Let's say we have the following complex number, $V$:
$V=V_i+jV_j=V_m\angle{V_a}$
Which type of representation is better in an NLP (polar or rectangular)?
The polar form leads to one nasty constraint and two linear constraints:
$(\alpha V_{m,1}^2 + \beta V_{m,1}V_{m,2}\cos{(V_{a,1}-V_{a,2})}) = X$
$V_{m,1}<=V_{max}$
$V_{m,2}<=V_{max}$
The rectangular form leads to three second order constraints:
$(\alpha' V_{i,1}^2 + \beta' V_{i,1}^2 + \gamma' V_{i,1}V_{j,2} + \lambda' V_{j,1}V_{i,2}) = X$
$V_{i,1}^2+V_{j,1}^2 \leq V_{max}^2$
$V_{i,2}^2+V_{j,2}^2 \leq V_{max}^2$
I asked some mathematicians in a conference and the said that it depends but in most cases the rectangular form is preferred because of avoiding the sinoidal term. What is so bad about the Sinosoidal terms that avoiding them is preferred> even though it might lead to more nonlinear constraints!