microgrid with series micro-source inverters (SMSI-MG) and the modular multilevel converter half-bridge series microgrid (MMC-MG), the HBCS-MG has the advantage of enhancing accessibility to clean energy while reducing the number of power switching devices [6,7].

In recent years, the number of types of loads in microgrids has increased, including inductive loads, capacitive loads, and others. These loads are affected by the material, environment and other factors, which give them fractional-order characteristics [8]. Concurrently, the filters in microgrids need to have high-frequency resolution and selectivity, and the incorporation of fractional-order filters can extend the performance of traditional filters and facilitate the processing of complex signals more effectively [9,10]. In addition, the DC-DC converters, PWM rectifiers and inverters that are commonly used in microgrids all have fractional-order characteristics. These components with fractional-order characteristics are referred to as fractional elements. Conventional microgrid models based on integer-order inductance and capacitance suffer from modeling errors in describing strong nonlinearities, time lags, and memory effects and cannot accurately reflect their true physical properties. Fractional-order calculus, with its memory, heredity, and cross-scale modeling capabilities, is able to capture the dynamic properties of the system at multiple time scales, which can more accurately describe the history-dependent and long-time-distance behavior of nonlinear dynamic systems and avoid the deficiencies of traditional integer-order models in multi-scale modeling. Thus, the establishment of fractional-order microgrid models helps to describe their characteristics more accurately.

In traditional PID controllers, the order of both the differential and integral components is one. Fractional-order PID controllers expand the control range compared to integer-order PID control, and the microgrid control can be more flexible and accurate by adjusting the fractional-order of the controller [11,12]. Compared with other nonlinear controllers such as sigmoid PID control, neuroendocrine PID control, and the BELBIC PID controller, etc., all the above controllers have higher degrees of freedom and anti-interference ability and can adapt well to nonlinear systems. However, the FOPID controller has more adjustable parameters, a simple and clear structure, and is easier to implement for complex systems that can be described by an accurate mathematical model. If it is combined with a neural network, it will have stronger adaptive properties. For example, the FOPID tuning tool for automatic voltage regulator using the marine predators algorithm, the FOPID controller for blood pressure regulation using genetic algorithm, and the fractional-order sliding mode control of a 4D memristive chaotic system have been studied, yielding a certain amount of research [13,14,15].

At present, the FOPID controller has been used in several fields of renewable energy (especially wind energy). For a wind energy conversion system (WECS) based on a hybrid excitation synchronous generator (HESG), the literature [16,17] examines the Robust Fractional-Order Control (CRONE controller), the H∞ regulator, and the FOPI controller. Their performance was analyzed, and the FOPI controller was proven to be superior to the other two. However, research on the application of FOPID controllers in microgrid control is still in its infancy, especially for the HBCS-MG system. Most existing research focused on the grid-connected mode, with all of them being controlled by integer-order PI controllers under the assumption of integer-order circuit models. Therefore, it is of great importance to study the fractional-order characteristics of each fractional element in the HBCS-MG and apply fractional-order based control to it.

The majority of existing studies on fractional-order converters have concentrated on DC-DC converters. The main modeling techniques used include the state-space averaging method, circuit averaging method, small-parameter equivalent method, and small-signal equivalent method [18]. Based on the aforementioned modeling methods, scholars have constructed a fractional-order model of the converter in the continuous [19], intermittent [20], and pseudo-continuous [21] models of the inductor current.

Research on fractional-order PWM rectifiers and inverters is still in its infancy. The single-phase fractional-order voltage source PWM rectifier (FOVSR) is modeled based on the Caputo definition, which reveals the effect of fractional-order inductance and fractional-order capacitance on the single-phase FOVSR [22]. In the case of three-phase PWM rectifiers, reference [23] combines the actual fractional-order characteristics of inductors and capacitors to model and analyze the three-phase voltage-source PWM rectifiers.

For fractional-order inverters, a circulating current suppression strategy, a submodule capacitor voltage suppression strategy, and a power feedforward double-closed-loop control strategy based on the fractional-order PID controller in the MMC converter were developed. However, the fractional characteristics of the components have not been incorporated into this model [24]. In view of the resonance problem of integer-order LCL filters in grid-connected inverter applications, colleagues have studied the control of fractional-order LCL filters and their single-phase grid-connected inverters. Their findings demonstrate that the resonance peaks can be effectively avoided by reasonable selection of the orders of inductors and capacitors [25]. In recent years, scholars have derived the complete fractional-order mathematical model of a single-phase fractional-order LCL grid-connected inverter. Their findings indicate that the fractional-order LCL filter can basically prevent resonance and that the addition of a grid-voltage fractional-order feedforward control strategy can reduce the grid-connected current harmonics [26]. On this basis, the fractional-order mathematical model of a three-phase grid-connected inverter in dq rotating coordinates was established, and a three-phase double-closed-loop decoupled fractional-order PI control in the rotating coordinate system was proposed [27].

All of the above studies have demonstrated the superiority of fractional-order modeling and fractional-order PID control of inverters. However, the existing studies on inverters have concentrated on the fractional-order modeling and control of grid-connected inverters, whereas the modeling and control of off-grid microgrids, with consideration of the fractional-order characteristics of the load, have not yet been addressed. In fact, off-grid conditions, environmental variations, high-frequency disturbances, and control strategies affect the fractional-order characteristics of nonlinear loads and filters within the microgrid system, resulting in load voltage fluctuations and power quality degradation. Additionally, the fractional-order controller can improve the control accuracy of the fractance. Therefore, it is necessary to study the effects of fractional-order variation of filters and nonlinear loads on the system under off-grid conditions.

In this paper, a simplified equivalent circuit calculation method for fractional-order inductors is proposed. Under the structure of the HBCS-MG off-grid system, the fractional-order characteristics of the LC filter and the resistive-inductive load are considered concurrently to establish the state-space model of the system in dq coordinates. A double FOPI decoupling structure is established, with the capacitor current as the inner-loop and the load voltage as the outer-loop, so as to obtain the complete fractional-order transfer function between the load voltage and the power supply of the system in the off-grid state. The introduction of the load fractional-order increases the complexity of the system transfer function and enhances the difficulty of response characterization. In conjunction with the stability margin and step response characteristics of the system and the simulation results, the effects of the fractional-order change of the fractional element and controller on the stability, dynamic response, and output characteristics of the system are analyzed. The findings of this study can inform the optimization of controller and filter parameters in the HBCS-MG off-grid operation mode, thereby improving the precision of the system control.

The major innovations of the article are presented below.

• A simplified equivalent circuit for a fractional-order inductor is proposed, which greatly reduces the computational complexity of its parameters.
• The existing studies on inverters are all concentrated on the fractional-order modeling and control of grid-connected inverters, this paper considered fractional-order characteristics of the resistive-inductive load in off-grid conditions.
• Based on the double fractional-order PI controller, the transfer function between the inverter port voltage and the load voltage is derived so as to establish the complete fractional state-space mathematical model of the system in dq coordinates.
• The impact of the fractional-order variation of the fractional-order PI controllers and the fractional elements on system performance in the frequency domain and time domain is described in detail, laying the foundation for optimizing system control.

2. HBCS-MG System Structure and Fractional Mathematical Model

2.1. HBCS-MG System Structure

The schematic diagram of the HBCS-MG in the off-grid mode is shown in Figure 1. Each phase contains K generating units, and each generating unit is connected in series via a micro-source half-bridge converter (MS-HBC).

In Figure 1, points A, B, and C are the upper output ports of the first MS-HBC of the three phases, point O is the common connection point of the lower output port of the last MS-HBC of the three phases, and point N is the star-connection point of the load equivalent circuit. Let X = A, B, and C represent the phase sequence of three phases. Assuming the symmetry of the three-phase filter, L_fα1 is the inductance of the filter inductor with the fractional-order of α₁; C_fβ is the capacitance of the filter capacitor with the fractional-order of β; L_XZα2 and R_X are the inductive and resistive components in each phase load, respectively, with the fractional-order of α₂. In off-grid mode, although the single-phase output voltage of the system is DC, the system is able to supply AC power to the load if it is connected in three-phase Y or ∆, due to the potential difference between points O and N. The choice of line voltage to supply the load has a higher voltage utilization, but for the purposes of simplified calculation, the circuit is equated to a three-phase Y connection.

The overall control block diagram of the system is shown in Figure 2. First, the given value of the load voltage in the dq coordinate system u_Cfd ** and u_Cfq ** is obtained through virtual synchronous generator (VSG) control. Then, a double-closed-loop FOPI control is employed, with the inner-loop for filter capacitor current control and the outer-loop for load voltage control. Finally, the modulation signals are distributed to each MS-HBC in accordance with the power demand to realize the system control.

2.2. Equivalent Circuit of the Fractional Element

The impedance of the fractional-order capacitor and inductor in the s-domain, based on Caputo’s definition, can be expressed as ^CZ_Cβ(s) and ^CZ_Lα(s), respectively [28], where

$${}^{\mathrm{C}}Z_{C\beta }\left(s\right)={\displaystyle \frac{U_{C\beta }\left(s\right)}{I_{C\beta }\left(s\right)}}={\displaystyle \frac{1}{s^{\beta }{C}_{\beta }}}$$

(1)

$${}^{\mathrm{C}}Z_{L\alpha }\left(s\right)={\displaystyle \frac{U_{L\alpha }\left(s\right)}{I_{L\alpha }\left(s\right)}}=s^{\alpha }{L}_{\alpha }$$

(2)

The Oustaloup algorithm can be approximated by a set of filters for the fractional-order operator s^γ. Specifically, in the amplitude-frequency domain, a given frequency band is divided equally into d parts, and a first-order filter with poles and zeros in each small frequency interval is used to approximate s^γ. The formula is as follows:

$$s^{\gamma }\approx H{\displaystyle \prod _{i=1}^{\mathrm{d}}\frac{s-{\omega }_i^{\prime }}{s-{\omega }_i}}$$

(3)

where ${\omega }_i^{\prime }$ and ${\omega }_i$ are the zeros and poles of the integer-order filter, respectively.

A distributed expansion of Equation (3) is given as Equation (4), in which the residue, poles, and individual terms in the distributed expansion equation of s^γ are represented by r_i, p_i, and g, respectively.

$$s^{\gamma }={\displaystyle \frac{r_1}{s+p_1}}+{\displaystyle \frac{r_2}{s+p_2}}+\dots +{\displaystyle \frac{r_{\mathrm{d}}}{s+p_{\mathrm{d}}}}+g$$

(4)

Let the fractional-order capacitor be C_β, and its equivalent circuit is the same as that in the literature [25], as shown for C_β in Figure 3a. The fractional-order inductor is denoted by L_α due to the complexity of the equivalent circuit of the fractional-order inductor in the existing literature. A new chain equivalent circuit of the fractional-order inductor is proposed in this paper, as shown in Figure 3b.

The impedance of the chain equivalent circuit of the fractional-order capacitor C_β and inductor L_α is derived from Figure 3:

$$Z_{C\beta }={\displaystyle \frac{1}{s^{\beta }{C}_{\beta }}}={\displaystyle \sum _{i=1}^{\mathrm{d}}\frac{1/C_i}{s+1/R_i{C}_i}}+R_0$$

(5)

$$Z_{L\alpha }=L_{\alpha }{s}^{\alpha }={\displaystyle \sum _{i=1}^{\mathrm{d}}\frac{R_{i}s}{R_i/L_i+s}}+L_{0}s$$

(6)

For the fractional-order capacitor, the distributed expansion of s^−β is first calculated, then s^−β is divided by C_β and compared with Equation (5) to obtain the size of each group of components in the chain equivalent circuit of the fractional-order capacitor, as C_i = C_β/r_i, R_i = 1/C_ip_i, and R₀ = g/C_β.

For the fractional-order inductor, a comparison of Equations (4) and (6) shows that the numerators differ by s. Therefore, the distributed expansion of s^α−1 is first obtained and then multiplied by L_αs, which is the distributed expansion of Z_Lα. Let the residue, poles, and individual terms in the distributed expansion equation of s^α−1 be t_i, q_i, and f, respectively. Multiplying s^α−1 by L_αs and comparing with Z_Lα gives the sizes of each group of integer-order elements in the chain equivalent circuit of fractional-order inductors R_i = L_αt_i, L_i = R_i/q_i, and L₀ = L_αf. This approach provides a more straightforward and computationally efficient alternative to the equivalent circuit calculation method for fractional-order inductors, as discussed in the literature [27,28].

The capacitor is selected with a capacitance of 20 μF and an order of β = 0.8. The inductor is selected with an inductance of 5 mH and an order of α = 0.9. The filter order is d = 5. The parameters of each component of the integer-order in the equivalent circuit are provided in

Table 1. Parameters of fractional-order capacitor and inductor in the equivalent circuit.

	d	1	2	3	4	5	R0/Ω	L0/mH
${}^{\mathrm{C}}Z_{C\beta }\left(s\right)$	Ri/Ω	47.7	931.4	1.8 × 104	3.4 × 105	1.2 × 107	5	/
	Ci/μF	5.8	1.2	24.6	50.8	56.7
${}^{\mathrm{C}}Z_{L\alpha }\left(s\right)$	Ri/Ω	1.26	0.07	3.9 × 10−3	2.1 × 10−4	1.2 × 10−5	/	1.99
	Li/mH	0.75	1.04	1.44	1.98	2.78

.

Figure 4 shows the numerical impedance and equivalent circuit impedance Bode plots of the aforementioned fractional element, according to the Caputo definition. A comparison of the Bode plot of the above fractional element reveals that the amplitude-frequency characteristics of the two are essentially identical within the 10⁻²–10⁴ frequency band. Consequently, the equivalent circuit of the fractional element used in this paper is considered sufficient to characterize its properties.

2.3. Fractional-Order Equivalent Circuit of the System

Let the filtered inductor current, capacitor current, load current, and load voltage in the three-phase system be i_XL = [i_AL i_BL i_CL]^T, i_XCf = [i_ACf i_BCf i_CCf]^T, i_XZ = [i_AZ i_BZ i_CZ]^T, and u_XCf = [u_ACf u_BCf u_CCf]^T, respectively. u_XN = [u_AN u_BN u_CN]^T is the potential difference between points X and N. The single-phase equivalent circuit of the system is shown in Figure 5. The effective value of the system line voltage is 220 V when the peak value of u_XCf is 180 V.

When the DC-side voltage of each MS-HBC is equal to u_dci, the expression for the output voltage of each phase u_XO, as shown in Figure 1, is given by Equation (1) [29]. Where M_i is the modulation of the ith MS-HBC, ω_c is the carrier frequency, ω_s is the modulation waveform frequency, θ_x is the phase shift angle, m is the factor of harmonics of the carrier, and n is the factor of harmonics of the modulation waveform.

$$\begin{array}{cc}{u}_{X\mathrm{o}}=\hfill & {\displaystyle \sum _{i=1}^K{u}_{Xi}}={\displaystyle \frac{K{u}_{\mathrm{dc}i}}{2}}+{\displaystyle \sum _{i=1}^K\frac{M_i{u}_{\mathrm{dc}i}}{2}\sin \left({\omega }_{s}t+{\theta }_x\right)}\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^K{\displaystyle \sum _{m=1,3,5,\dots }^{\infty }{\displaystyle \sum _{n=-\infty }^{\infty }\frac{2u_{\mathrm{dc}i}}{m\pi }{J}_n\left(\frac{m{M}_i\pi }{2}\right)\sin \left(\frac{m+n}{2}\pi \right)·}}} \cos \left[m{\omega }_{c}t+\left(i-1\right){\displaystyle \frac{2\pi }{N}}+n\left({\omega }_{s}t+{\theta }_x\right)\right]\hfill \end{array}$$

(7)

From the principle of the three-phase star circuit, u_XN can be obtained as follows in Equation (7).

$$u_{\mathrm{XN}}=\left[\begin{array}{c}{u}_{\mathrm{AN}}\\ u_{\mathrm{BN}}\\ u_{\mathrm{CN}}\end{array}\right]=\left[\begin{array}{ccc}{\displaystyle \frac{2}{3}}& -{\displaystyle \frac{1}{3}}& -{\displaystyle \frac{1}{3}}\\ -{\displaystyle \frac{1}{3}}& {\displaystyle \frac{2}{3}}& -{\displaystyle \frac{1}{3}}\\ -{\displaystyle \frac{1}{3}}& -{\displaystyle \frac{1}{3}}& {\displaystyle \frac{2}{3}}\end{array}\right]\left[\begin{array}{c}{u}_{\mathrm{Ao}}\\ u_{\mathrm{Bo}}\\ u_{\mathrm{Co}}\end{array}\right]$$

(8)

Neglecting the effect of harmonics, the calculation gives $u_{X\mathrm{N}}={\displaystyle \sum _{i=1}^N\frac{M_i}{2}{u}_{\mathrm{dc}i}\sin \left({\omega }_{s}t+{\theta }_x\right)}$.

From Figure 5, the state-space equations of the system under the three-phase stationary coordinate system are shown in Equation (9):

$$\left\{\begin{array}{l}{L}_{f{\alpha }_1}{s}^{{\alpha }_1}{i}_{XL}\left(s\right)=u_{X\mathrm{N}}\left(s\right)-u_{XCf}\left(s\right)\\ C_{f\beta }{s}^{\beta }{u}_{XCf}\left(s\right)=i_{XL}\left(s\right)-i_{XZ}\left(s\right)\\ L_{XZ{\alpha }_2}{s}^{{\alpha }_2}{i}_{XZ}\left(s\right)=u_{XCf}\left(s\right)-u_{RX}\left(s\right)\end{array}\right.$$

(9)

Taking phase A as an example, define G as the transfer function between u_ACf and u_AN, then G is obtained from Equation (9):

$$G={\displaystyle \frac{R_{\mathrm{A}}+L_{\mathrm{A}Z{\alpha }_2}{s}^{{\alpha }_2}}{L_{f{\alpha }_1}{L}_{\mathrm{A}Z{\alpha }_2}{C}_{f\beta }{s}^{{\alpha }_1+{\alpha }_2+\beta }+L_{f{\alpha }_1}{s}^{{\alpha }_1}+L_{\mathrm{A}Z{\alpha }_2}{s}^{{\alpha }_2}+R_{\mathrm{A}}{L}_{f{\alpha }_1}{C}_{f\beta }{s}^{{\alpha }_1+\beta }+R_{\mathrm{A}}}}$$

(10)

2.4. Fractional-Order Mathematical Model of the System in Rotating Coordinates

When the system is in the three-phase symmetrical case, the abc/dq transform matrix of the sine function is shown in Equation (11).

$$P={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}\sin {\omega }_{s}t& \sin \left({\omega }_{s}t-{\displaystyle \frac{2}{3}}\pi \right)& \sin \left({\omega }_{s}t+{\displaystyle \frac{2}{3}}\pi \right)\\ \cos {\omega }_{s}t& \cos \left({\omega }_{s}t-{\displaystyle \frac{2}{3}}\pi \right)& \cos \left({\omega }_{s}t+{\displaystyle \frac{2}{3}}\pi \right)\end{array}\right]$$

(11)

The formula for the fractional derivative of a trigonometric function, according to Caputo’s definition, is as Equation (12), where $D_t^{\gamma }f(t)$ is the γth derivative [24]:

$$D_t^{\gamma }\sin \left({\omega }_{s}t\right)=a^{\gamma }\sin \left({\omega }_{s}t+\gamma \pi /2\right)$$

(12)

$$D_t^{\gamma }\cos \left({\omega }_{s}t\right)=a^{\gamma }\cos \left({\omega }_{s}t+\gamma \pi /2\right)$$

(13)

Equations (12) and (13) are used to derive the fractional-order derivative of the trigonometric function in the inverse matrix of (11), which is then multiplied by P, yielding (14):

$$P{\displaystyle \frac{{\mathrm{d}}^{\gamma }\left(P^{-}\right)}{\mathrm{d}{t}^{\gamma }}}=\left[\begin{array}{cc}{\omega }_s{}^{\alpha }\cos {\displaystyle \frac{\gamma \pi }{2}}& -{\omega }_s{}^{\alpha }\sin {\displaystyle \frac{\gamma \pi }{2}}\\ {\omega }_s{}^{\alpha }\sin {\displaystyle \frac{\gamma \pi }{2}}& {\omega }_s{}^{\alpha }\cos {\displaystyle \frac{\gamma \pi }{2}}\end{array}\right]$$

(14)

If α = 1, then $P{\displaystyle \frac{{\mathrm{d}}^{\alpha }\left(P^{-}\right)}{\mathrm{d}{t}^{\alpha }}}=\left[\begin{array}{ll}0& -\omega \\ {\omega }^{\alpha }& 0\end{array}\right]$ is consistent with integer-order.

Let i_Ld/q, u_XNd/q, u_Cfd/q, i_Zd/q, u_Zd/q, and i_Cfd/q be the representations of i_XL, i_XN, u_XCf, i_XZ, u_RX, and i_XCf in dq coordinates, respectively. Combined with Equations (9) and (14), the state-space model of the HBCS-MG off-grid system in dq coordinates is shown in (15).

$$\left\{\begin{array}{l}{\displaystyle \frac{i_{L\mathrm{d}/\mathrm{q}}\left(s\right)}{G_{l1}}}=u_{X\mathrm{Nd}/\mathrm{q}}\left(s\right)-u_{Cf\mathrm{d}/\mathrm{q}}\left(s\right)\pm L_{f{\alpha }_1}{\omega }^{{\alpha }_1}\sin {\displaystyle \frac{{\alpha }_1\pi }{2}}{i}_{L\mathrm{q}/\mathrm{d}}\left(s\right)\\ {\displaystyle \frac{u_{Cf\mathrm{d}/\mathrm{q}}\left(s\right)}{G_c}}=i_{L\mathrm{d}/\mathrm{q}}\left(s\right)-i_{Z\mathrm{d}/\mathrm{q}}\left(s\right)\pm C_{f\beta }{\omega }^{\beta }\sin {\displaystyle \frac{\beta \pi }{2}}{u}_{Cf\mathrm{q}/\mathrm{d}}\left(s\right)\\ {\displaystyle \frac{i_{Z\mathrm{d}/\mathrm{q}}\left(s\right)}{G_{l2}}}=u_{Cf\mathrm{d}/\mathrm{q}}\left(s\right)-u_{R\mathrm{d}/\mathrm{q}}\left(s\right)\pm L_{Z{\alpha }_2}{\omega }^{{\alpha }_2}\sin {\displaystyle \frac{{\alpha }_2\pi }{2}}{i}_{\mathrm{Zq}/\mathrm{d}}\left(s\right)\end{array}\right.$$

(15)

G_l1, G_c, and G_l2 are as follows:

$$G_{l1}=1/\left(L_{f{\alpha }_1}{s}^{{\alpha }_1}+L_{f{\alpha }_1}{\omega }^{{\alpha }_1}\cos {\displaystyle \frac{{\alpha }_1\pi }{2}}\right)$$

(16)

$$G_c=1/\left(C_{f\beta }{s}^{\beta }+C_{f\beta }{\omega }^{\beta }\cos {\displaystyle \frac{\beta \pi }{2}}\right)$$

(17)

$$G_{l2}=1/\left(L_{Z{\alpha }_2}{s}^{{\alpha }_2}+L_{Z{\alpha }_2}{\omega }^{{\alpha }_2}\cos {\displaystyle \frac{{\alpha }_2\pi }{2}}\right)$$

(18)

3. Fractional-Order Double-Closed-Loop Control of the System

3.1. Fractional-Order Decoupling Control

As can be observed from the analysis of Equation (15), the system exhibits a nonlinear coupling phenomenon. In this paper, the load voltage feedforward control is employed to compensate for the effect of voltage fluctuations on the control of the system. The system control block diagram is shown in Figure 6, where G_rk (k = 1, 2 representing the inner- and outer-loops of the control system, respectively) is the transfer function of the FOPI controller, α_rk is the order of the integral term in the transfer function of the fractional-order PI controller, G_p is the SPWM gain, and u_Cfd ** and u_Cfq ** are the given values of the dq-axis output load voltage, obtained by VSG control.

The transfer function of the FOPI controller is G_rk, as shown in (19).

$$G_{rk}=K_{pk}+{\displaystyle \frac{K_{ik}}{s^{{\alpha }_{rk}}}}$$

(19)

The reference value of the dq-axis load voltage is as follows:

$${u_{Cf\mathrm{d}}}^{*}=\left({i_{Cf\mathrm{d}}}^{*}-i_{Cf\mathrm{d}}\right)G_{r1}{G}_p$$

(20)

$${u_{Cf\mathrm{q}}}^{*}=\left({i_{Cf\mathrm{q}}}^{*}-i_{Cf\mathrm{q}}\right)G_{r1}{G}_p$$

(21)

According to the left side of Equation (15), two new variables, M_d and M_q, are introduced:

$$M_{\mathrm{d}}=i_{L\mathrm{d}}\left(s\right)/G_{l1}$$

(22)

$$M_{\mathrm{q}}=i_{L\mathrm{q}}\left(s\right)/G_{l1}$$

(23)

The current inner-loop decoupling control is achieved when the equations ${u_{Cf\mathrm{d}}}^{*}=M_{\mathrm{d}}$ and ${u_{Cf\mathrm{q}}}^{*}=M_{\mathrm{q}}$ are satisfied. Similarly, the voltage outer-loop dq-axis decoupling is realized when Equation (24) is satisfied.

$${i_{Cf\mathrm{d}/\mathrm{q}}}^{*}={\displaystyle \frac{1}{G_c}}{u}_{Cf\mathrm{d}/\mathrm{q}}\left(s\right)=\left({u_{Cf\mathrm{d}/\mathrm{q}}}^{**}-u_{Cf\mathrm{d}/\mathrm{q}}\right)G_{r2}$$

(24)

Using the d-axis as an example, the control block diagram of the double-closed-loop control system after decoupling is shown in Figure 7.

3.2. Transfer Function of the Fractional-Order Double-Closed-Loop Control System

The transfer function of the system’s inner-loop current in dq coordinates is obtained from Figure 7:

$$G_{ib}={\displaystyle \frac{i_{Cf\mathrm{d}}}{{i_{Cfd}}^{*}}}={\displaystyle \frac{G_{r1}{G}_p{G}_{l1}}{s^{\beta }{C}_{f\beta }+G_{l1}+1/\left(R+s^{{\alpha }_2}{L}_{Z{\alpha }_2}\right)+G_{r1}{G}_p{G}_{l1}{C}_{f\beta }{s}^{\beta }}}$$

(25)

The transfer function of the system with the inner-loop closed and the outer-loop open is given by Equation (26).

$$\begin{array}{ll}{G}_{k1}& ={\displaystyle \frac{u_{Cf\mathrm{d}}}{{u_{Cf\mathrm{d}}}^{**}}}={\displaystyle \frac{G_{r1}{G}_{r2}{G}_p{G}_{l1}\left(R+s^{{\alpha }_2}{L}_{Z{\alpha }_2}\right)}{s^{\beta }{C}_{f\beta }\left(R+s^{{\alpha }_2}{L}_{Z{\alpha }_2}\right)+1+G_{l1}\left(R+s^{{\alpha }_2}{L}_{Z{\alpha }_2}\right)+G_{r1}{G}_p{G}_{l1}{s}^{\beta }{C}_{f\beta }\left(R+s^{{\alpha }_2}{L}_{Z{\alpha }_2}\right)}}\\ & =G_p{\displaystyle \frac{b_1{s}^{n{b}_1}+b_2{s}^{n{b}_2}+\dots +b_7{s}^{n{b}_7}+b_8}{a_1{s}^{n{a}_1}+a_2{s}^{n{a}_2}+\dots +a_8{s}^{n{a}_8}+a_9}}\end{array}$$

(26)

The transfer function of the voltage and current double-closed-loop control system is given by Equation (27).

$$\begin{array}{ll}{G}_{ub}& ={\displaystyle \frac{G_{r2}{G}_{r1}{G}_p{G}_{l1}}{s^{\beta }{C}_{f\beta }\left(1+G_{r1}{G}_p{G}_{l1}\right)+\frac{1}{R+s^{{\alpha }_2}{L}_{Z{\alpha }_2}}+G_{l1}+G_{r2}{G}_{r1}{G}_p{G}_{l1}}}\\ & =G_p{\displaystyle \frac{b_1{s}^{n{b}_1}+b_2{s}^{n{b}_2}+\dots +b_7{s}^{n{b}_7}+b_8}{a_1{s}^{n{a}_1}+a_2{s}^{n{a}_2}+\dots +a_8{s}^{n{a}_8}+G_p\left(c_1{s}^{n{c}_1}+c_2{s}^{n{c}_2}+\dots +c_7{s}^{n{c}_7}\right)+c_8}}\end{array}$$

(27)

Note: a_i, na_i, b_i, nb_i, c_i, and nc_i are listed in the Appendix A.

Equation (27) and Appendix A show that the full fractional-order mathematical model constructed has 15 orders, and 12 control parameters need to be adjusted, taking into account the impedance values of the three fractional reactance elements and the fractional-order, which is of some complexity. Therefore, it is of practical significance to accurately model and analyze the operating characteristics of the off-grid microgrid under different parameters, which can lay a foundation for the subsequent optimization of the controller parameters.

4. Effect of the Fractional-Order Controller on System Performance

To verify the accuracy of the fractional-order mathematical model of the HBCS-MG system in the off-grid state and analyze the effects of variations in α_rk on performance, this section presents both the characterization of the system transfer function and the electrical characteristics obtained from the simulation.

The power component of the simulation model is shown in Figure 1. The equivalent circuit of the fractional element based on the Oustaloup filter is used to construct the filter inductor and the inductive component of the load. The simulation time is 1 s, and the load change is divided into three steps, with the impedance angle of Z₃ being larger than Z₂. The simulation parameters are shown in

Table 2. Parameters of simulation circuit.

E	170 V	Time/s Load	0–0.4 Z1	0.4–0.7 Z2	0.7–1 Z3
Lf	5 mH	RX/Ω	4	2.1	1.4
Cf	20 μF	LXZ/mH	0	1.4	1.3

and

Table 3. The coefficients of fractional-order controller.

	Proportionality Coefficient	Integral Coefficient
Inner-loop PI controller	kp1 = 3	ki1 = 500
Outer-loop PI controller	kp2 = 2	ki2 = 500

.

4.1. Amplitude-Frequency Characteristics of α_rk Variation with the Outer-Loop Opened

The filter parameters are presented in Table 1. Under load Z₂, from Equation (26), the frequency domain characteristics of the system with α_r1 and α_r2 varied within the frequency range of 10⁻¹–10⁷ rad/s are shown in

Table 4. Outer-loop open system amplitude margin.

αr2 αr1	0.3	0.5	0.8	1	1.2	1.5	1.8
0.3	−16.3	−1.0	Inf	Inf	Inf	Inf	−34.7
0.5	−1.1	15.1	Inf	Inf	Inf	Inf	−20.1
0.8	Inf	Inf	Inf	Inf	Inf	−18.7	−7.0
1	Inf	Inf	Inf	Inf	Inf	−8.1	0.0
1.2	Inf	Inf	Inf	Inf	−10.3	0.3	6.7
1.5	Inf	Inf	−23.3	−12.6	−3.6	8.4	17.5
1.8	−38.9	−24.5	−11.9	−5.1	1.1	10.7	25.3

Note: Inf in the table indicates infinity.

and

Table 5. Outer-loop open system phase-angle margin.

αr2 αr1	0.3	0.5	0.8	1	1.2	1.5	1.8
0.3	−15.0	−1.5	33.7	36.5	36.8	36.9	36.9
0.5	−1.3	59.2	99.2	112.1	117.7	119.3	119.4
0.8	22.0	107.9	83.8	82.8	92.3	114.7	119.5
1	23.6	115.9	91.9	67.9	53.1	47.0	−0.3
1.2	23.7	118.4	105.7	67.7	32.5	−1.3	−45.9
1.5	23.8	119.0	114.8	92.9	22.5	−38.9	−85.1
1.8	23.8	119.0	116.1	105.1	−15.6	−79.5	−122.1

. When α_r1 = α_r2, the variation in the Bode plots of the transfer function G_k1 is shown in Figure 8.

By examining Figure 8 and Table 4 and Table 5 together, it is evident that in this case, when both α_r1 and α_r2 are small (<0.5) or large (>1.5), the amplitude margin or phase margin of G_k1 becomes negative. Therefore, the system’s stability must be evaluated in conjunction with its dynamic response characteristics. When either α_r1 or α_r2 falls between 0.5 and 1.5, no crossover frequency exists within the 10⁻¹–10⁷ rad/s range, indicating that the system remains stable.

4.2. Characterization of the System Step Response as α_rk Varies

The step response waveform of G_ub is plotted to obtain its peak value, peak value time, and steady-state time when α_rk varies, and the data are presented in

Table 6. The peak value of the step response when α_rk changes.

αr2 αr1	0.5	0.7	0.8	1	1.2	1.5	Trends
0.5	1.03	1.064	1.058	1.023	1.023	1.028	↑↓↑
0.7	1.047	1.111	1.105	1.089	1.075	1.068	↑↓
0.8	1.035	1.095	1.116	1.131	1.125	1.10.7	↑↓
1	1.011	1.055	1.089	1.189	1.245	1.246	↑
1.2	1.013	1.043	1.077	1.196	1.331	1.44	↑
1.5	1.017	1.039	1.061	1.152	1.333	NaN	↑
1.8	1.02	1.04	1.055	1.111	1.25	NaN	↑
trends	↑↓↑	↑↓	↑↓	↑↓	↑	↑

,

Table 7. The peak value time of the step response when α_rk changes.

αr2 αr1	0.5	0.7	0.8	1	1.2	1.5	Trends
0.5	0.0008	0.0014	0.0019	0.0058	0.0183	0.0502	↑
0.7	0.0014	0.0026	0.0037	0.0078	0.0174	0.0453	↑
0.8	0.0019	0.0036	0.0051	0.0104	0.0196	0.045	↑
1	0.0099	0.0078	0.0101	0.0178	0.0178	0.053	↑
1.2	0.0249	0.0203	0.0202	0.0279	0.0279	0.0713	↑
1.5	0.064	0.0543	0.0507	0.0504	0.0504	2.9071	↓↑
1.8	0.1287	0.113	0.106	0.0944	0.0944	2.7238	↓↑
trends	↑	↑	↑	↑	↓↑	↓↑

and

Table 8. The steady-state time of the step response when α_rk changes.

αr2 αr1	0.5	0.7	0.8	1	1.2	1.5	Trends
0.5	0.0014	0.0029	0.004	0.009	0.0248	0.2324	↑
0.7	0.0027	0.0056	0.0084	0.0194	0.1079	0.222	↑
0.8	0.0034	0.0085	0.0123	0.0463	0.104	0.2176	↑
1	0.002	0.0205	0.0252	0.0773	0.1159	0.3892	↑
1.2	0.0055	0.1528	0.1518	0.1276	0.2744	1.5546	↑
1.5	0.064	0.2865	0.278	0.2615	0.6758	NaN	↑
1.8	0.502	0.4849	0.8499	1.1981	3	NaN	↑
trends	↑	↑	↑	↑	↑	↑

, respectively. In the table, arrow ↑ indicates an upward trend, ↓ indicates an downward trend. NaN indicates inexistence.

As illustrated in Table 6, when either α_r1 or α_r2 is relatively small (α_rk < 0.7), and only one of them is increased, the peak value first increases, then decreases and increases again. However, the second increase is significantly smaller than the first. If either α_r1 or α_r2 is increased to some extent (α_rk ≥ 0.7), an increase in only one of them (change term < 1.5) will cause the peak to first increase and then decrease. If either α_r1 or α_r2 is further increased (α_r1 > 1 or α_r2 > 1.3), the peak will continue to increase with the change term until a point of instability is reached. Additionally, the impact of a simultaneous increase in α_rk on the peak value of the step response is more pronounced than that of an increase in a single term. From Table 7, it can be seen that the peak value time increases as α_r1 or α_r2 increases.

As illustrated in Table 8, the steady-state time increases as α_rk increases. When both α_r1 and α_r2 ≥ 1.4, the step response curve oscillates, and the system is in an unstable state, there is no peak value. In general, the impact of a change in either α_r1 or α_r2 on the peak value of the step response and steady-state time of the system is symmetrically distributed.

Taking αr1 = αr2 = 0.8 as an example under the step response, the overshooting amount is 11.6%, a reduction of 7.3% compared with the integer-order case, the peak time is 0.0051 s, a decrease of 0.013 s compared with the integer-order case, and the steady-state time is shortened by 0.065 s. It is shown that the fractional-order PI control outperforms the integer-order PI control.

4.3. Electrical Characterization of the System as α_rk Changes

The filter parameters are kept constant, and the simulation is carried out with α_r2 = 0.8 and α_r1 varied as an example. The variation of the system’s output voltage is shown in

Table 9. Load voltage as α_r1 increases.

	αr1	0.4	0.6	0.8	1.2	1.4	1.8
up/V	0 s	180.5	180.5	180.3	180.5	179.1	179.4
	0.4 s	185.8	192.7	185.2	185.3	185.5	185.5
	0.7 s	195.1	187.5	187.9	195.4	189.1	188.9
Z1	uw/V THD	180 1.27%	180 0.76%	180 0.74%	180 0.74%	180 0.75%	180 0.76%
Z2	uw/V THD	180 3.45%	180 3.47%	180 3.49%	180 3.49%	180 3.49%	180 3.49%
Z3	uw/V THD	180 3.58%	180 3.59%	180 3.58%	180 3.58%	180 3.61%	180 3.61%

.

Table 10. Load voltage as α_rk increases synchronously.

	αrk	0.3	0.5	0.8	1	1.2	1.5	1.8
up/V	0.4 s	unstable	186	185.2	185.6	186.6	186.6	205.2
	0.7 s	187.8	195	187.9	196.4	190.7	189.4	195.3
Z1	uw/V THD	184.3 126%	180 1.63%	180 0.76%	180 0.73%	180 0.73%	178.9 1.19%	unstable 5.33%
Z2	uw/V THD	180.4 3.57%	180 3.45%	180 3.47%	180 3.5%	180 3.5%	179.9 3.51%	unstable 5.24%
Z3	uw/V THD	180.1 3.86%	180 3.58%	180 3.59%	179.9 3.66%	180 3.65%	179.9 3.69%	unstable 5.73%

illustrates the impact of synchronous variation in α_rk on the load voltage. Where u_p denotes the peak value of the voltage at the instant of load switching, u_w denotes the amplitude of the steady-state voltage after load switching.

Table 9 illustrates that at 0.4 s, there is an increasing and then decreasing trend with increasing α_r1, which coincides with the peak of the step response. In the simulation, it should be noted that the maximum of u_p is observed at α_r1 = 0.6, while the peak of the step response is observed at α_r1 = 0.8. This is due to the inclusion of VSG control in the simulation, which permits a broader range of fractional-order adjustment for the controller. Moreover, modifying only α_r1 has a negligible impact on u_w and its total harmonic distortion (THD). Similarly, the simulation indicates that modifying only α_r2 has the same impact on u_w and its THD as modifying only α_r1. Therefore, this result is not presented again.

As can be seen from Table 10, different from the theoretical calculations where the peak value of the step response increases continuously with the increase of α_rk, the u_p at 0.4 s does not differ significantly at α_rk ∈ [0.5, 1.5], and it reaches its maximum at α_rk = 1.8. This is also due to the addition of the VSG control, which improves the stability of the system. The integer-order PI controller is not in an optimal state with regard to its voltage response under load disturbance. At 0.7 s, the peak value under integer-order PI control is 196.4 V, while when α_r1 = α_r2 = 0.8, the peak value of the load voltage is 187.9 V, a reduction of 4.2%. This indicates that the FOPI controller improves the control accuracy of the system.

When α_rk = 0.3, the THD of u_w under Z₁ load is 126%, which indicates an unstable state. In contrast, the system shows relatively stable behavior under Z₂ and Z₃ loads. When α_rk = 1.8, the stability of the system is significantly impaired, exceeding the regulatory limits of the VSG control and resulting in an unstable state. Additionally, the THD of Z₃ is higher than Z₁ and Z₂. Therefore, a moderate increase in α_rk when the impedance angle declines and a moderate decrease in α_rk when the impedance angle rises in the load will assist in reducing THD in the load voltage.

The waveform of the system load current as α_rk varies is shown in Figure 9, Figure 10 and Figure 11. From these figures, it can be seen that the current oscillation is mainly observed under a resistive load when α_rk = 0.3 and that the load current amplitude exhibits a sustained oscillation when α_rk = 1.8.

Figure 12 shows the variation of load power with α_rk. It can be seen that when α_rk is small, the active power variation under a resistive load is greater than that under a resistive-inductive load. As α_rk increases, the impact on the active power of the load also increases.

In conclusion, α_rk being either too low or too high will result in a deterioration of the system performance. If α_rk is set too low, it will mainly affect the voltage quality of resistive loads. It should be noted that α_rk has a relatively small effect on reactive power. Comparison of the system transfer function characteristics with the electrical characteristics proves the accuracy of the proposed model.

5. Effect of the Fractional-Order of the Fractance Element on System Performance

This section also analyzes the effect of variations α₁, β, and α₂ on the system performance, both in terms of the system transfer function characteristics and the electrical characteristics obtained from the simulation.

5.1. Effect of Changing α₁ on System Performance

5.1.1. Amplitude-Frequency Characteristics of α₁ Variation with the Outer-Loop Opened

When α_r1 = α_r2 = 0.8, β = 0.8, and α₂ = 0.9, the Bode plot of the system’s outer-loop open transfer function G_k1 as α₁ is varied is shown in Figure 13. Let the amplitude margin be defined as h, the phase-angle margin as γ, the crossover frequency as ω_x, and the cut-off frequency as ω_c. The typical data of the Bode plot characteristics are presented in

Table 11. Characteristics under the frequency domain as α₁ vary.

	0.5	0.8	1	1.1	1.2	1.5
h/(dB)	Inf	Inf	Inf	Inf	80	30
γ/(deg)	120	112	95	73	37	−125
ωx/(rad/s)	NaN	NaN	NaN	NaN	2.5 × 105	600/3.8 × 104
ωc/(rad/s)	338	304	299	299	265	57

Note: Inf in the table means infinity, and NaN means not present.

.

Combined with Figure 13 and Table 11, it can be seen that when α₂ and β are held constant and α₁ ≤ 1, with the increase of α₁, the amplitude remains relatively constant in the low-frequency band but gradually decreases in the high-frequency band. This is accompanied by a reduction in the cut-off frequency and phase-angle margin, as well as a phase decreasing and approaching −180°. When α₁ = 1.2, the phase-frequency curve is tangent to −180° in the high-frequency range, and although the stability margin decreases, the system remains stable.

When α₁ reaches a certain threshold (α₁ > 1.2), α₁ variations affect the amplitude-frequency curves in all frequency bands, and the phase-angle margin shifts from positive to negative. A phase traversal of −180° occurs in both the high and low-frequency bands, and the amplitude margin is reduced, resulting in an unstable system state.

5.1.2. Effect of Changing α₁ on the Dynamic Response Characteristics of the System

The step response waveform of the double-closed-loop system as α₁ changes is shown in Figure 14. From the step response waveform of the system as α₁ changes, it can be observed that an increase in α₁ results in a higher peak of the step response and a longer steady-state time.

5.1.3. Electrical Characterization of the System as α₁ Change

As α₁ is varied, the simulation demonstrates the variation of u_w, as shown in

Table 12. Load voltage as α₁ increases.

	α2	0.6	0.8	0.9	1	1.1	1.2
up/V	0.4 s	185.3	185.3	185.2	183.1	192.9	Unstable
	0.7 s	186.6	186.8	187.9	191.1	201
Z1 Steady-state	uw/V	180	180	180	180	180
	THD	24.1%	1.8%	0.76%	0.32%	0.17%
Z2 Steady-state	uw/V	180	180	180	180	180
	THD	111.2%	26.3%	3.47%	1.57%	1.15%
Z3 Steady-state	uw/V	179.4	180	180	180	NaN
	THD	110.8%	25.3%	3.59%	2.07%	112%

. It can be seen that as α₁ increases, the THD of the load voltage decreases while the u_w remains constant. When α₁ is increased to 1.2, the load voltage becomes unstable. The THD of u_w is at a minimum when α₁ = 1.1, indicating that the introduction of a fractional-order filter improves the stability of the system.

Figure 15 illustrates the 0.4 s local amplification waveform of the load current. It can be observed that an increase in α₁ results in a corresponding rise in the transient peak value of the load current, which is consistent with the peak value characteristic of the transfer function step response.

When α₁ = 1.1 and 1.2, the load current waveform of phase A is shown in Figure 16. It can be seen that when α₁ = 1.1, the grid requirements can be met under both the resistive load Z₁ and the resistive-inductive load Z₂. However, as the load and its power factor increase, the system becomes unstable. As α₁ is further increased (α₁ = 1.2), the load voltage cannot meet the grid requirements under either load. This indicates that reducing the load impedance angle will increase the adjustable range of α₁.

The load power waveform as α₁ changes is shown in Figure 17. From its local amplification curve, it can be observed that as α₁ increases (α₁ ∈ {0.6, 0.8, 1}), the peak of the load active power at 0.4 s increases and the response speed slows down. This indicates that the stability margin decreases, which is consistent with the theoretical analysis and proves the accuracy of the constructed mathematical model. As the load power factor increases, the impact of α₁ on the power output becomes more evident. Once α₁ ≥ 1.1, the active power output of the system is unable to align with the load demand.

In conclusion, if α₁ is either too high or too low, this will affect the power quality of the system. Reducing α₁ to an insufficient level will affect the filtering effectiveness of the system output voltage, while increasing α₁ to an excessive level will primarily affect the stability of the system output voltage. The effect of α₁ on resistive-inductive loads is more significant than on resistive loads, and the larger the impedance angle, the easier it is to cause system instability.

5.2. Effect of Changing β on System Performance

5.2.1. Amplitude-Frequency Characteristics of β Variation with the Outer-Loop Opened

When α₁ and α₂ are set to 0.9, all other circuit and control parameters remain unaltered. The Bode plot of the outer-loop open transfer function of the system as β is varied is shown in Figure 18. Typical data for the corresponding characteristic curves are presented in

Table 13. Characteristics under the frequency domain as β vary.

β	0.5	0.8	1	1.1	1.2	1.4
h/(dB)	Inf	Inf	Inf	16.4	6.8	−14.2
γ/(deg)	83.9	83.8	83.6	83.5	83.5	−55.4
ωx/(rad/s)	NaN	NaN	NaN	9.1 × 103	4.6 × 103	1.7 × 103
ωc/(rad/s)	609	610	613	618	628	1.9 × 103

.

Combined with Figure 18 and Table 13, when α₁ and α₂ are held constant, an increase in β has a negligible impact on the low-frequency band, which only affects the high-frequency band, and the cut-off frequency remains largely unaltered when β is less than 1.2. Given the minimal impact of the high-frequency band on the actual amplitude, β has a negligible influence on the stability margin of the system within the stabilization interval. Once β ≥ 1.4, the phase-angle margin becomes negative, and the stability of the system must be determined in conjunction with other characteristics.

5.2.2. Effect of Changing β on the Dynamic Response Characteristics of the System

The step response waveforms of the double-closed loop control system as β changes are illustrated in Figure 19. From the figure, it can be observed that the load step response characteristics are almost identical when the fractional-order of the capacitor is less than 1.4. When β is equal to 1.4, the oscillation frequency of the step response curve increases while the steady-state time remains unaltered. Consequently, the impact of modifying the fractional-order of the capacitor on the system’s output is found to be insignificant.

5.2.3. Electrical Characterization of the System as β Vary

As β is varied, the simulation shows the variation in the steady-state value of the system output voltage, as shown in

Table 14. Load steady-state voltage as β increases.

	β	0.6	0.8	0.9	1	1.2	1.4
up/V	0.4 s	184.7	185.2	185.2	186.3	185.4	186
	0.7 s	187.7	187.9	187.9	187.9	188.1	188.6
Z1 Steady-state	uw/V	180	180	180	180	180	180 0.17%
	THD	0.83%	0.76%	0.67%	0.46%	0.28%
Z2 Steady-state	uw/V	180	180	180	180	180	180 0.21%
	THD	27.4%	3.47%	1.63%	1.52%	0.29%
Z3 Steady-state	uw/V	179.4	180	180	180	180	180 0.4%
	THD	2.5e3%	3.59%	2.09%	1.5%	0.5%

. An examination of the data presented in Table 14 reveals that the value of β does not influence the steady-state value of the system load voltage. However, when β is set to a value of 0.6 or below, it has an impact on the quality of the load voltage waveform, although the amplitude remains stable. An increase in β facilitates an improvement in the quality of the load voltage.

The load current waveforms as β is varied are presented in Figure 20. As can be observed from the figure, the load current waveforms exhibit a high degree of overlap as β is varied. In conclusion, the alteration of the fractional-order of the filter capacitor has a negligible influence on the load steady-state current, and the simulation results are consistent with the theoretical analysis.

5.3. Effect of Changing α₂ on System Performance

5.3.1. Amplitude-Frequency Characteristics of α₂ Variation with the Outer Loop Opened

When β = 0.8, α₁ = 0.9, and α₂ is varied, the Bode plot of the transfer function G_k1 is shown in Figure 21. The characteristic data are given in

Table 15. Characteristics under the frequency domain as α₂ varies.

α2	0.5	0.8	1	1.1	1.2	1.4
h/(dB)	Inf	Inf	Inf	Inf	Inf	Inf
γ/(deg)	103.1	104.5	107.7	84.2	70.3	69
ωx/(rad/s)	NaN	NaN	NaN	NaN	NaN	NaN
ωc/(rad/s)	299.6	299.5	293.8	1.8 × 104	1.7 × 104	1.5 × 104

.

Combined with Figure 21 and Table 15, it can be seen that when α₂ ≤ 1, overall, the amplitude increases with α₂ in the low-frequency band. However, in the high-frequency band, the amplitude and phase essentially overlap, resulting in a negligible increase in the cut-off frequency and a minimal reduction in phase-angle margin. As α₂ increases, the stability margin is abruptly reduced after α₂ > 1, resulting in a decline in stability.

5.3.2. Effect of Changing α₂ on the Dynamic Response Characteristics of the System

Figure 22 shows the step response waveforms of the double-closed-loop system when changing α₂. From the local amplification waveform at the peak value in Figure 22, it can be observed that as α₂ continues to increase, the peak value of the system step response also gradually increases. Nevertheless, the extent of this alteration is considerably less than that of the influence exerted by α₁ on the peak value of the step response.

5.3.3. Electrical Characterization of the System as α₂ Varies

Table 16. Load steady-state voltage as α₂ increases.

	α2	0.6	0.8	0.9	1	1.2	1.4
up/V	0.4 s	190.5	185	185.2	185.1	183.1
	0.7 s	188.7	188.2	187.9	187.5	187.3
Z1 Steady-state	uw/V	180	180	180	180	180	180 0.76%
	THD	0.77%	0.75%	0.76%	0.75%	0.75%
Z2 Steady-state	uw/V	180	180	180	180	180	Unstable
	THD	27.4%	1.93%	3.47%	4.79%	6.34%
Z3 Steady-state	uw/V	179.4	180	180	180	180
	THD	2.5e3%	2.01%	3.59%	5.09%	9.54%

shows the variation in the steady-state value of the load voltage as α₂ changes. It can be seen that with α₂ = 0.8 as the reference, an increase or decrease in α₂ results in a reduction in the quality of the system output voltage. Furthermore, when α₂ is set to 1.4, the voltage of the resistive-inductive load becomes unstable.

Figure 23 illustrates the locally amplified waveform of the load current as α₂ varies. It can be observed that an increase in α₂ is equivalent to an increase in the load impedance in the microgrid, which consequently leads to a reduction in the load current.

The load power waveforms as α₂ varies are presented in Figure 24. It can be observed that as α₂ increases, there is a corresponding decrease in the active power demand of the microgrid, accompanied by an increase in the reactive power demand. When α₂ reaches a certain level, the load will release active power in reverse, while reactive power will continue to increase.

In conclusion, the alteration in α₂ affects the equivalent impedance of the inductive component within the load. An increase in α₂ results in a higher impedance and a larger power factor angle. Consequently, the load current, active power, and reactive power are all affected. Compare the effects of changes in α₁, β, and α₂ on the frequency domain characteristics and electrical characteristics of the system to demonstrate the correctness of the model developed.

6. Conclusions

This paper employs the HBCS-MG system as a background, considers the fractional-order characteristics of the filter inductor, filter capacitor and load inductive components, and adopts a double-closed-loop control strategy with FOPI controllers for the purpose of analyzing the effects of fractional-order changes in the controller and each fractional element on the output characteristics of the system.

The optimal fractional-order range of FOPI controllers and fractional elements is shown in

Table 17. The optimal fractional-order range of FOPI controllers and fractional elements.

	Z1		Z2		Z3
Variable	Stable Range	Optimum/ THD	Stable Range	Optimum/ THD	Stable Range	Optimum/ THD
αrk	0.5–1.2	1/0.73%	0.3–1.5	0.5/3.45%	0.3–1.5	0.5/3.58%
α1	0.7–1.1	1.1/0.17%	0.9–1.1	1.1/1.15%	0.9–1	1/2.07%
β	0.6–1.4	1.4/0.17%	0.8–1.4	1.4/0.21%	0.8–1.4	1.4/0.4%
α2	0.6–1.4	Same	0.8–1.1	0.8/1.93%	0.8–1	0.8/2.01%

.

The results demonstrate that the introduction of a FOPI controller into a double-closed-loop control system improves the accuracy of the control system. When α_rk is set to a low value, it mainly affects the voltage quality of the resistive load. The impact of the fractional-order changes of the three fractional elements, namely the filter inductor, the inductive load component and the filter capacitor, on the output voltage of the system diminishes in a sequential manner. If the controller parameters remain unchanged and the fractional-order of the filter inductance and filter capacitor is insufficient, the filtering effect will be reduced, although stability is maintained. If the fractional-order of the filter inductance and load inductance is excessive, the system will become unstable. Furthermore, the filter capacitance has a negligible impact on the output characteristics of the system. The adjustable range of the filter inductance fractional-order is less extensive under a resistive load than under an inductive load. An increase in the fractional-order of the load inductance will result in a reduction in active power and an increase in reactive power at the load. The comparison between the frequency domain characteristics and the electrical characteristics demonstrates the correctness of the complete fractional mathematical model.

7. Discussion

The use of fractional-order control helps to improve the accuracy of microgrid control. Although this study has provided a detailed analysis of the impact of changing the fractional-order of the controller and the fractional elements within the system on the output characteristics, the following challenges exist in its implementation.

(1) Fractional-order controllers involve non-integer-order calculus operations, which may lead to real-time problems in embedded systems or real-time control systems with limited computational resources.

(2) Traditional analog or digital controllers are usually based on integer-order differentiation, and direct implementation of fractional-order controllers requires special numerical computation methods.

(3) Because the fractional-order characteristics of components are usually related to environmental factors such as frequency and temperature, it is a challenge to accurately model and optimize the controller parameters under different operating conditions.