Chance-constrained Optimization Research Articles

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.

Optimal Operation of Power Systems With Energy Storage Under Uncertainty: A Scenario-Based Method With Strategic Sampling

The multi-period dynamics of energy storage (ES), intermittent renewable generation and uncontrollable power loads, make the optimization of power system operation (PSO) challenging. A multi-period optimal PSO under uncertainty is formulated using the chance-constrained optimization (CCO) modeling paradigm, where the constraints include the nonlinear energy storage and AC power flow models. Based on the emerging scenario optimization method which does not rely on pre-known probability distribution, this paper develops a novel solution method for this challenging CCO problem. The proposed method is computationally effective for mainly two reasons. First, the original AC power flow constraints are approximated by a set of learning-assisted quadratic convex inequalities based on a generalized least absolute shrinkage and selection operator (LASSSO). Second, considering the physical patterns of data and driven by the learning-based sampling, the strategic sampling method is developed to significantly reduce the required number of scenarios by different sampling strategies. The simulation results on IEEE standard systems indicate that 1) the proposed strategic sampling significantly improves the computational efficiency of the scenario-based approach for solving the chance-constrained optimal PSO problem, 2) the data-driven convex approximation of power flow can be promising alternatives of nonlinear and nonconvex AC power flow.

Stochastic Control and Pricing for Natural Gas Networks

We propose stochastic control policies to cope with uncertain and variable gas extractions in natural gas networks. Given historical gas extraction data, these policies are optimized to produce the real-time control inputs for nodal gas injections and for pressure regulation rates by compressors and valves. We describe the random network state as a function of control inputs, which enables a chance-constrained optimization of these policies for arbitrary network topologies. This optimization ensures the real-time gas flow feasibility and a minimal variation in the network state up to specified feasibility and variance criteria. Furthermore, the chance-constrained optimization provides the foundation of a stochastic pricing scheme for natural gas networks, which improves on a deterministic market settlement by offering the compensations to network assets for their contribution to uncertainty and variance control. We analyze the economic properties, including efficiency, revenue adequacy, and cost recovery, of the proposed pricing scheme and make them conditioned on the network design.

Computing Essential Sets for Convex and Nonconvex Scenario Problems: Theory and Application

The scenario approach is a general data-driven algorithm to chance-constrained optimization. It seeks the optimal solution that is feasible to a carefully chosen number of scenarios. A crucial step in the scenario approach is to compute the cardinality of essential sets, which is the smallest subset of scenarios that determine the optimal solution. This article addresses the challenge of efficiently identifying essential sets. For convex problems, we demonstrate that the sparsest dual solution of the scenario problem could pinpoint the essential set. For nonconvex problems, we show that two simple algorithms return the essential set when the scenario problem is nondegenerate. Finally, we illustrate the theoretical results and computational algorithms on security-constrained unit commitment (SCUC) in power systems. In particular, case studies of chance-constrained SCUC are performed in the IEEE 118-bus system. Numerical results suggest that the scenario approach could be an attractive solution to practical power system applications.

C-Vine Copula 기반의 풍력 발전 투입을 위한 확률론적 조류 계산

Owing to the increase in renewable energy, the power system is exposed to more variability and uncertainty, which increases the difficulty in analyzing the power systems. Thus, an accurate power system analysis considering the stochastic characteristics of renewable energy is required. In this paper, we propose a Monte-Carlo based probabilistic load flow method that takes into account the statistical dependencies between the power generation from the multiple wind sites. To this end, the vine copula method was employed to capture their dependencies and the marginal distribution of the power generation from the multiple wind sites is modeled using a 3-parameter Weibull distribution. In addition, we suggested the chance-constrained optimization problem for properly allocating a target wind power capacity to the buses. From the numerical experiment conducted on the IEEE 39-bus system, it has been demonstrated that the proposed method can more accurately identify the potential risk with the modeling of the dependencies between the power generation from the multiple wind sites as compared to the deterministic approach or probabilistic load flow based on the random sampling method.

Joint optimization of Volt/VAR control and mobile energy storage system scheduling in active power distribution networks under PV prediction uncertainty

Joint optimization of Volt/VAR control and mobile energy storage system scheduling in active power distribution networks under PV prediction uncertainty

Active Exploration by Chance-Constrained Optimization for Voltage Regulation with Reinforcement Learning

Voltage regulation in distribution networks encounters a challenge of handling uncertainties caused by the high penetration of photovoltaics (PV). This research proposes an active exploration (AE) method based on reinforcement learning (RL) to respond to the uncertainties by regulating the voltage of a distribution network with battery energy storage systems (BESS). The proposed method integrates engineering knowledge to accelerate the training process of RL. The engineering knowledge is the chance-constrained optimization. We formulate the problem in a chance-constrained optimization with a linear load flow approximation. The optimization results are used to guide the action selection of the exploration for improving training efficiency and reducing the conserveness characteristic. The comparison of methods focuses on how BESSs are used, training efficiency, and robustness under varying uncertainties and BESS sizes. We implement the proposed algorithm, a chance-constrained optimization, and a traditional Q-learning in the IEEE 13 Node Test Feeder. Our evaluation shows that the proposed AE method has a better response to the training efficiency compared to traditional Q-learning. Meanwhile, the proposed method has advantages in BESS usage in conserveness compared to the chance-constrained optimization.

Chance-constrained optimization under limited distributional information: A review of reformulations based on sampling and distributional robustness

• Chance-constrained programming is a powerful paradigm to model risk tolerances. • Sample average approximation leads to mixed-integer linear reformulations. • Distributionally robust chance-constrained programs model distribution ambiguity. • Scalable reformulations and decomposition enable solution of large-scale problems. • Mixed-integer conic formulations can be readily used with optimization solvers. Chance-constrained programming (CCP) is one of the most difficult classes of optimization problems that has attracted the attention of researchers since the 1950s. In this survey, we focus on cases when only limited information on the distribution is available, such as a sample from the distribution, or the moments of the distribution. We first review recent developments in mixed-integer linear formulations of chance-constrained programs that arise from finite discrete distributions (or sample average approximation). We highlight successful reformulations and decomposition techniques that enable the solution of large-scale instances. We then review active research in distributionally robust CCP, which is a framework to address the ambiguity in the distribution of the random data. The focal point of our review is on scalable formulations that can be readily implemented with state-of-the-art optimization software. Furthermore, we highlight the prevalence of CCPs with a review of applications across multiple domains.

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

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

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

Distributionally robust chance-constrained flexibility planning for integrated energy system

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

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

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