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The rapid development of digital economy has led to a dramatic increasement of internet data centers (IDCs), which consume a large amount of electricity. The diverse data-processing demands, high power consumption and workload uncertainty put forward a high requirement for the economical and secure operation of IDCs. As information technology (IT) devices are driven by direct current (DC), flexible substation (FS) has been gradually utilized to provide DC power for IDCs. This paper proposes robust operation strategies for minimizing IDC power consumption with FS integration. First, the linearized IDC power consumption model based on the technology of dynamic voltage and frequency scaling (DVFS) is proposed to describe the operating state of IT devices. Then, considering the FS-based coordinated operation of IDC, photovoltaic (PV) station and energy storage system (ESS), the deterministic power consumption minimization model of IDCs is established. Considering the workload uncertainty, the operation strategies based on distributionally robust optimization (DRO) for IDCs are further proposed. Finally, the effectiveness of proposed method is validated on a modified practical network with IDCs integrated. Results show that through the dispatch of workload and power flow regulation of FS, the power consumption and dropping workload of IDCs are effectively reduced.