A New Solution for Stochastic Optimal Power Flow: Combining Limit Relaxation with Iterative Learning Control

Due to the increase of intermittent renewable energy sources, it is becoming more important to consider renewable generation forecast error when solving the optimal power flow (OPF) problem. The stochastic OPF, which uses multiple forecast scenarios, generally leads to a lower cost compared to the standard deterministic OPF, which uses a single forecast. However, the stochastic OPF is computationally more demanding than the deterministic OPF. Both cost savings and computation times further increase when storage units or ramp-constrained generators are included, as they require solving a multi-period OPF problem. Our contribution is a hybrid method approaching the cost performance of the stochastic OPF while maintaining a computational burden close to the deterministic OPF. The method combines elements from both the stochastic and the deterministic OPF, and relies on Benders Cuts to interface them. Using a receding horizon approach over one year, we find that, based on eleven test cases, one version of our hybrid method leads to at least 70% of the cost improvement of the stochastic OPF, while the computation time increase is at most 40% of the stochastic OPF time increase. Furthermore, the computational advantage of our method increases with the system size. Two different versions of the method allow favoring of either the computational improvement or the cost improvement. We also identify directions for further improvement. Finally, our method can be used for more general problems in which one wishes to combine two models with different levels of complexity.

Optimal DG penetration rate planning based on S-OPF in active distribution network

An iterative learning control algorithm within prescribed input-output subspace

The increasing penetration of renewable energy sources (RES), particularly solar photovoltaic (PV) and wind power, introduces significant uncertainty into distribution networks, affecting voltage stability, power losses, and operational reliability. This paper proposes a stochastic optimal power flow (SOPF) framework that explicitly models renewable generation uncertainty in distribution systems. A probabilistic modeling approach based on scenario generation is adopted to represent the variability of RES output. The objective function minimizes expected operational cost and power losses while satisfying voltage, thermal, and generation constraints under uncertainty. The proposed stochastic formulation is solved using a scenario-based optimization technique integrated with a metaheuristic algorithm. Simulation results on the IEEE 33-bus distribution system demonstrate improved voltage profile, reduced expected losses, and enhanced system robustness compared to deterministic OPF approaches.

With the increasing penetration of intermittent renewable energy sources into the electric power grid, there is an emerging need to develop stochastic optimization methods to enable the reliable and efficient operation of power systems having a large fraction of their power supplied form uncertain resources. In this paper, we formulate the stochastic AC optimal power flow (OPF) problem as a two-stage stochastic program with robust constraints. This problem amounts to an infinite-dimensional nonconvex optimization problem. We develop a finite-dimensional inner approximation as a semidefinite program. Its solution yields an affine recourse policy that is guaranteed to be feasible for the stochastic AC OPF problem.

Recent research shows that non-convex AC OPF problem can be recast as a convex Semidefinite (SD) problem or Second Order Cone Programming (SOCP) problem. This paper presents a stochastic SOCP OPF (SSOCP-OPF) model for power systems connected to the wind farms. This is performed by reformulating the original non-convex OPF problem using more practical parameters of the power system. Finally, we obtain a convex optimization problem through some well-known approximations and an exact relaxation incorporating the stochastic nature of wind power. One of the advantage of convex SOCP problems, which are a general form of linear problems, is that they can be efficiently solved through Interior Point Methods (IPMs). The proposed OPF model takes advantage of both DC-OPF models (solution efficiency) and full AC-OPF models (solution accuracy). As an application of SSOCP-OPF model, we study the impact of wind power uncertainty on the transmission loss in the power systems. To evaluate the proposed stochastic model modified IEEE 30-bus test system is used. The optimization problem is coded in GAMS platform and solved using its embedded interior point optimizer MOSEK.

Due to the increasing hydrogen demands, a strong sense of commitment has recently been found to take advantage of the economic opportunities offered by power-to-hydrogen (P2H) units considering the high penetration of renewable energy sources (RESs). Deriving a market participation model for extracting green hydrogen with special attention to the grid code requirements is a fundamental challenge that has not yet been addressed. Motivated by this challenge, this paper presents a stochastic security-constrained optimal power flow (SSC-OPF) model to optimally allocate P2H units in renewable-dominated regional power grids. The main aim of the proposed planning model is to maximize the profit of power grid operators by extracting as much green hydrogen as possible and delivering it to the downstream industries. The presented model covers essential operational constraints, reserve adequacy issues, conservation voltage reduction, and uncertain behavior of demands and RESs to ensure the realistic operation of power grids. Moreover, the net present value of the proposed model is calculated to determine the profitability rate of using P2H units according to business models. The applicability of the proposed model is examined on the extended IEEE 30-bus and IEEE 118-bus test systems. The simulation results show that the use of P2H units in combination with RESs not only makes power grids more profitable but also improves the technical parameters.

This paper is concerned with optimal management of distributed energy resources (DERs) in distribution networks under uncertainty. The sources of uncertainty include user power consumption and intermittent renewable energy produced by photovoltaic (PV) units. The DERs enabling smart control laws mainly comprise photovoltaic inverters that produce or absorb reactive power, and storage units (batteries) that can be used to coordinate the photovoltaic units with the loads. A stochastic optimal power flow is formulated, where the control actions are expressed as affine functions of the uncertain quantities. Voltage security constraints are included as probabilistic constraints. Assuming that the uncertain quantities follow the Gaussian distribution, the resulting optimization problem is shown to be a second-order cone program. Numerical tests validate the probabilistic voltage specification.

Demand response (DR) can provide reserves in power systems but a fundamental challenge is that the amount of capacity available from DR is time-varying and uncertain. We propose a stochastic optimal power flow (OPF) formulation that handles uncertain energy from wind and uncertain reserves provided by DR. To handle the uncertainty, we formulate chance constraints and use a scenario based methodology to solve the stochastic OPF problem. This technique allows us to provide a-priori guarantees regarding the probability of constraint satisfaction. Additionally, we devise a strategy for the reserves, provided either by the generators or the loads, that could be deployed in real time operation. To evaluate the effectiveness of our methodology, we carry out a simulation based analysis on the IEEE 30-bus network. Our case studies show that optimizing over the reserves provided by DR, even though they are uncertain, results in lower total cost compared to the case where only generation side reserves are taken into account. We also carry out a Monte Carlo analysis to empirically estimate the probability of constraint satisfaction and demonstrate that it is within the theoretical limits.

Aiming for the median solution between cyber-intensive optimal power flow (OPF) solutions and subpar local control, this work advocates deciding inverter injection setpoints using deep neural networks (DNNs). Instead of fitting OPF solutions in a black-box manner, inverter DNNs are naturally integrated with the feeder model and trained to minimize a grid-wide objective subject to inverter and network constraints enforced on the average over uncertain grid conditions. Learning occurs in a quasi-stationary fashion and is posed as a stochastic OPF, handled via stochastic primal-dual updates acting on grid data scenarios. Although trained as a whole, the proposed DNN is operated in a master-slave architecture. Its master part is run at the utility to output a condensed control signal broadcast to all inverters. Its slave parts are implemented by inverters and are driven by the utility signal along with local inverter readings. This novel DNN structure uniquely addresses the small-big data conundrum where utilities collect detailed smart meter readings yet on an hourly basis, while in real time inverters should be driven by local inputs and minimal utility coordination to save on communication. Numerical tests corroborate the efficacy of this physics-aware DNN-based inverter solution over an optimal control policy.

In this paper, a new iterative learning algorithm is proposed for repetitive nonlinear systems. The control system employs a combination of state feedback and iterative learning control (ILC) in which the coefficients of states are learned similar to ILC methods. The control system is in a closed loop format both in iteration domain (because of ILC) and in time domain (because of feedback control) which improves the robustness of the conventional ILC. The convergence of the control algorithm is also proved. Finally, simulation results for a 5-DOF manipulator have been presented to illustrate that the proposed algorithm is more robust than a first order P type ILC method at the presence of white Gaussian noise as a nonrepeating disturbance.

The repetitive processing and large quantity of single product represented by 3C products are urgently needed. However, for current processing operations, previous processing data have not been used in the optimization of control input. In order to utilize previous processing data to facilitate the next process and avoid adverse effects caused by repetitive disturbance and noise, the idea of iterative learning was introduced to improve the accuracy of machining. On the control level, since it is difficult to obtain high accuracy by traditional feedback control when faced with complex trajectories, an open-loop iterative learning controller and a position loop feedback controller were introduced, which worked fast with good convergence effects. Aiming at reducing the influence of accidental error, step type iterative learning was put forward. The iteration mechanism was stopped when the accuracy converged to the allowable range so as to reduce computational complexity, store the current iterative part of the control input, and make constant value compensation. However, in simulation and experiment, it was found that after superposition of the iterative learning controller, the phenomenon of partial divergence of the system tracking error occurred. Therefore, the speed and acceleration characteristics of input trajectories in time domain and frequency domain were analyzed. High-frequency noise was introduced in frequency domain, which was found to be the cause of the abovementioned phenomenon, and high-frequency components were filtered to solve the problem. To further improve the accuracy of convergence and avoid filtering effective high-frequency information in some area, a switchable filter based on the analysis of the frequency characteristics of input trajectory was proposed. Through SIMULINK simulation and dSPACE experimental verification, it was proved that the iterative learning controller of modifying controlled quantity and filter based iterative learning control method are effective.

Iteration dependent interval based open‐closed‐loop iterative learning control for time varying systems with vector relative degree

Since they were first introduced by Arimoto, iterative learning control (ILC) methods have received lots of attention because of their effectiveness and briefness. In this paper, an open-closed-loop PID-type ILC scheme with bounded learning matrixes for the control of nonlinear time-varying systems are studied. Under a few reasonable restrictions that are very common, as in other ILC research papers on the convergence problem, sufficient conditions for guaranteeing the convergence of ILC systems are given and then proved with linear operator theory. Both the system and the ILC scheme described have very general forms. It is shown that the convergence condition is weaker than some known results for a similar ILC scheme.

We develop DeepOPF-Stoc as a neural network approach to efficiently solve the two-stage stochastic AC optimal power flow (AC-OPF) problem, which emphasizes reliable and cost-effective grid operation while accounting for uncertainties in renewable energy generation and load demand. We first identify a crucial partial permutation-invariance property of the mapping from load to first-stage solution. We then leverage this property in DeepOPF-Stoc to design (i) PPNN as a principled architectural framework that orchestrates standard neural network designs to learn such mappings, and (ii) a PPNN-aware sampling algorithm to prepare training data efficiently. DeepOPF-Stoc effectively addresses the dimensionality explosion issue inherent in vanilla designs, significantly reducing both the required number of neurons and the volume of training data. Our theoretical analysis confirms the universal approximation capability of PPNNs for our application. We further prove that the PPNN-aware sampling algorithm improves sampling efficiency by a factor of K! compared to uniform sampling, where K represents the number of second-stage scenarios. Simulation results on IEEE 118-bus and synthetic 793-bus test systems demonstrate the superiority of DeepOPF-Stoc over state-of-the-art alternatives. It achieves two orders of magnitude speedup compared to iterative solvers while generating feasible first-stage solutions with comparable second-stage out-of-sample feasibility and 0.95% cost difference.