Mixed-integer Semidefinite Programming Research Articles

Toward a scalable robust security-constrained optimal power flow using a proximal projection bundle method

Robust security-constrained optimal power flow (rSCOPF) aims to find the worst-case contingencies of alternating current optimal power flow (ACOPF) in power systems. With the rise of GPU architectures on the upcoming supercomputer architectures, optimization algorithms that rely on sparse linear algebra and indefinite linear systems are becoming increasingly hard to solve efficiently (e.g. interior-point method). To address this we revisit a maximin optimization formulation of the rSCOPF and the single-level mixed-integer semidefinite programming (MISDP) reformulation, which is obtained by taking the Lagrangian relaxation of the inner minimization ACOPF problem. In this paper, we focus on the development of a proximal projection bundle method (PPBM) for solving continuous relaxation node subproblems of the MISDP problem, based primarily on the well-known alternating direction method of multipliers. Cutting planes reminiscent of bundle method ideas are also applied in coordination with updates of the proximal parameter. The cutting-plane method can generate a large number of linear inequalities, leading to a large scale but decomposable quadratic programming (QP) subproblem that is amenable to GPUs. We present the numerical results on the IEEE 30, 57, 118, and 300-bus systems by using our PBMM method. We discuss the main computational bottleneck of our method, which is the time taken to solve each iteration of a QP subproblem instance of the PPBM, and how GPU architectures can accelerate this solution process.

A mixed-integer SDP solution approach to distributionally robust unit commitment with second order moment constraints

A power system unit commitment (UC) problem considering uncertainties of renewable energy sources is investigated in this paper, through a distributionally robust optimization approach. We assume that the first and second order moments of stochastic parameters can be inferred from historical data, and then employed to model the set of probability distributions. The resulting problem is a two-stage distributionally robust unit commitment with second order moment constraints, and we show that it can be recast as a mixed-integer semidefinite programming (MI-SDP) with finite constraints. The solution algorithm of the problem comprises solving a series of relaxed MI-SDPs and a subroutine of feasibility checking and vertex generation. Based on the verification of strong duality of the semidefinite programming (SDP) problems, we propose a cutting plane algorithm for solving the MI-SDPs; we also introduce a SDP relaxation for the feasibility checking problem, which is an intractable biconvex optimization. Experimental results on a IEEE 6-bus system are presented, showing that without any tunings of parameters, the real-time operation cost of distributionally robust UC method outperforms those of deterministic UC and two-stage robust UC methods in general, and our method also enjoys higher reliability of dispatch operation.

A regularized state estimation scheme for a robust monitoring of the distribution grid

In this paper, we propose a regularized state estimation (SE) scheme for the distribution grid. The ultimate goal is to track accurately the system state at a faster time scale according to the requirements of the new operational environment. The SE algorithm operates at two different time scales in which the set of available measurements are different. At the main time instants (every 15 min) the set of observations comprises SCADA measurements, pseudomeasurements and micro Phasor Measurement Units (μPMUs). In this case, we resort to a Regularized version of the Normal Equations based SE (R-NESE). In the intermediate time instants, only a reduced number of μPMU measurements is available. To circumvent observability issues, we exploit the fact that the voltage drop in adjacent buses is limited and, on that basis, a regularized weighted total variation estimation (WTVSE) problem is formulated. Then, the impact of in-line voltage regulators (IVLRs) is explicitly taken into consideration and that, forces us to decompose and solve the original SE problem for a number of smaller regions (D-WTVSE). The latter can be iteratively solved by resorting to the Alternating Direction Method of Multipliers (ADMM). Complementarily, we also present a μPMU placement method (μPP) in order to improve the conditioning of the R-NESE problem. This problem can be posed as a mixed integer semidefinite programming model (MISDP) and, thus, can be efficiently solved. The performance of the proposed scheme is numerically assessed on (mostly) a 95-bus distribution system for a number of realistic conditions of noise, load and photovoltaic generation profiles. A number of benchmarks are provided, as well.

Coordinating Multiple Sources for Service Restoration to Enhance Resilience of Distribution Systems

When a major outage occurs on a distribution system due to extreme events, microgrids, distributed generators, and other local resources can be used to restore critical loads and enhance resiliency. This paper proposes a decision-making method to determine the optimal restoration strategy coordinating multiple sources to serve critical loads after blackouts. The critical load restoration problem is solved by a two-stage method with the first stage deciding the post-restoration topology and the second stage determining the set of loads to be restored and the outputs of sources. In the second stage, the problem is formulated as a mixed-integer semidefinite program. The objective is maximizing the number of loads restored, weighted by their priority. The unbalanced three-phase power flow constraint and operational constraints are considered. An iterative algorithm is proposed to deal with integer variables and can attain the global optimum of the critical load restoration problem by solving a few semidefinite programs under two conditions. The effectiveness of the proposed method is validated by numerical simulation with the modified IEEE 13-node test feeder and the modified IEEE 123-node test feeder under plenty of scenarios. The results indicate that the optimal restoration strategy can be determined efficiently in most scenarios.

Robust Measurement Placement for Distribution System State Estimation

This paper considers the problem of placing a limited number of measurements to improve the estimation accuracy in distribution system state estimation (DSSE). In distribution systems, intermittent distributed energy resources and volatile loads will result in a wide variation of system operating conditions. The proposed measurement placement problem is to decide the optimal locations and types of measurements to be placed in the distribution systems that minimize the worst-case estimation errors for DSSE over different system operating conditions. Four indices of the estimation error covariance matrix are chosen as the criteria of accuracy. The proposed measurement placement problem is formulated as a mixed-integer semidefinite programming (MISDP) problem. To avoid the combinatorial complexity, a convex relaxation, followed by a local optimization method, is employed to solve the MISDP problem. The proposed problem and the effectiveness of the proposed solution method are numerically demonstrated on the 33-bus distribution system.

On construction of upper and lower bounds for the HOMO-LUMO spectral gap

In this paper we study spectral properties of graphs which are constructed from two given invertible graphs by bridging them over a bipartite graph. We analyze the so-called HOMO-LUMO spectral gap which is the difference between the smallest positive and largest negative eigenvalue of the adjacency matrix of a graph. We investigate its dependence on the bridging bipartite graph and we construct a mixed integer semidefinite program for maximization of the HOMO-LUMO gap with respect to the bridging bipartite graph. We also derive upper and lower bounds for the optimal HOMO-LUMO spectral graph by means of semidefinite relaxation techniques. Several computational examples are also presented in this paper.

Robust truss topology optimization via semidefinite programming with complementarity constraints: a difference-of-convex programming approach

The robust truss topology optimization against the uncertain static external load can be formulated as mixed-integer semidefinite programming. Although a global optimal solution can be computed with a branch-and-bound method, it is very time-consuming. This paper presents an alternative formulation, semidefinite programming with complementarity constraints, and proposes an efficient heuristic. The proposed method is based upon the convex-concave procedure for DC (difference-of-convex) programming. It is shown that the method can often find a practically reasonable truss design within the computational cost of solving some dozen of convex optimization subproblems.

Optimized Measurement Allocation for Power Distribution Systems Using Mixed Integer SDP

This paper presents a method for placement of metering systems in distribution grids taking into account the aspects of observability and state estimation accuracy. The measurement placement problem is conceptualized as the optimized allocation of a prespecified number of measurements in order to minimize the error variances of the state estimates either for observable or nonobservable networks. The proposed approach exploits the M-optimal experimental design technique to formulate the problem as a mixed integer semidefinite programming model. We demonstrate the efficacy of the proposed approach in terms of optimality and convergence speed through simulations on a 95-bus distribution grid.

A framework for solving mixed-integer semidefinite programs

Mixed-integer semidefinite programs (MISDPs) arise in many applications and several problem-specific solution approaches have been studied recently. In this paper, we investigate a generic branch-and-bound framework for solving such problems. We first show that strict duality of the semidefinite relaxations is inherited to the subproblems. Then solver components such as dual fixing, branching rules, and primal heuristics are presented. We show the applicability of an implementation of the proposed methods on three kinds of problems. The results show the positive computational impact of the different solver components, depending on the semidefinite programming solver used. This demonstrates that practically relevant MISDPs can successfully be solved using a general purpose solver.

Allocating Sensors and Actuators via Optimal Estimation and Control

We consider the problem of planning the location and size of sensors and actuators to achieve optimal dynamic performance. Using basic results from control and convex optimization, we formulate mixed-integer semidefinite programs for the actuator placement and sizing to obtain the linear quadratic regulator with the lowest cost, and the sensor placement to obtain the Kalman filter with the lowest error. The two formulations are nearly identical due to the duality of optimal linear control and estimation. We also pose similar problems in terms of observability and controllability, which result in smaller mixed-integer semidefinite programs. Since the mixed-integer semidefinite programing is not yet a mature technology, we also use greedy heuristics in conjunction with continuous semidefinite programming. The approach is demonstrated on two modern applications from power systems: the placement and sizing of energy storage for regulation and the placement of phasor measurement units for estimation.

User grouping and admission control for multi-group multicast beamforming in MIMO systems

In this paper, we consider the multi-group multicasting problem in a MIMO wireless cellular system, whereby the base station can generate multiple beampatterns to serve multiple groups simultaneously. Each user is interested in multiple multicast groups, but is only allowed to be assigned to one of them eventually during each transmission interval. We formulate it as a joint optimization problem of multicast beamformer design as well as user grouping, and admission control is also considered to account for the quality of service demands of individual users. To deal with this NP-hard problem, we adopt the big-M formulation method and the semidefinite relaxation scheme to reformulate it to a mixed-integer semidefinite programming problem. Three algorithms are developed to solve the obtained approximate problem with different computational costs. Simulation results are presented to demonstrate the performance of the proposed algorithms in terms of the number of admitted users, power consumption and computation time.

Unit commitment under uncertainty in AC transmission systemsviarisk averse semidefinite stochastic Programs

This paper addresses unit commitment under uncertainty of load and power infeed from renewables in alternating current (AC) power systems. Beside traditional unit-commitment constraints, the physics of power flow are included. To gain globally optimal solutions a recent semidefinite programming approach is used, which leads us to risk averse two-stage stochastic mixed integer semidefinite programs for which a decomposition algorithm is presented.

Maximization of the Spectral Gap for Chemical Graphs by means of a Solution to a Mixed Integer Semidefinite Program

Maximization of the Spectral Gap for Chemical Graphs by means of a Solution to a Mixed Integer Semidefinite Program

Robust Truss Topology Design with Beam Elements via Mixed Integer Nonlinear Semidefinite Programming

In this article, we propose a nonlinear semidefinite program (SDP) for the robust trusstopology design (TTD) problem with beam elements. Starting from the semidefinite formulation ofthe robust TTD problem we derive a stiffness matrix that can model rigid connections between beams.Since the stiffness matrix depends nonlinearly on the cross-sectional areas of the beams, this leads toa nonlinear SDP. We present numerical results using a sequential SDP approach and compare them toresults obtained via a general method for robust PDE-constrained optimization applied to the equationsof linear elasticity. Furthermore, we present two mixed integer semidefinite programs (MISDP), onefor the optimal choice of connecting elements, which is nonlinear, and one for the correspondingproblem with discrete cross-sectional areas.

Synthesizing Robust Communication Networks for Unmanned Aerial Vehicles With Resource Constraints

In this article, we address the problem of synthesizing communication networks for unmanned aerial vehicles (UAVs) in the presence of resource constraints. UAVs can be deployed as backbone nodes in ad hoc networks that can be central to civilian and military applications. The cost of operation of the network depends on the resources that are used such as the total power consumption associated with the network and the number of communication links in the network. The objective of the problem is to synthesize a communication network that maximizes connectivity subject to the cost of operation being within the specified budget for the resources. It is known that algebraic connectivity is a measure of robust connectivity and hence, it is chosen as an objective for optimization. We pose the network synthesis problem as a mixed-integer semidefinite program (MISDP): (1) provide an algorithm for computing optimal solutions using cutting plane methods; (2) develop lower bounds by posing the problem as a binary semidefinite program; and (3) construct feasible solutions using heuristics and estimate their quality. The network synthesis problem is a nondeterministic polynomial--time (NP)-hard problem. We provide some computational results to corroborate the performance of the proposed algorithms.

Reactive Power Optimization for Distribution Network with Distributed Generators Based on Semi-Definite Programming

Reactive power optimization for distribution network with distributed generators is a complicated nonconvex nonlinear mixed integer programming problem. This paper built a mathematical model of reactive power optimization for distribution network and a new method to solve this problem was proposed based on semi-definite programming. The original mathematical model was transformed and relaxed into a convex SDP model, to guarantee the global optimal solution within the polynomial times. Then the model was extended to a mixed integer semi-definite programming model with discrete variables when considering discrete compensation equipment such as capacitor banks. Global optimal solution of this model can be obtained by cutting plane method and branch and bound method. Numerical tests on the modified IEEE 33-bus system show this method is exact and can be solved efficiently.

An outer-approximation approach for information-maximizing sensor selection

This paper addresses information-maximizing sensor selection that determines a set of measurement locations providing the largest entropy reduction in the estimates of the state variables. A new mixed-integer semidefinite program (MISDP) formulation is proposed for this selection under the constraints resulting from communication limitations. This formulation employs binary variables indicating if the corresponding measurement location is selected, and ensures convexity of the objective function and linearity of the constraint functions by exploiting the linear equivalent form of a bilinear term involving binary variables. An outer-approximation algorithm is then developed for the MISDP formulation that obtains the global optimal solution by solving a sequence of mixed-integer linear programs for which reliable solvers are available. Numerical experiments verify the solution optimality and the computational effectiveness of the proposed algorithm by comparing it to branch-and-bound-based approaches with nonlinear programming relaxation. An example of sensor selection to track a moving target is considered to demonstrate the applicability of the proposed method and highlight its ability to handle quadratic constraints.

Branching on hyperplane methods for mixed integer linear and convex programming using adjoint lattices

We present branching-on-hyperplane methods for solving mixed integer linear and mixed integer convex programs. In particular, we formulate the problem of finding a good branching hyperplane using a novel concept of adjoint lattice. We also reformulate the problem of rounding a continuous solution to a mixed integer solution. A worst case complexity of a Lenstra-type algorithm is established using an approximate log-barrier center to obtain an ellipsoidal rounding of the feasible set. The results for the mixed integer convex programming also establish a complexity result for the mixed integer second order cone programming and mixed integer semidefinite programming feasibility problems as a special case. Our results motivate an alternative reformulation technique and a branching heuristic using a generalized (e.g., ellipsoidal) norm reduced basis available at the root node.

Global optimization of robust truss topology via mixed integer semidefinite programming

This paper discusses a global optimization method of robust truss topology under the load uncertainties and slenderness constraints of the member cross-sectional areas. We consider a non-stochastic uncertainty of the external load, and attempt to minimize the maximum compliance corresponding to the most critical load. A design-dependent uncertainty model in the external load is proposed in order to consider the variation of truss topology rigorously. It is shown that this optimization problem can be formulated as a 0–1 mixed integer semidefinite programming (0–1MISDP) problem. We propose a branch-and-bound method for computing the global optimal solution of the 0–1MISDP. Numerical examples illustrate that the topology of robust optimal truss depends on the magnitude of uncertainty. The presented method can provide global optimal solutions for benchmark examples, which can be used for evaluating the performance of any other local optimization method for robust structural optimization.

Clustering via minimum volume ellipsoids

We propose minimum volume ellipsoids (MVE) clustering as an alternative clustering technique to k-means for data clusters with ellipsoidal shapes and explore its value and practicality. MVE clustering allocates data points into clusters in a way that minimizes the geometric mean of the volumes of each cluster's covering ellipsoids. Motivations for this approach include its scale-invariance, its ability to handle asymmetric and unequal clusters, and our ability to formulate it as a mixed-integer semidefinite programming problem that can be solved to global optimality. We present some preliminary empirical results that illustrate MVE clustering as an appropriate method for clustering data from mixtures of "ellipsoidal" distributions and compare its performance with the k-means clustering algorithm as well as the MCLUST algorithm (which is based on a maximum likelihood EM algorithm) available in the statistical package R.