Modelling of energy service networks - Modelling Distributed Energy Resources in Energy Service Networks - page 44

Assuming the value given by each iteration is sufficiently close to its previous
value, it is possible to neglect expressions beyond the first derivative ( i.e. higher order
terms).

f ( x 0 ) x f ( x 0 + x ) = f ( x 0 ) + = 0 (3.55)

Therefore, the suggested variation x that should make f ( x 0 + x ) approach 0 is:

=− f ( x 0 ) − 1 f ( x 0 )

x (3.56)

where [ f ( x 0 )] − 1 is the matrix of first partial derivatives and popularly referred to as

the Jacobian ( J ).
If simplified, the above expression for the i -th iteration can be expressed as:

=− J i − 1 f ( x i 0 ) x i + 1

(3.57)

Once the variation in the state variables is calculated, the iterative approxima-
tion of the state variables can be updated as a function of its values of the previous
iteration plus the correction values of its following iteration:
x i + 1
0

x i 0 + x i + 1

=

(3.58)

The calculations are repeated as many times as necessary until x is within the
accepted tolerance value , such as stated in (3.31), (3.32) and (3.52) for electrical
and natural gas systems. Consequently, in order to apply load flow problems within
the context of the Newton-Raphson method, the relevant expressions for energy
service networks must be arranged in the form of equation (3.56).
3.3.2.1 The electrical system Jacobian matrix
For the electrical system load flow, the state variables of the system represented by
x 0 in (3.56) are the nodal voltage magnitudes and phase angles. In consequence, the
Newton-Raphson algorithm can be presented by the following vector relationship:

δ δθ − 1

P
Q

δP
δV /V θ

V / V

=− (3.59)

δQ
δθ

δQ
δV /V

The number of elements in the Jacobian matrix for electrical systems consists
of twice the number of PQ nodes plus the PV nodes minus the slack node. It is
important to clarify that the correction terms V are divided by V to compensate
for the fact that some Jacobian elements are multiplied by V . This fact yields simplified
calculations in the derivative terms.
The Jacobian matrix is formed by using entries H , N , J and L from the node
connectivity data; thus, taking (3.59) as a basis the expression becomes:
P
Q

HN
JL

θ
V / V

= (3.60)

As seen in section 3.1.3, terms in the electrical system regarding voltage, admit-
tance and current are employed to build the Jacobian matrix. Hence, (3.24)-(3.26)
are considered when element km connects nodes k to m . However, before elaborating