#### 4.5 Linear Shift Factors

The DC power ﬂow model can also be used to compute the sensitivities of branch ﬂows to
changes in nodal real power injections, sometimes called injection shift factors (ISF) or
generation shift factors [20]. These sensitivity matrices, also called power
transfer distribution factors or PTDFs, carry an implicit assumption about the
slack distribution. If is used to denote a PTDF matrix, then the element in
row and column , , represents the change in the real power ﬂow in
branch given a unit increase in the power injected at bus , with the assumption
that the additional unit of power is extracted according to some speciﬁed slack
distribution:

| (4.34) |

This slack distribution can be expressed as an vector of non-negative
weights whose elements sum to 1. Each element speciﬁes the proportion of the slack taken
up at each bus. For the special case of a single slack bus , is equal to the vector
. The corresponding PTDF matrix can be constructed by ﬁrst creating the
matrix

| (4.35) |

then inserting a column of zeros at column . Here and are obtained from
and , respectively, by eliminating their reference bus columns and, in the case of
, removing row corresponding to the slack bus.

The PTDF matrix , corresponding to a general slack distribution , can be
obtained from any other PTDF, such as , by subtracting from each column,
equivalent to the following simple matrix multiplication:

| (4.36) |

These same linear shift factors may also be used to compute sensitivities of branch
ﬂows to branch outages, known as line outage distribution factors or LODFs [21]. Given a
PTDF matrix , the corresponding LODF matrix can be constructed as
follows, where is the element in row and column , representing the change in
ﬂow in branch (as a fraction of the initial ﬂow in branch ) for an outage of
branch .

First, let represent the matrix of sensitivities of branch ﬂows to branch endpoint
injections, found by multplying the PTDF matrix by the node-branch incidence
matrix:

| (4.37) |

Here the individual elements represent the sensitivity of ﬂow in branch with
respect to injections at branch endpoints, corresponding to a simulated increase in ﬂow
in branch . Then can be expressed as

| (4.38) |

Matpower includes functions for computing both the DC PTDF matrix and the
corresponding LODF matrix for either a single slack bus or a general slack distribution
vector . See the help for makePTDF and makeLODF and Sections 9.5.5 and 9.5.4,
respectively, for details.