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### 5.2.3. Approximation of Dissipation Energy Component

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It is to be noted that the term in the energy expression corresponding to the dissipation component consists of a path dependent integral. This cannot be evaluated unless the system trajectory is known. Uyemura and colleagues [1] developed an approximation to this term based on a linear angle trajectory assumption. This approximation was then used by Athay and colleagues [2] and compared with the actual dissipation energy component evaluated along the trajectory. From the experience gained from numerous simulations, it has been found that for a first swing transient, this approximation is fairly accurate and has become standard practice in the TEF method to incorporate the effect of transfer conductances.

The approximation is derived as follows: let θ_{i}(t) and θ_{j}(t) represent the angular trajectories of machines i and j with respect to the time t, respectively. Let θ^{a} and θ^{b} represent the initial and final vectors, of angular positions for the n machines. The linear angle trajectory assumption is invoked between θ^{a} and θ^{b} and is expressed by