Combinatorial interaction testing (CIT) is an important technique for testing highly configurable software systems with demonstrated effectiveness in practice. The goal of CIT is to generate test cases covering the interactions of configuration options, under certain hard constraints. In this context, constrained covering arrays (CCAs) are frequently used as test cases in CIT. Constrained Covering Array Generation (CCAG) is an NP-hard combinatorial optimization problem, solving which requires an effective method for generating small CCAs. In particular, effectively solving t-way CCAG with t>=4 is even more challenging. Inspired by the success of automated algorithm configuration and automated algorithm selection in solving combinatorial optimization problems, in this paper, we investigate the efficacy of automated algorithm configuration and automated algorithm selection for the CCAG problem, and propose a novel, automated CCAG approach called AutoCCAG. Extensive experiments on public benchmarks show that AutoCCAG can find much smaller-sized CCAs than current state-of-the-art approaches, indicating the effectiveness of AutoCCAG. More encouragingly, to our best knowledge, our paper reports the first results for CCAG with a high coverage strength (i.e., 5-way CCAG) on public benchmarks. Our results demonstrate that AutoCCAG can bring considerable benefits in testing highly configurable software systems.