Mixed-integer Second-order Cone Programming Research Articles

Distributionally Robust Unit Commitment in Coordinated Electricity and District Heating Networks

Coordinated operations of electricity and district heating networks offer a potential for mitigating inherent variability of renewable energy sources (RES) in the ongoing transition to smart grids. This paper proposes a two-stage distributionally robust optimization (DRO) approach to determine the optimal day-ahead unit commitment in coordinated electricity and district heating networks with variable RES power output. The proposed formulation is to minimize the worst-case expected total cost over an ambiguity set comprising a family of probability distributions with given support and moments of RES power output. As such, the proposed DRO approach can overcome the limitations of stochastic programming in its inherent dependence of exact probability distributions along with a huge computational burden, but also becomes less conservative than classical robust optimization. The pertinent DRO model is eventually reformulated as a tractable mixed-integer second-order cone (SOC) programming after employing linear decision rules and the SOC duality. Simplified affine policies are utilized to further improve computational tractability and performance. Finally, case studies are conducted based on Barry Island electricity and district heating networks. The numerical results demonstrate the decision-making superiority of the proposed method as compared with deterministic, stochastic programming, and robust optimization approaches. They also validate the computational improvement of the proposed approach by employing simplified affine policies.

A multi-period scheduling strategy for ADN considering the reactive power adjustment ability of DES

Nowadays, the active distribution network (ADN) is facing great challenges on safe operation and coordinated scheduling. To resolve these issues, a novel multi-period active-reactive coordination scheduling strategy efficient solution method is proposed in this paper after addressing the uncertainty of DG and load by probabilistic scenario method. Firstly, based on the second-order cone relaxation (SOCR), the power flow equations are convexized and relaxed to transform the original model into mixed integer second-order cone programming (MISOCP). Then, the controllable resources such as on-load tap changer (OLTC), capacitor banks (CB) and ESS are accurately linearized, so the model is converted to 0–1 MISOCP problem. Finally, compared with the existing strategies, numerical tests on the modified IEEE 33-bus test system demonstrate 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 and the utilization of ESS. Meanwhile, the accuracy and efficiency of the 0–1 MISOCP model are verified through four operating schemes. In addition, the sensitivity analysis of adjustment step and action time of the discrete adjustment devices on optimization results is carried out in detail.

An MISOCP-Based Solution Approach to the Reactive Optimal Power Flow Problem

In this letter, we present an alternative mixed-integer non-liner programming formulation of the reactive optimal power flow (ROPF) problem. We utilize a mixed-integer second-order cone programming (MISOCP) based approach to find global optimal solutions of the proposed ROPF problem formulation. We strengthen the MISOCP relaxation via the addition of convex envelopes and cutting planes. Computational experiments on challenging test cases show that the MISOCP-based approach yields promising results compared to a semidefinite programming based approach from the literature.

Data‐driven distributionally robust joint planning of distributed energy resources in active distribution network

With the increasing penetration of distributed energy resources (DERs) in the active distribution network (ADN), how to enable joint planning of DERs under the uncertainty of distributed generations (DGs) has become a challenging problem. This study establishes a two-stage joint planning model considering doubly-fed induction generator, photovoltaics (PVs) with the ancillary services of PV inverter, distributed energy storage systems and different types of controllable loads in the ADN. To address the uncertainties of DGs, a two-stage data-driven distributionally robust planning model is constructed. The proposed model is solved in a `master and sub-problem' framework by column-and-constraint generation algorithm, where the master problem is to minimise the total cost and find the optimal planning decision under the worst probability distributions, and the sub-problem is to find the worst probability distribution of given uncertain scenarios. Besides, the original mixed-integer non-linear planning problem is converted into a mixed-integer second-order cone programming problem through second-order cone relaxation, Big-M and piecewise linearisation method. The numerical results based on 33-bus system verify the effectiveness of the proposed model.

Optimization of maintenance scheduling for offshore wind turbines considering the wake effect of arbitrary wind direction

An optimization model for offshore wind farm maintenance scheduling is presented, considering minimum maintenance costs and maximum power generation. For power generation, the wind speed at each tower site plays an important role that is impacted not only by the dynamically changing wake overlap area and the consequence of variation of wind direction but also by the relative position of the wake affected by the maintenance status. This paper combines the wake model with the maintenance status to accurately express the input wind speed of the wind turbine (WT) in each period. Because the optimization model includes complex dynamical coupling relationships and a number of nonlinear constraints, mixed integer second-order cone programming (MISOCP) are employed to address these issues. The MISOCP model is relaxed as a mixed integer linear programming (MILP) model to improve computational efficiency and the ε-constraint method is utilized to handle the multi-objective function. The proposed model and method are tested in a short-term maintenance case of an offshore wind farm. The numerical results demonstrate that the proposed approach can achieve sound economic benefits and provide comprehensive decision support.

Optimal generation scheduling in microgrids using mixed-integer second-order cone programming

In this article, a new model is proposed for the operation of microgrids (MGs) using optimal distribution feeder reconfiguration (DFR) and optimal generation scheduling (OGS). The distributed generation units, electric vehicles and demand response (DR) are modelled in the presented model. The DR programs consider load shifting, load curtailing and time-of-use programs. The proposed model is formulated as a mixed-integer second-order cone programming problem and solved by the GAMS software package. The total operating cost is taken into account as the objective function and several operational constraints are used in the optimization problem. The simulation results of the 33-bus MG are shown, where considering OGS along with DFR leads to a lower operating cost. In addition, the load shifting of a DR program can reduce the total operating cost by smoothing the load profile and reducing the cost of power purchases from the main grid.

Service-oriented robust worker scheduling with motivation effects

Due to gradual disappearance of global demographic dividend, the improvement of workforce efficiency becomes increasingly important. It has been commonly recognised that workforce motivation can largely stimulate the workers, especially in manufacturing systems. This paper investigates a worker scheduling problem under given work shifts with workforce motivation effects, where motivation effects are characterised by motivation coefficients of job processing times. We focus on the situation where motivation coefficients are uncertain due to various factors, and only the mean vector and covariance matrix are known. The objective is to maximise the service level, measured by the probability of ensuring no tardy jobs. We first propose a distributionally robust chance constrained formulation with a probabilistic objective function. Then an adapted sample average approximation (SAA) method and a heuristic, based on an approximated mixed integer second-order cone programming (MI-SOCP) model and the idea of problem decomposition, is developed. Numerical results show that the decomposition-based heuristic is more efficient. We also draw some managerial insights.

A Multi-Step Reconfiguration Model for Active Distribution Network Restoration Integrating DG Start-Up Sequences

The ever-increasing penetration of Distributed Generators (DGs) in distribution networks suggests to enable their potentials in better fulfilling the restoration objective. The objective of the restoration problem is to resupply the maximum energy of loads considering their priorities using minimum switching operations. Basically, it is desired to provide a unique configuration that is valid regarding the load and generation profiles along the entire restorative period. However, this unique configuration may not satisfy at the same time: I) the DG start-up requirements at the beginning of the restoration plan and II) the topological conditions that would allow the DG to provide later on the most efficient support for the supply of loads. Therefore, it is proposed in this paper to allow a limited number of reconfiguration steps according to the DG start-up requirements. In addition, this paper presents a novel formulation for the reconfiguration problem that accounts for partial restoration scenarios where the whole unsupplied area cannot be restored. The decision variables of the proposed multi-step restoration problem are: I) the line switching actions at each step of the reconfiguration process, II) the load switching actions during the whole restorative period and, III) the active/reactive power dispatch of DGs during the whole restorative period. A relaxed AC power flow formulation is integrated to the optimization problem in order to ensure the feasibility of the solution concerning the operational safety constraints. The overall model is formulated in terms of a mixed-integer second-order cone programming. Two simulation scenarios are studied in order to illustrate different features of the proposed strategy and to demonstrate its effectiveness particularly in the case of large-scale outages in distribution networks.

Operation of Distribution Network Considering Compressed Air Energy Storage Unit and Its Reactive Power Support Capability

Energy storage is playing an increasingly important role in power system operation due to its ability to shave the peak and fill the valley. Advanced adiabatic compressed-air energy storage (AA-CAES) is a clean and scalable energy storage technology and has attracted wide attention recently. This paper proposes a multi-state operation model of AA-CAES capturing the dynamic change of internal physical status. A stand-by state is introduced, at which AA-CAES works like a condenser that produces only reactive power. It brings two benefits. First, the facility is started up and shut down less frequently, which helps extend its service time. Second, it can provide continuous reactive power support all day long, unlike on-load tap changers or shunt capacitors whose voltage regulation strategy is discrete and suffers from a very low switching frequency. Taking AA-CAES into account, the optimal operation of a distribution network is formulated as a bi-objective dynamic optimal power flow problem. Nash bargaining is employed to extract a single Pareto optimal solution that fairly compromises two objectives. To solve the operation problem in a tractable manner, the bargaining objective and power flow constraints are convexified, giving rise to a mixed-integer second-order cone program. A feasibility recovery algorithm is developed to find a physically meaningful solution whenever the convex relaxation on power flow is inexact. Case studies on the IEEE 33-bus system demonstrate the effectiveness and advantages of the proposed method.

Profit-oriented distributionally robust chance constrained flowshop scheduling considering credit risk

Customer credit risk or payment probability, influenced by factors such as financial conditions and bank policies, has hindered fast Asia-Pacific economic growth. Besides, the working time is usually limited due to regulations and limited resources. Driven by profit, some jobs may be rejected on tactical level and the accepted jobs are scheduled on operational level, respecting the allowed working time. This paper studies a stochastic flowshop scheduling problem, assuming that only the mean and covariance matrix of uncertain payment probabilities and processing times are known. The objective is to maximise the profit level, i.e. the probability of the profit no less than the planned one, while controlling the risk of surpassing the limited working time. A new distributionally robust chance constrained model is proposed. The sample average approximation (SAA) method, the robust SAA method and a hierarchical approach, based on an approximated mixed integer second-order conic program, are developed. Numerical experiments show that the hierarchical approach is more efficient. Moreover, some managerial insights are drawn.

Robust chance-constrained programming approach for the planning of fast-charging stations in electrified transportation networks

In this paper, a bi-level programming model is established to address the planning issues of fast-charging stations in electrified transportation networks with the consideration of uncertain charging demands. The capacitated flow refueling location model is considered in the upper level to minimize the planning cost of fast-charging stations while the traffic assignment model is utilized in the lower level to determine the spatial and temporal distribution of plug-in electric vehicle flows over entire transportation networks. Such bi-level model unveils the inherent relationship among charging demands, electrical demands and the spatial and temporal distribution of plug-in electric vehicle flows. Robust chance constraints are formulated to characterize the service abilities of fast-charging stations under distribution-free uncertain charging demands, where the ambiguity set is constructed to estimate the potential values of the uncertainties based on their moment-based information, such that the robust chance constraints can exactly be reduced to mixed integer linear constraints. By introducing new variables, the bi-level model is then reformulated into a single-level mixed integer second-order cone programming model so as to be solved via off-the-shelf solvers, which guarantee the optimality of the solution. A case study is conducted to illustrate the effectiveness of the proposed planning model, which reveals three critical factors that significantly impact the planning outcomes.

Distributed Self-Healing Scheme for Unbalanced Electrical Distribution Systems Based on Alternating Direction Method of Multipliers

With the advent of Smart Grids and advanced communication technologies, the self-healing scheme has become a desirable function of the operation and planning of electrical distribution systems (EDSs). In the presence of a permanent fault, an optimized self-healing scheme minimizes the unsupplied demand while maintaining the faulted section of the network isolated. The service restoration of the self-healing scheme is a combinatorial optimization problem whose computational complexity grows exponentially with the number of binary variables. To resolve this issue, a distributed optimal service restoration strategy is developed based on the alternating direction method of multipliers (ADMM). The service restoration problem is formulated as a mixed-integer second-order cone programming (MISOCP) problem. The decision variables of the problem are the status of the remote-controlled switches, load zones and load shedding at each controllable demand. Operational constraints, such as current and voltage magnitude constraints, distributed generation (DG) capacity constraints and radial topology constraints, are respected in the optimization problem. Through the ADMM, the optimization problem is distributed among the zones of the EDS, without requiring a central controller. Two test systems, an unbalanced 44-node system and the IEEE 123-node system, were used to conduct case studies. Results show that the proposed method can provide optimal service restoration solutions in reasonable time without a central controller.

Pricing inertia and Frequency Response with diverse dynamics in a Mixed-Integer Second-Order Cone Programming formulation

Low levels of system inertia in power grids with significant penetration of non-synchronous Renewable Energy Sources (RES) have increased the risk of frequency instability. The provision of a certain type of ancillary services such as inertia and Frequency Response (FR) is needed at all times, to maintain system frequency within secure limits if the loss of a large power infeed were to occur. In this paper we propose a frequency-secured optimisation framework for the procurement of inertia and FR with diverse dynamics, which enables to apply a marginal-pricing scheme for these services. This pricing scheme, deduced from a Mixed-Integer Second-Order Cone Program (MISOCP) formulation that represents frequency-security constraints, allows for the first time to appropriately value multi-speed FR.

Distributionally Robust Chance-Constrained Generation Expansion Planning

This article addresses a centralized generation expansion planning problem, accounting for both long- and short-term uncertainties. The long-term uncertainty (demand growth) is modeled via a set of scenarios, while the short-term uncertainty (wind power generation) is described by a family of probability distributions with the same first- and second-order moments obtained from historical data. The resulting model is a distributionally robust chance-constrained optimization problem, which selects the conventional generating units to be built among predefined discrete options. This model includes a detailed representation of unit commitment constraints. To achieve computational tractability, we use a tight relaxation approach to convexify unit commitment constraints and solve the model with linear decision rules, resulting in a mixed-integer second-order cone program. It is observed that the proposed model exhibits better out-of-sample performance in terms of total expected system cost and its standard deviation compared to a chance-constrained model that assumes a Gaussian distribution of short-term uncertainty. A similar observation is made when comparing the proposed model against a chance-constrained program that uses empirical renewable power generation data with unknown type of distribution, recasting as either a robust optimization or a stochastic program.

A novel management scheme to reduce emission produced by power plants and plug-in hybrid electric vehicles in a smart microgrid

Recently, with the growth and development of distributed generation (DGs) and energy storage systems (ESSs), as well as smart control equipment, microgrids (MGs) have been developed. Microgrids are comprised of a limited number of constitutive parts, including loads, DGs, ESSs, and electric vehicles (EVs). This paper presents a novel scheme to manage active and reactive powers, based on DGs, ESSs, and EVs to reduce the total operation cost including power generation and emission costs. Simultaneous management of active and reactive power makes it possible to consider grid operation constraints together. In the proposed schedule, the vehicles are assumed to be plug-in hybrid electric vehicles. They are able to run both on gasoline and electricity. The proposed schedule incorporates the cost of the consumed fuel and generated pollution into the function of the MG. It is programmed using GAMS software as a mixed-integer second-order cone programming problem and is implemented on a test MG. Simultaneous management of active and reactive power sources can result in lower cost compared to separated scheduling.

Renewable generation capacity and reliability assessments for resilient active distribution networks based on time-sequence production simulation

The penetration of renewable generation will affect the energy utilization efficiency, economic benefit and reliability of the active distribution network (ADN). This paper proposes a time-sequence production simulation (TSPS) method for renewable generation capacity and reliability assessments in ADN considering two operational status: the normal status and the fault status. During normal operation, an optimal dispatch model is proposed to promote the renewable consumption and increase the economic benefit. When a failure occurs, the renewable generators are partitioned into islands for resilient power supply and reliability improvement. A novel dynamic island partition model is presented based on mixed integer second-order cone programming (MISOCP). The effectiveness of the proposed TSPS method is demonstrated in a standard network integrated with historical data of load and renewable generations.

Spatial-Data-Driven Drone Placement for Emergency Response

With recent technological advancements, there is a growing interest in deploying unmanned aerial vehicles (UAVs), commonly known as drones, in providing assistance, delivering supplies, and rescue operations during emergency situations. To ensure a quick response to a future emergency, it is crucial to strategically deploy the drones. In this work, we consider the drone placement problem as a facility location problem under the spatial uncertainty associated with future emergency locations. We propose a spatial-data-driven model to deal with the key challenges, namely the continuous region of potential emergencies, sparse historical incidents, and non-stationarity of spatial distribution. Specifically, our solution approach involves three steps - First, leveraging the historical data or forecasts about emergency locations, we partition the studied region into several disjoint convex subregions; From a reference spatial distribution obtained from data or forecasts, we then construct a spatial ambiguity set about the future spatial distribution such that it lies within a certain Wasserstein distance of the reference distribution; Finally, under L1 Wasserstein distance, we derive a robust optimization formulation for drone placement as a mixed-integer second-order cone program based on the vertices of those subregions. In our approach, the clustered pattern of spatial distribution can be easily incorporated and, more importantly, the challenges in dealing with a continuous region in facility optimization can be well circumvented via spatial ambiguity set modeling by adopting the robust optimization framework. In a case study on Arizona wildfire fighting, our numerical results suggest that our model produces robust drone base station locations that hedge against the spatial uncertainty, and outperformed the sample average approximation approach in terms of quick response and fair accessibility.

Interactive Equilibrium of Electricity-Gas Energy Distribution System and Integrated Load Aggregators Considering Energy Pricings: A Master-Slave Approach

Integration of electricity and gas distribution networks improves energy utilization and alleviates environmental pollution. In order to obtain a reasonable energy prices and optimize the day-ahead scheduling scheme under the premise of considering the interests of both supply and demand, an interactive equilibrium model considering electricity-gas energy distribution system and integrated load aggregators is established in this paper. In the upper level, the integrated energy distribution network (IEDN) maximizes the profit via solving the day-ahead optimal energy flow considering energy prices changing in the next day. In the lower level, integrated load aggregators (ILA) minimize charging costs, thus optimizing energy distribution based on integrated demand responses. By using the Karush-Kuhn-Tucker (KKT) optimality conditions and the duality theorem of linear programming, the master-slave game model is transformed into a single layer optimization problem. To solve this problem, second-order cone programming (SOCP) and penalty convex-concave procedure (PCCP) are used to deal with the non-convex functions of electricity-gas distribution system. Therefore, the proposed model is approximated by a mixed-integer second-order cone programming problem. Finally, case studies verify the effectiveness of the proposed model and method.

A Bi-Level Corrective Line Switching Model for Urban Power Grid Congestion Mitigation

The fast urbanization process in developing countries creates an electricity load surge that poses serious congestion problems to the urban power grid (UPG). The current practice of system operators is to reconfigure the topological structure of the 110-kV high voltage distribution network (HVDN) in an attempt to adjust the load distribution and eliminate the UPG congestion. However, HVDN reconfiguration is a large scale, non-convex and non-linear optimization problem that is intractable to be solved in an efficient manner. In order to reduce the computing cost and overcome the difficulty in the solving process, new partition rules are proposed to divide the HVDN into multiple smaller regions. The topological structure of each region can be optimized either independently or interdependently based on the condition of the UPG. The solving process is modeled as a bi-level iterative line switching problem. At the upper level, the operational boundary (OB) of each regional HVDN is calculated based on a set of convex ACOPF constraints. At the lower level, the line switching scheme of each regional HVDN is derived based on the corresponding mixed integer second-order cone programming (MISOCP) model considering the OB. Then, the line switching result is examined by the AC power flow to identify whether there is any violation or not. This iterative process continues until all the violations are corrected in the UPG. The proposed method was validated by a modified IEEE 58-node test system and a practical 157-node UPG system. The simulation results showed that the proposed method is able to achieve a higher computational efficiency than the traditional method so it is promising for online applications.

Privacy-Preserving Collaborative Operation of Networked Microgrids With the Local Utility Grid Based on Enhanced Benders Decomposition

In this paper, a privacy-preserving decision-making framework is developed for a collaborative operation of networked microgrids and the local utility grid, where the distribution system operator (DSO) interacts with microgrid master controllers (MCs) in a leader-followers fashion. First, the paper models the decision-making in the collaborative operation presented as an integrated mixed-integer second-order cone programming problem which satisfies various technical and operational requirements in a look-ahead manner. Then, the integrated optimization problem is decomposed into a master problem and a set of subproblems in order to decentralize the decision-making process of DSO and MCs. The master problem is solved by DSO for optimizing the local utility grid operation, and the subproblems are solved independently by MCs for optimizing the operation of individual microgrids. We propose an enhanced Benders decomposition algorithm to guide the iterations between the master problem and subproblems. In particular, an enhanced approach is introduced in the Benders decomposition algorithm for generating valid cutting planes from mixed-integer linear programming subproblems. Mathematically, the proposed algorithm can find the globally optimal solution after a finite number of iterations. Accordingly, DSO and MCs manage to collaborate on making operational decisions that would maximize the social welfare without compromising independent decision-making and privacy provisions. Numerical experiments are conducted on a simplified two-bus system and a modified IEEE 123-bus system. The results validate the efficiency, the robustness, and the scalability of the proposed collaborative decision-making framework in power systems.