Software-Defined Wide Area Networks (SD-WANs) are an emerging Software-Defined Network (SDN) technology to reinvent Wide Area Networks (WANs) for ubiquitous network interconnections in cloud computing, edge computing, and the Internet of Everything. The state-of-the-art overlay-based SD-WANs are simply conjunctions of Virtual Private Network (VPN) and SDN architecture to leverage the controllability and programmability of SDN, which are only applicable for specific platforms and do not comply with the extensibility of SDN. This paper motivates us to refactor traditional VPNs with SDN architecture by proposing an overlay-based SD-WAN solution named Software-Defined Virtual Private Network (SD-VPN). An SDN-based auto-constructed VPN model and its evaluating metrics are put forward to automatically construct overlay WANs by node placement and service orchestration of SD-VPN. Therefore, a joint placement algorithm of VPN nodes and algorithms for overlay WAN service loading and offloading are proposed for SD-VPN controllers. Finally, a three-layer SD-VPN system is implemented and deployed in actual network environments. Simulation experiments and system tests are conducted to prove the high-efficiency controllability, real-time programmability, and auto-constructed deployability of the proposed SD-VPN. Performance trade-off between SD-VPN control channels and data channels is evaluated, and SD-VPN controllers are proven to be extensible for other VPN protocols and advanced services.