Accurate day-ahead electricity price forecasting is crucial for both power companies' interests and systems' stability. However, it faces numerous challenges due to various uncertainties, especially the increasingly frequent extreme weather events. Australia’s National Electricity Market (ANEM) experiences the highest price volatility. To improve the forecasting accuracy in ANEM, this paper proposes a residual electricity price forecasting method with Kolmogorov-Arnold Networks (KAN) for day-ahead 30-min electricity price prediction. Unlike the Multi-Layer Perceptron (MLP), KAN includes learnable activation functions and shifts the previous learning from points to edges, making it more sensitive to features. Thus, KAN can capture more complicated nonlinearities of electricity prices in ANEM. Furthermore, we use discrete Fourier transform to create activation functions to reduce the effects of wild price swings and ensure a stable learning process. Our experimental results, based on actual electricity price data from the Australian Energy Market Operator, demonstrate that our method is more accurate than the traditional MLP approach.