Second-order Cone Programming Model Research Articles

Distributed generator-based distribution system service restoration strategy and model-free control methods

The rapid growth of distributed generator (DG) capacities has introduced additional controllable assets to improve the performance of distribution systems in terms of service restoration. Renewable DGs are of particular interest to utility companies, but the stochastic nature of intermittent renewable DGs could have a negative impact on the electric grid if they are not properly handled. In this study, we investigate distribution system service restoration using DGs as the primary power source, and we develop an effective approach to handle the uncertainty of renewable DGs under extreme conditions. The distribution system service restoration problem can be described as a mixed-integer second-order cone programming model by modifying the radial topology constraints and power flow equations. The uncertainty of renewable DGs will be modeled using a chance-constrained approach. Furthermore, the forecast errors and noises in real-time operation are solved using a novel model-free control algorithm that can automatically track the trajectory of real-time DG output. The proposed service restoration strategy and model-free control algorithm are validated using an IEEE 123-bus test system.

Optimal Restoration of Active Distribution Systems With Voltage Control and Closed-Loop Operation

This article presents a new mixed-integer second-order cone programming model for solving the restoration problem in active distribution systems considering the optimal control of distributed generators (DGs), voltage regulators (VRs), on-load tap changers (OLTCs), and capacitor banks (CBs). In contrast to most of the works found in the literature, temporary loops may be formed in the network during the restorative operation state. This operating condition allows restoring the service to more loads. In this way, the objective function of the problem minimizes (i) the total load not supplied, (ii) the number of switching operations, (iii) the changes in the statuses of the voltage control devices and in the dispatch of the DGs, and (iv) the number of basic loops formed in the system. Several tests are carried out using a 53-node system for single-fault and multiple-fault scenarios. The results obtained with the proposed approach outperform the solutions achieved when only radial configurations are allowed in the problem. Moreover, it is also verified that the voltage control allows for more efficient restoration schemes.

A Mixed-Integer Convex Model for the Optimal Placement and Sizing of Distributed Generators in Power Distribution Networks

The optimal placement and sizing of distributed generators is a classical problem in power distribution networks that is usually solved using heuristic algorithms due to its high complexity. This paper proposes a different approach based on a mixed-integer second-order cone programming (MI-SOCP) model that ensures the global optimum of the relaxed optimization model. Second-order cone programming (SOCP) has demonstrated to be an efficient alternative to cope with the non-convexity of the power flow equations in power distribution networks. Of relatively new interest to the power systems community is the extension to MI-SOCP models. The proposed model is an approximation. However, numerical validations in the IEEE 33-bus and IEEE 69-bus test systems for unity and variable power factor confirm that the proposed MI-SOCP finds the best solutions reported in the literature. Being an exact technique, the proposed model allows minimum processing times and zero standard deviation, i.e., the same optimum is guaranteed at each time that the MI-SOCP model is solved (a significant advantage in comparison to metaheuristics). Additionally, load and photovoltaic generation curves for the IEEE 69-node test system are included to demonstrate the applicability of the proposed MI-SOCP to solve the problem of the optimal location and sizing of renewable generators using the multi-period optimal power flow formulation. Therefore, the proposed MI-SOCP also guarantees the global optimum finding, in contrast to local solutions achieved with mixed-integer nonlinear programming solvers available in the GAMS optimization software. All the simulations were carried out via MATLAB software with the CVX package and Gurobi solver.

Coordinated scheduling of integrated power and gas grids in consideration of gas flow dynamics

Natural gas, hydrogen, methane, or their mixture play important roles in the energy systems, and the coordinated optimal scheduling for the integrated energy systems are requisite in consideration of complicated system characteristics. This paper focuses on mixed-integer second-order cone programming-based optimal scheduling for the electric grids with gas flow dynamics in consideration of AC power flow, unit commitment, and line switching. For the gas grids, differential continuity equations and differential momentum equations, representing gas flow dynamics, are discretized to a group of algebraic equations with the implicit trapezoidal rules. In the algebraic equations, nonconvex bilinear terms with integer variables, representing gas flow directions, are transformed into linear inequality constraints with McCormick envelopes, and quadratic terms are relaxed by the second-order cone (SOC) approach. For the electric grids, SOC relaxation-based AC power flow models are employed, and the line switching problem is modeled by the improved SOC relaxation approach with McCormick envelopes. The entire problem is established as a mixed-integer second-order cone programming model, and the impacts of gas flow dynamics on the feasible region compared to the steady-state region are discussed.

Big data network security index correlation measure based on the fusion of modified two order cone programming model

The existing technology can not meet the needs of large data network security index measurement in the era of big data. New theories and methods need to be explored to support the application of large data. In this paper, an improved second-order cone programming model (MTOCPM) is used as the data expression vector. The power spectral density feature of large data is extracted from a large number of noisy and fuzzy data. The second-order cone programming model of large data information flow is constructed by rough concept lattice method. Combining with the finite convergence of the second-order cone programming model algorithm, the reliability of each clustering sample is measured, and the accurate search of data clustering centre is realised. Experiments show that the method based on MTOCPM fusion can not only greatly reduce the communication cost between network nodes, but also improve the global measurement accuracy by about 10%.

A Thinned Irregular Array Synthesis Approach Based on Benders Decomposition

A thinned irregular tiled array synthesis approach is proposed for 5G base station array design. To fit the needs of multiusers, different sizes of vertical tiles with thinned elements ae considered. The tiling configuration and excitations are optimized to achieve high gain, low sidelobes, and wide-angle scanning simultaneously. A mixed integer second-order cone programming (MISOCP) model is obtained from convexity analysis and transformation. Besides, a generalized Benders decomposition algorithm is built to decouple the complex original problem into mixed integer linear programming (MILP) master problems and convex second-order cone programming (SOCP) subproblems, which is used to solve the large-scale problems efficiently. Irregular partition optimization is based on full-wave simulations of regular finite arrays. The joint effects of active VSWR and radiation patterns are considered to reduce the mismatch between element simulations and final performances. Numerical results show that the 5G array synthesized by the proposed approach is superior to those reported by the state-of-the-art methods in terms of both E- and H-plane scans and the number of T/R modules.

Reconfiguration of Radial Distribution Systems: An Efficient Mathematical Model

For a few decades, researchers try to reach a better solution by proposing new models or methods for distribution system reconfiguration (DSR) problem. This paper presents an efficient model for reconfiguration of radial distribution systems, aiming loss minimization. The proposed model can be easily solved by every commercial solvers and software. The proposed mixed-integer second-order cone programming (MISOCP) model was tested on 16-, 33-, 69-, 70-, 119-, and 136-bus test systems as well as Taiwan power company network (TPC). The results show high efficiency of the proposed model for reconfiguration of radial distribution systems.

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.

Distribution Network-Constrained Optimization of Peer-to-Peer Transactive Energy Trading Among Multi-Microgrids

This article proposes a two-level network-constrained peer-to-peer (P2P) transactive energy for multi-microgrids (MGs), which guarantees the distribution power network security and allows MGs to trade energy with each other flexibly. At the lower level, a P2P transactive energy is employed for multi-MGs to trade energy with each other. A multi-leader multi-follower (MLMF) Stackelberg game approach is utilized to model the energy trading process among MGs. We prove the existence and the uniqueness of the Stackelberg equilibrium (SE) and provide the closed-form expression for SE. For privacy concerns, we provide several distributed algorithms to obtain SE. At the upper level, the distribution system operator (DSO) reconfigures the distribution network based on the P2P transactive energy trading results by applying the AC optimal power flow considering the distribution network reconfiguration. If there are any network violations, DSO requests trading adjustments at the lower level for network security. We reformulate the DSO operation problem in a mixed-integer second-order cone programming (MISOCP) model and ensure its solution accuracy. Numerical results for a 4-Microgrid system, a modified IEEE 33-bus and 123-bus distribution power system show the effectiveness of the proposed transactive model and its solution technique.

A Second-Order Cone Programming Reformulation of the Economic Dispatch Problem of BESS for Apparent Power Compensation in AC Distribution Networks

The problem associated with economic dispatch of battery energy storage systems (BESSs) in alternating current (AC) distribution networks is addressed in this paper through convex optimization. The exact nonlinear programming model that represents the economic dispatch problem is transformed into a second-order cone programming (SOCP) model, thereby guaranteeing the global optimal solution-finding due to the conic (i.e., convex) structure of the solution space. The proposed economic dispatch model of the BESS considers the possibility of injecting/absorbing active and reactive power, in turn, enabling the dynamical apparent power compensation in the distribution network. A basic control design based on passivity-based control theory is introduced in order to show the possibility of independently controlling both powers (i.e., active and reactive). The computational validation of the proposed SOCP model in a medium-voltage test feeder composed of 33 nodes demonstrates the efficiency of convex optimization for solving nonlinear programming models via conic approximations. All numerical validations have been carried out in the general algebraic modeling system.

Second-Order Cone Approximation for Voltage Stability Analysis in Direct-Current Networks

In this study, the voltage stability margin for direct current (DC) networks in the presence of constant power loads is analyzed using a proposed convex mathematical reformulation. This convex model is developed by employing a second-order cone programming (SOCP) optimization that transforms the non-linear non-convex original formulation by reformulating the power balance constraint. The main advantage of the SOCP model is that the optimal global solution of a problem can be obtained by transforming hyperbolic constraints into norm constraints. Two test systems are considered to validate the proposed SOCP model. Both systems have been reported in specialized literature with 6 and 69 nodes. Three comparative methods are considered: (a) the Newton-Raphson approximation based on the determinants of the Jacobian matrices, (b) semidefinite programming models, and (c) the exact non-linear formulation. All the numerical simulations are conducted using the MATLAB and GAMS software. The effectiveness of the proposed SOCP model in addressing the voltage stability problem in DC grids is verified by comparing the objective function values and processing time.

The stochastic multi-modal hub location problem with direct link strategy and multiple capacity levels for cargo delivery systems

This paper introduces the stochastic multi-modal hub location problem with direct link strategy and multiple capacity levels for cargo delivery systems under demand uncertainty. For capturing the uncertain nature of demand, we present a stochastic optimization model to formulate this problem formally via the expected value and the chance-constrained programming techniques. Under some mild assumptions, we propose a computationally tractable approach to turn the original model into a crisp equivalent integer second-order cone programming model, which can be efficiently solved by CPLEX for only small instances. Hence, we design a memetic algorithm incorporating genetic search and local intensification for realistic size instances. Furthermore, we provide extended analyses of the original model by considering the mode-specific hub and incorporating the fixed transportation cost. To demonstrate the superiority of the proposed model and the effectiveness of the solution approach, we conduct a series of computational experiments based on Turkish network data set.

Graph-Based Second-Order Cone Programming Model for Resilient Feeder Routing Using GIS Data

One important task in power distribution system expansion planning is feeder routing (FR), which is to determine the optimal route from a medium voltage substation to load points and the optimal size of the conductors to be installed. This paper proposes a novel graph-based model for the resilient feeder routing problem using geographical information system (GIS). We show that incorporating GIS data enhances the FR solution optimality. Moreover, by introducing a representing graph, radiality of the planned network is guaranteed. A second-order cone programming (SOCP) model is used to model power flows through feeders. Since the representing graph ensures radiality, the SOCP model is exact. The uncertainty of rooftop solar generations and demand forecasting errors are taken into account, and a stochastic programming-based solution algorithm is developed. The proposed model represents practical aspects such as economic objectives (installation cost, power losses, resiliency), technical constraints (voltage limits, radiality constraint, reliability), and geographical constraints. The efficiency of the algorithm is demonstrated using three case studies: a small test system, a realistic one, and a synthetic large test system.

Economic Dispatch of Renewable Generators and BESS in DC Microgrids Using Second-Order Cone Optimization

A convex mathematical model based on second-order cone programming (SOCP) for the optimal operation in direct current microgrids (DCMGs) with high-level penetration of renewable energies and battery energy storage systems (BESSs) is developed in this paper. The SOCP formulation allows converting the non-convex model of economic dispatch into a convex approach that guarantees the global optimum and has an easy implementation in specialized software, i.e., CVX. This conversion is accomplished by performing a mathematical relaxation to ensure the global optimum in DCMG. The SOCP model includes changeable energy purchase prices in the DCMG operation, which makes it in a suitable formulation to be implemented in real-time operation. An energy short-term forecasting model based on a receding horizon control (RHC) plus an artificial neural network (ANN) is used to forecast primary sources of renewable energy for periods of 0.5h. The proposed mathematical approach is compared to the non-convex model and semidefinite programming (SDP) in three simulation scenarios to validate its accuracy and efficiency.

Service-oriented robust worker scheduling with motivation effects

Due to gradual disappearance of global demographic dividend, the improvement of workforce efficiency becomes increasingly important. It has been commonly recognised that workforce motivation can largely stimulate the workers, especially in manufacturing systems. This paper investigates a worker scheduling problem under given work shifts with workforce motivation effects, where motivation effects are characterised by motivation coefficients of job processing times. We focus on the situation where motivation coefficients are uncertain due to various factors, and only the mean vector and covariance matrix are known. The objective is to maximise the service level, measured by the probability of ensuring no tardy jobs. We first propose a distributionally robust chance constrained formulation with a probabilistic objective function. Then an adapted sample average approximation (SAA) method and a heuristic, based on an approximated mixed integer second-order cone programming (MI-SOCP) model and the idea of problem decomposition, is developed. Numerical results show that the decomposition-based heuristic is more efficient. We also draw some managerial insights.

Robust chance-constrained programming approach for the planning of fast-charging stations in electrified transportation networks

In this paper, a bi-level programming model is established to address the planning issues of fast-charging stations in electrified transportation networks with the consideration of uncertain charging demands. The capacitated flow refueling location model is considered in the upper level to minimize the planning cost of fast-charging stations while the traffic assignment model is utilized in the lower level to determine the spatial and temporal distribution of plug-in electric vehicle flows over entire transportation networks. Such bi-level model unveils the inherent relationship among charging demands, electrical demands and the spatial and temporal distribution of plug-in electric vehicle flows. Robust chance constraints are formulated to characterize the service abilities of fast-charging stations under distribution-free uncertain charging demands, where the ambiguity set is constructed to estimate the potential values of the uncertainties based on their moment-based information, such that the robust chance constraints can exactly be reduced to mixed integer linear constraints. By introducing new variables, the bi-level model is then reformulated into a single-level mixed integer second-order cone programming model so as to be solved via off-the-shelf solvers, which guarantee the optimality of the solution. A case study is conducted to illustrate the effectiveness of the proposed planning model, which reveals three critical factors that significantly impact the planning outcomes.

A Bi-Level Corrective Line Switching Model for Urban Power Grid Congestion Mitigation

The fast urbanization process in developing countries creates an electricity load surge that poses serious congestion problems to the urban power grid (UPG). The current practice of system operators is to reconfigure the topological structure of the 110-kV high voltage distribution network (HVDN) in an attempt to adjust the load distribution and eliminate the UPG congestion. However, HVDN reconfiguration is a large scale, non-convex and non-linear optimization problem that is intractable to be solved in an efficient manner. In order to reduce the computing cost and overcome the difficulty in the solving process, new partition rules are proposed to divide the HVDN into multiple smaller regions. The topological structure of each region can be optimized either independently or interdependently based on the condition of the UPG. The solving process is modeled as a bi-level iterative line switching problem. At the upper level, the operational boundary (OB) of each regional HVDN is calculated based on a set of convex ACOPF constraints. At the lower level, the line switching scheme of each regional HVDN is derived based on the corresponding mixed integer second-order cone programming (MISOCP) model considering the OB. Then, the line switching result is examined by the AC power flow to identify whether there is any violation or not. This iterative process continues until all the violations are corrected in the UPG. The proposed method was validated by a modified IEEE 58-node test system and a practical 157-node UPG system. The simulation results showed that the proposed method is able to achieve a higher computational efficiency than the traditional method so it is promising for online applications.

Risk-based dynamic generation and transmission expansion planning with propagating effects of contingencies

Transmission networks and generating units must be reinforced to satisfy the ever-increasing demand for electricity and to keep power system reliability within an acceptable level. According to the standards, the planned power system must be able to supply demand in the case of outage of a single element (N-1 security criteria), and the possibility of cascading failures must be minimized. In this paper, we propose a risk-based dynamic generation and transmission expansion planning model with respect to the propagating effect of each contingency on the power system. Using the concept of risk, post-contingency load-shedding penalty costs are obtained and added in the objective function to penalize high-risk contingencies more dominantly. The McCormick relaxation is tailored to alter the objective function into a linear format. To keep the practicality of the proposed model, a second-order cone programming model is applied for power flow representation, and the problem is modeled in a dynamic time frame. The proposed model is formulated as a mixed-integer second-order cone programming problem. The numerical studies on the RTS 24-bus test system illustrate the efficacy of the proposed model.

A Mixed-Integer Second-Order Cone Programming Algorithm for the Optimal Power Distribution of AC-DC Parallel Transmission Channels

For the controllability of the transmission power of DC transmission channels, the optimal power distribution (OPD) of AC-DC parallel transmission channels is an effective measure for improving the economic operation of an AC-DC interconnected power grid. A dynamic optimal power flow model for day-ahead OPD of AC-DC parallel transmission channels is established in this paper. The power flow equation constraints of an AC-DC interconnected power grid and the constraints of the discrete regulation requirement of the transmission power of DC channels are considered, which make the OPD model of the AC-DC parallel transmission channels a mixed-integer nonlinear non-convex programming (MINNP) model. Through a cone relaxation transformation and a big M method equivalent transformation, the non-convex terms in the objective function and constraints are executed with the convex relaxation, and the MINNP model is transformed into a mixed-integer second-order cone programming model that can be solved reliably and efficiently using the mature optimization solver GUROBI. Taking an actual large-scale AC-DC interconnected power grid as an example, the results show that the OPD scheme of the AC-DC parallel transmission channels obtained by the proposed algorithm can effectively improve the economical operation of an AC-DC interconnected power grid.

Robust disassembly line balancing with ambiguous task processing times

Disassembly line balancing problem (DLBP), which is to select disassembly process, open workstations and assign selected tasks to opened workstations, plays an important role in the recycling of End Of Life products. In real-world disassembly operations, task processing times are usually stochastic due to various factors. Most related works address the uncertain processing times by assuming that the probability distribution is known and the task processing times are independent of each other. In practice, however, it is difficult to get the complete distributional information and there is always underlying correlation between the uncertain processing times. This paper investigates the DLBP with partial uncertain knowledge, i.e. the mean and covariance matrix of task processing times. A new distributionally robust formulation with a joint chance constraint is proposed. To solve the problem, an approximated mixed integer second-order cone programming (MI-SOCP) model is proposed, and a two-stage parameter-adjusting heuristic is further developed. Numerical experiments are conducted, to evaluate the performance of the proposed method. We also draw some managerial insights and consider an extension problem.