#### 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.