Zero Duality Gap in Optimal Power Flow Problem

The optimal power flow (OPF) problem is nonconvex and generally hard to solve. We provide a sufficient condition under which the OPF problem is equivalent to a convex problem and therefore is efficiently solvable. Specifically, we prove that the dual of OPF is a semidefinite program and our sufficient condition guarantees that the duality gap is zero and a globally optimal solution of OPF is recoverable from a dual optimal solution. This sufficient condition is satisfied by standard IEEE benchmark systems with 14, 30, 57, 118 and 300 buses after small resistance (10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> per unit) is added to every transformer that originally assumes zero resistance. We justify why the condition might hold widely in practice from algebraic and geometric perspectives. The main underlying reason is that physical quantities such as resistance, capacitance and inductance, are all positive.

The optimal power flow (OPF) problem is a relevant subject for the secure and economic power systems operation. For instance, OPF can be used to reduce the power system technical losses. A reduction in 0.1% in the losses accounts for near 50 billion USD in cost savings. Likewise, OPF can be used along with power production forecast tools, for renewable energy sources, to evaluate their impact in the grid security operation planning. In an OPF problem, an objective function related with demand supply generation cost, power line losses, or violation limits is optimized subject to several system security constraints. These constraints are related with branch power flow limits, voltages limits, power injection limits, among others. The exact model of the objective function and constraints of the OPF problem is neither linear nor convex. In this paper it is presented the OPF problem in a Quadratically Constrained Quadratic Program (QCQP) approach. Objective function is based on a quadratic function of the bus voltages and constraints are formulated as Quadratic forms. A two-bus system is used to demonstrates the non-convexity of the OPF problem. Also, it is presented the Rank- 1 convex relaxation of the OPF problem which relaxes the QCQP model into a positive Semidefinite Programming (SDP) model. Once the OPF problem is relaxed into a SDP convex form, a global optimal solution can be obtained. An application example for the OPF problem is presented for the IEEE 14 system. The QCQP and the SDP results are compared and discussed.

A semidefinite programming (SDP) relaxation globally solves many optimal power flow (OPF) problems. For other OPF problems where the SDP relaxation only provides a lower bound on the objective value rather than the globally optimal decision variables, recent literature has proposed a penalization approach to find feasible points that are often nearly globally optimal. A disadvantage of this penalization approach is the need to specify penalty parameters. This paper presents an alternative approach that algorithmically determines a penalization appropriate for many OPF problems. The proposed approach constrains the generation cost to be close to the lower bound from the SDP relaxation. The objective function is specified using iteratively determined weights for a Laplacian matrix. This approach yields feasible points to the OPF problem that are guaranteed to have objective values near the global optimum due to the constraint on generation cost. The proposed approach is demonstrated on both small OPF problems and a variety of large test cases representing portions of European power systems.

A semidefinite programming (SDP) relaxation globally solves many optimal power flow (OPF) problems. For other OPF problems where the SDP relaxation only provides a lower bound on the objective value rather than the globally optimal decision variables, recent literature has proposed a penalization approach to find feasible points that are often nearly globally optimal. A disadvantage of this penalization approach is the need to specify penalty parameters. This paper presents an alternative approach that algorithmically determines a penalization appropriate for many OPF problems. The proposed approach constrains the generation cost to be close to the lower bound from the SDP relaxation. The objective function is specified using iteratively determined weights for a Laplacian matrix. This approach yields feasible points to the OPF problem that are guaranteed to have objective values near the global optimum due to the constraint on generation cost. The proposed approach is demonstrated on both small OPF problems and a variety of large test cases representing portions of European power systems.

The optimal power flow (OPF) problem determines an optimal operating point of the power network that minimizes a certain objective function subject to physical and network constraints. There has been a great deal of attention in convex relaxation of the OPF problem in recent years, and it has been shown that a semidefinite programming (SDP) relaxation can solve different classes of the nonconvex OPF problem to global or near-global optimality. Although it is known that the SDP relaxation is exact for radial networks under various conditions, solving the OPF problem for cyclic networks needs further research. In this paper, we propose sufficient conditions under which the SDP relaxation is exact for special but important cyclic networks. More precisely, when the objective function is a linear function of active powers, we show that the SDP relaxation is exact for odd cycles under certain conditions. Also, the exactness of the SDP relaxation for simple cycles of size 3 and 4 is proved under different technical conditions. The existence of rank-1 or -2 SDP solutions for weakly-cyclic networks is also proved in both lossy and lossless cases. In addition, when the objective function is an increasing function of reactive powers, we prove that the SDP relaxation is exact under certain conditions. This result justifies why the sum of reactive powers acts as a low-rank-promoting term for OPF. The findings of this paper provide intuition into the behavior of SDPs for different building blocks of cyclic networks.

Control of inverter-based microgrids is typically accomplished through a hierarchical control architecture, where the lower level controllers are designed to ensure convergence to a desired equilibrium determined by the upper level controller. Selecting the equilibrium can be cast as an optimal power flow (OPF) problem. The OPF problems associated with microgrid applications often involve non-monotone cost functions and nontrivial reactive power cost, posting new challenges for OPF problems that have not been completely addressed in the literature. In this paper, we formulate a distributed OPF problem motivated by microgrid control applications. We first derive new conditions on the exact SDP convex relaxation on the OPF problems with quadratic cost functions on the active and reactive power. For microgrids with tree topology, we show that under mild conditions on the reactive power bounds, a near global optimal solution can always be found for the OPF problem with those nontrivial cost functions. Such results are not only important for microgrid applications, but also provide new perspectives and insights for general OPF problems. The effectiveness of the proposed algorithm is demonstrated through numerical examples.

The optimal power flow (OPF) problem is a well-known non-convex optimization problem that aims to minimize the cost of electric power generation subject to consumer demand, the physics of power flow, and technological constraints. To find an optimal solution to this problem, local search techniques such as interior point methods are typically used. However, due to the non-convex nature of the problem, these methods are likely to result in a sub-optimal solution. The goal of this paper is to characterize the worst-case performance of local search on the OPF problem. To accomplish this, we formulate the OPF problem as a canonical quadratically-constrained quadratic program (QCQP). Then, we study the problem of finding the worst-case local minimum of this QCQP, which is non-convex and hard to solve in general. We find a relaxation of this problem into a semidefinite program (SDP) and show that it is exact for certain cases. Using some test cases which are known to have multiple local minima, we demonstrate the effectiveness of the proposed relaxation to bound the worst-case local minimum. We compare the obtained upper bound on local minima to the lower bound provided by the standard SDP relaxation of the OPF problem to understand how much SDP outperforms local search for a given problem.

Recently, there has been significant interest in convex relaxations of the optimal power flow (OPF) problem. A semidefinite programming (SDP) relaxation globally solves many OPF problems. However, there exist practical problems for which the SDP relaxation fails to yield the global solution. Conditions for the success or failure of the SDP relaxation are valuable for determining whether the relaxation is appropriate for a given OPF problem. To move beyond existing conditions, which only apply to a limited class of problems, a typical conjecture is that failure of the SDP relaxation can be related to physical characteristics of the system. By presenting an example OPF problem with two equivalent formulations, this paper demonstrates that physically based conditions cannot universally explain algorithm behavior. The SDP relaxation fails for one formulation but succeeds in finding the global solution to the other formulation. Since these formulations represent the same system, success (or otherwise) of the SDP relaxation must involve factors beyond just the network physics. The lack of universal physical conditions for success of the SDP relaxation motivates the development of tighter convex relaxations capable of solving a broader class of problems. Tools from polynomial optimization theory provide a means of developing tighter relaxations. We use the example OPF problem to illustrate relaxations from the Lasserre hierarchy for polynomial optimization and a related "mixed semidefinite/second-order cone programming" hierarchy.

The feasible region plays an important role in optimal power flow (OPF) problems. However, constructing the feasible region of general optimal power flow problems is a challenging task. In this paper, a trajectory-unified (TJU) method is developed to compute the feasible region of general large-scale OPF problems. This is the first attempt, to our knowledge, to develop a numerical method to compute feasible regions of OPF problems. In addition, the projection of the computed feasible region into a desired low-dimensional sub-space is presented. By employing the proposed TJU method, we compute the feasible region of a 9-bus and the IEEE 118-bus OPF problem. It is shown that the feasible region of a power system grows in size from light-loading conditions to medium-loading conditions while it shrinks in size from medium-loading conditions to heavy-loading conditions. This discovery of a geometric property of the feasible solution asserts the observations that OPF problems are generally easy to solve during medium-loading conditions but are generally difficult to solve during heavy-loading conditions.

The AC optimal power flow (OPF) problem is essential for the schedule and operation of power systems. Convex relaxation methods have been studied and used extensively to obtain an optimal solution to the OPF problem. When the exactness of convex relaxations is not guaranteed, it is important to recover a feasible solution for the convex relaxation methods. This paper presents an alternative convex optimisation (ACP) approach that can efficiently recover a feasible solution from the result of second‐order cone programming (SOCP) relaxed OPF in mesh networks. The OPF problem is first formulated as a difference‐of‐convex programming (DCP) problem, then efficiently solved by a penalty convex concave procedure (CCP). By using the solution of a tightened SOCP OPF as an initial point, the proposed algorithm is able to find a global or near‐global optimal solution to the AC OPF problem. Numerical tests show that the proposed method outperforms those semi‐definite programming (SDP) and quadratically constrained quadratic programming (QCQP)–based algorithms.

Recent years have witnessed the success of employing convex relaxations of the AC optimal power flow (OPF) problem to find global or near-global optimal solutions. The majority of the effort has focused on solving problem formulations where variables live in continuous spaces. Our focus here is in the extension of these results to the co-optimization of network topology and the OPF problem. We employ binary variables to model topology reconfiguration in the standard semidefinite programming (SDP) formulation of the OPF problem. This makes the problem non-convex, not only because the variables are binary, but also because of the presence of bilinear products between the binary and other continuous variables. Our proposed convex relaxation to this problem incorporates the bilinear terms in a novel way that improves over the commonly used McCormick approximation. We also address the exponential complexity associated with the discrete variables by partitioning the network graph in a way that minimizes the impact on the optimal value of the relaxation. As a result, the problem is broken down into several parallel mixed-integer problems, reducing the overall computational complexity. Simulations in the IEEE 118-bus test case demonstrate that our approach converges to solutions which are very close to the lower bound of the mixed-integer OPF problem.

This paper proposes a sequential convex optimization method to solve broader classes of optimal power flow (OPF) problems over radial networks. The non-convex branch power flow equation is decomposed as a second-order cone inequality and a non-convex constraint involving the difference of two convex functions. Provided with an initial solution offered by an inexact second-order cone programming relaxation model, this approach solves a sequence of convexified penalization problems, where concave terms are approximated by linear functions and updated in each iteration. It could recover a feasible power flow solution, which usually appears to be very close, if not equal, to the global optimal one. Two variations of the OPF problem, in which non-cost related objectives are optimized subject to power flow constraints and the convex relaxation is generally inexact, are elaborated in detail. One is the maximum loadability problem, which is formulated as a special OPF problem that seeks the maximal distance to the boundary of power flow insolvability. The proposed method is shown to outperform commercial nonlinear solvers in terms of robustness and efficiency. The other is the bi-objective OPF problem. A non-parametric scalarization model is suggested, and is further reformulated as an extended OPF problem by convexifying the objective function. It provides a single trade-off solution without any subjective preference. The proposed computation framework also helps retrieve the Pareto front of the bi-objective OPF via the $\varepsilon $ -constraint method or the normal boundary intersection method. This paper also discusses extensions for OPF problems over meshed networks based on the semidefinite programming relaxation method.

Recently, there has been significant interest in convex relaxations of the optimal power flow (OPF) problem. A semidefinite relaxation globally solves many OPF problems. However, there exist practical problems for which the semidefinite relaxation fails to yield the global solution. Conditions for the success or failure of the semidefinite relaxation are valuable for determining whether the relaxation is appropriate for a given OPF problem. To move beyond existing conditions, which only apply to a limited class of problems, a typical conjecture is that failure of the semidefinite relaxation can be related to physical characteristics of the system. By presenting an example OPF problem with two equivalent formulations, this paper demonstrates that physically based conditions cannot universally explain algorithm behavior. The semidefinite relaxation fails for one formulation but succeeds in finding the global solution to the other formulation. Since these formulations represent the same system, success (or otherwise) of the semidefinite relaxation must involve factors beyond just the network physics.

The presence of distributed generators with DC output power and the advancement in power electronics devices have motivated system planners and grid operators to move towards integration of DC microgrids into conventional AC grid. In this paper, we address the optimal power flow (OPF) problem in AC-DC networks. The goal of the AC-DC OPF problem is to jointly minimize the total electricity generation cost of the network and the cost of transferring active power from the AC grid to the DC microgrids. The optimization problem is subject to the power flow constraints, voltage magnitude limits, the limits of the network power lines, and the limits imposed by the power ratings of AC-DC power electronic converters. The formulated AC-DC OPF problem is shown to be nonlinear. We propose an approach to reformulate the AC-DC OPF problem as an equivalent traditional AC OPF problem. Due to the non-convexity of the AC OPF problem, we use convex relaxation techniques and transform the problem to a semidefinite program (SDP). We show that the relaxation gap is zero. That is the optimal solution of the non-convex and the transformed convex problems are equal. Simulation studies are performed on an IEEE 14-bus system connected to two 9-bus DC microgrids. We show that the sufficient condition for the zero relaxation gap is satisfied, and the proposed SDP approach enables us to find the global optimal solution efficiently.

This paper proposes a recursive conic approximation methodology to deal with the optimal power flow (OPF) problem in unbalanced bipolar DC networks. The OPF problem is formulated through a nonlinear programming (NLP) representation, where the objective function corresponds to the minimization of the expected grid power losses for a particular load scenario. The NLP formulation has a non-convex structure due to the hyperbolic equality constraints that define the current injection/absorption in the constant power terminals as a function of the powers and voltages. To obtain an approximate convex model that represents the OPF problem in bipolar asymmetric distribution networks, the conic relation associated with the product of two positive variables is applied to all nodes with constant power loads. In the case of nodes with dispersed generation, a direct replacement of the voltage variables for their expected operating point is used. An iterative solution procedure is implemented in order to minimize the error introduced by the voltage linearization in the dispersed generation sources. The 21-bus grid is employed for all numerical validations. To validate the effectiveness of the proposed conic model, the power flow problem is solved, considering that the neutral wire is floating and grounded, and obtaining the same numerical results as the traditional power flow methods (successive approximations, triangular-based, and Taylor-based approaches): expected power losses of 95.4237 and 91.2701 kW, respectively. To validate the effectiveness of the proposed convex model for solving the OPF problem, three combinatorial optimization methods are implemented: the sine-cosine algorithm (SCA), the black-hole optimizer (BHO), and the vortex search algorithm (VSA). Numerical results show that the proposed convex model finds the global optimal solution with a value of 22.985 kW, followed by the VSA with a value of 22.986 kW. At the same time, the BHO and SCA are stuck in locally optimal solutions (23.066 and 23.054 kW, respectively). All simulations were carried out in a MATLAB programming environment.