# Parameter Design of Half-Bridge Converter Series Y-Connection Microgrid Grid-Connected Filter Based on Improved PSO-LSSVM

Authors: Xinggui Wang; Jinjian Li; Qun Guo; Hailiang Wang; Yingjie Ding
Journal: International Transactions on Electrical Energy Systems
 (2023)

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## Abstract

The half-bridge converter series Y-connection microgrid (HCSY-MG) is a new type of series microgrid. In order to reduce the harmonic content in HCSY-MG grid-connected current and at the same time simplify the parameter design process of the LCL filter, this study proposed an LCL filter parameter design method based on an improved particle swarm optimization-least squares support vector machine (PSO-LSSVM) by analyzing the harmonic characteristics of the HCSY-MG grid-connected current. In addition, to enhance the convergence speed of PSO-LSSVM, the inertia factor during its parameters’ update is made to adjust adaptively according to the direction of two consecutive parameter changes to constitute an improved PSO-LSSVM. Through simulation and comparative analysis, it is demonstrated that the improved PSO-LSSVM can enhance the convergence speed; the proposed filter parameter design method can effectively reduce the harmonic content in the HCSY-MG grid-connected current and is simpler and more comprehensive than the existing design method.


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## Body

## 1. Introduction

In the context of global environmental pollution and primary energy shortage, microgrid, which was proposed at the beginning of the 21st century, has received unprecedented attention [1, 2]. The microgrid structures that today can be classified as AC, DC, and hybrid AC-DC are based on the main bus supply [3]. Among them, the AC microgrids are the most widely used and valuable for research because they supply the same power as the electric system. However, the parallel access method used by each grid-connected converter in the AC microgrid generates a large amount of loop current within the system, which makes the utilization of renewable energy low, and also, the stability and power quality of the system is poor [4, 5].In order to solve the inherent problems in traditional AC microgrids, some scholars have proposed series structured microgrids with the topology, including series micropower grids and modular multilevel converter microgrids [6, 7]. Due to the special topology of the series microgrid, its harmonics are mainly distributed around integer multiples of the equivalent carrier frequency, so most of the existing parameter design methods for grid-connected filters will no longer be applicable. This study will take a half-bridge converter series Y-connection microgrid (HCSY-MG) as the research object and propose a grid-connected LCL filter parameter design method applicable to series-connected microgrids to simplify the design process while reducing the harmonic content in the grid-connected current. The HCSY-MG system has the advantages of simple control, no loop current, and higher utilization of renewable energy due to the topology of Y-connection, so it is meaningful to study the parameter design method of its grid-connected filter and can provide some references for the grid-connected operation of the series microgrid.In microgrids, active and passive power filters are generally used to reduce the amplitude of harmonics in the system. In the literature [8–10], corresponding active filter optimization design methods are proposed for different types of microgrids, which effectively improve the power quality of the AC busbar. Compared to the active filter, the passive filter is more suitable for scenarios with a higher power. Because the HCSY-MG system already contains a large number of electric power components, the use of an active filter will further increase the difficulty of controlling the system; this study will use the passive LCL grid-connected filter. The LCL filter is widely used as the interface between a microgrid converter and power system because of its better filtering effect. However, the topology of its third-order system makes the design of component parameters a research focus and difficulty [11]. Regarding the parametric design methods of LCL filters, there are mainly traditional design methods, graphical analysis methods, optimized parameters methods, and multivariate system of equations methods. Among them, the conventional design method for LCL filter parameters is given by literature [11], where the parameters of each element are designed sequentially subjected to constraints. Through theoretical analysis, literature [12, 13] proposed a graph containing the LCL filter parameters with the filtering effects and then used graphical analysis to obtain a parameter design and optimization method. In literature [14], a method based on a multivariate system of equations is given to obtain the design values of each element parameter by solving the LCL filter parametric equation, which has a strong generality. In addition to the abovementioned three methods, the most commonly used LCL filter parameter design method is the parameter optimization method. For example, there are minimization of energy storage element values, minimization of damping resistance losses, and minimization of total inductance values [15–17]. The literature [18] proposed a design approach based on multiobjective optimization, where the objectives considered include a filter inductance ratio to minimize total filter inductance and filter admittance to meet grid regulation and characteristic impedance for low current stress of switch stack. Regarding optimization algorithms, the genetic algorithm and its improvements, adaptive weight particle swarm algorithms, as well as adaptive multipopulation-modified nondominated ranking genetic algorithm combined with criteria importance interrelationship and similarity ranking preference technique have been successfully applied to the optimal design of parameters for grid-connected LCL filters [19–21]. Besides, a combination of the previous methods can be used to make the designed LCL filter parameters better. For example, based on the causes of harmonics generated by the converter, literature [22] developed an estimation model of the current total harmonic distortion (THD), and the values of the parameters of each element of the LCL filter were obtained with the current THD minimum as the optimization objective. Literature [23] based on the traditional graphical method and using the Bode diagram of the main transfer function as an entry point, the relationship between each component parameter of the filter and the design requirement was analyzed to first determine the approximate range of values for each parameter and then determine the optimal values by the Gray wolf optimization algorithm. Literature [24] analyzed the high-frequency harmonics of inverter output voltage in detail and constructed multiobjective optimization function that included five optimization objectives. Then, combined with the particle swarm optimization, a screening method is proposed to realize the multiobjective optimization process of designing LCL filter parameters.For the mentioned problems that require experience, iterative trials, and optimization, machine learning models can achieve better solutions through their ability to extract deep and high-order information from the data. The least square support vector machine (LSSVM) belonging to machine learning has been successfully applied to short-term load forecasting in the power system, photovoltaic output power forecasting, and orbital circuit fault diagnosis due to its ability to show superior modeling and prediction in applications with small sample size, nonlinearity, and low dimensionality and can be improved according to specific objects [25–27].In summary, most of the existing LCL filter parameter design methods are proposed based on common two-level converters, which are not necessarily applicable to the HCSY-MG system or even to the series-connected microgrids, so an LCL filter parameter design method applicable to the HCSY-MG system is studied in this article. Then, the parameter design method of an HCSY-MG grid-connected LCL filter based on improved PSO-LSSVM proposed in this study is based on the analysis of the grid-connected current characteristics of the HCSY-MG system, which utilized the PSO-LSSVM algorithm in machine learning and made its inertia factor adaptively adjusted during particle velocity update to obtain an improved PSO-LSSVM model. The LCL filter design method proposed in this study can flexibly adjust the input vector to be more suitable for the target system according to the actual application while avoiding the iterative trial of relying on engineering experience, which can make the design process more convenient. Through simulation, it is verified that the LCL filter designed by this method can effectively improve the power quality of grid-connected current, reduce the harmonic content in grid-connected current, and meet the corresponding constraint requirements.


## 2. System Structure and Grid-Connected Current Characteristics

### 2.1. Structure of the HCSY-MG System

The HCSY-MG system is based on the existing series microgrid and uses generation submodule (GM) as the basic unit, which is connected in a series structure with Y-connection to form a three-phase converter link and then connected to the AC bus through the LCL grid-connected filter. The topology of the GM is shown in Figure1, and the topology of the HCSY-MG is shown in Figure 2.Figure 1
Topology structure of GM. (a) Photovoltaic GM topology and (b) wind power GM topology.
(a)(b)Figure 2
Topology structure of HCSY-MG.In Figure1, ES is the energy storage, R is the energy dissipation resistor, C is the voltage regulator capacitor, udc is the half-bridge converter DC side bus voltage, idc is the half-bridge converter output current, and S1,2 is the power switching device. In the GM topology, both the ES and C are used to stabilize the bus voltage on the DC side of the microsource, where the ES also provides some damping for the system to ensure stable and safe operation of the HCSY-MG system. When the microsource output power is greater than the rated output power, the remaining energy can be stored in ES to avoid waste. When the microsource output power is less than the rated output power, the energy stored in ES can be released to compensate for the lack of microsource output power. In addition, the C mitigates the effect of small fluctuations in microsource output power on the HCSY-MG system output and the effect of small fluctuations in load or grid power on the microsource side. In Figure 2, PCC is the point of common coupling.From Figure2, the HCSY-MG system consists of N GMs per phase, and the three-phase converter link includes a total of 3N GMs. This system can adjust the system output voltage according to the actual application through adjusting the number of converter link output levels by flexibly selecting the number of GMs in each phase. By equating the N series GMs of each phase as the controlled voltage source, the equivalent circuit of the system in a grid-connected mode can be obtained as shown in Figure 3.Figure 3
System equivalent circuit of the HCSY-MG grid-connected mode.In Figure3, O is the connection point of the three-phase series converter link, uXO is the output voltage of phase X (X = A, B, and C) of the series converter link, respectively, Rl is the line equivalent resistance of each phase of the series converter link, Lc is the converter-side filter inductor, Rf is the filter damping resistor, Cf is the filter capacitor, Lg is the grid-side filter inductor, Rg is the grid line equivalent resistance, ieX is the grid current of each phase X, eX is the grid voltage of each phase X, and N is the grid neutral point.


### 2.2. Analysis of Grid-Connected Current Characteristics

When the HCSY-MG system is in stable operation, it is known from the system topology that the double Fourier expansion of the output voltage of theith GM under the carrier phase shifting Sine pulse width modulation (CPS-SPWM) strategy for phase A, for example, can be given by [28](1)uAi=udcAi2+MudcAi2sinω0t+φA+∑m=1∞∑n=−∞∞2udcAimπJnMmπ2sinm+nπ2cosmωct+i−12πN+nω0t+φA,where uAi is the output voltage of the ith GM of phase A, udcAi is the DC chain voltage of the ith GM of phase A, M denotes the modulation depth, ω0 is the modulating wave angular frequency, φA is the initial phase angle of the phase A modulated wave, m is the harmonic order of the carrier wave (m = 1, 2, …), n is the harmonic order of the reference wave (n = 0, ±1, ±2, …), Jnx refers to the Bessel coefficient of order n and argument x, ωc is the angular frequency of the triangular carriers, and N is the total number of microsources per phase.So, the output voltage of phaseA, uAO, can be expressed as(2)uAO=∑i=1NuAi=udcA2+M⋅udcA2sinω0t+φA+∑m=N,2N,⋯∞∑n=−∞∞2udcAmπJnMmπ2sinm+nπ2cosmωct+nω0t+φA,where udcA=∑i=1NudcAi is the equivalent DC bus voltage of phase A on the converter-side of the system.LetφA=0, φB=−2π/3, and φC=2π/3, then the output voltage uAB can be expressed as(3)uAB=uAO−uBO=udcA2−udcB2+MudcA2+udcB4⋅sinω0t+34M⋅udcB⋅cosω0t+∑m=N,2N,...∞∑n=−∞∞2a2+b2mπJnMmπ2sinm+nπ2sinmωct+nω0t+ψ,where a=udcBsinmθ−2πn/3; ψ=arctanb/a; b=udcA−udcBcosmθ−2πn/3; θ is the carrier phase shift angle.At this point, letudcA=udcB=udcC=udc, θ=0∘, then the voltage uAB can be expressed as(4)uAB=uAO−uBO=32Mudcsinω0t+π6+∑m=N,2N,...∞∑n=−∞∞−1n4udcmπJnMmπ2sinm+nπ2sinnπ3sinmωct+nω0t−nπ3.When the local load or parallel network adopts the Δ-connection method, the connection point directly receives theuAB, uBC, and uCA. If the current follows the voltage, the output current of phase A of the system can be expressed as(5)iea=Ieasinω0t+π6+φ+ieah,where Iea is the amplitude of the fundamental component of the output current, φ is the power factor angle, and ieah is the high-frequency harmonic component.When the local load or parallel network adopts the Y-connection method, the connection point does not directly receive theuAB, uBC, and uCA. At this point, through the KVL equation, the output voltage can be expressed as(6)uAN=2uAO−uBO−uCO3=uAB−uCA3,uBN=2uBO−uAO−uCO3=uBC−uAB3,uCN=2uCO−uAO−uBO3=uCA−uBC3,where uXN is the voltage between the output point of phase X (X = A, B, C) of the series converter link and the local load or the neutral point of the power system.Furthermore, continuing with the example of phaseA, the voltage uAN can be expressed as(7)uAN=uANf+uANh=12Mudcsinω0t+∑m=N,2N,...∞∑n=±1,±3,...∞4udcmπJnMmπ2sinm+nπ2cosnπ3sinmω0t+nωct,where uANf and uANh are the fundamental and harmonic components of uAN, respectively.So, the fundamental component of the output voltageuAN can be expressed as(8)uANf=12Mudcsinω0t=uAOf−12udc,where uAOf is the fundamental component of uAO.If the current follows the voltage, then the output current of phaseA of the system at this time can be expressed as(9)iea=Ieasinω0t+φ+ieah.From the aforementioned mathematical expressions, the characteristics of the grid-connected current of the HCSY-MG system can be obtained as follows:(1)
The frequency of the grid-connected current harmonics can be expressed asmωc+nω0t, i.e., the harmonics mainly occur around integer multiples of the equivalent carrier frequency(2)
Whenn is an even number, there are no harmonics of an integer multiple of the equivalent carrier frequency in the grid-connected current(3)
When the equivalent DC bus voltages of each phase are not equal, the harmonics of the grid-connected current will appear near the equivalent carrier frequency multiple, and the amplitude of the harmonic current is positively correlated with the difference of the equivalent DC bus voltages(4)
In the same spectrum, the harmonic component of the grid-connected current decreases as the equivalent carrier frequency increases(5)
The equivalent carrier frequency can be increased by increasing the number of microsourcesN or by increasing the carrier frequency fc, thus reducing the harmonic content in the grid-connected currentRegarding the harmonic content of grid-connected current in a microgrid, the IEEE Std. 1547-2003 standard requires that the current THD should be less than 1.5% [29]. Therefore, to make the grid-connected current THD of the HCSY-MG system meet the requirement, the LCL filter parameter will be designed using the improved PSO-LSSVM algorithm in this study.


## 2.1. Structure of the HCSY-MG System

The HCSY-MG system is based on the existing series microgrid and uses generation submodule (GM) as the basic unit, which is connected in a series structure with Y-connection to form a three-phase converter link and then connected to the AC bus through the LCL grid-connected filter. The topology of the GM is shown in Figure1, and the topology of the HCSY-MG is shown in Figure 2.Figure 1
Topology structure of GM. (a) Photovoltaic GM topology and (b) wind power GM topology.
(a)(b)Figure 2
Topology structure of HCSY-MG.In Figure1, ES is the energy storage, R is the energy dissipation resistor, C is the voltage regulator capacitor, udc is the half-bridge converter DC side bus voltage, idc is the half-bridge converter output current, and S1,2 is the power switching device. In the GM topology, both the ES and C are used to stabilize the bus voltage on the DC side of the microsource, where the ES also provides some damping for the system to ensure stable and safe operation of the HCSY-MG system. When the microsource output power is greater than the rated output power, the remaining energy can be stored in ES to avoid waste. When the microsource output power is less than the rated output power, the energy stored in ES can be released to compensate for the lack of microsource output power. In addition, the C mitigates the effect of small fluctuations in microsource output power on the HCSY-MG system output and the effect of small fluctuations in load or grid power on the microsource side. In Figure 2, PCC is the point of common coupling.From Figure2, the HCSY-MG system consists of N GMs per phase, and the three-phase converter link includes a total of 3N GMs. This system can adjust the system output voltage according to the actual application through adjusting the number of converter link output levels by flexibly selecting the number of GMs in each phase. By equating the N series GMs of each phase as the controlled voltage source, the equivalent circuit of the system in a grid-connected mode can be obtained as shown in Figure 3.Figure 3
System equivalent circuit of the HCSY-MG grid-connected mode.In Figure3, O is the connection point of the three-phase series converter link, uXO is the output voltage of phase X (X = A, B, and C) of the series converter link, respectively, Rl is the line equivalent resistance of each phase of the series converter link, Lc is the converter-side filter inductor, Rf is the filter damping resistor, Cf is the filter capacitor, Lg is the grid-side filter inductor, Rg is the grid line equivalent resistance, ieX is the grid current of each phase X, eX is the grid voltage of each phase X, and N is the grid neutral point.


## 2.2. Analysis of Grid-Connected Current Characteristics

When the HCSY-MG system is in stable operation, it is known from the system topology that the double Fourier expansion of the output voltage of theith GM under the carrier phase shifting Sine pulse width modulation (CPS-SPWM) strategy for phase A, for example, can be given by [28](1)uAi=udcAi2+MudcAi2sinω0t+φA+∑m=1∞∑n=−∞∞2udcAimπJnMmπ2sinm+nπ2cosmωct+i−12πN+nω0t+φA,where uAi is the output voltage of the ith GM of phase A, udcAi is the DC chain voltage of the ith GM of phase A, M denotes the modulation depth, ω0 is the modulating wave angular frequency, φA is the initial phase angle of the phase A modulated wave, m is the harmonic order of the carrier wave (m = 1, 2, …), n is the harmonic order of the reference wave (n = 0, ±1, ±2, …), Jnx refers to the Bessel coefficient of order n and argument x, ωc is the angular frequency of the triangular carriers, and N is the total number of microsources per phase.So, the output voltage of phaseA, uAO, can be expressed as(2)uAO=∑i=1NuAi=udcA2+M⋅udcA2sinω0t+φA+∑m=N,2N,⋯∞∑n=−∞∞2udcAmπJnMmπ2sinm+nπ2cosmωct+nω0t+φA,where udcA=∑i=1NudcAi is the equivalent DC bus voltage of phase A on the converter-side of the system.LetφA=0, φB=−2π/3, and φC=2π/3, then the output voltage uAB can be expressed as(3)uAB=uAO−uBO=udcA2−udcB2+MudcA2+udcB4⋅sinω0t+34M⋅udcB⋅cosω0t+∑m=N,2N,...∞∑n=−∞∞2a2+b2mπJnMmπ2sinm+nπ2sinmωct+nω0t+ψ,where a=udcBsinmθ−2πn/3; ψ=arctanb/a; b=udcA−udcBcosmθ−2πn/3; θ is the carrier phase shift angle.At this point, letudcA=udcB=udcC=udc, θ=0∘, then the voltage uAB can be expressed as(4)uAB=uAO−uBO=32Mudcsinω0t+π6+∑m=N,2N,...∞∑n=−∞∞−1n4udcmπJnMmπ2sinm+nπ2sinnπ3sinmωct+nω0t−nπ3.When the local load or parallel network adopts the Δ-connection method, the connection point directly receives theuAB, uBC, and uCA. If the current follows the voltage, the output current of phase A of the system can be expressed as(5)iea=Ieasinω0t+π6+φ+ieah,where Iea is the amplitude of the fundamental component of the output current, φ is the power factor angle, and ieah is the high-frequency harmonic component.When the local load or parallel network adopts the Y-connection method, the connection point does not directly receive theuAB, uBC, and uCA. At this point, through the KVL equation, the output voltage can be expressed as(6)uAN=2uAO−uBO−uCO3=uAB−uCA3,uBN=2uBO−uAO−uCO3=uBC−uAB3,uCN=2uCO−uAO−uBO3=uCA−uBC3,where uXN is the voltage between the output point of phase X (X = A, B, C) of the series converter link and the local load or the neutral point of the power system.Furthermore, continuing with the example of phaseA, the voltage uAN can be expressed as(7)uAN=uANf+uANh=12Mudcsinω0t+∑m=N,2N,...∞∑n=±1,±3,...∞4udcmπJnMmπ2sinm+nπ2cosnπ3sinmω0t+nωct,where uANf and uANh are the fundamental and harmonic components of uAN, respectively.So, the fundamental component of the output voltageuAN can be expressed as(8)uANf=12Mudcsinω0t=uAOf−12udc,where uAOf is the fundamental component of uAO.If the current follows the voltage, then the output current of phaseA of the system at this time can be expressed as(9)iea=Ieasinω0t+φ+ieah.From the aforementioned mathematical expressions, the characteristics of the grid-connected current of the HCSY-MG system can be obtained as follows:(1)
The frequency of the grid-connected current harmonics can be expressed asmωc+nω0t, i.e., the harmonics mainly occur around integer multiples of the equivalent carrier frequency(2)
Whenn is an even number, there are no harmonics of an integer multiple of the equivalent carrier frequency in the grid-connected current(3)
When the equivalent DC bus voltages of each phase are not equal, the harmonics of the grid-connected current will appear near the equivalent carrier frequency multiple, and the amplitude of the harmonic current is positively correlated with the difference of the equivalent DC bus voltages(4)
In the same spectrum, the harmonic component of the grid-connected current decreases as the equivalent carrier frequency increases(5)
The equivalent carrier frequency can be increased by increasing the number of microsourcesN or by increasing the carrier frequency fc, thus reducing the harmonic content in the grid-connected currentRegarding the harmonic content of grid-connected current in a microgrid, the IEEE Std. 1547-2003 standard requires that the current THD should be less than 1.5% [29]. Therefore, to make the grid-connected current THD of the HCSY-MG system meet the requirement, the LCL filter parameter will be designed using the improved PSO-LSSVM algorithm in this study.


## 3. LCL Grid-Connected Filter Characteristics and Constraints

### 3.1. LCL Grid-Connected Filter Characteristics

The LCL grid-connected filter topology with a damping resistor is shown in Figure4.Figure 4
LCL grid-connected filter topology with a damping resistor.In Figure4, ui is the input voltage, io is the output grid-connected current, and uo is the output voltage. Its open-loop transfer function can be obtained from Figure 4:(10)Gs=RfCfs+1LcLgCfs3+Lc+LgRfCfs2+Lc+Lgs,where s is the frequency domain variable.LetRf  = 0 in the previous equation to obtain the open-loop transfer function of the LCL filter without the damping resistor. The Bode diagram of the LCL grid-connected filter is shown in Figure 5.Figure 5
Bode diagram of the LCL grid-connected filter.As can be seen from Figure5, the LCL filter amplitude response curve without the damping resistor has a spike at the resonance point, while the phase response has a −180° change, which can easily lead to the instability of the grid-connected system and thus affect the safe and reliable operation of the power system. The damping resistor can cut the resonance spike in the amplitude response curve to a certain extent and avoid the −180° change in the phase response, i.e., avoid the instability of the grid-connected system and guarantee the safe and reliable operation of the power system.


### 3.2. LCL Grid-Connected Filter Constraints

The current flowing through the converter-side filter inductorLc generates a ripple current on the operation of the switch device, which affects the effectiveness of the LCL filter. The ripple current ∆imax can be expressed as(11)∆imax=ui4fequLc,where fequ is the equivalent carrier frequency.The introduction of the damping resistor also brings some additional power losses. To ensure the filter performance, the power loss of the whole filter needs to be less than 5% of the rated active power. The power loss of the filter can be given by [30](12)Ploss=Rfuo2Rf2+1/ωequ2Cf2,where Ploss is the power loss of the filter and ωequ is the equivalent carrier angular frequency.The total inductance of the filter can effectively reduce the harmonic current content of the grid, but too large inductance will make the filter response to the current weak, so the total inductance of the filter needs to be reasonably selected. The constraint on the total filter inductanceL can be given by [31](13)ui43∆imax⋅fequ≤L≤umax2−ui2/3ωequimax,where L=Lc+Lg, umax, and imax are the peak values of output voltage and current, respectively.


## 3.1. LCL Grid-Connected Filter Characteristics

The LCL grid-connected filter topology with a damping resistor is shown in Figure4.Figure 4
LCL grid-connected filter topology with a damping resistor.In Figure4, ui is the input voltage, io is the output grid-connected current, and uo is the output voltage. Its open-loop transfer function can be obtained from Figure 4:(10)Gs=RfCfs+1LcLgCfs3+Lc+LgRfCfs2+Lc+Lgs,where s is the frequency domain variable.LetRf  = 0 in the previous equation to obtain the open-loop transfer function of the LCL filter without the damping resistor. The Bode diagram of the LCL grid-connected filter is shown in Figure 5.Figure 5
Bode diagram of the LCL grid-connected filter.As can be seen from Figure5, the LCL filter amplitude response curve without the damping resistor has a spike at the resonance point, while the phase response has a −180° change, which can easily lead to the instability of the grid-connected system and thus affect the safe and reliable operation of the power system. The damping resistor can cut the resonance spike in the amplitude response curve to a certain extent and avoid the −180° change in the phase response, i.e., avoid the instability of the grid-connected system and guarantee the safe and reliable operation of the power system.


## 3.2. LCL Grid-Connected Filter Constraints

The current flowing through the converter-side filter inductorLc generates a ripple current on the operation of the switch device, which affects the effectiveness of the LCL filter. The ripple current ∆imax can be expressed as(11)∆imax=ui4fequLc,where fequ is the equivalent carrier frequency.The introduction of the damping resistor also brings some additional power losses. To ensure the filter performance, the power loss of the whole filter needs to be less than 5% of the rated active power. The power loss of the filter can be given by [30](12)Ploss=Rfuo2Rf2+1/ωequ2Cf2,where Ploss is the power loss of the filter and ωequ is the equivalent carrier angular frequency.The total inductance of the filter can effectively reduce the harmonic current content of the grid, but too large inductance will make the filter response to the current weak, so the total inductance of the filter needs to be reasonably selected. The constraint on the total filter inductanceL can be given by [31](13)ui43∆imax⋅fequ≤L≤umax2−ui2/3ωequimax,where L=Lc+Lg, umax, and imax are the peak values of output voltage and current, respectively.


## 4. Improved PSO-LSSVM Algorithm

### 4.1. The PSO-LSSVM Algorithm

In the analysis of existing LCL filter parameter design methods and in the process of obtaining training samples, it was found that the attenuation of the harmonic amplitude of the gird-connected current by the LC branch in the LCL filter is a nonlinear process. Also, because the number of training samples obtained by the simulation in this study is small and the LSSVM can obtain better results in the field of modeling and prediction of nonlinear data with a small sample size, LSSVM is used as the basic algorithm in this study. In addition, the use of the PSO algorithm to optimize the parameters affecting the anti-interference and generalization of LSSVM models has been widely used means. For these reasons, the PSO-LSSVM has good compatibility with the problem discussed in this study, so this algorithm is chosen as the core algorithm of this study.Given a sample setxj,yj,j=1,2,...,q, where xj is the input vector, yj is the output vector, and q is the size of the sample set, then the optimization model of LSSVM can be given by [32](14)minJω,ξ=12ω2+12P∑j=1qξj2,s.t.yj=ωϕxj+b+ξj,j=1,2,...,q,where ω is the weight vector, b is the bias, P is the error penalty factor, ξj is the relaxation variable, and ϕ⋅ is the nonlinear mapping function.By solving the previous optimization model, the final regression model of LSSVM can be expressed as(15)fx=∑j=1qλjKx,xj+b,where λj is the Lagrange multiplier and Kx,xj is the kernel function.To ensure the generalization of the regression model, the radial basis function is chosen as the kernel function, which can be expressed as(16)Kx,xj=exp−x−xj2σ2,where σ is the kernel parameter.From the above standard type of LSSVM, it can be seen that the kernel parameterσ and the error penalty factor P will affect the anti-interference and generalization of the model, so they need to be optimized. Currently, the PSO algorithm is commonly used to optimize the previous two parameters of LSSVM, namely, the PSO-LSSVM model. The velocity and position of the particles in the PSO optimization algorithm are updated in the following way:(17)vt+1=αvt+β1⋅rand0,1pbestt−xt+β2⋅rand0,1gbestt−xt,xt+1=xt+vt,where vt is the speed of the particle, α is the inertia factor, β1 is the self-learning factor, rand0,1 is a random number between (0, 1), pbestt is the current optimal position of the particle, xt is the current position of the particle, β2 is the global-learning factor, and gbestt is the current global optimal position of the particle.


### 4.2. The Inertia Factor Adaptive Adjustment Strategy of Improved PSO-LSSVM

In this study, to balance the global-search and self-search capability of PSO-LSSVM while improving the model convergence speed, an inertia factor adaptive adjustment strategy has been used to make the inertia factor adjust its size according to the difference between two consecutive update directions. The inertia factor adaptive adjustment strategy can be expressed as(18)α′=D⋅α,∆v>0,d⋅α,∆v<0,where α′ is the inertia factor after updating, D is the rate of inertia increase, and d is the rate of inertia decrease.PSO-LSSVM is already a mature model in which the PSO algorithm is widely used and can achieve the global optimality. In addition, the adaptive adjustment of inertia can give the improved PSO-LSSVM model the ability to flush out the current optimal value and find the global optimal value. Therefore, the proposed improved PSO-LSSVM in this study can reach the global optimum.In summary, when the maximum number of iterations is maxgen, the flowchart of the PSO-LSSVM algorithm for adaptive inertia is shown in Figure6.Figure 6
Flowchart of the PSO-LSSVM algorithm for adaptive inertia.


## 4.1. The PSO-LSSVM Algorithm

In the analysis of existing LCL filter parameter design methods and in the process of obtaining training samples, it was found that the attenuation of the harmonic amplitude of the gird-connected current by the LC branch in the LCL filter is a nonlinear process. Also, because the number of training samples obtained by the simulation in this study is small and the LSSVM can obtain better results in the field of modeling and prediction of nonlinear data with a small sample size, LSSVM is used as the basic algorithm in this study. In addition, the use of the PSO algorithm to optimize the parameters affecting the anti-interference and generalization of LSSVM models has been widely used means. For these reasons, the PSO-LSSVM has good compatibility with the problem discussed in this study, so this algorithm is chosen as the core algorithm of this study.Given a sample setxj,yj,j=1,2,...,q, where xj is the input vector, yj is the output vector, and q is the size of the sample set, then the optimization model of LSSVM can be given by [32](14)minJω,ξ=12ω2+12P∑j=1qξj2,s.t.yj=ωϕxj+b+ξj,j=1,2,...,q,where ω is the weight vector, b is the bias, P is the error penalty factor, ξj is the relaxation variable, and ϕ⋅ is the nonlinear mapping function.By solving the previous optimization model, the final regression model of LSSVM can be expressed as(15)fx=∑j=1qλjKx,xj+b,where λj is the Lagrange multiplier and Kx,xj is the kernel function.To ensure the generalization of the regression model, the radial basis function is chosen as the kernel function, which can be expressed as(16)Kx,xj=exp−x−xj2σ2,where σ is the kernel parameter.From the above standard type of LSSVM, it can be seen that the kernel parameterσ and the error penalty factor P will affect the anti-interference and generalization of the model, so they need to be optimized. Currently, the PSO algorithm is commonly used to optimize the previous two parameters of LSSVM, namely, the PSO-LSSVM model. The velocity and position of the particles in the PSO optimization algorithm are updated in the following way:(17)vt+1=αvt+β1⋅rand0,1pbestt−xt+β2⋅rand0,1gbestt−xt,xt+1=xt+vt,where vt is the speed of the particle, α is the inertia factor, β1 is the self-learning factor, rand0,1 is a random number between (0, 1), pbestt is the current optimal position of the particle, xt is the current position of the particle, β2 is the global-learning factor, and gbestt is the current global optimal position of the particle.


## 4.2. The Inertia Factor Adaptive Adjustment Strategy of Improved PSO-LSSVM

In this study, to balance the global-search and self-search capability of PSO-LSSVM while improving the model convergence speed, an inertia factor adaptive adjustment strategy has been used to make the inertia factor adjust its size according to the difference between two consecutive update directions. The inertia factor adaptive adjustment strategy can be expressed as(18)α′=D⋅α,∆v>0,d⋅α,∆v<0,where α′ is the inertia factor after updating, D is the rate of inertia increase, and d is the rate of inertia decrease.PSO-LSSVM is already a mature model in which the PSO algorithm is widely used and can achieve the global optimality. In addition, the adaptive adjustment of inertia can give the improved PSO-LSSVM model the ability to flush out the current optimal value and find the global optimal value. Therefore, the proposed improved PSO-LSSVM in this study can reach the global optimum.In summary, when the maximum number of iterations is maxgen, the flowchart of the PSO-LSSVM algorithm for adaptive inertia is shown in Figure6.Figure 6
Flowchart of the PSO-LSSVM algorithm for adaptive inertia.


## 5. Parameter Design of the LCL Grid-Connected Filter Based on Improved PSO-LSSVM

The steps for the design of LCL grid-connected filter parameters based on the improved PSO-LSSVM are as follows:(1)
Based on the constraint related to the ripple current, the design value of the converter-side filter inductor is found(2)
According to the improved PSO-LSSVM, the design values of the filter capacitor and grid-side filter inductor are obtained(3)
According to the power loss constraints of the LCL filter, the design value of the damping resistor is found(4)
The other constraints are checked for the designed LCL filter(5)
If the constraint is satisfied, the designed LCL filter is the desired one; if the constraint is not satisfied, we return to step (1) and redesign


### 5.1. Converter-Side Filter Inductor Design

The characteristics of the LCL filter show that the ripple current flowing through the filter inductor on the converter-side affects the performance of the filter. The expression of the ripple current is shown in (11), which contains the relevant parameter for the converter-side filter inductor. Then, the converter-side filter inductor can be expressed as(19)Lc=ui4fequ∆imax.Let the equivalent DC chain voltageudci=80V, microsource number N = 4, and the carrier frequency fc=1kHz of the HCSY-MG system. Then, according to (19), the converter-side filter inductance can be calculated as Lc=1mH, so the design value of Lc is chosen as 1mH.


### 5.2. Parameter Design Based on Improved PSO-LSSVM

#### 5.2.1. Determination of Model Input and Output Vectors

Combining the analysis of the LCL grid-connected filter characteristics in Section3 and the analysis of harmonic characteristics of the grid-connected current of the HCSY-MG system in Section 2.2, the input vectors of the filter parameter design model based on improved PSO-LSSVM are chosen as the equivalent DC chain voltage, converter output voltage, equivalent carrier frequency, high-frequency harmonic current content (including fequ−5fn, fequ−3fn, fequ−fn, fequ+fn, fequ+3fn, and fequ+5fn), current THD allowable value, and desired resonant frequency, in 11 groups. The output vectors are selected as the grid-side filter inductor value and filter capacitor value, in 2 groups. Here, fequ and fn are the equivalent carrier frequency and fundamental frequency, respectively.


#### 5.2.2. Data Preprocessing and Model Evaluation Metrics

From the correlation analysis in Section3, it is clear that the variables in the input and output vectors have large order-of-magnitude differences. To ensure the accuracy of model building and the homogeneity of the data, the data need to be normalized using a linear transformation.To measure the effectiveness of the improved PSO-LSSVM algorithm and the feasibility and effectiveness of the filter parameter design method based on the improved PSO-LSSVM, the fitness function is selected as(20)g=12fxi−yi2.In addition, the root mean square error (RMSE) and the mean relative error (MRE) were selected as the network evaluation criteria.


#### 5.2.3. Validation of Improved Strategies

Based on the designed converter-side filter inductor, the feasibility and effectiveness of the improved PSO-LSSVM are verified by obtaining training sample sets through simulation.According to the description related to the improved PSO-LSSVM model in Section4, the basic parameters are selected as the self-learning factor β1=1.5 and the global-learning factor β2=1.7, and the maximum number of iterations is set to 60.The effectiveness of the proposed improved PSO-LSSVM model in improving the convergence speed is first verified by comparison. Under the same model parameters and training sample set, the PSO-LSSVM fitness function curves using the inertia factor adaptive adjustment strategy and the common strategy are shown in Figure7. The corresponding network evaluation metrics are shown in Table 1.Figure 7
PSO-LSSVM adaptation function curves before and after improvement.Table 1
PSO-LSSVM network evaluation metrics before and after improvement.
ModelsRMSEMRE (%)Training time (s)PSO-LSSVM0.17255.91453.81Improvement PSO-LSSVM0.17035.81694.27From the change curve of the fitness function in Figure7, it can be seen that when the inertia factor in the particle velocity update formula adopts an adaptive adjustment strategy, the value of the fitness function can be stabilized at a smaller value sooner and the convergence of the model can be accelerated, which in turn ensures the applicability of the model in the case of a small training sample set.As can be seen from the data in Table1, when the inertia factor is adjusted with the adaptive adjustment strategy, it does not have a large impact on the error of the model, but it can slightly improve the accuracy of the model. In addition, since the adaptive inertia factor adjustment strategy adjusts the inertia of the particle velocity update after each iteration, there is a certain extension of the model training time.


#### 5.2.4. Filter Capacitor and Grid-Side Filter Inductor Design

Based on the previous improved PSO-LSSVM model, the input vectors are set as shown in Table2.Table 2
Improved PSO-LSSVM model input vector.
NumbersInput vector nameInput vector value1Equivalent DC chain voltage320 V2Converter output voltage160 V3Equivalent carrier frequency4 kHz4fequ−5fn harmonic content1%5fequ−3fn harmonic content0.5%6fequ−fn harmonic content1%7fequ+fn harmonic content1%8fequ+3fn harmonic content0.5%9fequ+5fn harmonic content1%10THD of current1.5%11Desired resonant frequency3 kHzThen, the corresponding output vectors are obtained by the improved PSO-LSSVM as shown in Table3.Table 3
Improved PSO-LSSVM model output vector.
NumbersOutput vector nameOutput vector value1Grid-side filter inductor value0.124 mH2Filter capacitor value18.426μFConsidering the actual component values, the design value of the grid-side filter inductor designed value is 0.1 mH and the capacitor designed value is 20μF.


### 5.3. Design of the Damping Resistor

The designed value of the damping resistor can be obtained from Subsection3.2 by the filter power loss, i.e., equation (12). Let the Ploss=5%P, then the value of the damping resistor is calculated to be Rf=0.48Ω. Therefore, the design value of the damping resistor is chosen as 0.5 Ω.The verification of whether the designed LCL filter satisfies the relevant constraints will be performed later.


## 5.1. Converter-Side Filter Inductor Design

The characteristics of the LCL filter show that the ripple current flowing through the filter inductor on the converter-side affects the performance of the filter. The expression of the ripple current is shown in (11), which contains the relevant parameter for the converter-side filter inductor. Then, the converter-side filter inductor can be expressed as(19)Lc=ui4fequ∆imax.Let the equivalent DC chain voltageudci=80V, microsource number N = 4, and the carrier frequency fc=1kHz of the HCSY-MG system. Then, according to (19), the converter-side filter inductance can be calculated as Lc=1mH, so the design value of Lc is chosen as 1mH.


## 5.2. Parameter Design Based on Improved PSO-LSSVM

### 5.2.1. Determination of Model Input and Output Vectors

Combining the analysis of the LCL grid-connected filter characteristics in Section3 and the analysis of harmonic characteristics of the grid-connected current of the HCSY-MG system in Section 2.2, the input vectors of the filter parameter design model based on improved PSO-LSSVM are chosen as the equivalent DC chain voltage, converter output voltage, equivalent carrier frequency, high-frequency harmonic current content (including fequ−5fn, fequ−3fn, fequ−fn, fequ+fn, fequ+3fn, and fequ+5fn), current THD allowable value, and desired resonant frequency, in 11 groups. The output vectors are selected as the grid-side filter inductor value and filter capacitor value, in 2 groups. Here, fequ and fn are the equivalent carrier frequency and fundamental frequency, respectively.


### 5.2.2. Data Preprocessing and Model Evaluation Metrics

From the correlation analysis in Section3, it is clear that the variables in the input and output vectors have large order-of-magnitude differences. To ensure the accuracy of model building and the homogeneity of the data, the data need to be normalized using a linear transformation.To measure the effectiveness of the improved PSO-LSSVM algorithm and the feasibility and effectiveness of the filter parameter design method based on the improved PSO-LSSVM, the fitness function is selected as(20)g=12fxi−yi2.In addition, the root mean square error (RMSE) and the mean relative error (MRE) were selected as the network evaluation criteria.


### 5.2.3. Validation of Improved Strategies

Based on the designed converter-side filter inductor, the feasibility and effectiveness of the improved PSO-LSSVM are verified by obtaining training sample sets through simulation.According to the description related to the improved PSO-LSSVM model in Section4, the basic parameters are selected as the self-learning factor β1=1.5 and the global-learning factor β2=1.7, and the maximum number of iterations is set to 60.The effectiveness of the proposed improved PSO-LSSVM model in improving the convergence speed is first verified by comparison. Under the same model parameters and training sample set, the PSO-LSSVM fitness function curves using the inertia factor adaptive adjustment strategy and the common strategy are shown in Figure7. The corresponding network evaluation metrics are shown in Table 1.Figure 7
PSO-LSSVM adaptation function curves before and after improvement.Table 1
PSO-LSSVM network evaluation metrics before and after improvement.
ModelsRMSEMRE (%)Training time (s)PSO-LSSVM0.17255.91453.81Improvement PSO-LSSVM0.17035.81694.27From the change curve of the fitness function in Figure7, it can be seen that when the inertia factor in the particle velocity update formula adopts an adaptive adjustment strategy, the value of the fitness function can be stabilized at a smaller value sooner and the convergence of the model can be accelerated, which in turn ensures the applicability of the model in the case of a small training sample set.As can be seen from the data in Table1, when the inertia factor is adjusted with the adaptive adjustment strategy, it does not have a large impact on the error of the model, but it can slightly improve the accuracy of the model. In addition, since the adaptive inertia factor adjustment strategy adjusts the inertia of the particle velocity update after each iteration, there is a certain extension of the model training time.


### 5.2.4. Filter Capacitor and Grid-Side Filter Inductor Design

Based on the previous improved PSO-LSSVM model, the input vectors are set as shown in Table2.Table 2
Improved PSO-LSSVM model input vector.
NumbersInput vector nameInput vector value1Equivalent DC chain voltage320 V2Converter output voltage160 V3Equivalent carrier frequency4 kHz4fequ−5fn harmonic content1%5fequ−3fn harmonic content0.5%6fequ−fn harmonic content1%7fequ+fn harmonic content1%8fequ+3fn harmonic content0.5%9fequ+5fn harmonic content1%10THD of current1.5%11Desired resonant frequency3 kHzThen, the corresponding output vectors are obtained by the improved PSO-LSSVM as shown in Table3.Table 3
Improved PSO-LSSVM model output vector.
NumbersOutput vector nameOutput vector value1Grid-side filter inductor value0.124 mH2Filter capacitor value18.426μFConsidering the actual component values, the design value of the grid-side filter inductor designed value is 0.1 mH and the capacitor designed value is 20μF.


## 5.2.1. Determination of Model Input and Output Vectors

Combining the analysis of the LCL grid-connected filter characteristics in Section3 and the analysis of harmonic characteristics of the grid-connected current of the HCSY-MG system in Section 2.2, the input vectors of the filter parameter design model based on improved PSO-LSSVM are chosen as the equivalent DC chain voltage, converter output voltage, equivalent carrier frequency, high-frequency harmonic current content (including fequ−5fn, fequ−3fn, fequ−fn, fequ+fn, fequ+3fn, and fequ+5fn), current THD allowable value, and desired resonant frequency, in 11 groups. The output vectors are selected as the grid-side filter inductor value and filter capacitor value, in 2 groups. Here, fequ and fn are the equivalent carrier frequency and fundamental frequency, respectively.


## 5.2.2. Data Preprocessing and Model Evaluation Metrics

From the correlation analysis in Section3, it is clear that the variables in the input and output vectors have large order-of-magnitude differences. To ensure the accuracy of model building and the homogeneity of the data, the data need to be normalized using a linear transformation.To measure the effectiveness of the improved PSO-LSSVM algorithm and the feasibility and effectiveness of the filter parameter design method based on the improved PSO-LSSVM, the fitness function is selected as(20)g=12fxi−yi2.In addition, the root mean square error (RMSE) and the mean relative error (MRE) were selected as the network evaluation criteria.


## 5.2.3. Validation of Improved Strategies

Based on the designed converter-side filter inductor, the feasibility and effectiveness of the improved PSO-LSSVM are verified by obtaining training sample sets through simulation.According to the description related to the improved PSO-LSSVM model in Section4, the basic parameters are selected as the self-learning factor β1=1.5 and the global-learning factor β2=1.7, and the maximum number of iterations is set to 60.The effectiveness of the proposed improved PSO-LSSVM model in improving the convergence speed is first verified by comparison. Under the same model parameters and training sample set, the PSO-LSSVM fitness function curves using the inertia factor adaptive adjustment strategy and the common strategy are shown in Figure7. The corresponding network evaluation metrics are shown in Table 1.Figure 7
PSO-LSSVM adaptation function curves before and after improvement.Table 1
PSO-LSSVM network evaluation metrics before and after improvement.
ModelsRMSEMRE (%)Training time (s)PSO-LSSVM0.17255.91453.81Improvement PSO-LSSVM0.17035.81694.27From the change curve of the fitness function in Figure7, it can be seen that when the inertia factor in the particle velocity update formula adopts an adaptive adjustment strategy, the value of the fitness function can be stabilized at a smaller value sooner and the convergence of the model can be accelerated, which in turn ensures the applicability of the model in the case of a small training sample set.As can be seen from the data in Table1, when the inertia factor is adjusted with the adaptive adjustment strategy, it does not have a large impact on the error of the model, but it can slightly improve the accuracy of the model. In addition, since the adaptive inertia factor adjustment strategy adjusts the inertia of the particle velocity update after each iteration, there is a certain extension of the model training time.


## 5.2.4. Filter Capacitor and Grid-Side Filter Inductor Design

Based on the previous improved PSO-LSSVM model, the input vectors are set as shown in Table2.Table 2
Improved PSO-LSSVM model input vector.
NumbersInput vector nameInput vector value1Equivalent DC chain voltage320 V2Converter output voltage160 V3Equivalent carrier frequency4 kHz4fequ−5fn harmonic content1%5fequ−3fn harmonic content0.5%6fequ−fn harmonic content1%7fequ+fn harmonic content1%8fequ+3fn harmonic content0.5%9fequ+5fn harmonic content1%10THD of current1.5%11Desired resonant frequency3 kHzThen, the corresponding output vectors are obtained by the improved PSO-LSSVM as shown in Table3.Table 3
Improved PSO-LSSVM model output vector.
NumbersOutput vector nameOutput vector value1Grid-side filter inductor value0.124 mH2Filter capacitor value18.426μFConsidering the actual component values, the design value of the grid-side filter inductor designed value is 0.1 mH and the capacitor designed value is 20μF.


## 5.3. Design of the Damping Resistor

The designed value of the damping resistor can be obtained from Subsection3.2 by the filter power loss, i.e., equation (12). Let the Ploss=5%P, then the value of the damping resistor is calculated to be Rf=0.48Ω. Therefore, the design value of the damping resistor is chosen as 0.5 Ω.The verification of whether the designed LCL filter satisfies the relevant constraints will be performed later.


## 6. Simulation Verification

Based on the abovementioned method, a grid-connected simulation model is built based on the obtained HCSY-MG system parameters and the parameters of each element of the LCL grid-connected filter, to validate the results and conclusions drawn.


### 6.1. Verification of Grid-Connected Current Characteristics

First, the grid-connected current characteristics of the HCSY-MG system are verified. The grid-connected current waveform and its spectrum when the LCL filter is not added to the HCSY-MG grid-connected system are shown in Figure8.Figure 8
Waveform of the grid-connected current without a filter and its frequency spectrum. (a) Waveform of the grid-connected current and (b) frequency spectrum of the grid-connected current.
(a)(b)From Figure8(a), it can be seen that the grid-connected current has some sinusoidal characteristics, but the overall quality is poor. Figure 8(b) can reflect the following characteristics: (i) the grid-connected current does not contain harmonics at an integer multiple of the equivalent carrier frequency; (ii) the harmonic currents are mainly distributed near the equivalent carrier frequency multiple; (iii) the harmonic content decreases with the increase of the equivalent carrier frequency. This proves the correctness of the analysis of the harmonic characteristics of the grid-connected current of the HCSY-MG system in Section 2.2. In addition, the grid-connected current THD = 13.11%, which is greater than the 1.5% required by IEEE Std. 1547-2003 standard, so it cannot be directly connected to the grid.


### 6.2. LCL Filter Performance Verification

Then, the feasibility of the proposed filter parameter design method based on improved PSO-LSSVM is verified.We set the LCL grid-connected filter with each element parameter as described in Section5.2, i.e., Lc=1mH, Lg=0.1mH, Cf=20μF, and Rf=0.5Ω, to obtain the corresponding filter transfer function as(21)Gs=RfCfs+1LcLgCfs3+Lc+LgRfCfs2+Lc+Lgs=10×10−6s+12×10−12s3+11×10−9s2+1.1×10−3s.The Bode diagram corresponding to the transfer functionG(s) is shown in Figure 9.Figure 9
LCL filter diagram after parameter design.From the LCL filter Bode plot magnitude response curve in Figure9, it can be seen that the LCL filter designed based on the improved PSO-LSSVM algorithm can effectively suppress high-frequency harmonics. From the phase response curve, it can be seen that the designed filter has a phase change of −135° at the resonant frequency and does not have a change of −180° which can easily lead to system instability, so the design of each element parameter of the filter is reasonable and does not make the gird-connected system unstable.The LCL filter was connected to the HCSY-MG grid-connected system, and the waveforms of the grid-connected current and its spectrum are obtained as shown in Figure10.Figure 10
The waveform of the grid-connected current with a filter and its spectrum. (a) Waveform of the grid-connected current and (b) frequency spectrum of the grid-connected current.
(a)(b)From Figure10(a), it can be seen that the power quality of the grid-connected current is better. Figure 10(b) shows that the current THD is reduced to 1.13%, which meets the conditions required in IEEE Std. 1547-2003 standard, the harmonic current content near 4 kHz is less than 1%, and the harmonic current content near 8 kHz is extremely low, reaching a negligible level. Therefore, by comparing Figures 8 and 10, it can be seen that the LCL filter designed based on the method proposed in this study can effectively improve the power quality of the grid-connected current and reduce the harmonic content in the grid-connected current.


### 6.3. LCL Filter Constraint Verification

According to the design steps in Section5, it is necessary to verify whether the designed LCL filter satisfies the corresponding constraints. In addition to the constraints already verified in the previous section, the remaining conditions are to be verified as follows.


#### 6.3.1. Resonant Frequency

According to the design of each element parameter of the LCL filter in Section5, the corresponding resonant frequency of the filter can be calculated as(22)fres=12πLc+LgLcLgCf=3734Hz.For the resonant frequencyfres, the general constraint can be expressed as(23)10f0≤fres≤fequ.Comparing equations (22) and (23), it is found that the designed filter satisfies the constraint of the resonant frequency.


#### 6.3.2. Total Inductance

According to the constraint of (13) for the total inductance value, it can be calculated as 0.58mH≤L=1.1mH≤346mH. Therefore, the total inductance value L = 1.1 mH satisfies the relevant constraint.Simulation and verification results show that the LCL grid-connected filter design method based on the improved PSO-LSSVM algorithm proposed in this study can effectively reduce the harmonic content in the grid-connected current without destabilizing the HCSY-MG grid-connected system and meet the corresponding constraint requirements.


### 6.4. LCL Filter Contrast Analysis

To illustrate the advantages of the proposed LCL grid-connected filter parameter design method for the HCSY-MG system over the existing methods and its applicability to the HCSY-MG system, the comparative analysis of the previous methods is necessary. The LCL filter parameters designed by different methods and the corresponding parameters of the HCSY-MG system grid-connected current are shown in Table4, where the target of optimized parameter method is the minimization of energy storage element values of the literature [19].Table 4
The LCL filter parameter designed by different methods and the corresponding parameter HCSY-MG system grid-connected current.
Lc (mH)LgmHCf (μF)FundamentalTHD (%)fr (Hz)Design method in this paper10.12010.421.133734Traditional design method10.51010.41.72758Graphical analysis method10.4610.421.753845Optimized parameter method10.5510.41.833900Multivariate system of equation method1.20.6109.81.382517Compared with the traditional design method, the design method proposed in this study does not need to rely on the iterative trial of engineering experience. Compared with the graphical analysis method, the design method proposed in this study also avoids iterative trial and does not require a complicated graphing process to obtain the desired filter parameters. The optimized parameter method can only consider one or more objectives and cannot comprehensively consider the requirements of the whole system, while the method proposed in this study can measure the characteristics of the whole system comprehensively with the help of the machine learning model, i.e., the improved PSO-LSSVM, and thus derive more reasonable filter parameters. Although the multivariate system of the equations’ method represents the system characteristics by a system of equations, the resulting system of equations is difficult to solve, while the method proposed in this study only requires the data obtained from the HCSY-MG system operation to design reasonable filter parameters easily.Comparing the design results, it can be seen that the total inductance of the LCL filter designed based on the method proposed in this study is the smallest, and the grid-connected current THD is also smaller; although the filter capacitance is larger, it still meets the corresponding constraint requirements. In addition, the method proposed in this study is more applicable to the HCSY-MG system, and also, the input and output vector of the improved PSO-LSSVM can be flexibly adjusted according to the actual application objects.


### 6.5. Sensitivity Analysis

When the parameters of the HCSY-MG system change, the power quality parameters of its grid-connected current will also change. The LCL filter parameter design method for the HCSY-MG system used in this study uses the grid-connected current characteristics of the HCSY-MG system as the training samples for the improved PSO-LSSVM model, and even the parameters of the HCSY-MG system change, the corresponding training samples will also change. Therefore, when the parameters of the HCSY-MG system change, with the modeling and prediction capability of the improved PSO-LSSVM model, the results of the LCL grid-connected filter parameters designed based on the method proposed in this study will also be designed in accordance with the system requirements.The improved PSO-LSSVM used in the LCL filter parameters design method proposed in this study consists of three core components, which are the PSO optimization algorithm, the LSSVM model, and the inertia factor adaptive algorithm. Among them, the inertia factor adaptive algorithm is an innovation in this study. Many analyses of the sensitivity of the parameters of the PSO optimization algorithm have been made in the existing literature, as well as a sensitivity analysis of the LSSVM model has been given in the literature [32]. So, a corresponding sensitivity analysis of the inertia factor adaptive algorithm will be given as follows.The description of increasing rateD and decreasing rate d in Subsection 4.2 shows that these two parameters only affect the convergence speed of the improved PSO-LSSVM when they are varied within a reasonable range of values and do not have a large impact on the design results and training time. To verify this conjecture, the improved PSO-LSSVM network evaluation metrics and training time, the corresponding filter design values, and the HCSY-MG grid-connected current power quality parameters for different values of D and d are summarized in Table 5. The results for the fundamental component and THD are obtained according to the inductance and capacitance values that match the actual situation.Table 5
Sensitivity analysis of differentD and d.
D, d1.1, 0.41.1, 0.61.1, 0.81.5, 0.41.5, 0.61.5, 0.81.7, 0.41.7, 0.61.7, 0.8Training time (s)4.314.284.184.244.274.264.324.274.30RMSE, MRE0.1965, 6.3760.1786, 5.92870.4106, 6.54870.1891, 6.14120.1703, 5.81690.2013, 6.21480.5162, 7.10470.1862, 6.10640.1807, 6.2303LgmH, Cf (μF)0.102, 20.2080.146, 17.7120.217, 14.7840.145, 19.4810.124, 18.4260.134, 19.3860.267, 21.4200.162, 20.7140.126, 18.711Fund., THD10.42, 1.13%10.15, 1.05%10.06, 1.12%10.13, 1.05%10.42, 1.13%10.13, 1.05%10.52, 2.11%10.27, 1.07%10.42, 1.13%As can be seen from the table, too large or too small rates may cause large errors in the improved PSO-LSSVM modeling and prediction results, but they do not have a large impact on the training time. The previous validation results are the same as the theoretical analysis. The reason that too large or too small rates make the improved PSO-LSSVM modeling and prediction results to have large errors may be the inability to converge to the global optimal solution with a finite number of iterations constraint. When the global optimal solution cannot be obtained by the improved PSO-LSSVM, it will make the designed LCL filter fail to meet the requirements of the HCSY-MG grid-connected operation. Therefore, the values ofD and d chosen in this study are the values for the minimum case of RMSE and MRE, i.e., D = 1.5 and d = 0.6.


## 6.1. Verification of Grid-Connected Current Characteristics

First, the grid-connected current characteristics of the HCSY-MG system are verified. The grid-connected current waveform and its spectrum when the LCL filter is not added to the HCSY-MG grid-connected system are shown in Figure8.Figure 8
Waveform of the grid-connected current without a filter and its frequency spectrum. (a) Waveform of the grid-connected current and (b) frequency spectrum of the grid-connected current.
(a)(b)From Figure8(a), it can be seen that the grid-connected current has some sinusoidal characteristics, but the overall quality is poor. Figure 8(b) can reflect the following characteristics: (i) the grid-connected current does not contain harmonics at an integer multiple of the equivalent carrier frequency; (ii) the harmonic currents are mainly distributed near the equivalent carrier frequency multiple; (iii) the harmonic content decreases with the increase of the equivalent carrier frequency. This proves the correctness of the analysis of the harmonic characteristics of the grid-connected current of the HCSY-MG system in Section 2.2. In addition, the grid-connected current THD = 13.11%, which is greater than the 1.5% required by IEEE Std. 1547-2003 standard, so it cannot be directly connected to the grid.


## 6.2. LCL Filter Performance Verification

Then, the feasibility of the proposed filter parameter design method based on improved PSO-LSSVM is verified.We set the LCL grid-connected filter with each element parameter as described in Section5.2, i.e., Lc=1mH, Lg=0.1mH, Cf=20μF, and Rf=0.5Ω, to obtain the corresponding filter transfer function as(21)Gs=RfCfs+1LcLgCfs3+Lc+LgRfCfs2+Lc+Lgs=10×10−6s+12×10−12s3+11×10−9s2+1.1×10−3s.The Bode diagram corresponding to the transfer functionG(s) is shown in Figure 9.Figure 9
LCL filter diagram after parameter design.From the LCL filter Bode plot magnitude response curve in Figure9, it can be seen that the LCL filter designed based on the improved PSO-LSSVM algorithm can effectively suppress high-frequency harmonics. From the phase response curve, it can be seen that the designed filter has a phase change of −135° at the resonant frequency and does not have a change of −180° which can easily lead to system instability, so the design of each element parameter of the filter is reasonable and does not make the gird-connected system unstable.The LCL filter was connected to the HCSY-MG grid-connected system, and the waveforms of the grid-connected current and its spectrum are obtained as shown in Figure10.Figure 10
The waveform of the grid-connected current with a filter and its spectrum. (a) Waveform of the grid-connected current and (b) frequency spectrum of the grid-connected current.
(a)(b)From Figure10(a), it can be seen that the power quality of the grid-connected current is better. Figure 10(b) shows that the current THD is reduced to 1.13%, which meets the conditions required in IEEE Std. 1547-2003 standard, the harmonic current content near 4 kHz is less than 1%, and the harmonic current content near 8 kHz is extremely low, reaching a negligible level. Therefore, by comparing Figures 8 and 10, it can be seen that the LCL filter designed based on the method proposed in this study can effectively improve the power quality of the grid-connected current and reduce the harmonic content in the grid-connected current.


## 6.3. LCL Filter Constraint Verification

According to the design steps in Section5, it is necessary to verify whether the designed LCL filter satisfies the corresponding constraints. In addition to the constraints already verified in the previous section, the remaining conditions are to be verified as follows.


### 6.3.1. Resonant Frequency

According to the design of each element parameter of the LCL filter in Section5, the corresponding resonant frequency of the filter can be calculated as(22)fres=12πLc+LgLcLgCf=3734Hz.For the resonant frequencyfres, the general constraint can be expressed as(23)10f0≤fres≤fequ.Comparing equations (22) and (23), it is found that the designed filter satisfies the constraint of the resonant frequency.


### 6.3.2. Total Inductance

According to the constraint of (13) for the total inductance value, it can be calculated as 0.58mH≤L=1.1mH≤346mH. Therefore, the total inductance value L = 1.1 mH satisfies the relevant constraint.Simulation and verification results show that the LCL grid-connected filter design method based on the improved PSO-LSSVM algorithm proposed in this study can effectively reduce the harmonic content in the grid-connected current without destabilizing the HCSY-MG grid-connected system and meet the corresponding constraint requirements.


## 6.3.1. Resonant Frequency

According to the design of each element parameter of the LCL filter in Section5, the corresponding resonant frequency of the filter can be calculated as(22)fres=12πLc+LgLcLgCf=3734Hz.For the resonant frequencyfres, the general constraint can be expressed as(23)10f0≤fres≤fequ.Comparing equations (22) and (23), it is found that the designed filter satisfies the constraint of the resonant frequency.


## 6.3.2. Total Inductance

According to the constraint of (13) for the total inductance value, it can be calculated as 0.58mH≤L=1.1mH≤346mH. Therefore, the total inductance value L = 1.1 mH satisfies the relevant constraint.Simulation and verification results show that the LCL grid-connected filter design method based on the improved PSO-LSSVM algorithm proposed in this study can effectively reduce the harmonic content in the grid-connected current without destabilizing the HCSY-MG grid-connected system and meet the corresponding constraint requirements.


## 6.4. LCL Filter Contrast Analysis

To illustrate the advantages of the proposed LCL grid-connected filter parameter design method for the HCSY-MG system over the existing methods and its applicability to the HCSY-MG system, the comparative analysis of the previous methods is necessary. The LCL filter parameters designed by different methods and the corresponding parameters of the HCSY-MG system grid-connected current are shown in Table4, where the target of optimized parameter method is the minimization of energy storage element values of the literature [19].Table 4
The LCL filter parameter designed by different methods and the corresponding parameter HCSY-MG system grid-connected current.
Lc (mH)LgmHCf (μF)FundamentalTHD (%)fr (Hz)Design method in this paper10.12010.421.133734Traditional design method10.51010.41.72758Graphical analysis method10.4610.421.753845Optimized parameter method10.5510.41.833900Multivariate system of equation method1.20.6109.81.382517Compared with the traditional design method, the design method proposed in this study does not need to rely on the iterative trial of engineering experience. Compared with the graphical analysis method, the design method proposed in this study also avoids iterative trial and does not require a complicated graphing process to obtain the desired filter parameters. The optimized parameter method can only consider one or more objectives and cannot comprehensively consider the requirements of the whole system, while the method proposed in this study can measure the characteristics of the whole system comprehensively with the help of the machine learning model, i.e., the improved PSO-LSSVM, and thus derive more reasonable filter parameters. Although the multivariate system of the equations’ method represents the system characteristics by a system of equations, the resulting system of equations is difficult to solve, while the method proposed in this study only requires the data obtained from the HCSY-MG system operation to design reasonable filter parameters easily.Comparing the design results, it can be seen that the total inductance of the LCL filter designed based on the method proposed in this study is the smallest, and the grid-connected current THD is also smaller; although the filter capacitance is larger, it still meets the corresponding constraint requirements. In addition, the method proposed in this study is more applicable to the HCSY-MG system, and also, the input and output vector of the improved PSO-LSSVM can be flexibly adjusted according to the actual application objects.


## 6.5. Sensitivity Analysis

When the parameters of the HCSY-MG system change, the power quality parameters of its grid-connected current will also change. The LCL filter parameter design method for the HCSY-MG system used in this study uses the grid-connected current characteristics of the HCSY-MG system as the training samples for the improved PSO-LSSVM model, and even the parameters of the HCSY-MG system change, the corresponding training samples will also change. Therefore, when the parameters of the HCSY-MG system change, with the modeling and prediction capability of the improved PSO-LSSVM model, the results of the LCL grid-connected filter parameters designed based on the method proposed in this study will also be designed in accordance with the system requirements.The improved PSO-LSSVM used in the LCL filter parameters design method proposed in this study consists of three core components, which are the PSO optimization algorithm, the LSSVM model, and the inertia factor adaptive algorithm. Among them, the inertia factor adaptive algorithm is an innovation in this study. Many analyses of the sensitivity of the parameters of the PSO optimization algorithm have been made in the existing literature, as well as a sensitivity analysis of the LSSVM model has been given in the literature [32]. So, a corresponding sensitivity analysis of the inertia factor adaptive algorithm will be given as follows.The description of increasing rateD and decreasing rate d in Subsection 4.2 shows that these two parameters only affect the convergence speed of the improved PSO-LSSVM when they are varied within a reasonable range of values and do not have a large impact on the design results and training time. To verify this conjecture, the improved PSO-LSSVM network evaluation metrics and training time, the corresponding filter design values, and the HCSY-MG grid-connected current power quality parameters for different values of D and d are summarized in Table 5. The results for the fundamental component and THD are obtained according to the inductance and capacitance values that match the actual situation.Table 5
Sensitivity analysis of differentD and d.
D, d1.1, 0.41.1, 0.61.1, 0.81.5, 0.41.5, 0.61.5, 0.81.7, 0.41.7, 0.61.7, 0.8Training time (s)4.314.284.184.244.274.264.324.274.30RMSE, MRE0.1965, 6.3760.1786, 5.92870.4106, 6.54870.1891, 6.14120.1703, 5.81690.2013, 6.21480.5162, 7.10470.1862, 6.10640.1807, 6.2303LgmH, Cf (μF)0.102, 20.2080.146, 17.7120.217, 14.7840.145, 19.4810.124, 18.4260.134, 19.3860.267, 21.4200.162, 20.7140.126, 18.711Fund., THD10.42, 1.13%10.15, 1.05%10.06, 1.12%10.13, 1.05%10.42, 1.13%10.13, 1.05%10.52, 2.11%10.27, 1.07%10.42, 1.13%As can be seen from the table, too large or too small rates may cause large errors in the improved PSO-LSSVM modeling and prediction results, but they do not have a large impact on the training time. The previous validation results are the same as the theoretical analysis. The reason that too large or too small rates make the improved PSO-LSSVM modeling and prediction results to have large errors may be the inability to converge to the global optimal solution with a finite number of iterations constraint. When the global optimal solution cannot be obtained by the improved PSO-LSSVM, it will make the designed LCL filter fail to meet the requirements of the HCSY-MG grid-connected operation. Therefore, the values ofD and d chosen in this study are the values for the minimum case of RMSE and MRE, i.e., D = 1.5 and d = 0.6.


## 7. Conclusion

The LCL grid-connected filter parameter design method proposed in this study is firstly for a new type of series microgrid which is the HCSY-MG system. Then, a new method is proposed to design the HCSY-MG grid-connected filter parameters using the improved PSO-LSSVM, in which the power quality of the grid-connected current is used as the input vector of the improved PSO-LSSVM, and the parameters of the LC in the filter are taken as the output vector to obtain the LCL filter parameters that meet the design requirements according to the HCSY-MG system operating characteristics and relevant constraints.The main work and innovations of this study are as follows: (1) Based on the HCSY-MG mathematical model of grid-connected operation, the characteristics of the grid-connected current of the system are analyzed and obtained. (2) Combining with the application context, an improved PSO-LSSVM is proposed by adaptively adjusting the size of the inertia factor by (18) according to the direction of the two consecutive particle velocity in the PSO-LSSVM. (3) Based on the previous research work, a parameter design method of the LCL grid-connected filter for the HCSY-MG system based on the improved PSO-LSSVM is proposed.The positive points of the proposed work are as follows: (1) The grid-connected current characteristics of the HCSY-MG system, a new type of series microgrid, are analyzed to provide a reference for the subsequent grid-connected operation and control of the system. (2) The LCL grid-connected filter parameter design method based on the improved PSO-LSSVM is proposed for the HCSY-MG system, which reduces the THD of grid-connected current from 13.11% to 1.13% and effectively improves the power quality of the grid-connected current of the HCSY-MG system. The proposed method provides a new idea and method for the design of grid-connected filter parameters of microgrids. (3) For the application context of this study, an improved PSO-LSSVM with adaptive change of the inertia factor is proposed, which improves the convergence speed of the model and enriches the application of PSO-LSSVM.



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*Source: 9534004-2023-02-24.xml*