Modelling of energy service networks - Modelling Distributed Energy Resources in Energy Service Networks - page 45

Table 3.3 Non-diagonal Jacobian elements present

based on node classification

PV node m PQ node m

PV node k H km H km , N km

PQ node k H km , J km H km , N km , J km , L km

the Jacobian matrix and based on node classification, it must be established which
entries will be required.
For the diagonal elements, the matrix can have one or four terms, depending on
the classification of node k being either a PQ or a PV node. If node k is a PV node,
then the reactive power will be unknown; this makes term Q k unnecessary and
discards entries J kk and L kk . Similarly, since voltage magnitude is fixed in this type
of node, entry N kk can be omitted. Hence, a generator node has the single entry H kk .
Now, if node k is a load node, all four elements H kk , N kk , J kk and L kk are considered.
On the other hand, the non-diagonal elements require either one, two or four entries,
depending on the type of node(s) they are linked to in the network. In summary,
Table 3.3 categorises and establishes the alternatives that might arise when building
km entries [176].
Overall, two cases may arise when building first partial derivatives: k = m and

k = m , also commonly known, respectively, as the self and mutual Jacobian terms.

For case k = m :

δP k
δθ k =−

V k B kk −

H kk = Q Tk (3.61)

δP k
δV k /V k =

V k G kk +

=

N kk P Tk (3.62)

δQ k
δθ k =−

V k G kk +

J kk = P Tk (3.63)

δQ k
δV k /V k =−

V k B kk +

L kk = Q Tk (3.64)

For case k = m :

δP k
δθ m =

H km = f k a km − e k b km (3.65)

δP k
δV m /V m =

N km = e k a km + f k b km (3.66)

δQ k
δθ m =−

J km = e k a km − f k b km =− N km (3.67)

δQ k
δV m /V m =

L km = f k a km − e k b km = H km (3.68)