Mixed-integer Second-order Cone Programming Research Articles

Digital twin‐based online resilience scheduling for microgrids: An approach combining imitative learning and deep reinforcement learning

AbstractStrong uncertainty of renewables puts high demands on the fast response of flexibility resources and resilience‐oriented optimal scheduling for microgrids (MGs). Digital twins (DT) technology based on data‐driven methods is a potential solution to this problem. A DT‐based online resilience scheduling framework for MGs is designed in this study. Based on the proposed framework, a hybrid sequential‐parallel combination method of imitation learning (IL) and deep reinforcement learning (DRL) is proposed to develop the optimal scheduling strategy for MGs. First, a mixed integer second‐order cone programming (MISOCP) model is adopted to behave as an expert to generate decision demonstrations corresponding to operation scenarios of MGs. Then, IL and deep deterministic policy gradient (DDPG) are combined by (1) sequential pretrain and finetune and (2) parallel experience storing and sampling two steps to learn the optimal policy for MGs. Expert demonstrations from the MISOCP model are utilized to pretrain a deep neural network model by IL and initialize the policy network of DDPG to avoid it learning from scratch. Moreover, an expert replay buffer is introduced specifically to avoid forgetting the well‐behaved expert experience and to further accelerate the training process. A 6‐bus MG test system case study demonstrates the effectiveness and scalability of the proposed approach.

Mixed-Integer Second-Order Cone Programming Reformulations of a Fractional 0-1 Program for Task Assignment

Fractional 0-1 programming is a subfield of nonlinear integer optimization in which the objective is to optimize the sum of ratios of affine functions subject to a set of linear constraints. It is well-known that fractional 0-1 programs can be formulated as mixed-integer linear programs. Recently, several alternative mixed-integer second-order cone programming reformulations have been proposed for fractional 0-1 programs. These reformulations, which can be solved directly by standard commercial solvers, have been reported to be efficient for certain types of problems. In this paper, we consider a task assignment problem with respect to preferences, where the objective is to maximize total weighted satisfaction while maintaining a fair distribution. The problem’s mathematical model turns out to be a fractional 0-1 program. We investigate three mixed-integer second-order cone programming reformulations thereof, and we compare, by means of a computational study, the performance of these reformulations with a benchmark mixed-integer linear programming formulation that was proposed and analyzed in the literature before. The latter, namely the mixed-integer linear programming formulation, turns out to be significantly better for the problem in question.

The Cost and Benefit of Enhancing Cybersecurity for Hybrid AC/DC Grids

As critical interfaces of AC grids and DC grids inside a hybrid AC/DC grid, the voltage-sourced-converter (VSC) has been demonstrated to be vulnerable to false data injection (FDI) cyber-attacks. As a result, the cyber-attack-induced AC grid frequency deviations and DC grid voltage deviations threaten the secure operation. To enhance cybersecurity in a not only feasible but also cost-effective manner, this paper proposes a cost-benefit-based cyber-defense strategy for a hybrid AC/DC grid. First, this paper establishes a spatial-temporal dual cyber-attack evaluation model, in which the cyber-attack-induced frequency and voltage deviations are modelled in both a spatially and temporally dual manner. Then, the proposed cost-benefit-based cyber-defense strategy is modelled as a VSC commitment problem to achieve the trade-off between maximizing the cyber-defense benefits and minimizing the cyber-defense costs. The VSC commitment problem is then mathematically convexified into a mixed-integer second-order cone programming (MISOCP) problem, which could be efficiently solved in an event-triggered manner against unfolding cyber-attack events. Simulation results on a test hybrid AC/DC grid verified the feasibility and the cost-effectiveness of the proposed cyber-defense strategy.

Distributionally Robust Unit Commitment Considering Unimodality-Skewness Information of Wind Power Uncertainty

The deepening penetration of renewable energy sources (RESs) into power grids imposes remarkable challenges that merit special attention. In this paper, we work out a distributionally robust chance-constrained (DRCC) day-ahead unit commitment problem while the stochasticity and variability arising from RESs are taken into consideration. The developed framework optimizes the expected objective function and stipulates that the DRCCs will hold within a moment-based ambiguity set, which additionally incorporates the unimodality information and mode skewness. The employed ambiguity set can not only account for the wind power uncertainty but also expressly alleviate the conservatism compared with the single moment metric. By deriving efficient mathematical reformulations to tackle the expected objective function and DRCCs, the proposed model is reduced to an exact mixed-integer second-order cone programming problem that can readily be implemented. Numerical experiments are carried out on two representative test systems to demonstrate the effectiveness and efficiency of the suggested methodology.

Reactive power optimization in active distribution systems with soft open points based on deep reinforcement learning

In the context of the increasing penetration level of photovoltaic energy, its intermittence and randomness bring challenges such as voltage over-limit and increased network loss to the distribution network. The paper proposes a real-time two-time-scale voltage regulation method with soft open point (SOP) to solve the reactive power optimization problem. To obtain hourly scheduling strategies for on-load tap changers and switchable capacitor banks, the centralized optimization problem is formulated as a mixed integer second-order cone programming in the first stage. In the second stage, the active distribution system is divided into multiple subsystems according to the proposed partitioning method, and the SOP is regulated to adjust the voltage in real-time through an effective control strategy and local measurement information of the subsystems. The proposed method takes full account of the randomness of photovoltaic, and can conduct distributed real-time voltage regulation in the active distribution system based on the rapid and continuous response of SOP. Compared with traditional methods, this method has lower communication costs, better real-time performance, and does not dependent on an accurate power flow model. Finally, the effectiveness of the proposed method is verified by the IEEE 33-bus and IEEE 123-bus simulation examples.

Optimal Probabilistic Allocation of Photovoltaic Distributed Generation: Proposing a Scenario-Based Stochastic Programming Model

The recent developments in the design, planning, and operation of distribution systems indicate the need for a modern integrated infrastructure in which participants are managed through the perceptions of a utility company in an economic network (e.g., energy loss reduction, restoration, etc.). The penetration of distributed generation units in power systems are growing due to their significant influence on the key attributes of power systems. As a result, the placement, type, and size of distributed generations have an essential role in reducing power loss and lowering costs. Power loss minimization, investment and cost reduction, and voltage profile improvement combine to form a conceivable goal function for distributed generation allocation in a constrained optimization problem, and they require a complex procedure to control them in the most appropriate way while satisfying network constraints. Such a complex decision-making procedure can be solved by adjusting the dynamic optimal power flow problem to the associated network. The purpose of the present work is to handle the distributed generation allocation problem for photovoltaic units, attempting to reduce energy and investment costs while accounting for generation unpredictability as well as load fluctuation. The problem is analyzed under various scenarios of solar radiation through a stochastic programming technique because of the intense uncertainty of solar energy resources. The formulation of photovoltaic distributed generation allocation is represented as a mixed-integer second-order conic programming problem. The IEEE 33-bus and real-world 136-bus distribution systems are tested. The findings illustrate the efficacy of the proposed mathematical model and the role of appropriate distributed generation allocation.

Optimal AC/DC Distribution Systems Expansion Planning From DSO's Perspective Considering Topological Constraints

As the integration of DC distributed resources increases in recent years, hybrid AC/DC topology emerges as a promising alternative in the planning of distribution systems. This paper presents an AC/DC optimal expansion planning of distribution networks from Distribution System Operator's perspective with a special focus on grid architecture through Integer Programming-based topological constraints. The proposed formulation aims to determine the optimal location and sizing of converters and to select lines type (AC or DC) and state (closed or open). The problem is formulated as a mixed integer second-order conic programming model that guarantees global optimality and verifiable exactness by using off-the-shelf solvers. A multi-scenario approach with a Wasserstein distance-based scenario reduction is applied to deal with resource uncertainty. Case studies based on a 13 bus and IEEE 33 bus distribution networks were used to verify the effectiveness of the model. The results show that a hybrid architecture can improve the efficiency of the grid through optimal active power flow and reduce the total network costs.

Virtual Network Function Placement Model Considering Both Availability and Probabilistic Protection for Service Delay

This paper proposes a virtual network function (VNF) placement model considering both availability and probabilistic protection for the service delay to minimize the service deployment cost. Both availability and service delay are key requirements of services; a service provider handles the VNF placement problem with the goal of minimizing the service deployment cost while meeting these and other requirements. The previousworks do not consider the delay of each route which the service can take when considering both availability and delay in the VNF placement problem; only the maximum delay was considered. We introduce probabilistic protection for service delay to minimize the service deployment cost with availability. The proposed model considers that the probability that the service delay, which consists of networking delay between hosts and processing delay in each VNF, exceeds its threshold is constrained within a given value; it also considers that the availability is constrained within a given value. We develop a two-stage heuristic algorithm to solve the VNF placement problem; it decides primary VNF placement by solving mixed-integer second-order cone programming in the first stage and backup VNF placement in the second stage. We observe that the proposed model reduces the service deployment cost compared to a baseline that considers the maximum delay by up to 12%, and that it obtains a feasible solution while the baseline does not in some examined situations.

Multi-Timescale Coordinated Control With Optimal Network Reconfiguration Using Battery Storage System in Smart Distribution Grids

This paper focuses on the global optimality of the relaxed solutions of a multi-timescale co-optimization problem. The proposed co-optimization framework involves the multi-timescale co-optimization of distribution feeder reconfiguration with the optimal dispatch of traditional voltage regulating devices and utility-scale distributed energy resources. The on-load tap changer (OLTC) is scheduled on an hourly basis while the potential of fast-acting battery energy storage system and photovoltaic inverters is exploited by dispatching them on a 20-min basis. The optimal switching plan is computed on a daily basis. The proposed multi-timescale co-optimization model is formulated as a mixed-integer second-order cone program to achieve global optimum. The objective is to reduce power losses and improve load balancing among feeders. The proposed co-optimization framework satisfies the grid security constraints by employing the accurate DistFlow branch equations and exact linearization of the OLTC model. To ensure radiality, the limitation of widely used spanning tree constraints is addressed by combining them with single-commodity flow constraints. The simulation results demonstrate the feasibility (hence global optimality) of the relaxed solutions computed by the co-optimization framework.

Distributionally Robust Energy Management for Islanded Microgrids With Variable Moment Information: An MISOCP Approach

The ever-increasing penetration of renewable energy sources (RESs) has brought tremendous challenges to power system energy management problems. In this paper, we develop a distributionally robust (DR) joint chance-constrained microgrid energy management model while the uncertainties stemming from RESs are embedded. The proposed framework minimizes the expected cost function and ensures that the DR joint chance constraints (CCs) will satisfy for any distribution over a promising ambiguity set, which is designed based on the variable moments. The employed ambiguity set can describe the uncertainties more accurately compared with the commonly-used moment metric (i.e., fixed moments). By deriving equivalent second-order cone programming (SOCP) reformulations of the expected objective function and introducing effective approximation methods such as the optimized Bonferroni approximation to deal with the DR joint CCs, the proposed model is finally reduced to a tractable mixed-integer SOCP problem that can readily be solved. Case studies are conducted on a representative test system to corroborate the effectiveness of the suggested approach.

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.

Distributed coordinated planning for cross-border energy system: An ADMM-based decentralized approach

Cross-border interconnection and trade (CBIT) can effectively leverage the differences in resource endowments and energy demand between regions and countries, making cross-border cooperation a potential option for addressing the high penetration of variable renewable energy. This paper presents centralized and decentralized planning approaches for the cross-border energy transmission expansion planning (TEP) problem. The proposed centralized mixed-integer second-order cone programming (MISOCP) model collaboratively optimizes the expansion of national energy networks, while considering the impact of CBIT. To protect the privacy of energy data in each country, the alternating direction method of multipliers (ADMM) algorithm is extended to decompose the centralized optimization formulation into mutually independent sub-optimization issues, and adaptively adjust the penalty parameters and Lagrangian multipliers to improve convergence. This study also quantifies the effect of cross-border cooperation in such co-expansion problems as a comparative consideration. Numerical simulations on the modified cross-border integrated energy network validates the correctness and effectiveness of the proposed model and solution methodology. The results show that cross-border cooperation can reduce a country's internal energy network congestion, thus postponing the expansion investment decisions. It also shows that the distributed coordinated planning is more efficient in terms of lower cost and renewable generation curtailment that than independent optimization method. Therefore, cross-border collaboration brings economic, environmental, and renewable integration benefits to interconnected countries.

Optimal resilience enhancement dispatch of a power system with multiple offshore wind farms considering uncertain typhoon parameters

With the increasing penetration of offshore wind power capacity into power systems, extreme typhoon events have been severely threatening the secure operation of a power system with multiple offshore wind farms (OWFs). To alleviate this problem, this paper proposes an optimal resilience enhancement dispatch (ORED) framework consisting of three stages: preventive control, emergency response, and rapid restoration. Combining the wind field model of typhoon and the probability distribution model of typhoon parameters, a Wasserstein distance-based ambiguity set (AS) is constructed for describing the uncertainty of typhoons. In addition, a distributionally robust chance constraint-based ORED (DRCC-ORED) model, which considers uncertain typhoon parameters, is established for a power system with multiple OWFs. A conditional value-at-risk (CVaR) approximation method is used to transform the DRCCs with random variables into linear constraints, and the expectation operation of the cost function under the worst probability distribution is reformulated. Further, the proposed bi-level DRCC-ORED model is transformed into a single-level mixed-integer second-order cone programming model, which can be efficiently solved by the commercial solver GUROBI. Case studies on the modified IEEE 39-bus system with an OWF and an actual provincial power system with four OWFs demonstrate the high effectiveness of the proposed model.

Network-Constrained Transactive Control for Multi-Microgrids-Based Distribution Networks With Soft Open Points

Different from most transactive control studies only focusing on economic aspect, this paper develops a novel network-constrained transactive control (NTC) framework that can address both economic and secure issues for a multi-microgrids-based distribution network considering uncertainties. In particular, we innovatively integrate a transactive energy market with the novel power-electronics device (i.e., soft open point) based AC power flow regulation technique to improve economic benefits for each individual microgrid and meanwhile ensure the voltage security of the entire distribution network. In this framework, a dynamic two-timescale NTC model consisting of slow-timescale pre-scheduling and real-time corrective scheduling stages is formulated to work against multiple system uncertainties. Moreover, the original bilevel game problems are transformed into a single-level mixed-integer second-order cone programming problem through KKT conditions, duality, linearization and relaxation techniques to avoid iterations of transitional methods, so as to improve the solving efficiency. Finally, numerical simulations on modified 33-bus and 123-bus test systems with multi-microgrids verify the effectiveness of the proposed framework.

部材の密集度を考慮した骨組構造物の大域的トポロジー最適化

It is known that the topology optimization (TopOpt) problem of frame structures can be formulated as a mixed-integer second-order cone programming problem (MISOCP), when the compliance is minimized and the cross-section of each member is selected from a set of predetermined candidates. However, when considering the constructability of real-world building structures, it is insufficient to only minimize compliance. In this paper, in order to control the density of members, the constraints of the number of members connected to each node are additionally considered into TopOpt by MISOCP treated in the previous study.

Modeling data-driven adaptive distributionally robust equilibrium last mile relief network under centrality metric

This paper studies a variant of the two-stage last mile relief network problem by introducing a centrality metric objective including the distance centrality metric, the demand centrality value-at-risk and the residual supplies centrality metric. Under the scenario-based design, the demand in each scenario is described as a random fuzzy variable whose probability distribution is ambiguous. In order to characterize its ambiguousness, this paper designs a Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm based on sample data to construct a data-driven ambiguity set containing the support and mean, which not only realizes the visualization of the ambiguous probability distribution, but also provides a finite set of scenarios for the uncertain demand. Given the constructed data-driven ambiguity set, this paper proposes a data-driven adaptive distributionally robust equilibrium programming (DDADREP) method for the last mile relief network problem (LMRNP). In view of its computational difficulty, the safe approximation, duality technique and credibility measure theory are used to approximate it as a mixed integer second-order cone programming with significant computational advantages. Finally, the proposed model is validated with two real-world cases, and its effectiveness as well as advantages is demonstrated by comparison with the classical robust optimization and the nominal equilibrium optimization, respectively.

Hub Network Design Problem with Capacity, Congestion, and Stochastic Demand Considerations

Our study introduces the hub network design problem with congestion, capacity, and stochastic demand considerations (HNDC), which generalizes the classical hub location problem in several directions. In particular, we extend state-of-the-art by integrating capacity acquisition decisions and congestion cost effect into the problem and allowing dynamic routing for origin-destination (OD) pairs. Connecting strategic and operational level decisions, HNDC jointly decides hub locations and capacity acquisitions by considering the expected routing and congestion costs. A path-based mixed-integer second-order cone programming (SOCP) formulation of the HNDC is proposed. We exploit SOCP duality results and propose an exact algorithm based on Benders decomposition and column generation to solve this challenging problem. We use a specific characterization of the capacity-feasible solutions to speed up the solution procedure and develop an efficient branch-and-cut algorithm to solve the master problem. We conduct extensive computational experiments to test the proposed approach’s performance and derive managerial insights based on realistic problem instances adapted from the literature. In particular, we found that including hub congestion costs, accounting for the uncertainty in demand, and whether the underlying network is complete or incomplete have a significant impact on hub network design and the resulting performance of the system. Funding: This work was supported by Türkiye Bilimsel ve Teknolojik Araştırma Kurumu [Grant 218M520]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/trsc.2022.0112 .

Robust homecare service capacity planning

We study a home healthcare planning problem under the demand uncertainty, where the service type (authorization) and capacity are the first-stage decision and the homecare resource allocation is the second-stage decision which adapts to the demand realizations. We model the problem using an adaptive robust optimization technique where we construct a budget uncertainty set of demand using the well-known Mahalanobis Distance. We analyze the impact of the authorization, capacity decisions as well as the budget (robustness level) of the Mahalanobis uncertainty set onto the worst-case revenue. To solve the model, we develop a Benders decomposition algorithm that solves a pair of a mixed-integer second-order cone program (MISOCP) and a mixed integer linear program (MILP) in each iteration, both can be handled by off-the-shelf MIP solvers, with finite-step convergence. We also develop an affine approximation approach that directly solves one instance of MISOCP. Finally, sufficient numerical studies demonstrate the effectiveness of our model and the solution approaches.

Analysis of strategic bidding of a DER aggregator in energy markets through the Stackelberg game model with the mixed-integer lower-level problem

An open electricity market environment provides sufficient conditions for a distributed energy resource (DER) aggregator to participate in the day-ahead electricity market. Therefore, a one-leader multi-follower Stackelberg game model is developed to analyze the strategic bidding behavior of a DER aggregator and one of the lower-level problems is to check the security of the distribution system (DS), which is elaborated by the mixed-integer second-order cone programming (MISOCP) model considering the active power loss and discrete controls in the DS. Moreover, we first use the Karush–Kuhn–Tucker (KKT)-based reformulation approach to transform the model into a one-leader one-follower model. However, the objective function of the upper-level model cannot be completely linearized by using the strong dual theorem because the active power loss of the DS is considered in the lower-level security check problem. Therefore, we propose a piecewise linear approximation approach based on the data-driven algorithm to linearize the upper-level objective function. Moreover, we apply multivariate linear surrogation to transform this one-leader one-follower model into a single-level MISOCP model. Finally, we verify the correctness of the proposed algorithm by comparing it with the particle swarm optimization (PSO) algorithm in a small integrated transmission and distribution (T&D) system and validate the effectiveness of the proposed algorithm in a practical integrated T&D system.

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.