Real-Time Precise Orbit Determination (RTPOD) of Low-Earth-Orbit (LEO) satellites can greatly expand their ability to perform complex scientific missions, such as real-time environment monitoring, maneuver control and satellite autonomous navigation. In this paper, the model of real-time kinematic precise orbit determination is introduced. We present a conception that the LEO Priori Orbit Constraint (POC) is used in the process of RTPOD for the sake of improving the accuracy, convergence speed and stability. The broadcast ephemeris, predicted part of ultra-rapid ephemeris and real-time precise ephemeris are adopted respectively to propose 6 different RTPOD solutions, which are then demonstrated and analyzed using the observations from Swarm A/B/C satellites during 7 days. The results show that the accuracy is improved in turn by using broadcast ephemeris, IGU and IGC ephemeris. Moreover, adding POC can further enhance the result while using the same ephemeris. The IGC+POC solution using the priori orbit with a 1m standard deviation reaches an accuracy of 6.12cm, 5.55cm and 4.98cm in the radial, along and cross component, respectively, which is comparable to the post-processing kinematic POD. Analyses based on different priori orbits indicate that the ideal priori orbit should appear less noise and long-term systematic biases, and short-term systematic biases show little influence on constraint results. Furthermore, adding POC can remarkably speed up the convergence. The convergence of IGC solution needs about 31min on average, whereas the average convergence time after adding POC with a 1m standard deviation is about only 4min, which is beneficial to the fast re-convergence after the occurrences of cycle slip, loss of lock and communication link interruption, and is of great significance in practical application scenarios.