Учебники / 0841558_16EA1_federico_milano_power_system_modelling_and_scripting

Example 6.2 Maximization of the Distance to Voltage Collapse

The following optimization problem is implemented to represent system security through the use of voltage stability conditions, based on what was proposed in [43, 46, 47]:

where z = (μ, θ, v, qG, θc, vc, qcG, pG, pL).

In this case, a second set of power flow equations and constraints with a superscript c is introduced to represent the system at the limit or critical conditions associated with the loading margin μ that drives the system to its maximum loading condition. The critical power flow equations gc can present a line outage. The maximum or critical loading point could be either associated with a thermal or bus voltage limit or a voltage stability limit (collapse point) corresponding to a system singularity (saddle-node bifurcation) or system controller limits like generator reactive power limits (limit induced bifurcation) [39, 259]. Thus, for the current and maximum loading conditions, the generator and load powers are defined as follows:

pcG = (1 + μ + kGc )pG pcL = (1 + μ)pL

where kGc represents a scalar variable which distributes system losses associated only with the solution of the critical power flow equations in proportion to the power injections obtained in the solution process (distributed slack bus

model). It is assumed that the losses corresponding to the maximum loading level defined by μ are distributed among all generators.

For the sake of example, consider the Lagrangian function L associated to problem (6.26) with all inequalities transformed into equalities through the vector of slack variables s as in (6.16).

−ρTc gc(θc, vc, qcG, μ, pG, pL)

−πμmin (μ − μmin − sμmin )

− πTmax (φmax − φji − sφmax )

φji ji ji

− πTc max (φmax − φ − sφc max )

φij ij ijc ij

− πTc max (φmax − φ − sφc max )

φji ji jic ji

−πTvmax (vmax − v − svmax )

−πTvmin (v − vmin − svmin )

−πTvc max (vmax − vc − svc max )

−πTvc min (vc − vmin − svc min )

where ρ and ρc Rng , and all the other π (πk > 0, k) correspond to the Lagrangian multipliers. The s variables have to satisfy the non-negativity condition s > 0.

6.3Nonlinear Programming Solvers

As indicated in the previous section, the problem of finding a local minimizer of (6.7) or (6.16) is equivalent to solve (6.10)-(6.14) or (6.17)-(6.22), respectively. These are sets of nonlinear equalities and inequalities. The main challenges from the solution method viewpoint are twofold:

1.The inequalities constraints (6.12) and (6.14) or (6.21) and (6.22) complicate considerably the solution process.

2.The conditions (6.10) or (6.17) contain the Jacobian matrices ϕz and gz and hz . Thus any solution method that involves the calculation of Lzz (such as the any Newton’s method), implies setting up the Hessian matrices ϕzz and gzz and hzz . In case of power flow equations, calculating gzz is not a trivial task.

In the following sections, only two solution methods are described, namely the reduced gradient method and the primal-dual interior point method.

6.3.1Generalized Reduced Gradient Method

The generalized reduced gradient (GRG) has been one of first methods used in power system analysis [80]. The GRG method is also used in well-assessed solvers such as CONOPT [81]. This method works for constrained nonlinear problems and resembles the solution approach of the simplex method used for linear programming [96]. The main idea of the GRG method is to divide the variables z into two subsets, one of basic (or dependent) variables and one of non-basic (or independent) variables. In mathematical terms, basic variables are those variables that are unequivocally determined once the vector of nonbasic variables is assigned. To define basic and non-basic variables is generally easy in physical problems. As a matter of fact, according to Sections 1.4 and 6.2, z = [yT , ηT ]T . Thus y are the basic variables and η are the non-basic ones. For example, in the standard optimal power flow problem, y are bus voltages, while η are the generator active powers.

In order to describe the reduced gradient method, consider for simplicity the following optimization problem:

How to handle inequalities is explained later on. The reduced gradient r(η) of (6.7), with r(η) : Rnη → Rnη , is defined as:

Di erentiating g(y, η), and assuming that the current point (y(i), η(i)) is feasible and satisfies g(y(i), η(i)) = 0, one has:

After solving the restoration move, new constraints may become binding, and the vector ga has to be updated before solving the next projection step.

Example 6.3 Continuation Power Flow as Reduced Gradient

Method

This example states the formal analogy between the reduced gradient method described above and the homotopy predictor-corrector method described in Section 5.4 of Chapter 5. A similar proof was originally presented in [37] and further formalized in [21].

The analogy can be shown straightforwardly by observing that in the continuation power flow analysis, the non-basic variable is the loading level μ and that the objective function is ϕ = −μ. Since ϕy = 0 and ϕμ = −1, the reduced gradient (6.31) becomes:

Hence, the reduced gradient r coincides with the tangent vector τ used for the predictor step (e.g., projection move) discussed in Section 5.4. As for the restoration move discussed above, it is just another version of the correctors steps described the same Section 5.38. Moreover, the PV generator reactive power limits discussed in Example 5.4 are inequality constraints. Whenever a reactive power limit (e.g., qGhmax) is reached, the constraint qGh ≤ qGhmax becomes binding and the vector of y is updated to include the voltage magnitude vh at the PV generator bus. Alternatively, if vh is already defined as a variable, the constraint vh = vGhref is removed from g.

6.3.2Interior Point Method

Although Interior Points Methods (IPMs) have been formalized in late sixties [95], the nineties are the dawn of most IPM-based applications for power system analysis [8, 111, 148, 209, 311, 352]. IPM-based OPF problems proved to be robust, especially in large networks, as the number of iterations increase slightly with the number of constraints and network size.

In particular, in [251, 311, 312], the authors present a comprehensive investigation of the use of primal-dual IPM for nonlinear problems, and describe the application of Newton’s direction and Mehrotra’s predictor-corrector to the OPF. The latter allows reducing the number of iterations to obtain the final solution. Both methods are described in this section.

The main idea of the primal-dual IPM discussed in [95] is the introduction of logarithmic barrier function that allows incorporating inequality constraints in the objective function. In this way, inequalities are implicitly taken into account. The objective function modified by means of the logarithmic barrier function is:

Newton directions obtained from (6.48) or (6.49) are generally su cient to lead to convergence. However, sparse matrix factorization is the most time consuming operation of the whole algorithm. Thus, it is worth looking for methods that allows reducing the iterations and, consequently, the number of factorizations. In this vein, the Mehrotra’s predictor-corrector method is a good option. The Mehrotra’s method consists in computing variable directions in two steps but only needs one factorization of the matrices in (6.48) or (6.49), thus leading to a computational burden similar to the standard Newton’s directions. Details on the Mehrotra’s method can be found in [190].

Predictor Step: The predictor step is obtained as follows:

The prediction provided by (6.52) is also called the a ne-scaling direction. Using this direction is possible to estimate a new barrier parameter value μˆ(pi). How to update the barrier parameter is explained in the following subsection.

Corrector Step: The corrector step is obtained as follows:

where the term μˆ(pi)e is called centering direction and helps the current point keep away from the boundary of the feasible region, and Sp πp is called corrector direction and in some measure compensates nonlinearity not taken into account in the a ne-scaling direction.

Since the matrix Lξξ in (6.52) and (6.53) is the same, only one factorization is needed and thus the corrector step does not suppose a relevant extra computing time. Nevertheless, the variable directions obtained using the Mehrotra’s

method allow reducing the number of iterations with respect to Newton’s directions.

Updating Variables

The new primal and dual variables are computed based on the previously computed directions:

where α(Pi) (0, 1] and α(Di) (0, 1] are the step length parameters for the primal and dual variables, respectively. The maximum values of the step lengths can be estimated using the following heuristic rules:

where γ (0, 1) is a safety factor that ensure that the next point satisfies the strict positivity condition. A typical value for the safety factor is γ = 0.99995. In NLP problems, such as the optimal power flow, primal and dual variables are interdependent due to the dual feasibility conditions. In this case, to use the same step length for both primal and dual variables can help obtain convergence:

However, separate step lengths have proved to work well in [111].

Reducing the Barrier Parameter

The barrier parameter μˆ(i) has to be updated (and hopefully reduced) at each iteration. The new value of the barrier parameter is computed based on the complementarity gap ˆ(i) that is the residual of the complementarity conditions:

The complementarity gap ˆ(i) → 0 as the primal variables approach a local minimizer z → z . Then, the new barrier parameter is computed as: