Mixed-integer Convex Programming Research Articles

Resilience enhancement of cyber-physical distribution systems via mobile power sources and unmanned aerial vehicles

Mobile power sources (MPS) and unmanned aerial vehicles (UAV) are critical and flexible resources to enhance the resilience of cyber–physical distribution systems. However, they are usually independently planned and dispatched. In this paper, considering the cyber–physical interdependence, topology reconfiguration and planned distribution generator islanding, an allocation and dispatch strategy of MPSs and UAVs is proposed. Before an event, a two-stage stochastic optimization based allocation model is built to pre-position MPSs and UAVs considering the uncertainty of events. After the event, a dispatch model is proposed to identify routings of MPSs and UAVs to restore electricity services. Note that both the models are mixed-integer nonlinear three-dimensional (3D) problems. As the optimal service radius and height of a UAV are independent with other variables, these two models are decomposed into two parts, i.e., one part to calculate the optimal service radius and height, and the other to identify resilience enhancement strategy. Then the two models are transformed into a mixed-integer convex programming solved by Progressive Hedging algorithm and a mixed-integer second-order cone programming, respectively. The effectiveness is verified on the modified IEEE 33-node and 123-node test systems. Numerical results highlight the necessity of co-optimizing MPSs and UAVs on cyber–physical distribution systems.

Spreading code optimization for low-earth orbit satellites via mixed-integer convex programming

Optimizing the correlation properties of spreading codes is critical for minimizing inter-channel interference in satellite navigation systems. By improving the codes’ correlation sidelobes, we can enhance navigation performance while minimizing the required spreading code lengths. In the case of low-earth orbit (LEO) satellite navigation, shorter code lengths (on the order of a hundred) are preferred due to their ability to achieve fast signal acquisition. Additionally, the relatively high signal-to-noise ratio in LEO systems reduces the need for longer spreading codes to mitigate inter-channel interference. In this work, we propose a two-stage block coordinate descent (BCD) method which optimizes the codes’ correlation properties while enforcing the autocorrelation sidelobe zero property. In each iteration of the BCD method, we solve a mixed-integer convex program over a block of 25 binary variables. Our method is applicable to spreading code families of arbitrary sizes and lengths, and we demonstrate its effectiveness for a problem with 66 length-127 codes and a problem with 130 length-257 codes.

Two-stage robust programming model for active distribution network with a high proportion of renewable energy

To reduce operating costs and obtain the optimal location and capacity of renewable energy access, a two-stage robust planning model is studied for active distribution networks with a high proportion of renewable energy access. To build a function to minimize annual comprehensive costs, the constraints of power flow, branch power, and node voltage of the active distribution network are studied. Considering the characteristics and properties of power flow safety constraints, the mixed integer second-order cone transformation is used to construct a mixed integer convex programming model, which is convenient for efficiently obtaining the global optimal solution. To avoid the influence of uncertainty, a two-stage robust model is constructed. The explicit maximum value method is used to obtain the solution results that satisfy the planning model of all uncertain problems. Experiments show that the model can obtain the optimal access position and capacity of renewable energy. After application, it can reduce the annual comprehensive cost of the distribution network and ensure the stable and safe operation of the active distribution network.

Multi-Objective Mixed-Integer Convex Programming Method for the Siting and Sizing of UPFCs in a Large-Scale AC/DC Transmission System

Multi-Objective Mixed-Integer Convex Programming Method for the Siting and Sizing of UPFCs in a Large-Scale AC/DC Transmission System

Distributionally robust hierarchical multi-objective voltage risk management for unbalanced distribution systems

The large-scale integration of renewable energy generators (REGs) is key to reducing carbon emissions. However, the intermittency of REG outputs poses operational risk challenges on distribution systems (DSs). This paper proposes a novel Wasserstein-based distributionally robust assessment for the voltage risk index that represents both likelihood and non-linearly increasing severity of voltage violations. We propose a hierarchical DRO model for risk management that integrates a multi-objective distributionally robust optimization (DRO) at the distribution system operator (DSO) level and a basic DRO at the user level. Then, a systematic solution method is developed, where the proposed model is equivalently reformulated into tractable bilevel mixed-integer convex programs. The effectiveness, advantages, and scalability of the proposed method are validated on the IEEE 33-bus system and unbalanced IEEE-123-bus system.

Mixed-integer trajectory optimization with no-fly zone constraints for a hypersonic vehicle

This paper focuses on mixed-integer trajectory optimization of no-fly zones avoidance for a hypersonic vehicle. When there are multiple no-fly zones, the trajectory optimization problem has many local optimal solutions, and the performance of the solution depends on the selection of an initial guess. Essentially, the selection of the initial guess can be abstracted as a discrete path decision. To deal with this issue, this paper first integrates the discrete path decision and continuous trajectory optimization for no-fly zones avoidance and models these as a mixed-integer optimal control problem (MIOCP). As the corner stone, a creative directed graph is established to model the path decision problem, which is then formulated by integer constraints. Combined with the minimum effort optimal control problem (Problem 1), we form the MIOCP (Problem 2), thus integrating the path decision and trajectory optimization to improve global performance. Additionally, to solve the MIOCP, an effective iterative mixed-integer convex programming (IMICP) algorithm is customized by adding two ad-hoc techniques, relaxation and a proximity term, thus remedying the artificial infeasibility. Simulation results show that this approach is practically independent of the initial guess and has better performance than existing methods.

Optimal predictive control of phase change material-based energy storage in buildings via mixed-integer convex programming

This paper describes a mixed-integer convex programming-based control strategy to optimize the operation of phase change material-based energy storage integrated in building supply ducts. To improve the numerical feasibility, the original nonlinear control problem is pre-conditioned and transformed to a mixed-integer convex program through convexification of the cooling system control model, mixed-integer reformulation of the phase change material dynamics and discretization of the supply airflow. The overall control framework was leveraged toward development of two model predictive control strategies to optimally charge/discharge the phase change material storage, through supply air temperature reset, and the building passive thermal mass, via scheduling of the zone air temperature setpoint. These strategies were tested and compared to two baseline control strategies using a simulation case study over three summer days. Test results show that using the phase change material energy storage alone, energy cost savings of 2.9% and peak demand reduction of 46.7% could be achieved, compared to a conventional fixed-supply air temperature and zone air temperature night setup control strategy; when both the active (phase change material) and passive (building thermal mass) storage capacities are utilized, the savings potentials could increase to 8.4% for the energy cost and 65% for the demand charge. • Integration of PCM energy storage in supply ducts is effective in reducing demand. • Predictive control of PCM storage offers significant utility cost savings potential. • Charging/discharging of PCM thermal storage results in efficiency losses. • Combining active and pass storage can provide the maximum load shifting potential.

End-to-end delay guaranteed Service Function Chain deployment: A multi-level mapping approach

Network Function Virtualization (NFV) enables service providers to maximize the business profit via resource-efficient QoS provisioning for customer requested Service Function Chains (SFCs). In recent applications, end-to-end delay is one of the crucial QoS requirements that need to be guaranteed in SFC deployment. Whereas it has been considered in the literature, accurate and comprehensive modeling of the delay in the problem of maximizing the service provider’s profit has not yet been addressed. In this paper, at first, the problem is formulated as a mixed-integer convex programming model. Then, by decomposing the model, a two-level mapping algorithm is developed, that at the first level, maps the functions of the SFCs to virtual network function instances; and in the second level, deploys the instances in the physical network. Simulation results show that the proposed approach achieves a near-optimal solution in comparison to the performance bound obtained by the optimization model, and also superiors the existing solutions.

Hub location problem considering spoke links with incentive-dependent capacities

This study proposes a hub-and-spoke (HS) network in which the capacity of a spoke link depends on the incentive provided to the transporters. The incentives influence the capacities of the spoke links, which in turn affect the hub selections and flow assignment. A mixed integer convex programming model is developed to optimize the proposed HS network. The model minimizes total transport cost by locating p hubs, routing flows between origin–destination (OD) pairs and determining the incentives on spoke links. A customized outer-approximation algorithm is presented to provide exact solutions to the developed model, which is difficult to solve optimally within acceptable computational times for large-scale instances. An approximation algorithm based on the stepwise linearization method is then developed to provide suboptimal solutions. The developed model and algorithms are validated in computational experiments with different problem scales. The experiments demonstrate that the proposed HS network tends to route the flows on more spoke links, balance the transport costs on the potential paths between OD pairs and serve more demands with lower transport costs, when compared with the HS network with fixed capacities.

Resilience promotion of active distribution grids under high penetration of renewables using flexible controllers

Flexible voltage controllers (FVCs), such as voltage regulation transformers (VRTs) and shunt capacitor banks (SCBs), can play a significant role in enhancing the resilience of active distribution systemS (ADS) equipped with large-scale renewable distributed generation. In this paper, a comprehensive resilience response framework is presented, which provides preventive and emergency actions both before and after hurricane events. In this study, a preventive action identifies the on/off state of dispatchable distributed generation (DDG) units and discrete settings of FVCs for both before and after a hurricane event, whereas an emergency action includes the redispatch of DDG unit, load curtailment, and voltage regulation after a hurricane event. The core of the proposed framework is a trilevel max–min robust optimization (MMRO) model, which enhances the resilience of the ADS operation problem against multistage extreme N − k contingencies and renewable power generation uncertainty. The proposed MMRO model is formulated as a mixed-integer convex programming model by relaxing nonconvex AC power flow equations into mixed-integer second-order cone relaxation (MISOCR) equations. Finally, to test the effectiveness of the proposed trilevel MMRO model, we conduct experiments on a modified IEEE-33/69 bus distribution system. The numerical results show that the proposed trilevel MMRO model is an effective model for enhancing the resilience of ADSs.

Convexification of Queueing Formulas by Mixed-Integer Second-Order Cone Programming: An Application to a Discrete Location Problem with Congestion

Mixed-integer second-order cone programs (MISOCPs) form a novel class of mixed-integer convex programs, which can be solved very efficiently as a result of the recent advances in optimization solvers. This paper shows how various performance metrics of M/G/1 queues can be modeled by different MISOCPs. To motivate the reformulation method, it is first applied to a challenging stochastic location problem with congestion, which is broadly used to design socially optimal service systems. Three different MISOCPs are developed and compared on different sets of benchmark test problems. The new formulations efficiently solve very large-size test problems that cannot be solved by the two existing methods developed based on linear programming within reasonable time. The superiority of the conic reformulation method is next shown over a state-space decomposition method recently used to solve an assignment problem in queueing systems. Finally, the general applicability of the method is shown for similar optimization problems that use queue-theoretic performance measures to address customer satisfaction and service quality.

A More Scalable Mixed-Integer Encoding for Metric Temporal Logic

The state-of-the-art in optimal control from timed temporal logic specifications, including Metric Temporal Logic (MTL) and Signal Temporal Logic (STL), is based on Mixed-Integer Convex Programming (MICP). The standard MICP approach is sound and complete, but struggles to scale to long and complex specifications. Drawing on recent advances in trajectory optimization for piecewise-affine systems, we propose a new MICP encoding for finite transition systems that significantly improves scalability to long and complex MTL specifications. Rather than seeking to reduce the number of variables in the MICP, we focus instead on designing an encoding with a tight convex relaxation. This leads to a larger optimization problem, but significantly improves branch-and-bound solver performance. In simulation experiments involving a mobile robot in a grid-world, the proposed encoding can reduce computation times by several orders of magnitude.

Iterative LP-Based Methods for the Multiperiod Optimal Electricity and Gas Flow Problem

In light of the increasing coupling between electricity and gas networks, this paper introduces two novel iterative methods for efficiently solving the multiperiod optimal electricity and gas flow (MOEGF) problem. The first is an iterative MILP-based method and the second is an iterative LP-based method with an elaborate procedure for ensuring an integral solution. The convergence of the two approaches is founded on two key features. The first is a penalty term with a single, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">automatically tuned</i> , parameter for controlling the step size of the gas network iterates. The second is a sequence of supporting hyperplanes and halfspaces for controlling the convergence of the electricity network iterates. Moreover, the two proposed algorithms use as a warm start the solution from a novel polyhedral relaxation of the MOEGF problem, for a noticeable improvement in computation time as compared to a cold start. Unlike the first method, which invokes a branch-and-bound algorithm to find an integral solution, the second method implements an elaborate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">steering</i> procedure that guides the continuous variables to take integral values at the solution. Numerical evaluation demonstrates that the two proposed methods can converge to high-quality <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">feasible</i> solutions in computation times at least <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">two orders of magnitude faster</i> than both a state-of-the-art nonlinear branch-and-bound (NLBB) MINLP solver and a mixed-integer convex programming (MICP) relaxation of the MOEGF problem. The experimental setup consists of five test cases, three of which involve the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">real</i> electricity and gas transmission networks of the state of Victoria with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">actual</i> linepack and demand profiles.

The Mixed-Observable Constrained Linear Quadratic Regulator Problem: the Exact Solution and Practical Algorithms

This paper studies the problem of steering a linear system subject to state and input constraints towards a goal location that may be inferred only through noisy partial observations. We assume mixed-observable settings, where the system's state is fully observable and the environment's state defining the goal location is only partially observed. In these settings, the planning problem is an infinite-dimensional optimization problem where the objective is to minimize the expected cost. We show how to reformulate the control problem as a finite-dimensional deterministic problem by optimizing over a trajectory tree. Leveraging this result, we demonstrate that when the environment is static, the observation model piecewise, and cost function convex, the original control problem can be reformulated as a Mixed-Integer Convex Program (MICP) that can be solved to global optimality using a branch-and-bound algorithm. The effectiveness of the proposed approach is demonstrated on navigation tasks, where the goal location should be inferred through noisy measurements.

Risk Constrained Energy Efficient Optimal Operation of a Converter Governed AC/DC Hybrid Distribution Network With Distributed Energy Resources and Volt-VAR Controlling Devices

Increasing penetration of direct current (dc) based distributed energy resources and dc loads in the conventional alternating current (ac) network necessitate the deployment of ac/dc hybrid distribution networks (HDNs). In view with the development of advanced energy management policy for ac/dc HDNs, unlike previous literatures, this article proposes a risk constrained energy efficient management algorithm by merging load shifting (LS) and conservation voltage reduction (CVR) techniques. The optimization framework aims to simultaneously minimize both true and conditional risk or conditional value at risk values of the expected energy cost under uncertain solar power generation, load demand, and upper grid energy price. In contrast with the available stochastic optimization process, in this article, two-point estimation strategy is employed in place of Monte Carlo simulation for scenario generation from the probability density functions of the uncertain parameters to reduce computational exertion. The proposed centralized optimization framework is initially developed as mixed integer nonconvex programming but to avoid computation complexity, the nonlinear components are replaced by their linear counterparts. Later, a new solution process named successive mixed integer linear programming (s-MILP) is proposed to obtain the optimal decisions for deployment of LS and CVR through smart inverters and volt-VAR controlling devices. Efficacy of the proposed technique is demonstrated on modified IEEE 33 bus ac/dc HDN and the most energy efficient operation is found by merging LS and CVR. Simulation outcomes prove fast and near optimal convergence of the s-MILP compared to conventional second-order conic programming relaxed mixed integer convex programming and piecewise linearization based MILP. Further, to assess the impact of network size on the solution time and optimality, the proposed advanced distribution network management systems strategy is employed on 132 bus ac/dc HDN.

Partially distributed outer approximation

This paper presents a novel partially distributed outer approximation algorithm, named PaDOA, for solving a class of structured mixed integer convex programming problems to global optimality. The proposed scheme uses an iterative outer approximation method for coupled mixed integer optimization problems with separable convex objective functions, affine coupling constraints, and compact domain. PaDOA proceeds by alternating between solving large-scale structured mixed-integer linear programming problems and partially decoupled mixed-integer nonlinear programming subproblems that comprise much fewer integer variables. We establish conditions under which PaDOA converges to global minimizers after a finite number of iterations and verify these properties with an application to thermostatically controlled loads and to mixed-integer regression.

Optimal Building Thermal Load Scheduling for Simultaneous Participation in Energy and Frequency Regulation Markets

This paper presents an optimal scheduling solution for building thermal loads that simultaneously participate in the wholesale energy and frequency regulation markets. The solution combines (1) a lower-level regulation capacity reset strategy that identifies the available regulation capacity for each hour, and (2) an upper-level zone temperature scheduling algorithm to find the optimal load trajectory with a minimum net electricity cost. In the supervisory scheduling strategy, piece-wise linear approximations of representative air-conditioning equipment behaviors, derived from an offline analysis of the capacity reset mechanism, are used to predict the cooling power and regulation capacity; and a mixed-integer convex program is formulated and solved to determine the optimal control actions. In order to evaluate the performance of the developed control solution, two baseline strategies are considered, one with a conventional night setup/back control and the other utilizing an optimization procedure for minimizing the energy cost only. Five-day simulation tests were carried out for the various control strategies. Compared to the baseline night setup/back strategy, the energy-priority controller led to a 26% lower regulation credit and consequentially caused a net cost increase of 2%; the proposed bi-market control solution was able to increase the regulation credit by 118% and reduce the net electricity cost by 14%.

Real-time autonomous dynamic reconfiguration based on deep learning algorithm for distribution network

In this work, a novel real-time autonomous dynamic reconfiguration (ADR) method is proposed to reduce the cost of power loss and switch action of distribution network based on the deep learning (DL) algorithm. The proposed ADR method can be decision-making from the historical control dataset and the real-time system state. To obtain the historical control dataset efficiently, the heuristic rule is first introduced to improve the mixed integer convex programming method for making reconfiguration strategies to support grid operate in cost effective manner. To learn the reconfiguration strategy from the historical dataset, a long-short term memory network (LSTM) is presented to provide the effective training on the historical control dataset, and a switch action function considering real-time differences of operation cost is formulated, which can be combined with the trained LSTM model to achieve ADR. Case studies on the IEEE 33-bus system and a realistic Taiwan power company (TPC) 84-bus system verify that the ADR can obtain an ideal reconfiguration solution in the order of milliseconds and has high robustness.

Randomized Constraints Consensus for Distributed Robust Mixed-Integer Programming

In this article, we consider a network of processors aiming at cooperatively solving mixed-integer convex programs subject to uncertainty. Each node only knows a common cost function and its local uncertain constraint set. We propose a randomized, distributed algorithm working under asynchronous, unreliable, and directed communication. The algorithm is based on a local computation and communication paradigm. At each communication round, nodes perform two updates: 1) A verification in which they check—in a randomized fashion—the robust feasibility of a candidate optimal point, and 2) an optimization step in which they exchange their candidate basis (the minimal set of constraints defining a solution) with neighbors and locally solve an optimization problem. As a main result, we show that processors can stop the algorithm after a finite number of communication rounds (either because verification has been successful for a sufficient number of rounds or because a given threshold has been reached) so that candidate optimal solutions are consensual. The common solution has proven to be—with high confidence—feasible and, hence, optimal for the entire set of uncertainty except a subset having an arbitrarily small probability measure. We show the effectiveness of the proposed distributed algorithm using two examples: a random, uncertain mixed-integer linear program and a distributed localization in wireless sensor networks. The distributed algorithm is implemented on a multicore platform in which the nodes communicate asynchronously.

Exact Logit-Based Product Design

The share-of-choice product design (SOCPD) problem is to find the product, as defined by its attributes, that maximizes market share arising from a collection of customer types or segments. When customers follow a logit model of choice, the market share is given by a weighted sum of logistic probabilities, leading to the logit-based share-of-choice product design problem. At first glance, this problem appears hopeless: one must optimize an objective function that is neither convex nor concave, over an exponentially-sized discrete set of attribute combinations. In this paper, we develop an exact methodology for solving this problem based on modern integer, convex and conic optimization. At the heart of our methodology is the surprising result that the logit-based SOCPD problem can be exactly reformulated as a mixed-integer convex program. Using both synthetic problem instances and instances derived from real conjoint data sets, we show that our methodology can solve large instances to provable optimality or near optimality in operationally feasible time frames.