Mixed-integer Second-order Cone Programming Research Articles

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.

A VSC-Based BESS Model for Multi-Objective OPF Using Mixed Integer SOCP

This paper presents a new model of voltage source converter (VSC) based battery energy storage systems (BESSs) that interface with power grids. A VSC-based BESS is made up of a series connection of a VSC, its connecting transformer, and a BESS. The VSC allows a BESS to generate both active and reactive powers in all four quadrants. The proposed model captures the coupling between active power, reactive power, and the voltage of a BESS. In addition, the proposed model explicitly describes the relationship between the control configuration of pulsewidth modulation (PWM) VSC and the power output of the BESS. Therefore, the proposed model possesses unparalleled control capabilities in the operational parameters of both the ac and dc sides of the converter. By incorporating such a model into the active-reactive optimal power flow (A-R-OPF) formulation, we can not only optimize the active and reactive power output of the BESS in power systems, but also understand how the optimal powers are generated by setting the operational parameters of both the BESS and PWM-VSC. To solve the A-R-OPF problem with the proposed model, we propose a sequence of strong relaxations to transform the problem into a mixed-integer second-order cone programming problem. Such a formulation is amendable for efficient solutions using off-the-shelf solvers. Case studies on the IEEE benchmark systems show that more than 17.73% of power losses in transmission lines and more than 0.961% of interface losses can be reduced by using the proposed model in comparison to the traditional BESS model.

Service restoration of active distribution systems with increasing penetration of renewable distributed generation

The increasing penetration of renewable generation poses several challenges to the operation of distribution networks under emergency conditions. This study proposes a service restoration method that exploits active management of the distribution network. The proposed active network management (ANM) method considers the coordinated control of the available switches, distributed generation, and the operation of on-load tap changer (OLTC) to determine the optimal service restoration plan. The objectives of the proposed model are the minimisation of (i) the out of service area considering customer priorities, (ii) the number of switch operations, (iii) the number of tap changes of the OLTC, and (iv) the injected power from the substations considering the variation of load demand and renewable generation. The service restoration problem is formulated as a mixed-integer second-order cone programming problem, which is efficiently solved by commercial branch and bound solvers. Results on a 135-bus distribution system and a 540-bus distribution system highlight the importance and the benefits of incorporating ANM into the solution of the service restoration problem.

Offline-Channel Planning in Smart Omnichannel Retailing

Problem definition: Observing the retail industry inevitably evolving into omnichannel, we study an offline-channel planning problem that helps an omnichannel retailer make store location and location-dependent assortment decisions in its offline channel to maximize profit across both online and offline channels, given that customers’ purchase decisions depend on not only their preferences across products but also their valuation discrepancies across channels, as well as the hassle costs incurred. Academic/practical relevance: The proposed model and the solution approach extend the literature on retail channel management, omnichannel assortment planning, and the broader field of smart retailing/cities. Methodology: We derive parameterized models to capture customers’ channel choice and product choice behaviors, and customize a corresponding parameter estimation approach employing the expectation-maximization method. To solve the NP-hard optimization model, we develop a tractable mixed-integer second-order conic programming (MISOCP) reformulation and explore the structural properties of the reformulation to derive strengthening cuts in closed-form. Results: We numerically validate the efficacy of the proposed solution approach and demonstrate the parameter estimation approach. We further draw managerial insights from the numerical studies using real data sets. Managerial implications: We verify that omnichannel retailers should provide location-dependent offline assortments. In addition, our benchmark studies reveal the necessity and significance of jointly determining offline store locations and assortments, as well as of incorporating the online channel while making offline-channel planning decisions.

Stochastic optimal transmission switching considering the correlated wind power

Optimal transmission switching (OTS) is widely used in mitigating transmission congestion and reducing power losses. However, with large-scale renewable energy integration to the grid, making a decision for the randomness characteristic of renewable energy (e.g. wind and solar power) is complicated in OTS. In this study, a novel stochastic optimal power flow-based point estimation method (PEM) is presented to model the uncertainties of wind power and load in OTS. Polynomial normal transformation is introduced to handle the correlations between random variables, and percentile matching method is employed to obtain the coefficients of polynomial normal transformation, which can avoid the integral operation. Moreover, the expected value of power flow obtained by the PEM is applied to the presented model to control the line overload risk. Two indexes, namely, population sample mean and mean of population sample standard deviation, are proposed to investigate the effect of correlations on OTS strategies. The proposed model is finally transformed into a mixed-integer second-order cone programming to consider bus voltage and reactive power, and the modified IEEE RTS 24-bus system and IEEE 118 system are presented to test the model.

Multi-period Robust Dispatch of Active Distribution Network Based on Scene Analysis Method

With the large-scale access of intermittent renewable energy, the active distribution networks (ADN) is facing great challenges on achieving full consumption and efficient use of intermittent renewable energy on the premise of ensuring safe operation. To resolve these issues, a novel multi-period robust dispatch strategy for ADN based on scene analysis method is proposed in this paper. Then, the second-order cone relaxation (SOCR) is used to transform the original model into a mixed integer second-order cone programming (SOCP), which gains as much improvement as possible in computational performance at the expense of as few losses as possible in accuracy. Finally, numerical test on the modified IEEE 33-bus test system demonstrates that on the premise of realizing the maximum utilization of DG, the proposed strategy can also achieve the purposes of energy-saving, loss reduction and improvement in voltage level.

Retail market equilibrium and interactions among reconfigurable networked microgrids

In this paper, a retail market equilibrium among a set of competing players, i.e., retailers and Microgrids (MGs), considering Dynamic Line Rating (DLR) constraint and demand response (DR) actions is modeled. In this regard, a multi-follower-type bi-level optimization scheduling for decision making of Distribution System Operator (DSO) and reconfigurable networked MGs with inherently conflicting objectives is presented. In this model, the upper-level player of the Main Problem (MP) minimizes the total cost from DSO’s perspective, while in the lower-level of the MP, networked MGs compete with retailers in the retail electricity market as an interior Sub Problem (SP). In the upper-level of the SP the profit of each MG is maximized, where in the lower-level of the SP the social welfare of the retail market clearing process is maximized. Since the resulting bi-level problem is nonlinear and non-convex, it is transformed into a single-level Mixed-Integer Second-Order Cone Programming (MISOCP) problem using Karush–Kuhn–Tucker (KKT) optimality conditions and linearization techniques. The effectiveness of the proposed model is investigated on a real-test system in both grid-connected and islanded modes under different scenarios.

Integrated demand response for a load serving entity in multi-energy market considering network constraints

The rapid development of an integrated energy system makes it difficult for traditional power market to adapt to the trend of multi-energy interactions. Therefore, a tri-layer multi-energy day-ahead market structure and operation mechanism, allowing the simultaneous trading of electricity, heat and natural gas, are proposed in this paper. Concentrating on the profit of the load serving entity in this market, the optimal transaction strategy based on the integrated demand response is explicitly modeled in detail. In particular, the physical constraints of the power distribution network, natural gas network and district heating network are strictly considered. To address the nonlinear and nonconvex problems in the distribution network and natural gas network, the mixed-integer second-order cone programming method and piecewise linearization process are used. Furthermore, a novel conditional value at risk approach is proposed to address the uncertain forecasted market prices, so that the risk can be mitigated. Compared with the traditional electricity market, the LSE can earn a higher profit in the proposed market, and the integrated demand response program enhances the potential of multi-energy peak load shifting. Finally, the effectiveness of the proposed method has been verified on an integrated energy system with IEEE 33-bus power system, an 11-node gas system and a 6-node heat system. A set of comparative cases verify the necessity for the IES to keep the balance between the market economy and network security operation.

Optimal Operational Scheduling of Reconfigurable Multi-Microgrids Considering Energy Storage Systems

This paper proposes an optimal operational scheduling of a reconfigurable multi-microgrid (MG) distribution system complemented by demand response programs and Energy Storage Systems (ESSs) in an uncertain environment. Since there is a set of competing players with inherently conflicting objectives in the system under study such as the Distribution System Operator (DSO) and MG owners, a one-leader multi-follower-type bi-level optimization model is proposed. In this framework, the upper-level player as a leader minimizes the total cost from DSO’s point of view, while the lower-level players as multi-followers maximize the profit of MG owners. Since the resulting model is a non-linear bi-level optimization problem, it is transformed into a single-level mixed-integer second-order cone programming problem through Karush–Kuhn–Tucker conditions. The satisfactory performance of the proposed model is investigated on a real-test system under different scenarios and working conditions.

Analytical Approach for Active Distribution Network Restoration Including Optimal Voltage Regulation

The ever-increasing utilization of sensitive loads in the industrial, commercial, and residential areas in distribution networks requires enhanced reliability and quality of supply. This can be achieved, thanks to self-healing features of smart grids that already include the control technologies necessary for the restoration strategy in case of a fault. In this paper, an analytical and global optimization model is proposed for the restoration problem. A novel mathematical formulation is presented for the reconfiguration problem reducing the number of required binary variables while covering more practical scenarios compared to the existing models. The considered self-healing actions besides the network reconfiguration are the nodal load-rejection, the tap setting modification of voltage regulation devices (incl. OLTCs, SVR, and CBs), and the active/reactive power dispatch of DGs. The voltage dependency of loads is also considered. Thus, the proposed optimization problem determines the most efficient restoration plan minimizing the number of de-energized nodes with the minimum number of self-healing actions. The problem is formulated as a Mixed-Integer Second Order Cone Programming (MISOCP) and solved using the Gurobi solver via the MATLAB interface YALMIP. A real 83–node distribution network is used to test and verify the presented methodology.

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.

Day-Ahead Dispatch of Integrated Electricity and Natural Gas System Considering Reserve Scheduling and Renewable Uncertainties

For secure operation of integrated electricity and natural gas system (IEGS), reserve is a useful support to manage renewable uncertainties and N − 1 contingencies. Thus, a day-ahead economic dispatch model of IEGS with reserve scheduling is presented in this paper. Considering the uncertainty of gas flow direction, a novel second-order cone (SOC) relaxation of Weymouth equation is designed to address the nonconvexity. Then, the proposed robust nonconvex model is mathematically transformed into a solvable mixed integer second-order cone programming (MISOCP) problem. To guarantee the tightness of SOC relaxation and achieve accurate dispatch solutions, MISOCP results are corrected accordingly by the multi-slack-node gas flow calculation with the Newton–Raphson method. Numerical cases are performed on IEEE 39-bus-15-node and IEEE 118-bus-40-node test IEGSs, demonstrating the proposed approach is feasible and effective for exact IEGS day-ahead dispatch and the proposed MISOCP model can provide a more economic dispatch solution with a shorter computational time than conventional MISOCP and mixed integer linear programming (MILP) models.

Joint planning of distributed generation and electric vehicle charging stations considering real-time charging navigation

With the popularity of intelligent mobile terminals, as well as the improvement in wireless communication technologies, the application of real-time charging navigation for electric vehicles (EVs) has met a booming development in recent years. Accordingly, more and more EV owners tend to make their charging decisions depending on the guidance from navigation systems, which to some extent transforms the EV charging demands into spatially dispatchable resources in the operation of distribution systems. To involve this spatially dispatchable characteristic at the planning stage of distribution systems, and deploy power devices in a cost-effective way, a comprehensive optimization model concerning the joint planning of distributed generators (DGs) and electric vehicle charging stations (EVCSs) is proposed in this paper from the perspective of a social planner. The proposed joint planning model is embedded with the spatial scheduling problem of EV charging demands, and focuses on the optimum of relevant social costs. To relieve the complexity of the optimization model, an exact second-order cone programming (SOCP) relaxation is utilized to transform the proposed model into the type of mixed integer second-order cone programming (MISOCP), which can be efficiently solved in polynomial time. Finally, a practical urban area coupled with the 33-bus distribution system is used as the test system to implement the proposed approach, and the corresponding allocation schemes as well as annualized social costs are minutely presented. The numerical comparison with traditional Voronoi diagram-based planning methods reveals the significant economic benefits achieved through considering real-time charging navigation technologies.

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.

Placement and Sizing of Inverter-Based Renewable Systems in Multi-Phase Distribution Networks

This paper develops a tractable formulation for optimal placement and sizing of inverter-based renewable systems in multi-phase distribution networks. The goal of the formulation is to minimize the cost of inverter installation, average power import, and average distributed generation curtailment. Three-phase and single-phase inverter models are presented that preserve the underlying mappings between renewable uncertainty to power injection. The uncertainty of distributed generators (DGs) and loads are characterized by a finite set of scenarios. Linear multi-phase power flow approximations are used in conjunction with scenario reduction techniques to arrive at a tractable two-stage stochastic formulation for optimal DG placement and sizing. First-stage decisions are locations for DG deployment and capacity sizes, and second-stage decisions include DG real power curtailment, reactive power support, as well as feeder voltage profile. The resulting formulation is a mixed-integer second-order cone program and can be solved efficiently either by existing optimization solvers or by relaxing the binary variables to the [0,1] interval. Simulation studies on standard multi-phase IEEE test feeders promise that optimal stochastic planning of DGs reduces costs during validation, compared to a scheme where uncertainty is only represented by its average value.

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.

Distribution-Free Stochastic Closed-Loop Supply Chain Design Problem with Financial Management

Financial flow is an important part of supply chain management (SCM) and increasingly playing a crucial role as the amount of global trade increases. Reasonable and scientific financial operation is necessary in closed-loop supply chain management, especially when customer demand is uncertain. However, financial flow, which may lead to an increase in effectiveness, has rarely been considered in the literature. In this paper, we present a closed-loop supply chain design with financial management problem, which is tackled as a stochastic programming model with ambiguity demand set. The main contributions of this work include: (i) A joint chance constrained programming model is proposed to maximize the total profit, and (ii) financial flow and uncertain demand are both taken into consideration. According to the characteristic of the problem, we chose four approaches, namely sample average approximation (SAA), enhanced sample average approximation (ESAA), Markov approximation (MA), and mixed integer second-order conic program (MI-SOCP). Computational experiments were conducted to compare the adopted methods, and 10,000 scenarios were generated to examine the reliability of the methods. Numerical results revealed that the Markov approximation approach can achieve more reliable solutions.

Data-Driven Risk-Averse Two-Stage Optimal Stochastic Scheduling of Energy and Reserve With Correlated Wind Power

This paper proposes a data-driven optimization method to solve the integrated energy and reserve dispatch problem with variable and correlated renewable energy generation. The proposed method applies the kernel density estimation to establish an ambiguity set of continuous multivariate probability distributions and the optimization model for the integrated dispatch is formulated as a combination of stochastic and robust optimization problems. First, a risk-averse two-stage stochastic optimization model is formulated to hedge the distributional uncertainty. Next, the second-stage worst case expectation is evaluated, using the equivalent model reformulation, as a combination of conditional value-at-risk (CVaR) and the extreme cost in the worst case scenario. The CVaR is calculated using a scenario-based stochastic optimization problem. After describing the wind power correlation in ellipsoidal uncertainty sets, the robust optimization problem for finding the worst case cost is cast into a mixed-integer second-order cone programming problem. Finally, the column-and-constraint generation method is employed to solve the proposed risk-averse two-stage problem. The proposed method is tested on the 6-bus and IEEE 118-bus systems and validated by comparing the results with those of conventional stochastic and robust optimization methods.

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.