Second-order Cone Programming Model Research Articles

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.

Charging Network Planning for Electric Bus Cities: A Case Study of Shenzhen, China

In 2017, Shenzhen replaced all its buses with battery e-buses (electric buses) and has become the first all-e-bus city in the world. Systematic planning of the supporting charging infrastructure for the electrified bus transportation system is required. Considering the number of city e-buses and the land scarcity, large-scale bus charging stations were preferred and adopted by the city. Compared with other EVs (electric vehicles), e-buses have operational tasks and different charging behavior. Since large-scale electricity-consuming stations will result in an intense burden on the power grid, it is necessary to consider both the transportation network and the power grid when planning the charging infrastructure. A cost-minimization model to jointly determine the deployment of bus charging stations and a grid connection scheme was put forward, which is essentially a three-fold assignment model. The problem was formulated as a mixed-integer second-order cone programming model, and a “No R” algorithm was proposed to improve the computational speed further. Computational studies, including a case study of Shenzhen, were implemented and the impacts of EV technology advancements on the cost and the infrastructure layout were also investigated.

Expansion planning of active distribution system considering multiple active network managements and the optimal load-shedding direction

In this paper, a new expansion planning model for an active distribution system considering multiple active network management schemes is proposed, where demand management considering the optimal load-shedding direction is innovatively introduced into the planning model. This model optimizes the investment decisions of all the infrastructures as well as the corresponding operational strategies in a coordinated manner. Therefore, the final solution is identified by the optimal type, location and size of the feeders, distributed generators, capacitor banks, static var compensators, voltage regulators, energy storage systems, and transformer substations. To incorporate the optimal load reduction into active network management, the optimal load-shedding direction is first introduced, and an iterative method is developed to compute it. On this basis, the coordination planning model of an active distribution system is established, and the operational strategy of all the active management devices is integrated into the expansion planning problem of an active distribution system. By applying the second-order cone relaxation method, the original mixed-integer nonlinear model is converted into a mixed-integer second-order cone programming model that be efficiently solved. Finally, a two-stage optimization algorithm is proposed to obtain the final solutions. Simulation results based on a 37-bus system and a 122-bus system corroborate the effectiveness of the proposed model.

Stochastic optimal dispatch of PV/wind/diesel/battery microgrids using state‐space approximate dynamic programming

In the operation of microgrids (MGs), the stochastic production of solar/wind, the discrete variables of photovoltaic (PV) inverter's auxiliary service state and diesel generators’ (DGs’) off–on state generally need to be considered, and a stochastic mixed-integer non-linear non-convex programming (MINNP) model is established for the optimal dispatch of MGs. In this model, the expected value of the sum of DGs’ operation as well as start-up cost, the network-loss cost and the PV inverter's auxiliary service cost, is considered as the objective function. The stochastic MINNP model is transformed into a stochastic mixed-integer second-order cone programming (MISOCP) model to reduce the computational complexity. The state-space approximate dynamic programming algorithm is adopted to solve the stochastic MISOCP model. In the algorithm, based on the approximate value functions of typical states that are computed according to the Markov decision process, solving the optimisation model of multiple periods is executed by solving each period's optimisation model one by one to improve the computational efficiency. Meanwhile, parallel computing is executed to greatly improve the efficiency of the proposed algorithm. Test results on two modified IEEE-33 bus and IEEE-123 bus islanded MGs with PV/wind/diesel/battery demonstrate the correctness and efficiency of the proposed model and algorithm.

Microgrids Energy Management Using Robust Convex Programming

This paper presents an energy management system (EMS) for single-phase or balanced three-phase microgrids via robust convex optimization. Along a finite planning horizon, the solution provided by the proposed microgrids EMS remains feasible under adverse conditions of random demands and renewable energy resources. The proposed model is represented as a convex mixed-integer second-order cone programming model. Two operation modes are considered: grid-connected and isolated. In grid-connected mode, the proposed EMS minimizes the costs of energy imports, dispatches of distributed generation (DG) units, and the operation of the energy storage systems. In isolated mode, the proposed EMS minimizes the unsupplied demand considering consumer priorities. Global robustness of the proposed mathematical model is adjusted using a single parameter $\zeta $ . The robustness of the solutions provided by the robust EMS are assessed using the Monte Carlo simulation method. In this case, DG units are set to operate in frequency and voltage droop control to support network fluctuations. Simulations are deployed using a microgrid with 136-nodes and several distributed energy resources. Results showed that the proposed model is suitable for the short-term microgrids energy management system. The robustness of the final solution was directly proportional to the operational costs, and it can be effectively controlled by the proposed parameter $\zeta $ for both operation modes. When compared to stochastic approaches, the proposed formulation proved to be more flexible and less time-consuming.

Conic Programming Based Day-ahead Optimization Dispatch in Distribution Network Containing Large-scaled Distributed Generation and Electric Vehicle

Regarding day-ahead optimization dispatch in distribution network containing large-scaled distributed generations (DGs) and electric vehicles (EVs), this paper proposes an optimization dispatch method based on conic programming. A model for dispatch that can optimize the EV aggregator charging power and the output of DGs is established. The proposed model takes the constrains of EV charging energy and charging power into consideration. In addition, DG can curtail the active power in the model to guarantee the operation safety. Then nonlinear optimization dispatch model are transformed and relaxed into second-order cone programming model to be solved. The optimization dispatch method proposed in this paper is verified by the IEEE33 node test system.

Optimal ESS allocation in distribution network using accelerated generalised Benders decomposition

With the advancement of energy storage technologies, installing an energy storage system (ESS) in a distribution network has become a new solution to accommodate more and more distributed renewable generations. In this study, the optimal allocation of distributed ESSs is studied to maximise the benefit of the distribution system operator. The ESS allocation problem is divided into two stages: the mixed integer investment problem as the first stage and the optimal operation problems considering daily charging/discharging schedule of ESSs as the second stage. To tackle the uncertainties of distributed generation output and base load, typical days (scenarios) are firstly obtained by the clustering method and thereafter the operation problems include a number of scenarios, each with the corresponding possibility. Then each second stage problem is relaxed to a second-order cone programming model. To efficiently solve the whole problem considering multiple scenarios, the generalised Benders decomposition (GBD) algorithm is adopted, which is further accelerated by relaxing and rebinding integer constraints. Numerical experiments are conducted on a 17-bus test system to demonstrate the effectiveness of the proposed method. Additionally, comparisons between different algorithms are performed to verify the merits of the proposed acceleration method with respect to the original GBD and the branch-and-bound algorithm.

Optimal sizing and placement of energy storage systems and on-load tap changer transformers in distribution networks

The large-scale deployment of distributed energy resources will produce reverse power flows, voltage, and congestion problems in the distribution networks. This paper proposes a novel optimization model to support distribution system operators planning future medium voltage distribution networks characterized by high penetration of behind-the-meter distributed energy resources. The optimization model defines the optimal mix, placement, and size of on-load tap charger transformers and energy storage devices with the objectives of mitigating network technical problems and minimizing both investment and operation costs. The proposed optimization model relaxes the non-convex formulation of the optimal power flow to a constrained second-order cone programming model and exactly linearizes the non-linear model of the on-load tap changer transformer via binary expansion scheme and big-M method. These two transformations reduce the computational burden of the optimization allowing it to be applicable to real-scale distribution grids, as demonstrated by the results. The numerical results also show that the joint optimization of energy storage devices and on-load tap changer transformers produces a more affordable and flexible planning strategy than the individual optimization of the technologies.

Parameterized Modeling and Planning of Distributed Energy Storage in Active Distribution Networks

In recent years, distributed energy storage (DES) has experienced rapid growth and has been widely applied in active distribution networks (ADNs). Owing to the close correlation between the characteristics and the application scenarios, DES modeling needs to be parameterized separately for various application demands. In this paper, a parameterized model for optimal DES planning in ADNs is proposed. The typical scenarios for DES planning are generated by the clustering technique, containing the patterns of load demand, wind turbine output and photovoltaic output. Secondly, an optimal planning model of DES considering parameterized characteristics is established, which is essentially a mixed integer non-linear optimization problem. Then, the model is converted to a mixed-integer second-order cone programming model, which can be solved efficiently by available commercial software. Finally, case studies on the modified IEEE 33-node system and IEEE 123-node system verify the efficiency of the proposed method, and the effects of DES planning are validated by two evaluation indexes.

Mixed-integer second-order cone programming model for bus route clustering problem

Bus route clustering problem (BRCP) concerns the assignment of bus routes to different boarding locations of a bus station with the objective of minimizing passenger waiting time. In this study, we formulate the BRCP as a mixed-integer second-order cone program (MISOCP). Simulations are conducted, in which the MISOCP model is applied to a major bus station in Hong Kong based on the network of actual bus routes. Experiments are tested for large size instances under different scenarios. Results show that the complexity of the BRCP is highly dependent on the overlapping degree of bus networks, while other factors, including the number of bus routes, destinations, and boarding locations have a joint effect; the influence is instance-specific based on different overlapping topologies of bus route networks.

Dynamic Coordinated Active–Reactive Power Optimization for Active Distribution Network with Energy Storage Systems

This paper proposes a coordinated active–reactive power optimization model for an active distribution network with energy storage systems, where the active and reactive resources are handled simultaneously. The model aims to minimize the power losses, the operation cost, and the voltage deviation of the distribution network. In particular, the reactive power capabilities of distributed generators and energy storage systems are fully utilized to minimize power losses and improve voltage profiles. The uncertainties pertaining to the forecasted values of renewable energy sources are modelled by scenario-based stochastic programming. The second-order cone programming relaxation method is used to deal with the nonlinear power flow constraints and transform the original mixed integer nonlinear programming problem into a tractable mixed integer second-order cone programming model, thus the difficulty of problem solving is significantly reduced. The 33-bus and 69-bus distribution networks are used to demonstrate the effectiveness of the proposed approach. Simulation results show that the proposed coordinated optimization approach helps improve the economic operation for active distribution network while improving the system security significantly.

Active splitting strategy searching approach based on MISOCP with consideration of power island stability

Active splitting control utilizes real-time decision and system-level splitting to prevent cascading blackouts and to maintain power supply under severe disturbances. Splitting strategy searching (SSS) is one of the most crucial issues in active splitting control for deciding “where to split”. SSS determines the splitting surface in real time to properly divide the asynchronous generators into isolated islands with an optimal control effect. In this paper, an SSS approach that focuses on island stability is presented. The proposed SSS approach is designed to ensure a rational stability margin and regulation ability on each island during and after the transient process of system splitting. This method includes the active/reactive power flow feasibility constraints and voltage/angle stability constraints in the steady state, as well as the frequency response capability constraints in the transient process. By considering the island stability constraints in the SSS, the proposed approach can avoid the splitting strategies with poor stability performance. Therefore, the major advantage of the proposed approach is ensuring better island static and transient stability during and after the splitting control. In addition, the entire model is formulated as a mixed-integer second-order cone programming (MISOCP) model. Thus, it can be rapidly solved by using commercial optimization solvers. Numerical simulation of a realistic provincial power system in central China demonstrates the validity of the proposed approach and the necessity of considering the island stability issues.

Energy management in distribution systems, considering the impact of reconfiguration, RESs, ESSs and DR: A trade-off between cost and reliability

The distribution network operator is usually responsible for improvement of efficiency and reliability of the network. This paper proposes a framework to demonstrate the impact of renewable energy sources (RESs), energy storage systems (ESSs), demand response (DR) and reconfiguration on the optimal sharing of energy. The proposed model determines the optimal locations of RESs, ESSs and DR in the distribution network to minimize simultaneously the cost of energy procurement and energy not supplied. A multi-objective optimization problem is formulated with a mixed-integer second-order cone programming model and ε-constraint method is used to generate Pareto optimal solutions. The network reconfiguration is also considered to optimize the power flow by changing the network topology. The proposed model is implemented on the IEEE standard 33-bus radial test system, and solved by General Algebraic Modeling System (GAMS) optimization software. According to the simulation results, the proposed framework is beneficial both from the reliability and economic perspectives.

Operational Strategy Based Evaluation Method of Distributed Energy Storage System in Active Distribution Networks

Abstract Distributed energy storage system (DESS) is an effective way in adapting to increasing penetration of distributed generations (DGs) and will play an important role in the operation of active distribution networks (ADNs). However, the high capital cost and limited life cycles restrict the further development of DESS. In this paper, an operational strategy based method for economic evaluation of DESS is proposed. Firstly, rate of return is set as the economic evaluation index. Then, the economic operation model is established. By applying convex relaxation, the original nonlinear programming (NLP) model is converted into an effectively solved second-order cone programming (SOCP) model. Finally, the effectiveness of the economic evaluation method is verified on the modified IEEE 33-node test feeder.

A Combined Central and Local Voltage Control Strategy of Soft Open Points in Active Distribution Networks

With the increasing penetration of distributed generation (DG), the risk of voltage violations in active distribution networks (ADNs) has become a major concern for the system operator. Soft open point (SOP) is a flexible power electronic device which can realize accurate active and reactive power flow control. This paper proposes a combined central and local operation strategy of SOPs to realize voltage control in ADNs. The active power of SOPs is centrally adjusted based on the information and forecasting throughout the network, which aims to maintain the voltage within the limits in the global optimization. And the local control of reactive power based on real-time measurements can rapidly respond to the frequent voltage violations caused by the fluctuations of DG outputs. The potential benefits of SOPs are fully explored to reduce power losses and improve voltage profile of ADNs. By applying convex relaxation, the original mixed-integer nonlinear programming (MINLP) model is converted into an effectively solved mixed-integer second-order cone programming (MISOCP) model. Case studies on the PG&E 69-node distribution system are conducted to verify the effectiveness of the proposed method.

Second-order cone programming relaxation-based multi-period reactive power optimization with VSC-Type HVDC and DFIG

This paper presents a second order cone programming (SOCP) formulation of the dynamic reactive power optimization (DROPF) problem for the voltage source converter (VSC)-based high-voltage direct current (HVDC) transmission network, considering reactive power support of the doubly fed induction generator (DFIG). In this model, the reactive power support of DFIG, VAR compensators, the position of tap-changer and the states of VSC converters are formulated as continuous and discrete decision variables. By using approximation techniques, the original non-convex optimisation model is converted into a mixed-integer second-order cone programming model. Later, the SOCP formulation can be solved using many standard optimization packages. Then a DC system with VSC technology is modelled in the IEEE 30-bus example system. The SOCP formulation of AC-DC DROPF is applied to the modified IEEE 30-bus example system and the results are discussed.

Coordination of generation, transmission and reactive power sources expansion planning with high penetration of wind power

High penetration of renewable energy is one of the important feature of future power systems. However, the randomness and intermittence of renewable energy have brought new challenges to planners. To accommodate the uncertainty and variability of wind power, an innovative scenario-based stochastic model that incorporates generation, transmission, and reactive power planning is presented, considering spinning reserve, unit ramp rate, and current output of units. To consider reactive power flow and voltage, a second-order cone programming model based on relaxed AC optimal power flow is introduced. Moreover, unit status and optimal transmission switching are implemented to promote wind consumption, reduce congestion, and attain a more flexible topology. The proposed methodology organically combines operational constraints with planning issues. Finally, two modified test systems with high wind power integration are utilized to illustrate the effectiveness of the presented model.

Distributionally robust parallel machine ScheLoc problem under service level constraints

Abstract Scheduling-Location (ScheLoc) problem is relatively new and has received a lot attention. Such a problem focuses on integrating the tactical-level decision (i.e., selecting locations for machines) and the operational-level decision (i.e., schedule of the jobs). Especially, the release time depends on the distance between the job storage location and its processing machine, as each job should be transported to the machine location for processing. The existing researches considering the ScheLoc problem are limited, and most of them focus on the deterministic settings. In practice, job processing times are usually uncertain, and the probability distribution of the uncertain processing times may not be exactly estimated, due to various factors. This work investigates a parallel machine ScheLoc problem under stochastic processing times with only partial distributional information (i.e., the mean and covariance matrix), to minimize the cost for operating machines and control the service level at the same time. The service level in this work is measured by the probability of ensuring no tardy job. For the problem, a distributionally robust formulation is first proposed, in which the service level is restricted with a joint chance constraint. By applying a popular approximation method, an approximated mixed integer second-order cone programming (MI-SOCP) model is then developed. A case study is conducted and reported, to illustrate the applicability of the MI-SOCP model.

Joint reconfiguration of feeders and allocation of capacitor banks in radial distribution systems considering voltage-dependent models

This paper presents a study about operation planning for radial distribution systems using a mixed-integer second-order cone programming formulation. The proposal consists of a complete model for solving the problems of the reconfiguration of feeders and allocation of capacitor banks simultaneously, with the objective of achieving a minimum overall investment, in both fixed and switchable capacitor banks, and operational costs, related to the cost of energy losses. The approach includes a voltage-dependent load representation, as well as a precise representation of the operation of capacitor banks, which is also voltage dependent. The use of a mixed-integer second-order cone programming model guarantees convergence to optimality using existing optimization software. A strategy to reduce the number of candidate nodes of the system for capacitor banks allocation is also presented. Several tests are performed using the 69-node distribution system and a real 2313-node distribution system, demonstrating the effectiveness of the proposal. The results indicate that better solutions can be obtained by performing the simultaneous optimization of the reconfiguration and allocation of capacitor banks than by solving the two problems sequentially, which may even lead to infeasible solutions. It is also verified that both the load and capacitor banks models have an influence on the problem’s solution.

Decentralized Energy Management of Networked Microgrid Based on Alternating-Direction Multiplier Method

With the ever-intensive utilization of distributed generators (DGs) and smart devices, distribution networks are evolving from a hierarchal structure to a distributed structure, which imposes significant challenges to network operators in system dispatch. A distributed energy-management method for a networked microgrid (NM) is proposed to coordinate a large number of DGs for maintaining secure and economic operations in the electricity-market environment. A second-order conic programming model is used to formulate the energy-management problem of an NM. Network decomposition was first carried out, and then a distributed solution for the established optimization model through invoking alternating-direction method of multipliers (ADMM). A modified IEEE 33-bus power system was finally utilized to demonstrate the performance of distributed energy management in an NM.