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In this paper, the model of Archimedes wave swing (AWS) based wave energy conversion (WEC) system is proposed where a new coordinate transformation for the linear permanent magnet generator between the abc frame of reference and the dq0 frame of reference is proposed and the model of the LPMG in dq0 reference frame suitable for power system dynamic and stability analysis is derived for the first time. Since the output power and the induced voltage both vary, full-scale back-to-back converters are employed to connect the WEC to the power grid in order to maintain a constant active power and terminal voltage at the system side. The controllers for the generator side converter and the grid side converter of the back-to-back converters are also developed. Simulations are performed, when WEC operates under conditions such as no-load, supplying an isolated R-C load and integration with a power system, to simulate the dynamics of the WEC and verify the effectiveness of the model derived and the controllers developed. Comparison between the dynamic responses simulated and those obtained in the field test demonstrates that using the proposed dq0 model, the dynamics of the WEC system can be simulated properly. Using the generator side converter controller, the maximum power can be extracted from the wave while the loss in the LPMG can be minimized. Furthermore, the WEC can keep the output power at the power grid converter side constant and maintain the terminal voltage constant with the grid side converter controller. These are the desired features for the WEC to operate in parallel with the power grid. The effectiveness of the controller proposed is also validated under small disturbance.