Joint Chance Constraints Research Articles

Optimizing the economic dispatch of weakly-connected mini-grids under uncertainty using joint chance constraints

AbstractIn this paper, we deal with a renewable-powered mini-grid, connected to an unreliable main grid, in a Joint Chance Constrained (JCC) programming setting. In several rural areas in Africa with low energy access rates, grid-connected mini-grid system operators contend with four different types of uncertainties: forecasting errors of solar power and load; frequency and outages duration from the main-grid. These uncertainties pose new challenges to the classical power system’s operation tasks. Three alternatives to the JCC problem are presented. In particular, we present an Individual Chance Constraint (ICC), Expected-Value Model (EVM) and a so called regular model that ignores outages and forecasting uncertainties. The JCC model has the capability to guarantee a high probability of meeting the local demand throughout an outage event by keeping appropriate reserves for Diesel generation and battery discharge. In contrast, the easier to handle ICC model guarantees such probability only individually for different time steps, resulting in a much less robust dispatch. The even simpler EVM focuses solely on average values of random variables. We illustrate the four models through a comparison of outcomes attained from a real mini-grid in Lake Victoria, Tanzania. The results show the dispatch modifications for battery and Diesel reserve planning, with the JCC model providing the most robust results, albeit with a small increase in costs.

Risk-averse distributionally robust optimization for construction waste reverse logistics with a joint chance constraint

The uncertainty of the amount of construction and demolition waste (CDW) generation affects the CDW reverse logistics network service level. We investigate a CDW reverse logistics network location-routing problem considering uncertainties. To minimize the total social cost, a two-stage risk-averse distributionally robust optimization model is developed, which aims to optimize the location and number of CDW disposal facilities and the CDW transportation scheme. We introduce the mean-conditional value at risk measure and a joint chance constraint into our model to consider the government’s risk aversion. The above model is approximated as a standard second-order cone programming model (SOCP) considering a special case. To exactly solve the SOCP, we design an outer approximation algorithm. Several performance tests and a case study are proposed, and sensitivity analysis provides helpful managerial insights.

Joint chance‐constrained coordinated scheduling for electricity‐heat coupled systems considering hydrogen storage

AbstractStrong interdependence between power production and heat supply from combined power and heat units could restrict the wind power integration. Deploying hydrogen storage, typically formed by the electrolyser, hydrogen tank, and fuel cell, could be a promising measure to accommodate surplus wind power and facilitate the coordinated operation of power and heat. The authors consider coordinated scheduling under wind power uncertainty for an electricity‐heat coupled system with hydrogen storage and propose a distributionally robust joint chance‐constrained coordinated scheduling (DRJCC‐CS) model, where the waste heat from the electrolyser and fuel cell is incorporated. The authors design distributionally robust joint chance constraints to account for wind power uncertainty related constraints and derive their tractable inner and outer mixed‐integer convex approximations. Consequently, the proposed DRJCC‐CS model is casted into a mixed‐integer convex programme. Case studies are conducted to demonstrate the effectiveness of the proposed DRJCC‐CS method.

Multi-source coordinated low-carbon optimal dispatching for interconnected power systems considering carbon capture

The overall electricity consumption of electrolytic aluminum and ferroalloy loads is significant, and some of these loads have dispatch potential that can be used to locally absorb wind power while reducing dependence on conventional thermal power. To characterize the uncertainty of wind power, a fuzzy set of wind power forecasting error probability distribution based on the Wasserstein distance was first established, and the approximate radius of the fuzzy set was corrected under extreme scenarios. By introducing joint chance constraints, the inequalities of uncertain variables were established at the lowest confidence level to improve the reliability of the model. Next, a two-stage distributed robust optimal scheduling model for source-load coordination was developed. In the first stage, wind power forecasting information was fully utilized to schedule the electrolytic aluminum load and optimize unit commitment. In the second stage, the uncertainty of wind power output was considered to schedule the ferroalloy load and optimize unit output. The model was approximately transformed into a mixed-integer linear programming problem and solved using a sequential algorithm. The IEEE 24-bus system was used for case validation. The validation results show that the model can effectively improve wind power absorption capacity, reduce overall operating costs, and achieve a balance between low carbon emissions and robustness.

Revenue stacking of BESSs in wholesale and aFRR markets with delivery guarantees

Battery Energy Storage Systems (BESSs) provide a crucial solution for mitigating the challenges posed by renewable energy intermittency, concurrently driving down energy costs within wholesale markets. To harness their potential, innovative business models are required for optimal BESS operation and revenue maximisation. This paper addresses this need by proposing a novel revenue stacking approach for the participation in Day-Ahead and automatic Frequency Restoration Reserve (aFRR) markets. By considering the uncertainty in the activation of aFRR events, the proposed model provides real-time delivery guarantees for a given reliability level, while maximising the revenues. The model is built over a novel characterisation of the uncertainty and a tight reformulation of the joint chance-constraints that drive energy and capacity guarantees. The effectiveness of the proposed approach is demonstrated through a case study based on real data from the Belgian market, yielding a 17.3% increase in profits over the individual chance-constraints approach.

Robust optimal power flow considering uncertainty in wind power probability distribution

This paper proposes an optimal power flow model that takes into account the uncertainty in the probability distribution of wind power. The model can schedule controllable generators under any possible distribution of wind power to ensure the safe and economic operation of the system. Firstly, considering the incompleteness of historical wind power data, the paper models the uncertainty of wind power using second-order moments of probability distribution and their fluctuation intervals. Subsequently, a robust optimal power flow model based on probability distribution model and joint chance constraints is established. The Lagrangian duality theorem is then employed to eliminate random variables from the optimization model, transforming the uncertainty model into a deterministic linear matrix inequality problem. Finally, a convex optimization algorithm is used to solve the deterministic problem, and the results are compared with traditional chance-constrained optimal power flow model. The feasibility and effectiveness of the proposed method are validated through case study simulations.

A chance-constrained optimization approach integrating project scheduling and material ordering to manage the uncertain material supply

To deal with the impact of uncertain material supply on the implementation of projects, we investigate a problem of concurrently finding a robust baseline schedule and a material ordering plan under different levels of uncertainty. Assuming the material warehouses are restricted, a chance-constrained model of the integrated resource-constrained project scheduling and material ordering problem with a confidence level (1−θ) is thus formulated. To tackle the difficult joint chance constraints, we firstly transform the chance-constrained model into a scenario-based integer programming model by the sample average approximation approach. Based on dominance relationship among scenarios, we then derive an exact branch-and-bound algorithm and a sampling-based genetic algorithm (SGA) to solve the model. For the branch-and-bound algorithm, a tailor-made branching scheme and four pruning rules are developed to accelerate the search process. For the SGA, two sampling methods and a local search procedure are presented to boost solution quality. Furthermore, extensive numerical experiments are carried out to test the performance of the proposed algorithms. Specifically, 4608 instances with four different confidence levels and six different sample sizes are generated. Then the Taguchi method is employed to calibrate the parameters of the SGA. Besides, the numerical results demonstrate that the proposed branch-and-bound algorithm can solve more instances with shorter CPU times than the CPLEX solver, and the solution quality of the proposed SGA is far better than that of two existing meta-heuristics and two variants of SGA. Finally, a sensitivity analysis for the key parameters of the model is conducted. By considering the uncertain material supply, limited warehouse capacity and various confidence levels, our research contributes to the effective decision on project scheduling and material ordering in real-world contexts.

Microgrid Formation and Real-Time Scheduling of Active Distribution Networks Considering Source-Load Stochasticity

The post-disruption microgrid (MG) formation and the subsequent scheduling are resilience-enhancing measures for active distribution networks (ADNs) against disastrous events. This paper proposes an integrated MG formation and scheduling solution, considering stochastic loads and distributed generators (DGs). Specifically, a first-stage MG formation model and a sec-ond-stage MG scheduling model are proposed to flexibly identify MG boundaries and restore as many critical loads as possible. The MG formation problem is formulated over multiple time steps to address time-varying generation outputs of intermittent DGs. Moreover, in the MG scheduling problem, the stochasticity of load demands and DG outputs is modeled as joint chance constraints (JCCs) to describe the possibility of security violations in the over-all system. An efficient approximation method is next developed to convexify the JCCs into solvable second-order cones. The interac-tion between the MG formation and scheduling by the model predictive control implementation achieves a responsive solution for enhancing the emergency support capability of ADNs by incor-porating power forecasts and real-time scheduling. The performance of the proposed method is verified in a modified real-world 136-node test feeder under catastrophic failures.

An Integrated Predictive Maintenance and Operations Scheduling Framework for Power Systems Under Failure Uncertainty

Maintenance planning plays a key role in power system operations under uncertainty as it helps providers and operators ensure a reliable and secure power grid. This paper studies a short-term condition-based integrated maintenance planning with operations scheduling problem while considering the possible unexpected failure of generators as well as transmission lines. We formulate this problem as a two-stage stochastic mixed-integer program with failure scenarios sampled from the sensor-driven remaining lifetime distributions of the individual system elements whereas a joint chance constraint consisting of Poisson Binomial random variables is introduced to account for failure risks. Because of its intractability, we develop a cutting-plane method to obtain an exact reformulation of the joint chance constraint by proposing a separation subroutine and deriving stronger cuts as part of this procedure. We also derive a second-order cone programming-based safe approximation of this constraint to solve large-scale instances. Furthermore, we propose a decomposition algorithm implemented in parallel fashion for solving the resulting stochastic program, which exploits the features of the integer L-shaped method and the special structure of the maintenance and operations scheduling problem to derive valid and stronger sets of optimality cuts. We further present preprocessing steps over transmission line flow constraints to identify redundancies. To illustrate the computational performance and efficiency of our algorithm compared with more conventional maintenance approaches, we design a computational study focusing on a weekly plan with daily maintenance and hourly operational decisions involving detailed unit commitment subproblems. Our computational results on various IEEE instances demonstrate the computational efficiency of the proposed approach with reliable and cost-effective maintenance and operational schedules. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0154 .

Maximizing the survival probability in a cash flow inventory problem with a joint service level constraint

This paper investigates a multi-period stochastic cash flow inventory problem with the aim of maximizing the long-term survival probability, which may be the objective of some retailers especially in periods of economic distress. Demand in each period is stochastic and can be non-stationary. In order to avoid too many lost sales under this objective, we introduce a joint chance constraint requiring the probability of no stockouts during the planning horizon to be higher than a specified service level. We develop a scenario-based model and a sample average approximation (SAA) model to solve the problem. A statistical upper bound on the survival probability based on SAA is provided and we discuss upper and lower bounds for the problem based on stochastic dynamic programming. We also propose a rolling horizon approach with service rate updating to test the out-of-sample performance of the two stochastic models and solve problems with long planning horizons. We test the two methods in large numerical tests and find that the rolling horizon approach together with the stochastic models can solve realistically sized problems in reasonable time.

Random games under normal mean–variance mixture distributed independent linear joint chance constraints

In this paper, we study an n player game where the payoffs as well as the strategy sets are defined using random variables. The payoff function of each player is defined using expected value function and his/her strategy set is defined using a linear joint chance constraint. The random constraint vectors defining the joint chance constraint are independent and follow normal mean–variance mixture distributions. For each player, we reformulate the joint chance constraint in order to prove the existence of a Nash equilibrium using the Kakutani fixed-point theorem under mild assumptions. We illustrate our theoretical results by considering a game between two competing firms in financial market.

Sample-Adaptive Robust Economic Dispatch With Statistically Feasible Guarantees

The high penetration of renewable energy brings significant uncertainty to the power grids. Taking economic dispatch (ED) as an example, the inaccurate prediction of renewable energy generations dramatically increases the dispatch cost and risks the power grid's reliable operation. The accurate distribution knowledge of the renewable generations enables modeling the ED as stochastic programming with joint chance constraints, which various classical methods can tackle. However, in practice, such distribution knowledge is inaccessible, and we can only observe samples from some unknown distribution. This makes conducting effective ED solely based on the observed samples challenging. It is particularly true when we need to handle the joint chance constraints. To tackle these challenges, we introduce the notions of statistical feasibility and statistically feasible ED to guarantee the satisfaction of the joint chance constraints. Specifically, we first propose a sample-adaptive robust optimization (RO) to decouple the joint constraints. We then identify that the inaccurate uncertainty set leads to RO's conservativeness, and then reconstruct the constraint-specific uncertainty sets. We design the corresponding sample-adaptive reconstruction-based RO (ReconRO) based on the reconstructed uncertainty sets to further enhance the ED's effectiveness.

A Two-Step Approach to Wasserstein Distributionally Robust Chance- and Security-Constrained Dispatch

This paper considers a security constrained dispatch problem involving generation and line contingencies in the presence of the renewable generation. The uncertainty due to renewables is modeled using joint chance-constraint and the mismatch caused by contingencies and renewables are handled using reserves. We consider a distributionally robust approach to solve the chance-constrained program. We assume that samples of the uncertainty are available. Using them, we construct a set of distributions, termed ambiguity set, containing all distributions that are close to the empirical distribution under the Wasserstein metric. The chance constraint is imposed for all distributions in the ambiguity set to form the distributionally robust optimization problem. This problem is nonconvex and computationally heavy to solve exactly. We adopt a two-step approach to find an approximate solution. In the first step, we construct a polyhedral set in the space of uncertainty that contains enough mass under all distributions in the ambiguity set. This set is constructed by solving several two-dimensional distributionally robust problems. In the second step, we solve a linear robust optimization problem where the uncertain constraint is imposed for all uncertainty values lying in the polyhedral set. We demonstrate the scalability and robustness of our method using numerical experiments.

Multiple Joint Chance Constraints Approximation for Uncertainty Modeling in Dispatch Problems

Multiple Joint Chance Constraints Approximation for Uncertainty Modeling in Dispatch Problems

Data-driven dynamic optimization for real-time parking reservation considering parking unpunctuality

This article examines a real-time parking reservation service that dynamically allocates a limited number of parking slots (which can be reused) to serve parking requests in a crowded area. The study takes into account two stochastic factors: (i) random arrivals of parking requests and (ii) driver parking unpunctuality, which are seldom simultaneously considered in the existing literature. This study formulates the dynamic parking resource allocation problem as a joint chance-constrained model to maximize the expected total revenue over a finite horizon. Due to the complex nature of the model considering the two stochastic factors, the study proposes a data-driven approach using joint chance constraint decomposition and sample average approximation to approximate the model to a deterministic mixed-integer programming model. Furthermore, a stratified ranked set sampling method is introduced to construct a high-quality sample set as the input to the deterministic model, and valid inequalities are designed to accelerate the solution process. Numerical experiments based on real-world data are conducted to validate the performance of the proposed approach under multiple parking scenarios and provide findings.

A robust optimization approach for inventory management with limited-time discounts and service-level requirement under demand uncertainty

A multi-period inventory management problem is considered for a retailer offering limited-time discounts and having a joint service-level requirement over the discount periods under demand uncertainty. In each period, the retailer, only knowing a reference value and the bounds of the uncertain demand, has to determine the order quantity of the product before demand realizations to maximize the total profit over the planning horizon while achieving the required service level over the discount periods. A budgeted uncertainty set is adopted to accommodate demand uncertainty and a joint chance constraint is used to formulate the service-level requirement. The retailer problem is formulated as an affinely adjustable robust chance-constrained model and is transformed into a linear program. A double-layer iterative approach is proposed to solve the problem. The outer layer uses a posteriori method to update the budget coefficient. The inner layer solves the affinely adjustable robust chance-constrained model with the updated budget coefficient, and obtains the updated order quantities. A case study based on real data demonstrates that the proposed approach outperforms other approaches by better balancing the average realized profit and the realized service level. Moreover, the results show that the retailer can obtain a larger average realized profit by gradually than abruptly changing the prices.

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.

Convexity of linear joint chance constrained optimization with elliptically distributed dependent rows

In this paper, we study the convexity of the linear joint chance constraints. We assume that the constraint row vectors are elliptically distributed. Further, the dependence of the rows is modeled by a family of Archimedean copulas, namely, the Gumbel–Hougaard copulas. Under mild assumptions, we prove the eventual convexity of the feasibility set.

Distributionally robust chance-constrained programming for multi-period emergency resource allocation and vehicle routing in disaster response operations

Emergency resource allocation and vehicle routing are the most essential and inseparable response actions in emergency management after disasters. In particular, disaster response operations are significantly affected by high uncertainty and incomplete dynamic information of demand and risk. For this purpose, we construct a risk-based ambiguity set for modeling the distributional uncertainty in the demand and describing the coupling relationship between the demand and risk in different periods (e.g., secondary disasters strike). Against this background, we present two distributionally robust chance-constrained programming (DRCCP) models with both individual and joint chance constraints for multi-period emergency resource allocation and vehicle routing problem under demand distributional ambiguity. For DRCCP with individual chance constraint, we can derive the computationally tractable reformulation of the proposed model with a safe approximation index. For DRCCP with joint chance constraint, we can use Bonferroni’s approximation to obtain a set of tractable individual chance constraints. As for the solution method, we first decompose the original model into the emergency resource allocation and vehicle routing subproblems, and then develop an efficient adaptive large neighborhood search (ALNS) algorithm. We evaluate the performance of the proposed ALNS heuristic algorithm on a hypothetical instance set and show that the ALNS algorithm is capable of producing high-quality solutions within a reasonable computing time. We also conduct a real case study of the Wenchuan earthquake in China to demonstrate the superiority of the proposed DRCCP approach in a comparative perspective. In addition, we provide some possible extensions of the considered problem. Finally, we explore the managerial insights that may be useful for the disaster response operation.

Computationally Efficient Data-Driven Joint Chance Constraints for Power Systems Scheduling

Although data-driven nonparametric Joint Chance Constraints (JCCs) may lead to more reliable decision-making than individual chance constraints, their computational complexity is a major bottleneck. This paper presents computationally efficient data-driven nonparametric joint chance-constrained programming for multi-interval power systems management. Reserve and transmission line constraints are modeled as data-driven JCCs. Piecewise uniform kernel functions incorporate historical data of uncertain parameters into optimization. Data-driven nonparametric JCCs are modeled as a product of integrated kernel functions. Two approaches are proposed to linearize data-driven nonparametric JCCs. i) The noncontinuous kernel function is linearized with Special Ordered Sets of type 1 (SOS1) variables. ii) A tight convex envelope of multilinear monomial terms, which appear due to the product of kernel functions, is approximated by an optimization subproblem making the scheduling problem bi-level optimization. The continuity and linearity of the lower-level convex envelope approximation subproblem allow replacing it with optimality conditions to form a single-level scheduling problem. Simulation results show the tightness of the proposed linearization approaches and the computational efficiency of data-driven JCC programming.