Line Search Techniques For Optimal Multiplier Newton Raphson For Ill-Conditioned Or Stressed

Networks

Mazhar Ali PhD. Power Systems (Student)

Mathematical Modeling: Internet & Power Systems Preliminary project Presentation

Problem Statement

1.  Networks are being operated in a closer proximity of their stability limits.

2.  Conventional Newton Raphson (NR) doesn’t convergence (due to stressed loading condition).

3.  In this case the convention NR might take several iterations (In other words, it will not converge quadratically) or might not converge at all.

4.  One can address this problem with step size and can compute an optimal step size or optimal multiplier at each NR iteration.

(Ensuring the convergence and will always able to find line approximation for NR.

Goals:

1.  Developing an NR power flow algorithm for Cartesian and polar coordinates.

2.  Formulating a Mathematical model for optimal multiplier computations. 3.  Proposing some approximation for Hessian Matrix (as described in the

following text) for fast computation. 4.  Testing IEEE- Networks (Solvable, Unsolvable, Stressed and Un-Stressed

Networks) & comparison with normal NR. 5.  Making some significant conclusions and comparison with existing

approaches.

Newton Raphson

Approximate ! ! with 1st Order Taylor series about !! : a straight line

! !! + !"!" !! !!!! − !! = 0!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Update Step : ∆!! = [!"!" !! ]!!!(!!) !!!! = !! − !∆!!

Jacobian Matrix=! = [!"!" !! ]!! ; Here: != Step Size

Minimization of threshold: Until ∆!!!! < !ℎ!"#ℎ!"#!!"! ∆!!(!!!!) < !ℎ!"#ℎ!"#

!

Power Flow Model  !Polar!Formulation:!

!!" = !!" + !!" !and!!! = !! ∠!! !

!!!"#$% ! = ! !! !! !!" cos !! − !! + !!" sin !! − !! !!

!!!+ !!"#$ ,!!!! − !!"#,!!!! = 0!

!!

!!!"#$% ! = ! !! !! !!" sin !! − !! + !!" cos !! − !! !!

!!!!+ !!"#$ ,!!!! − !!"#,!!!! = 0!

!Rectangular!Formulation:!

!!" = !!" + !!" !and!!!! = !!! + j!!! !!

!!!"#$ = [!

!!!!!!(!!"!!! − !!"!!!)+ !!!(!!"!!! + !!"!!!)]+ !! ,! − !! ,! = 0

!!!"#$ = [!

!!!!!!(!!"!!! − !!"!!!)− !!!(!!"!!! + !!"!!!)]+ !! ,! − !! ,! = 0

PV Buses:

!!!"#$ ! ! = !!! ! + !!! ! − !! !"#$%&%#'! = 0

!!!

Mathematical Model ! ! = 0

Here !! ∈ ℝ!! ,! is the vector of system variables described in more detail below, !(!):!ℝ! !×!ℝ!! → !ℝ!! !is the

system of nonlinear equations representing the power flow equations.

We can derive expression for optimal Multiplier ! at iteration (!) of!! ! , by expanding equation in !!(!) in a second order Taylor series expansion:

!! !!! ≈ !! ! ! + !!!∆! ! ≈ !!! ! ! + ∇!"!! ! ! ! ∆! ! + 12 ∆! ! ! ∇!!! !!! ! ! ! ∆!(!)

!Minimization!Problem:!!

!! !!! = ! !! ! ! + !! ∇!"!! ! ! ! ∆! ! + !!

2 ∆! ! ! ∇!!! !!! ! ! ! ∆!(!)

!(!) = !! ! !!! ! ………!! ! !

!(!) = arg !min! !12 ! !!! ! ! !!!

= arg !min! !12 ! !!! ! !

∇!!! !!! ! =

!!!!(!)!!!!

!!!!(!)!!!!! … !!!!(!)

!!!!!!!!!(!)!!!!!

!!!!(!)!!!! … ⋮

⋮!!!!(!)!!!!!

⋮!!!!(!)!!!!!

⋱…

⋮!!!!(!)!!!!

!

Considerations: