Учебники / 0841558_16EA1_federico_milano_power_system_modelling_and_scripting

7.2.3Analysis in the Z-Domain

The state matrix in (7.16) leads to the computation of the eigenvalues in the S-domain, i.e., the system is stable if {λh} < 0, h = 1, 2, . . . , nx. To compute the eigenvalues in the Z-domain can lead to some numeric advantages as discussed in the following subsection. The Z-domain can also ease the visualization of sti systems since, in the Z-domain, if the system is stable, all the eigenvalues are inside the unit circle [147]. For obtaining the Z-domain eigenvalues, a bi-linear transformation is performed:

where χ is a weighting factor that, based on heuristic considerations, can be set χ = 8. Computing AZ is more expensive than AS but using AZ can be useful for fastening the determination of the maximum amplitude eigenvalue (e.g., by means of a power method), especially in case of unstable equilibrium points with only one eigenvalue outside the unit circle.

Example 7.4 Eigenvalues of the IEEE 14-Bus System in the Z-Domain

Figure 7.4 shows the eigenvalue analysis for the IEEE 14-bus tests system in the Z-domain. The figure also shows two dotted curves indicating the

Fig. 7.4 Eigenvalues of the IEEE 14-bus system in the Z-domain

associated with λk , then ν(i) → νk for i → ∞. Finally, the eigenvalue λk is computed as:

The rationale of the power method is relatively simple and is worth being briefly outlined. Assume that the initial vector ν(0) can be written as the linear combination of all eigenvectors νh of the matrix A:

where, by hypothesis, ck = 0. Assume also that A is diagonalizable and can be written as N ΛN −1.6 Then, at the ith iteration:

6 A similar proof holds if A is decomposed into its Jordan’s canonical form.

7.3.2Inverse Iteration

The inverse method is a variant of the power method described above and allows finding the eigenvalue λk and the associated eigenvector νk if a good

˜

estimation λk of the eigenvalue is known.

By way of introduction, observe that to find the minimum eigenvalue of a given matrix A, it su ces to apply the power method iteration (7.37) to A−1. In fact, the eigenvalues of A−1 are the inverse of the eigenvalues of A. Clearly, this algorithm works only if the matrix A is not singular.

The inverse iteration consists in applying the power method iteration (7.37)

− ˜ −1 − ˜ −1

to the matrix (A λk In) . In fact, the eigenvalues of (A λk In) are

− ˜ −1 − ˜ −1 − ˜ −1 ˜

(λ1 λk ) , . . . , (λk λk ) , . . . , (λk λn) . Thus, if λk is su ciently close

− ˜ −1 − ˜ −1

to λh, then (λk λk ) is the biggest eigenvalue of (A λk In) , which is the necessary condition for the power method to converge.

7.3.3Rayleigh’s Iteration

The Rayleigh’s iteration is an improvement of the inverse iteration. One

where (7.45) is called the Rayleigh’s quotient. The convergence characteristics of this method are generally better (i.e., cubically) than the inverse iteration.

Example 7.5 Inverse and Rayleigh’s Iterations for the IEEE 14-Bus System

For the sake of example, consider the matrix AZ defined in (7.35), ν(0) =

˜(0) − −5

[1, 1, . . . , 1] and λk = 0.9 and a tolerance = 10 . The inverse iteration converges to the eigenvalue −0.9545 in 13 iterations, while the Rayleigh’s iteration in 5.

From the scripting viewpoint, both inverse and Rayleigh’s iterations require practically the same code. For example, the inverse iteration is as follows:

from cvxopt.umfpack import linsolve from cvxopt.base import matrix

from cvxopt.blas import dotu

b = matrix(1, (system.DAE.nx, 1), ’d’)

mold = 99999 m0 = -0.9 iteration = 0 while 1:

linsolve(As - m0*In, b) nor = (dotu(b, b))**0.5 b = b/nor

m = dotu(b, As*b)/dotu(b, b) if abs(m - mold) < 1e-5:

break mold = m

iteration += 1

To obtain the Rayleigh’s iteration, it su ces to substitute the 12th line of the code above with:

m0 = m = dotu(b, As*b)/dotu(b, b)

7.4Power Flow Modal Analysis

Beside small-signal stability analysis, eigenvalues and eigenvectors can also be used for assessing sensitivities. In particular, an interesting approach is evaluating the modal analysis for the power flow Jacobian matrix [104, 211, 354].

Let us consider the classical power flow model defined in Section 4.3 (e.g., constant PQ loads and constant PV generators). After solving the power flow analysis, the Jacobian matrix can be easily computed. If the Newton’s method is used, the Jacobian matrix is available as byproduct of the solution algorithm. The eigenvalue analysis is performed on a reduced matrix, as follows. Recalling (4.45), the Jacobian matrix can be divided into four sub-matrices:

In case of the classical power flow model, one can associate a physical meaning to each sub-matrix, since load and generator powers are constant. In fact, consider the linearization of the power flow equations with constant power injections:

The basic assumption of [104] is to consider that p ≈ 0. This is reasonable if one is interested only in the relationship between reactive powers and bus voltage magnitudes. Furthermore, recalling the assumptions of the fast-decoupled power flow (see Section 4.4.7 of Chapter 4), the pθ world is quite decoupled from the qv one. Thus, one can define a reduced power flow Jacobian matrix as follows:

it follows that the modal sensitivity for each eigenvalue λk is:

where BP Q is the set of PQ load buses. If λh > 0, then an increase in the injected reactive power leads to a bus voltage increase, which is the normal situation for systems with inductive branches and loads. If λk < 0, the voltage decreases if the reactive power injected at the bus increases. This is considered an unstable behavior (at least for inductive systems), and actually it is typical of power flow solutions of the lower part of the nose curve (see Section 5.1 of Chapter 5). A special case is λk = 0 that can be viewed as an infinite sensitivity of the voltage with respect to the reactive power. Actually, this case corresponds to a saddle-node bifurcation.

Example 7.6 Power Flow Modal Analysis for the IEEE 14-Bus System

Figure 7.5 and Table 7.3 show the results of the modal analysis of the power flow Jacobian matrix as well as the participation factors for the IEEE 14-bus test system. In this case study, only static power flow data are considered, i.e., loads are modelled as constant PQ and generators as constant PV or slack.

Figure 7.5 and Table 7.3 only show 9 eigenvalues. In fact, the 14-bus system system has 5 generators that maintain constant the voltage at the bus where they are connected. As expected, all eigenvalues are positive, thus indicating that ∂v/∂q > 0 at all load buses.

7.4.1Singular Value Decomposition

The Singular Value Decomposition (SVD) is an important kind of matrix factorization and has several applications, for example, for computing the