Continuation Power Flow

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

(5.1)

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

(5.2)

are restructured as

f (x,λ) = g(x) − λb = 0

(5.3)

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

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

(5.4)

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

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