Order Cone Research Articles

An optimal transmission line switching and bus splitting heuristic incorporating AC and N-1 contingency constraints

Optimal transmission line switching and/or bus splitting is shown to contribute in relieving congestion and reducing the operation cost by rerouting power flows throughout the network. Although bus splitting may be as powerful as line switching in congestion mitigation and is typically considered a smaller disturbance compared with line switching, it has received less attention in the literature in part due to the more complicated node-breaker modeling requirement. In this paper, an optimal transmission line switching and bus splitting heuristic is presented to minimize the operation cost while respecting AC and N-1 contingency constraints. We present a two-level solution method where switching decisions are made in the upper level problem formulated as a mixed integer second order cone programming master problem, while the resulting network topology is checked against AC and N-1 contingency constraints in lower level subproblems. Line switching and bus splitting are modeled as switching actions assuming double-bus double-breaker substation arrangements where all elements at a substation, including generators, loads, lines and shunt elements, are given switches to connect to either of the busbars if the respective substation is split. We also introduce additional constraints to model a breaker-and-a-half substation scheme. Furthermore, a pre-screening step is presented to limit the search space of the problem, thus accelerating the solution process. We demonstrate the application of the proposed method on IEEE standard test systems.

Active lateral earth pressure of geosynthetic-reinforced retaining walls with inherently anisotropic frictional backfills subjected to strip footing loading

In this paper, a detailed numerical study is conducted to evaluate the lateral earth pressure acting on geosynthetic-reinforced retaining walls with an anisotropic granular backfill subjected to strip footing loadings. To this end, the well-established lower bound theory of limit analysis coupled with the robust second order cone programming (SOCP) and the finite element discretization method is exploited and implemented in the stability analysis of reinforced retaining structures. For the finite element limit analysis, a number of constraints associated with the lower-bound axioms are satisfied, including element equilibrium, discontinuity equilibrium, boundary conditions and the yield criterion enforcement. By adopting second-order cone programming (SOCP) optimization, the nonlinear Mohr-coulomb failure criterion is simulated using three nodal auxiliary variables defined as functions of nodal stresses generated at each point. In addition, the primal-dual interior-point algorithm is adopted to gain the optimal solution for the unknown stress variables in the SOCP optimization problem. Accordingly, the contribution of soil inherent anisotropy to the influence ofa number of parameters on the lateral earth pressure is thoroughly examined. It was observed that as the anisotropy ratio increases (the horizontal friction angle decreases) and the number of reinforcement layers decreases, the coefficient of active earth pressure increases in all cases of geosynthetic-reinforced retaining structure. Decreasing the number of reinforcement layers in the retained backfill will be translated into an equivalent softened material; hence, increasing the active earth pressure coefficient. In addition, the increase in the anisotropy ratio leads to the overall decrease in the shear strength of the backfill soil, causing the retaining structure to reach the limit state earlier at smaller displacements, thus giving rise to the increase in the coefficient of active lateral earth pressure. The rate of increase in the coefficient of active earth pressure with anisotropy ratio grows with the increase in the foundation width and load intensity and the decrease in the foundation-wall distance.

Differential equations with multiple sign changing convolution coefficients

For continuous functions [Formula: see text] and [Formula: see text], which are allowed to change sign, we consider the nonlocal differential equation [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] represents the finite convolution of the functions [Formula: see text] and [Formula: see text]. A model case is the equation [Formula: see text] The existence of at least one positive solution to these problems subjected to a variety of boundary conditions is studied. Due to the use of a nonstandard order cone we are able to achieve our results without having to assume that the coefficients [Formula: see text] and [Formula: see text] are strictly positive.

Partition-based distributionally robust optimization via optimal transport with order cone constraints

In this paper we wish to tackle stochastic programs affected by ambiguity about the probability law that governs their uncertain parameters. Using optimal transport theory, we construct an ambiguity set that exploits the knowledge about the distribution of the uncertain parameters, which is provided by: (1) sample data and (2) a-priori information on the order among the probabilities that the true data-generating distribution assigns to some regions of its support set. This type of order is enforced by means of order cone constraints and can encode a wide range of information on the shape of the probability distribution of the uncertain parameters such as information related to monotonicity or multi-modality. We seek decisions that are distributionally robust. In a number of practical cases, the resulting distributionally robust optimization (DRO) problem can be reformulated as a finite convex problem where the a-priori information translates into linear constraints. In addition, our method inherits the finite-sample performance guarantees of the Wasserstein-metric-based DRO approach proposed by Mohajerin Esfahani and Kuhn (Math Program 171(1–2):115–166. https://doi.org/10.1007/s10107-017-1172-1, 2018), while generalizing this and other popular DRO approaches. Finally, we have designed numerical experiments to analyze the performance of our approach with the newsvendor problem and the problem of a strategic firm competing à la Cournot in a market.

Nonlocal differential equations with concave coefficients of convolution type

For a continuous function A:R→R the existence of at least one positive solution of the nonlocal differential equation −A∫01(g∘u)(s)dsu′′(t)=λf(t,u(t)),t∈(0,1)is considered in the specific case in which g is a concave, strictly increasing continuous function. As a model case we consider −A∫01|u(s)|qdsu′′(t)=λf(t,u(t)),t∈(0,1),where 0<q<1. The results rely on the use of a nonstandard order cone together with an application of topological fixed point theory. Due to this the coefficient function A is neither required to be strictly positive nor monotone nor satisfy any particular growth condition. As a consequence of the theory developed for the preceding problem we also provide results for the nonlocal equation −A((b∗(g∘u))(1))u′′(t)=λf(t,u(t)),t∈(0,1)where t↦(b∗(g∘u))(t) is a finite convolution.

A Full-Newton Step Interior Point Method for Fractional Programming Problem Involving Second Order Cone Constraint

Some efficient interior-point methods (IPMs) are based on using a self-concordant barrier function related to the feasibility set of the underlying problem.Here, we use IPMs for solving fractional programming problems involving second order cone constraints. We propose a logarithmic barrier function to show the self concordant property and present an algorithm to compute $\varepsilon-$solution of a fractional programming problem. Finally, we provide a numerical example to illustrate the approach.

Cell-Free Massive MIMO: Joint Maximum-Ratio and Zero-Forcing Precoder With Power Control

Cell-free massive multiple-input multiple-output (MIMO) system is a promising architecture for next generation wireless systems by deploying a very large number of distributed access points (APs), which simultaneously serve a smaller number of user equipments (UEs) over the same time-frequency resources. It guarantees uniformly good service at high spectral efficiency with simple linear precoding techniques and max-min power control. In this article, we propose a new joint maximum-ratio and zero-forcing (JMRZF) precoding scheme, where part of APs are combined to perform centralized zero-forcing (ZF), while other APs apply simple maximum-ratio transmission (MRT). Our proposed precoder offers an adaptable trade-off between the spectral efficiency and front-haul signalling overhead. A corresponding AP subset selection scheme is also proposed which is based on large-scale fading coefficients. A closed-form expression for the achievable spectral efficiency of our proposed scheme is derived, which represents a generalized result including both fully distributed MRT and fully centralized ZF cases. Based on this closed-form expression, max-min power control is formulated and solved via the second order cone and first order methods. The former can obtain the global optimal solution, but its computational complexity is very high. On the other hand, the latter technique is sub-optimal, yet, it has very low computational complexity. Hence, it is suitable for large-scale cell-free massive MIMO systems with hundreds or thousands of APs and users. Numerical results show that our proposed JMRZF scheme can substantially outperform the local precoding schemes, even when a small part of APs are combined to deploy ZF and is implementable even when each AP has very few antennas. In addition, it is shown that our max-min power controls improves the spectral efficiency significantly, compared to the uniform power control scheme.

Optimal Operation for Integrated Electricity and Natural Gas Systems Considering Dynamic Gas Linepack

With the increasing penetration of gas-fired turbines, the efficient and safe operation of power systems depends on the sufficient supply of natural gas. Hence, we proposed a coordinated optimal operation model of integrated electricity and natural gas systems (IENs). The gas linepack is considered by applying dynamic flow models. The non-convex Weymouth equality is converted to a second-order cone inequality constraint and a concave constraint that can be linearized by Taylor expansion. The initial operation point is obtained by optimizing the steady flow model. We proposed an algorithm based on the sequential strategy to solve this mixed-integer second order cone programming (MISOCP). The case study demonstrates that the proposed method can solve the MISOCP effectively and the operation cost can be reduced by taking full use of the gas linepack that is calculated by the dynamic gas flow model.

A rigorous numerical formulation for upper bound analysis of reinforced soils using second order cone programming

A rigorous upper bound formulation is established for reinforced soils considering both the tensile rupture of the reinforcement and the relative slippage of the reinforcement-soil interface. To represent its finite tensile strength and negligible compressive strength, a novel strategy is proposed to calculate the plastic dissipation rate of the reinforcement without the incorporation of stress variables. Plastic dissipation rates of the soil, the reinforcement and their interfaces are obtained using only kinematic variables and all flow rules are expressed in terms of linear constraints and second order cones. The solution domain is then discretized using linear strain elements for the soil and constant strain elements for the reinforcement and the interface. Numerical examples are given to show the accuracy of the present formulation. The effect of design parameters such as the tensile strength, the length and the location of the reinforcement is discussed.

A control optimization model for CVaR risk of distribution systems with PVs/DSs/EVs using Q-learning powered adaptive differential evolution algorithm

Distributed generation and energy storage brings opportunities and risks of distribution systems, such as greater power loss and abnormal voltage fluctuation. A Q-learning powered optimization model for these risk control of is presented in this paper. Considering the uncertainties of output power of distributed generation systems, charging and discharging power of electric vehicles and energy storage devices, a CVaR-based energy risk control model for distribution system with renewable energy is presented to determine the control value of output power of distributed generation systems. Q-learning powered adaptive differential evolution algorithm is used to solve the proposed optimization problem. Second order cone programming is used to simplify the objective function and constraints of the optimization model, and Q-learning driven adaptive differential evolution algorithm is used to enhance the ability of solving, simplify the calculation, and make the solution faster and more stable. The feasibility and applicability of the proposed model and algorithm are verified by simulating IEEE-118 distribution system.

A dual simplex-type algorithm for the smallest enclosing ball of balls

We define the notion of infimum of a set of points with respect to the second order cone. This problem can be showed to be equivalent to the minimum ball containing a set of balls problem and to the maximum intersecting ball problem, as well as others. We present a dual algorithm which can be viewed as an extension of the simplex method to solve this problem.

A topological approach to a class of one-dimensional Kirchhoff equations

We consider, for q &gt; 1 q&gt;1 , the one-dimensional Kirchhoff-type problem − A ( ∫ 0 1 ( u ′ ( s ) ) q d s ) u ( t ) a m p ; = λ f ( u ( t ) ) , t ∈ ( 0 , 1 ) u ( 0 ) a m p ; = 0 u ′ ( 1 ) a m p ; = 0 \begin{equation} \begin {split} -A\left (\int _0^1\big (u’(s)\big )^q\ ds\right )u(t)&amp;=\lambda f\big (u(t)\big )\text {, }t\in (0,1)\\ u(0)&amp;=0\\ u’(1)&amp;=0\notag \end{split} \end{equation} and demonstrate the existence of at least one positive solution to this problem. The main contribution is to show that by using a nonstandard order cone together with topological fixed point theory much weaker conditions than usual can be imposed on the coefficient function A A .

Stochastic Expansion Planning of Various Energy Storage Technologies in Active Power Distribution Networks

This work aims to minimize the cost of installing renewable energy resources (photovoltaic systems) as well as energy storage systems (batteries), in addition to the cost of operation over a period of 20 years, which will include the cost of operating the power grid and the charging and discharging of the batteries. To this end, we propose a long-term planning optimization and expansion framework for a smart distribution network. A second order cone programming (SOCP) algorithm is utilized in this work to model the power flow equations. The minimization is computed in accordance to the years (y), seasons (s), days of the week (d), time of the day (t), and different scenarios based on the usage of energy and its production (c). An IEEE 33-bus balanced distribution test bench is utilized to evaluate the performance, effectiveness, and reliability of the proposed optimization and forecasting model. The numerical studies are conducted on two of the highest performing batteries in the current market, i.e., Lithium-ion (Li-ion) and redox flow batteries (RFBs). In addition, the pros and cons of distributed Li-ion batteries are compared with centralized RFBs. The results are presented to showcase the economic profits of utilizing these battery technologies.

Chance-constrained optimization for integrated local energy systems operation considering correlated wind generation

Energy hubs, which integrate multiple energy vectors through converters, can enhance the value of Integrated Local Energy Systems (ILES) via increased flexibility and reduced costs. However, uncertain renewable energy and the non-convex, non-linear properties of energy flows complicate the modelling and operation of energy hub systems. This paper develops chance-constrained optimization methods for planning and operation of energy hub systems under uncertainty. The non-linear formulations of power and gas flows are relaxed by convexification methods, leading to a formulation of Second Order Cone Problem (SOCP), which can be efficiently solved to global optimality. The correlation between geographically close wind generators connected to the hub systems is modelled by establishing their relation using Gaussian copula. The proposed chance-constrained optimization is demonstrated on a six-hub system within a multi-vector energy distribution network with 7 electrical buses and 7 gas nodes. The value of different levels of system integration through the installation of energy hubs is investigated. The results show that by combining system integration via energy hubs with chance constrained operation, the proposed method can reduce operating costs and increase renewable energy yields, thereby benefitting hub system operators and customers with reduced energy infrastructure investment and energy costs.

Peripheral stability of circular tunnels in anisotropic undrained clay

In the present analysis, solutions have been obtained for evaluating the minimum support pressure to be offered by means of lining for maintaining peripheral stability of circular tunnels constructed in clay taking into account anisotropic undrained shear strength parameters. The finite element approach based on lower bound theorem of limit analysis was implemented for performing the computations and second order cone programming technique has been applied for optimizing the magnitude of support pressure. The solutions obtained in this analysis can be used for computing the support pressure for tunnels driven in anisotropic normally consolidated (NC) or lightly over consolidated (LOC) clay for linear increase in undrained shear strength with depth, and anisotropic heavily over consolidated (HOC) clay where undrained shear strength remains constant with depth. The anisotropic shear strength of clay substantially influences the required support pressure, that is, the support pressure computed using anisotropy strength parameters has been found to be higher for tunnel driven in NC/LOC clay and lower or higher for tunnel driven in HOC clay than that determined on the basis of isotropic strength parameters.

Exact solution approaches for the workload smoothing in assembly lines

In this paper, the problem of minimizing the smoothness index for an assembly line given a fixed cycle time and the number of workstations is studied. This problem which is known as the workload smoothing line balancing problem (WSLBP) is a mixed-integer quadratic programming problem. Until recently, this problem has only been tackled using heuristic approaches. Recently, there have been some attempts to solve this problem exactly using mixed-integer linear programming (MILP). The MILP formulations, however, are not usually capable of solving large size problem instances. In this paper, the aim is to solve the WSLBP using mathematical programming formulations by using off-the-shelf solvers. Differently from the literature some non-MILP formulations are also considered for the problem. For this purpose, three MILP formulations, one from the literature, and two non-MILP formulations are compared. The two non-MILP formulations include a mixed-integer second order cone programming formulation and a constraint programming model. The superiority of the non-MILP formulations over the considered MILP formulations is experimentally shown.

Solving variational inequalities and cone complementarity problems in nonsmooth dynamics using the alternating direction method of multipliers

AbstractThis work presents a numerical method for the solution of variational inequalities arising in nonsmooth flexible multibody problems that involve set‐valued forces. For the special case of hard frictional contacts, the method solves a second order cone complementarity problem. We ground our algorithm on the Alternating Direction Method of Multipliers (ADMM), an efficient and robust optimization method that draws on few computational primitives. In order to improve computational performance, we reformulated the original ADMM scheme in order to exploit the sparsity of constraint jacobians and we added optimizations such as warm starting and adaptive step scaling. The proposed method can be used in scenarios that pose major difficulties to other methods available in literature for complementarity in contact dynamics, namely when using very stiff finite elements and when simulating articulated mechanisms with odd mass ratios. The method can have applications in the fields of robotics, vehicle dynamics, virtual reality, and multiphysics simulation in general.

Stability of unsupported circular tunnels in anisotropic normally and over consolidated saturated clay

The stability of long unsupported tunnel excavated in normally consolidated (NC) and heavily over consolidated (HOC) saturated clay under undrained condition has been assessed. The analysis was performed using finite element formulation of lower bound limit analysis in conjunction with second order cone programming. The influence of both anisotropy and non-homogeneity in undrained shear strength of NC clay and only anisotropy in undrained shear strength of HOC clay was examined. A dimensionless stability factor is defined to assess the tunnel stability in terms of self weight and undrained shear strength of soil mass, and tunnel cover. The variation of stability factor as a function of anisotropic and non-homogeneous strength parameters of soil, surcharge on the ground surface and tunnel cover was established. The anisotropy and non-homogeneity in undrained strength of clay have been found to have significant effect on the stability of tunnels. With reduction in the values of anisotropic and non-homogeneous strength parameters, the magnitude of stability factor decreases.

A mixed integer programming formulation for the stochastic lot sizing problem with controllable processing times

In this study, we address the capacitated stochastic lot-sizing problem under α service level constraints. We assume that processing times can be decreased in return for compression cost that follows a convex function. We consider this problem under the static uncertainty strategy suggesting to determine replenishment plans at the beginning of the planning horizon. We develop an extended mixed integer programming (MIP) formulation built on a predefined piecewise linear approximation. Then, we adopt the so-called dynamic cut generation approach to be able to use the proposed MIP formulation with no prior approximation of the cost function. Also, we demonstrate how to extend the dynamic cut generation approach to consider the exact inventory cost in the objective function. We show the computational performance of the proposed MIP model with the dynamic cut generation approach in an extensive numerical study where second order cone programming formulations developed in the literature are used as benchmark. The results reveal that the proposed MIP model deployed with the dynamic cut generation yields a superior computational performance as compared to the benchmark formulations especially when the order of compression cost function is higher.

Alternate second order conic program reformulations for hub location under stochastic demand and congestion

In this paper, we study the single allocation hub location problem with capacity selection in the presence of congestion at hubs. Accounting for congestion at hubs leads to a non-linear mixed integer program, for which we propose 18 alternate mixed integer second order conic program (MISOCP) based reformulations. Based on our computational studies, we identify the best MISOCP-based reformulation, which turns out to be 20–60 times faster than the state-of-the-art. Using the best MISOCP-based reformulation, we are able to exactly solve instances up to 50 nodes in less than half-an-hour. We also theoretically examine the dimensionality of the second order cones associated with different formulations, based on which their computational performances can be predicted. Our computational results corroborate our theoretical findings. Such insights can be helpful in the generation of efficient MISOCPs for similar classes of problems.