Chance Constrained Research Articles

Condition-Based Maintenance for Wind Farms Using a Distributionally Robust Chance Constrained Program

Condition-Based Maintenance for Wind Farms Using a Distributionally Robust Chance Constrained Program

Data-driven stochastic AC-OPF using Gaussian process regression

At present, electricity generation is responsible for more than a quarter of the greenhouse gas emissions in the US. Integrating significant amounts of renewable energy sources into a power grid is probably the most accessible way to reduce carbon emissions from power grids and slow down climate change. Unfortunately, the most accessible renewable power sources, such as wind and solar, are highly intermittent and thus bring a lot of uncertainty to power grid operation and challenge existing optimization and control policies. The chance-constrained (CC) alternating current optimal power flow (AC-OPF) framework finds the minimum cost of generation dispatch, maintaining the power grid operation within security limits with a prescribed probability. Unfortunately, the AC-OPF problem’s chance-constrained extension is non-convex, computationally challenging, and requires knowledge of system parameters and also needs additional assumptions to be made about the behavior of renewable generation probability distribution. Known linear and convex approximations to the above problems, though tractable, are too conservative for operational practice and do not consider uncertainty in system parameters. This paper presents an alternative data-driven approach for solving the stochastic AC-OPF problem, based on Gaussian process regression (GPR) to close this gap. The Gaussian process (GP) approach learns a simple yet non-convex data-driven approximation to the AC power flow equations that can incorporate uncertain inputs. The latter is then used to determine the solution of CC-OPF efficiently, by accounting for both input and parameter uncertainty. The practical efficiency of the proposed approach using different approximations for GP-uncertainty propagation is validated and illustrated using a number of standard IEEE test cases.

Robust flexibility needs assessment with bid matching framework for distribution network operators

For market-based procurement of low-voltage flexibility, DSOs identify the amount of flexibility needed for resolving probable distribution network (DN) voltage and thermal congestion. A framework is required to avoid over or under-procurement of flexibility in the presence of uncertainty. To this end, we propose a scenario-based robust chance-constrained (CC) day-ahead flexibility needs assessment (FNA) framework. The CC level is analogous to the risk DSO is willing to take in flexibility planning. Multi-period optimal power flow is performed to calculate the amount of flexibility needed to avoid network issues. Future uncertainties are considered as multiple scenarios, which are utilized to solve the flexibility needs assessment optimal power flow (FNA-OPF) problem. The FNA tool calculates the temporal and locational ramp-up and ramp-down flexibility needs of the DN. We also propose a Pareto optimal mechanism for selecting CC level to reduce flexibility needs while reducing DN congestion. Zonal clustering of a DN feeder is performed using electrical distance as a measure, with spatial partitioning and silhouette coefficient. Further, a zonal and nodal bid matching algorithm is proposed, where alternate flexibility bids in the same zone are identified in case nodal matching is not exact, compared to the calculated FNA. The proposed bid matching algorithm assists the system operator in bid selection by avoiding rerunning power flow evaluations. The zonal and nodal bid matching leads to more than 43% additional reduction of probable network incidents compared to naive nodal bid matching. Furthermore, we also calculate FNA for a real German DN with 646 nodes and 331 consumers.

Capturing delays in response of emergency services in Delhi

Response time of an ambulance plays a significant role in pre-hospital care. The absence of response standards in India has made it challenging for emergency services to provide efficient and timely pre-hospital services. Also, traffic congestion in a city like Delhi may prove detrimental to a patient who needs urgent transport. Since travel times fluctuate throughout the day, solving an ambulance location problem with average travel times would not suffice. Therefore, the Gaussian Mixture Model (GMM) has been used to capture the variability in travel time between each origin-destination pair for Delhi's vast transportation network. This variation in travel times and delays in the dispatch of an ambulance (pre-trip delay) has been incorporated into the traditional double standard model as a chance constraint. The study thus builds three different variations of model, one being deterministic and the other two stochastic with probabilistic response times. These models are referred to as the Chance Constrained Double Standard Model (cc-DSM), Chance Constrained Double Standard Stochastic Model (cc-DSSM) and Double Standard Stochastic Model (DSSM). This study shows the similarity of previously used relocation approaches with the current approach and highlights the difference between vehicle busyness concept from the concept of multiple coverage.

Distributionally Robust Chance Constrained Optimization for Providing Flexibility in an Active Distribution Network

In this paper, we propose a distributionally robust chance constrained (DRCC) optimization problem for the operation of an active distribution network (ADN). The ADN’s operator uses the proposed problem to centrally optimize the dispatch plan of his resources, namely photovoltaic (PV) and battery energy storage (BES) systems, and to participate in wholesale real/reactive power and flexibility markets. We model the uncertainties in the problem by knowing a set of probability distributions, i.e., an ambiguity set. The uncertainties include production capability of PV systems, end-users’ consumption, requested flexibility by the external network’s operator, and voltage magnitude at the point of common coupling (PCC). The resulting formulation is a DRCC optimization problem for which a solution methodology based on freely available solvers is presented. We evaluate the performance of proposed solution in the numerical results section by comparing it with two benchmark models based on stochastic and chance constrained (CC) optimization.

Stochastic Steady-State Security Assessment for Power System Operation by Chance Constrained Optimal Power Flow-Based Load Shedding

The operational uncertainty of the power system is significantly increased by the high penetration of renewable energy and diversification of load behavior, which have aroused the novel research on stochastic steady-state security assessment (SSSA) urgently when the operational condition gets worse and contingency happens more frequently. First of all, a novel contingency analysis model has been proposed based on chance constrained optimal power flow (CCOPF) in this work, which is able to give a comprehensive consideration on conventional generator outage, uncertainty of system operation and load shedding of interruptible loads (ILs). Moreover, the proposed model is very practical and can be solved much efficiently by an iteration method so that the purpose of secure system operation after contingency can be better realized. Second, to further enhance the steady-state security of the system, a novel pricing method for ILs has been developed here. In addition, a composite contingency ranking index has been also established to reflect the severity of generator outage more accurately. Finally, extensive study results based on a modified IEEE-39 test system have demonstrated the validity of the proposed model, which can hold a good balance amongst security, economy, controllability, and maneuverability.

ALSO-X and ALSO-X+: Better Convex Approximations for Chance Constrained Programs

In a chance constrained program (CCP), decision makers seek the best decision whose probability of violating the uncertainty constraints is within the prespecified risk level. As a CCP is often nonconvex and is difficult to solve to optimality, much effort has been devoted to developing convex inner approximations for a CCP, among which the conditional value-at-risk ([Formula: see text]) has been known to be the best for more than a decade. This paper studies and generalizes the [Formula: see text], originally proposed by Ahmed, Luedtke, SOng, and Xie in 2017 , for solving a CCP. We first show that the [Formula: see text] resembles a bilevel optimization, where the upper-level problem is to find the best objective function value and enforce the feasibility of a CCP for a given decision from the lower-level problem, and the lower-level problem is to minimize the expectation of constraint violations subject to the upper bound of the objective function value provided by the upper-level problem. This interpretation motivates us to prove that when uncertain constraints are convex in the decision variables, [Formula: see text] always outperforms the [Formula: see text] approximation. We further show (i) sufficient conditions under which [Formula: see text] can recover an optimal solution to a CCP; (ii) an equivalent bilinear programming formulation of a CCP, inspiring us to enhance [Formula: see text] with a convergent alternating minimization method ([Formula: see text]); and (iii) an extension of [Formula: see text] and [Formula: see text] to distributionally robust chance constrained programs (DRCCPs) under the [Formula: see text]Wasserstein ambiguity set. Our numerical study demonstrates the effectiveness of the proposed methods. Funding: This work was supported by the Division of Civil, Mechanical and Manufacturing Innovation [Grant 2046426]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2021.2225 .

Chance‐constrained model predictive control a reformulated approach suitable for sewer networks

AbstractIn this work, a revised formulation of chance‐constrained (CC) model predictive control (MPC) is presented. The focus of this work is on the mathematical formulation of the revised CC‐MPC, and the reason behind the need for its revision. The revised formulation is given in the context of sewer systems, and their weir overflow structures. A linear sewer model of the Astlingen benchmark sewer model is utilized to illustrate the application of the formulation, both mathematically and performance‐wise through simulations. Based on the simulations, a comparison of performance is done between the revised CC‐MPC and a comparable deterministic MPC, with a focus on overflow avoidance, computation time, and operational behavior. The simulations show similar performance for overflow avoidance for both types of MPC, while the computation time increases slightly for the CC‐MPC, together with operational behaviors getting limited.

Examination of the Chance Constrained Optimal WT Penetration Level in Distorted Distribution Network with Wind Speed and Load Uncertainties

Harmonic penetration can be problematic by the growing interconnection of Wind Turbines (WTs) in distribution networks. Since the active power outputs of WTs and loads in the distribution system have uncertainties, the optimal WT penetration level problem can be considered to have a stochastic nature. In this study, this problem is taken into account in the stochastic optimization method with the consideration of uncertainties in wind speed and distribution network load profile. Chance constraint programming is taken into account in the determination of optimal WT penetration levels by applying the Genetic Algorithm (GA) along with Monte Carlo Simulation (MCS). The harmonic power flow analysis based on the decoupled harmonic load flow approach is employed in the distorted distribution network. Chance constraints are considered for the harmonic issues such as the Total Harmonic Distortion of Voltage (VTHD), Individual Harmonic Distortion of Voltage (VIHDh), and Root Mean Square of Voltage (VRMS).

Chance-Constrained Based Voltage Control Framework to Deal with Model Uncertainties in MV Distribution Systems

In this paper, a chance-constrained (CC) framework is developed to manage the voltage control problem of medium-voltage (MV) distribution systems subject to model uncertainty. Such epistemic uncertainties are inherent in distribution system analyses given that an exact model of the network components is not available. In this context, relying on the simplified deterministic models can lead to insufficient control decisions. The CC-based voltage control framework is proposed to tackle this issue while being able to control the desired protection level against model uncertainties. The voltage control task disregarding the model uncertainties is firstly formulated as a linear optimization problem. Then, model uncertainty impacts on the above linear optimization problem are evaluated. This analysis defines that the voltage control problem subject to model uncertainties should be modelled with a joint CC formulation. The latter is accordingly relaxed to individual CC optimizations using the proposed methods. The performance of proposed CC voltage control methods is finally tested in comparison with that of the robust optimization. Simulation results confirm the accuracy of confidence level expected from the proposed CC voltage control formulations. The proposed technique allows the system operators to tune the confidence level parameter such that a tradeoff between operation costs and conservatism level is attained.

Uncertain Time-Resource-Cost Trade-Off Models for Construction Project Schedule

Construction project schedule is to bring the completion time and the total cost into balance by allocating resources. In order to shorten the completion time and decrease the total cost as far as possible, three time-resource-cost trade-off models for construction project under uncertain environment, including expected value (E-V) model, chance-constrained (C-C) model and chance maximization (C-M) model, were built by using uncertain theory. The duration time of activities, the allocation time of material resources and human resources are considered as uncertain variables in the three models. The E-V model was to shorten the expected completion time under the chance constraint of cost. The C-C model was to obtain the minimum completion time under the chance constraints of completion time and cost. The C-M model was to maximize the chance that the completion time is not lager than the due date under the chance constraint of cost. For the convenience of calculation, the three models were changed into crisp equivalent form. A practical case study was provided to show the application process of three models and the results can be obtained by using the genetic algorithm. Finally, some extended construction project scheduling models are introduced.

Sewer orientated framework for ensemble‐based chance‐constrained model predictive control

AbstractIn this work, we present a framework for ensemble‐based (E) chance‐constrained (CC) model predictive control (MPC) in sewer systems. The framework considers the availability of ensemble forecasts and the difficulties with propagation of distributions; through distribution estimation. Utilizing a case study of the sewer network of the city of Aarhus in Denmark, the performance of the ECC‐MPC framework is evaluated through simulations. The evaluations were based on linear models of the case study and compare the ECC‐MPC performance with the performance of CC‐MPC. Based on the simulations, it was found that the ECC‐MPC performed comparable to the performance of the CC‐MPC, not only in the context of overflow and outflow but also with respect to behavior in response to changes in different aspects of forecast uncertainties. Regarding the aspects, it was found that expectation offset biases in the forecast were affecting the performance of the CC‐ and ECC‐MPC the most. While other aspects only had a reduced effect on the performances, within the ranges tested. With the comparable performances, it was found that ECC‐MPC would work as an alternative approach to CC‐MPC.

Research on Optimal Configuration of Pumped Storage Power Station Capacity Considering the Peak Regulation Capacity Ratio of Microgrid System - Based on Chance Constrained Programming

Research on Optimal Configuration of Pumped Storage Power Station Capacity Considering the Peak Regulation Capacity Ratio of Microgrid System - Based on Chance Constrained Programming

Taylor Approximation for Chance Constrained Optimization Problems Governed by Partial Differential Equations with High-Dimensional Random Parameters

Taylor Approximation for Chance Constrained Optimization Problems Governed by Partial Differential Equations with High-Dimensional Random Parameters

A two-stage two-layer optimization approach for economic operation of a microgrid under a planned outage

Microgrids (MGs) are a promising and viable solution for sustainable cities and community development. However, it is challenging to achieve the economic operation of MGs due to the uncertainties of generation, load demand, and outage occurrence. Consequently, this paper proposes a novel two-stage two-layer optimization approach to minimize the total operation cost of an MG under a high dimension of uncertainties of generation, load demand, and planned outages. In the upper layer, the binary chance-constrained (CC) for outage planning under the n-1 criterion is developed considering the outage uncertainty. Then, the CC for optimal day-ahead scheduling is introduced to deal with the uncertainties of generation and load demand. Since the prediction error means are not perfectly known, we propose an efficient second order cone formulation via the distributionally robust optimization method to solve the MG economic operation problem. The lower layer performs real-time optimization to compensate for the power deviation and determines a fair benefit sharing for the use of the external devices. To validate the proposed approach, we conducted simulations with different uncertainty levels and outage scenarios. The simulation results illustrate the effectiveness of the proposed approach, resulting in a remarkable operational cost reduction.

Chance-Constrained Water Pumping to Manage Water and Power Demand Uncertainty in Distribution Networks

Water pumping in drinking water distribution networks (WDNs) can be treated as a flexible load in the power distribution network (PDN). In this article, we formulate an optimization problem to minimize the electricity costs associated with pumping subject to WDN and PDN constraints. In practice, both water and power demands are uncertain and pumps should be scheduled to ensure that pump operation does not violate either networks’ constraints for nearly all possible uncertainty realizations. To address this problem, we formulate a chance-constrained (CC) optimization problem that simultaneously determines pumping schedules along with the parameters of real-time control policies that can be used to respond to water and power demand forecast errors. We use approximations and relaxations along with the scenario approach for CC programming to reformulate the optimization problem into a convex deterministic problem. We demonstrate the performance of the approach through case studies and also explore the impact of the relaxations, an approach to improve computational tractability, and tradeoffs associated with the way in which we define the cost of real-time control actions. We find that optimal scheduling and real-time control of water pumping can effectively manage water and power demand uncertainty, meaning water demand is satisfied and both the WDN and PDN operate within their limits; however, the approach is conservative leading to high reliability at high cost.

An Approach to Chance Constrained Problems Based on Huge Data Sets Using Weighted Stratified Sampling and Adaptive Differential Evolution

In this paper, a new approach to solve Chance Constrained Problems (CCPs) using huge data sets is proposed. Specifically, instead of the conventional mathematical model, a huge data set is used to formulate CCP. This is because such a large data set is available nowadays due to advanced information technologies. Since the data set is too large to evaluate the probabilistic constraint of CCP, a new data reduction method called Weighted Stratified Sampling (WSS) is proposed to describe a relaxation problem of CCP. An adaptive Differential Evolution combined with a pruning technique is also proposed to solve the relaxation problem of CCP efficiently. The performance of WSS is compared with a well known method, Simple Random Sampling. Then, the proposed approach is applied to a real-world application, namely the flood control planning formulated as CCP.

Risk-Based Procurement Strategy for Electricity Retailers: Different Scenario-Based Methods

In mid-term planning, the objective of an electricity retailer is to procure the electricity energy to supply its demands, while efficient control of the financial risks under uncertainties is as vital as maximizing profit. To meet these challenges, this paper proposes and compares different stochastic-based risk measures including minimax regret (MMR), chance-constrained (CC), and conditional value-at-risk (CVaR) criteria, which are useful tools for assessing the risk levels, considering whether a retailer is a risk-taking or a risk-averse participant. To obtain an optimal procurement strategy, robust versions of implemented approaches are applied. To show the real performance of the proposed procurement solutions, postoptimization procedures with different price behaviors are performed to carry out experimental results on the basis of realistic data. The results show that for the same robustness degree of strategy, CVaR is less conservative than MMR and CC is less conservative than CVaR.

Pool & Discard Algorithm for Chance Constrained Optimization Problems

In this paper, we describe an effective algorithm for handling chance constrained optimization problems, called the Pool & Discard algorithm. The algorithm utilizes the scenario approximation framework for chance constrained optimization problems, and the warm-start and problem modification features of modern solvers. The exploitation of the problem structure and efficient implementation allows us to considerably speed up the computations, especially for large instances, when compared with conventional methods.

Multi-period planning of multi-energy microgrid with multi-type uncertainties using chance constrained information gap decision method

In this paper, we study the multi-period planning problem of multi-energy microgrids considering the long-term uncertainty (i.e., the declining trend of battery storage investment cost) and the short-term uncertainty (i.e., renewable energy generation and electrical/heat load). We first present the joint deterministic multi-period planning approach for multi-energy microgrid coupling electricity and heat carriers. Then, an information gap decision (IGD)-based multi-energy microgrid multi-period planning model dealing with the long-term uncertainty is proposed, and the proposed model is further converted into a mixed integer linear planning (MILP) IGD-based planning model. Next, to coordinate the long-term uncertainty and the short-term uncertainty in multi-energy microgrid planning problems, we develop a chance constrained (CC) IGD-based multi-period planning model and then convert such model into a MILP CC-IGD equivalence. Finally, the strengthened bilinear Benders decomposition (SBBD) algorithm is adopted to efficiently solve our proposed MILP CC-IGD model for large-scale multi-energy microgrid planning problems. Our numerical results demonstrate the advantage of the joint planning of electricity and heat supply systems in multi-energy microgrids. Case studies verify the effectiveness of considering multi-type uncertainties in multi-energy microgrid planning, especially the declining trend uncertainty of battery storage investment cost. Experimental results also show that the SBBD algorithm is more efficient on computing our proposed MILP CC-IGD model compared to commercial solvers, such as CPLEX.