Column-and-constraint Generation Research Articles

Distributionally robust disaster relief planning under the Wasserstein set

We study a two-stage natural disaster management problem modeled as a stochastic program, where the first stage consists of a facility location problem, deciding where to open facilities and pre-allocate resources such as medical and food kits, and the second stage is a fixed-charge transportation problem, routing resources to affected areas after observing a disaster. Our model has binary variables present in both stages. Due to the lack of data, classical stochastic programming approaches may be ill-suited, and we propose a two-stage distributionally robust formulation with a Wasserstein ambiguity set, where we consider distributions consistent with historical data and a tunable parameter to control the level of risk aversion. We develop a tailored column-and-constraint generation (CCG) algorithm to solve an extensive reformulation, where scenarios are iteratively generated. We handle the presence of binary variables in the second stage by leveraging the structure of our support set and second-stage problem, and provide conditions under which the optimal value of the latter is concave with respect to the intensity of the disaster, leading to an efficient scenario generation procedure. We also show that our results extend to the case where the second stage is a fixed-charge network flow problem. We perform extensive computational experiments demonstrating the computational advantage of our method over classical CCG implementations on synthetic instances, and illustrate the benefits of our approach on a popular case study from the literature of hurricane threats on the Gulf of Mexico in the United States.

Energy management of multi-microgrid system with renewable energy using data-driven distributionally robust optimization

ABSTRACT By clustering multiple microgrids (MGs), a multi-microgrid (MMG) system plays a significant role in integrating a large amount of renewable generation. However, the large-scale utilization of renewable energy also brings uncertainties to MMG energy management. In this work, a novel comprehensive bi-level MMG energy management model considering uncertainties of renewable energy is first developed which includes unit commitment (UC) problem and other models in the lower MG level. Then considering the non-convexity of the bi-level problem, a decomposition method called analytical target cascading (ATC) is employed to deal with it by decentralization. With regard to the energy management of each individual MG, a two-stage distributionally robust model is developed which describes the uncertainties from probability distribution of renewable generation with a metric-based ambiguity set. Moreover, an efficient solution scheme based on column and constraint generation (CCG) algorithm and a decomposition method without duality is designed to handle the MG energy management problem. Finally, we implement extensive experiments to corroborate the effectiveness of the proposed approach. Particularly, the optimal solution from the proposed method can attain 2.98% less cost compared with that from robust optimization (RO) method and achieve 86.17% less energy trading amount than that in independent mode.

Data-Driven Distributionally Robust Optimization-Based Coordinated Dispatching for Cascaded Hydro-PV-PSH Combined System

The increasing penetration of photovoltaic (PV) and hydroelectric power generation and their coupling uncertainties have brought great challenges to multi-energy’s coordinated dispatch. Traditional methods such as stochastic optimization (SO) and robust optimization (RO) are not feasible due to the unavailability of accurate probability density function (PDF) and over-conservative decisions. This limits the operational efficiency of the clean energies in cascaded hydropower and PV-enriched areas. Based on data-driven distributionally robust optimization (DRO) theory, this paper tailors a joint optimization dispatching method for a cascaded hydro-PV-pumped storage combined system. Firstly, a two-step model for a Distributed Renewable Optimization (DRO) dispatch is developed to create the daily dispatch plan, taking into account the system’s complementary economic dispatch cost. Furthermore, the inclusion of a complementary norm constraint is implemented to restrict the confidence set of the probability distribution. This aims to identify the optimal adjustment scheme for the day-ahead dispatch schedule, considering the adjustment cost associated with real-time operations under the most unfavorable distribution conditions. Utilizing the MPSP framework, the Column and Constraint Generation (CCG) algorithm is employed to resolve the two-stage dispatch model. The optimal dispatch schedule is then produced by integrating the daily dispatch plan with the adjustive dispatch scheme. Finally, the numerical dispatch results obtained from an actual demonstration area substantiate the effectiveness and efficiency of the proposed methodology.

Robust post-disaster repair crew dispatch for distribution systems considering the uncertainty of switches

After natural disasters’ strike, the remote-controlled switch plays a key role in network reconfiguration to restore the critical load. However, the status of these switches is hardly acquired in case of cyber malfunctions. This paper proposes a robust repair crew dispatch approach for distribution systems considering the uncertainty of remote-controlled switches. The proposed approach is formulated as a tri-level robust optimization model to obtain the repair crew dispatch strategy applicable to possible fault scenarios. The system operator determines the repair crew dispatch scheme at the first level. At the second level, the real-time status of switches under different combinations of faults and repair processes is modeled and the worst fault scenario is calculated. Based on the real-time status of switches, the operator utilizes distributed generators and network reconfiguration to restore loads. A nested column-and–constraint generation (NC&CG) algorithm is customized for the proposed robust model to achieve the optimal solution. Simulation results on the IEEE-33 node distribution test system and the IEEE 123-node system show the proposed approach can effectively reduce the maximum system loss and the NC&CG algorithm can significantly reduce the computing time.

Distributionally robust production and replenishment problem for hydrogen supply chains

This study investigates a two-stage production and replenishment problem for the hydrogen supply chain (HSC), where the uncertainty of demand and the risk of pipeline disruptions are simultaneously considered. We model this problem by adopting a distributionally robust optimization (DRO) approach. Before realizing the demands, the decision-maker decides the hydrogen production, storage, and truck-transported replenishment policies. Then, after demands and pipeline working status are realized, the pipeline-transported replenishment policy is determined in response to the first-stage decisions. The replenishment policies are determined under differentiated service level requirements. In addition, we characterize the uncertainty and correlation of demands through a factor-based model approach to accommodate the uncertainty of the distribution of demand factors. To solve the proposed model, we propose a column and constraint generation (CCG) algorithm, which can terminate in finite iterations. Through theoretical analysis and numerical experiments, we verify that the proposed model outperforms classical benchmark models in terms of total costs and are more robust. Finally, we examine the impact of the replenishment structure on the performance of the HSC and the computational efficiency of the proposed CCG algorithm.

A multi-agency coordination resource allocation and routing decision-making problem: A coordinated truck-and-drone DSS for improved wildfire detection coverage

This study proposes a novel coordinated truck-drone system as a decision support system to improve fire suppression operations. The problem in this study is formulated as a bi-objective mathematical model to minimize total monitoring cost and time, considering trucks as mobile drone depots and that drones can fly at various altitudes and access hard-to-reach areas. In addition, time and cost parameters in the mathematical model are deemed uncertain, rendering the model more realistic. Accelerated Benders' decomposition (ABD) is utilized to solve the model rapidly. Furthermore, a column-and-constraint generation (CCG) algorithm is employed, which is an effective method for solving scenario-based models under uncertainty. Two criteria are subsequently used to evaluate and compare proposed robust optimization approaches (risk averse and min-max relative regret). The results indicate that the proposed system can assist firefighters in determining the optimal number of drones and trucks to patrol and the time and cost required to visit all areas. Moreover, the findings demonstrate that the min-max relative regret approach can outperform other methods during the hot season when the risk of wildfire is elevated. In contrast, when the fire risk is lower during the cold season, time and cost can be effectively managed, and a risk-averse strategy can be implemented. Finally, the proposed solution framework can facilitate optimal strategic organizational decision planning.

Optimal design and operation of islanded multi-microgrid system with distributionally robust optimization

Renewable generation has been increasingly utilized recently, and multi-microgrid (MMG) system shows great potential in absorbing renewable energy. In this paper, a two-stage distributionally robust model is proposed for the optimal design and operation of islanded MMG which is seldom studied. Considering the uncertainty of renewable generation, the first-stage design strategy and second-stage operation decision are co-optimized by minimizing the investment cost and operation cost. To model the uncertainty of renewable power, a distance-based ambiguity set is employed to capture the unknown probability distribution. Then the two-stage problem is solved by a column and constraint generation (CCG) based method. Simulation experiments and results with an islanded MMG system demonstrate the effectiveness and performance of the proposed model.

A tri-level optimization model for facility location-protection problem considering design and redesign decisions under disruption

Purpose Fortification-interdiction models provide system designers with a broader perspective to identify and protect vital components. Based on this concept, the authors examine how disruptions impact critical supply systems and propose the most effective protection strategies based on three levels of decision-makers. This paper aims to investigate location and fortification decisions at the first level. Moreover, a redesign problem is presented in the third level to locate backup facilities and reallocate undisrupted facilities following the realization of the disruptive agent decisions at the second level. Design/methodology/approach To address this problem, the authors develop a tri-level planner-attacker-defender optimization model. The model minimizes investment and demand satisfaction costs and alleviates maximal post-disruption costs. While decisions are decentralized at different levels, the authors develop an integrated solution algorithm to solve the model using the column-and-constraint generation (CCG) method. Findings The model and the solution approach are tested on a real supply system consisting of several hospitals and demand areas in a region in Iran. Results indicate that incorporating redesign decisions at the third level reduces maximum disruption costs. Originality/value The paper makes the following contributions: presenting a novel tri-level optimization model to formulate facility location and interdiction problems simultaneously, considering corrective measures at the third level to reconfigure the system after interdiction, creating a resilient supply system that can fulfill all demands after disruptions, employing a nested CCG method to solve the model.

Two-stage robust optimization of regional integrated energy systems considering uncertainties of distributed energy stations

For a regional integrated energy system (RIES) composed of an energy supply network and distributed energy station, the uncertainty of distributed photovoltaic (PV) output and the fluctuation of various loads pose significant challenges to the stability of system operation and the accuracy of optimal scheduling. In order to enhance the operational reliability of regional integrated energy systems and reduce the impact of photovoltaic and load uncertainties on distributed energy stations, this study proposes robust optimization method of regional integrated energy systems that takes into account the uncertainty of the distributed energy station. First, the regional integrated energy system is divided into an upper electric-gas energy supply network and a lower distributed energy station. The upper model mainly realizes energy transmission, while the lower model is a two-stage robust optimization model of distributed energy stations in the form of min–max–min, which mainly realizes flexible energy supply of different types of energy. Then, the lower two-stage robust optimization model is simplified and solved using a column and constraint generation (CCG) algorithm. After that, an alternating direction method of multipliers (ADMM) is used to solve the upper and lower models of the regional integrated energy system, and the solution scale is reduced while ensuring the correlation between the energy transmission network and the distributed energy stations. Finally, a test example is provided to illustrate the effectiveness and usefulness of the proposed method. It follows from simulation results that the robust optimization method can effectively reduce the instability of the system operation caused by uncertainty factors and improve the system’s anti-interference ability, and in addition, systems with high penetration levels of photovoltaic output will benefit more from robust optimization.

Bidding strategy for virtual power plants with the day-ahead and balancing markets using distributionally robust optimization approach

Virtual power plant (VPP) coordinates the energy consumption or production of its components and trades power in both day-ahead market (DAM) and balancing market (BM) to maximize operating margins, where consists of intermittent distributed generation, energy storage devices, and flexible demand. Due to the uncertainty of electricity prices and wind power output and imbalance penalties, VPP bidding is risky. Meanwhile, both traditional stochastic optimization (SO) and robust optimization (RO) algorithms have certain limitations and shortcomings in dealing with wind power output uncertainties. Therefore, a two-stage distribution robust optimization (DRO) model is proposed in this paper for determining the optimal bidding strategy for VPP participation in the energy market and combining L1 norm with L∞ norm to simultaneously constrain the confidence set of uncertain probability distributions. The column-and-constraint generation (CCG) algorithm is used to solve it. The robustness and feasibility of the proposed model are verified by a case study.

Fast Solving Method for Two-Stage Multi-Period Robust Optimization of Active and Reactive Power Coordination in Active Distribution Networks

This paper considers constraints on the cycle number of energy storage systems (ESSs), travel distances of switchable capacitors reactors (SCRs), on load tap changers (OLTCs) and step voltage regulators (SVRs). The characteristics and operational constraints of these equipments are properly formulated with binaries and big-M method, obtaining desirable linear descriptions. The branch flow equations of distribution systems for active and reactive power coordination are constructed. Considering physical constraints, a multi-period mixed-integer deterministic second-order conic programming (SOCP) to minimize network losses is formulated. Then, considering uncertainties of loads and active power of renewable distributed generators (DGs), a two-stage robust model is developed. A directly iteratively solving method between the first and second stage models is proposed. In contrast to the traditional column-and-constraint generation (CCG) method, increases to variables and constraints are not needed to solve the first stage model. To solve the second stage multi-period model, only the model of each single period needs to be solved first. Then their objective function values are accumulated, and the worst scenarios of each period are concatenated. As a result, the solving complexity is greatly reduced. The capabilities of the proposed method are validated by three simulation cases. It is found that the computational rate using the proposed method is significantly enhanced with much less computer storage. Specifically, the simulation results of 4, 33 and 69-bus systems indicate that the computing rate of the proposed method is about 28 times higher than CCG method when 8 iterations are performed. Meanwhile, the precision of optimization results is also improved.

Mitigating the Impacts of Uncertain Geomagnetic Disturbances on Electric Grids: A Distributionally Robust Optimization Approach

Severe geomagnetic disturbances (GMDs) increase the magnitude of the electric field on the Earth’s surface (E-field) and drive geomagnetically-induced currents (GICs) along the transmission lines in electric grids. These additional currents can pose severe risks, such as current distortions, transformer saturation and increased reactive power losses, each of which can lead to system unreliability. Several mitigation actions (e.g., changing grid topology) exist that can reduce the harmful GIC effects on the grids. Making such decisions can be challenging, however, because the magnitude and direction of the E-field are uncertain and non-stationary. In this paper, we model uncertain E-fields using the distributionally robust optimization (DRO) approach that determines optimal transmission grid operations such that the worst-case expectation of the system cost is minimized. We also capture the effect of GICs on the nonlinear AC power flow equations. For solution approaches, we develop an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">accelerated</i> column-and-constraint generation (CCG) algorithm by exploiting a special structure of the support set of uncertain parameters representing the E-field. Extensive numerical experiments based on “epri-21” and “uiuc-150” systems, designed for GMD studies, demonstrate (i) the computational performance of the accelerated CCG algorithm, (ii) the superior performance of distributionally robust grid operations that satisfy nonlinear, nonconvex AC power flow equations and GIC constraints, in comparison with standard stochastic programming-based methods during the out-of-sample testing.

Integrated Planning of Cyber-Physical Active Distribution System Considering Multidimensional Uncertainties

The integration of cyber technologies is transforming the power distribution network into a cyber-physical active distribution system (CPADS), and brings more flexibility to its operation. However, various uncertainties from the cyber space and physical space are emerging and coupled with each other, which results in significant challenges for the distribution network planning and operation. Thus, a bilevel integrated planning model accounting for multidimensional uncertainties from both the cyber space and physical space is proposed for the CPADS in this paper. With the goal of minimizing the annual investment and operation costs, the model comprehensively considers the location and selection of active-management elements (e.g., electrical storage systems, capacitor banks, and switches), investment constraints, power flow constraints, and operation constraints for all the active-management elements, etc. Moreover, based on the typical cyber system failure scenario represented by information link failure (ILF), a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N-k</i> uncertainty set considering the failure probability distribution is proposed, while the uncertainties of renewable energy and load demand are characterized by box-like uncertainty sets. Finally, the proposed planning model is recast into a multi-stage robust optimization model which can be divided into a master problem and two-step subproblems (SP-1 and SP-2) to be solved by the column and constraint generation (CCG) algorithm. Case studies and comparative analysis on the PG 69 distribution system are performed to verify the effectiveness of the proposed method.

Two-stage stochastic robust optimization model of microgrid day-ahead dispatching considering controllable air conditioning load

The wide application of wind power generation and photovoltaic (PV) power generation has greatly increased the uncertainty of power grid. Under the background of smart grid, peak shaving can be carried out with the help of users' equipment. This paper studies the cooling efficiency of air conditioning from the heat transfer process between room and outdoor, and establishes the air conditioning load control model. Based on this, a stochastic robust day-ahead dispatching model for microgrid is proposed, taking cost minimization as objective function. The day-ahead dispatching is considered as the first stage which considering the arrangement of electricity purchasing and generating of thermal power units. The real-time dispatching is regarded as the second stage. Based on the results of the first stage, real-time dispatching of the various situations of the renewable energy output can be carried out. The decision variables are air conditioning control and ESS charge and discharge arrangement. Then the results are fed back to the first stage, and the cost is minimized by iteration. In this paper, the double-layer problem in the second stage is transformed into a single-layer problem by mathematical method, and the Column-and-Constraint Generation (CCG) algorithm is used to solve the two-stage stochastic robust optimization model. Finally, based on the data of a microgrid in a central city, the effectiveness of the method is verified, and the economic superiority of the day-ahead dispatching model at stochastic and robust level is discussed.

Robust Unit Commitment for Minimizing Wind Spillage and Load Shedding With Optimal DPFC

The distributed power flow controller (DPFC) has a positive effect of UC problem on the network side based on its ability to manage capacity of power flow. This study presents a novel two-stage robust model to optimize the status of the generator and location–allocation of the DPFC, while simultaneously considering wind and load uncertainties. The column-and-constraint generation (CCG) method is utilized to solve the two-stage problem into the master problem and the subproblem iteratively. The optimal status of the generator and location of the DPFC can be easily obtained with the master problem, and the dispatch solution and compensation level of the DPFC are solved in the subproblem. We conduct the IEEE 24 bus system to verify the performance of the proposed procedure. There are effects on wind spillage/load shedding and generator dispatch scheduling planning once the DPFC is injected. Detailed simulation results illustrate the effect of the proposed approach.

Guest Editorial: Enhancing hosting capability for renewable energy generation in active distribution networks

Guest Editorial: Enhancing hosting capability for renewable energy generation in active distribution networks

Distributionally robust co‐optimization of the demand‐side resources and soft open points allocation for the high penetration of renewable energy

The new technologies, such as soft open points (SOPs) and demand response (DR), offer original approaches to stabilize the system operation and accommodate the high penetration of distributed generators (DGs). However, the applied effects of these technologies are closely affected by the uncertainties of the consumer's willingness and the outputs of DGs. To solve the problems caused by the uncertainty of the resources in active distribution system, this paper proposes a distributionally robust co-optimization model for the demand side resources and SOPs, which realizes the combination of investment economy and operation robustness. Next, using equivalent substitution and polyhedron linearization technique, this paper proposes the mathematical method which transforms the original non-linear programming model into a mixed-integer linear programming model, and obtains the solution to the distributionally robust model with column and constraint generation (CCG) algorithm. Finally, the effectiveness of the proposed model is verified with the sample from a power distribution system in northern China. The result demonstrates that the co-optimization model can realize complementary advantages of the demand side resources and SOPs, which can not only guarantee the economy of scheme, but also improve the accommodation of DGs.

A distributed calculation method for robust day-ahead scheduling of integrated electricity-gas systems

The urgency to address the uncertainty in the day-ahead scheduling (DAS) of integrated electricity and natural gas systems (IEGSs) has been highlighted, and the robust optimization has been proven to be an effective method. However, the gas system and electricity system are generally operated by different utilities in practice, i.e., gas system operator (GSO) and electricity system operator (ESO), which poses a considerable challenge to solve this two-stage mixed-integer nonlinear DAS model due to the limit on information sharing. In this paper, a novel distributed calculation method is proposed to solve the robust DAS model of IEGSs in a decentralized way. First, a mixed-integer master problem and a non-convex max–min subproblem can be obtained from the original DAS model by employing the column and constraint generation (CCG) method. Afterwards, based on the limited information exchange between ESO and GSO, an innovative decomposition method is proposed for the master problem to obtain the optimal DAS solution, and then a novel inner CCG algorithm is presented for the subproblem to guarantee the security of IEGSs. By doing so, the robust DAS of IEGS is decomposed into several mixed-integer linear programming (MILP) problems, and GSO and ESO can coordinately solve these MILPs with limited information sharing. Last, numerical tests on two IEGSs verify the superiority of the proposed method in hedging against the wind uncertainty and achieving the solution optimality.

Multi-Period Fast Robust Optimization for Partial Distributed Generators (DGs) Providing Ancillary Services

Distributed generators providing auxiliary service are an important means of guaranteeing the safe and economic operation of a distribution system. In this paper, considering an energy storage system (ESS), switchable capacitor reactor (SCR), step voltage regulator (SVR), and a static VAR compensator (SVC), a two-stage multi-period hybrid integer second-order cone programming (SOCP) robust model with partial DGs providing auxiliary service is developed. If the conic relaxation is not exact, a sequential SOCP is formulated using convex–concave procedure (CCP) and cuts, which can be quickly solved. Moreover, the exact solution of the original problem can be recovered. Furthermore, in view of the shortcomings of the large computer storage capacity and slow computational rate for the column and constraint generation (CCG) method, a method direct iteratively solving the master and sub-problem is proposed. Increases to variables and constraints to solve the master problem are not needed. For the sub-problem, only the model of each single time period needs to be solved. Then, their objective function values are accumulated, and the worst scenarios of each time period are concatenated. As an outcome, a large amount of storage memory is saved and the computational efficiency is greatly enhanced. The capability of the proposed method is validated with three simulation cases.

Robust day-ahead coordinated scheduling of multi-energy systems with integrated heat-electricity demand response and high penetration of renewable energy

As fossil fuels dry up and the proportion of renewable energy increases, multi-energy system (MES) becomes an effective way to realize renewable energy accommodation. High penetration of renewable energy brings challenges to the operation of power system, such as frequent curtailment of wind and the uncertainty of renewable energy threaten the security of the system. This paper proposes a two-stage robust scheduling model of MES considering integrated heat-electricity demand response (DR) to derive robust operation decisions. The proposed robust model minimizes the operation costs under base-case scenario, while guaranteeing all operation constraints are met in any scenarios. Power-to-Gas (P2G) equipment which converts surplus wind and photovoltaic generation into natural gas is included to effectively reduce curtailment of renewable generation. Furthermore, integrated heat-electricity DR, fully utilizing the couplings among various systems, is proposed to decrease operation costs and enhance system security against uncertainties. Column-and-Constraint Generation (CCG) method is used to effectively solve the proposed robust model. Numerical results in MATLAB indicate that robust optimization ensures secure operation of the system, and the integrated heat-electricity DR could promote the accommodation of renewable energy as well as enhance economic benefits and robustness of the system.