5 Continuation Power Flow

Continuation methods or branch tracing methods are used to trace a curve given an initial point on the curve. These are also called predictor-corrector methods since they involve the prediction of the next solution point and correcting the prediction to get the next point on the curve.

Consider a system of n  nonlinear equations g (x ) = 0,x ∈ ℝn  . By adding a continuation parameter λ  and one more equation to the system, x  can be traced by varying λ  . The resulting system f(x,λ) = 0  has n + 1  dimensions. The additional equation is a parameterized equation which identifies the location of the current solution with respect to the previous or next solution.

The continuation process can be diagrammatically shown by (5.1).

(  j  j) Predictor   j+1  j+1  Corrector ( j+1  j+1)
 x ,λ   −− −−−→  (ˆx   ,ˆλ   ) −−− −−→   x   ,λ

where, (xj,λj)  represents the current solution at step j  , (ˆxj+1,ˆλj+1)  is the predicted solution for the next step, and   j+1  j+1
(x   ,λ   )  is the next solution on the curve.

Continuation methods are employed in power systems to determine steady state stability limits [22] in what is called a continuation power flow29 . The limit is determined from a nose curve where the nose represents the maximum power transfer that the system can handle given a power transfer schedule. To determine the steady state loading limit, the basic power flow equations

       [           inj ]
g(x) =   P (x) − P inj   = 0,
         Q (x) − Q

are restructured as

f (x,λ) = g(x) − λb = 0

where x ≡ (Θ, Vm)  and b  is a vector of power transfer given by

    [   inj      inj ]
b =    Ptarget − P base  .
      Qintjarget − Qinbjase

The effects of the variation of loading or generation can be investigated using the continuation method by composing the b  vector appropriately.

 5.1 Parameterization
 5.2 Predictor
 5.3 Corrector
 5.4 Step Length Control
 5.5 Event Detection and Location
 5.6 runcpf
  5.6.1 CPF Callback Functions
  5.6.2 CPF Example