Chance-constrained Optimization Model Research Articles

Closed-loop supply chain network design with price-greenness-sensitive demand: A distributionally robust chance-constrained optimization approach

In response to the government’s heightened focus on recycling and remanufacturing, as well as the growing awareness among consumers about environmental security, manufacturing companies are currently required to establish efficient closed-loop supply chain networks in order to improve their socialreputation and competitive advantage. This study investigates the optimization of a Closed-Loop Supply Chain (CLSC) network that involves multiple products, multiple periods, and uncertain returns, which also considers the influence of many factors, such as carbon cap-and-trade policy, raw part procurement discounts, and facility capacity constraints, on the supply chain. Simultaneously, customer demand is sensitive to both product pricing and product greenness, and product greenness can be improved by investing in emission reduction technologies. To address the uncertainty in the returns, we propose a two-stage distributionally robust chance-constrained optimization model, which is transformed into a mixed integer linear programming model. To efficiently address the complex problem, we designan improved Benders decomposition (IBD) algorithm. The experimental results confirm that the IBD algorithm has significant advantages when compared to the Benders decomposition algorithm. Additionally, this study conducted a sensitivity analysis on key parameters and proposed operation suggestions of practical importance.

Modeling 3E-S sustainable development problem by an ambiguous chance constrained optimization method

The environmental-energy-economic-social (3E-S) sustainable development problem aims to balance the environment, energy, and economy upon which human society critically relies. However, it is further complicated by various ambiguous input parameters about greenhouse gas (GHG) emissions, electricity consumption, gross domestic product (GDP), and unemployment rate. To address this complexity, this paper develops an ambiguous chance constrained optimization model with multiple chance soft constraints for optimizing the 3E-S sustainable development problem by strategically allocating labor across key sectors. Tractable formulations of the proposed model are derived by constructing three types of ambiguity sets for input parameters with ambiguous probability distributions. The first is to construct support-mean ambiguous distribution sets with little known information, the second is to extend support-mean ambiguity sets to mean-covariance ambiguity sets by incorporating interactive relationships among sectors, and the third is to extend support-mean ambiguity sets to inner-outer ambiguity sets with Wasserstein metric from globalized distributionally robust perspective. The availability and computability of the proposed model have been illustrated by planning India’s 3E-S sustainability for the year 2030. The results suggest that (i) the proposed method exhibits the capability to offer flexible policy measures; (ii) India should prioritize the development of the Mining and quarrying sector, the Transport, storage, communication, and services sector, and the Constructions sector; (iii) India needs to make greater efforts for promoting GDP while controlling GHG emissions and energy consumption; (iv) The policy measures and objectives are influenced by tolerance levels, support intervals, interactive relationships among sectors, globalized sensitivity, and Wasserstein ball size.

Stochastic home care transportation with dynamically prioritized patients: An integrated facility location, fleet sizing, and routing approach

We study a home health care (HHC) problem that is characterized by prioritized patients and uncertain demands. In practice, HHC supply chain networks often struggle to meet high demand due to a shortage of service vehicles. Additionally, disruptions caused by natural calamities and pandemics (e.g., COVID-19) further compound these challenges, necessitating the consideration of real-life characteristics such as patient priorities, infrastructure locations, and transportation of medical supplies with uncertain demands. To formulate the problem, we propose a multi-depot and multi-period chance-constrained optimization model with precedence constraints, assuming that the demand quantities for medical supplies are random variables. Since patients’ medical conditions vary in severity, the priority of each patient is translated into a time-dependent potential healthcare cost that changes dynamically over the planning horizon. The solution to the proposed model determines the optimal locations for the base Mobile Health Facilities (MHFs) and the fleet size of HHC vehicles, and generates scheduling and routing plans to visit patients within specified time windows. We propose a unique three-phase solution approach, integrated with stochastic simulation, to address the problem. We then assess the robustness of the proposed model based on a realistic case of HHC service provision in Hong Kong and explore the optimal values for two model parameters, namely the Vehicle Threshold Index and the MHF Threshold Index. The performance evaluation tests show that the proposed solution method is efficient and effective for solving real-world problems.

Distributionally robust chance-constrained optimization for the integrated berth allocation and quay crane assignment problem

In this paper, we are the first to propose a distributionally robust chance-constrained (DRCC) optimization model for the integrated berth allocation and quay crane assignment problem (BACAP) in container terminals. In contrast to the classical deterministic BACAP model, we consider the arrival time to be uncertain due to the frequent arrival delays in ports. We then impose a chance constraint that the service time must start after the uncertain arrival time with a probability of at least 1−ϵ, where 1−ϵ represents the target service level in container terminals (ϵ represents the target risk tolerance of the port manager). Under the moment-based ambiguity set, we reformulate the DRCC model into a mixed integer semi-definite programming (MISDP) model. Additionally, we develop an efficient decomposition branch-and-bound algorithm to solve the MISDP model and obtain the exact solution. Fortunately, a special case of the DRCC model arises when the mean and covariance utilized in the ambiguity set are precise, allowing for the transformation of the DRCC model into a mixed integer programming (MIP) model. This conversion significantly reduces the complexity of the problem. Impressively, the solving time of the MISDP model with the decomposition branch-and-bound algorithm is comparable to that of the transformed MIP model. The numerical results show that our model can achieve a schedule with high service at low cost. Meanwhile, we have made an intriguing discovery that the correlation between the target risk tolerance and the actual service level can be depicted as a staircase function regardless of the datasets. This finding offers crucial insights for port management, enabling them to strike a balance between the cost and actual service level by determining an appropriate target service level.

Data-Driven Chance-Constrained Schedule Optimization of Cascaded Hydropower and Photovoltaic Complementary Generation Systems for Shaving Peak Loads

The coordinated scheduling of cascade hydropower with photovoltaic (PV) power stations can significantly improve the utilization rate of delivery transmission lines. However, the inherent uncertainty associated with photovoltaic (PV) forecasts challenges the reliable and economic operation of the complementary energy system. Against this background, in this paper, a day-ahead, chance-constrained scheduling for cascaded hydro–photovoltaic complementary generation systems (CHPSs) considering the transmission capacity is proposed. Firstly, the uncertainty of PV forecast errors is simulated by a probability density function fitted using kernel density estimation with historical sampling data. Then, a chance-constrained optimization model considering peak-shaving demands of the receiving-end power grid is developed to determine the day-ahead optimal schedules of CHPSs. Also, complex hydraulic coupling and unit operation constraints of cascade hydropower are considered in the proposed model. To deal with the nonlinear and stochastic constraints, an efficient linearization method is adopted to transform the proposed model into a mixed-integer linear programming (MILP) problem. Finally, the effectiveness and feasibility are verified by case studies. The results show that the day-ahead schedule optimized by the proposed method can fully balance peak-shaving and photovoltaic accommodation while considering photovoltaic output uncertainty.

An optimal capacity allocation method for integrated energy systems considering uncertainty

The randomness and fluctuation of wind power output will cause certain waste in capacity allocation of integrated energy system. Therefore, a robust chance constrained optimization model is proposed to solve the shortcomings of the traditional model in wind power output analysis, such as weak reliability and conservative results. First, the uncertainty of wind power output is analyzed. Second, a robust chance constrained optimization model combining stochastic programming and robust optimization is established to deal with the uncertainty of the output power of the integrated energy system objectively. Finally, the results are compared with the existing integrated energy system capacity configuration, and the validity of the model is verified. The results show that the robust chance constrained optimization model proposed in this paper can effectively reduce the capacity cost by 38.2% while ensuring the robustness of the system in wind power uncertainty analysis.

A distributionally robust production planning model for maximizing customer satisfaction with budget and carbon emissions constraints

This paper investigates a new stochastic green production planning problem without full knowledge about the probability distribution on customer demands. The only information regarding the demand is the means and covariance matrix. The goal of the study is to provide an eco-conscious manufacturer with solution methods capable of yielding, within a reasonable amount of computation time, a robust solution that maximizes customer satisfaction while taking into account the constraints of budgets and periodic carbon caps spontaneously set out of environmental awareness. We formulate the problem into a novel distributionally robust chance-constrained optimization model under demand uncertainty set. To effectively solve the model, two heuristics are proposed. The widely used sample average approximation (SAA) scheme is first deployed as a benchmark method. However, the computation time increases rapidly with the problem size. In some instances, even feasible solutions cannot be found within a reasonable amount of computation time. We then develop an approximated mixed-integer reformulation on the basis of second order cone program (SOCP). A real-world case is solved and extensive numerical experiments show that the SOCP heuristic is superior over the SAA heuristic in terms of computation time while the solution quality is only slightly lower in most of instances. Finally, sensitivity analysis on budgets, periodic carbon caps, the maximum budget overrun risk level, initial inventory levels, and the magnitude of demand uncertainty are conducted and useful managerial insights are derived.

Chance-constrained optimization of distributed power and heat storage in integrated energy networks

Energy storage is widely recognized as an effective alternative to match the instantaneous imbalance between distributed renewable energy sources and loads. However, suitable installation sites and capacities are necessarily determined in integrated energy networks. This paper constructs a two-layer optimization model of integrated power and heat networks to obtain the optimal installation nodes and capacities of electric energy storage (EES) and thermal energy storage (TES) units. In the optimization model, the interchange of power to heat with storage units is collaborated to unitize distributed renewable energy sources and decrease comprehensive costs, including investment cost of storage units and operation cost. A chance-constrained optimization model is proposed to transform the uncertain variables of electric and heat loads into a deterministic optimization problem. The impacts of confidence levels involved in load probability distributions on the optimal nodes and capacities are analyzed. The results demonstrate that the optimal installation nodes are determined by the distributions of distributed power sources and load ratios in networks. The increased confidence level globally raises the optimal installation capacities of EES and TES. When the confidence level increases from 0.50 to 0.99, the utilization rate of wind power increases by 3.91 %, whereas the comprehensive cost increases by 22.34 %. The hybrid integrations of EES and TES consume more uncertain renewable power and improve the economic performances by the interchange of power to heat.

Enhancing Flexibility at the Transmission-Distribution Interface With Power Flow Routers

Distribution systems are becoming more active and provide more power support for transmission systems via their connection points at the transmission-distribution (T-D) interfaces. This paper proposes a chance constrained optimization model to estimate the flexibility area at a T-D interface. In this model, network flexibility is for the first time involved and facilitated by introducing power flow routers (PFRs) into the distribution system. The chance-constrained flexibility area (CCFA) given by the proposed model provides a description of the aggregated power dispatchability of distribution system with the guarantee of low risks of voltage violations under renewable uncertainties. We exploit the mechanism of network flexibility and reveal that it significantly reduces the voltage standard deviations, which cannot be achieved by node power flexibility. To evaluate the CCFA, an efficient two-layer iteration algorithm is designed. The inner layer finds the boundary point of CCFA along a given direction for active and reactive power. The outer layer determines those key directions to be included in the estimation as the corresponding points capture the features of the CCFA boundary. Case studies show that PFRs significantly enlarge the CCFA, which reveals that network flexibility is an important supplement to node power flexibility in T-D interaction.

Distributionally Robust Frequency Constrained Scheduling for an Integrated Electricity-Gas System

Power systems are shifted from conventional bulk generation toward renewable generation. This trend leads to the frequency security problem due to the decline of system inertia. On the other hand, natural gas-fired units are frequently scheduled to provide operational flexibility due to their fast adjustment ability. The interdependence between the power system and natural gas system is thus intensified. In this context, this paper considers the frequency constrained scheduling problem from the perspective of an integrated electricity-gas system under wind power uncertainty. A distributionally robust (DR) joint chance-constrained optimization model is proposed to co-optimize the unit commitment and virtual inertia provision from wind farm systems. This model incorporates both frequency constraints and natural gas system (NGS) operational constraints and addresses the wind power uncertainty by designing DR joint chance constraints. It is shown that this model admits a mixed-integer second-order cone programming. Case studies demonstrate that the proposed method can provide a highly reliable and computationally efficient solution and show the importance of incorporating NGS operational constraints in the frequency constrained scheduling problem.

A Robust Strategy for Leveraging Soft Open Points to Mitigate Load Altering Attacks

Power system cybersecurity is emerging as a critical and urgent problem to the energy sector due to the ongoing power grid modernization initiative. Load altering attack (LAA) is an important category of cyberattacks on the modern power systems, in which the attackers may damage the grid by viciously altering the remotely controllable loads (RCL) that are not properly protected. In order to mitigate the impacts of LAAs on the distribution systems, the promising soft open point (SOP) technology is deployed in this study. A two-stage optimization framework is proposed for the optimal installation and operation of SOPs for defending the distribution systems against LAAs. A chance-constrained optimization model is developed to guarantee the confidence level of the proposed two-stage model of SOPs in mitigating the impacts of LAAs. Further, a Wasserstein metric based distributionally robust chance-constrained (DRCC) optimization method is developed to ensure the robustness of the proposed model against the ambiguity of the empirical probability distribution in practice. Case studies were performed on a 69-bus test system to validate the proposed method. The results of case studies show that the proposed framework is able to mitigate the impacts of LAAs on distribution systems with the installation of SOPs. By applying the DRCC optimization method, the proposed model manages to keep satisfactory confidence levels under the ambiguous probability distributions in the case studies.

Optimization of booster chlorination for water distribution system under dual uncertainties

ABSTRACT Chlorine is generally used as disinfectant to sustain water quality in water distribution system (WDS). To prevent the occurrence of disinfectant by-product as well as control the growth of microorganism, the residual chlorine concentration in WDS should be kept between acceptable intervals. However, the chlorine concentration was uncertain due to chlorine decay, imprecise measure, hydraulic and topological conditions. In addition, nodal response coefficients to unit disinfectant mass in the left-hand-side of the constraints were also uncertain, which is affected by uncertain pipe diameter, roughness coefficients, nodal demand, bulk decay coefficient, and wall decay coefficient, etc. Therefore, in this paper a fuzzy chance-constrained optimization model under dual uncertainties was proposed and applied to optimize the chlorine injection mass to maintain chlorine in WDS distributed uniformly with consideration of randomness of chlorine concentration limits as well as uncertainty in response coefficient. The total injection mass decreases with the uncertainty level under possibility of dominance and necessity of dominance indices, while increases with the uncertainty level under possibility of strict dominance and necessity of strict dominance indices, which is effective in chlorine injection management.

The impact of optimally dispatched energy storage devices on electricity price volatility

A high level of electricity price volatility increases financial risks and safety issues in power system operations. Storage devices as an efficient solution to mitigating price volatility have attracted extensive attention. In this paper, the mechanism of how storage devices impact price volatility is investigated at first. Through analyzing the connection between an economic dispatch problem and its Lagrange dual, we reveal that the capacity and charge/discharge power of a storage device installed at a node have an aggregate impact on the range of the price changes of that node. A model to calculate the sensitivity of price volatility to storage device parameters is proposed. To determine the optimal nodes to place storage devices and the optimal storage capacities to constrain the price change levels, this paper develops a chance-constrained optimization model, and a scenario-based approach is adopted to solve the problem. The IEEE 24-bus system is employed in the case studies, associated with Gurobi and Julia. Simulation studies show that if there is one node with the highest price change all the time, the price volatility must have the highest sensitivity to the storage device on that node. However, in the case where there are multiple nodes having the highest price changes at different times, a node with high price change does not necessarily mean high sensitivity. The simulation studies verify that the chance-constrained optimization model is effective to suppress excess price changes.

Aggregate Flexibility of Virtual Power Plants With Temporal Coupling Constraints

The virtual power plant (VPP) provides an efficient way to manage a bunch of distributed energy resources (DERs). In order to participate in the system operation and power market as a whole, the estimation of VPP power flexibility is indispensable. However, the temporal coupling constraints make it intractable to aggregate the power flexibility directly. In this paper, we formulate the flexibility evaluation problem as a chance constrained optimization model incorporated the randomness of renewables. To make it tractable, the evaluated flexibility is represented by the combination of a virtual generator and a virtual battery which is decomposed by a robust optimization algorithm. Then, a novel high-dimensional polytope based bound shrinking method is proposed to estimate the parameters of the virtual generator and virtual battery. The numerical test results show the proposed method obtain much more precise results with lower conservativeness than those of existing methods. The disaggregation feasibility of the evaluation results is also verified.

Optimal Reserve and Energy Scheduling for a Virtual Power Plant Considering Reserve Activation Probability

With the increasing share of variable and limitedly predictable renewable energy in power systems worldwide, ensuring reserve capacity to maintain the balance of supply and demand becomes more important. On the other hand, the development of the virtual power plant model (VPP) allows renewable sources and energy storage to participate in reserve service. This paper addresses the optimal reserve bidding strategy problem of a VPP comprising of renewable energy resources (RESs), energy storage systems (ESSs), and several customers. The VPP participates in balance capacity (BC), day-ahead (DA), and intra-day (ID) markets. The scheduling problem is formulated as a two-stage chance-constrained optimization model taking the uncertainty of RESs production, load consumption, and probability of reserve activation into account. The response of VPP after its reserve capacity is called and generated is also considered to increase the operational flexibility of VPP. The proposed model is implemented on a test VPP system, and the effects of RESs sizing, ESSs sizing, and the probability of reserve activation are analyzed. Results indicate that the proposed model can perform well under real-world conditions.

Distributionally robust chance-constrained flexibility planning for integrated energy system

Inflexible combined heat and power (CHP) plants and uncertain wind power production result in excess power in distribution networks, which leads to inverse power flow challenging grid operations. Power-to-X facilities such as electrolysers and electric boilers can offer extra flexibility to the integrated energy system. In this regard, we aim to jointly determine the optimal Power-to-X facility sizing and integrated energy system operations in this study. To account for wind power uncertainties, a distributionally robust chance-constrained model is developed to characterize wind power uncertainties using ambiguity sets. Linear decision rules are applied to analytically express real-time recourse actions when uncertainties are exposed, which allows the propagation of wind power uncertainties to gas and heat systems. Accordingly, the developed three-stage distributionally robust chance-constrained model is converted into a computationally tractable single-stage mixed-integer conic model. A case study validates the effectiveness of introducing the electrolyser and electric boiler into the integrated energy system, with respect to the decreased system cost, expanded CHP plant flexibility and reduced inverse power flow. The developed distributionally robust optimization model exhibits better effectiveness and robustness compared to a chance-constrained optimization model assuming wind forecast errors follow Gaussian distribution. Detailed profit analysis reveals that although the overall system cost is minimized, the profit is distributed unevenly across various stakeholders in the system. The profit mainly falls with the wind power plants, which therefore are most motivated to make investments in the flexibility resources. Other parties rely on additional policies such as bilateral contracts with the wind power plants to gain incentives to invest. The findings from this study can be used to motivate policy-makers to make proper regulations to incentivize investments in flexibility resources and establish a more reliable power grid.

A Fuzzy Credibility-Based Chance-Constrained Optimization Model for Multiple-Objective Aggregate Production Planning in a Supply Chain under an Uncertain Environment

A Fuzzy Credibility-Based Chance-Constrained Optimization Model for Multiple-Objective Aggregate Production Planning in a Supply Chain under an Uncertain Environment

Lagrangian relaxation based heuristics for a chance-constrained optimization model of a hybrid solar-battery storage system

We develop a stochastic optimization model for scheduling a hybrid solar-battery storage system. Solar power in excess of the promise can be used to charge the battery, while power short of the promise is met by discharging the battery. We ensure reliable operations by using a joint chance constraint. Models with a few hundred scenarios are relatively tractable; for larger models, we demonstrate how a Lagrangian relaxation scheme provides improved results. To further accelerate the Lagrangian scheme, we embed the progressive hedging algorithm within the subgradient iterations of the Lagrangian relaxation. We investigate several enhancements of the progressive hedging algorithm, and find bundling of scenarios results in the best bounds. Finally, we provide a generalization for how our analysis extends to a microgrid with multiple batteries and photovoltaic generators.

Uncertainty-fully-aware coordinated dispatch of integrated electricity and heat system

The power system operators are facing significant challenges in the operation due to the increasing penetration level of renewable energy sources (RESs). The flexibility from the district heating system (DHS) is attracting considerable interest to deal with RES uncertainty. This paper formulates a distributionally robust chance-constrained (DRCC) optimization model of the integrated electricity and heating system (IEHS) dispatch to hedge the uncertainty of RES and exploit the flexibility of the DHS. In particular, the uncertainty from the electrical power system (EPS) is propagated to the DHS so that both systems can respond to the uncertainty of RES. The uncertainty of the wind power is modeled by an ambiguity set, which defines a family of probability distributions with the same first and second-moment property. Real-time regulation actions of both the EPS and DHS to respond to the wind power forecast errors are modeled through the data-driven affine control policies. To achieve computational tractability, the proposed DRCC model is reformulated as a second-order cone program (SOCP). The simulation results tested on the integrated six-bus and seven-node system demonstrate that the proposed DRCC model outperforms the chance-constraints dispatch based on Gaussian distribution for the secure operation of the IEHS.

Rationalizing ex situ collection of reproductive materials for endangered livestock breed conservation

Improvements in ex situ storage of genetic and reproductive materials offer an alternative for endangered livestock breed conservation, but collections should be optimized cost-effectively to avoid duplication, and with reference to the sustainability of in situ breeds. We developed a multi-period chance-constrained optimization model to rationalize collections of endangered livestock breeds at risk of in situ extinction in Spain. The model configures collections by determining the least-cost optimal collection locations, timing and material quantities (semen doses). A decision variable defining an “acceptable level of risk” allows decision makers to specify tolerable levels of in situ breed endangerment when taking ex situ collection and storage decisions. Using data from 18 gene banks we demonstrate how collections can be rationalized, and derive cost curves relating marginal ex situ collection cost and accepted probability of in situ extinction covering the period 2018 to 2060. The modelling framework can be replicated in countries seeking to rationalize ex situ collections under limited conservation budgets and uncertain in situ extinction risks.