This paper considers a structural-factor approach to modeling high-dimensional time series. We decompose individual series into trend, seasonal, and irregular components and consider common factors for the irregular components. For analyzing many time series, we employ a time polynomial for the trend and a linear combination of trigonometric series for the seasonal component. A new factor model is then proposed for the irregular components to simplify the modeling process and to achieve parsimony. We propose a Bayesian Information Criterion to consistently determine the order of the polynomial trend and the number of trigonometric functions. A test statistic is used to determine the number of common factors. The convergence rates for the estimators of the trend and seasonal components and the limiting distribution of the test statistic are established under the setting that the number of time series tends to infinity with the sample size, but at a slower rate. We use simulation to study the performance of the proposed analysis in finite samples and apply the proposed approach to model weekly PM2.5 data observed at 15 monitoring stations on Taiwan.