Second-order Cone Programming Method Research Articles

Demand Response Power-Gas Interconnection Energy System and Power Metering Based on SOGA

Under the double background of global energy crisis and environmental pollution, China vigorously develops renewable energy while accelerating the construction of energy Internet. Taking demand response into account, this paper proposes the optimal design of power-gas interconnection energy system and power metering, and establishes the optimization research model of IEGES (integrated electricity-gas energy system) with demand response. Firstly, the structure model of the electric-gas interconnection energy system with CHP (Cogeneration, combined heat and power) as the core is constructed, and the energy conversion relationship and different energy flow directions of the coupling equipment are expounded from three aspects. The natural gas source point, pipeline equation, power side branch equation, voltage and current equation are modeled and sorted out, and the square term in the equation is linearized by second-order cone programming method, and the mixed integer nonlinear programming problem is transformed into mixed integer linear programming problem. A single objective genetic algorithm with “elite strategy” was selected to solve the equipment capacity optimization problem of IEGES system with system economy as the optimization objective. After a long time of parameter combination attempts, the current population size is 30, the number of iterations is 600, the crossover rate is 0.8, and the heritability is 0.3. The above parameters can obtain better convergence results on the basis of considering the operation time. Finally, a stochastic optimization method of energy Internet considering integrated demand response and uncertainty of wind power is proposed, which aims to meet the energy demand of end users while minimizing the operating cost of the system. The comprehensive demand response strategy including internal and external demand response is considered in the model. Internal demand response is realized by adjusting the internal operation mode of EH, while external demand response is implemented by the end user’s active response, and the load is time-shifted or interrupted under the guidance of external signals.

A multi-product and multi-period supply chain network design problem with price-sensitive demand and incremental quantity discount

Demand is an important factor in supply chain network design (SCND). Among the many factors that affect demand, the impact of price cannot be underestimated. This study, for the first time, provides enterprises with direct multi-period pricing and demand decision solutions in SCND by simultaneously considering the impact of price-sensitive demand and incremental quantity discounts on the network. In this way, a mixed-integer nonlinear programming model is established based on the relationships between price and demand functions to maximize the overall profit of the supply chain. Due to the nonlinear characteristics of the model, this study proves the existence of the optimal solution of the problem and the feasibility of implementing the following two algorithms by analyzing the nature of the problem. The first algorithm is based on the second-order cone programming (SOCP) method, and the second algorithm is based on the outer-approximation method. The results of numerical experiments at different scales show that the SOCP method can obtain the optimal solution for small and medium-scale experiments. In contrast, the outer-approximation method can find approximate optimal solutions within a reasonable time for large-scale experiments. The results confirm that considering incremental quantity discount can significantly enhance the profitability of supply chain networks in long-term planning. Finally, some future research directions in the field of SCND are discussed.

Mixed-integer second-order cone programming method for active distribution network

Developing a novel type of power system is an important means of achieving the “dual carbon” goals of achieving peak carbon emissions and carbon neutrality in the near future. Given that the distribution network has access to a wide range of distributed and flexible resources, reasonably controlling large-scale and adjustable resources is a critical factor influencing the safe and stable operation of the active distribution network (ADN). In light of this, the authors of this study propose a mixed-integer second-order cone programming method for an active distribution network by considering the collaboration between distributed, flexible resources. First, Monte Carlo sampling is used to simulate the charging load of electric vehicles (EVs), and the auto regressive moving average (ARMA) and the scenario reduction algorithms (SRA) based on probability distance are used to generate scenarios of the outputs of distributed generation (DG). Second, we establish an economical, low-carbon model to optimize the operation of the active distribution network to reduce its operating costs and carbon emissions by considering the adjustable characteristics of the distributed and flexible resources, such as on-load tap changer (OLTC), devices for reactive power compensation, and EVs and electric energy storage equipment (EES). Then, the proposed model is transformed into a mixed-integer second-order cone programming (SOCP) model with a convex feasible domain by using second-order cone relaxation (SOCR), and is solved by using the CPLEX commercial solver. Finally, we performed an arithmetic analysis on the improved IEEE 33-node power distribution system, the results show that ADN’s day-to-day operating costs were reduced by 47.9% year-on-year, and carbon emissions were reduced by 75.2% year-on-year. The method proposed in this paper has significant effects in reducing the operating cost and carbon emissions of ADNs, as well as reducing the amplitude of ADN node voltages and branch currents.

Integrated optimization of driving strategy and energy management for hybrid diesel multiple units

Hybrid diesel multiple units (HDMU) are considered an effective way to reduce fuel consumption and pollution emissions in regional non-electrified railways. To minimize the fuel consumption of HDMU, this paper formulates an integrated optimization of driving strategy and energy management. By introducing convex relaxation and linearization, we propose a second-order cone programming method to solve this integrated optimization. Case studies of multiple scenarios validate the effectiveness and robustness of the proposed method. In addition, the simulation results show that compared with the sequential optimization with the step-by-step improvement of driving strategy and energy management, the integrated optimization considers the battery characteristics when optimizing the driving strategy and reduces the electric braking power of the train, thus allowing the battery to absorb more braking energy and reduce fuel consumption.

Dynamic reconfiguration of AC/DC hybrid distribution network considering non-ideal linear BES model

The charging and discharging process of battery energy storage (BES) directly affects its cycle aging. Therefore, considering the cycle aging cost and demand response of BES, this paper proposes a dynamic reconfiguration method of AC/DC hybrid distribution network. By sampling the operation area of charging and discharging power, the BES model in the non-ideal state is reconstructed into a linear model in the convex envelope set of sampling points. On this basis, the cycle aging cost models based on the cumulative throughput and discharge depth are established respectively, and the AC/DC hybrid distribution network reconfiguration model with the minimum comprehensive cost is constructed, and the mixed integer second-order cone programming method is used to solve it. The effectiveness and economy of the proposed reconfiguration method for AC/DC hybrid distribution network are verified by simulation on a modified 33-bus and 70-bus system.

Combined load bearing capacity of rigid piles embedded in a cross-anisotropic clay deposit using 3D finite element lower bound

In this study, an iterative-based three-dimensional finite element lower bound in association with the second-order cone programming method is adopted to evaluate the limit load of a single pile embedded in cross-anisotropic soils under general loading condition. The lower bound solutions of the pile embedded in an anisotropic soil deposit can be found by formulating the element equilibrium, equilibrium of shear and normal stresses along discontinuities, boundary conditions, yield function, and optimizing the objective function through the second-order cone programming method in conjunction with an iterative-based update procedure. A general loading condition is considered to profile the expansion of the safe load in the vertical-horizontal-moment (V–H–M) space. The results of this study are compared and validated against three different cases including an isotropic lateral loading, anisotropic end bearing capacity, and a pile embedded in an isotropic soil deposit under general loading condition. A parametric study is conducted to evaluate the impact of different influencing factors. It was found that the effect of anisotropy on the variation of lateral limit load of a single pile is more pronounced than the corresponding vertical and bending moment limit loads, whereas the interface properties have more significant effects on the vertical and bending moment limit loads in comparison to the lateral limit load.

Operation Optimization Method of Distribution Network with Wind Turbine and Photovoltaic Considering Clustering and Energy Storage

The problem of distribution network operation optimization is diversified and uncertain. In order to solve this problem, this paper proposes a method of distribution network operation optimization considering wind-solar clustering, which includes source load and storage. Taking the total operating cost as the objective function, it includes network loss cost, unit operating cost, and considers a variety of constraints such as energy storage device constraints and demand response constraints. This paper aims to optimize the operation according to different wind-solar clustering scenes to improve the economy of distribution network. Taking the 365-day wind-solar output curves as the research object, K-means clustering is carried out, and the best k value is obtained by elbow rule. The second-order cone programming method and solver are used to solve the optimization model of each typical scenario, and the operation optimization analysis of each typical scenario obtained by clustering is carried out. Taking IEEE33 system and local 365-day wind-solar units output scenes as examples, the period is 24 h, which verifies the effectiveness of the proposed method. The proposed method has guiding significance for the operation optimization of distribution network.

A CVaR optimization method for priority of hesitant fuzzy preference relation with chance constraint

In this paper, we establish a chance constrained model for the priority of hesitant fuzzy preference relation based on the idea of statistical distribution for preference information as stochastic variables with unknown distribution. Inspired by the idea of conditional value-at-risk (CVaR) robust optimization, a deterministic convex reformulation is proposed for tackling the chance constrained problem. The existing state-of-the-art methods usually assume that the probability density function of preference information is known a priori, such as Gaussian distribution. However, it is generally over-conservatism. On the contrary, our proposed method provides a tractable second-order cone (SOC) reformulation for the chance constrained problem with the first and second moments, which is easy to handle and calculate. We also analyze the weight acquisition problem of hesitant fuzzy preference relation with unknown distribution preference using the SOC programming method, and obtain the priority weight with its approximately equivalent computationally tractable conic optimization model. A case study is conducted which shows that the proposed method achieves a good general conclusion by comparing it with the optimization method under Gaussian distribution. In addition, this method can also get better decision support for incomplete preference information.

Sharpe-Ratio Portfolio in Controllable Markov Chains: Analytic and Algorithmic Approach for Second Order Cone Programming

The Sharpe ratio is a measure based on the theory of mean variance, it is the measure of the performance of a portfolio when the risk can be measured through the standard deviation. This paper suggests a Sharpe-ratio portfolio solution using a second order cone programming (SOCP). We use the penalty-regularized method to represent the nonlinear portfolio problem. We present a computationally tractable way to determining the Sharpe-ratio portfolio. A Markov chain structure is employed to represent the underlying asset price process. In order to determine the optimal portfolio in Markov chains, a new hybrid optimization programming method for SOCP is proposed. The suggested method’s efficiency and efficacy are demonstrated using a numerical example.

Day-Ahead Optimal Dispatch for Active Distribution Network Considering Probability Model of Controllable Distributed Generation

In this paper, the probabilistic model of the controllable distributed generation in active distribution network is developed and applied to the daily stochastic optimal dispatch. The probabilistic characteristics of photovoltaic power generation system with active control capability are explored, and the relationship between the reference value of active power and its cumulative distribution function and mean value is obtained. The active power probability model of wind power generation system is improved according to the actual wind speed power curve. By fully utilizing the inverter capacity and coordinating active power, the reactive power of distributed generation is actively controlled under the constraint of power factor. Then considering the chance constraints, a daily optimal scheduling model for active distribution network with the goal of minimizing the operating cost of distribution network is developed, and the constraints that can calculate the charge and discharge times of the energy storage system are designed. The chance constrained programming is solved by the heuristic method, and the deterministic optimization steps are solved by the second-order cone programming method, respectively. The probabilistic power flow method based on stochastic response surface method is utilized to test chance constraints. Finally, the modified IEEE33 node distribution system example shows that the obtained models and algorithms are correct and can meet the requirements of safe and economic operation.

Penetration trajectory optimization for the hypersonic gliding vehicle encountering two interceptors

The penetration trajectory optimization problem for the hypersonic gliding vehicle (HGV) encountering two interceptors is investigated. The HGV penetration trajectory optimization problem considering the terminal target area is formulated as a nonconvex optimal control problem. The nonconvex optimal control problem is transformed into a second-order cone programming (SOCP) problem, which can be solved by state-of-the-art interior-point methods. In addition, a penetration strategy that only requires the initial line-of-sight (LOS) angle information of the interceptors is proposed. The convergent trajectory obtained by the proposed method allows the HGV to evade two interceptors and reach the target area successfully. Furthermore, a successive SOCP method with a variable trust region is presented, which is critical to balance the trade-off between time consumption and optimality. Finally, the effectiveness and performance of the proposed method are verified by numerical simulations.

Economic dispatch for a regionally integrated heat and electricity system with variable pipeline flow direction

Traditional models of heating networks are not able to take into account the changes in the value and the direction of pipeline mass flow in a multi-heat-source district heating network or the time delay characteristics of the network with constant temperature and variable flow operation. A day-ahead dispatch model for regional integrated heating and electricity system based on improved second-order cone programming method is therefore proposed. We first construct a heating network transmission model with variable value and direction of pipeline mass flow, and then use a convex relaxation to transform this nonlinear model into a mixed-integer second-order cone programming model. A hydraulic time-delay correction model is then constructed to reflect the dynamic characteristics of the heating network under variable mass flow conditions. Finally, we establish a model of an electric boiler with thermal storage to improve the flexibility of the distribution network. Case studies are used to show that the optimised results are more economical in terms of both the variable value and direction of mass flow. The proposed programming method retains the variability in the flow direction and achieves precise relaxation without the need to solve for the initial value. We also verify that the hydraulic time delay characteristics have a strong influence on the optimised results of the regionally integrated heat–electricity system under conditions of constant temperature and variable flow. In addition, an electric boiler with thermal storage can reduce renewable energy curtailment and effectively improve operational flexibility.

PV Inverter Reliability-Constrained Volt/Var Control of Distribution Networks

The increasing penetration of photovoltaic (PV) systems promotes utilization of PV inverters for volt/var control (VVC) in distribution networks. However, PV inverters are vulnerable and their reliability is one of the most critical concerns for sustainable PV energy utilization. To enhance PV inverter reliability, this paper proposes a PV inverter reliability-constrained VVC method of distribution networks under uncertainties. Firstly, a full reliability analysis procedure for PV inverters is developed to estimate the lifetimes of core devices inside inverters. Secondly, considering the impacts of VVC on PV inverter reliability, reliability constraints are proposed based on regulation of inverter apparent power and application of a PV curtailment scheme. Thirdly, a multi-objective PV inverter reliability-constrained VVC optimization model is proposed, which aims to minimize network power loss and PV curtailed power, simultaneously. Accordingly, the nonconvex model is relaxed by a second-order cone programming method, and the uncertainties such as PV power generation and loads are addressed by a scenario-based stochastic optimization method. The proposed VVC method is tested on the 33-bus and 69-bus distribution networks, and the simulation results verify high efficiency of the proposed method in energy loss reduction and inverter reliability enhancement.

Optimal Capacity Allocation of Energy Storage in Distribution Networks Considering Active/Reactive Coordination

Energy storage system (ESS) has been advocated as one of the key elements for the future energy system by the fast power regulation and energy transfer capabilities. In particular, for distribution networks with high penetration of renewables, ESS plays an important role in bridging the gap between the supply and demand, maximizing the benefits of renewables and providing various types of ancillary services to cope the intermittences and fluctuations, consequently improving the resilience, reliability and flexibility. To solve the voltage fluctuations caused by the high permeability of renewables in distribution networks, an optimal capacity allocation strategy of ESS is proposed in this paper. Taking the life cycle cost, arbitrage income and the benefit of reducing network losses into consideration, a bilevel optimization model of ESS capacity allocation is established, the coordination between active/reactive power of associate power conversion system is considered, and the large scale nonlinear programming problem is solved using genetic algorithm, simulated annealing and mixed integer second-order cone programming method. The feasibility and effectiveness of the proposed algorithm have been verified.

Three dimensional undrained bearing capacity analysis of laterally loaded pile in heterogeneous marine deposits

In this study, the ultimate undrained bearing capacity of a laterally loaded pile, embedded in heterogeneous marine deposits exhibiting linear and bi-linear undrained shear strength profiles was calculated by putting into practice the three-dimensional finite element lower bound limit analysis in conjunction with the second-order cone programming method. The lower bound finite element bearing capacity of a laterally loaded single pile can be found by formulating the element equilibrium, discontinuity equilibrium, boundary conditions, yield function, and optimization of the objective function through these constraints by second-order cone programming method. A parametric study is also conducted to evaluate the most effective factors including the dimensionless pile embedded length ratio, soil strength parameter, heterogeneity factors, transformation depth and soil–pile interface adhesion. The findings of the current study have been compared and validated with the results reported in the literature and presented in the form of tables and figures. The results show that that the effects of heterogeneity, linear and bi-linear variations of undrained shear strength with depth are more meaningful for soft marine clay in comparison to stiff clay and the curves of heterogeneous to homogeneous bearing capacity ratio reach to the unit value by increasing the dimensionless cohesion factor.

An Inexact Augmented Lagrangian Method for Second-Order Cone Programming with Applications

In this paper, we adopt the augmented Lagrangian method (ALM) to solve convex quadratic second-order cone programming problems (SOCPs). Fruitful results on the efficiency of the ALM have been estab...

Coordination of Active and Reactive Power in Active Distribution Networks Based on Successive Linear Approximation

Active distribution network (ADN) is an important part of the future power market, which undertakes the mission of accommodating high penetration distributed generation. However, traditional dispatch mode of the distribution system cannot accommodate high-penetrated renewable energy. In this paper, a novel optimal scheduling model is developed to optimize the accommodation of distributed renewable energy by coordinating active and reactive power. The proposed model takes capacitor bank (CB), energy storage systems (ESS) and static var generator (SVG) as management devices, optimize the day-ahead dispatchable resources of active distribution system in a centralized manner. Different from second-order cone programming method that cannot cooperate with the maximum DG penetration objective, we apply the successive linear approximation method to transform the original non-convex and nonlinear optimal dispatch model with AC balance equation into a mixed-integer linear programming (MILP) model, so that the model can be solved efficiently and accurately by commercial optimization software. A real distribution system in Nantong is employed to verify the proposed model.

An improved solution method via the pole-transformation process for the maximum-crossrange problem

The maximum-crossrange problem is difficult to solve rapidly and stably because of highly nonlinear dynamics and nonconvex constraints. In this paper, an improved successive second-cone programming method via the pole-transformation process is proposed. The main contribution of this work is twofold. Firstly, the pole-transformation coordinate frame is defined, and the original problem is reformulated by the pole-transformation process to avoid the linearization of downrange and crossrange angles. The terminal state constraint is relaxed to an inequality constraint, which improves the stability performance of the solution method. Secondly, an adaptively adjustment method of the trust-region contraction coefficient is proposed to improve the convergence performance of the solution method. The optimal solution of the original problem is obtained by the improved successive second-order cone programming method. The numerical experiments verify the effectiveness of the pole-transformation process, and the convergence and stability performances of the improved solution method proposed in this paper.

Spiral-diving trajectory optimization for hypersonic vehicles by second-order cone programming

Spiral maneuvering targets are well known to be very difficult to intercept due to its high maneuverability and unpredictability. This paper focuses on the optimization of a spiral-diving trajectory for a hypersonic vehicle to strike a stationary target. Second-order cone programming (SOCP), a subclass of convex optimization, is applied to achieve this optimization task. First, based on a detailed analysis of spiraling nature, the dynamics are partially reconstructed to better formulate the spiraling motion. Then, a nonconvex optimal control problem is formulated with the maximum impact velocity as a performance index. Constraints on the states, controls, and terminal conditions are helping to shape a feasible and practical spiraling trajectory. This nonconvex problem is convexified and subsequently discretized in a suitable form so that it can be easily solved in polynomial time using the existing primal-dual interior-point algorithm. In addition, a relaxation technique is used to convexify control constraints and is theoretically proved to be valid. The high reliability and efficiency of the successive SOCP method are verified by numerical examples and comparisons with other methods.

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.