Second-order Cone Research Articles

Nash Bargaining-Based Coordinated Frequency-Constrained Dispatch for Distribution Networks and Microgrids

As the penetration of distributed renewable energy continues to increase in distribution networks, the traditional scheduling model that the inertia and primary frequency support of distribution networks are completely dependent on the transmission grid will place enormous regulatory pressure on the transmission grid and hinder the active regulation capabilities of distribution networks. To address this issue, this paper proposes a coordinated optimization method for distribution networks and microgrid clusters considering frequency constraints. First, the confidence interval of disturbances was determined based on historical forecast deviation data. On this basis, a second-order cone programming model for distribution networks with embedded frequency security constraints was established. Then, microgrids performed economic dispatch considering the reserves requirement to provide inertia and primary frequency support, and a stochastic optimization model with conditional value-at-risk was built to address uncertainties. Finally, a cooperative game between the distribution network and microgrid clusters was established, determining the reserve capacity provided by each microgrid and the corresponding prices through Nash bargaining. The model was further transformed into two sub-problems, which were solved in a distributed manner using the ADMM algorithm. The effectiveness of the proposed method in enhancing the operational security and economic efficiency of the distribution networks is validated through simulation analysis.

Modeling EV dynamic wireless charging loads and constructing risk constrained operating strategy for associated distribution systems

Dynamic wireless charging (DWC) is an emerging technology that enables the charging of electric vehicles (EVs) while they are in motion. However, previous load modeling methods have not thoroughly explored the detailed analysis of DWC load characteristics. Existing research only considers the single-node supply mode for dynamic wireless charging roads (DWCRs), and the assessment of operational risks arising from the uncertain DWC loads has not been addressed. This paper begins by conducting an equivalent circuit analysis of a typical EV DWC system with multiple segmented coils. We present a more accurate trapezoidal power model for a single EV. Subsequently, we model the aggregated EV DWC load, accounting for traffic flow and headway using Poisson and negative exponential distribution functions, respectively. In the operation process, we consider a multi-node supply mode for DWCRs. To address the inaccuracy of long-term predictions, we propose a rolling optimization model to coordinate DWC and renewables with heterogeneous uncertainties by introducing a risk metric to manage potential uncertain risks. The proposed optimization model is transformed into a mixed-integer second-order cone programming (MISOCP) problem after convex relaxation. Finally, we conduct case studies to validate the proposed methods.

Enhancing offshore parcel delivery efficiency through vessel-unmanned aerial vehicle collaborative routing

Offshore oil and gas production is vital for global energy supply, but it faces logistical challenges due to the high costs and inefficiencies of traditional supply methods. This paper introduces the vessel-unmanned aerial vehicle (UAV) routing problem with multiple visits in a single flight (VURP-M), a novel logistical model that addresses aimed at enhancing the replenishment of offshore platforms with small, essential items. The VURP-M allows the UAV to perform multiple visits during a single flight, optimising the delivery process. To tackle the VURP-M, we propose two mixed-integer second-order cone programs that capture the problem's complexities. Given its NP-hard nature, we employ an adaptive large neighbourhood search (ALNS) method, featuring a segmented initialisation process and problem-specific operators guided by a rule-based mechanism to improve solution efficiency. The ALNS formulates an initial solution by solving a travelling salesman problem to create a giant tour, which is then used to group sequential targets into multiple UAV flights. The subsequent optimal resolution of a SOCP determines the take-off and landing points for each flight. Subsequently, the ALNS refines the initial solution through destroy and repair operators, enhancing the search for superior sequences and allocation schemes. The effectiveness of our approach is demonstrated through a real-world case study and numerical experiments on random instances featuring up to 100 platforms. The results offer implications of the collaborative vessel-UAV model for the offshore logistics industry.

Robustifying Conditional Portfolio Decisions via Optimal Transport

We propose a data-driven portfolio selection model that integrates side information, conditional estimation, and robustness using the framework of distributionally robust optimization. Conditioning on the observed side information, the portfolio manager solves an allocation problem that minimizes the worst-case conditional risk-return tradeoff, subject to all possible perturbations of the covariate-return probability distribution in an optimal transport ambiguity set. Despite the nonlinearity of the objective function in the probability measure, we show that the distributionally robust portfolio allocation with a side information problem can be reformulated as a finite-dimensional optimization problem. If portfolio decisions are made based on either the mean-variance or the mean-conditional value-at-risk criterion, the reformulation can be further simplified to second-order or semidefinite cone programs. Empirical studies in the U.S. equity market demonstrate the advantage of our integrative framework against other benchmarks. Funding: The material in this paper is based on work supported by the Air Force Office of Scientific Research [Award FA9550-20-1-0397]. Additional support is gratefully acknowledged from the National Science Foundation [Grants 1915967, 1820942, and 1838676], the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2016-05208], and the China Merchant Bank. V. A. Nguyen gratefully acknowledges the generous support from the Chinese University of Hong Kong [Improvement on Competitiveness in Hiring New Faculties Funding Scheme] and the Chinese University of Hong Kong [Direct Grant 4055191]. S. Wang is partially supported by the National Natural Science Foundation of China [Grant 72371022]. Finally, this research was enabled in part by support provided by Compute Canada. Supplemental Material: The computer code and data that support the findings of this study and the online appendix are available within this article’s supplemental material at https://doi.org/10.1287/opre.2021.0243 .

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.

Electricity arbitrage for mobile energy storage in marginal pricing mechanism via bi-level programming

The rapid growth of renewable energy is integrated into the distribution system. The creation of both new market mechanisms and investment models has critical effects on the economics and security of the distribution market. Mobile energy storage has been used to increase the resilience of distribution grids due to their advantages in mobility and flexibility, which offer electricity arbitrage options for merchant investments. This paper presents a bi-level optimization framework based on location marginal pricing settlement of mobile energy storage financial rights revenue in active distribution systems. In the developed framework, the upper-level problem determines the optimal capacity, routing, and dispatching of mobile energy storage to maximize revenue in the liberalized electricity market. The lower-level problem performs the joint optimization of energy and reserve market clearing as well as renewable energy capacity optimization based on the alternating current optimal power flow model. The non-convexity of the line flow and location marginal pricing is linearized via second-order cone relaxation and Karush-Kuhn-Tucker. The bi-level programming is reformatted as mathematical programming with equilibrium constraints. Further, the revenue risk due to renewable energy uncertainty is measured via conditional value-at-risk. Finally, the effectiveness of the optimization framework and solution method is verified using the modified IEEE-33 test system. The results show that mobile energy storage promotes renewable energy and reduces distribution network costs by 2.3%.

Flexibility-Constrained Energy Storage System Placement for Flexible Interconnected Distribution Networks

Configuring energy storage systems (ESSs) in distribution networks is an effective way to alleviate issues induced by intermittent distributed generation such as transformer overloading and line congestion. However, flexibility has not been fully taken into account when placing ESSs. This paper proposes a novel ESS placement method for flexible interconnected distribution networks considering flexibility constraints. An ESS siting and sizing model is formulated aiming to minimize the life-cycle cost of ESSs along with the annual network loss cost, electricity purchasing cost from the upper-level power grid, photovoltaic (PV) curtailment cost, and ESS scheduling cost while fulfilling various security constraints. Flexible ramp-up/-down constraints of the system are added to improve the ability to adapt to random changes in both power supply and demand sides, while a fluctuation rate of net load constraints is also added for each bus to reduce the net load fluctuation. The nonconvex model is then converted into a second-order cone programming formulation, which can be solved in an efficient manner. The proposed method is evaluated on a modified 33-bus flexible distribution network. The simulation results show that better flexibility can be achieved with slightly increased ESS investment costs. However, a large ESS capacity is needed to reduce the net load fluctuation to low levels, especially when the PV capacity is large.

Calculation of Distribution Network PV Hosting Capacity Considering Source–Load Uncertainty and Active Management

The access of a high proportion of photovoltaic (PV) will change the energy structure of the distribution network (DN), resulting in a series of safety operation risks. This paper proposes a two-stage PV hosting capacity (PVHC) calculation model to assess the maximum PVHC, considering the uncertainty and active management (AM). Firstly, we employ a robust optimization model to characterize the uncertainty of sources and loads in DN with PV and analyze the worst-case scenarios for PVHC. Subsequently, we construct a PVHC calculation model that takes into account AM, and convert the model into a mixed-integer second-order cone (MISOC) model using linearization techniques. Finally, we apply “heuristic optimization + CPLEX solver” to solve the model and introduce overvoltage and overcurrent indices to analyze the safety of the DN under PV limit access. Case studies are carried out on the IEEE 33-bus system and a practical case. Results show that (1) only the uncertainty that reduces the load or increases the output efficiency will affect PVHC; (2) for DN limited by overvoltage, AM can better improve PVHC; however, for DN limited by maximum transmission power, the effect of AM is low; (3) for most DN, SVC can improve PVHC, but the effect is modest. And network reconfiguration can significantly increase PVHC on the system with poor branch network, even reaching 150% of the original PVHC.

A shakedown oriented topology optimization algorithm utilizing second-order cone programming (SOCP) and its application in spacecraft structure design

A shakedown oriented topology optimization algorithm utilizing second-order cone programming (SOCP) and its application in spacecraft structure design

Optimal operation of three-phase unbalanced active distribution system based on semidefinite relaxation and convex-concave procedure

The three-phase distribution optimal power flow (D-OPF) problem has attracted much attention in recent years due to practical requirements for managing distributed energy resources and multi-energy customers in unbalanced active distribution systems. To deal with the strong nonconvexity of the three-phase OPF problem, this paper develops the semidefinite programming (SDP) formulation of three-phase OPF problems considering the mutual coupling in multi-phase networks. Instead of directly dropping the rank-one constraints, a sequential convex optimization algorithm based on the convex-concave procedure (CCP) is proposed for recovering a feasible and optimal solution of three-phase power flow. The proposed iterative algorithm not only has high computational efficiency but also guarantees the optimality when the semidefinite relaxation or second-order cone relaxation method is inexact. Case studies on various IEEE testing distribution systems verify that the proposed algorithm recovers the feasible and optimal power flow solution. It also exhibits a relatively low number of iterations and short solving time in large-scale IEEE testing systems.

EV charging load forecasting and optimal scheduling based on travel characteristics

Accurate prediction of EV charging load distribution is not only the basis of evaluating the capacity of distribution network to receive EV, but also the necessary premise to promote the research of charging station planning and vehicle-network interaction. Based on this, firstly, according to the travel characteristics of electric private car and taxi users, this paper constructs the models of speed-flow and EV power consumption per mileage, thus, the real-time changes of vehicle speed and power consumption per mileage are simulated. Secondly, consider factors such as charging price, time, and power consumption along the way, as well as multiple sources of information such as traffic networks, vehicles, public quick charging stations, and distribution networks, the OD matrix analysis method is introduced to simulate EV moving characteristics for EV trip assignment matrix, and the EV charging load forecasting system with real-time interaction of multi-source information is established. Finally, the second-order cone optimization method is used to optimize the space-time distribution of regional charging load. The model and method proposed in this paper are more consistent with the actual situation and can improve the accuracy of prediction.

Reformulation and Enhancement of Distributed Robust Optimization Framework Incorporating Decision-Adaptive Uncertainty Sets

Distributionally robust optimization (DRO) is an advanced framework within the realm of optimization theory that addresses scenarios where the underlying probability distribution governing the data is uncertain or ambiguous. In this paper, we introduce a novel class of DRO challenges where the probability distribution of random variables is contingent upon the decision variables, and the ambiguity set is defined through parameterization involving the mean and a covariance matrix, which also depend on the decision variables. This dependency makes DRO difficult to solve directly; therefore, first, we demonstrate that under the condition of a full-space support set, the original problem can be reduced to a second-order cone programming (SOCP) problem. Subsequently, we solve this second-order cone programming problem using a projection differential equation approach. Compared with the traditional methods, the differential equation method offers advantages in providing continuous and smooth solutions, offering inherent stability analysis, and possessing a rich mathematical toolbox, which make the differential equation a powerful and versatile tool for addressing complex optimization challenges.

The Low-Carbon Path of Active Distribution Networks: A Two-Stage Model from Day-Ahead Reconfiguration to Real-Time Optimization

The integration of renewable energy sources and distributed energy storage systems increasingly complicates the operation of distribution networks, while stringent carbon reduction targets demand low-carbon operational strategies. To address these complexities, this paper introduces a two-stage model for reconfiguring distribution networks and ensuring low-carbon dispatch. Initially, second-order cone programming is employed to minimize losses in the network. Subsequently, the outputs of renewable energy and energy storage systems are optimized using the mantis search algorithm (MSA) to achieve low-carbon dispatch, with the network’s carbon potential as the evaluation metric. The proposed model demonstrates a significant reduction in average active power loss by 34.85%, a decrease in daily carbon emissions by 509.97 kg, and a reduction in carbon emission costs by 17.24%, thereby markedly enhancing the economic and social benefits of grid operations.

Robust distribution networks reconfiguration considering the improvement of network resilience considering renewable energy resources

The integration of renewable energy sources into smart distribution grids poses substantial challenges in maintaining grid stability, efficiency, and reliability due to their inherent variability and intermittency. This study addresses these challenges by proposing a novel two-level optimization model aimed at enhancing operational efficiency and robustness in smart distribution grids. The model synergistically integrates renewable energy sources, energy storage systems, electric vehicles, and demand-side management through a dynamic reconfiguration approach. It employs a robust optimization framework combined with a two-stage second-order cone optimization model to manage real-time operations and strategic grid reconfiguration. Key findings from simulations on the IEEE 33 and 69-bus networks underscore the model’s effectiveness. In the 33-bus system, implementing the demand response program led to a significant reduction in power losses, from 0.64 MW to 0.52 MW, and improved voltage stability, with the minimum voltage increasing from 0.970 to 0.980 p.u. Similarly, in the 69-bus system, power losses decreased from 0.85 MW to 0.79 MW, and voltage stability improved, with the minimum voltage rising from 0.962 to 0.972 p.u. The model also demonstrated reduced energy procurement needs, showcasing its impact on enhancing grid efficiency and reliability. These results highlight the model’s potential for advancing smart grid management strategies, offering significant improvements in operational performance and stability under varying demand conditions.

A multi-objective robust dispatch strategy for renewable energy microgrids considering multiple uncertainties

The demand for low-carbon transformations and the uncertainty of renewable energy sources and loads present significant challenges for the optimal dispatch of microgrid. This study proposed a multi-objective robust dispatch strategy to reduce the risks associated with the uncertainty of renewable energy source output and loads while promoting low-carbon and economical microgrid operation. The economic emission dispatch problem for a microgrid was formulated as a multi-objective robust dual-layer optimization model. Consequently, a high-dimensional adjustable linear polyhedral uncertainty set was proposed to describe the uncertainty of renewable energy sources and loads. This study transformed the original model into an easy-to-solve single-layer second-order cone programming optimal power flow optimization model by employing second-order cone relaxation and duality transformation. Thereafter, a synthetic membership function was proposed to determine the optimal compromise solution. To determine the charging and discharging statuses of the battery storage system and the electricity traded between the microgrid and the external power grid, a battery storage system control strategy based on time-of-use electricity prices and real-time power flow calculations was proposed. Simulations conducted on a modified IEEE-30 bus system demonstrated that the proposed strategy effectively reduced the economic costs and carbon emissions of the microgrid by 8.23 % and 2.43 %, respectively.

Optimal operation of multi-energy carriers considering energy hubs in unbalanced distribution networks under uncertainty

This article presents a two-stage stochastic programming model to address the dispatching scheduling problem in an energy hub, considering an unbalanced active low-voltage (LV) network. A three-phase version of the second-order cone relaxation of DistFlow AC optimal power flow (AC-OPF) is employed to incorporate unbalanced network constraints, while the objective minimizes the Local Energy Community (LEC) operational cost. The model results have been validated using OpenDSS, encompassing energy losses, voltage levels, and active/reactive power. Likewise, a comparative analysis between the three-phase model and the traditional single-phase model, using a modified version of the IEEE European LV Test Feeder as a case study, reveals interesting differences, such that the single-phase model underestimates voltage limits during photovoltaic (PV) system operation and overestimates energy purchased from the main grid, compared with the three-phase model. Similarly, the comparison results reveal that discrepancies between the single and three-phase models intensify as the power injected from PV systems rises. This notably impacts the total energy purchased from the grid, battery operation, and the satisfaction of thermal consumption through electricity. Finally, while the three-phase model offers valuable insights into security levels for voltage and grid energy purchase, its longer computational time makes it more suitable for strategic use rather than daily operational frameworks.

Flexibility-Oriented AC/DC Hybrid Grid Optimization Using Distributionally Robust Chance-Constrained Method

With the increasing integration of stochastic sources and loads, ensuring the flexibility of AC/DC hybrid distribution networks has become a pressing challenge. This paper aims to enhance the operational flexibility of AC/DC hybrid distribution networks by proposing a flexibility-oriented optimization framework that addresses the growing uncertainties. Notably, a comprehensive evaluation method for operational flexibility assessment is first established. Based on this, this paper further proposes a flexibility-oriented operation optimization model using the distributionally robust chance-constrained (DRCC) method. A customized solution method utilizing second-order cone relaxation and sample average approximation (SAA) is also introduced. The results of case studies indicate that the flexibility of AC/DC hybrid distribution networks is enhanced through sharing energy storage among multiple feeders, adaptive reactive power regulation using soft open points (SOPs) and static var compensators (SVCs), and power transfer between feeders via SOPs.

Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a new type of kernel functions

Kernel functions are essential for designing and analyzing interior-point methods (IPMs). They are used to determine search directions and reduce the computational complexity of the interior point method. Currently, IPM based on kernel functions is one of the most effective methods for solving LO [1,20], second-order cone optimization (SOCO) [2], and symmetric optimization (SO) and is a very active research area in mathematicalprogramming. This paper presents a large-update primal-dual IPM for SDO based on a new bi-parameterized hyperbolic kernel function. Then we proved that the proposed large-update IPM has the same complexity bound as the best-known IPMs for solving these problems. Taking advantage of the favorable characteristics of the kernel function, we can deduce that the iteration bound for the large update method is $\mathcal{O}\left( \sqrt{n}\log n\log\dfrac{n}{\varepsilon }\right) $ when a takes a special value utilizing the favorable properties of the kernel function. These theoretical results play an essential role in the design and analysis of IPMs for CQSCO [7] and the Cartesian$\ P_{\ast }\left( \kappa \right) $-SCLCP [8]. The proximity function has never been used. In order to validate the efficacy of our algorithm and verify the effectiveness of our algorithm, examples aregiven to illustrate the applicability of our main results, and we compare our numerical results with some alternatives presented in the literature.

Solving Two-stage Quadratic Multiobjective Problems via Optimality and Relaxations

This paper focuses on the study of robust two-stage quadratic multiobjective optimization problems. We formulate new necessary and sufficient optimality conditions for a robust two-stage multiobjective optimization problem. The obtained optimality conditions are presented by means of linear matrix inequalities and thus they can be numerically validated by using a semidefinite programming problem. The proposed optimality conditions can be elaborated further as second-order conic expressions for robust two-stage quadratic multiobjective optimization problems with separable functions and ellipsoidal uncertainty sets. We also propose relaxation schemes for finding a (weak) efficient solution of the robust two-stage multiobjective problem by employing associated semidefinite programming or second-order cone programming relaxations. Moreover, numerical examples are given to demonstrate the solution variety of our flexible models and the numerical verifiability of the proposed schemes.

Collaborative Optimization for Active Distribution Network with Soft Open Point and Energy Storage

Background: The increasing use of high-penetration distributed clean energy has led to voltage fluctuations in the distribution network that surpass safety limits and pose challenges to economic and low-carbon operation. Traditional voltage regulation devices are unable to meet the real-time high-precision voltage control needs when encountering frequent fluctuations due to physical limitations. The Soft Open Point (SOP) can provide continuous reactive power to achieve rapid voltage regulation. Objective: We propose a collaborative optimization approach that fully considers the regulatory capabilities of SOP and energy storage to eliminate voltage violations and reduce network losses. Methods: This study introduces an active distribution network reactive power voltage optimization approach that incorporates intelligent SOP and energy storage, combining conventional devices (such as on-load tap changers and capacitor banks) with contemporary devices (such as SOP, distributed generators, and energy storage) to establish synergy. A coordinated optimization model was developed, considering various operational constraints to minimize network losses and regulate the voltage of distribution network nodes. By using linearization and quadratic relaxation, the original non-convex mixed-integer non-linear optimization model was transformed into a Mixed-Integer Second-Order Cone Programming (MISOCP) model to meet the requirements of rapid voltage regulation. Results: A case study was conducted on the IEEE 33-node test system, showing that the proposed approach effectively reduces network losses and eliminates voltage violations. Conclusion: The feasibility and effectiveness of the proposed methodology have been validated, confirming its ability to ensure the safe and economic operation of active distribution networks.