#### Design example of LCL filter

A calculation method of ripple current was given in the design of a single inductance filter in the previous article, and another method of calculation of ripple current will be introduced here. Also considering that the prototype adopts the unipolar SPWM strategy, the expression of the bridge arm output voltage U_{inv} is

Among them, M is the modulation ratio; N is the ratio of the carrier frequency to the fundamental frequency of the power grid; J_{n}(x) is the Bessel function of the first kind, and the expressions are shown in equations (1.2) to (1.3) respectively.

From formula (1.4), it can be obtained that the ratio of each harmonic in uinv to the fundamental wave is (1.5).

U_{L}=ω_{0}(L_{1}+L_{2})·I_{rated}﹤10%U_{g} (1.4)

Among them, M and N are calculated by the parameters in the table of Figure 1: M=0.778, N=300, and the harmonic spectrum of u_{inv} is obtained as shown in Figure 2.

In the case of unipolar SPWM strategy, the harmonics of the inverter bridge arm output voltage uinv are mainly distributed around the switching frequency and multiples of the switching frequency, and the largest content is the (N±1) harmonic. Since the filter’s attenuation ability to harmonics increases proportionally with the increase of the harmonic order, it is only necessary to suppress the (N-1) order of dominant harmonics to the range specified by the standard during filter design. The simplified LCL circuit considering only the (N-1) order of dominant harmonics is shown in Figure 3. Since the grid basically does not contain (N-1) order of harmonics, the grid can be regarded as a short-circuit state at this time.

1. Consider the current ripple content on the inverter side

From equation (1.4) and the previous analysis, it is known that the inductor current i_{L1} on the inverter side mainly contains N±1 harmonics. Consider here to suppress the N-1 harmonics to 10% of the rated current Irated, that is

From Figure 3, the effective value of the (N-1) sub-harmonic current in the inductor current i_{L1} on the inverter side is:

Among them, ω_{N-1}=2π(N-1) f_{0}, which is the (N-1) harmonic angular frequency. Because

the above formula can be simplified as

Knowing from the above formula, the magnitude of the current ripple on the inverter side is mainly determined by the value of L_{1}, and has little to do with L_{2 }and C_{1}. According to formula (1.6) and formula (1.8), the value range of L_{1} can be obtained:

Substituting the relevant parameters into the above formula, we get: L>0.43mH.

After determining the value range of the inductance L_{1}, the magnetic core is selected for inductance design. The magnetic core should be selected from a magnetic material with low high-frequency loss and an appropriate structure, such as sendust, ferrosilicon and other materials. In order to reduce the size, weight, and cost of the filter as much as possible, and make full use of the magnetic core, the total loss (that is, the sum of the iron loss and the copper loss) is as small as possible, and the final value of L_{1 }can be determined. In this example, combined with the ferrosilicon core specifications of a certain company, L_{1} can be selected as 0.6mH.

2. Consider the current ripple content of the grid

According to Figure 3, the (N-1) harmonic content in the grid current i_{g} is:

After the value of L_{1} is determined, the approximate value of the N-1 current ripple flowing through L_{1} can be obtained according to the formula (1.8), and then the formula (1.8) is substituted into the formula (1.10) to obtain

According to the restrictions on the grid current harmonics of the photovoltaic system in the IEEE Std929-2000 standard, the harmonic content above the 33rd order in the grid current must be less than 0.3% of the grid current rating, that is

According to formula (1.12) and formula (1.11), and considering that ω_{N-1}^{2}L_{2}C_{1}>1 when the switching frequency is 15kHz, the value range of the product of L_{2} and C_{1} can be obtained as

Substituting the relevant parameters into the above formula: L_{2}C_{1}> 2.83 (mH·μF).

3. Consider the total inductance voltage drop of the filter

According to formula (1.4), under rated working conditions, the impedance voltage drop U_{L} generated by the inductance of the LCL filter should be less than 10% of the grid voltage. Substituting relevant parameters into equation (1.4), and taking into account that L_{1} must be at least greater than 0.43mH in step 1, so the value of L_{2} is limited to

4. Consider the reactive power absorbed by capacitor C1

The greater the capacitance value, the more reactive power it generates, and the greater the current through L1 and the switch tube, the greater the loss. According to formula (1.15), the value range of the capacitor can be obtained:

5. Consider the parameter optimization plan and finally determine the parameters

The approximate value range of the LCL filter parameters is obtained above, and then an optimization scheme will be selected to finally determine the filter parameters. First, consider the influence of filter parameters on the suppression effect of grid background harmonics as an example to introduce the optimal selection of LCL filter parameters.

Take the double closed-loop feedback control scheme of grid current and capacitor current (the control scheme is shown in Figure 4) as an example to analyze the influence of LCL filter parameters on the suppression effect of grid background harmonics. The following analysis methods are also applicable to other current control schemes, so I won’t repeat them here.

In Figure 4, k_{C} is the feedback coefficient of capacitive current proportional feedback; G_{f}(s) is the grid voltage feedback coefficient. Generally, proportional feedforward is used, and G_{f}(s)=1 when full proportional feedforward is used;G_{C}(s) is the current regulator function, as shown in formula (1.16), the PI regulator parameters can be selected according to formula (1.17).

In the control structure of Fig. 4, the grid voltage proportional feedforward compensation is used to suppress the influence of grid background harmonics on the grid current. When using proportional feedforward, the transfer function from u_{g} to i_{g} is as shown in equation (1.18):

In the above formula, except for L2, C1 and kc, the other parameters are known. It can be seen from the above formula that under the condition of certain other parameters, the larger L2 and the smaller C1, the better the suppression effect on the harmonic influence of the power grid. From equation (1.10), when L1 has been determined, the content of (N-1) sub-ripples in the grid current is only related to the product of L2 and C1. And as the product of L2 and C1 increases, the harmonic content of the grid current decreases. However, the increase in the product of L2 and C1 will increase the size, weight, cost, and loss of the filter. Assume that the product of L2 and C1 is a fixed constant σ, namely

Substituting formula (1.19) into formula (1.18), the transfer function from u_{g} to i_{g} is obtained as:

The amplification factor of the system to the grid voltage is

The above formula is a function related to L_{2}, σ, k_{C} and ω, where ω=2πf, which is the harmonic angular frequency.

Figure 5 shows the variation of the system’s harmonic amplification coefficient to the grid with the inductance value L2 when k_{C}=0, σ=3×10^{-9}. The 3rd, 7th and 23rd harmonics along the direction of the arrows in the figure are respectively the 3rd, 7th and 23rd harmonics. In the low frequency band (3rd to 23rd), the harmonic attenuation ability gradually weakens as the number of harmonics increases. In addition, when the product σ of L2 and C1 is constant, as L2 increases and C1 decreases, the harmonic amplification factor drops sharply. When L2 increases to a certain value, the downward trend slows down.

Figure 6 shows the relationship between the 11th harmonic amplification factor of the power grid and L2 under different capacitance current feedback coefficients kc when σ = 3×10^{-9} (here, the 11th harmonic is taken as an example, and the other harmonics are similar). In the figure, the directions c along the arrows are 0, 3, 6, and 9, respectively. When L2 is constant, as k_{C} increases, the harmonic amplification factor increases. However, the change trend of the harmonic amplification coefficient with L2 is similar to that shown in Figure 5. First, it drops sharply, and then the downward trend gradually slows down.

Figure 7 shows the relationship between the 11th (550Hz) harmonic amplification factor of the power grid and the change of L2 under different σ conditions when kc=0. In the figure, σ along the direction of the arrow are 3×10-9, 4×10-9, 5×10-9 and 6×10-9 respectively. When L2 is constant, increasing σ is equivalent to increasing C1 . When L2 is constant, with the increase of σ, that is, with the increase of C1, the harmonic amplification factor increases, but the trend of the change of the harmonic amplification factor with L2 is similar to Figure 5.

At present, the performance of high-frequency capacitors can be very good, and within a certain range, as the capacitance value increases, the volume and cost of the capacitor increase relatively small; as the inductance increases, the volume, weight, cost, and loss will increase significantly. Therefore, when the ability to suppress the grid current ripple is constant, that is, when the product of L2 and C1 is constant, the larger the value of C1, the smaller the value of L2, the better. However, as can be seen from Figures 5 to 7, as L2 increases and C1 decreases, the system’s ability to attenuate grid harmonics gradually increases. Therefore, from the perspective of suppressing the influence of harmonics on the power grid, it is hoped that the larger the value of L2, the smaller the value of C1, the better. In addition, after the value of L2 is greater than 0.3mH, the downward trend of the power grid harmonic amplification coefficient begins to gradually slow down. Considering a compromise, the value of L2 is 0.3-0.5mH. After determining the range of the inductance L2, select the magnetic core model for inductance design; the same as the design of the inductance L1, based on reducing the volume, weight, cost and making full use of the magnetic core as much as possible, the final L2 is 0.36mH.

Substituting the value of L2 into formula (1.13), the value range of C1 is obtained: C1>7.9μF. In addition, from the perspective of suppressing the harmonics of the power grid, the smaller C1 is, the better, so C1 takes 8μF.

6. Verify the resonance frequency

Substitute the final values of the filter parameters into equation (1.22) for verification. If the resonance frequency restriction conditions are met, the selected parameters are feasible; if not, the values of L2 and C1 should be adjusted appropriately. Like formula (1.23), when L2 takes 0.36mH and C1 takes 8μF, the resonant frequency restriction condition is met.

Know the value of L1 and the minimum product of L2C1 (2.83mH·μF) from the first four restriction conditions, and the maximum value of capacitor C1 (16μF). The total energy storage W_{energy} of the filter can be expressed as a function of the capacitance value C1:

Among them, the product of L2C1 is selected according to the minimum value obtained from the grid current ripple limit, L1 takes the known value obtained from the previous design, and the others are the known parameters in the table in Figure 1.

The relationship between the total stored energy and the capacitance can be drawn through the known parameters, as shown in Figure 8.

As can be seen in Figure 8, when the capacitance value is 5.5μF, the total energy storage is the smallest, so C1 takes 5.5μF, and the grid-side inductance is 0.51mH. Also verify that the resonance frequency is:

The final value of each parameter of the LCL filter based on the optimization principle of the total energy storage is

Inverter side inductance L_{1}: 0.6mH

Grid side inductance L_{2}: 0.51 mH

Filter capacitor C_{1}: 55μF