Page 36 - IJEEE-2023-Vol19-ISSUE-1

32 | Basim T. Kadhem

The controller reduction approach uses the same concept. .456 = K"-. .-+ %..+ - .%*. .% L (15)
0
Based on the Schur balanced model reduction process [15], our

approach involves model and controller reduction. In this where Ar, Br, and Cr are the reduced-order plant's state space

instance, the reduction goal is described as follows. matrices for the design (Dr = 0). In comparison to the original
kth -order reduced model computation From an nth-order
complete model, Gr(s) = Cr(sI - Ar)-1Br + Dr such that G(s) = ? 29th-order designed controller, Fig. 8 shows the singular value

(sI - ?)-1 ? + D plot for a range of decreased size controller options. For a

?@ - @.?, = 2 ?/0123* D/ (11) variety of reduced size controller options, Fig. 9 also shows the

error bound (the infinity norm of the difference between full

and reduced controller).

where si denotes the Hankel singular values of G(j?), i.e., the

square roots of the eigenvalues of their controllability and

observability grammians

D/ = EF/(57) (12)

where li (PQ)is the ith largest eigenvalue of PQ and P, Q are

the solutions of the following Lyapunov equalities:
5"( + "5 + %%( = 0 (controllability grammian) (13)

7" + "J7 + *J* = 0 (absorbability grammian) (14)

Note that whereas Ar, Br, Cr, and Dr are the state space

matrices of the reduced-order model ?, ?, ?, and D are those

of the full order model G(s). The use of numerical techniques,

such as the Krylov subspace-based technique, may be

necessary in situations where there are many state variables

(i.e., more than 1000), as the analytical techniques by

themselves will not be sufficient [22].

For the purpose of designing controllers, a lower order

model is obtained using the best Hankel norm approximation

technique. It provides the best reduced order model for the Fig.7. LTR method at plant input for various q parameter
values.
given order [16–18] by minimizing the difference between the

nominal and reduced order models' frequency responses.

Given that it is numerically efficient and that the provided error

bound can be used as a criterion to set the order of the

reduction, it meshes nicely with the LQG design.

V. CENTRALIZED CONTROL METHOD Fig.8. Singular value plot of controller approximation.

A. Application of LTR The 14th order controller is nearly indistinguishable from
the original 29thorder (full order), as can be seen in Fig. 8,
The LTR technique for q = 1, 5, 10, 100, and 1000 is while deterioration begins to occur as the order is further
shown in Fig.7 and (10). To illustrate the qualities of reduced. This is further supported by Fig. 9, which
high-quality sensor equipment, the measurement noise demonstrates how the error bound significantly increases after
covariance is chosen to be quite low [22]. For the sake of selecting the 14th order controller. From Fig. 9, it can be
suitable comparison, the controller that produced the results in
Fig. 7 is the full 29th order. For the remainder of the design, q
was fixed at 10, resulting in a sufficient recovery within the
interest frequency range and a quicker roll-off at high
frequencies than q=1000.

B. LQG Controller Reduction

The complete order controller in this system is of 29th
order. The planned LQG controller is of the 14th order, which is
equivalent to the order of the design (reduced) plant, and its
transfer function is shown in equation (15). The controller size
should be further minimized while still meeting the necessary
damping ratios for the torsional modes for the entire power
network model. The Schur balanced reduction approach,
which is covered in Section IV [15], is used for the reduction
process.