Chance-constrained Optimal Power Flow Research Articles

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.

Difference-of-Convex approach to chance-constrained Optimal Power Flow modelling the DSO power modulation lever for distribution networks

The increasing expansion of renewable energy sources leads to the growth of uncertainty in the distribution network operation. Short-term operational planning performed by distribution system operators should evolve to address those new operating conditions, in particular to allow the efficient utilization of different flexibility levers. In this work, the use of a chance-constrained Alternating Current Optimal Power Flow (AC-OPF) is proposed to model the operational planning problem, considering the activation of several levers such as power modulation and power curtailment. The correlation between the renewable generation profiles and loads is considered via a joint probability constraint approximated with scenarios. The main novelty of the present manuscript is the adoption of a Difference-of-Convex approach that allows to solve the obtained optimization problem without convexification or linearization of the core OPF equations. Furthermore, the approach yields a natural and embarrassingly parallelizable scenario decomposition. The method starts with a reformulation of the model as a Difference-of-Convex optimization problem, and then a proximal bundle method algorithm is applied to solve it. The proposed methodology is tested in a 33 bus distribution network with 11 different values for the safety level defining the probability constraint, ranging from 0.75 to 1.

GP CC-OPF: Gaussian Process based optimization tool for Chance-Constrained Optimal Power Flow

GP CC-OPF: Gaussian Process based optimization tool for Chance-Constrained Optimal Power Flow

Deep-Quantile-Regression-Based Surrogate Model for Joint Chance-Constrained Optimal Power Flow With Renewable Generation

Joint chance-constrained optimal power flow (JCC-OPF) is a promising tool for managing distributed renewable generation uncertainties. However, existing works are usually based on power flow equations, which require accurate network parameters that may be unobservable in many distribution systems. To address this issue, this paper proposes a learning-based surrogate model for JCC-OPF with renewable generation. This model equivalently converts joint chance constraints into quantile-based forms. Two multi-layer perceptrons are trained based on special loss functions to predict the quantile of constraint violations and expected power loss. By reformulating these two MLPs into mixed-integer linear constraints, we can replicate the JCC-OPF without network parameters. Two pre-processing steps, i.e., data augmentation and calibration, are further developed to improve its performance. The former trains a simulator to generate more training samples for enhancing the prediction accuracy of MLPs. The latter designs a positive parameter based on empirical prediction errors to calibrate the outputs of MLPs so that feasibility can be guaranteed. Numerical experiments based on the IEEE 33- and 123-bus systems validate that the proposed model can achieve desirable feasibility and optimality simultaneously with no need for network parameters.

Learning Reactive Power Control Polices in Distribution Networks Using Conditional Value-at-Risk and Artificial Neural Networks

Scalable coordination of photovoltaic (PV) inverters, considering the uncertainty in PV and load in distribution networks (DNs), is challenging due to the lack of real-time communications. Decentralized PV inverter setpoints can be achieved to address this issue by capitalizing on the abundance of data from smart utility meters and the scalable architecture of artificial neural networks (ANNs). To this end, we first use an offline, centralized data-driven conservative convex approximation of chance-constrained optimal power flow (CVaR-OPF) in which conditional value-at-risk (CVaR) is used to compute reactive power setpoints of PV inverter, taking into account PV and load uncertainties in DNs. Following that, an artificial neural network (ANN) controller is trained for each PV inverter to emulate the optimal behavior of the centralized control set- points of PV inverter in a decentralized fashion. Additionally, the voltage regulation performance of the developed ANN controllers is compared with other decentralized designs (local controllers) developed using model-based learning (regression-based controller), optimization (affine feedback controller), and case-based learning (mapping) approaches. Numerical tests using real-world feeders corroborate the effectiveness of ANN controllers in voltage regulation and loss minimization.

Fair operating envelopes under uncertainty using chance constrained optimal power flow

The significant penetration of rooftop solar photovoltaics (PV) is pushing low-voltage distribution networks to the brink of their technical limits, which increasingly requires distribution system operators (DSO) to limit power exports. The easiest but also the bluntest instrument is to impose a blanket constraint, which is suboptimal and not in the best interest of PV owners. Alternatively, the DSO can compute PV export limits, also called operating envelopes, using AC optimal power flow (OPF) in real-time. However, this assumes full observability of customer demand, which is unrealistic in practice. To address this gap, we propose a bi-level framework under uncertainty to compute operating envelopes that are robust against uncertainty. At the upper level, the DSO solves a deterministic OPF using day-ahead prediction of customer demand and PV generation. The linearised OPF solution is then used in a chance-constrained OPF problem, resulting in a second-order cone model, which could be solved efficiently. At the lower level, prosumers optimise battery operation against the operating envelopes submitted by the DSO day-ahead. The proposed operating envelopes are fair in the sense that all prosumers are subject to the same export limit irrespective of their location in the network.

Non-parametric Joint Chance-Constrained OPF via Maximum Mean Discrepancy Penalization

The chance-constrained optimal power flow (CC-OPF) has gained prominence due to increased uncertainty in the power system. However, solving CC-OPF for general uncertainty distribution classes is challenging due to lack of analytical formulation of probabilistic constraints and cost-complexity trade-off issues. This work proposes a novel joint chance-constrained optimal power flow (JCC-OPF) via maximum mean discrepancy (MMD) penalization to obtain a probabilistically feasible low-cost solution. The idea is to view the JCC-OPF problem as a distribution matching problem. The MMD quantifies the distance between two probability distributions embedded into reproducing kernel Hilbert space (RKHS) and thus provides an efficient way to minimize the distance between distributions. The RKHS embedding, also called kernel mean embedding (KME), is a non-parametric method that does not require any information about the random injection’s distribution while performing the embedding. Furthermore, the proposed method is based on a point-wise evaluation of the constraint functions and has the same complexity as a deterministic OPF problem. The proposed penalization-based formulation handles JCC directly and does not require the conversion of joint chance constraints into individual ones. Simulations on IEEE 24-Bus, 30-Bus, and 57-Bus systems validate the proposed method’s non-parametric nature and ability to obtain a probabilistically feasible solution. Benchmarking results against existing approaches indicate the better computational performance of the proposed method.

Chance-Constrained OPF in Droop-Controlled Microgrids With Power Flow Routers

High penetration of renewable generation poses challenges to power system operation due to its uncertain nature. In droop-controlled microgrids, the voltage volatility induced by renewable uncertainties is aggravated by the high droop gains. This paper proposes a chance-constrained optimal power flow (CC-OPF) problem with power flow routers (PFRs) to better regulate the voltage profile in microgrids. PFR refers to a general type of network-side controller that brings more flexibility to the power network. Comparing with the normal CC-OPF that relies on node power flexibility only, the proposed model introduces a new dimension of control from power network to enhance system performance under renewable uncertainties. Adopting a partial linearization method and an iterative algorithm allows us to address the CC-OPF problem by iteratively solving a subproblem. Since the inclusion of PFRs complicates the subproblem and makes common solvers no longer applicable directly, a semidefinite programming relaxation is used to transform each subproblem into a convex form. The proposed method is verified on a modified IEEE 33-bus system and the results show that PFRs significantly reduce the standard deviations of voltage magnitudes and contribute to mitigating the voltage volatility, which makes the system operate in a more economic and secure way.

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.

Alternating Direction Method of Multipliers for Solving Joint Chance Constrained Optimal Power Flow Under Uncertainties

The increasing penetration of renewable energy sources in power systems introduces additional load fluctuations. Lack of awareness of them may lead to a high risk of system failure. Many existing approaches mitigate this issue by considering individual chance constraints (CCs) for the physical limitation of the system. To guarantee that all the operational constraints are satisfied with a predetermined probability, this paper uses joint chance constraints to formulate the optimal power flow (OPF) problem. Additionally, to ensure the scalability, this paper presents an alternating direction method of multipliers (ADMM) with convex optimization subproblems to solve the joint chance-constrained (JCC) OPF, where the computational burden is reduced. At last, to avoid making assumptions about the uncertainties, the CCs are approximated with a sample-based approach.

Chance-Constrained OPF: A Distributed Method With Confidentiality Preservation

This paper focuses on the global chance-constrained optimal power flow problem of a multi-regional interconnected grid. In this global problem, however, multiple regional independent system operators (ISOs) participate in the decision-making process, raising the need for distributed but coordinated approaches. Most notably, due to the regulation and security concerns, regional ISOs may refuse to share confidential information with others, including generation cost, load data, system topologies, and line parameters. Accordingly, this paper proposes a distributed CC-OPF method with strict confidentiality preservation, which theoretically enables regional ISOs to determine the exact global optimal dispatchable generations within their regions without disclosing confidential data. The proposed method neither requires parameter tunings nor has convergence issues. Results from a number of test cases show that the proposed method has considerably high accuracy, regardless of the system size, load level, and amount of regions, thereby outperforming the state-of-the-art obfuscation approaches.

Chance-constrained optimal power flow based on a linearized network model

In this study, the focus is put on improving the solution process for a chance-constrained alternating current (AC) optimal power flow model. Firstly, a novel chance-constrained optimal power flow model is proposed and introduced based on a linearized network model with explicit bounds on voltage magnitude, reactive power, and apparent power flow, which can achieve a desirable computation performance from the perspective of modeling. In particular, the linearization imposed on the apparent power flow will induce joint chance constraints, making the deterministic transformation of the chance constraints challenging to perform. Thus, this paper adopts an improved Boole’s inequality to address this issue. In a further step, as a substitute to the analytical reformulation method and the Monte Carlo simulation method, the three-point estimation and the Cornish-Fisher series expansion are combined to efficiently conduct uncertainty evaluation on the AC power flow recovered solution, while ensuring all chance constraints in the stochastic scenarios are satisfied. If any violation probability exceeds the given value, the corresponding constraint bounds will be tightened, and an updated deterministic linearly-constrained model needs to be solved again. This process is repeated until all the convergence conditions are reached. Case studies on two test systems verified the characteristics/advantages of the proposed chance-constrained optimal power flow modeling and solution approach.

Combining model-based and model-free methods for stochastic control of distributed energy resources

Modern distribution systems are experiencing a fast transformation with the growing penetration of distributed energy resources (DERs). Along with the economic and environmental benefits of DERs, challenges arise to address the uncertainties caused by their inherent volatility. If properly coordinated, however, DERs have the potential to provide the controllability that grid operators need. In the paper, we propose a hierarchical control framework that combines the model-based and model-free methods for stochastic DER control in distribution systems. The upper-level scheduler considers a chance-constrained optimal power flow problem (model-based) that schedules DER setpoints to minimize the operational cost and maintain the operating reserve. The lower-level distributed DER controllers absorb real-time disturbances and uncertainties using the extremum seeking control (model-free) to achieve grid objectives. The combination of model-based and model-free methods allows us to take the advantages of both methods to effectively manage the uncertainty in distribution systems. The proposed work is demonstrated on the IEEE 13-node feeder.

Chance-Constrained Optimal Power Flow of Integrated Transmission and Distribution Networks With Limited Information Interaction

Considering the separate management of transmission networks (TNs) and distribution networks (DNs), this article proposes a chance-constrained optimal power flow (CCOPF) formulation for integrated transmission and distribution (I-T&D) networks and its solution algorithm with limited information interaction. The uncertainties of loads and renewable generations are considered in the formulation, which guarantees that generations, power flows, and voltage magnitudes in both TNs and DNs remain within their bounds with a predefined probability. A double-iterative solution algorithm is proposed to solve CCOPF, of which the inner-iteration is applied to solve a deterministic OPF of I-T&D networks while the outer-iteration is to repeatedly update uncertain margins around a forecasted solution to converge finally. Particularly, the heterogeneous decomposition algorithm is applied to solve the deterministic OPF, and an I-T&D-power-flow-based two-point estimation method is proposed to calculate uncertainty margins. The overall solution algorithm is realized based on boundary information exchange between TNs and DNs, where the data and model privacy of TNs and DNs are well-preserved. Numerical experiments demonstrate the accuracy and efficiency of the proposed solution algorithm.

Online Learning for Network Constrained Demand Response Pricing in Distribution Systems

Flexible demand response (DR) resources can be leveraged to accommodate the stochasticity of some distributed energy resources. This paper develops an online learning approach that continuously estimates price sensitivities of residential DR participants and produces such price signals to the DR participants that ensure a desired level of DR capacity. The proposed learning approach incorporates the dispatch decisions on DR resources into the distributionally robust chance-constrained optimal power flow (OPF) framework. This integration is shown to adequately remunerate DR resources and co-optimize the dispatch of DR and conventional generation resources. The distributionally robust chance-constrained formulation only relies on empirical data acquired over time and makes no restrictive assumptions on the underlying distribution of the demand uncertainty. The distributional robustness also allows for robustifying the optimal solution against systematically misestimating empirically learned parameters. The effectiveness of the proposed learning approach is shown via numerical experiments. The paper is accompanied by the code and data supplement released for public use, see [27].

Chance-Constrained AC Optimal Power Flow: A Polynomial Chaos Approach

As the share of renewables in the grid increases, the operation of power systems becomes more challenging. The present paper proposes a method to formulate and solve chance-constrained optimal power flow while explicitly considering the full nonlinear ac power flow equations and stochastic uncertainties. We use polynomial chaos expansion to model the effects of arbitrary uncertainties of finite variance, which enables to predict and optimize the system state for a range of operating conditions. We apply chance constraints to limit the probability of violations of inequality constraints. Our method incorporates a more detailed and a more flexible description of both the controllable variables and the resulting system state than previous methods. Two case studies highlight the efficacy of the method, with a focus on satisfaction of the ac power flow equations and on the accurate computation of moments of all random variables.

Joint Chance Constraints in AC Optimal Power Flow: Improving Bounds Through Learning

This paper considers distribution systems with a high penetration of distributed, renewable generation and addresses the problem of incorporating the associated uncertainty into the optimal operation of these networks. Joint chance constraints, which satisfy multiple constraints simultaneously with a prescribed probability, are one way to incorporate uncertainty across sets of constraints, leading to a chance-constrained optimal power flow problem. Departing from the computationally heavy scenario-based approaches or approximations that transform the joint constraint into conservative deterministic constraints; this paper develops a scalable, data-driven approach which learns operational trends in a power network, eliminates zero-probability events (e.g., inactive constraints), and accurately and efficiently approximates bounds on the joint chance constraint iteratively. In particular, the proposed framework improves upon the classic methods based on the union bound (or Boole’s inequality) by generating a much less conservative set of single chance constraints that also guarantees the satisfaction of the original joint constraint. The proposed framework is evaluated numerically using the IEEE 37-node test feeder, focusing on the problem of voltage regulation in distribution grids.

Optimal Load Ensemble Control in Chance-Constrained Optimal Power Flow

Distribution system operators (DSOs) world-wide foresee a rapid roll-out of distributed energy resources. From the system perspective, their reliable and cost effective integration requires accounting for their physical properties in operating tools used by the DSO. This paper describes a decomposable approach to leverage the dispatch flexibility of thermostatically controlled loads (TCLs) for operating distribution systems with a high penetration level of photovoltaic resources. Each TCL ensemble is modeled using the Markov decision process (MDP). The MDP model is then integrated with a chance constrained optimal power flow that accounts for the uncertainty of photovoltaic resources. Since the integrated optimization model cannot be solved efficiently by existing dynamic programming methods or off-the-shelf solvers, this paper proposes an iterative spatio-temporal dual decomposition algorithm (ST-D2). We demonstrate the merits of the proposed integrated optimization and ST-D2 algorithm on the IEEE 33-bus test system.

Distributionally Robust Chance-Constrained Optimal Power Flow Assuming Unimodal Distributions With Misspecified Modes

Chance constrained optimal power flow (CC-OPF) formulations have been proposed to minimize operational costs while controlling the risk arising from uncertainties like renewable generation and load consumption. To solve CC-OPF, we often need access to the (true) joint probability distribution of all uncertainties, which is rarely known in practice. A solution based on a biased estimate of the distribution can result in poor reliability. To overcome this challenge, recent work has explored distributionally robust chance constraints, in which the chance constraints are satisfied over a family of distributions called the ambiguity set. Commonly, ambiguity sets are only based on moment information (e.g., mean and covariance) of the random variables; however, specifying additional characteristics of the random variables reduces conservatism and cost. Here, we consider ambiguity sets that additionally incorporate unimodality information. In practice, it is difficult to estimate the mode location from the data and so we allow it to be potentially misspecified. We formulate the problem and derive a separation-based algorithm to efficiently solve it. Finally, we evaluate the performance of the proposed approach on a modified IEEE-30 bus network with wind uncertainty and compare with other distributionally robust approaches. We find that a misspecified mode significantly affects the reliability of the solution and the proposed model demonstrates a good trade-off between cost and reliability.

Chance-constrained optimal power flow with non-parametric probability distributions of dynamic line ratings

Compared to Seasonal Line Rating (SLR), Dynamic Line Rating (DLR) allows for higher power flows on overhead transmission lines, depending on the actual weather conditions. Nevertheless, the potential of DLR has to be traded off against the additional uncertainty associated with varying ratings. This paper proposes a DC-Optimal Power Flow (DCOPF) algorithm that accounts for DLR uncertainty by means of Chance-Constraints (CC). The goal is to determine the optimal day-ahead dispatch taking the cost of reserve procurement into account. The key contribution of this paper consists in considering both non-parametric predictive distributions of DLR and the combined wind power uncertainty in the optimization problem. Our results highlight the benefits of DLR in wind-dominated power systems, assuming typical risk aversion levels in the line rating estimation.