An Evolutionary Annealing–Simplex Method for Inductance Value Selection for LCL Filters ^†

Abstract

Grid-connected voltage-source inverters (VSIs) have evolved over the past years for the interconnection of renewable energy sources with the grid to satisfy the increased electrical load demand and utilize clean carbon-free energy. However, VSIs suffer a significant drawback of switching harmonics caused by the inverter switches. To reduce the amount of current distortion injected into the utility grid and meet harmonic constraints as defined by power quality standards, an LCL filter is often used. Although extensive literature describes assorted design procedures for an LCL filter, all these methods cannot guarantee an optimal solution, because no direct mathematical function between the LCL filter parameters and the total harmonic distortion (THD) exists. Presented in this paper is the application of an evolutionary annealing–simplex method to select the inductance values in the LCL filter. This method is compared to other methods previously presented in the literature: a systematic method and an inductance grid-search method based on a calculated distribution of total harmonic distortion for various inductance values. The goal is to select the smallest overall inductance values to ensure lower manufacturing costs while maintaining the filter performance. Simulation and experimental results verify that at least a 40% reduction in overall inductance can be achieved with the proposed method while simultaneously guaranteeing the filtering performance.

1. Introduction

Nowadays, the worldwide tendency to integrate renewable energy into the utility grid has been driven by both political and environmental concerns. The advantages brought by renewable energy are its environmentally friendly nature and low fuel cost. Therefore, there has been a growing demand for renewable energy sources as a viable solution to the inadequacies of conventional power plants. Annual electricity generation from wind and solar power has been proliferating. Moreover, by 2040, renewable energy is projected to generate an equal amount of electricity as from coal and natural gas [1].

Grid-connected voltage-source inverters (VSIs) are widely used for energy conversion from a DC source to a utility grid [2]. VSIs are becoming more sophisticated and making a growing contribution to the grid management. To reduce the amount of current distortion injected into the utility grid and meet harmonic constraints as defined by standards such as IEEE-519 [3] and IEEE-1547 [4], a passive power filter is required between a VSI and the grid. Traditional use of the L filter as a simple first-order filter is considered bulky and inefficient when it is necessary to comply with harmonic standards [5]. Even though L filters have a much simpler structure and require basic control techniques, an increased filter inductance to attenuate current harmonics accompanied by a high switching frequency makes the filter configuration less adequate for high-power applications. Therefore, recent research on higher-order filters has been popular to overcome such limitations. In comparison to the conventional L filter, the LCL filter not only achieves better harmonics reduction in the filter output current but also lowers the overall cost with smaller values of inductance.

There are various state-of-the-art methods for designing and optimizing the LCL filter parameters. The authors in [6] categorized these methods in the following two aspects:

• The LCL filter parameter selection is based on a detailed analytical circuit model approach.
• The LCL filter parameter selection is based on a conceptual approach.

Reference [7] presents a typical analytical circuit approach. It introduces an iterative design procedure for LCL filters for VSIs. The analytical expression of the inverter harmonic voltages by Bessel functions is applied to design the filter parameters for defined maximum grid-current harmonics. This type of approach can also be found in [8,9,10]. The circuit model provides a precise point of view to design the filter. However, the complexity of this type of approach makes it less popular than the following conceptual approaches.

Conceptual approaches for LCL filter design have also been studied in [11,12,13,14]. These methods use the system’s parameters as known conditions, then follow a step-by-step procedure to yield the LCL filter parameters that limit the total harmonic distortion (THD) of the current injected into the grid, satisfying the IEEE standard requirements. The conceptual approaches are intuitive and efficient and have also been applied and proven effective by various industry or laboratory applications. However, because no direct mathematical function exists between the LCL filter parameters and the THD, all these methods cannot guarantee an optimal solution. Other filter parameter combinations can also potentially meet the IEEE standard. For an LCL filter, compared to the values of capacitors and resistors, the inductance values of the two inductors have a more dominant impact on the filtering performance, the manufacturing cost, and the overall power losses. It is necessary and beneficial to investigate the inductance value combinations of an LCL filter thoroughly. References [15,16,17,18,19] have also discussed several optimized design methods for LCL filters. However, there is no comparison in simulation, hardware experiment, and manufacturing cost between the proposed optimal design method and a widely applied design method such as [14].

Our previous research showed the distribution relationship of the THD with varying inductance values and proposed the inductance grid search (IGS) design method [20]. However, generating a precise THD distribution requires enormous data points and is time-consuming. In this paper, an evolutionary annealing–simplex (EAS) algorithm is proposed to assist the inductance determination efficiently with less total inductance cost as well. Specifically, initial inductance values are determined based on the existing procedure [12], and candidate values are rendered following the simplex method initialization. The corresponding THD output of each candidate is then simulated and the worst one is replaced by generating a new point that reflects the centroid point determined by other candidates. Such processing can be treated as a gradient-free downhill search that aims to find parameters to lower the THD. Simulation and experimental results are performed to verify the algorithm’s performance.

The contribution of this work is:

• The development of a proposed methodology based on an evolutionary annealing–simplex algorithm to significantly lower manufacturing costs with at least a 40% reduction in inductance values.
• The proposed method requires a reasonable computation power and requires less time than other stochastic grid search methods.
• The performance of the proposed evolutionary annealing–simplex algorithm is compared with two other literature methods for LCL filter inductance selection via both simulations and hardware experiments.

The structure of this paper is formulated as follows: Section 1 is the introduction to various LCL filter design approaches. Section 2 proposes an improved LCL filter design method based on a widely used conventional approach. A detailed introduction of our proposed EAS algorithm is included as well. Section 3 presents LCL filter design examples with a conventional systematic design method, an inductance grid search design method, and an EAS design method, respectively. Simulation results are presented in this section. In Section 4, three sets of LCL filters are built and tested. Their performance are compared via the experimental results. Section 5 summarizes and provides the conclusion of the paper.

2. Design Methods for the LCL Filter

Figure 1 illustrates the per-phase model of an LCL filter with wye-connected capacitors mentioned in [14], where the left side is connected to an inverter and the right side is connected to the grid. $v_i$ is the inverter output voltage and $v_g$ the grid voltage. $L_1$ is the inverter-side inductor, $L_2$ is the grid-side inductor, $R_1$ and $R_2$ are parasitic resistances of the inductors, and $C_f$ is a capacitor with a damping resistor $R_f$. The parameters $L_1$, $L_2$, $C_f$, and $R_f$ of an LCL filter are selected using the procedures presented in the following sections to guarantee that harmonics in the current injected into the grid $i_g$ fulfill the IEEE requirements in [3,4].

2.1. Systematic Design Method

Figure 2 shows the step-by-step LCL filter design procedure proposed in [12], where rated power $P_B$, grid frequency $f_g$, inverter switching frequency $f_{sw}$, DC voltage source voltage $V_{DC}$, and grid voltage $V_g$ are all regarded as known input data. The goal of this design procedure is to yield the previous four LCL filter parameters from these input data to suppress the harmonic components in the current injected into the utility grid $i_g$.

First, the base impedance and capacitance are calculated with the rated working conditions and grid information in (1) and (2),

$$Z_b=\frac{3V_g^2}{P_B}$$

(1)

$$C_b=\frac{1}{2\pi f_g{Z}_b}$$

(2)

Then, the filter capacitance is determined by referring to a percentage of the base capacitance $C_b$. A design factor of 5% is chosen considering that the maximum power factor variation seen by the grid is 5%, resulting in,

$$C_f=0.05C_b$$

(3)

Next, the inductor design considers the maximum peak-to-peak current ripple of the rated current $i_g$, which is determined by the rated working condition of the inverter system and defined as follows:

$$i_g=\frac{\sqrt{2}{P}_B}{3V_g}$$

(4)

A 10% ripple of the rated current for the design parameters is given by:

$$\mathrm{\Delta }{I}_{Lmax}=0.1i_g$$

(5)

The inverter-side inductor $L_1$ is determined by:

$$L_1=\frac{V_{DC}}{6f_{sw}\mathrm{\Delta }{I}_{Lmax}}$$

(6)

The grid-side inductor $L_2$ is determined with the required attenuation factor $k_a$, which is expected to be 20%:

$$L_2=\frac{\sqrt{\frac{1}{k_a^2}}+1}{C_f{\left(2\pi f_{sw}\right)}^2}$$

(7)

In the end, the damping resistor $R_f$ is related to the resonant angular frequency of the filter ${\omega }_{res}$, which is given by:

$${\omega }_{res}=\sqrt{\frac{L_1+L_2}{L_1{L}_2{C}_f}}$$

(8)

$$10f_g<f_{res}<0.5f_{sw}$$

(9)

The resonant frequency $f_{res}$ must satisfy Equation (9). If so, the damping resistor $R_f$ can be determined by:

$$R_f=\frac{1}{3{\omega }_{res}{C}_f}$$

(10)

However, there is no direct relationship described between the THD and filter parameters in this systematic design method. Moreover, some calculations require setting values based on experience, such as the required current ripple percentage to calculate the inverter-side inductor $L_1$. The choice of the required attenuation factor $k_a$ value also affects the grid-side inductor $L_2$. This inexplicit and experience-dependent design procedure cannot guarantee optimally designed parameters for LCL filters and, in some cases, can be relatively conservative. There are other parameters that achieve the same or a better level of performance while having reduced inductance values and less overall cost.

2.2. Inductance Grid Search Design Method

For an LCL filter, the inverter-side inductor $L_1$ and the grid-side inductor $L_2$ not only have the dominant role in filtering current but also contribute to the majority of the cost and difficulties in manufacturing the filter. Considering the complexity of a grid-connected inverter system, it is very difficult to find an explicit mathematical function to describe the impact of $L_1$ and $L_2$ on the THD. However, with the assistance of model-based simulation tools such as Simulink or PSCAD, we can determine how $L_1$ and $L_2$ affect the THD by varying $L_1$ and $L_2$ and calculating the THD with a simulation. Figure 3 shows a grid-connected inverter simulation system in Simulink.

Figure 4 shows the distribution of the THD with the initial data point calculated by the existing design method in [12]. Parameters $L_1$ and $L_2$ are varied in the same order of magnitude as the initial data point, and the THD is generated through Simulink. Simultaneously increasing $L_1$ and $L_2$ gives the best filter performance, but it also significantly increases the cost. Point A in Figure 4 is generated by the systematic design method. It has a relatively larger $L_1$ value and smaller $L_2$ value. However, there are other points in this figure that have the same level or even better filtering performance, such as Point B. From the inductance value standpoint, it has a much smaller $L_1$ value and slightly larger $L_2$ value; however, the overall inductance and manufacturing cost decrease significantly. During the simulation process, the resonant frequency of the LCL filter with every combination of $L_1$ and $L_2$ is generated with (8). Only the combinations which meet the requirements of (9) are kept.

This inductance grid search (IGS) design method takes advantage of the assistance from the state-of-the-art high-speed simulation tool. For every particular case, it generates the detailed THD distribution and provides numerous qualified inverter-side inductor and grid-side inductor combinations to consider.

3. Proposed Evolutionary Annealing–Simplex Design Method

As Figure 4 shows, there are plenty of existing candidate points to obtain a lower THD output with less inductance cost. However, such a grid search strategy is time-consuming. In this section, we introduce our proposed evolutionary annealing–simplex (EAS) algorithm, which is used to rapidly determine the inductance values for an LCL filter.

In general, we assume the THD is a nonlinear function of two variables, $L_1$ and $L_2$, and treat the variable determination as an optimization problem as follows:

$$\arg \underset{L_1,L_2}{min}THD;\left(L_1,L_2\right)$$

(11)

Gradient-based methods are widely used to solve optimization problems [21]. However, it is hard to apply them, because they require derivative information for the optimization, which is absent in our case due to the lack of a clear mathematical relationship.

Thus, in this paper, a gradient-free search strategy was introduced to optimize (11), according to the Nelder–Mead simplex method and the simulated annealing (SA) algorithm in [22,23]. The essence of the Nelder–Mead simplex method is an evolving pattern of $n+1$ points, which are the vertices of a simplex, spanning an n-dimensional space. The simplex explores the feasible space either by reflecting, contracting, or expanding away from the worst vertex or by shrinking toward the best vertex. An appropriate sequence of such movements converges to the nearest local minimum. However, this is a heuristic search method that can converge to nonstationary points.

The simulated annealing (SA) algorithm was presented in [23]. It is an analogy of the thermodynamic process. For slowly cooled thermodynamical systems, nature can find the minimum state of energy, while the system may end in an amorphous state of higher energy if it is cooled quickly, which is expressed by the Boltzmann probability distribution:

$$p\left(E\right)\approx exp(-E/kT)$$

(12)

The energy of a system in thermal equilibrium at any given temperature T is probabilistically distributed among all different states E. The system may switch to a new energy state after certain times of random search, irrespective of whether it is higher or lower, which can prevent the local minimum trap and be in favor of finding a better, more global minimum. However, such a process is often time-consuming and sacrifices efficiency.

Our EAS design method is based on a controllable random search concept, which is a combination of the simplex and SA methods. In each loop, instead of the simple random search, we apply the simplex-based gradient-free search strategy, which is an approximation of the gradient descent method, for each initialization to ensure the best candidate we can reach during the search. In addition, a controllable random initialization inspired by the SA algorithm can effectively prevent us from the local minimum trap. Furthermore, compared with the grid search shown in Figure 4, such processing can rapidly find a suitable candidate point meeting our demand. Let the inductor combinations be denoted as:

$$l_n=\{L_{1_n},L_{2_n}\}$$

(13)

The initial combination $l_0$ was chosen based on reference [12], with the $L_1$ value one order magnitude smaller. The specific processing steps of the simplex algorithm can be described as:

Step 1: An initial simplex $S=l_1,l_2,l_3$ is formulated by randomly selecting its vertices from an initial point $l_0$, where function values $TH{D}_1$, $TH{D}_2$, and $TH{D}_3$ are determined via simulation. Sort the vertices $l_1$, $l_2$, and $l_3$ so that $TH{D}_1$, $TH{D}_2$, and $TH{D}_3$ are in ascending order.

Step 2: Suppose $l_3$ is the worst vertex with the highest $TH{D}_3$. A new point $l_r$ is generated by reflecting away from $l_3$ via,

$$l_r=g+(0.5+u)(g-l_3)$$

(14)

where g is the centroid of the subset $S-l_3$, and u is a uniform random number. Simulate to get $TH{D}_r$.

Step 3: If $TH{D}_r<TH{D}_1$:

• Calculate the extended point $l_e$ via:

  $$l_e=g+(0.25+0.5u)(l_r-g)$$

  (15)
• If $TH{D}_e<TH{D}_1$, replace the worst point with the extension point $l_3=l_e$, $TH{D}_3=TH{D}_e$.
• If $TH{D}_e>TH{D}_1$, replace the worst point with the reflection point $l_3=l_r$, $TH{D}_3=TH{D}_r$.

Step 4: If $TH{D}_r>TH{D}_1$:

• If $TH{D}_r<TH{D}_2$ or $TH{D}_3$, replace the worst point $l_3=l_r,TH{D}_3=TH{D}_r$
• If $TH{D}_r>TH{D}_2$:

  (a)

  If $TH{D}_r>TH{D}_3$, calculate the contracted point $l_c$, $TH{D}_c$

  $$l_c=g-(0.25+0.5u)(g-l_3).$$

  (16)

  Simulate to get $TH{D}_c$:

  • If $TH{D}_c>TH{D}_3$, shrink the simplex towards the best vertex $l_1$, such that:

    $$l_{new}=l_c-(0.25+0.5u)(l_c-l_1).$$

    (17)
  • If $TH{D}_c<TH{D}_3$, replace the worst point $l_3=l_c,TH{D}_3=TH{D}_c$.

  (b)

  If $TH{D}_r<TH{D}_3$, replace the worst point $l_3=l_r$, $TH{D}_3=TH{D}_r$.

Step 5: If the THD is not lower than the required value, which means the stopping condition is not satisfied, the algorithm returns to step 2 and continues searching. The following Figure 5 illustrates the algorithm flow chart for the proposed design method.

Figure 6 indicates the beginning steps of the optimized design approach, with the green dot as the initial point. The initial point has a combination of relatively small inductor values, and an initial simplex is generated with its vertices randomly selected from the initial point. For each point, the inductor combination is used as the LCL filter parameters and simulated to acquire the THD. The THD is used to evaluate and determine the next point. The simplex moves along the direction of decreasing THD.

Figure 7 illustrates the overall search process for the optimized design approach. X and Y are the inductance values of $L_1$ and $L_2$, accordingly. Z is the THD value of the grid-injected current. The stopping condition can be set when the THD is lower than a certain desired level. An upper limit for the appointed $L_1$ or $L_2$ can also be applied.

This evolutionary annealing–simplex (EAS) design method can not only find smaller inductors than the systematic design method but also dramatically decreases the simulation time needed compared to the inductance grid search method and significantly increases design efficiency.

4. Simulation Results for LCL Filter Design Examples

In this section, all three previously mentioned grid-connected LCL filter design methods were utilized to select the inductance values for the LCL filter with a wye-connected capacitor configuration. The system specifications are shown in

Table 1. Test system parameters.

f	Grid frequency	60 Hz
$V_g$	Grid phase voltage	120 V RMS
$P_B$	Rated power	5 kW
$f_{sw}$	Switching frequency	10 kHz
$V_{dc}$	DC source voltage	400 V

.

With the systematic design method in [12], a step-by-step procedure was followed to calculate the parameters of the LCL filter as follows:

• The base impedance and capacitance were calculated with (1) and (2). The value of the damping capacitor was 15 μF, determined by Equation (3).
• With a 10% allowed ripple, Equations (5) and (6) yielded an inverter-side inductance $L_1$ of $3.4$ mH.
• With the required attenuation factor $k_a$ set as 20%, using Equation (7), the grid-side inductance $L_2$ was found to be $0.1$ mH.
• With the known values of $L_1$, $L_2$, and $C_f$, the resonant frequency was calculated by (8) to be $f_{res}=4169.4$ Hz, which met the requirement in (9).
• Equation (10) gave the damping resistor value as 0.85 Ω.

With the system parameters shown in Table 1 as inputs, the step-by-step procedure was programmed as a MATLAB function. The filter parameters were calculated within seconds. The performance of the LCL filter was evaluated via simulation.

A simulation of the grid-connected inverter system was conducted with MATLAB and the Simulink Power System Toolbox. The computer used in the simulation was equipped with a 12,600 K CPU, running at the performance core clock frequency of 5.1 GHz. The simulation model is shown in Figure 3, with a sampling time and a step size of 0.5 μs. The inverter was controlled in a stationary dq frame via a Park transformation and the power reference was set as a 5 kW active power. The reference currents were calculated by [24] with $P_{ref}$ = 5 kW and $Q_{ref}=0$. The Park transformation was also applied to the grid voltage $v_g$ to transform it into components $v_d$ and $v_q$. The reference currents were tracked by PI controllers. The THD of the current injected into the grid $i_g$ was measured when the system reached a steady state. By the conventional conceptual design approach, the LCL filter with the parameters shown in the first column of

Table 2. LCL filter parameters.

	Systematic Method	IGS Method	EAS Method
$L_1$	$3.4$ mH	$1.5$ mH	$1.45$ mH
$L_2$	$0.1$ mH	$0.6$ mH	$0.25$ mH
$C_f$	15 μF	15 μF	15 μF
$R_f$	0.85 Ω	0.85 Ω	0.85 Ω
Simulated THD	1.8%	1.6%	1.5%
Consumed Time	2 s	8931 s	158 s

yielded a THD of $i_g$ for 1.8%.

As mentioned previously, even though this set of LCL filter parameters was not the optimal selection, its parameters and performance can still be used as a baseline to determine the search range for the inductance grid search design method. The focus of the IGS design method was on searching for other $L_1$ and $L_2$ combinations that not only fulfilled the filtering requirement but also lowered the overall cost. The values of the damping capacitor and resistor were unchanged from the systematic design method. In our case, the search range for $L_1$ was from 1 mH to 4 mH with a unit increment of $0.1$ mH. For $L_2$, the search range was from $0.1$ mH to 1 mH with 30 data points. For every unique combination of $L_1$ and $L_2$, the simulation process was repeated and every time, the THD of $i_g$ was recorded.

Figure 4 presents the distribution of THDs with 900 data points based on the chosen range. Point A on the left is the inductor combination from the systematic design method. The tendency between the varying inductor values and the THD is well illustrated and can be used as a guide to choosing suitable LCL filter parameters. However, considering the complexity of the grid-connected inverter system in Simulink, generating such a distribution of THD is bulky and time-consuming.

The evolutionary annealing–simplex design method can start with a very small set of $L_1$ and $L_2$ combinations. The initial inductor value combination $l_0$ was chosen based on the result of the systematic design method, with the value of $L_1$ one order of magnitude smaller. In our particular case, the initial point was set as $l_0$ = {0.3 mH, 0.1 mH)} The vertices of an initial simplex S were randomly selected within the range of double $l_0$, and u was a random number between zero and one. The search continued until the THD was lower than both previous methods.

Based on the different methods, the selected inductance values were designed and simulated with the corresponding THD values listed in Table 2. Their performance is tested and compared via hardware experiments in the following section. The EAS method achieved the best performance with the least overall inductance value compared to the systematic and IGS methods..

Table 3. Applied EAS parameters and settings.

Inverter-side inductance of the initial point $l_0$	0.3 mH
Grid-side inductance of the initial point $l_0$	$0.1$ mH
Inverter-side inductance increment	$0.1$ mH
Grid-side inductance increment	$0.1$ mH
Iterations	9
Population size	27
Stopping criteria	($THD$ < 1.5%)
Max iteration	50

shows the EAS parameters and setting applied to this case study.

5. Experimental Data and Performance Analysis

Three custom-made LCL filters designed by the three methods were tested and validated in a 5 kW three-phase grid-connected inverter prototype test system under the same condition. The test system consisted of an NHR 9410 grid simulator, emulating a 120 V Vac system. An AgileSwitch 100 kW DC-AC inverter, with a switching frequency of 10,000 Hz, was controlled in the stationary dq frame to achieve a 5 kW active power control, using a control algorithm programmed and executed by a dSPACE DS1202 real-time interface (RTI) platform. An NHR 9210 battery test system was used as a DC source with a DC link voltage set at 400 V. A Tektronix MDO3024 oscilloscope was used to measure the THD with a built-in power quality software package.

Three custom-made LCL filters were constructed using the inductor values in

Table 4. Inductor parameters.

Inductance (mH)	1.5 mH	0.6 mH	0.5 mH	0.25 mH	0.1 mH
Core type	77102-A7	77442-A7	77442-A7	77442-A7	77102-A7
Number of stacks	3	3	3	2	1
Wire	AWG 12	AWG 12	AWG 12	AWG 12	AWG 12
Number of turns	107	33	30	26	23

. Design software in [25] assisted in selecting suitable toroid cores from magnetics and calculating the number of turns for inductors. All inductors in Table 4 were hand-made and validated by an AP300 frequency response analyzer to guarantee that the desired inductance values were satisfied. The systematic method filter consisted of an extra RL series 2.3 mH three-phase line reactor from MTE in series with 0.6 mH and 0.5 mH inductors to achieve 3.4 mH in total. All components used for the three custom-made LCL filters are shown in Figure 8.

In the hardware experiment, the main focus was on the current injected into the utility grid $i_g$ with the same system configuration to deliver an identical rated power. Three filters were connected in the system sequentially. The current waveforms were captured and their THDs were recorded to evaluate and compare the performance of the three filters. For the LCL filter from the systematic design method, the experimental results are shown in Figure 9 and Figure 10.

The experimental results of the LCL filter designed by the inductance grid search design method are shown in Figure 11 and Figure 12.

The experimental results of the LCL filter designed by the evolutionary annealing–simplex design method are shown in Figure 13 and Figure 14.

The experimental results indicated that all three filters fulfilled the design requirement, suppressed the harmonics generated by the inverter, and functioned well under the same test setup. The slight performance difference reflected the THD for $i_g$ and verified the simulation results listed in Table 2. Even though both the IGS and EAS design methods increased the grid-side inductor $L_2$, the inverter-side inductor $L_1$ dropped from the 3 mH level to the 1 mH level. This would broaden the selection of magnetic cores and lower difficulties in inductor manufacturing. In addition, at least 40% of overall inductance reduction was achieved, which would also significantly reduce the cost of the LCL filter.

6. Conclusions

Presented in this paper is an evolutionary annealing–simplex method to select the inductance values for an LCL filter applied in the grid-connected voltage-source inverter system, aiming to lower the inductance values and reduce the cost of manufacturing. The inductance values from a systematic method were utilized as a starting point. The IGS design method generated the detailed distribution of THD, from which the parameters of the LCL filter could be chosen. The IGS method is time-consuming; thus, an EAS design method was proposed based on a controllable random search, and it significantly reduced the computation time compared to that of the IGS method. The comparison of three filters was performed in simulation and hardware experiments. Both IGS and EAS methods can be applied in addition to the widely used systematic design method in LCL filter design, achieving at least 40% of overall inductance reduction, better performance, and less cost in industrial applications.