The integration of renewable energy sources, particularly photovoltaic (PV) systems, into the utility grid relies heavily on power electronic converters, primarily grid-tied inverters. These inverters generate high-frequency switching harmonics that must be attenuated to meet stringent grid codes (e.g., IEEE 1547). Among various filter topologies, the LCLLC filter has emerged as a superior candidate for grid-tied inverter applications. It inherits the high-frequency attenuation characteristics of the LCL filter while incorporating a series-tuned branch (Lf-Cf) to create a notch at the switching frequency, effectively suppressing dominant harmonics. The performance, cost, and physical size of this filter are critically dependent on its parameter set (L1, L2, C, Lf, Cf). Traditional design methods often rely on sequential, conservative rules-of-thumb, leading to suboptimal solutions that are either oversized (increasing cost and loss) or inadequately damped. Single-objective optimization methods simplify the problem but necessitate difficult and often arbitrary weighting of competing goals like performance, cost, and loss. Classical multi-objective algorithms, such as standard Multi-Objective Particle Swarm Optimization (MOPSO), can struggle with convergence and population diversity, potentially settling in local optima. This paper presents a comprehensive methodology for the optimal design of LCLLC filter parameters for a three-phase PV grid-tied inverter. We formulate a constrained multi-objective optimization problem with three key targets: minimizing current ripple, maximizing harmonic attenuation, and minimizing total cost/loss. To solve this problem effectively, a novel improved multi-objective particle swarm algorithm incorporating a Tri-Competitive mechanism and a hierarchical Crisscross strategy (TCCMOPSO) is proposed. The method is validated through detailed simulation of a PV grid-tied inverter system in MATLAB/Simulink, demonstrating its correctness and superiority over conventional approaches.
System Modeling and Problem Formulation
The typical structure of a two-level, three-phase voltage source inverter (VSI) with an LCLLC filter connected to the grid is considered. The primary function of this grid-tied inverter is to convert DC power from the PV array into high-quality AC power synchronized with the utility grid. The LCLLC filter is placed between the inverter bridge and the grid connection point.
LCLLC Filter Transfer Function
The transfer function from the inverter output voltage \(U_{inv}(s)\) to the grid current \(I_g(s)\) is fundamental for analyzing filter performance and stability. Deriving from the circuit topology, the transfer function \(G(s)\) is given by:
$$
G(s) = \frac{I_g(s)}{U_{inv}(s)} = \frac{L_f C_f s^2 + 1}{a_5 s^5 + a_3 s^3 + a_1 s}
$$
where the coefficients are:
$$ a_5 = L_1 L_2 L_f C_f C $$
$$ a_3 = L_1 L_2 (C_f + C) + L_f C_f (L_1 + L_2) $$
$$ a_1 = L_1 + L_2 $$
This fifth-order model reveals the resonant characteristics of the filter, which must be carefully managed through parameter selection and damping strategies.
Optimization Objectives for the Grid-Tied Inverter Filter
The design of the filter for the grid-tied inverter must balance multiple, often conflicting, engineering requirements. We define three core objective functions.
1. Minimization of Inverter-side Current Ripple (\(\mathbf{f_1}\)):
The ripple current in the inverter-side inductor \(L_1\) is directly related to the switching frequency \(f_{sw}\), DC-link voltage \(U_{DC}\), and the value of \(L_1\). A smaller ripple reduces conduction and core losses in \(L_1\). The ripple current ratio \(\lambda\) is defined as:
$$ \lambda = \frac{\Delta i_{max}}{I_g} = \frac{U_{DC}}{8 I_g f_{sw} L_1} $$
Thus, the first objective is:
$$ \min f_1 = \lambda $$
2. Maximization of Harmonic Attenuation (\(\mathbf{f_2}\)):
The primary purpose of the filter in the grid-tied inverter is to attenuate switching harmonics. The harmonic attenuation ratio from the inverter-side current \(I_1(j\omega_n)\) to the grid current \(I_g(j\omega_n)\) at the n-th harmonic frequency \(\omega_n = 2\pi n f_0\) is:
$$ H_{atten}(j\omega_n) = \left| \frac{I_g(j\omega_n)}{I_1(j\omega_n)} \right| = \frac{|1 – L_f C_f \omega_n^2|}{|1 – (L_f C_f + L_2 C + L_2 C_f)\omega_n^2 + L_2 L_f C_f C \omega_n^4|}
$$
For effective filtering, especially at high frequencies (e.g., above the 35th harmonic as per many grid standards), this ratio should be as small as possible. We define the second objective as the sum of attenuation ratios for high-order harmonics (n=35 to 50 in this study):
$$ \min f_2 = D_x = \sum_{n=35}^{50} H_{atten}(j\omega_n)
$$
3. Minimization of Total Cost and Loss (\(\mathbf{f_3}\)):
The total cost is strongly correlated with the physical volume and material of inductors and capacitors. Furthermore, the total filter capacitance contributes to reactive power generation at the fundamental frequency, leading to increased apparent power and losses. Therefore, a combined cost-loss function \(W\) is defined, which penalizes both total inductance and total capacitance. A scaling factor is introduced to balance the numerical ranges of inductance (mH) and capacitance (μF).
$$ W = k_L (L_1 + L_2 + L_f) + k_C (C + C_f) $$
Typically, \(k_L = 1 \times 10^3\) and \(k_C = 1 \times 10^6\) are used. Thus, the third objective is:
$$ \min f_3 = W $$
Design Constraints for the Grid-Tied Inverter Application
Practical implementation of the filter in a grid-tied inverter imposes several constraints on the parameters.
C1: Ripple Current Constraint. The ripple current ratio is typically limited to 10-30%.
$$ L_1 \ge \frac{U_{DC}}{8 f_{sw} \lambda_{max} I_g} \quad \text{where } \lambda_{max} = 0.2 \text{ (20%)}
$$
C2: Reactive Power Constraint. The reactive power generated by the filter capacitors at the fundamental frequency should not exceed a small percentage (e.g., 5%) of the rated active power \(P_0\).
$$ C_{total} = C + C_f \le \frac{0.05 P_0}{2\pi f_0 U_g^2}
$$
C3: Resonance Frequency Constraints. The resonant frequency of the main LCL-like part (excluding the trap branch) must be placed safely between 10 times the grid frequency and half the switching frequency to avoid interference with control bandwidth and switching harmonics.
$$ 10 f_0 \le f_r = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2}{L_1 L_2 (C + C_f)}} \le 0.5 f_{sw}
$$
C4: Total Inductance Constraint. The total voltage drop across the filter inductors at the fundamental frequency should be limited (e.g., <10% of grid voltage).
$$ L_{total} = L_1 + L_2 + L_f \le \frac{0.10 U_g^2}{2\pi f_0 P_0}
$$
C5: Notch Frequency Constraint. The series resonant branch (Lf, Cf) must be tuned precisely to the switching frequency to create the desired attenuation notch.
$$ f_{sw} = \frac{1}{2\pi \sqrt{L_f C_f}}
$$
The complete multi-objective optimization problem for the grid-tied inverter LCLLC filter design is therefore:
$$
\begin{aligned}
& \underset{L_1, L_2, C, L_f, C_f}{\text{minimize}}
& & \mathbf{F} = [f_1, f_2, f_3]^T \\
& \text{subject to}
& & \text{C1, C2, C3, C4, C5}
\end{aligned}
$$
Tri-Competitive Crisscross Multi-Objective PSO (TCCMOPSO)
To solve the complex, constrained, multi-objective problem defined above, we propose an enhanced variant of the MOPSO algorithm. Standard MOPSO can suffer from premature convergence and loss of population diversity. The TCCMOPSO algorithm introduces two key mechanisms to overcome these limitations.
1. Fitness Assignment using ISDE+ Indicator
Maintaining a balance between convergence and diversity is crucial. We employ a hybrid fitness metric, ISDE+, which combines the convergence-oriented \(I_{\epsilon+}\) indicator and the diversity-promoting Shift-based Density Estimation (SDE). For a particle \(\mathbf{x}\), its fitness is calculated as the minimum ISDE+ distance to any particle \(\mathbf{y}\) in the current population \(P\) that dominates it or has a better aggregate objective sum \(SB(\mathbf{x}) = \sum_{i=1}^M f_i(\mathbf{x})\).
$$
\text{Fitness}(\mathbf{x}) = \min_{\substack{\mathbf{y} \in P_{\mathbf{x}} \\ \mathbf{y} \neq \mathbf{x}}} \left\{ \sqrt{ \sum_{i=1}^{M} \left( \max\{0, f_i(\mathbf{y}) – f_i(\mathbf{x})\} \right)^2 } \right\}
$$
where \(P_{\mathbf{x}} = \{ \mathbf{y} \in P : SB(\mathbf{y}) < SB(\mathbf{x}) \}\). A lower fitness value is better.
2. Tri-Competitive Particle Update Strategy
After each iteration, the population \(P\) (size \(N\)) is divided into an elite archive \(G\) (size \(\alpha\)) selected via non-dominated sorting and crowding distance, and a remainder set \(P^*\). Instead of a global best guide, particles in \(P^*\) are updated through a competitive tournament:
- Three particles are randomly selected from \(P^*\).
- Their fitness values are compared, ranking them as Champion (best fitness), Runner-up, and Third-place.
- The Champion particle learns from a randomly selected elite particle from \(G\):
$$ \mathbf{V}_{w}(t+1) = \mathbf{r}_1 \otimes \mathbf{V}_{w}(t) + \mathbf{r}_2 \otimes (\mathbf{X}_{r}(t) – \mathbf{X}_{w}(t)) $$
$$ \mathbf{X}_{w}(t+1) = \mathbf{X}_{w}(t) + \mathbf{V}_{w}(t+1) $$ - The Runner-up particle then learns from the updated Champion:
$$ \mathbf{V}_{m}(t+1) = \mathbf{r}_3 \otimes \mathbf{V}_{m}(t) + \mathbf{r}_4 \otimes (\mathbf{X}_{w}(t+1) – \mathbf{X}_{m}(t)) $$
$$ \mathbf{X}_{m}(t+1) = \mathbf{X}_{m}(t) + \mathbf{V}_{m}(t+1) $$ - The Third-place particle learns from the updated Runner-up:
$$ \mathbf{V}_{l}(t+1) = \mathbf{r}_5 \otimes \mathbf{V}_{l}(t) + \mathbf{r}_6 \otimes (\mathbf{X}_{m}(t+1) – \mathbf{X}_{l}(t)) $$
$$ \mathbf{X}_{l}(t+1) = \mathbf{X}_{l}(t) + \mathbf{V}_{l}(t+1) $$
Here, \(\mathbf{r}_1\) to \(\mathbf{r}_6\) are vectors of random numbers uniformly distributed in [0,1], and \(\otimes\) denotes element-wise multiplication. This hierarchical learning structure accelerates convergence toward the Pareto front while preserving directional diversity.
3. Hierarchical Crisscross Search Strategy
To further enhance exploration and escape local optima, a crisscross search is applied in a hierarchical manner after the competitive update. This involves two operations:
Horizontal Crossover (HC): Performed between particles of similar rank. For two distinct parent particles \(\mathbf{X}_{a}\) and \(\mathbf{X}_{b}\) in the same hierarchical group (e.g., elite group or updated non-elite group), two offspring are generated:
$$
\begin{aligned}
\mathbf{MHC}_{a,d} &= r_1 \cdot \mathbf{X}_{a,d} + (1-r_1) \cdot \mathbf{X}_{b,d} + c_1 \cdot (\mathbf{X}_{a,d} – \mathbf{X}_{b,d}) \\
\mathbf{MHC}_{b,d} &= r_2 \cdot \mathbf{X}_{b,d} + (1-r_2) \cdot \mathbf{X}_{a,d} + c_2 \cdot (\mathbf{X}_{b,d} – \mathbf{X}_{a,d})
\end{aligned}
$$
where \(d\) is the dimension index, \(r_1, r_2 \in [0,1]\), and \(c_1, c_2 \in [-1,1]\) are random numbers.
Vertical Crossover (VC): Operates on different dimensions within a single particle to exploit potential variable dependencies. For a particle \(\mathbf{X}_a\), two dimensions \(d_1\) and \(d_2\) are crossed:
$$ \mathbf{MVC}_{a, d_1} = R \cdot \mathbf{X}_{a,d_1} + (1-R) \cdot \mathbf{X}_{a,d_2} $$
where \(R \in [0,1]\). Note: VC requires parameters to be normalized to a common range (e.g., [0,1]) before operation.
The hierarchical application involves three rounds:
- Round 1 (Elite-Champion HC): Horizontal crossover between particles in the elite archive \(G\) and the updated Champion particles.
- Round 2 (Runner-up-Third-place HC): Horizontal crossover between updated Runner-up and Third-place particles.
- Round 3 (Population-wide VC): Vertical crossover applied to the entire merged population.
From the combined pool of original particles, updated particles, and crossover-generated offspring, the best \(N\) particles are selected based on non-domination and crowding distance to form the next generation.
The complete flowchart of the TCCMOPSO algorithm for optimizing the grid-tied inverter filter parameters involves initialization, fitness evaluation, elite selection, tri-competitive update, hierarchical crisscross search, and environmental selection, iterating until a termination criterion is met.
Simulation Results and Analysis
A simulation model of a three-phase PV grid-tied inverter with an LCLLC filter was built in MATLAB/Simulink to validate the proposed design method. System parameters are: DC link voltage \(U_{DC} = 700V\), grid phase voltage \(U_g = 220V\) (rms), grid frequency \(f_0 = 50Hz\), switching frequency \(f_{sw} = 10kHz\), and rated output power \(P_0 = 10kW\). The TCCMOPSO algorithm was configured with a population size \(N=120\), maximum iterations \(I=200\), and elite archive size \(\alpha=9\).
Pareto-Optimal Solutions from TCCMOPSO
The algorithm successfully generated a set of Pareto-optimal solutions, representing different trade-offs among the three objectives. A selection of these optimal parameter sets is presented in Table 1.
| Sol. # | L1 (mH) | L2 (mH) | C (μF) | Lf (mH) | Cf (μF) | Ltotal (mH) | Ctotal (μF) | Cost W | THD (%) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 0.63 | 1.25 | 0.20 | 0.78 | 0.32 | 2.66 | 0.52 | 3.18 | 0.86 |
| 2 | 1.69 | 1.25 | 0.29 | 0.97 | 0.26 | 3.91 | 0.55 | 4.46 | 0.50 |
| 3 | 1.72 | 0.96 | 0.50 | 0.48 | 0.53 | 3.16 | 1.03 | 4.19 | 0.52 |
| 4 | 0.52 | 1.30 | 0.30 | 0.12 | 2.12 | 1.94 | 2.42 | 4.36 | 1.06 |
| 5 | 2.81 | 1.11 | 0.20 | 0.30 | 0.85 | 4.22 | 1.05 | 5.27 | 0.94 |
| 6 | 1.71 | 0.54 | 0.75 | 0.60 | 0.42 | 2.85 | 1.17 | 4.02 | 0.55 |
| 7 | 0.97 | 0.31 | 0.86 | 0.87 | 0.29 | 2.15 | 1.15 | 3.30 | 1.06 |
| 8 | 1.65 | 1.20 | 0.21 | 0.88 | 0.29 | 3.73 | 0.50 | 4.23 | 0.53 |
| 9 | 1.69 | 0.23 | 0.51 | 0.28 | 0.88 | 2.20 | 1.39 | 3.59 | 0.82 |
All solutions satisfy the grid code requirement of THD < 5%. Solution #6 was chosen as a balanced compromise, offering very low THD (0.55%) with a moderate cost index \(W = 4.02\). The corresponding waveforms for the inverter-side current and the grid current using this parameter set are shown in Figure 1. The inverter-side current exhibits significant switching ripple (THD ~3.06%), which is effectively filtered by the optimized LCLLC filter, resulting in a smooth, sinusoidal grid current with a THD of only 0.55%.
Comparative Performance Analysis
To benchmark the proposed TCCMOPSO method, the filter design problem was also solved using a standard Genetic Algorithm (GA), a single-objective PSO (weighted-sum approach), and a conventional MOPSO algorithm. The best-compromise solutions from each method are summarized in Table 2.
| Algorithm | L1 (mH) | L2 (mH) | C (μF) | Lf (mH) | Cf (μF) | Ltotal (mH) | Ctotal (μF) | Cost W | Grid Current THD (%) |
|---|---|---|---|---|---|---|---|---|---|
| GA | 0.76 | 0.66 | 1.75 | 0.34 | 0.75 | 1.76 | 2.50 | 4.26 | 1.08 |
| PSO | 1.20 | 0.89 | 0.47 | 0.36 | 1.94 | 2.45 | 2.41 | 5.86 | 1.28 |
| MOPSO | 1.02 | 1.04 | 0.56 | 0.08 | 3.28 | 2.14 | 3.84 | 5.98 | 1.22 |
| TCCMOPSO | 1.71 | 0.54 | 0.75 | 0.60 | 0.42 | 2.85 | 1.17 | 4.02 | 0.55 |
The comparative analysis reveals clear advantages of the proposed method for the grid-tied inverter application:
- Superior Filtering Performance: TCCMOPSO achieves the lowest grid current THD (0.55%), significantly better than GA (1.08%), PSO (1.28%), and MOPSO (1.22%). The harmonic spectra confirm that the TCCMOPSO-designed filter provides deeper attenuation at the switching frequency and its multiples.
- Optimal Cost-Performance Trade-off: The cost function \(W\) for the TCCMOPSO solution is 4.02, which is lower than both PSO (5.86) and MOPSO (5.98), and comparable to GA (4.26). Crucially, TCCMOPSO achieves this low cost while also delivering the best THD performance, whereas the GA solution with similar cost has over double the THD.
- Reduced Power Loss: Since the reactive power loss is proportional to the total capacitance \(C_{total}\), the TCCMOPSO design, with the smallest \(C_{total}\) (1.17 μF), will inherently have the lowest fundamental-frequency capacitive loss among all compared designs. This is a critical efficiency advantage for the grid-tied inverter system.
The grid current waveforms from all four methods, plotted together, visually demonstrate the smoother sinusoidal output achieved by the TCCMOPSO-optimized filter. The convergence characteristics of the TCCMOPSO algorithm also showed a more rapid and stable approach to the Pareto front compared to standard MOPSO, attributed to its competitive and crisscross operators enhancing exploration and exploitation.
Conclusion
This paper has addressed the critical challenge of optimal parameter design for LCLLC filters in three-phase photovoltaic grid-tied inverters. By formulating the design problem as a constrained multi-objective optimization with three practical goals—minimizing current ripple, maximizing harmonic attenuation, and minimizing combined cost/loss—a more holistic and efficient design framework is established. The proposed Tri-Competitive Crisscross Multi-Objective Particle Swarm Optimization (TCCMOPSO) algorithm effectively solves this problem by improving population diversity, convergence speed, and the ability to escape local optima through its novel update and search strategies. Simulation results for a 10 kW grid-tied inverter system validate the methodology. The TCCMOPSO-derived LCLLC filter parameters not only ensure grid current THD (0.55%) far below the standard limit but also achieve a superior balance between performance and economic/reactive power loss indicators when compared to designs from GA, PSO, and standard MOPSO algorithms. This method provides a powerful and practical tool for engineers to design high-performance, cost-effective filters for modern renewable energy grid-tied inverters.