Second-order Cone Programming Relaxation Research Articles

A comprehensive optimization mathematical model for wind solar energy storage complementary distribution network based on multi-regulatory devices under the background of renewable energy integration

In the context of global energy transformation and sustainable development, integrating and utilizing renewable energy effectively have become the key to the power system advancement. However, the integration of wind and photovoltaic power generation equipment also leads to power fluctuations in the distribution network. The research focuses on the multifaceted challenges of optimizing the operation of distribution networks. It explores the operation and control methods of active distribution networks based on energy storage and reactive power compensation equipment. The stable operation of the distribution network is analyzed under the conditions of wind and photovoltaic integration, with a particular focus on precise regulation to address the limitations of existing methods. Afterwards, the study proposes an improvement plan that combines on load tap changer transformers and reactive power compensation equipment to solve complex power balance problems through second-order cone programming relaxation method. The results of numerical analysis show that the constructed mathematical model maintains a stable voltage of 1 to 1.1 pu at distribution network nodes within 24 h. Especially during peak hours from 15:00 to 24:00, it remains normal without any abnormal fluctuations when the control equipment is not added. These results confirm that precise regulation of multiple devices ensures voltage stability and avoids low or high voltage issues.

Iterative convex relaxation of unbalanced power distribution system integrated multi-energy systems

This study describes a multi-period, multi-energy operation of a three-phase unbalanced Electric Distribution System (EDS) that coordinates with a District Heating System (DHS) and a Natural Gas (NG) distribution system. The electrical subsystem problem is formulated as a bi-level programming problem, with levels 1 and 2 solving the subsystem’s linearized and relaxed nonlinear versions, respectively. Second-Order Cone Programming (SOCP) and polyhedral relaxations are availed to circumvent the non-convexities involved in the subsystems. The SOCP relaxation for an unbalanced power distribution network is inexact; also, the subsystems applying polyhedral relaxation may not generate a meaningful solution as the relaxations are not tight. Therefore, a solution recovery algorithm is developed for each subsystem to recover a feasible solution from their relaxed counterparts. The successive bound tightening algorithm employing a solution recovery procedure is proposed for each subsystem, improving solution quality and strengthening the relaxations with the desired computational efficiency. The proposed solution strategy to optimize the Multi-Energy System (MES) operation cost is verified on the three-phase IEEE-13 and IEEE-123 bus systems, each coordinating with a 30-node DHS and a 6-bus NG network. The results analyses demonstrate that the proposed solution strategy efficiently achieves an optimal solution, reducing maximum relaxation error below 0.1% for each subsystem. The proposed strategy delivered a 14.58% reduction in real power losses and a 12.73% decrease in phase voltage unbalance rate for the EDS in MES. Furthermore, a 1.96% decrease in operational cost demonstrates the techno-economic benefits of the proposed strategy.

Low carbon and economic optimal operation of electricity-gas integrated energy system considering demand response

Integrated Energy System is an important direction for the transformation of the power industry. IES can give full play to the complementary role of multiple energy sources and serve to achieve the industry's energy saving and emission reduction goals. Firstly, this paper establishes a P2G refined operation model considering two stages of hydrogen production from electrolytic water and hydrogen methanation. Secondly, based on the analysis of the IES load characteristics, the multi-energy loads are divided into longitudinal demand responses where the same energy type can be transferred and reduced in time, and horizontal demand responses where different energy types can be substituted for each other, and the corresponding mathematical models are established. Finally, the reward and penalty ladder carbon trading mechanism are used to quantify the carbon trading cost, and the IES low-carbon economic optimization model integrating the Power-to-gas refinement model and demand response is constructed. The model was subjected to second-order cone programming relaxation and incremental segmented linearization methods before being solved using the CPLEX commercial solver. Simulation analyses were performed in an IES consisting of an IEEE 33-node distribution network and a 24-node gas distribution network. The results show that considering demand response can fully exploit the flexible coordination performance of the load side, resulting in a 12.6% reduction in total system cost and a 38.8% reduction in carbon emissions.

Security region of inverter-interfaced power systems: Existence, expansion, and application

Since the increasing integration of inverter-based resources leads to a transition to inverter-interfaced power systems, the additional physical limits of inverters raise concerns for secure operation, which needs to be not only ensured but also enhanced. This study, therefore, investigates how to geometrically ensure and enlarge the security region of single-inverter-interfaced power systems, followed by establishing the inverter-security-constrained optimal power flow for multi-inverter-interfaced power systems. Therein, the inverter over-current and over-modulation/over-voltage limits are visualized by projecting cones into circles, ellipses, and a quadratic curve in different control variable spaces. First, for single-inverter-interfaced power systems, this study proposes a closed-form sufficient condition that guarantees the security region existence under different control modes. Then, this study quantifies and enlarges the security region size by leveraging the counter-intuitive complementary impacts of inverter parameters and the opposite impacts of system parameters. As an extension, an inverter-security-constrained optimal power flow using second-order cone programming relaxation is formulated for multi-inverter-interfaced power systems, in which this study proposes a closed-form sufficient condition to ensure a non-empty and non-decreasing security region. Numerical results of a single-inverter-interfaced power system and a modified IEEE 14 bus multi-inverter-interfaced power system verify the proposed security region properties.

Fast Inverter Control by Learning the OPF Mapping Using Sensitivity-Informed Gaussian Processes

Fast inverter control is a desideratum towards the smoother integration of renewables. Adjusting inverter injection setpoints for distributed energy resources can be an effective grid control mechanism. However, finding such setpoints optimally requires solving an optimal power flow (OPF), which can be computationally taxing in real time. Previous works have proposed learning the mapping from grid conditions to OPF minimizers using Gaussian processes (GPs). This GP-OPF model predicts inverter setpoints when presented with a new instance of grid conditions. Training enjoys closed-form expressions, and GP-OPF predictions come with confidence intervals. To improve upon data efficiency, we uniquely incorporate the sensitivities (partial derivatives) of the OPF mapping into GP-OPF. This expedites the process of generating a training dataset as fewer OPF instances need to be solved to attain the same accuracy. To further reduce computational efficiency, we approximate the kernel function of GP-OPF leveraging the concept of random features, which is neatly extended to sensitivity data. We perform sensitivity analysis for the second-order cone program (SOCP) relaxation of the OPF, whose sensitivities can be computed by merely solving a system of linear equations. Extensive numerical tests using real-world data on the IEEE 13-and 123-bus feeders corroborate the merits of GP-OPF.

Convex Optimization of Cyberattacks Overflowing Multiple Lines in Cyber-Physical Power Systems

Cyberattacks designed to overflow transmission or distribution lines may lead to cascading failures or blackouts in smart grids. Accurately modeling these attacks is the first step in designing appropriate countermeasures. Therefore, this article aims at power system vulnerability analysis from the operator’s perspective, i.e., identifying the scenarios, where false data injection (FDI) cyberattacks can lead to overflow of multiple lines. A nonlinear optimization model is developed for cyberattacks that can bypass the state-estimation and result in congestion of multiple lines in smart grids. Unlike existing line congestion attacks that focused on dc state-estimation, the proposed model is based on FDIs on targeted set of buses to overflow targeted set of lines by bypassing ac state-estimation. Considering ac state-estimation in the attack model generalizes the proposed model for both transmission and distribution-level power grids. The nonconvex model is then converted to a convex formulation using second-order cone programming relaxation, and global optimality conditions are discussed. Several case studies are considered to validate the effectiveness of the proposed attack model using IEEE 118-bus benchmark.

An Enhanced Second-Order Cone Programming-Based Evaluation Method on Maximum Hosting Capacity of Solar Energy in Distribution Systems with Integrated Energy

In order to adjust to the change of the large-scale deployment of photovoltaic (PV) power generation and fully exploit the potentialities of an integrated energy distribution system (IEDS) in solar energy accommodation, an evaluation method on maximum hosting capacity of solar energy in IEDS based on convex relaxation optimization algorithm is proposed in this paper. Firstly, an evaluation model of maximum hosting capacity of solar energy for IEDS considering the electrical-thermal comprehensive utilization of solar energy is proposed, in which the maximization of PV capacity and solar collector (SC) capacity are fully considered. Secondly, IEDS’s potential in electricity, heat, and gas energy coordinated optimization is fully exploited to enhance the hosting capacity of solar energy in which the electric distribution network, heating network, and natural gas network constraints are fully modeled. Then, an enhanced second-order cone programming (SOCP)-based method is employed to solve the proposed maximum hosting capacity model. Through SOCP relaxation and linearization, the original nonconvex nonlinear programming model is converted into the mixed-integer second-order cone programming model. Meanwhile, to ensure the exactness of SOCP relaxation and improve the computation efficiency, increasingly tight linear cuts of distribution system and natural gas system are added to the SOCP relaxation. Finally, an example is given to verify the effectiveness of the proposed method. The analysis results show that the maximum hosting capacity of solar energy can be improved significantly by realizing the coordination of an integrated multi-energy system and the optimal utilization of electricity, heat, and gas energy. By applying SOCP relaxation, linearization, and adding increasingly tight linear cuts of distribution system and natural gas system to the SOCP relaxation, the proposed model can be solved accurately and efficiently.

A new SOCP relaxation of nonconvex quadratic programming problems with a few negative eigenvalues

We present a new second order cone programming (SOCP) relaxation of nonconvex quadratic programs with a few negative eigenvalues (NQP-r-NE) by employing the difference of convex (DC) decomposition and simultaneous matrix diagonalization together. The auxiliary variables are bounded from above by one more convex quadratic constraint in the proposed SOCP relaxation than that in the classical SOCP relaxation provided by Kim and Kojima (2001). We prove that the proposed SOCP relaxation is strictly tighter than the classical SOCP relaxation under certain circumstances, especially when there exists one constraint matrix being positive definite in the primal problem. Three types of numerical experiments including the large-scale NQP-r-NE problem with the number of negative eigenvalues r≤20, the optimal spectrum sharing problem in MIMO cognitive radio networks and the two-trust-region subproblem are provided to illustrate that the proposed SOCP relaxation could achieve a better lower bound than that of the classical SOCP relaxation. Moreover, when the number of variables is larger than 250, the proposed SOCP relaxation shows its great superiority both in the bound quality and computing efficiency compared to the classical SOCP relaxation.

Service restoration for resilient distribution system with coordination of battery swapping stations and distributed generations

Due to sufficient energy storage resources and central dispatch, the battery swapping station (BSS) offers a promising solution for the service restoration and resilience enhancement of the distribution system (DS). This paper proposes a joint service restoration strategy for the DS by coordinating BSSs and distributed generations (DGs) with distribution network reconfiguration. Firstly, the spatio-temporal distribution of battery swapping demand for electric vehicles (EVs) is analyzed. On this basis, the participation of the BSS in the DS service restoration is investigated, and the model of restoration capacity for the BSS is established, which can describe the power supply ability of the BSS to support the DS restoration. Secondly, a mathematical model for the resilient DS restoration with coordination between BSSs and DGs is established, aiming to minimize the total system cost by optimally allocating BSSs and DGs during the restoration stage. The second-order cone programming (SOCP) relaxation is also adopted to transform the proposed restoration model into a mixed-integer convex formulation, which can be tractably solved. Finally, numerical simulations are conducted based on the modified IEEE 33-bus distribution test system. The results demonstrate the effectiveness of the proposed strategy for the service restoration and resilience enhancement of the DS.

Second-Order Cone Programming Relaxation of Stealthy Cyberattacks Resulting in Overvoltages in Cyber-Physical Power Systems

Deployment of photovoltaic generation in smart grids has introduced overvoltage problems. Cyberattackers can take advantage of this situation and cause intentional overvoltages in distribution systems using stealthy false data injections (FDIs). To protect distribution grids from these FDIs, the first step is to model cyberattacks that can cause overvoltages. An accurate nonlinear optimization model is developed for stealthy FDI cyberattacks that can result in overvoltages on targeted nodes in distribution grids without being detected by bad data detection algorithms. Compared to the existing research, which mainly focused on dc state estimation for transmission systems, the proposed model formulates cyberattacks that cause overvoltages using ac state estimation in distribution grids. The proposed nonlinear model is then approximated to a convex optimization model using second-order cone programming relaxation formulation to guarantee that a global optimum exists for the model. Several case studies are considered to validate the effectiveness of the proposed attack model. It is shown that by limiting the FDIs on targeted buses to 20% of their nominal load, multiple buses can experience severe overvoltages in a distribution grid.

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.

An Approximate Mixed-Integer Convex Model to Reduce Annual Operating Costs in Radial Distribution Networks Using STATCOMs

The problem of optimal siting and sizing of distribution static compensators (STATCOMs) is addressed in this research from the point of view of exact mathematical optimization. The exact mixed-integer nonlinear programming model (MINLP) is decoupled into two convex optimization sub-problems, named the location problem and the sizing problem. The location problem is addressed by relaxing the exact MINLP model, assuming that all the voltages are equal to 1∠0∘, which allows obtaining a mixed-integer quadratic programming model as a function of the active and reactive power flows. The solution of this model provides the best set of nodes to locate all the STATCOMs. When all the nodes are selected, it solves the optimal reactive power problem through a second-order cone programming relaxation of the exact optimal power flow problem; the solution of the SOCP model provides the optimal sizes of the STATCOMs. Finally, it refines the exact objective function value due to the intrinsic non-convexities associated with the costs of the STATCOMs that were relaxed through the application of Taylor’s series expansion in the location and sizing stages. The numerical results in the IEEE 33- and 69-bus systems demonstrate the effectiveness and robustness of the proposed optimization problem when compared with large-scale MINLP solvers in GAMS and the discrete-continuous version of the vortex search algorithm (DCVSA) recently reported in the current literature. With respect to the benchmark cases of the test feeders, the proposed approach reaches the best reductions with 14.17% and 15.79% in the annual operative costs, which improves the solutions of the DCVSA, which are 13.71% and 15.30%, respectively.

Optimal Flow of MVDC Shipboard Microgrids With Hybrid Storage Enhanced With Capacitive and Resistive Droop Controllers

Hybrid storage system composed of battery energy storage systems (BESSs) and supercapacitors is a promising solution to mitigate the high frequency power fluctuations of pulsed power loads (PPLs) in medium voltage DC (MVDC) shipboard power systems. Due to the presence of multiple storage units, a power sharing algorithm needs to be considered within the energy management system (EMS) in the shipboard power system. In this paper, an optimal power flow problem is formulated for MVDC shipboard power systems with hybrid energy storage systems. Battery energy storage systems (BESSs) and conventional generation units are enhanced with virtual resistive droop controllers to share the steady-state power fluctuations. Supercapacitors are enhanced with virtual capacitive droop controllers to share the high frequency fluctuations in the load. The optimal flow accounts for minimizing the operational cost of the generation units in the shipboard power system considering the constraints of network, load balance, voltage profile, and power/energy limits. Second-order cone programming (SOCP) relaxation is used to approximate the nonconvexity of the optimal flow formulation and necessary conditions for global optimality of the solution are discussed. Results confirm the effectiveness of the proposed formulation in load support and managing the energy between storage units.

Topology-Cognizant Optimal Power Flow in Multi-Terminal DC Grids

In this paper, we propose a topology-cognizant optimal power flow (OPF) paradigm with additional safety constraints for multi-terminal direct current (MTDC) grids. The resulting formulation is concerned with the optimization of controller set-points, i.e., voltage and power levels at each bus, and the switching status of transmission lines that collectively referred to as grid topology. A pair of additional safety constraints are integrated into the problem formulation to prevent voltage violations caused by power fluctuations in between two controller set-point updates. Searching for a grid topology that offers more efficient operation leads to a mixed-integer nonlinear program (MINLP) which is computationally challenging due to: i) Non-convex power flow equations, (ii) Non-convex converter loss equations, and iii) Binary variables accounting for the operational status of transmission lines. Non-convexities of power flows and converter loss equations are tackled by means of a mixed-integer second-order cone programming (MISOCP) relaxation, while the optimal switching status of transmission lines are determined via a branch-and-bound search. Numerical results for the modified IEEE 14, 30, and 57-bus systems are used to verify the merits of the proposed method. Furthermore, this method is experimentally validated using the CIGRE B4 DC grid benchmark in a real-time hardware-in-the-loop platform.

An SOCP Estimator for Hybrid RSS and AOA Target Localization in Sensor Networks.

This work addresses the problem of target localization in three-dimensional wireless sensor networks (WSNs). The proposed algorithm is based on a hybrid system that employs angle of arrival (AOA) and received signal strength (RSS) measurements, where the target’s transmit power is considered as an unknown parameter. Although both cases of a known and unknown target’s transmit power have been addressed in the literature, most of the existing approaches for unknown transmit power are either carried out recursively, or require a high computational cost. This results in an increased execution time of these algorithms, which we avoid in this work by proposing a single-iteration solution with moderate computational complexity. By exploiting the measurement models, a non-convex least squares (LS) estimator is derived first. Then, to tackle its nonconvexity, we resort to second-order cone programming (SOCP) relaxation techniques to transform the non-convex estimator into a convex one. Additionally, to make the estimator tighter, we exploit the angle between two vectors by using the definition of their inner product, which arises naturally from the derivation steps that are taken. The proposed method not only matches the performance of a computationally more complex state-of-the-art method, but it outperforms it for small N. This result is of a significant value in practice, since one desires to localize the target using the least number of anchor nodes as possible due to network costs.

Intelligent Reflecting Surface-Aided Joint Processing Coordinated Multipoint Transmission

This article investigates intelligent reflecting surface (IRS)-aided multicell wireless networks, where an IRS is deployed to assist the joint processing coordinated multipoint (JP-CoMP) transmission from multiple base stations (BSs) to multiple cell-edge users. By taking into account the fairness among cell-edge users, we aim at maximizing the minimum achievable rate of cell-edge users by jointly optimizing the transmit beamforming at the BSs and the phase shifts at the IRS. As a compromise approach, we transform the non-convex max-min problem into an equivalent form based on the mean-square error method, which facilities the design of an efficient suboptimal iterative algorithm. In addition, we investigate two scenarios, namely the single-user system and the multiuser system. For the former scenario, the optimal transmit beamforming is obtained based on the dual subgradient method, while the phase shift matrix is optimized based on the Majorization-Minimization method. For the latter scenario, the transmit beamforming matrix and phase shift matrix are obtained by the second-order cone programming and semidefinite relaxation techniques, respectively. Numerical results demonstrate the significant performance improvement achieved by deploying an IRS. Furthermore, the proposed JP-CoMP design significantly outperforms the conventional coordinated scheduling/coordinated beamforming coordinated multipoint (CS/CB-CoMP) design in terms of max-min rate.

New phase‐changing soft open point and impacts on optimising unbalanced power distribution networks

Three-phase unbalanced conditions in distribution networks are conventionally caused by load imbalance, asymmetrical fault conditions of transformers and impedances of three phases. The uneven integration of single-phase distributed generation (DG) worsens the imbalance situation. These unbalanced conditions result in financial losses, inefficient utilisation of assets and security risks to the network infrastructure. In this study, a phase-changing soft open point (PC-SOP) is proposed as a new way of connecting soft open points (SOPs) to balance the power flows among three phases by controlling active power and reactive power. Then an operational strategy based on PC-SOPs is presented for three-phase four-wire unbalanced systems. By optimising the regulation of SOPs, optimal energy storage systems dispatch and DG curtailment, the proposed strategy can reduce power losses and three-phase imbalance. Second-order cone programming (SOCP) relaxation is utilised to convert the original non-convex and non-linear model into an SOCP model which can be solved efficiently by commercial solvers. Case studies are conducted on a modified IEEE 34-node three-phase four-wire system and the IEEE 123-node test feeder to verify the effectiveness, efficiency and scalability of the proposed PC-SOP concept and its operational strategy.

Exploiting aggregate sparsity in second-order cone relaxations for quadratic constrained quadratic programming problems

Among many approaches to increase the computational efficiency of semidefinite programming (SDP) relaxation for nonconvex quadratic constrained quadratic programming problems (QCQPs), exploiting the aggregate sparsity of the data matrices in the SDP by Fukuda et al. [Exploiting sparsity in semidefinite programming via matrix completion I: General framework, SIAM J. Optim. 11(3) (2001), pp. 647–674] and second-order cone programming (SOCP) relaxation have been popular. In this paper, we exploit the aggregate sparsity of SOCP relaxation of nonconvex QCQPs. Specifically, we prove that exploiting the aggregate sparsity reduces the number of second-order cones in the SOCP relaxation, and that we can simplify the matrix completion procedure by Fukuda et al. in both primal and dual of the SOCP relaxation problem without losing the max-determinant property. For numerical experiments, nonconvex QCQPs from the lattice graph and pooling problem are tested as their SOCP relaxations provide the same optimal value as the SDP relaxations. We demonstrate that exploiting the aggregate sparsity improves the computational efficiency of the SOCP relaxation for the same objective value as the SDP relaxation, thus much larger problems can be handled by the proposed SOCP relaxation than the SDP relaxation.

A SOCP Relaxation for Cycle Constraints in the Optimal Power Flow Problem

This article presented a convex relaxation approach for the optimal power flow problem. The proposed approach leveraged the second-order cone programming (SOCP) relaxation to tackle the non-convexity within the feasible region of the power flow problem. Recovering an optimal solution that is feasible for the original non-convex problem is challenging for networks with cycles. The main challenge is the lack of convex constraints to present the voltage angles within a cycle. This article aims to fill this gap by presenting a convex constraint enforcing the sum of voltage angles over a cycle to be zero. To this end, the higher-order moment relaxation matrix associated with each maximal clique of the network is formed. The elements of this matrix are utilized to form a convex constraint enforcing the voltage angle summation over each cycle. To keep the computation burden of leveraging the higher-order moment relaxation low, a set of second-order cone constraints are applied to relate the elements of the higher-order moment relaxation matrix. The case study presented the merit of this work by comparing the solution procured by the introduced approach with other relaxation schemes.

Optimal Reconfiguration of DC Networks

In this paper, we consider the problem of optimizing voltage set points and switching status of components in direct current power networks subject to physical and security constraints. The problem is cast as a mixed-integer nonlinear programming with two sources of computational complexity: i) Non-convex power flow equations, and ii) The presence of binary variables accounting for the on/off status of network components. A strengthened second-order cone programming (SOCP) relaxation is developed to tackle the non-convexity of power flow equations, and a branch-and-bound search is employed for determining optimal network configurations. The efficacy of the proposed method in optimizing the operation while mitigating contingencies is experimentally validated in a real-time hardware-in-the-loop environment using IEEE benchmark data.