Chance-constrained Optimization Research Articles

Scheduling of Energy Hub Resources Using Robust Chance-Constrained Optimization

This paper develops a robust chance-constrained model for handling the uncertainties of generation and consumption in multi-carrier energy hubs. The proposed model incorporates corresponding loading factors for each type of electrical, heating, and cooling loads. This is done to assess the maximum loadability of the whole system. In this respect, the chance-constrained approach is implemented for the feasibility assessment of the operation problem with uncertainties. The uncertainties which are assumed here include the forecast errors of electrical, heating, and cooling load demands, and the volatile solar power generation. The overall problem formulation is developed in the mixed-integer linear programming (MILP) framework. The standard chance-constrained approach is converted to a deterministic optimization model by utilizing the Big M method. The main objective of the proposed model is to maximize the loadability index with uncertainties while addressing the permissible risk index of the decision-maker. The studied energy hub comprises electrical, heating, and cooling loads, and the energy flow technique is adopted in this paper to model the load balance equations. The simulation results are presented for different scenarios while addressing features of the proposed model for the summer and winter seasons. Furthermore, the developed model is evaluated for different scenarios and a comparison is made with the information-gap decision theory (IGDT) method.

Data-Driven Tuning for Chance-Constrained Optimization: Two Steps Towards Probabilistic Performance Guarantees

Parameters involved in the formulation of optimization problems are often partially unknown or random. A popular way to mitigate the effect of uncertainty is using joint chance constraints, which guarantee constraint satisfaction with high probability but are challenging to solve. In this letter, we analyze an approach for joint chance constrained problems that involves iteratively tuning problem parameters. We first show that existing naive approaches to tuning can lead to solutions without feasibility guarantees. We then introduce a two-step approach, where a tuning-based solution generation step is followed by an a posteriori solution verification step. A main challenge of the two-step approach is to guarantee that the solution generated in the first step has a high probability of being verified as feasible in the second step. We therefore analyze how the relationship between the feasibility criteria used in each step impacts the probability of obtaining a feasible solution. We demonstrate our results in a numerical case study of the optimal power flow problem.

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.

A novel approach for solving decision-making problems with stochastic linear-fractional models

Stochastic chance-constrained optimization has a wide range of real-world applications. In some real-world applications, the decision-maker has to formulate the problem as a fractional model where some or all of the coefficients are random variables with joint probability distribution. Therefore, these types of problems can deal with bi-objective problems and reflect system efficiency. In this paper, we present a novel approach to formulate and solve stochastic chance-constrained linear fractional programming models. This approach is an extension of the deterministic fractional model. The proposed approach, for solving these types of stochastic decision-making problems with the fractional objective function, is constructed using the following two-step procedure. In the first stage, we transform the stochastic linear fractional model into two stochastic linear models using the goal programming approach, where the first goal represents the numerator and the second goal represents the denominator for the stochastic fractional model. The resulting stochastic goal programming problem is formulated. The second stage implies solving stochastic goal programming problem, by replacing the stochastic parameters of the model with their expectations. The resulting deterministic goal programming problem is built and solved using Win QSB solver. Then, using the optimal value for the first and second goals, the optimal solution for the fractional model is obtained. An example is presented to illustrate our approach, where we assume the stochastic parameters have a uniform distribution. Hence, the proposed approach for solving the stochastic linear fractional model is efficient and easy to implement. The advantage of the proposed approach is the ability to use it for formulating and solving any decision-making problems with the stochastic linear fractional model based on transforming the stochastic linear model to a deterministic linear model, by replacing the stochastic parameters with their corresponding expectations and transforming the deterministic linear fractional model to a deterministic linear model using the goal programming approach

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.

Chance-constrained optimization for integrated local energy systems operation considering correlated wind generation

Energy hubs, which integrate multiple energy vectors through converters, can enhance the value of Integrated Local Energy Systems (ILES) via increased flexibility and reduced costs. However, uncertain renewable energy and the non-convex, non-linear properties of energy flows complicate the modelling and operation of energy hub systems. This paper develops chance-constrained optimization methods for planning and operation of energy hub systems under uncertainty. The non-linear formulations of power and gas flows are relaxed by convexification methods, leading to a formulation of Second Order Cone Problem (SOCP), which can be efficiently solved to global optimality. The correlation between geographically close wind generators connected to the hub systems is modelled by establishing their relation using Gaussian copula. The proposed chance-constrained optimization is demonstrated on a six-hub system within a multi-vector energy distribution network with 7 electrical buses and 7 gas nodes. The value of different levels of system integration through the installation of energy hubs is investigated. The results show that by combining system integration via energy hubs with chance constrained operation, the proposed method can reduce operating costs and increase renewable energy yields, thereby benefitting hub system operators and customers with reduced energy infrastructure investment and energy costs.

Wasserstein distributionally robust chance-constrained optimization for energy and reserve dispatch: An exact and physically-bounded formulation

In the context of transition towards sustainable, cost-efficient and reliable energy systems, the improvement of current energy and reserve dispatch models is crucial to properly cope with the uncertainty of weather-dependent renewable power generation. In contrast to traditional approaches, distributionally robust optimization offers a risk-aware framework that provides performance guarantees when the distribution of uncertain parameters is not perfectly known. In this paper, we develop a distributionally robust chance-constrained optimization with a Wasserstein ambiguity set for energy and reserve dispatch, and provide an exact reformulation. While preserving the exactness, we then improve the model by enforcing physical bounds on the uncertainty space, resulting in a bilinear program. We solve the resulting bilinear model with an iterative algorithm which is computationally efficient and has convergence guarantee. A thorough out-of-sample analysis is performed to compare the proposed model against a scenario-based stochastic program. We also compare the performance of the proposed exact reformulation against an existing approximate technique in the literature, built upon a conditional-value-at-risk measure. We eventually show that the proposed physically-bounded exact reformulation outperforms the other methods by achieving a cost-optimal yet reliable trade-off between reserve procurement and load curtailment.

Electronic companion- Adjustable and Distributionally Robust Chance-Constrained Optimization for Providing Flexibilities in an Active Distribution Network

Electronic companion- Adjustable and Distributionally Robust Chance-Constrained Optimization for Providing Flexibilities in an Active Distribution Network

Distributed stochastic model predictive control for systems with stochastic multiplicative uncertainty and chance constraints

The rapid development of technology and economy has led to the development of chemical processes, large-scale manufacturing equipment, and transportation networks, with their increasing complexity. These large systems are usually composed of many interacting and coupling subsystems. Moreover, the propagation and perturbation of uncertainty make the control design of such systems to be a thorny problem. In this study, for a complex system composed of multiple subsystems suffering from multiplicative uncertainty, not only the individual constraints of each subsystem but also the coupling constraints among them are considered. All the constraints with the probabilistic form are used to characterize the stochastic natures of uncertainty. This paper first establishes a centralized model predictive control scheme by integrating overall system dynamics and chance constraints as a whole. To deal with the chance constraint, based on the concept of multi-step probabilistic invariant set, a condition formulated by a series of linear matrix inequality is designed to guarantee the chance constraint. Stochastic stability can also be guaranteed by the virtue of nonnegative supermartingale property. In this way, instead of solving a non-convex and intractable chance-constrained optimization problem at each moment, a semidefinite programming problem is established so as to be realized online in a rolling manner. Furthermore, to reduce the computational burdens and amount of communication under the centralized framework, a distributed stochastic model predictive control based on a sequential update scheme is designed, where only one subsystem is required to update its plan by executing optimization problem at each time instant. The closed-loop stability in stochastic sense and recursive feasibility are ensured. A numerical example is employed to illustrate the efficacy and validity of the presented algorithm in this study.

UAV Trajectory Planning With Probabilistic Geo-Fence via Iterative Chance-Constrained Optimization

Chance-constrained optimization provides a promi- sing framework for solving control and planning problems with uncertainties, due to its modeling capability to capture randomness in real-world applications. In this paper, we consider a UAV trajectory planning problem with probabilistic geo-fence, building on the chance-constrained optimization approach. In the considered problem, randomness of the model, such as the uncertain boundaries of geo-fences, is incorporated in the formulation. By solving the formulated chance-constrained optimization with a novel sampling based solution method, the optimal UAV trajectory is achieved while limiting the probability of collision with geo-fences to a prefixed threshold. Furthermore, to obtain a totally collision-free trajectory, i.e., avoiding the collision not only at the discrete time-steps but also within the entire time horizon, we build on the idea of an iterative scheme. That is, to iterate the solving of the chance-constrained optimization until the collision with probabilistic geo-fence is avoided at any time within the time horizon. At last, we validate the effectiveness of our method via numerical simulations.

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.

Distributionally Robust Chance-Constrained Backscatter Communication-Assisted Computation Offloading in WBANs

Implementing wireless body area networks (WBANs) is very challenging, due to limited power supply, inadequate computation capability, and imperfect channel state information (CSI). In this paper, we propose a hybrid offloading scheme with backscatter communication (BackCom) under imperfect CSI, where each sensor firstly receives radio frequency (RF) energy and then offloads body data task via low-power BackCom to the access point (AP) for edge computing. Aiming to minimize the end-to-end system latency, we jointly optimize the computation speed of AP for processing computation tasks, the power of the signal transmitted by the AP, and the power reflection coefficient under energy and data rate chance constraints. To solve the proposed distributionally robust chance-constrained optimization problem, we approximate chance constraints by the Bernstein-type-inequality (BTI) method and Conditional value-at-risk (CVaR) method in the Gaussian distribution and arbitrary distribution of channel estimation errors, respectively. To tackle the NP-hard problem efficiently, the original problem can be decomposed into two subproblems, which are solved by successive linear programming and iterative algorithm, respectively. Simulation results show that the CVaR method outperforms the other methods for the non-Gaussian CSI mismatch, and the Bernstein method is more suitable for the Gaussian distribution of CSI errors.

An interval chance-constrained programming-based optimization model for carbon capture, utilization, and storage system planning

Carbon capture, utilization, and storage (CCUS) are widely regarded as a crucial technological option for industrial large-scale carbon dioxide (CO2) emissions reduction. However, high-cost and uncertainties hinder the widespread application of CCUS technology. In this study, an interval-chance-constrained programming-based optimization model was proposed to address random probability distributions, interval values, complex interactions, and the dynamics of capacity expansion issues.The model was applied to a CCUS project in China. A set of violation probability levels (0.01, 0.05, 0.1, and 0.2) were designed to reflect system costs and risk levels. And then the solutions for system costs, capacity expansion, and operating schemes under four violation probability levels (pi’) could be generated. The results revealed that the model could ensure the highest reliability and largest CO2 storage under pi’ = 0.01. At this probability level, the amount of CO2 storage would range from 4972.05–5429.75 kilotons per annum (ktpa), the CCUS system cost would be highest at $166.57 million, and the net system benefits would be slightly less at $105.91 million. If policymakers strive to achieve the net system benefits of the project, the highest net system benefits would be achieved under pi’ = 0.05. At this probability level, the net system benefits would increase to $135.45 million, the system cost would reduce to $138.62 million, but the total amount of CO2 storage would decrease to between 4090.01 and 4653.24 ktpa, which would entail a high risk of system violation. These findings enable policymakers to determine the trade-offs among system reliability, CO2 reduction, and the benefits of the project. The modeling approach can also address interactions among CCUS activities and the dynamics of facility expansion issues as well as help policymakers develop adaptive operational strategies. This study enriches CCUS research through an interval chance-constrained optimization modeling approach for CCUS system management under multiple uncertainties.

An Efficient Neurodynamic Approach to Fuzzy Chance-constrained Programming

In both practical applications and theoretical analysis, there are many fuzzy chance-constrained optimization problems. Currently, there is short of real-time algorithms for solving such problems. Therefore, in this paper, a continuous-time neurodynamic approach is proposed for solving a class of fuzzy chance-constrained optimization problems. Firstly, an equivalent deterministic problem with inequality constraint is discussed, and then a continuous-time neurodynamic approach is proposed. Secondly, a sufficient and necessary optimality condition of the considered optimization problem is obtained. Thirdly, the boundedness, global existence and Lyapunov stability of the state solution to the proposed approach are proved. Moreover, the convergence to the optimal solution of considered problem is studied. Finally, several experiments are provided to show the performance of proposed approach.

A Comprehensive Network Restoration Model for Active Distribution Network Considering Forecast Uncertainty

The active distribution network management (ADNM) equipped by active distribution networks (ADNs) can enhance the resilience of the network after failure. This paper proposes a novel comprehensive post-event network restoration model and its chance-constrained variant for ADN that considering the dispatch of distributed generation (DG), energy storage system (ESS), demand response (DR), static var compensator (SVC), and network reconfiguration to fully investigate the potential of the ADNM in network restoration. In this paper, the line failure, bus failure, and the uncertainty from load and DG output forecast error are considered. Specifically, the line and bus failures are modeled by an enhanced fictitious network technique, while the uncertainty of load and DG output forecast is modeled by the chance-constrained optimization. The power flow is described by a linearized DistFlow model, and thus the deterministic and chance-constrained network restoration models are programmed by the mixed-integer linear programming (MILP). The proposed deterministic and chance-constrained restoration models are tested on the IEEE 33 bus system. Results demonstrate the effectiveness of the proposed deterministic and chance-constrained network restoration model.

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