Mixed-integer Second-order Cone Programming Research Articles

Applications of Stochastic Mixed-Integer Second-Order Cone Optimization

Second-order cone programming problems are a tractable subclass of convex optimization problems that can be solved using polynomial algorithms. In the last decade, stochastic second-order cone programming problems have been studied, and efficient algorithms for solving them have been developed. The mixed-integer version of these problems is a new class of interest to the optimization community and practitioners, in which certain variables are required to be integers. In this paper, we describe five applications that lead to stochastic mixed-integer second-order cone programming problems. Additionally, we present solution algorithms for solving stochastic mixed-integer second-order cone programming using cuts and relaxations by combining existing algorithms for stochastic second-order cone programming with extensions of mixed-integer second-order cone programming. The applications, which are the focus of this paper, include facility location, portfolio optimization, uncapacitated inventory, battery swapping stations, and berth allocation planning. Considering the fact that mixed-integer programs are usually known to be NP-hard, bringing applications to the surface can detect tractable special cases and inspire for further algorithmic improvements in the future.

Resilience-Oriented Distributed Load Restoration Method for Integrated Power Distribution and Natural Gas Systems

This study proposes a distributed load restoration method to effectively model and solve the load restoration problem of integrated power distribution and natural gas systems (IDGSs) after faults for resilient operation. The IDGS load restoration problem is devised as a mixed-integer second-order cone programming (MISOCP) model to minimize the total load loss cost of IDGS. This model considers distribution network reconfiguration and the coupling effect between the power distribution system (PDS) and natural gas system (NGS). Moreover, a bidirectional gas flow model is used to offer more flexibility for load restoration when the NGS is faulty. By considering the independent management and information privacy of each energy subsystem, the proposed model is solved using an alternating direction method of multipliers (ADMM) algorithm in a distributed manner. Furthermore, an adaptive-ADMM algorithm is used for convergence acceleration. Finally, numerical studies are conducted to validate the proposed methods effectiveness.

Multi-objective Dynamic Reconfiguration for Urban Distribution Network Considering Multi-level Switching Modes

The increasing integration of photovoltaic generators (PVGs) and the uneven economic development in different regions may cause the unbalanced spatial-temporal distribution of load demands in an urban distribution network (UDN). This may lead to undesired consequences, including PVG curtailment, load shedding, and equipment inefficiency, etc. Global dynamic reconfiguration provides a promising method to solve those challenges. However, the power flow transfer capabilities for different kinds of switches are diverse, and the willingness of distribution system operators (DSOs) to select them is also different. In this paper, we formulate a multi-objective dynamic reconfiguration optimization model suitable for multi-level switching modes to minimize the operation cost, load imbalance, and the PVG curtailment. The multi-level switching includes feeder-level switching, transformer-level switching, and substation-level switching. A novel load balancing index is devised to quantify the global load balancing degree at different levels. Then, a stochastic programming model based on selected scenarios is established to address the uncertainties of PVGs and loads. Afterward, the fuzzy c-means (FCMs) clustering is applied to divide the time periods of reconfiguration. Further-more, the modified binary particle swarm optimization (BPSO) and Cplex solver are combined to solve the proposed mixed-in-teger second-order cone programming (MISOCP) model. Numerical results based on the 148-node and 297-node systems are obtained to validate the effectiveness of the proposed method.

An MISOCP-Based Decomposition Approach for the Unit Commitment Problem with AC Power Flows

Unit Commitment (UC) and Optimal Power Flow (OPF) are two fundamental problems in short-term electric power systems planning that are traditionally solved sequentially. The state-of-the-art mostly uses a direct current (DC) approximation of the power flow equations. However, utilizing the DC approach in the UC-level may lead to infeasible or suboptimal generator commitment schedules for the OPF problem. In this paper, we aim to simultaneously solve the UC Problem with alternating current (AC) power flow equations, which combines the challenging nature of both UC and OPF Problems. Due to the highly nonconvex nature of the AC flow equations, we utilize the mixed-integer second-order cone programming (MISOCP) relaxation of the UC Problem as the basis of our solution approach. The MISOCP relaxation is utilized for finding both a lower bound and a candidate generator commitment schedule. Once this schedule is obtained, we solve a multi-period OPF problem to obtain feasible solutions for the UC problem with AC power flows. For smaller instances, we develop two different algorithms that exploit the recent advances in the OPF literature and obtain high-quality feasible solutions with provably small optimality gaps. For solving larger instances, we develop a Lagrangian decomposition based approach that yields promising results.

Optimal Energy Reserve Scheduling in Integrated Electricity and Gas Systems Considering Reliability Requirements

With the growing interdependence between the electricity system and the natural gas system, the operation uncertainties in either subsystem, such as wind fluctuations or component failures, could have a magnified impact on the reliability of the whole system due to energy interactions. A joint reserve scheduling model considering the cross-sectorial impacts of operation uncertainties is essential but still insufficient to guarantee the reliable operation of the integrated electricity and natural gas system (IEGS). Therefore, this paper proposes a day-ahead security-constrained unit commitment (SCUC) model for the IEGS to schedule the operation and reserve si-multaneously considering reliability requirements. Firstly, the multi-state models for generating units and gas wells are established. Based on the multi-state models, the expected unserved energy cost (EUEC) and the expected wind curtailment cost (EWC) criteria are proposed based on probabilistic methods considering wind fluctuation and random failures of components in IEGS. Furthermore, the EUEC and EWC criteria are incorporated into the day-ahead SCUC model, which is noncon-vex and mathematically reformulated into a solvable mixed-integer second-order cone programming (MISOCP) problem. The proposed model is validated using an IEEE 30-bus system and Belgium 20-node natural gas system. Numerical results demonstrate that the proposed model can effectively schedule the energy reserve to guarantee the reliable operation of the IEGS considering the multiple uncertainties in different subsystems and the cross-sectorial failure propagation.

Bi-Level Strategic Bidding Model of Gas-fired Units in Interdependent Electricity and Natural Gas Markets

The increasing penetration of renewables tends to demand more flexibility from gas-fired units to maintain system reliability, which significantly strengthens the interdependency between the electricity and the natural gas markets. To this end, intra-day gas market trading may be necessary, and a completely deregulated natural gas market may emerge in the future. As the interdependency between the electricity and the natural gas markets keeps growing, this paper proposes a bi-level strategic bidding model to study the market behaviors of the gas-fired unit, which simultaneously acts as a strategic power producer in the electricity market and a strategic gas buyer in the natural gas market. The upper-level problem maximizes the profit of the gas-fired unit, and the two lower-level problems respectively model the market-clearings of the electricity and the natural gas markets. The upper and lower levels interact through locational marginal prices of electricity and natural gas. The proposed bi-level optimization model is converted to a mathematical problem with equilibrium constraint (MPEC). Then it is transformed into a mixed-integer second-order-cone programming (MISOCP) for efficient calculation. Numerical results demonstrate the proposed methodology is capable of mimicking the market-clearing process and strategic behaviors of the gas-fired unit in the interdependent energy markets effectively.

Controlled Islanding Strategy Considering Uncertainty of Renewable Energy Sources Based on Chance-constrained Model

Controlled islanding plays an essential role in preventing the blackout of power systems. Although there are several studies on this topic in the past, no enough attention is paid to the uncertainty brought by renewable energy sources (RESs) that may cause unpredictable unbalanced power and the observability of power systems after islanding that is essential for back-up black-start measures. Therefore, a novel controlled islanding model based on mixed-integer second-order cone and chance-constrained programming (MISOCCP) is proposed to address these issues. First, the uncertainty of RESs is characterized by their possibility distribution models with chance constraints, and the requirements, e. g., system observ-ability, for rapid back-up black-start measures are also considered. Then, a law of large numbers (LLN) based method is employed for converting the chance constraints into deterministic ones and reformulating the non-convex model into convex one. Finally, case studies on the revised IEEE 39-bus and 118-bus power systems as well as the comparisons among different models are given to demonstrate the effectiveness of the proposed model. The results show that the proposed model can result in less unbalanced power and better observability after islanding compared with other models.

Distributionally Robust Optimal Dispatch of Virtual Power Plant Based on Moment of Renewable Energy Resource

Virtual power plants can effectively integrate different types of distributed energy resources, which have become a new operation mode with substantial advantages such as high flexibility, adaptability, and economy. This paper proposes a distributionally robust optimal dispatch approach for virtual power plants to determine an optimal day-ahead dispatch under uncertainties of renewable energy sources. The proposed distributionally robust approach characterizes probability distributions of renewable power output by moments. In this regard, the faults of stochastic optimization and traditional robust optimization can be overcome. Firstly, a second-order cone-based ambiguity set that incorporates the first and second moments of renewable power output is constructed, and a day-ahead two-stage distributionally robust optimization model is proposed for virtual power plants participating in day-ahead electricity markets. Then, an effective solution method based on the affine policy and second-order cone duality theory is employed to reformulate the proposed model into a deterministic mixed-integer second-order cone programming problem, which improves the computational efficiency of the model. Finally, the numerical results demonstrate that the proposed method achieves a better balance between robustness and economy. They also validate that the dispatch strategy of virtual power plants can be adjusted to reduce costs according to the moment information of renewable power output.

Optimal Scheduling for Flexible Regional Integrated Energy System with Soft Open Point

The Regional Integrated Energy System (RIES) has brought new modes of development, utilization, conversion, storage of energy. The introduction of Soft Open Point (SOP) and the application of Power to Gas (P2G) technology will greatly deepen the coupling of the electricity-gas integrated energy system, improve the flexibility and safety of the operation of the power system, and bring a deal of benefits to the power system. On this background, an optimal dispatch model of RIES combined cold, heat, gas and electricity with SOP is proposed. Firstly, RIES architecture with SOP and P2G is designed and its mathematical model also is built. Secondly, on the basis of considering the optimal scheduling of combined cold, heat, gas and electricity, the optimal scheduling model for RIES was established. After that, the original model is transformed into a mixed-integer second-order cone programming model by using linearization and second-order cone relaxation techniques, and the CPLEX solver is invoked to solve the optimization problem. Finally, the modified IEEE 33-bus system is used to analyze the benefits of SOP, P2G technology and lithium bromide absorption chillers in reducing system network loss and cost, as well as improving the system’s ability to absorb wind and solar and operating safety.

An integrated integer programming model with a simulated annealing heuristic for the carrier vehicle traveling salesman problem

We investigate a carrier vehicle traveling salesman problem (CVTSP) to visit n-targets in Euclidean space. Each target is visited by a smaller vehicle (e.g., UAV) consecutively, launched from and landed to a carrier. The smaller vehicle and the carrier must synchronize during both launching and landing. We present a mixed-integer second-order cone programming model with TSP-like constraints to determine the best target visiting sequence and the associated launch and land coordinates. A simulated annealing algorithm is implemented to shorten the computational time. The computational experiments indicate the simulated annealing algorithm calculates good CVTSP solutions within a reasonable amount of time.

Offline-Channel Planning in Smart Omnichannel Retailing

Problem definition: Observing the retail industry inevitably evolving into omnichannel, we study an offline-channel planning problem that helps an omnichannel retailer make store location and location-dependent assortment decisions in its offline channel to maximize profit across both online and offline channels, given that customers’ purchase decisions depend on not only their preferences across products but also, their valuation discrepancies across channels, as well as the hassle costs incurred. Academic/practical relevance: The proposed model and the solution approach extend the literature on retail-channel management, omnichannel assortment planning, and the broader field of smart retailing/cities. Methodology: We derive parameterized models to capture customers’ channel choice and product choice behaviors and customize a corresponding parameter estimation approach employing the expectation-maximization method. To solve the proposed optimization model, we develop a tractable mixed integer second-order conic programming reformulation and explore the structural properties of the reformulation to derive strengthening cuts in closed form. Results: We numerically validate the efficacy of the proposed solution approach and demonstrate the parameter estimation approach. We further draw managerial insights from the numerical studies using real data sets. Managerial implications: We verify that omnichannel retailers should provide location-dependent offline assortments. In addition, our benchmark studies reveal the necessity and significance of jointly determining offline store locations and assortments, as well as of incorporating the online channel while making offline-channel planning decisions.

Optimal participation of ADN in energy and reserve markets considering TSO-DSO interface and DERs uncertainties

The flexibility of distribution networks continues to thrive due to the increasing installment of distributed energy resources (DERs). In addition to meeting the load demand in active distribution networks (ADNs), DERs can also provide energy and reserve for the upper-layer grid (i.e., sub-transmission network) at their connection node by participating in the energy and reserve markets. This paper proposes a novel optimal participation model of ADN in energy and reserve markets that takes into account the uncertainties of DERs and the interface of transmission system operator (TSO) and distribution system operator (DSO). The problem of DSO’s strategic behavior is formulated as a stochastic bi-level optimization model. The upper-level model indicates the market clearing of ADNs managed by the DSO and the lower-level model represents the energy and reserve market clearing of the upper-layer grid managed by the TSO. The nonlinear bi-level model is converted into a mathematical program with equilibrium constraint (MPEC) model, and then the mixed-integer second order cone programming (MISOCP) model based on Karush-Kuhn-Tucker conditions and strong duality theory. The effectiveness of the proposed model on improving the economy of ADN and the utilization rate of DERs is validated by numerical studies.

Spatial–temporal optimal dispatch of mobile energy storage for emergency power supply

Mobile energy storage (MES) is a typical flexible resource, which can be used to provide an emergency power supply for the distribution system. However, it is inevitable to consider the complicated coupling relations of mobile energy storage, transportation network, and power grid, which can cause issues of complex modeling and low efficiency. To address that, this paper proposes a mobile energy storage dispatch model to minimize the load curtailment. The framework of rolling optimization is established to update the optimal strategy with the real-time factors, including grid, transportation, and MES. To increase solving efficiency, the model is extended to a mixed integer second-order cone programming model. Finally, the simulation results demonstrate the effectiveness of the proposed method in various scenarios.

Economic dispatch for a regionally integrated heat and electricity system with variable pipeline flow direction

Traditional models of heating networks are not able to take into account the changes in the value and the direction of pipeline mass flow in a multi-heat-source district heating network or the time delay characteristics of the network with constant temperature and variable flow operation. A day-ahead dispatch model for regional integrated heating and electricity system based on improved second-order cone programming method is therefore proposed. We first construct a heating network transmission model with variable value and direction of pipeline mass flow, and then use a convex relaxation to transform this nonlinear model into a mixed-integer second-order cone programming model. A hydraulic time-delay correction model is then constructed to reflect the dynamic characteristics of the heating network under variable mass flow conditions. Finally, we establish a model of an electric boiler with thermal storage to improve the flexibility of the distribution network. Case studies are used to show that the optimised results are more economical in terms of both the variable value and direction of mass flow. The proposed programming method retains the variability in the flow direction and achieves precise relaxation without the need to solve for the initial value. We also verify that the hydraulic time delay characteristics have a strong influence on the optimised results of the regionally integrated heat–electricity system under conditions of constant temperature and variable flow. In addition, an electric boiler with thermal storage can reduce renewable energy curtailment and effectively improve operational flexibility.

Coordinated multiobjective optimization of the integrated energy distribution system considering network reconfiguration and the impact of price fluctuation in the gas market

An integrated energy distribution system (IEDS) includes an energy hub, distribution network (DN), and heat distribution network (HN), which play an essential role in improving energy utilization and achieving multi-energy complementarity. However, the price uncertainty of the domestic natural gas market will bring challenges to the operation economy and safety of the IEDS in the future. A dispatch model of the integrated energy distribution system based on mixed-integer second-order cone programming (MISOCP) is proposed to increase the robustness of the system. First, considering the flexible topology of the distribution network, the distribution network is reconfigured to minimize the operation cost and improve the operating conditions of the system. Second, taking the fluctuation interval of the natural gas price into consideration, an interval multiobjective optimization problem is proposed to optimize the expected average operation cost and cost deviation. Finally, the optimal solution is selected by the epsilon constraint method and fuzzy multiweight technology. The dispatch results of the method in this paper are studied in four cases, and a sensitivity analysis is conducted to verify the feasibility of the method.

Cost-optimal operation and recovery method for power distribution systems considering multiple flexible resources and logistics restrictions

Extreme weather events do great harm to power distribution systems, causing widespread outages and huge economic losses. It is necessary to investigate the operation and recovery methods for power distribution systems against extreme weather events with small probability and high risk. Accordingly, this paper proposes a cost-optimal operation and recovery method for power distribution systems considering multiple flexible resources and logistics restrictions. The method is developed as a two-stage model to obtain the optimal cost consisting of lost load cost and network loss cost. In the first stage before the disaster, the pre-allocation of repair crews combines multiple flexible resources to cope with the upcoming extreme weather event. In the second stage after the disaster, repair crews coordinate with flexible resources to quickly recover the lost load of the system. The cooperative scheduling of repair crews and flexible resources in the method is restricted by the logistics of transportation networks. Based on the network loss, the two-stage model is formulated as a mixed integer second-order cone programming. The case study conducted on a modified IEEE 33-bus test system demonstrates the effectiveness of the method proposed in this paper.

Autonomous truck scheduling for container transshipment between two seaport terminals considering platooning and speed optimization

This study proposes an autonomous truck (AT) based scheduling problem for container transshipment between two seaport terminals, in which ATs can travel in platoons with short inter-vehicle distance. The following vehicles in a platoon can save fuel consumption because of the reduced air resistance. Based on a fuel consumption function with respect to travel speed and platoon formation, we formulate a mixed-integer second-order-cone programming (MISOCP) model to minimize the total operation cost. To address the extra computational challenges caused by the requirements of platoon coordination, we develop a column-generation-based heuristic method in which two types of columns representing AT schedules and platoon schemes are generated and added into the master problem. According to the practice of container transshipment between two port terminals in Singapore, we conduct extensive numerical experiments and evaluate the performance of the proposed method up to the instance of 50 requests in a two-hour time limit. The results show that the proposed CG heuristic can outperform the state-of-the-art solver by around 10% for the instances with 40–50 requests.

Service-oriented distributionally robust lane reservation

Most existing lane reservation studies usually consider a static transportation network with assuming constant link travel times. However, in reality the link travel times are highly uncertain due to various factors such as weather, accidents, road maintenance, intersections, etc. Moreover, the precise link travel time probability distribution is usually difficult to be obtained. This paper studies a new stochastic bus lane reservation problem with partial link travel time information, i.e., only the mean and covariance matrix are known. The objective is to maximize the bus service level measured by the probability of the event that all lines are jointly scheduled on time. For the problem, we formulate a service-oriented distributionally robust optimization model. Its complexity is shown to be NP-hard. To solve the problem, a sample average approximation (SAA)-based method is first adapted. Since the SAA-based approach is computational expensive, a new approximated mixed integer second-order cone programming (MI-SOCP)-based approach is developed. Computational results on a real-life case show that the proposed MI-SOCP-based approach can efficiently obtain satisfactory solutions of high quality. Besides, our results indicate that the proposed model and algorithm can provide better solutions with higher service level, as compared with general stochastic models with known distributions and without considering service levels.

Chance Constrained Scheduling and Pricing for Multi-Service Battery Energy Storage

This paper studies optimal day-ahead scheduling of grid-connected batteries that simultaneously provide three services: 1) load shifting, 2) real-time balancing, and 3) primary frequency control (PFC). The uncertainties of load and frequency are incorporated in the cost-minimizing scheduling problem via chance constraints. The resulting chance-constrained problem is then reformulated into a mixed-integer second-order cone program (MISOCP) that can be solved by commercial solvers. However, it is computationally formidable to obtain the globally optimal solution to the MISOCP due to the big problem size. To obtain a suboptimal solution quickly, a heuristic based on penalty alternating direction method (PADM) is developed to solve the MISOCP. Fixing the integer solution returned by the heuristic, we adopt the duality of the second-order cone program (SOCP) to price the three services in the local market. Theoretical analysis of the market equilibrium, individual rationality, and balanced budget is given. Real-world data of load, frequency, and price in the French grid is used in simulation. The results show that the proposed heuristic is computationally efficient, and the pricing results can guarantee a positive utility for each of the batteries, incentivizing them to provide services.

Resilience-Motivated Distribution System Restoration Considering Electricity-Water-Gas Interdependency

A major outage in the electricity distribution system may affect the operation of water and natural gas supply systems, leading to an interruption of multiple services to critical customers. Therefore, enhancing resilience of critical infrastructures requires joint efforts of multiple sectors. In this paper, a distribution system service restoration method considering the electricity-water-gas interdependency is proposed. The objective is to provide electricity, water, and natural gas supplies to critical customers in the desired ratio according to their needs after an extreme event. The operational constraints of electricity, water, and natural gas networks are considered. The characteristics of electricity-driven coupling components, including water pumps and gas compressors, are also modeled. Relaxation techniques are applied to nonconvex constraints posed by physical laws of those networks. Consequently, the restoration problem is formulated as a mixed-integer second-order cone program, which can readily be solved by the off-the-shelf solvers. The proposed method is validated by numerical simulations on electricity-water-gas integrated systems, developed based on benchmark models of the subsystems. The results indicate that considering the interdependency refines the allocation of limited generation resources and demonstrate the exactness of the proposed convex relaxation.