52 Handb ook of Big Data
(m −r) ×n matrix representing the dynamics of noncharacteristic generators; x is an r ×n
matrix and can be calculated from Equation 4.5; and κ
ξ
is an r × r square matrix; κ
¯
ξ
is an (m − r) × r matrix. Normally, κ
ξ
is invertible. We have two different approaches to
finding the approximate linear relations between
¯
ξ and ξ. The first approach is to solve the
following overdetermined equation:
¯
ξ = Cξ (4.8)
where C is an (m − r) × r matrix and can be determined by the least-squares method,
namely, C =
¯
ξ[(ξξ
T
)
−1
ξ]
T
. Another approach is to use the approximate linear relations in
Equation 4.8. According to Equation 4.8, we have
ξ ≈ κ
ξ
x (4.9)
and
¯
ξ ≈ κ
¯
ξ
x (4.10)
Premultiplying κ
−1
ξ
on both sides of Equation 4.9 yields
x ≈ κ
−1
ξ
ξ (4.11)
Substituting Equation 4.11 into 4.10 yields
¯
ξ ≈ κ
¯
ξ
κ
−1
¯
ξ
ξ (4.12)
Equation 4.8 or 4.12 establishes the approximate linear relations between the rotor angle
dynamics of characteristic generators and that of noncharacteristic generators. The dyna-
mics of all generators in the original system then can be reconstructed by using only the
dynamic responses from characteristic generators.
4.2.1.2.4 Generalization to High-Or der Models
In classical models, it is assumed that the magnitude of the generator internal voltage E
is
constant, and only its rotor angle δ changes after a disturbance. In reality, with the generator
excitation system, E
will also respond dynamically to the disturbance. The dynamics of E
can be treated in the same way as the rotor angle δ in the above-mentioned model reduction
method to improve the reduced model, except thatthesetofcharacteristic generators needs
to be determined from δ.Thisway,bothδ and E
of noncharacteristic generators will be
represented in the reduced model using those of the characteristic generators.
4.2.1.2.5 Online Application of the DEAR Method
For offline studies, the DEAR process can be performed at different conditions and operating
points of the target system (external area) to obtain the corresponding reduced models.
For online applications, however, computational cost may be very high if SVD has to be
calculated every time the system configuration changes. A compromise can be made by
maintaining a fixed set of characteristic generators, which is determined by doing SVDs
for multiple scenarios offline and taking the super set of the characteristic generators from
each scenario. During real-time operation of the system, the approximation matrix C from
Equation 4.8 used for feature reconstruction, is updated (e.g., using the recursive least-
squares method) based on a few seconds data right after a disturbance. This way, SVD is
not needed every time after a different disturbance occurs.
4.2.1.3 Case Study
In this section, the IEEE 145-bus, 50-machine system [5] in Figure 4.1 is investigated. There
are 16 and 34 machines in the internal and external areas, respectively. Generator 37 at