Fractional-order LCL lters: principle, frequency characteristics, and their analysis

 In this paper, a fractional-order LCL (FOLCL) filter is constructed by introducing fractional-order inductors (FOIs) and fractional-order capacitors (FOCs) to replace the inductors and capacitors in a traditional integer-order LCL (IOLCL) filter, respectively. The principle and frequency characteristics of an FOLCL filter are systematically studied, and five important properties are derived and demonstrated in-depth. One of the most important achievements is that we discover the necessary and sufficient condition for the existence of resonance for an FOLCL filter, that is, the sum of the order of the FOIs and the FOC is equal to 2, which provides a theoretical basis for avoiding the resonance of an FOLCL filter effectively in design. The correctness of the theoretical derivation and analysis are verified by digital simulation. Compared with an IOLCL filter, an FOLCL filter presents more flexible and diverse operating characteristics and has a broader application prospect.

Fractional-order inductors (FOIs) and fractional-order capacitor (FOCs) [12,13] are the basic building elements of fractional-order circuits. The mathematical models of an FOI and an FOC can be expressed as [6][7][8] where, L is the inductance of an FOI, and C is the From equation (1) we know that in addition to the traditional parameter (such as L or C), the fractional-order element introduces an additional parameter, namely order, which makes the element and system more complex and more difficult to analyze on the one hand, but on the other hand, it increases the flexibility and degree of freedom and brings in immense versatility towards design and application. For more information about the operating characteristics of FOIs and FOCs, please refer to [3, 6-8, 12, 13].
The traditional integer-order LCL (IOLCL) filter, which consists of two inductors L1 and L2, and one capacitor C, is an important circuit widely applied in power electronic converters, such as be used to a grid-connected inverter [37][38][39]. The IOLCL filter plays a key role in filtering high-order harmonics in the AC side voltage and improving the quality of the current injected into the grid [37][38][39]. Compared with L filters, IOLCL filters can achieve a better high-frequency harmonic attenuation effect. However, the IOLCL filter has an inherent resonant peak, which will destroy the stability of the system when the system is not designed and controlled properly [37][38][39].
An FOC is introduced to replace the integer-order capacitor in an IOLCL filter to form an LC α L filter in [40]. Theoretical analysis and numerical simulations show that the LC α L filter provides wider bandwidth to mitigate higher-order resonant frequencies than its integer-order counterpart. The circuit and mathematical model of a fractional-order LCL (FOLCL) filter are preliminarily given in [41]. However, the detailed theoretical derivation and analysis of the frequency characteristics are missing. Besides, we find the resonant frequency expression 1 2 given in the paper is incorrect, and the correct expression of resonant frequency will be derived later in this paper.
Based on the Caputo fractional calculus [1], this paper deal with the circuit constructing, mathematical modeling, and frequency characteristic analysis of an FOLCL filter, aiming to extend the LCL filter from integer-order domain to fractional-order domain, and to systematically reveal the principle and frequency characteristics of an FOLCL filter. The rest of this paper is organized as follows: the circuit, mathematical models of an FOLCL filter are given in Section 2. In Section 3, the frequency characteristics of an FOLCL filter are studied in-depth, and five important properties of an FOLCL filter are systematically presented. In Section 4, digital simulations are performed to verify the correctness of the theoretical derivation and analysis. Finally, conclusions are drawn in Section 5.

The circuit and mathematical models of an FOLCL filter
By introducing the FOIs and the FOC to replace the inductors and the capacitor in an IOLCL filter, respectively, the main circuit of an FOLCL filter can be obtained as shown in Fig. 1(a). Where, the FOLCL filter is used for the AC side filtering of a grid-connected inverter. 1 L denotes the inductance of the input side FOI By performing the Laplace transform on equation (2), the s-domain expression of an FOLCL filter can be obtained as According to equation (3), we obtain the model structure of an FOLCL filter, as shown in Fig. 1 According to equation (3) and Fig. 1(b), the output current expression of an FOLCL filter can be obtained as In order to simplify the analysis, let 1 system, so the transfer function from the input voltage i () u s to the output current 2 () i s can be obtained, that is, the transfer function of an FOLCL filter is From equation (6), we know that an FOLCL filter is a fractional-order system with order 2  , where ,( 2) 0,


. Up to now, there is no report on the frequency characteristics of this type of fractional-order system. According to equation (6), the frequency domain model of an FOLCL filter can be expressed as According to equation (17), we further obtain the following resonant frequency property of an FOLCL filter: Property 1 reveals the condition for the existence of the resonance of an FOLCL filter essentially and provides a direct criterion for judging whether an FOLCL filter has a resonant peak. For example, since both  and  are 1, that is 2    , thus an IOLCL filter has a resonant peak definitely. Furthermore, property 1 provides a theoretical basis for avoiding the resonance of an FOLCL filter effectively in design. When an FOLCL filter has a resonance peak, it is easy to cause system instability [37][38][39]. For an IOLCL filter, an additional passive or active damper is usually introduced to eliminate resonance and improve the stability against variations [37,38]. However, the former will lead to additional loss and weaken the attenuation ability of high-frequency harmonics, and the latter will increase the complexity of controller design.
Whereas, for an FOLCL filter, we can make 2    by properly choosing the order of the FOIs and the FOC, so as to effectively avoid the resonance peak fundamentally and improve the stability of the system without an additional damper.

The corner frequency and log magnitudefrequency characteristic of an FOLCL filter
To analyze the corner frequency of an FOLCL filter, in equation (19). From Fig. 2 we know that when Substituting equations (18) and (19) into equation (10), the magnitude-frequency characteristic of an FOLCL filter can be expressed as According to the value range of  , equation (20) can be further expressed as For the first formula of equation (22) .
According to equation (23), the log magnitudefrequency characteristic and its asymptotic slope of an FOLCL filter when t1  can be obtained as For the second formula of equation (22) , According to equation (26), the log magnitudefrequency characteristic and its asymptotic slope of an FOLCL filter when t1   can be obtained as For the first formula of equation (29) According to equation (30), we obtain For the second formula of equation (29) , According to (33), we obtain (3) A summary of the corner frequency and log magnitude-frequency characteristic Based on the above theoretical derivation, we conclude that the corner frequency of an FOLCL filter when ( . In conclusion, the log magnitude-frequency characteristics of an FOLCL filter can be further summarized as follows. The property 3 is very important for analyzing the filtering performance of an FOLCL filter. From property 3, it can be seen that the asymptote slope of the log magnitude-frequency characteristic of an FOLCL filter in the low-frequency band is only determined by the order  of the FOIs, and is independent of the order  of the FOC; while the asymptote slope in the high-frequency band is affected by both  and  . Considering that , and the asymptote slope in the low-frequency band and the high-frequency band are c 20dB de  and 60dB dec  , respectively.

The phase-frequency characteristic of an FOLCL filter
It is known to all that for an complex number where, 2 arctan( ) 2 yx From equation (37), the first term of equation (11) To facilitate the derivation, let the center frequency According to equation (37), the second term in equation (42) (41) and (44), the phasefrequency characteristics of an FOLCL filter can be summarized as follows.

The phase crossover frequency and gain margin of an FOLCL filter
It is well known that the frequency at which the open-loop phase-frequency characteristic intersects the horizontal line   is usually referred to as the phase crossover frequency, and denoted as g  . For an FOLCL filter, according to equation (40)  , then the right side of equation (46) is greater than 0, that is, there is a real number solution for equation (46) and a phase crossover frequency. To sum up, the phase crossover frequency characteristic of an FOLCL filter can be summarized as follows.

The gain crossover frequency and phase margin of an FOLCL filter
As is known to all, the frequency 1.
By sorting out equation (50), we obtain The above formula is a very complicated nonlinear equation, it is very difficult to find the analytical solution of the gain crossover frequency c  of an FOLCL filter.
In order to obtain the trend graph of the change of c  with the order of the FOIs and the order of the FOC, we successively used the fsolve function, the particle swarm optimization algorithm [42], and the differential evolution (DE) algorithm [43] to solve equation (49) based on MATLAB software and find the DE algorithm has the best solution effect and relatively stable results.  that an FOLCL filter has a resonance peak when 2   , which makes the log magnitude-frequency characteristic curve cross the 0dB line many times. This phenomenon will be further verified in the following digital simulations.
Besides, according to property 3 and Fig. 3, we know: (1) Except for the case of 2   , as the order  of the FOIs increases, the asymptotic slope of the log magnitude -frequency characteristic curve in the lowfrequency band increases correspondingly, and the gain crossover frequency decreases.
(2) Considering that the order  of the FOC does not affect the low-frequency asymptotic slope of the log magnitude -frequency characteristic curve of an FOLCL filter, except for the case of 2   , changing the order  theoretically will not affect the gain crossover frequency.
The formula for calculating the phase margin of an FOLCL filter is the same as that of an IOLCL filter, namely gi c

Simulation results and analysis
In order to verify the correctness of the theoretical derivation, analyze the frequency characteristics of an FOLCL filter, and further explore the influence of the orders of the FOIs and the FOC on the frequency characteristics of an FOLCL filter, this paper establishes the digital model of an FOLCL filter and perform simulations based on MATLAB software. The main circuit parameters of an FOLCL filter as the same as in Fig. 3 are employed, and the orders of the FOIs and the FOC will be given later.

The frequency characteristic simulation curves of an FOLCL filter
(1) The frequency characteristic curves when  is constant and  changes Simulation condition I: let the order  of the FOIs be 0.8, 1.0 and 1.2 respectively. The order  of the FOC in each case is increased from 0.6 to 1.4 at intervals of 0.2.
The Bode diagram of an FOLCL filter based on Simulation condition I as shown in Fig. 4 and the corresponding frequency characteristic indicators are summarized in Table I. Where Nan means that there is no such indicator, inf denotes that the indicator is infinite.

The analysis of the frequency characteristics of an FOLCL Filter
From Fig. 4, Fig. 5, Table I, and Table II 28867.5rad s, L L L L C     and the measured resonant frequency from the curves is 28867 rad/s. Obviously, the two frequencies are basically equal, and the slight error is due to the fact that the curves as shown in the Bode diagram is connected by discrete points. In conclusion, the correctness of property 1 and property 2 have been verified.
(2) In each sub-graphs of Fig. 4, when the order  of the FOIs is constant and the order  of the FOC changes, the low-frequency bands of the five log magnitude-frequency characteristic curves overlap and the high-frequency bands diverge, which indicates that the order  of the FOC does not affect the log magnitudefrequency characteristic of an FOLCL filter in the lowfrequency band, but it will affect the characteristic in the high-frequency band. The asymptotic slope of the log magnitude-frequency characteristic curve of an FOLCL On the whole, the correctness of property 4 has been verified.
(4) The measured value of the phase crossover frequencies g  shown in Table I and Table II Fig. 4. In general, the correctness of the theoretical derivation and analysis of the frequency characteristics of an FOLCL filter is verified by digital simulation, and the influences of the order  of the FOIs and the order  of the FOC on the frequency characteristics of an FOLCL filter has been highlighted.

Conclusion
On the basis of constructing the main circuit of a more general LCL filter namely an FOLCL filter and establishing its mathematical models, this paper pioneered the theoretical analysis method for the frequency characteristics of an FOLCL filter, and systematically analyzed and summarized the frequency characteristics of an FOLCL filter for the first time. Five important properties of an FOLCL filter are derived and demonstrated in-depth, which are the most important theoretical achievements in this paper. The main differences of frequency characteristic between an IOLCL filter and an FOLCL filter are summarized in Table III.

Resonance peak
Exists a resonance peak Exists a resonance peak when 2

 
The necessary and sufficient condition for the existence of a resonance peak is 2  .  Compared with an IOLCL filter, since the extra parameters of  and  are included, although an FOLCL filter is more complicated to analyze, it has many novel behaviors and a wider operating range, which brings great flexibility and advantages to the application. The research in this paper provides a theoretical basis for the design of an FOLCL filter.
By appropriately selecting the values of  and  , the FOLCL filter which can better coordinate filtering performance and system stability can be obtained. In the future, an FOLCL filter is expected to apply to replace the integer-order LCL filter in traditional integer-order power electronic systems (such as inverters and rectifiers), and to build fractional-order power electronic systems with better performance Besides, the frequency characteristic analysis method of an FOLCL filter presented in this paper fills the gap in the research of the frequency characteristic of a general fractional system with 2  -order (where , ( 2) 0,  ), and has good theoretical significance.