Modelling embedded technologies in energy service networks - Modelling Distributed Energy Resources in Energy Service Networks - page 55

By separating the active and reactive power flow terms from the element des-
cribed above in (4.7), the following expressions are deduced:

2 2 G km − |

=+|

V k |

| |

V k || V m ||

|

G km cos ( θ k − θ m + P km t t τ )

− | V k || V m || t | B km sin ( θ k − θ m + τ ) (4.8)

2 2 B km − |

Q km =−| V k | | t | V k || V m || t | G km sin ( θ k − θ m + τ )

+ | V k || V m || t | B km cos ( θ k − θ m + τ ) (4.9)

In a similar fashion to (4.5)-(4.7), the complex power travelling from node m
to node k can be determined:

V m ( I mk ) ∗ = I km ) ∗

S mk = V m ( − (4.10)

V m ( V k − V m ) ∗ y km =− V m ) ∗ y km

S mk =− V m ( V k | t | ∠ τ − (4.11)

|

τ ) ( G km −

2

S mk = V m | − | V m || V k || t | ∠ ( θ m − θ k − jB km ) (4.12)

Term (4.12) is analogous to term (4.7); therefore, the active and reactive power
flows going in direction m to k are described as:
P mk

2 G km − |

=+| V m | V m || V k || t | G km cos ( θ m − θ k − τ )

− | V m || V k || t | B km sin ( θ m − θ k − τ ) (4.13)

2 B km − |

Q mk =−| V m | V m || V k || t | G km sin ( θ m − θ k − τ )

+ | V m || V k || t | B km cos ( θ m − θ k − τ ) (4.14)

The losses in element km for active power are obtained by summing terms (4.8)
and (4.13). Similarly, adding (4.9) and (4.14) yields the reactive power difference.

2 2 G km + | 2 G km

P Lkm =+| V k | | t | V m |

2 G km | V k || V m ||

|

− t cos ( θ k − θ m + τ ) (4.15)

2 2 2 ) B km

|

V k |

| |

+ | V m |

Q Lkm =− ( t

+

2 B km | V k || V m || t | cos ( θ k − θ m + τ ) (4.16)

and phase
shift angle τ variables have the capability of altering the losses in element km .
Since most equipment connected to the electricity system generates or absorbs
reactive power, it is important to clarify the meaning of variable Q Lkm . Devices
which store energy by virtue of a 'magnetic field' produced by a flow of current are
said to absorb reactive power, while those which store energy by virtue of 'electric
fields' are said to generate reactive power. Hence, the value of the reactive power in
element km has a significant meaning based on the above rules:
where if Q Lkm is > 0, then reactive power is required by element km
if Q Lkm is < 0, then reactive power is provided by element km
Further details are given in Appendix E on the first and second partial deriva-
tives for active, reactive and loss flow calculation occurring at OLTC element km .

As it can be seen from (4.15) and (4.16), both the tap magnitude | t |