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The paper proposes a novel nonlinear acoustic echo cancellation approach based on adaptive second-order Volterra structures designed to increase the convergence rate of the conventional Volterra filter. Depending on the envelope of the resulting error's absolute value or, if available, on the local noise levels from the enclosure, the variable step sizes are weighted according to the decrease of the residual error. The efficiency of the proposed method is tested in a loudspeaker-enclosure-microphone setup modeled using measured linear and quadratic kernels. The effectiveness of the proposed algorithm based on normalized least-mean-square updates is then compared to the Normalized Least-Mean-Square adaptive second-order Volterra filter. The comparison is carried out for different input signals in terms of Echo Return Loss Enhancement. The conducted simulations show that the proposed method offers an increased convergence rate for the same steady-state error.