Учебники / 0841558_16EA1_federico_milano_power_system_modelling_and_scripting

Script 6.1 Interior Point Method

The following Python code implement the IPM described above including both Newton’s direction method and the Mehrotra’s predictor-corrector step. As usual, the system.Device class provides the interface between the IPM algorithm and system devices as discussed in Script 3.2 of Chapter 3. The matrices system.DAE.Oz and system.DAE.Hes are ϕz and Lzz , respectively.

import system

from cvxopt.base import matrix, spmatrix, sparse, spdiag, div, mul, log from cvxopt.umfpack import solve, symbolic, numeric

from cvxopt.blas import dotu

def opf():

#setup dimensions, variables, vectors and matrices exec system.Device.setup opf

Zz = spmatrix([], [], [], (system.DAE.ng, system.DAE.ng), ’d’)

#parameters

iteration = 1

mui = system.OPF.sigma/system.DAE.nh

msg = ’Iter. = %3d, mui = %s, |dz| = %s, |g(z)| = %s, |dOF| = %s’

#initial guess of primal, dual and slack variables exec system.Device.call opfguess

exec system.Device.call obj

system.DAE.s += 1e-6 # avoid zero slack variables system.DAE.pi = div(mui, system.DAE.s)

system.DAE.rho[:system.Bus.n] = matrix(1, (system.Bus.n, 1), ’d’)

#primal dual interior-point method

while 1:

# compute g(z), h(z, s), and Jacobian and Hessian matrices exec system.Device.call opf

system.DAE.h += system.DAE.s

s = mul(system.DAE.pi, system.DAE.s) - mui

Lz = system.DAE.Oz + (system.DAE.Gz.T)*system.DAE.rho + \

(system.DAE.Hz.T)*system.DAE.pi

Hp = spdiag(div(system.DAE.pi, system.DAE.s))

# reduced system

Lr = sparse([[system.DAE.Hes + system.DAE.Hz.T*(Hp*system.DAE.Hz), \ system.DAE.Gz], [system.DAE.Gz.T, Zz]])

if iteration <= 2: F = symbolic(Lr) try:

C = numeric(Lr, F) except ValueError:

# unexpected refactorization of Langragian Jacobian matrix F = symbolic(Lr)

C = numeric(Lr, F)

iteration += 1

if iteration > system.OPF.max iter: break

def step length(Ds, Dp):

ratio1 = [1.0/system.OPF.gamma] + \

[-s/d for s, d in zip(system.DAE.s, Ds) if d < 0] ratio2 = [1.0/system.OPF.gamma] + \

[-s/d for s, d in zip(system.DAE.pi, Dp) if d < 0]

alpha P = system.OPF.gamma*min(ratio1) alpha D = system.OPF.gamma*min(ratio2)

return alpha P, alpha D

Example 6.4 Optimal Power Flow Analysis for the IEEE 14-Bus System

This example shows the results of the primal-dual nonlinear IPM applied to the IEEE 14-bus system. The demand is considered inelastic and equal to the base case while generator costs are quadratic and, in order to provide more “freedom degrees”, also synchronous compensators are assumed to be standard generators with an active power output. The cost function data and variable limits are taken from [363] and are given in Appendix D. At the local minimizer, the objective function is 8081.55 e/h. Tables 6.1, 6.2, 6.3 and 6.4 show generator active and reactive powers, bus voltages and power flow results, respectively. As expected, dual variables are positive only if a limit is binding. Transmission line limits are not imposed.

Table 6.1 Optimal power flow results for the IEEE 14-bus system: power supplies

Table 6.2 Optimal power flow results for the IEEE 14-bus system: generator reactive powers

Table 6.3 Optimal power flow results for the IEEE 14-bus system: bus voltages

100.0000 0.9400 1.0391 1.0600 0.0000 −0.2309

110.0000 0.9400 1.0460 1.0600 0.0000 −0.2285

120.0000 0.9400 1.0448 1.0600 0.0000 −0.2362

130.0000 0.9400 1.03995 1.0600 0.0000 −0.2371

140.0000 0.9400 1.0239 1.0600 0.0000 −0.2491

In the OPF report, the LMP (Locational Marginal Prices) column are dual variables ρp, while the NCP column indicates Nodal Congestion Prices that are computed as follows. Using the decomposition formula for LMPs proposed in [353], one has:

ˆ

where h represents the inequality constraints as defined in (6.44), and πymax and πymin are the dual variables associated with the lower and upper limits of such inequality constraints.

Figure 6.2 shows the behavior of the convergence tests of the IPM method during the 8 iterations required to find the local minimizer with the given tolerance. The quantities used for the convergence tests are μˆ(i), z(i) ∞,g(z(i)) ∞, and Δϕ(z(i)). Each quantity is normalized with respect to the

152 6 Optimal Power Flow Analysis

Table 6.4 Optimal power flow results for the IEEE 14-bus system: bus power injections

Fig. 6.2 Convergence behavior of IPM for the IEEE 14-bus system. The upper plot is obtained using the Newton’s direction method, the lower plot using the Mehrotra’s predictor-corrector method

value of the first iteration. As expected, the Mehrotra’s predictor-corrector method converges in less iterations than the Newton’s direction method.

6.4Summary of IPM Parameters

This section summarizes most relevant parameters related to the interior point method.

Solver method: The methods described in this chapter are the GRG and the IPM. Of course, several other methods for nonlinear programming are available [24]. It is important to note that, since the OPF problem is non-convex,4 in general only local minima can be found. Furthermore, due to idiosyncrasies of nonlinearity, it is di cult to say a priori if a solver method performs well for a given NLP problem.

Tolerance 1: Tolerance used for checking the convergence of equalities g(z(i)). Typical value is 1 = 10−4.

Tolerance 2: Tolerance used for checking the convergence of variable increments z(i). Typical value is 2 = 10−2 1.

Tolerance μ: Threshold of the barrier parameter μˆ(i) used for defining the convergence of the IPM. Typical value is μ = 10−12.

Initial guess type: The initial guess can be obtained as a solution of a preliminary power flow analysis or as a flat start (e.g., middle value between the upper and lower limits).

Safety factor γ: The safety factor γ (0, 1) is used for estimating a good step length and ensuring the strict positivity condition. A typical value for the safety factor is 0.99995.

Maximum number of iterations: The maximum number of iterations before stopping the solver method if convergence is not reached. For IPM, a reasonable iteration limit is 50.

Method used for computing variable directions: Various methods are available for computing variable directions for the IPM. The most common is the Newton’s direction. The Mehrotra’s predictor-corrector method provides more precise directions with a negligible additional computational e ort. Higher order methods (such as composite Newton’s method) can be also used [312].

Centering parameter σ(i): The centering factor σ(i) (0, 1). The initial value of the centering factor can be fixed to σ(0) = 0.2.

versa.

Chapter 7

Eigenvalue Analysis

This chapter describes various aspects of eigenvalues analysis. Section 7.1 provides the background of modal analysis using a simple one-machine infinite-bus example. Section 7.2 describes the small signal stability analysis for dynamical power systems and outlines the properties of equilibrium points, including bifurcation points commonly shown by power systems, participation factors and the Z-domain transformation. Section 7.3 describes a variety of methods for computing a reduced set of eigenvalues and introduces the singular value decomposition. Section 7.4 describes the modal analysis applied to the power flow Jacobian matrix. Finally, Section 7.5 summarizes most relevant concepts related to eigenvalue analysis.

7.1Background

Consider the one-machine infinite-bus system of Figure 7.1. Assuming that the machine dynamics are described by the classical model, system equations are as follows:

where Ωb is the synchronous frequency in rad/s, ωs is the synchronous speed (ωs = 1 pu), H is the rotor inertia constant, pm is the mechanical power, pe(δ) is the electrical power defined as:

and xeq is the equivalent series reactance composed of the internal machine reactance, of the transmission line reactance and of the Thevenin reactance of the infinite bus, i.e., xeq = xd + xL + xTh.

F. Milano: Power System Modelling and Scripting, Power Systems, pp. 155–178. springerlink.com c Springer-Verlag Berlin Heidelberg 2010

Fig. 7.2 Equilibrium points of the OMIB system

As it is well known, for δ [0, π], the dynamical system (7.1)-(7.2) has two equilibrium points. Figure 7.2 shows these equilibrium points assuming e = v = pm = 1.0 pu and xeq = 0.5 pu.

Given the vector of state variables x = (δ, ω), the equilibrium points are:

xA = (0.5236, 1) xB = (2.6180, 1)

The most important property of an equilibrium point is whether it is stable or unstable. For the simple OMIB system, the stability of the points xA and xB can be easily determined using virtual variations, as follows.

Point xA: Assume that the system is steady-state at the equilibrium point xA and that the angle δ is perturbed by a small positive quantity, say