Mixed-integer Second-order Cone Programming Research Articles

Privacy-Preserving Optimal Dispatch for an Integrated Power Distribution and Natural Gas System in Networked Energy Hubs

Energy hubs (EHs) are deployed to supply multi-energy demands economically in the integrated energy systems while considering physical and operational constraints of individual energy systems. This paper presents a decentral framework for the optimal dispatch of integrated power distribution and natural gas system (IDGS) in networked energy hubs. The IDGS operator (IDGSO) is charged with delivering a constrained optimal dispatch while preserving the privacy of participants’ data. First, a comprehensive optimization model is proposed in this paper using the mixed-integer second-order cone programming and sequential second-order cone (SOC) algorithms to guarantee the exactness of SOC relaxation. In this optimization process, we consider advanced control methods, such as remotely controlled switches and static var compensators, and propose two improvements including initial point setting and a special type of cutting plane to obtain a fast and feasible solution. Next, we apply Benders decomposition for decentralizing the decision-making process in accordance with the practical leader–follower relationship between IDGSO and EHOs (EH operators) and preserving private operation data of individual EHOs. Finally, we present numerical case studies, which illustrate the effectiveness of the proposed iterative model and solution algorithm for enhancing the operation of EHs.

Coordinating dynamic network reconfiguration with ANM in active distribution network optimisation considering system structure security evaluation

The growing penetration of renewable energy resources in distribution networks demands for more active management tools. As a new controllable resource, dynamic network reconfiguration (DNR) can improve the flexibility of active distribution network, and in turn decreasing the operation cost of the distribution network and mitigating renewable distributed generation (RDG) curtailment. In this study, an optimisation model coordinating DNR with active network management (ANM) strategies is established, and the system structure security is evaluated. Considering the coordination of scheduling resources such as DNR switches, RDG active/reactive outputs, demand response and static var compensator, the model aims to minimise the comprehensive operation cost of distribution network and improve the consumption rate of RDG while satisfying the distribution network operation constraints. The model transformation method based on the second-order cone relaxation and variable substitution linearisation is proposed, and the original no-convex optimisation model is transformed into the mixed-integer second-order cone programming problem. Finally, the extended IEEE 33-node distribution network is utilised to conduct simulation calculation, and the results demonstrate the validity of the proposed model and its transformation method.

Unit Commitment With Gas Network Awareness

Recent changes in the fuel mix for electricity generation and, in particular, the increase in Gas-Fueled Power Plants (GFPP), have created significant interdependencies between the electrical power and natural gas transmission systems. However, despite their physical and economic couplings, these networks are still operated independently, with asynchronous market mechanisms. This mode of operation may lead to significant economic and reliability risks in congested environments as revealed by the 2014 polar vortex event experienced by the northeastern United States. To mitigate these risks, while preserving the current structure of the markets, this paper explores the idea of introducing gas network awareness into the standard unit commitment model. Under the assumption that the power system operator has some (or full) knowledge of gas demand forecast and the gas network, the paper proposes a tri-level mathematical program where natural gas zonal prices are given by the dual solutions of natural-gas flux conservation constraints and commitment decisions are subject to bid-validity constraints that ensure the economic viability of the committed GFPPs. This tri-level program can be reformulated as a single-level Mixed-Integer Second-Order Cone program which can then be solved using a dedicated Benders decomposition. The approach is validated on a case study for the Northeastern United States [1] that can reproduce the gas and electricity price spikes experienced during the early winter of 2014. The results on the case study demonstrate that gas awareness in unit commitment is instrumental in avoiding the peaks in electricity prices while keeping the gas prices to reasonable levels.

Mixed‐integer second‐order cone programming framework for optimal scheduling of microgrids considering power flow constraints

Microgrids have been recognised as one of the most promising but challenging research topics over the last decade. The optimal energy scheduling problem is regarded as the most essential aspect in the tertiary level control in microgrids. However, most existing centralised or distributed scheduling models only focus on the logical and dynamical feature of microgrids’ operation or the non-linear power flow constraint, which would overact the system performance without considering appropriate unit commitment requirements. Moreover, applying decomposition and iteration technics to complex scheduling problems would encounter convergence issues. To address this concern, this study presents a mixed integer linear reformulation to characterise the operation of different controllable devices and convex relaxation techniques to cope with non-linear power flow constraints, leading to a mixed integer second-order cone programming framework in a concordant yet computationally efficient pattern, capturing non-linear, logical and dynamical properties of the optimal energy scheduling problem in microgrids. The effectiveness of the proposed framework is validated on the IEEE 33-bus distribution network with both grid-connected and islanded modes.

Conic relaxation approaches for equal deployment problems

An important problem in the breeding of livestock, crops, and forest trees is the optimum selection of genotypes that maximize genetic gain. The key constraint in the optimal selection is a convex quadratic constraint that ensures genetic diversity, so that optimal selection can be cast as a second-order cone programming (SOCP) problem. Yamashita et al. (2015) exploit the structural sparsity of the quadratic constraints and reduce the computation time drastically while attaining the same optimal solution.This paper is concerned with a special case of equal deployment (ED), in which we solve the optimal selection problem with the constraint that contribution of genotypes must either be a fixed size or zero. This involves a form of combinatorial optimization, and the ED problem can be described as a mixed-integer SOCP problem.In this paper, we discuss conic relaxation approaches for the ED problem based on LP (linear programming), SOCP, and SDP (semidefinite programming). We then propose a steepest-ascent method that combines the solution obtained from the conic relaxation with a concept from discrete convex optimization, in order to acquire an approximate solution for the ED problem in a practical time. From numerical tests, we observed that among the LP, SOCP, and SDP relaxation problems, SOCP gave a suitable solution from the viewpoint of the optimality and computation time. The steepest-ascent method starting from the SOCP solution provides high-quality solutions much faster than other existing methods that have been widely used for optimal selection.

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

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

Optimal Scheduling of Water Distribution Systems

With dynamic electricity pricing, the operation of water distribution systems (WDS) is expected to become more variable. The pumps moving water from reservoirs to tanks and consumers can serve as energy storage alternatives if properly operated. Nevertheless, optimal WDS scheduling is challenged by the hydraulic law, according to which the pressure along a pipe drops proportionally to its squared water flow (WF). The optimal water flow (OWF) task is formulated here as a mixed-integer nonconvex problem incorporating flow and pressure constraints, critical for the operation of fixed-speed pumps, tanks, reservoirs, and pipes. The hydraulic constraints of the OWF problem are subsequently relaxed to second-order cone constraints. To restore feasibility of the original nonconvex constraints, a penalty term is appended to the objective of the relaxed OWF. The modified problem can be solved as a mixed-integer second-order cone program, which is analytically shown to yield WDS-feasible minimizers under certain sufficient conditions. Under these conditions, by suitably weighting the penalty term, the minimizers of the relaxed problem can attain arbitrarily small optimality gaps, thus providing OWF solutions. Numerical tests using real-world demands and prices on benchmark WDS demonstrate the relaxation to be exact even for setups where the sufficient conditions are not met.

Data-adaptive robust unit commitment in the hybrid AC/DC power system

This paper proposes the data-adaptive robust optimization for the optimal unit commitment in the hybrid AC/DC power system. With the convexified branch flow model interconnecting the AC and DC power grid, the unit commitment problem in the hybrid AC/DC power system is formulated as a mixed-integer second-order cone programming. Considering the temporal and spatial correlations of multiple wind farms, the data-adaptive uncertainty set and the extreme scenarios are introduced to reformulate the robust optimization. The column-and-constraints generation algorithm is adopted to solve the multi-scenario problem. Case studies in the modified IEEE 14-bus system and the Henan provincial power system demonstrate the applicability of the proposed model. Comparative results with the pure AC power system show the improvement of the flexibility by DC interconnections. Both the operational cost and the times of generator startup/shutdown are reduced. The regulation capability of the DC lines can be fully utilized to cope with the uncertainties introduced by wind power.

Towards Resilient Operation of Multimicrogrids: An MISOCP-Based Frequency-Constrained Approach

High penetration of distributed energy resources (DERs) is transforming the paradigm in power system operation. The ability to provide electricity to customers while the main grid is disrupted has introduced the concept of microgrids with many challenges and opportunities. Emergency control of dangerous transients caused by the transition between the grid-connected and island modes in microgrids is one of the main challenges in this context. To address this challenge, this paper proposes a comprehensive optimization and real-time control framework for maintaining frequency stability of multi-microgrid networks under an islanding event and for achieving optimal load shedding and network topology control with AC power flow constraints. The paper also develops a strong mixed-integer second-order cone programming (MISOCP)-based reformulation and a cutting plane algorithm for scalable computation. We believe this is the first time in the literature that such a framework for multi-microgrid network control is proposed, and its effectiveness is demonstrated with extensive numerical experiments.

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.

Multistage large-scale charging station planning for electric buses considering transportation network and power grid

With the applications of electric buses (e-buses), potential solutions to problems related to infrastructures for charging e-buses are emerging. This study particularly focused on large-scale fast charging-station planning for e-buses in the public transportation electrification process, according to the characteristics of e-bus operation and plug-in fast charging mode. We conducted an interdisciplinary study to optimize planning jointly under the transportation system and power grid. In addition to capturing the spatiality of the e-bus charging service network, we further considered temporality in order to conduct long-term planning in view of the continuously growing e-bus charging demand. A spatial-temporal model, which determines the sites and sizes of e-bus charging stations, was proposed and the strategies for multistage infrastructure planning were put forward. The model was equivalently transformed into a mixed-integer second-order cone programming with high computational efficiency. The model and the multistage planning strategies were justified through a series of numerical experiments. A case study of Shenzhen, China was implemented and the robustness of the model to plan changes was studied.

Multi-facility green Weber problem

Locating facilities to satisfy the demands of customers is a strategic decision for a distribution system. In this article, we study the multi-facility green Weber problem (MF-GWP), an extension of the classical multi-facility Weber problem, that considers environmental concerns in a distribution system in the context of a planar facility location problem. In the MF-GWP, the vehicles are sent directly from the facilities to the assigned customers to satisfy their demands. Each customer has a deadline and the vehicles serving the customer must arrive at the location of the customer no later than the deadline. The MF-GWP determines the locations of p facilities on the plane, p > 1, allocations of customers to the facilities, and the speeds of the distribution vehicles so as to minimize the total amount of CO2 emission in the distribution system. We formulate this problem as a mixed integer second order cone programming (MISOCP) problem. This formulation turns out to be weak and therefore only small size instances can be solved to optimality within four hours. For larger size instances, a local search heuristic is proposed and some well-known heuristics developed for the multi-facility Weber problem, namely “location-allocation”, “transfer follow-up”, and “decomposition” are adapted for the MF-GWP. We use second order cone programming (SOCP) and the proposed MISOCP formulation as subproblems within the heuristics. We provide our computational experiments to compare the proposed solution methods in terms of solution quality and time. The results show that within a fixed computational time, even though the location-allocation heuristic is able to make more replications, the improvement heuristics considered, i.e., transfer or transfer followed by decomposition, usually find better solutions while using less number of replications. We also investigate how the total amount of CO2 emitted by distribution vehicles changes with respect to the number of facilities located. We argue that in several real life applications from different sectors including aviation and robotics, MF-GWP and its extensions or modifications can be used to reduce the CO2 emission or energy consumption. As an illustrative example, we show the applicability of the MF-GWP within an assembly line system, where the stations are fed by dedicated rail-guided vehicles.

Optimal Portfolio of Distinct Frequency Response Services in Low-Inertia Systems

A reduced level of system inertia due to renewable integration increases the need for cost-effective provision of ancillary services, such as Frequency Response (FR). In this paper we propose a closed-form solution to the differential equation describing frequency dynamics, which allows to obtain frequency-security algebraic constraints to be implemented in optimisation routines. This is done while considering any finite number of FR services with distinguished characteristics, such as different delivery times and activation delays. The problem defined by these frequency-security constraints can be formulated as a Mixed-Integer Second-Order Cone Program (MISOCP), which can be efficiently handled by off-the-shelf conic optimisation solvers. This paper also takes into account the uncertainty in inertia contribution from the demand side by formulating the frequency-security conditions as chance constraints, for which an exact convex reformulation is provided. Finally, case studies highlighting the effectiveness of this frequency-secured formulation are presented.

Multi-period Market Operation of Transmission-Distribution Systems Based on Heterogeneous Decomposition and Coordination

The integration of shiftable/curtailment distribution generators (DGs) along with quick-response storage has not only increased the transaction’s flexibility but also puzzled the bidding willingness of transmission-connected market players (TMPs). In this paper, the method of heterogeneous decomposition and coordination (HGDC) is applied to decompose the integrated transmission-distribution market framework into a bi-level problem with a transmission wholesale market master problem and several distribution retail market subproblems in a decentralized organization structure. The price-based bidding willingness of demand-side resources’ (DSRs’) aggregator is simulated considering the relation between distribution system operator’s (DSO’s) operation modes and its equivalent market transactive price. Besides the traditional mixed-integer linear programming (MILP) model, the active reconfiguration model of DSOs based on mixed-integer second-order conic programming (MI-SOCP) is proposed to rearrange its operation switch status and elaborate its operation cost according to the market transaction. Multi-period optimal operation modes could be obtained through an HGDC-based iteration process by coordinating the transmission system operator (TSO) with DSOs and considering the market energy equilibrium and reserve requirements for security considerations. Karush-Kuhn-Tucker (KKT) conditions are used to testify the optimality and convergence of the bi-level model in theory. The T5-3D33 case is employed to illustrate the efficiency of the proposed model and method.

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.

Chance-constrained optimal dispatch of integrated electricity and natural gas systems considering medium and long-term electricity transactions

A novel stochastic optimal dispatch model is proposed considering medium and long-term electricity transactions for a wind power integrated energy system using chance constrained programming. The electricity contract decomposition problem is introduced into the day-ahead optimal dispatch plan formulation progress. Considering the case that decomposition results may be not executable in the dispatch plan, a coordinated optimization strategy based on the Lagrange multiplier is proposed to eliminate the non-executable electric quantity. At the same time, the uncertainties and correlation of wind power are considered in the dispatch model, and the original stochastic dispatch problem is transformed into a mixed integer second-order cone programming problem using second-order cone relaxation and sample average approximation approach. Case study results demonstrate the validity of the proposed method.

Multi-Timescale Voltage Stability-Constrained Volt/VAR Optimization With Battery Storage System in Distribution Grids

This paper proposes multi-timescale voltage stability-constrained volt/VAR optimization to improve the steady-state stability and security of smart distribution grids. It aims to coordinate the post-contingency slow-timescale corrective actions provided by an on-load tap changer with fast-timescale corrective actions provided by photovoltaic inverters and battery energy storage system. This multi-timescale operation exploits the difference between short- and long-term emergency ratings of line flows and voltage magnitudes usually defined by utilities for contingency cases. Global optimum is achieved by formulating the problem as mixed-integer second-order cone program. To justify the investment in battery storage, the proper coordination between base case state-of-charge and storage dynamic reserve margin required for post-contingency corrective actions is performed. Also, the steady-state voltage stability margin is computed through detailed voltage stability analysis. Simulation results demonstrate the effectiveness of the proposed multi-timescale corrective actions in maintaining the steady-state stability and security under contingency cases.

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.

Efficient Distribution System Optimal Power Flow With Discrete Control of Load Tap Changers

Load tap changers (LTCs) are commonly used for voltage control in distribution systems. The mechanical relays on LTCs are controlled in discrete steps; hence, distribution optimal power flow (DOPF) formulations require LTCs to be modeled with integer variables. Integer variables are generally ignored or relaxed and rounded off to reduce the computational burden in DOPF models. However, in this paper, we highlight and overcome complex infeasibility issues caused by rounding methods. Moreover, recent advances in convex optimization have improved the computational tractability of the DOPF problem. In this paper, we develop and analyze numerical efficiency of different but exact formulations for incorporating LTCs as integer control assets in a relaxed second-order cone program (SOCP) version of the DOPF model. The resulting formulation becomes a mixed-integer SOCP (MISOCP), which is computationally tractable compared to mixed-integer non-linear program (MINLP). To recover the exact optimal solution in the proposed MISOCP-DOPF model, a McCormick relaxation is employed within a sequential bound-tightening algorithm. We show that solutions obtained from MISOCP-DOPF are always ac feasible. Thus, the MISOCP-DOPF yields optimal and realistic ac solutions, and is validated in large distribution feeders and is compared to MINLP counterpart.