Stochastic Optimal Power Flow Problem Research Articles

Parallel interior-point solver for block-structured nonlinear programs on SIMD/GPU architectures

ABSTRACT We investigate how to port the standard interior-point method to new exascale architectures for block-structured nonlinear programs with state equations. Computationally, we decompose the interior-point algorithm into two successive operations: the evaluation of the derivatives and the solution of the associated Karush-Kuhn-Tucker (KKT) linear system. Our method accelerates both operations using two levels of parallelism. First, we distribute the computations on multiple processes using coarse parallelism. Second, each process uses SIMD/GPU accelerators locally to accelerate the operations using fine-grained parallelism. The KKT system is reduced by eliminating the inequalities and the state variables from the corresponding equations. We demonstrate our method's capability on the supercomputer Polaris, a testbed for the future exascale Aurora system. Each node is equipped with four GPUs, a setup amenable to our two-level approach. Our experiments on the stochastic optimal power flow problem show that the reduction method is 50x faster than the sparse linear solver HSL MA57 running in serial on the CPU, and 6x faster than Pardiso running in parallel on CPU on the same number of processes.

A solution to multi Objective Stochastic Optimal Power Flow problem using mutualism and elite strategy based Pelican Optimization Algorithm

Wind Energy is rapidly growing into one of the most popular energy options for power generation. However, the unpredictability of wind necessitates the employment of a distribution function when describing the wind environment. In the depiction of wind speeds, the two significant parameters of the Weibull distribution, such as the scale parameter and the shape parameter are often utilized. Therefore, it is essential to choose the most appropriate approach for estimating these parameters. Furthermore, the uncertainty in delegated power from variable wind speeds makes system management challenging. In this paper, the Weibull distribution parameters were determined using the Pelican Optimization Algorithm (POA). POA has gained extensive popularity among the research community and is currently used to address a wide range of optimization problems. However, the exploration and exploitation are not properly balanced in this algorithm. To overcome this problem a novel hybrid algorithm namely Mutualism phase based POA (MPOA) is proposed with the help of mutualism phase of Symbiotic Organisms Search (SOS) algorithm. To demonstrate the efficacy and robustness MPOA is applied on multi-objective Optimal Power Flow (MO-OPF) problems. For this study IEEE 30 bus standard system and modified IEEE 30 bus system consisting of two wind farms are taken. The purpose of this work is to examine the impact of changing the Weibull parameters, reverse and penalty cost coefficients on the overall generation cost while considering all the security aspects.

Integrating renewable energy and V2G uncertainty into optimal power flow: A gradient bald eagle search optimization algorithm with local escaping operator

AbstractHere, a new approach is proposed for solving the optimal power flow (OPF) problem in transmission networks using a Gradient Bald Eagle Search Algorithm (GBES) with a Local Escaping Operator (LEO). The method takes into account uncertainty of the renewable energy sources (wind energy and photovoltaic systems) and Vehicle‐to‐Grid (V2G) in the stochastic OPF problem. To improve the efficiency of the proposed technique and enhance its local exploitation capability, the LEO method's selection features are utilized. Monte Carlo methods are employed to estimate the generation costs of the renewable sources and PEVs and study their feasibility. The uncertainty of the renewable sources and PEVs is represented by Weibull, lognormal, and normal probability distribution functions (PDFs). The GBES approach is experimentally compared with well‐known meta‐heuristics using twenty‐three different test functions, and the results indicate its superiority over BES and other recently developed algorithms. Furthermore, the proposed method's effectiveness is evaluated using IEEE 30‐bus test system under various scenarios, and the simulation results demonstrate that it can effectively address OPF issues considering renewable energy sources and V2G, providing superior optimal solutions compared to other algorithms.

Convex Stochastic Approaches for the Optimal Allocation of Distributed Energy Resources in AC Distribution Networks with Measurements Fitted to a Continuous Probability Distribution Function

This paper addresses the optimal stochastic allocation of distributed energy resources in distribution networks. Typically, uncertain problems are analyzed in multistage formulations, including case generation routines, resulting in computationally exhaustive programs. In this article, two probabilistic approaches are proposed–range probability optimization (RPO) and value probability optimization (VPO)–resulting in a single-stage, convex, stochastic optimal power flow problem. RPO maximizes probabilities within a range of uncertainty, whilst VPO optimizes the values of random variables and maximizes their probabilities. Random variables were modeled with hourly measurements fitted to the logistic distribution. These formulations were tested on two systems and compared against the deterministic case built from expected values. The results indicate that assuming deterministic conditions ends in highly underestimated losses. RPO showed that by including ±10% uncertainty, losses can be increased up to 40% with up to −72% photovoltaic capacity, depending on the system, whereas VPO resulted in up to 85% increases in power losses despite PV installations, with 20% greater probabilities on average. By implementing any of the proposed approaches, it was possible to obtain more probable upper envelopes in the objective, avoiding case generation stages and heuristic methods.

Stochastic optimal power flow analysis of power system with renewable energy sources using Adaptive Lightning Attachment Procedure Optimizer

The integration of several renewable energy sources (RESs), such as photovoltaic generation (PVG) and wind turbine generation (WTG), introduces a significant amount of uncertainty for the optimal planning of the electrical power systems. The optimal power flow (OPF) problem became more difficult to solve when RESs uncertainties were taken into account. In this research, the stochastic OPF problem is solved using an Adaptive Lightning Attachment Procedure Optimizer (ALAPO) with the integration of RESs. The ALAPO is proposed in this study to improve both the exploration and exploitation process of the conventional Lightning Attachment Procedure Optimizer (LAPO). In order to create a number of scenarios that account for the uncertainties associated with load demand, wind speed, and solar irradiation, the scenario based reduction method is utilized. Further, to determine the optimal setting of active power of generators, tap setting of the transformers, reactive power of the capacitor banks, and the voltage profile at the generator buses for each scenario of RESs the objective function considered in this study is to minimize the power loss, voltage deviation, and voltage stability index of the power system. The effectiveness of the proposed algorithm is tested on IEEE 57-bus system. The results in this study show that the proposed ALAPO is superior for solving the OPF with the integration of RESs compared to the conventional methods like LAPO, Particle Swarm Optimization (PSO), Sand Cat Swarm Optimizer (SCSO), Gray Wolf Optimizer (GOW), Whale Optimization Algorithm (WOA), Black Widow Optimization Algorithm (BWO), and Capuchin Search Algorithm (CapSA).

A Gradient-Based Approach for Solving the Stochastic Optimal Power Flow Problem with Wind Power Generation

Although wind power generation improves decarbonization of the electricity sector, its increasing penetration poses new challenges for power systems planning, operation and control. In this paper, we propose a solution approach for Stochastic Optimal Power Flow (SOPF) models under uncertainty in wind power generation. Two complicating issues are handled: i) difficulties imposed by probability density functions used to formulate wind power costs and their derivatives; ii) the non-differentiability of the cost function for thermal units. Due to such issues, SOPF models cannot be solved by gradient-based approaches and have been solved by meta-heuristics only. We obtain exact analytical expressions for the first and second order derivatives of wind power costs and propose a technique for handling non-differentiability in thermal costs. The equivalent SOPF model that results from such recasting is a differentiable NLP problem which can be solved by efficient gradient-based algorithms. Finally, we propose a modified log-barrier primal-dual interior/exterior-point method for solving the equivalent SOPF model which, differently from meta-heuristic approaches, is able to calculate important dual variables such as energy prices. Our approach, which is applied to the IEEE 30-, 57- 118- and 300-bus systems, strongly outperforms a meta-heuristic approach in terms of computation times and optimality.

Optimal Power Flow Solution of Wind-Integrated Power System Using Novel Metaheuristic Method

Wind energy is particularly significant in the power system today since it is a cheap and clean power source. The unpredictability of wind speed leads to uncertainty in devolved power which increases the difficulty in wind energy system operation. This paper presents a stochastic optimal power flow (SCOPF) for obtaining the best scheduled power from wind farms while lowering total operational costs. A novel metaheuristics method called Aquila Optimizer (AO) is used to address the SCOPF problem due to its highly nonconvex and nonlinear nature. Wind speed is represented by the Weibull probability distribution function (PDF), which is used to anticipate the cost of wind-generated power from a wind farm based on scheduled power. Weibull parameters that provide the best match to wind data are estimated using the AO approach. The suggested wind generation cost model includes the opportunity costs of wind power underestimation and overestimation. Three IEEE systems (30, 57, and 118) are utilized to solve optimal power flow (OPF) using the AO method to prove the accuracy of this method, and results are compared with other metaheuristic methods. With six scenarios for the penalty and reverse cost coefficients, SCOPF is applied to a modified IEEE-30 bus system with two wind farms to obtain the optimal scheduled power from the two wind farms which reduces total operation cost.

Hybrid power systems with emission minimization: Multi-objective optimal operation

The demand of electrical power and energy is exponentially increased with the advancement of science and technology in past decades. To fulfil this increasing demand of energy, the burden on conventional fossil fuel based units has continuously been raised, which in turn has severe consequences on the environment. A clean pollution free and healthy environment necessitates the reduction in the use of conventional fossil fuel based units. In this context, integration of non-conventional and renewable power generation facilities with conventional power generators, for instance, hybrid power system is gaining attention. However, the intermittent posed by renewable power generation facilities may make the scheduling of generators and power system operation extremely challenging. These issues and its possible solution have been addressed in this proposed study. An attempt has been made in the evaluation of an optimum generation scheduling and coordination among different hybrid power system configurations in the form of wind-thermal, hydro-thermal-wind and hydro-thermal-wind-solar systems. This stochastic optimal power flow problem of the proposed system has been formulated in a multi-objective optimization framework while incorporating real-time operational constraints. A mathematical model, operational analysis and comparative evaluation between the optimum operational paradigms obtained with a modified bacteria foraging algorithm for wind-thermal, hydro-thermal-wind and hydro-thermal-wind-solar systems have been implemented. The effectiveness and promising solutions of hydro-thermal-wind-solar in response to fulfill the operational objectives like cost effective operation, minimization of transmission loss, emission and voltage variation over other hybrid configurations in multi-objective stochastic optimal power flow environment as implemented in IEEE30 bus power system has been demonstrated.

Stochastic second-order-cone complementarity problems: expected residual minimization formulation and its applications

This paper considers a class of stochastic second-order-cone complementarity problems (SSOCCP), which are generalizations of the noticeable stochastic complementarity problems and can be regarded as the Karush–Kuhn–Tucker conditions of some stochastic second-order-cone programming problems. Due to the existence of random variables, the SSOCCP may not have a common solution for almost every realization . In this paper, motivated by the works on stochastic complementarity problems, we present a deterministic formulation called the expected residual minimization formulation for SSOCCP. We present an approximation method based on the Monte Carlo approximation techniques and investigate some properties related to existence of solutions of the ERM formulation. Furthermore, we experiment some practical applications, which include a stochastic natural gas transmission problem and a stochastic optimal power flow problem in radial network.

Hybrid Stochastic-Deterministic Multiperiod DC Optimal Power Flow

Stochastic optimal power flow (OPF) formulations that minimize the expected operating costs over forecast scenarios generally result in lower costs than the standard (deterministic) OPF problem for power systems with significant forecast error, for example, from renewable energy sources. However, this type of stochastic OPF problem is more computationally demanding than the deterministic OPF, and even more so when storage units or ramp-constrained generators are included, as they require solving a multi-period OPF problem. We propose a hybrid method approaching the cost performance of the stochastic OPF problem and the computational burden of the deterministic OPF problem. Our method decomposes the problem into stochastic and deterministic subproblems, and relies on Benders’ Cuts to interface them. We present three versions of the method, which achieve different cost/computational burden trade-offs. The versions can be parametrized so that they scale well with the problem dimension. For one of the versions, we develop a multi-dimensional formulation of the Sandwich Algorithm , which is used to iteratively approximate a convex function. Through a case study using a 118-bus system, we find that our hybrid method achieves between 25% and 81% of the cost improvement of the stochastic OPF, but requires only 12 to 41% of the increased computation time required by the stochastic OPF. Both the hybrid method and the multi-dimensional Sandwich Algorithm can be used for problems outside of the field of power systems.

Real-Time Energy Management in Microgrids

Energy management in microgrids is typically formulated as an offline optimization problem for day-ahead scheduling by previous studies. Most of these offline approaches assume perfect forecasting of the renewables, the demands, and the market, which is difficult to achieve in practice. Existing online algorithms, on the other hand, oversimplify the microgrid model by only considering the aggregate supply-demand balance while omitting the underlying power distribution network and the associated power flow and system operational constraints. Consequently, such approaches may result in control decisions that violate the real-world constraints. This paper focuses on developing an online energy management strategy (EMS) for real-time operation of microgrids that takes into account the power flow and system operational constraints on a distribution network. We model the online energy management as a stochastic optimal power flow problem and propose an online EMS based on Lyapunov optimization. The proposed online EMS is subsequently applied to a real-microgrid system. The simulation results demonstrate that the performance of the proposed EMS exceeds a greedy algorithm and is close to an optimal offline algorithm. Lastly, the effect of the underlying network structure on energy management is observed and analyzed.

A Stochastic Optimal Power Flow Problem With Stability Constraints—Part I: Approximating the Stability Boundary

Stochastic optimal power flow can provide the system operator with adequate strategies for controlling the power flow to maintain secure operation under stochastic parameter variations. One limitation of stochastic optimal power flow has been that only line flows have been used as security constraints. In many systems voltage stability and small-signal stability also play an important role in constraining the operation. In this paper we aim to extend the stochastic optimal power flow problem to include constraints for voltage stability as well as small-signal stability. This is done by approximating the voltage stability and small-signal stability constraint boundaries with second-order approximations in parameter space. Then we refine methods from mathematical finance to be able to estimate the probability of violating the constraints. In this first part of the paper, we derive second-order approximations of stability boundaries in parameter space. In the second part, the approximations will be used to solve a stochastic optimal power flow problem.

A Stochastic Optimal Power Flow Problem With Stability Constraints—Part II: The Optimization Problem

Stochastic optimal power flow can provide the system operator with adequate strategies for controlling the power flow to maintain secure operation under stochastic parameter variations. One limitation of stochastic optimal power flow has been that only limits on line flows have been used as stability constraints. In many systems voltage stability and small-signal stability also play an important role in constraining the operation. In this paper we aim to extend the stochastic optimal power flow problem to include constraints for voltage stability as well as small-signal stability. This is done by approximating the voltage stability and small-signal stability constraint surfaces with second-order approximations in parameter space. Then we refine methods from mathematical finance to be able to estimate the probability of violating the constraints. In this, the second part of the paper, we look at how Cornish-Fisher expansion combined with a method of excluding sets that are counted twice, can be used to estimate the probability of violating the stability constraints. We then show in a numerical example how this leads to an efficient solution method for the stochastic optimal power flow problem.

An Implementation of the Stochastic OPF Problem

This paper presents an algorithm for solution of the stochastic optimal power flow (OPF) problem. The stochastic OPF problem minimizes the total cost of a base-case operating state plus the expected cost of recovery from contingencies such as line or generator outages. The algorithm presented is based on a cutting plane method that decomposes the OPF problem into smaller subproblems, which can be solved quickly and efficiently. The solutions of the individual subproblems are coordinated by the addition of linear constraints to a master problem. Typical results show a slight increase in the cost of the base-case operating state but a significant reduction in the expected cost of contingency recovery.

Stochastic optimal load flow using a combined quasi-Newton and conjugate gradient technique

This paper treats the problem of optimal power flow in electric power systems using a method which includes the effects of uncertain variables in formulating the problem for an all thermal electric power system. The method assumes that the system power demand is random and is normally distributed with zero mean and unit variance. The equality constraints associated with the formulation are the power flow equations in polar form utilizing the nodal admittance matrix approach. The non-linear programming technique of Powell's penalty function is used to allow the incorporation of inequality constraints in the formulation. The resulting stochastic optimal power flow problem reduces to one of solving a set of non-linear equations. Here we use a technique that combines the quasi-Newton method and the conjugate gradient method in what is referred to as the CONMIN algorithm. Computational implementation results involving four IEEE standard test networks are given to demonstrate the validity of this method.