This paper describes an application of an optimal power flow, solved by a direct interior point (IP) method, to restore system solvability. Using the P-Q load representation, power flow unsolvability occurs when, for a given set of active and reactive bus injections, the power flow equations have no real solution. The set of control actions in the IP algorithm includes rescheduling of active power of generators, adjustments on terminal voltage of generators, tap changes on LTC transformers, and as a last resort, minimum load shedding. The IP formulation allows observation of the impact of each control optimization in system solvability. Also described is a framework to calculate probabilistic indicators of solvability problems taking into account the probability of contingencies. The role of control optimization is illustrated in a real 11-bus power system and the probabilistic approach is applied to a 1600-bus power system derived from the Brazilian South/Southeast/Central West system.

Application of interior point methods to power flow unsolvability

This paper describes an approach to calculate voltage collapse related bulk reliability indices as well as their impact on the adequacy reliability indices of power systems. In this approach, the adequacy analysis of each selected system state is carried out in two steps. In the first, the minimum load shedding to restore system solvability is computed, neglecting operational constraints such as bus voltage levels and circuit power flows. In the second, the additional minimum load curtailment to alleviate any operating limit violations is calculated. In each step, optimal power flow is solved by a direct interior point algorithm, and related reliability indices are evaluated. The proposed methodology can be used in both enumeration and Monte Carlo methods. Case studies with a practical 11-bus network (the Brazilian Reduced Mato Grosso system) and with a 1600-bus network derived from the Brazilian South/Southeast/Central West system are presented and discussed.

The effects of voltage collapse problems in the reliability evaluation of composite systems

The calculation of power system maximum loadability consists in determining, from a point for which the power flow has a solution, the maximum increase of system load until the system reaches the point of collapse. Therefore, it is a measurement of the margin or distance to the voltage collapse. This problem is related to the computation of margins with respect to bifurcations. Under certain conditions, saddle-node bifurcations occur when the Jacobian matrix associated with the power flow balance equations becomes singular. These points belong to the surface which defines the frontier of the power flow solvability region. Outside this region, the power flow is unsolvable, i.e. the power flow balance equations have no real solutions. In turn, power flow solvability problems can be handled by a special purpose AC optimal power flow (OPF) which is solved by an interior point (IP) algorithm. In this paper, this special IP OPF is introduced into a probabilistic simulator, allowing the calculation of probabilistic indicators of system margins or maximum loadability, taking into account random forced outages of generators, lines and transformers. Mathematical formulation and numerical examples are presented and discussed.

Assessment of maximum loadability in a probabilistic framework

Optimal power flow (OPF) is a non-linear and non-convex problem that seeks the optimization of a power system operation point to minimize the total generation costs or transmission losses. This study proposes an OPF model considering current margins in radial networks. The objective function of this OPF model has an additional term of current margins of the line besides the traditional transmission losses and generations costs, which contributes to thermal stability margins of power systems. The model is a reformulated bus injection model with clear physical meanings. Second order cone program (SOCP) relaxations for the proposed OPF are made, followed by the over-satisfaction condition guaranteeing the exactness of the SOCP relaxations. A simple 6-node case and several IEEE benchmark systems are studied to illustrate the efficiency of the developed results.

SOCP Relaxations of Optimal Power Flow Problem Considering Current Margins in Radial Networks

The optimal power flow (OPF) is a non-linear and non-convex problem that seeks the optimization of a power system operation point to minimize the total generation cost or transmission loss. In this paper, an optimal power flow formulation for radial networks is proposed from the bus injection model. By inducing the branch variables in the model, the relaxation is based on, instead of a big network structure, small branch matrices. Through two steps, it transforms the quartic and non-convex constraints into some linear equations and convexifies the other constraints which cannot be linearized. In the first step, we focus on the equation constraints and reformulate the equations with branch variables. In the second step, we apply SOCP relaxations to inequality constraints. This way, the OPF problem is a convex optimization problem that the local optimum solution is the global optimum solution. The relaxation is tight and equals to the former OPF problem in the radial network in the case study with the standard IEEE benchmark systems. By comparing the simulating time, we find this method calculates faster than OPF with SDP relaxation method.

An Optimal Power Flow Formulation with SOCP Relaxation in Radial Network

This paper presents a method which applies an efficient optimal power flow model to composite power system reliability analysis. It is based on the contingency enumerative approach. The optimal power flow is the premier tool for contingency effects analysis. The method used here incorporates the process of economic dispatch, observes the operating constraints and determines tlie amount of load curtailment after exhaustion of all other options. The optimization problem is solved either with the simplex method or with an interior point algorithm. Tlie perforniiance evaluation of the optimal powcr flow model for a large power system is provided. Finally, the reliability computations are illustrated w i h a simple example.

https://ieeexplore.ieee.org/iel4/5511/15000/00686867.pdf

Feng Xia

Papers

Optimal Power Flow Application to Composite Power System Reliability Analysis