Statistical approximations of vibration power flow for complex structures with uncertainties

Reliability-based design optimization (RBDO) is an effective method for structural optimization due to its ability to take into consideration uncertainties in design variables. Performance measure approach (PMA) based methods are commonly utilized to evaluate the probabilistic constraints of RBDO problems. The advanced mean value (AMV) method is a very commonly used due to its simpleness and effectiveness. However, the AMV method sometimes produces unstable and inefficient results for concave and highly nonlinear limit-state functions. In order to improve robustness and efficiency, many methods have been developed, for example, chaos control based and conjugate gradient-based methods. These methods lead to more stable results as compared with the AMV approach but they are inefficient for use in complex and convex limit-state functions. The RBDO of structural components is often a difficult issue due to complicated constraints. In this paper, a novel hybrid approach, referred to as “hybrid gradient analysis (HGA)” is introduced for the evaluation of both convex and concave constraint functions in RBDO. The HGA method combines AMV and conjugate gradient analysis (CGA). The robustness, simpleness and effectiveness of the proposed HGA method are compared with various PMA methods aimed at reliability such as AMV, chaos control (CC), conjugate mean value (CMV), modified chaos control (MCC), hybrid mean value (HMV) and CGA methods by means of several nonlinear convex/concave limit-state functions and structural RBDO problems. Reliability analysis and RBDO results point out that the HGA approach introduced here is more effective and robust than the well-known approaches.

In probabilistic structural analysis, the performance or response functions usually are implicitly defined and must be solved by numerical analysis methods such as finite element methods. In such cases, the most commonly used probabilistic analysis tool is the mean-based, second-moment method which provides only the first two statistical moments. This paper presents a generalized advanced mean value (AMV) method which is capable of establishing the distributions to provide additional information for reliability design. The method requires slightly more computations than the second-moment method but is highly efficient relative to the other alternative methods. In particular, the examples show that the AMV method can be used to solve problems involving non-monotonic functions that result in truncated distributions.

This paper proposes a novel probabilistic power flow (PPF) algorithm considering generation dispatch. The intent is to examine the influence of uncertainty due to photovoltaic (PV) generation resources and loads. PV generation uncertainty may have a significant impact on transmission systems since this resource is easily influenced by changing environmental conditions. However, it is prudent to include generation dispatch in the PPF algorithm since the dispatching strategy compensates for PV generation injections and influences the uncertainty in the results. The proposed PPF algorithm is based on the cumulant method. The Gram-Charlier expansion is applied to approximate the distribution of probabilistic variables. Furthermore, this paper also proposes a probabilistic optimal power dispatching strategy which considers uncertainty problems in the economic dispatch and optimizes the expected value of the total cost with the overload probability as a constraint. The Arizona area of the Western Electricity Coordinating Council (WECC) system is used to test the proposed PPF algorithm and compare results with Monte Carlo simulation (MCS).

Wind farms commonly cluster in regions rich in wind resources. Thus, correlation of wind speeds from different wind farms should not be ignored when modeling a power system with large wind energy penetration. This paper proposes a probabilistic optimal power flow (POPF) technique based on the quasi-Monte Carlo simulation (QMCS) considering the correlation of wind speeds using copula functions. In this paper, a copula function is used to model the dependent structure of random wind speeds and their forecast errors. QMCS is employed in the sampling procedure to reduce computation burden. The proposed method is applied in probabilistic power flow (PPF). Furthermore, the PPF is used in the POPF problem that aims at minimizing the expectation and downside risk of fuel cost simultaneously. Simulation studies are conducted on a modified IEEE 118-bus power system with wind farms integrated in two areas, and the results show that the accuracy and efficiency are improved by the proposed method.

As PV penetration in the power distribution network is growing on an unprecedented scale, probabilistic power flow analysis is becoming crucial to assess the health of network operation. State-of-the-art probabilistic analysis relies on Stochastic Response Surface Method and Gaussian Copula for including PV sources correlation. However, it has been observed how in the presence of complex statistical distributions of injected PV powers, such a standard approach can provide inaccurate evaluations of the output variable distributions. In this article, we carefully investigate the origins of such a drawback and propose a novel implementation flow that overcomes the problem. In order to check the correctness of the proposed methodology, the results obtained with the Stochastic Response Surface Method are compared with Monte Carlo simulations. Several investigations, such as voltage uncertainties, network health and probability of violating quality constraints are conducted on two test networks.

Due to the limited joint position space and the consideration of reducing the moment of inertia and vibration for industrial robot, the design optimization for rotate vector (RV) reducer is becoming a new and urgent problem in industry. Currently, the existing researches focus on deterministic design optimization, which may cause unreliable designs without considering uncertainties. Therefore, the study focuses on the implementation of reliability-based design optimization (RBDO) to the RV reducer. The aim is to make the RV reducer smaller in size while ensuring a higher reliability. Firstly, a modified advanced mean value (MAMV) method is proposed to improve efficiency and robustness of the advanced mean value method, which encounters inefficiency and numerical instability for the concave or highly nonlinear performance measure functions. Secondly, a mathematical model of RBDO for the RV reducer is established. Thirdly, the proposed MAMV method is integrated into double-loop method to optimize the established mathematical model of RBDO with different target reliability. The results show that the proposed MAMV method is efficient compared with other methods. In addition, the volume of the RV reducer is correspondingly reduced by 9.44%, 7.89%, and 5.66% compared with that before optimization when the target reliabilities are 99.38%, 99.87%, and 99.98%.

Based on Monte Carlo and interior point method, this paper presents a probabilistic optimal power flow analysis method for the virtual power plant (VPP) containing photovoltaic (PV) generation, which can take into account both the randomness and correlation of PV generators' power outputs and loads at different locations. An optimal power flow model of VPP is firstly given to minimize the operation cost of VPP. The inverse transformation method and Cholesky decomposition technique are used to simulate the correlations between PV power outputs and loads, which obey Beta and Normal distributions respectively. Finally, the modified IEEE 30-bus test system is used to demonstrate the effectiveness and application of the presented method in the probabilistic optimal power flow analysis of VPP.

The natural gas system and electricity system are coupled tightly by gas turbines in an integrated energy system. The uncertainties of one system will not only threaten its own safe operation but also be likely to have a significant impact on the other. Therefore, it is necessary to study the variation of state variables when random fluctuations emerge in the coupled system. In this paper, a multi-slack-bus model is proposed to calculate the power and gas flow in the coupled system. A unified probabilistic power and gas flow calculation, in which the cumulant method and Gram–Charlier expansion are applied, is first presented to obtain the distribution of state variables after considering the effects of uncertain factors. When the variation range of random factors is too large, a new method of piecewise linearization is put forward to achieve a better fitting precision of probability distribution. Compared to the Monte Carlo method, the proposed method can reduce computation time greatly while reaching a satisfactory accuracy. The validity of the proposed methods is verified in a coupled system that consists of a 15-node natural gas system and the IEEE case24 power system.

The robustness and efficiency of inverse reliability methods are important issues in reliability-based design optimization (RBDO) using performance measure approach (PMA). The adaptive modified chaos control (ACC), step length adjustment (SA), and relaxed mean value (RMV) methods were recently implemented to improve the robustness and efficiency of PMA. In this paper, a limited descent mean value (LDMV) method is proposed to improve robustness and efficiency of inverse reliability analysis for either convex or concave probabilistic constraints. The LDMV formula is dynamically adjusted by an adaptive step size based on the advanced mean value method (AMV). The robustness and efficiency of the ACC, SA, RMV, and proposed LDMV methods are compared through six nonlinear performance functions. The results illustrated a similar robust performance between LDMV against RMV and FSL methods and superior to the ACC method. The proposed LDMV improves the robustness and efficiency of the inverse first-order reliability method in comparison with existing reliability methods.

This paper presents a general statistical treatment for power flow analysis in a complex structure. The nominal power flow is computed using the characteristic constraint (CC) modes of the component structures. These CC modes are built from component mode synthesis (CMS) of finite element models, and they form a highly-reduced order basis for capturing the motion of the interface between components — and thus the power flow. The power flow statistics are then calculated over an ensemble of systems due to the variation of structural system parameters. Each modal response is expanded in a basis of quadrature polynomials or locally linear interpolation (LLI) functions in these uncertain parameters. The system equations for the reduced order CMS model with uncertainties are derived using Galerkin's method. These equations have the same form as the nominal power flow equations, but the number of equations is increased by a factor equal to the order of the polynomials (when using quadrature polynomials) or the number of nodes in the parameter space (when using LLI functions). Thus, the ensemble-averaged power flow can be calculated by solving this new set of equations, instead of using Monte Carlo simulations. A cantilever plate and a L-shaped plate are used as example structures for the demonstration of these statistical approximations. In general, this statistical treatment provides efficient and accurate modeling of parameter uncertainties, which is critical for mid-frequency vibration analysis.

This paper develops a computationally efficient algorithm which speeds up the probabilistic power flow (PPF) problem by exploiting the inherently low-rank nature of the voltage profile in electrical power distribution networks. The algorithm is accordingly termed the Accelerated-PPF (APPF), since it can accelerate “any” sampling-based PPF solver. As the APPF runs, it concurrently generates a low-dimensional subspace of orthonormalized solution vectors. This subspace is used to construct and update a reduced order model (ROM) of the full nonlinear system, resulting in a highly efficient simulation for future voltage profiles. When constructing and updating the subspace, the power flow problem must still be solved on the full nonlinear system. In order to accelerate the computation of these solutions, a Neumann expansion of a modified power flow Jacobian is implemented. Applicable when load bus injections are small, this Neumann expansion allows for a considerable speed up of Jacobian system solves during the standard Newton iterations. APPF test results, from experiments run on the full IEEE 8500-node test feeder, are finally presented.

Probabilistic optimal power flow analysis for optimal deployment of Unified power flow controller in renewable integrated hydro power systems using marine predators algorithm

To evaluate the impact of the randomness and correlation of photovoltaic (PV) and load on AC/DC systems with a multiport current flow controller (M-CFC), this paper proposes a probabilistic optimal power flow calculation for AC/DC systems based on a nonparametric kernel density estimation. First, according to the M-CFC model, the DC power flow calculation method with M-CFC was deduced, and its influence on line loss was analyzed. Second, a nonparametric kernel density estimation with an adaptive bandwidth was used to accurately describe the probability distribution of the PV and load, and correlation samples of the PV and load were obtained by the mixed copula function. Then an optimization model that considers system loss and static security was established, and a fast nondominated sorting genetic algorithm based on the elite strategy (NSGA-II) was used to calculate the multi-objective probability optimal power flow of the AC/DC system. Finally, a case study was performed on the modified IEEE39 bus system using measured PV and load data. We verified the nonparametric kernel density estimation with an adaptive bandwidth can better adapt to random component uncertainty, and M-CFC can improve the static security of the system.

In the analysis of the vibration response and structure-borne vibration transmission between elements of a complex structure, statistical energy analysis (SEA) or finite element analysis (FEA) are generally used. However, an alternative method is using vibrational power flow techniques which can be especially useful in the mid frequencies between the optimum frequency regimes for FEA and SEA. Power flow analysis has in general been used on one-dimensional beam-like structures or between structures with point joints. In this paper, the power flow technique is extended to two-dimensional plate like structures joined along a common edge without frequency or spatial averaging the results, such that the resonant response of the structure is determined. The power flow results are compared to results obtained using FEA at low frequencies and SEA at high frequencies. The agreement with FEA results is good but the power flow technique has an improved computational efficiency. Compared to the SEA results the power flow results show a closer representation of the actual response of the structure.

This report first analysis quasi static power flow for photovoltaic power plant. Then, fault analysis for different symmetrical and asymmetrical fault is done. The system used is IEEE 39 bus New England system that a Photovoltaic farm with 500 MW power is connected to bus 24. Newton-Raphson Power flow method is used with Gauss-Jordan elimination method for matrix inversion in this power flow analysis. Different operation condition including faults are also considered for fault at bus 16. The results are validated with simulation in MATLAB software.