Rabih A. Jabr

Radial distribution load flow using conic programming

This paper shows that the load flow problem of a radial distribution system can be modeled as a convex optimization problem, particularly a conic program. The implications of the conic programming formulation are threefold. First, the solution of the distribution load flow problem can be obtained in polynomial time using interior-point methods. Second, numerical ill-conditioning can be automatically alleviated by the use of scaling in the interior-point algorithm. Third, the conic formulation facilitates the inclusion of the distribution power flow equations in radial system optimization problems. A state-of-the-art implementation of an interior-point method for conic programming is used to obtain the solution of nine different distribution systems. Comparisons are carried out with a previously published radial load flow program by R. Cespedes

Minimum Loss Network Reconfiguration Using Mixed-Integer Convex Programming

This paper proposes a mixed-integer conic programming formulation for the minimum loss distribution network reconfiguration problem. This formulation has two features: first, it employs a convex representation of the network model which is based on the conic quadratic format of the power flow equations and second, it optimizes the exact value of the network losses. The use of a convex model in terms of the continuous variables is particularly important because it ensures that an optimal solution obtained by a branch-and-cut algorithm for mixed-integer conic programming is global. In addition, good quality solutions with a relaxed optimality gap can be very efficiently obtained. A polyhedral approximation which is amenable to solution via more widely available mixed-integer linear programming software is also presented. Numerical results on practical test networks including distributed generation show that mixed-integer convex optimization is an effective tool for network reconfiguration.

Robust self-scheduling under price uncertainty using conditional value-at-risk

In a deregulated power industry, power producing companies bid in the hour-ahead and day-ahead power markets in an attempt to maximize their profit. For a successful bidding strategy, each power-producing company has to generate bidding curves derived from an optimal self-schedule. This self-schedule is commonly obtained from a profit-maximizing optimal power flow model based on predicted locational marginal prices (LMPs). However, at the time of self-scheduling, the predicted values of the LMPs are largely uncertain. Therefore, it is desired to produce robust self-schedules that can be used to lessen the risk resulting from exposure to fluctuating prices. In portfolio optimization theory, methods of risk management include Value-at-Risk (VaR) and conditional Value-at-Risk (CVaR). CVaR is known to be a more consistent measure of risk than VaR. In fact, whilst CVaR is the mean excess loss, the VaR provides no indication on the extent of losses that might be suffered beyond the amount indicated by this measure. This research proposes a method for robust self-scheduling based on CVaR. It will be shown that polynomial interior-point methods can be used to obtain the robust self-schedules from a second-order cone program. The obtained schedules provide a compromise solution between maximum profit and minimum risk. Simulation results on a standard IEEE bus test system will be used to demonstrate the scheduling model based on CVaR.

Statistical Representation of Distribution System Loads Using Gaussian Mixture Model

This paper presents a probabilistic approach for statistical modeling of the loads in distribution networks. In a distribution network, the probability density functions (pdfs) of loads at different buses show a number of variations and cannot be represented by any specific distribution. The approach presented in this paper represents all the load pdfs through Gaussian mixture model (GMM). The expectation maximization (EM) algorithm is used to obtain the parameters of the mixture components. The performance of the method is demonstrated on a 95-bus generic distribution network model.

Adjustable Robust OPF With Renewable Energy Sources

This paper presents an adjustable robust optimization approach to account for the uncertainty of renewable energy sources (RESs) in optimal power flow (OPF). It proposes an affinely adjustable robust OPF formulation where the base-point generation is calculated to serve the forecast load which is not balanced by RESs, and the generation control through participation factors ensures a feasible solution for all realizations of RES output within a prescribed uncertainty set. The adjustable robust OPF framework is solved using quadratic programming with successive constraint enforcement and can coordinate the computation of both the base-point generation and participation factors. Numerical results on standard test networks reveal a relatively small increase in the expected operational cost as the uncertainty level increases. In addition, solutions of networks that include both uncertain wind generation and Gaussian distributed demand are shown to have less cost and a higher level of robustness as compared to those from a recent robust scheduling method.

Exploiting Sparsity in SDP Relaxations of the OPF Problem

This letter presents a framework for exploiting sparsity in primal-dual interior-point based semidefinite programming (SDP) solutions of the optimal power flow (OPF) problem. It is shown that a formulation based on positive semidefinite matrix completion results in a drastic reduction in computational effort.

Optimal Power Flow Using an Extended Conic Quadratic Formulation

Recent research has shown that the load flow equations describing the steady-state conditions in a meshed network can be placed in extended conic quadratic (ECQ) format. This paper presents a study of the implementation of the new load flow equations format in an optimal power flow (OPF) program which accounts for control devices such as tap-changing transformers, phase-shifting transformers, and unified power flow controllers. The proposed OPF representation retains the advantages of the ECQ format: 1) it can be easily integrated within optimization routines that require the evaluation of second-order derivatives, 2) it can be efficiently solved for using primal-dual interior-point methods, and 3) it can make use of linear programming scaling techniques for improving numerical conditioning. The ECQ-OPF program is employed to solve the economic dispatch and active power loss minimization problems. Numerical testing is used to validate the proposed approach by comparing against solution methods and results of standard test systems.

Robust Transmission Network Expansion Planning With Uncertain Renewable Generation and Loads

This paper presents a robust optimization approach for transmission network expansion planning (TNEP) under uncertainties of renewable generation and load. Unlike conventional stochastic programming, the proposed approach does not require knowledge of the probability distribution of the uncertain net injections; rather the uncertainties of the net injections are specified by a simple uncertainty set. The solution algorithm is exact and produces expansion plans that are robust against all possible realizations of the net injections defined in the uncertainty set; it is based on a Benders decomposition scheme that iterates between a master problem that minimizes the cost of the expansion plan and a slave problem that minimizes the maximum curtailment of load and renewable generation. The paper demonstrates that when adopting the dc load flow model, both the master and the dual slave can be formulated as mixed-integer linear programs for which commercial solvers exist. Numerical results on several networks with uncertainties in their loads and renewable generation show that the proposed approach produces solutions that are superior to those from two recent techniques for robust TNEP design.

Distribution Voltage Control Considering the Impact of PV Generation on Tap Changers and Autonomous Regulators

The uptake of variable megawatts from photovoltaics (PV) challenges distribution system operation. The primary problem is significant voltage rise in the feeder that forces existing voltage control devices such as on-load tap-changers and line voltage regulators to operate continuously. The consequence is the deterioration of the operating life of the voltage control mechanism. Also, conventional non-coordinated reactive power control can result in the operation of the line regulator at its control limit (runaway condition). This paper proposes an optimal reactive power coordination strategy based on the load and irradiance forecast. The objective is to minimize the number of tap operations so as not to reduce the operating life of the tap control mechanism and avoid runaway. The proposed objective is achieved by coordinating various reactive power control options in the distribution network while satisfying constraints such as maximum power point tracking of PV and voltage limits of the feeder. The option of voltage support from PV plant is also considered. The problem is formulated as constrained optimization and solved through the interior point technique. The effectiveness of the approach is demonstrated in a realistic distribution network model.

A primal-dual interior point method for optimal power flow dispatching

In this paper the solution of the optimal power flow dispatching (OPFD) problem by a primal-dual interior point method is considered. Several primal-dual methods for OPF have been suggested, all of which are essentially direct extensions of primal-dual methods for linear programming. The aim of the present work is to enhance convergence through two modifications: a filter technique to guide the choice of the step length and an altered search direction in order to avoid convergence to a nonminimizing stationary point. A reduction in computational time is also gained through solving a positive definite matrix for the search direction. Numerical tests on standard IEEE systems and on a realistic network are very encouraging and show that the new algorithm converges where other algorithms fail.