Somayeh Sojoudi

Optimal charging of plug-in hybrid electric vehicles in smart grids

Plug-in hybrid electric vehicles (PHEVs) play an important role in making a greener future. Given a group of PHEVs distributed across a power network equipped with the smart grid technology (e.g. wireless communication devices), the objective of this paper is to study how to schedule the charging of the PHEV batteries. To this end, we assume that each battery must be fully charged by a pre-specified time, and that the charging rate can be time-varying at discrete-time instants. The scheduling problem for the PHEV charging can be augmented into the optimal power flow (OPF) problem to obtain a joint OPF-charging (dynamic) optimization. A solution to this highly nonconvex problem optimizes the network performance by minimizing the generation and charging costs while satisfying the network, physical and inelastic-load constraints. A global optimum to the joint OPF-charging optimization can be found efficiently in polynomial time by solving its convex dual problem whenever the duality gap is zero for the joint OPF-charging problem. It is shown in a recent work that the duality gap is expected to be zero for the classical OPF problem. We build on this result and prove that the duality gap is zero for the joint OPF-charging optimization if it is zero for the classical OPF problem. The results of this work are applied to the IEEE 14 bus system.

Physics of power networks makes hard optimization problems easy to solve

We have recently observed and justified that the optimal power flow (OPF) problem with a quadratic cost function may be solved in polynomial time for a large class of power networks, including IEEE benchmark systems. In this work, our previous result is extended to OPF with arbitrary convex cost functions and then a more rigorous theoretical foundation is provided accordingly. First, a necessary and sufficient condition is derived to guarantee the solvability of OPF in polynomial time through its Lagrangian dual. Since solving the dual of OPF is expensive for a large-scale network, a far more scalable algorithm is designed by utilizing the sparsity in the graph of a power network. The computational complexity of this algorithm is related to the number of cycles of the network. Furthermore, it is proved that due to the physics of a power network, the polynomial-time algorithm proposed here always solves every full AC OPF problem precisely or after two mild modifications.

Convex Relaxation for Optimal Power Flow Problem: Mesh Networks

This paper is concerned with a fundamental resource allocation problem for electrical power networks. This problem, named optimal power flow (OPF), is nonconvex due to the nonlinearities imposed by the laws of physics, and has been studied since 1962. We have recently shown that a convex relaxation based on semidefinite programming (SDP) is able to find a global solution of OPF for IEEE benchmark systems, and moreover this technique is guaranteed to work over acyclic (distribution) networks. The present work studies the potential of the SDP relaxation for OPF over cyclic (transmission) networks. Given an arbitrary weakly-cyclic network with cycles of size 3, it is shown that the injection region is convex in the lossless case and that the Pareto front of the injection region is convex in the lossy case. It is also proved that the SDP relaxation of OPF is exact for this type of network. Moreover, it is shown that if the SDP relaxation is not exact for a general mesh network, it would still have a low-rank solution whose rank depends on the structure of the network. Finally, a heuristic method is proposed to recover a rank-1 solution for the SDP relaxation whenever the relaxation is not exact.

Exactness of Semidefinite Relaxations for Nonlinear Optimization Problems with Underlying Graph Structure

This work is concerned with finding a global optimization technique for a broad class of nonlinear optimization problems, including quadratic and polynomial optimization problems. The main objective of this paper is to investigate how the (hidden) structure of a given real/complex-valued optimization problem makes it easy to solve. To this end, three conic relaxations are proposed. Necessary and sufficient conditions are derived for the exactness of each of these relaxations, and it is shown that these conditions are satisfied if the optimization problem is highly structured. More precisely, the structure of the optimization problem is mapped into a generalized weighted graph, where each edge is associated with a weight set extracted from the coefficients of the optimization problem. In the real-valued case, it is shown that the relaxations are all exact if each weight set is sign definite and in addition a condition is satisfied for each cycle of the graph. It is also proved that if some of these conditi...

Competitive equilibria in electricity markets with nonlinearities

This paper is concerned with the existence of competitive equilibria in electricity markets with nonconvex network constraints and nonlinear cost/utility functions. It is assumed that each self-interested market participant shares limited information with the Independent Service Operator (ISO). A necessary and sufficient condition is obtained to guarantee the existence of a competitive equilibrium in the context of economic dispatch. It is shown that a competitive equilibrium may exist, even when the duality gap is nonzero for the optimal power flow (OPF) problem. However, the Lagrange multipliers for the power balance equations in the OPF problem are indeed a correct set of market-clearing prices in presence of no duality gap, which is the case for IEEE systems with 14, 30, 57, 118 and 300 buses. In the case of zero duality gap for the OPF problem, a dynamic pricing scheme is proposed to enable the ISO to find the correct locational marginal prices in polynomial time. Finally, under the assumption that there are a sufficient number of phase shifters in the power system, it is proved that a competitive equilibrium always exists if the Lagrange multipliers associated with the power balance equations are all positive.

Convex relaxation for optimal power flow problem: Mesh networks

This paper is concerned with the optimal power flow (OPF) problem. We have recently shown that a convex relaxation based on semidefinite programming (SDP) is able to find a global solution of OPF for IEEE benchmark systems, and moreover this technique is guaranteed to work over acyclic (distribution) networks. The present work studies the potential of the SDP relaxation for OPF over mesh (transmission) networks. First, we consider a simple class of cyclic systems, namely weakly-cyclic networks with cycles of size 3. We show that the success of the SDP relaxation depends on how the line capacities are modeled mathematically. More precisely, the SDP relaxation is proven to succeed if the capacity of each line is modeled in terms of bus voltage difference, as opposed to line active power, apparent power or angle difference. This result elucidates the role of the problem formulation. Our second contribution is to relate the rank of the minimum-rank solution of the SDP relaxation to the network topology. The goal is to understand how the computational complexity of OPF is related to the underlying topology of the power network. To this end, an upper bound is derived on the rank of the SDP solution, which is expected to be small in practice. A penalization method is then applied to the SDP relaxation to enforce the rank of its solution to become 1, leading to a near-optimal solution for OPF with a guaranteed optimality degree. The remarkable performance of this technique is demonstrated on IEEE systems with more than 7000 different cost functions.

On the exactness of semidefinite relaxation for nonlinear optimization over graphs: Part II

This work is concerned with finding a global optimization technique for a broad class of nonlinear optimization problems, including quadratic and polynomial optimizations. The main objective of this two-part paper is to investigate how the (hidden) structure of a given real/complex-valued optimization makes the problem easy to solve. To this end, three conic relaxations are proposed. Necessary and sufficient conditions are derived for the exactness of each of these relaxations, and it is shown that these conditions are satisfied if the optimization is highly structured. More precisely, the structure of the optimization is mapped into a generalized weighted graph, where each edge is associated with a weight set extracted from the coefficients of the optimization. In the real-valued case, it is shown that the relaxations are all exact if each weight set is sign definite and in addition a condition is satisfied for each cycle of the graph. It is also proved that if some of these conditions are violated, the relaxations still provide a low-rank solution for weakly cyclic graphs. In the complex-valued case, the notion of “sign definite complex sets” is introduced for complex weight sets. It is then shown that the relaxations are exact if each weight set is sign definite (with respect to complex numbers) and the graph is acyclic. In part II of the paper, the complex-valued case is further studied for cyclic graphs and moreover the application of this two-part paper in power system is thoroughly discussed.

Robust efficiency and actuator saturation explain healthy heart rate control and variability

Significance Reduction in human heart rate variability (HRV) is recognized in both clinical and athletic domains as a marker for stress or disease, but previous mathematical and clinical analyses have not fully explained the physiological mechanisms of the variability. Our analysis of HRV using the tools of control mathematics reveals that the occurrence and magnitude of observed HRV is an inevitable outcome of a controlled system with known physiological constraints. In addition to a deeper understanding of physiology, control analysis may lead to the development of timelier monitors that detect control system dysfunction, and more informative monitors that can associate HRV with specific underlying physiological causes. The correlation of healthy states with heart rate variability (HRV) using time series analyses is well documented. Whereas these studies note the accepted proximal role of autonomic nervous system balance in HRV patterns, the responsible deeper physiological, clinically relevant mechanisms have not been fully explained. Using mathematical tools from control theory, we combine mechanistic models of basic physiology with experimental exercise data from healthy human subjects to explain causal relationships among states of stress vs. health, HR control, and HRV, and more importantly, the physiologic requirements and constraints underlying these relationships. Nonlinear dynamics play an important explanatory role––most fundamentally in the actuator saturations arising from unavoidable tradeoffs in robust homeostasis and metabolic efficiency. These results are grounded in domain-specific mechanisms, tradeoffs, and constraints, but they also illustrate important, universal properties of complex systems. We show that the study of complex biological phenomena like HRV requires a framework which facilitates inclusion of diverse domain specifics (e.g., due to physiology, evolution, and measurement technology) in addition to general theories of efficiency, robustness, feedback, dynamics, and supporting mathematical tools.

Low-rank solutions of matrix inequalities with applications to polynomial optimization and matrix completion problems

This paper is concerned with the problem of finding a low-rank solution of an arbitrary sparse linear matrix inequality (LMI). To this end, we map the sparsity of the LMI problem into a graph. We develop a theory relating the rank of the minimum-rank solution of the LMI problem to the sparsity of its underlying graph. Furthermore, we propose two graph-theoretic convex programs to obtain a low-rank solution. The first convex optimization needs a tree decomposition of the sparsity graph. The second one does not rely on any computationally-expensive graph analysis and is always polynomial-time solvable. The results of this work can be readily applied to three separate problems of minimum-rank matrix completion, conic relaxation for polynomial optimization, and affine rank minimization. The results are finally illustrated on two applications of optimal distributed control and nonlinear optimization for electrical networks.

Convexification of optimal power flow problem by means of phase shifters

This paper is concerned with the convexification of the optimal power flow (OPF) problem. We have previously shown that this highly nonconvex problem can be solved efficiently via a convex relaxation after two approximations: (i) adding a sufficient number of virtual phase shifters to the network topology, and (ii) relaxing the power balance equations to inequality constraints. The objective of the present paper is to first provide a better understanding of the implications of Approximation (i) and then remove Approximation (ii). To this end, we investigate the effect of virtual phase shifters on the feasible set of OPF by thoroughly examining a cyclic system. We then show that OPF can be convexified under only Approximation (i), provided some mild assumptions are satisfied. Although this paper mainly focuses on OPF, the results developed here can be applied to several OPF-based emerging optimization problems for future electrical grids.