A Design and Mathematical Modeling of the Novel L(LCL)₃ Hybrid Low-Pass Filter for Power Grid-Connected Solar PV Systems

Abstract

In the face of depleting natural resources, the efficient use of available energy sources is becoming increasingly important in reducing operational costs while satisfying ever-tighter pollution regulations. This paper presents the complete mathematical design framework and component-calculation methodology for the L(LCL)₃ filter, a novel high-order hybrid low-pass filter topology developed for the integration of utility-scale solar photovoltaic (PV) systems into high-voltage grid networks. The L(LCL)₃ filter extends the established LCL and LLCL filter architectures by incorporating three resonant trap circuits tuned to the inverter switching frequency (fsw=10 kHz), its second harmonic (20 kHz), and a third attenuation band between four series inductors (L1,L2,L3,L4) distributed in a 0.40:0.25:0.20:0.15 ratio. The design employs a rigorous per-unit system analysis using HVDC-class constraints: total inductance ≤ 8% of base inductance, total capacitance ≤ 3% of base capacitance (limiting reactive power to ≤ 3 MVAr), and current ripple ≤ 20% of base current. For a 100 MVA, 33 kV reference system with fgrid = 50 Hz and fsw=10 kHz, complete component calculations yield: L1 = 1.108 mH,L2 = 0.693 mH,L3 = 0.554 mH,L4 = 0.416 mH; C1 = 1.3 µF,C2 = 0.36 µF,C3 = 0.11 µF and damping network parameters Rd1 = 6.0 Ω,Rd2 = 4.0 Ω, Rd3 = 3.0 Ω, Rd4 = 2.0 Ω. The filter achieves a damping ratio ζ = 0.196 and quality factor Q = 2.55, ensuring well-damped operation with a primary resonance at 8,011 Hz. Resonance analysis confirms trap frequencies of 10,078 Hz (Trap 1) and 20,396 Hz (Trap 2) with less than 2% design error. MATLAB/Simulink simulation confirmed the analytical claims of grid-side THD 1.3% and primary resonance frequency within 3% of the analytically derived values, with filter efficiency exceeding 99.0% under rated operating conditions. The mathematical framework is presented in sufficient detail for direct replication and adaptation to other power ratings.