Second-order Cone Research Articles

Collaborative Pricing Strategy of Distribution Network and Electric Vehicle Aggregator Based on Integral System

Scaled EVs as a new type of load have great potential to participate in the standby market. However, the uncertainty of EV users' participation in aggregator regulation affects the optimal strategy for aggregators to participate in the market. Aiming at the uncertainty of users' willingness and the conflict of interest between aggregators and the distribution network, this paper proposes an optimal scheduling strategy of master-slave game for EV aggregators to improve users' willingness with a point system. First, the second-order cone optimal current model of the distribution network is established, and then the distribution network node marginal tariff is obtained. Second, the aggregator revenue model with energy storage is established, and the price-guided demand response model is proposed, while the aggregator point system is established to guide the charging behavior of EV users, so that the cooperative operation optimization problem between the distribution grid and the aggregator can be solved, and the economic interests among EV users, EV aggregators and distribution grid operators can be effectively balanced.

Distributionally robust chance-constrained kernel-based support vector machine

Support vector machine (SVM) is a powerful model for supervised learning. This article addresses the nonlinear binary classification problem using kernel-based SVM with uncertainty involved in the input data specified by the first- and second-order moments. To achieve a robust classifier with small probabilities of misclassification, we investigate a distributionally robust chance-constrained kernel-based SVM model. Since the moment information in the original problem becomes unclear/unavailable in the feature space via kernel transformation, we develop a data-driven approach utilizing empirical moments to provide a second-order cone programming (SOCP) reformulation for efficient computation. To speed up the required computations for solving large-size problems in higher dimensional space and/or with more sampling points involved in estimating empirical moments, we further design an alternating direction multipliers-based algorithm for fast computations. Extensive computational results support the effectiveness and efficiency of the proposed model and solution method. Results on public benchmark datasets without any moment information indicate that the proposed approach still works and, surprisingly, outperforms some commonly used state-of-the-art kernel-based SVM models.

A distributionally robust optimization approach for the potassium fertilizer product transportation considering transshipment through crossdocks

Potassium fertilizer is an essential input for agricultural productivity, and plays a critical role in various plant processes, influencing water uptake, enzyme activation, and photosynthesis. The efficient delivery of potassium fertilizer products to the end-users, typically farmers and agricultural enterprises, is of utmost importance. Due to the long traveling distance between the supplier and customers, decision makers of supplier normally set up multiple crossdocks to aid in transition and storage of potassium fertilizer products. In this situation, making an optimal transportation plan as well as the inventory level of crossdocks will directly enhance the efficiency and effectiveness of long-distance transportation. In this study, we model this problem as a dynamic transportation problem with transshipment considering the uncertain time-varying customers’ demand. To characterize the demand information, we first construct an ambiguity set of customers’ demand based in limited historical data and then develop a distributionally robust optimization-based (DRO) framework to optimize the transportation plan and related inventory level of crossdocks simultaneously. We also propose a general approach to overcome the computational challenges of DRO by transforming the original DRO into a second-order cone programming based on duality theory. Additionally, we introduce linear decision rules to adjust the optimization strategy based on the new observed demand, thus lead the model to handle the dynamic information flow in real time. In case study, all involved data are collected or derived from a real potassium fertilizer company located at Western China. The results show our model reduces the cost by 18.07% compared to stochastic model (sample average approximation), indicating a significant effectiveness of our model on the improvement of delivery efficiency and cost saving in real dynamic logistic systems. Also, we conclude the managerial insight that decision-makers should develop a comprehensive strategy, including improving communication to ensure order status updates, planning rationally to evenly distribute orders, and proactively allocating resources to meet operational demands.

Reactive power optimization for power distribution networks using mixed-integer Second-order cone programming

Reactive power optimization for power distribution networks using mixed-integer Second-order cone programming

Distribution Network Restoration Strategy Considering Mobile Energy Storage Systems

Abstract The increasing occurrence frequency of disruptive extreme weather events and the growing share of distributed resources further enhance the importance of restoration of distributed resource-assisted distribution networks. However, most distributed sources are not spatially flexible in scheduling, severely impacting the restoration performance of distribution networks. The emergence of mobile energy storage systems (MESS) provides temporally and spatially flexible resources. Based on the mobility of mobile energy storage, this paper proposes a distribution network restoration strategy considering MESS and prioritized restoration of critical loads. The restoration strategy is established aiming at maximizing weighted loads while satisfying multiple constraints. Specific operating characteristics of MESS are modeled and incorporated into the restoration model. The established restoration model is then reformulated into a mixed-integer second-order cone program. Case studies on a modified IEEE 33-node system verify the proposed restoration strategy.

Convex Approach to Real-Time Multiphase Trajectory Optimization for Urban Air Mobility

Electric vertical takeoff and landing technology has emerged as a promising solution to alleviate ground transportation congestion. However, the limited capacity of onboard batteries enforces hard time constraints for vehicle operations in a complex urban environment. Therefore, a constrained, real-time trajectory optimization method is necessary to enable safe and energy-efficient vehicle flight. Unfortunately, most existing trajectory optimization methods suffer from low computational efficiency, unpredictable convergence processes, and strong dependence on a good initial trajectory. In this paper, we employ a successive convex programming approach to solve the multiphase minimum-energy-cost electric vertical takeoff and landing vehicle trajectory optimization problem for urban air mobility applications. The main contribution is the development and validation of two novel convexification methods that find approximate optimal solutions to the multiphase nonlinear trajectory optimization problem in real time by solving a sequence of convex subproblems. Specifically, the first method transforms the original nonlinear problem into a sequence of second-order cone programming problems through a convenient change of variables and lossless convexification, while the second approach achieves a similar goal without the need for control convexification. The resulting convex subproblems can be solved reliably in real time by advanced interior point methods. The proposed methods are demonstrated through numerical simulations of two different urban air mobility scenarios and compared with the solution obtained from GPOPS-II, a state-of-the-art general-purpose optimal control solver. Our proposed sequential convex programming methods can obtain near-optimal solutions with faster computational speed than GPOPS-II.

A dynamic reconfiguration model and method for load balancing in the snow-shaped distribution network

The snow-shaped distribution network (SDN) is a cable distribution network composed of eight (or six) 10 kV feeders from four (or three) substations in a regular connection. Compared with the traditional 10 kV distribution network, SDN can support a wide range of load transfer among six or eight feeders. Aiming at the problem of load spatial-temporal unbalanced condition caused by the integration of distributed generators (DGs) and different load types in different feeders, this paper proposes a dynamic reconfiguration strategy for load balancing in SDN considering DGs and energy storage system (ESS). Firstly, the basic structure of SDN is analyzed and the power flow model for its dynamic reconfiguration is developed. Secondly, the dynamic reconfiguration optimization model for load balancing in SDN considering DGs and ESS is proposed to utilize the load transfer capability to mitigate the load unbalanced condition and reduce active power loss. Thirdly, the original non-convex model is converted into a mixed-integer second-order cone programming (MISOCP) model by applying the second-order cone relaxation and the big-M method, which is solved by CPLEX solver. Finally, the effectiveness of the proposed model and method are verified by an actual case in Tianjin and IEEE 33-node system. The analysis results show that the proposed method can significantly alleviate the load unbalanced spatial-temporal distribution and improve the economic efficiency by regulating the operation of SDN including ESS optimization and dynamic reconfiguration.

Chance constrained optimal power-water flow: An iterative algorithm assisted by deep neural networks

The optimal power-water flow (OPWF) is a fundamental tool for operating integrated power and water networks. To address uncertainties in renewable energy and load demands, researchers have used chance-constrained models to investigate OPWF under uncertainty. However, existing work has limitations, such as oversimplified network representations, intractable problem formulations, and high computational burdens. This paper presents a chance-constrained OPWF formulation that incorporates detailed network constraints and stochastic uncertainties from wind speeds and electricity/water demands. By leveraging the concept of quantiles of stochastic variables and convexification techniques, the model is recast into a deterministic mixed-integer second-order cone programming (MISOCP) problem with uncertain margins. To efficiently solve this problem, we propose a novel iterative algorithm assisted by deep neural networks (DNNs). The algorithm integrates two DNNs: one designed to solve power and water flow equations, accelerating the probabilistic evaluation of uncertain margins; and the other to predict pipeline flows and refine variable ranges, facilitating a warm-start approach to the MISOCP model. Case studies conducted on the IEEE 123-node feeder coupled with a 67-node water distribution network validate the two DNNs and the overall iterative algorithm. The findings demonstrate that the proposed DNN-assisted approach effectively balances system security and economy. Compared to existing algorithms, the relative error of the optimization solution is less than 0.03%, with a speedup ratio exceeding 200.

Improving interdependent urban power and gas distribution systems resilience through optimal scheduling of mobile emergency supply and repair resources

Urban electric power and natural gas systems, serving as the most critical facilities that provide fundamental support to people's daily life, have been increasingly interdependent in their operation over the past decades. This growing interdependency presents great challenges and demands for enhancing the resilience of the interdependent systems, particularly driven by the recent cold weather events. In this paper, we propose a comprehensive strategy for scheduling various types of emergency resources, including mobile and stationary generators, repair crews, and fuel tankers, with the goal of enhancing the resilience of interdependent power and gas distribution systems in the aftermath of disasters. In particular, the strategy incorporates the routing of mobile resources to orchestrate their dynamic moving behaviors. Fuel supply issue for mobile and stationary distributed generators is also addressed through modeling the fuel exchange processes among generators, fuel tankers, and locations. Furthermore, we present a model of repair process to fully capture the real-world repair activities, allowing varying numbers of repair crews to work in a cooperative way. The proposed strategy is formulated into a tractable mixed-integer second-order cone problem. Finally, case studies are carried out to demonstrate the effectiveness of the strategy in enhancing the resilience of the interdependent power and gas distribution systems.

Fast Low-Sidelobe Pattern Synthesis Using the Symmetry of Array Geometry.

Array pattern synthesis with low sidelobe levels is widely used in practice. An effective way to incorporate sensor patterns in the design procedure is to use numerical optimization methods. However, the dimension of the optimization variables is very high for large-scale arrays, leading to high computational complexity. Fortunately, sensor arrays used in practice usually have symmetric structures that can be utilized to accelerate the optimization algorithms. This paper studies a fast pattern synthesis method by using the symmetry of array geometry. In this method, the problem of amplitude weighting is formulated as a second-order cone programming (SOCP) problem, in which the dynamic range of the weighting coefficients can also be taken into account. Then, by utilizing the symmetric property of array geometry, the dimension of the optimization problem as well as the number of constraints can be reduced significantly. As a consequence, the computational efficiency is greatly improved. Numerical experiments show that, for a uniform rectangular array (URA) with 1024 sensors, the computational efficiency is improved by a factor of 158, while for a uniform hexagonal array (UHA) with 1261 sensors, the improvement factor is 284.

Decentralized energy market integrating carbon allowance trade and uncertainty balance in energy communities

With sustained attention to carbon neutrality, the personal carbon trading (PCT) scheme has been embraced as an auspicious paradigm for scaling down carbon emissions. To facilitate the simultaneous clearance of energy and carbon allowance inside the energy community while hedging against uncertainty, a joint trading framework is proposed in this article. The energy trading is implemented in a peer-to-peer (P2P) manner without the intervention of a central operator, and the uncertainty trading is materialized through procuring reserve of conventional generators and flexibility of users. Under the PCT scheme, carbon allowance is transacted via a sharing mechanism. Possible excessive carbon emissions due to uncertainty balance are tackled by obliging renewable agents to procure sufficient carbon allowances, following the consumption responsibility principle. A two-stage iterative method consisting of tightening McCormick envelope and alternating direction method of multipliers (ADMM) is devised to transform the model into a mixed-integer second-order cone program (MISOCP) and to allow for a fully decentralized market-clearing procedure. Numerical results have validated the effectiveness of the proposed market model.

Active Distribution Network Expansion Planning Based on Wasserstein Distance and Dual Relaxation

In the future, a high proportion of distributed generations (DG) will be integrated into the distribution network. The existing active distribution network (ADN) planning methods have not fully considered multiple uncertainties, differentiated regulation modes or the cost of multiple types of interconnection switches. Meanwhile, it is difficult to solve large-scale problems at small granularity. Therefore, an expansion planning method of ADN considering the selection of multiple types of interconnection switches is proposed. Firstly, a probability distribution ambiguity set of DG output and electrical-load consumption based on the Wasserstein distance is established for dealing with the issue of source-load uncertainty. Secondly, a distributionally robust optimization model for collaborative planning of distribution network lines and multiple types of switches based on the previously mentioned ambiguity set is established. Then, the original model is transformed into a mixed integer second-order cone programming (SOCP) model by using the convex relaxation method, the Lagrangian duality method and the McCormick relaxation method. Finally, the effectiveness of the proposed method is systematically verified using the example of Portugal 54. The results indicate that the proposed method raises the annual net profit by nearly 5% compared with the traditional planning scheme and improves the reliability and low-carbon nature of the planning scheme.

Optimal Power Flow in electrical grids based on power routers

In recent years, there has been an exploration of innovative electric grid concepts centered around power routers, devices capable of controlling power flows as desired. One of which is the Power Router Grid, a novel high-controllable network design fully based on power routers. Its ability to arbitrarily change power flows inside controllable lines raises the challenge of determining the optimal operation within such systems. This paper aims to push further the analysis of the operation and benefits of such networks by proposing a novel optimization model tailored for power router grids based on the most recent literature for the convexification of optimal power flows. Moreover, it demonstrated that radial-based models can be expanded for grids meshed through power routers. The model presented is also convex in the form of a second-order cone, ensuring a global optimum for diverse grid configurations. It has been implemented in Python using the PYOMO modeling language and applied to three case studies investigating the effects of power router operation modes on the grid’s optimal operation and associated costs. Results show that the definition of the power router grid design and the ports operation mode must be carefully decided when demand uncertainty of over 25% is taken into consideration. Moreover, controlling power flow near lines of higher impedance can lead up to a 21% increase in line losses.

Demand Response Power-Gas Interconnection Energy System and Power Metering Based on SOGA

Under the double background of global energy crisis and environmental pollution, China vigorously develops renewable energy while accelerating the construction of energy Internet. Taking demand response into account, this paper proposes the optimal design of power-gas interconnection energy system and power metering, and establishes the optimization research model of IEGES (integrated electricity-gas energy system) with demand response. Firstly, the structure model of the electric-gas interconnection energy system with CHP (Cogeneration, combined heat and power) as the core is constructed, and the energy conversion relationship and different energy flow directions of the coupling equipment are expounded from three aspects. The natural gas source point, pipeline equation, power side branch equation, voltage and current equation are modeled and sorted out, and the square term in the equation is linearized by second-order cone programming method, and the mixed integer nonlinear programming problem is transformed into mixed integer linear programming problem. A single objective genetic algorithm with “elite strategy” was selected to solve the equipment capacity optimization problem of IEGES system with system economy as the optimization objective. After a long time of parameter combination attempts, the current population size is 30, the number of iterations is 600, the crossover rate is 0.8, and the heritability is 0.3. The above parameters can obtain better convergence results on the basis of considering the operation time. Finally, a stochastic optimization method of energy Internet considering integrated demand response and uncertainty of wind power is proposed, which aims to meet the energy demand of end users while minimizing the operating cost of the system. The comprehensive demand response strategy including internal and external demand response is considered in the model. Internal demand response is realized by adjusting the internal operation mode of EH, while external demand response is implemented by the end user’s active response, and the load is time-shifted or interrupted under the guidance of external signals.

Distributed restoration for integrated electricity-gas-heating energy systems with an iterative loop scheme

Electricity, gas and heating networks are increasingly coupled to form an integrated energy system (IES), which is beneficial in improving the operational performance of the system during power outages. Generally, IES involves multiple energy operators, making the centralized framework impracticable. To this end, a novel distributed restoration method for IES is proposed in this paper. The IES restoration problem is formulated as a mixed-integer second-order cone programming (MISOCP) model to minimize the total load losses, which considers the coupling effects among the electricity, gas and heating networks. Then, considering the independent management and information privacy of different energy networks, the distributed restoration model is formulated by decomposing shared boundary information using the alternating direction method of multipliers (ADMM). For the sake of the non-convexity caused by binary load state variables, an iterative loop scheme is introduced in ADMM procedure. This allows us to solve a relaxed distributed model in the first stage, and perform iterative loops in the second stage, which can improve the convergence of ADMM and make the distributed solution feasible. Finally, numerical studies are conducted on 33-7-6-node IES and 123-24-32-node IES to validate the effectiveness and feasibility of the proposed method.

Generating linear, semidefinite, and second-order cone optimization problems for numerical experiments

The numerical performance of algorithms can be studied using test sets or procedures that generate such problems. This paper proposes various methods for generating linear, semidefinite, and second-order cone optimization problems. Specifically, we are interested in problem instances requiring a known optimal solution, a known optimal partition, a specific interior solution, or all these together. In the proposed problem generators, different characteristics of optimization problems, including dimension, size, condition number, degeneracy, optimal partition, and sparsity, can be chosen to facilitate comprehensive computational experiments. We also develop procedures to generate instances with a maximally complementary optimal solution with a predetermined optimal partition to generate challenging semidefinite and second-order cone optimization problems. Generated instances enable us to evaluate efficient interior-point methods for conic optimization problems.

A mathematical formulation and a tabu search heuristic for the joint vessel-UAV routing problem

Motivated by offshore inspection practices, we investigate a joint vessel-unmanned aerial vehicle (UAV) routing problem where vessels and UAVs work coordinately to perform inspection tasks. The problem is an extension of the mothership-drone routing problem by further considering multiple vessels and realistic constraints such as temporal-spatial coordination, service time, and inspection cycle. The goal is to minimize overall operational costs, including fixed vessel cost, and vessel and UAV routing costs. Decision-making involves task assignment, vessel route determination, and UAV take-off and landing locations. We formulate the problem as a mixed-integer second-order cone program. Addressing its NP-hard nature, we develop an enhanced tabu search algorithm (TS-RC), which incorporates two innovative mechanisms to reduce the computational burden. The first is to determine UAV take-off/landing points based on a new constructive procedure. The second is assessing neighborhood solutions using an approximate procedure. Results on a real-world case demonstrate a 10.29% reduction in operational costs compared to a classic vessel routing model. Moreover, numerical experiments on random instances with up to four vessels and 39 tasks demonstrate the performance of the proposed TS-RC method.

Mixed-integer linear programming for computing optimal experimental designs

The problem of computing an exact experimental design that is optimal for the least-squares estimation of the parameters of a regression model is considered. We show that this problem can be solved via mixed-integer linear programming (MILP) for a wide class of optimality criteria, including the criteria of A-, I-, G- and MV-optimality. This approach improves upon the current state-of-the-art mathematical programming formulation, which uses mixed-integer second-order cone programming. The key idea underlying the MILP formulation is McCormick relaxation, which critically depends on finite interval bounds for the elements of the covariance matrix of the least-squares estimator corresponding to an optimal exact design. We provide both analytic and algorithmic methods for constructing these bounds. We also demonstrate the unique advantages of the MILP approach, such as the possibility of incorporating multiple design constraints into the optimization problem, including constraints on the variances and covariances of the least-squares estimator.

Flexible allocation and optimal configuration of multi-level energy exploitation units for heterogeneous energy systems considering resource distribution

The energy systems are evolving into heterogeneous energy systems (HESs) with complicated integration of sources, networks, loads, and storage towards a decarbonized future of human beings. This paper develops the flexible allocation and optimal configuration of multi-level energy exploitation units (MEEUs) for the HES considering resource distribution including local energy conditions and actual coupled relationship of various energy networks. A bi-level optimal planning model of MEEUs for the HES considering initial investment cost, carbon emission cost, operation cost and line loss is developed to figure out the optimal location, structure, configuration and corresponding operation strategy. To address the non-convex and non-linear model, a novel hybrid solution method considering the mixing characteristics of mixed-integer second-order cone programming (MISOCP) model and quadratic non-convex model is proposed. The solution method adopts MISOCP model for figuring out the optimal location, structure and configuration of MEEUs and quadratic non-convex model for optimal operation strategy of MEEUs, which not only improves the computational efficiency but also ensures the solution accuracy. Finally, the related case studies are conducted to compare the proposed MEEU and distributed energy storage unit (DESU) in terms of energy consumption, renewable energy absorbency, carbon emission and line loss. The results verify the significant improvement of the comprehensive energy efficiency, the investment potentiality and the environmental benefits obtained by the proposed planning model.

Optimal allocation strategy of SOP in flexible interconnected distribution network oriented high proportion renewable energy distribution generation

Focusing on the issues of high distributed generation in the distribution network, this paper presents an SOP capacity allocation method for flexible interconnected distribution networks based on sensitivity analysis. Firstly, the improved sensitivity analysis is used to determine the best SOP installation position; Then, the number of SOPs is set in advance, and the location and capacity of SOPs are optimized using the second-order cone programming algorithm, with the minimum annual cost of distribution network as the optimization goal. Finally, this optimization method is verified on the modified IEEE 33-node distribution system model. The results show that the application of the optimization strategy proposed in this paper can significantly reduce the annual operating cost of distribution network and improve voltage stability, and the proposed model can effectively adjust the location and capacity of SOP access, thus improving the economic benefits and reliability of active distribution network.