The multiple solution analysis of probabilistic load flow

Quadratic probabilistic load flow with linearly modelled dispatch

Because of uncertainties of renewable generation resources and load, efficient tools are required for load flow analysis in power systems. Many of the existing papers assume a set of given probability density functions (PDFs) to model uncertainties and develop parametric probabilistic load flow tools. However, the uncertainties might not fall in any standard class of PDFs. Thus, nonparametric tools are needed. This paper presents a method for nonparametric probabilistic load flow analysis to determine PDFs of the load flow outputs. The proposed method is based on the mean value first order saddle point approximation. For a system with $\boldsymbol {n}$ random variables, $\boldsymbol {(n+1)}$ load flow calculations are utilized to establish first order Taylor series expansion and then saddle point approximation is adopted to determine the probabilistic characteristic of desired output variables. The proposed nonparametric estimator provides accurate results while needing a reasonable computational effort. The estimator needs only a limited available data set of random variables without requiring information whether the data set belongs to a certain class of parametric distribution functions. Furthermore, both probability and cumulative distribution functions of load flow outputs are directly established without using integral or differential operators. The proposed method is tested on the IEEE 14-bus and the IEEE 118-bus test systems and promising results are obtained.

Presents an application of a general homotopy method to the computation of the characteristics of a power system such as PV curves. The load flow equations are derived from a set of nonlinear differential and algebraic equations. A homotopy continuation method is used for following the characteristics of the power system. The stability of the load flow solutions is discussed by deriving the variational equation. >

The paper extends the methods of evaluating the continuity in supplying electric power systems using comparative methods of the probabilistic load flow and the stochastic load flow. In the first case a binomial distribution for the units generators and a normal one for the loads are accepted. Branch outages are simulated by fictitious power injections at the corresponding nodes. The loss of load probability and expected energy not supplied are estimated with Gauss Laplace integral functions. The results are compared with the ones obtained through a reliability simulation with DigSILENT Power Factory application.

This paper presents a Probabilistic Load Flow (PLF) procedure that allows the impact of distributed generation (DG) on the transmission system to be assessed. By means of the Monte Carlo technique, several system states are simulated, varying the power injection of DG. The load flow solution is performed by the Newton-Raphson method and a distributed slack bus formulation is adopted, in order to distribute the active power mismatch (created by the random generation) on all the thermal units in the system. Tests are carried out both on a small system and on the Italian network. Results lead to conclude that the probabilistic approach to the power system analysis provides more indications than the traditional deterministic techniques, especially considering a high penetration of DG.

Normal-transformation-based probabilistic load flow with correlated wind and load forecast errors

Advanced Probabilistic Load Flow Technique in Consideration of Non-Gaussianity and Nodal Correlation of Input Variables

Probabilistic load flow (PLF) calculation, as a fundamental tool to analyze transmission system behavior, has been studied for decades. Despite a variety of available methods, existing PLF approaches rarely take system control into account. However, system control, as an automatic buffer between the fluctuations in random variables and the variations in system variables (e.g., nodal voltages), has a significant impact on the final PLF result. To consider control actions' influence, this paper proposes the first analytical PLF method for the transmission grid that takes into account primary and secondary frequency controls. This method is based on a high-precision linear power flow model, whose precision is even further improved in this paper by an original correction approach. This paper also proves that if the joint probability distribution (JPD) of random variables is expressed by a Gaussian mixture model (GMM), then the JPD of system variables is an infinite GMM. By leveraging this proposition, the proposed method generates the joint PLF of the whole system at the quasi-steady and steady states. The high accuracy and satisfactory efficiency of this method are verified on multiple test systems with different percentages of renewables.

An intrusive spectral method of probabilistic load flow (PLF) is proposed in the paper, which can handle the uncertainties arising from renewable energy integration. Generalized polynomial chaos (gPC) expansions of dependent random variables are utilized to build a spectral stochastic representation of PLF model. Instead of solving the coupled PLF model with a traditional, cumbersome method, a modified stochastic Galerkin (SG) method is proposed based on the P-Q decoupling properties of load flow in power system. By introducing two pre-calculated constant sparse Jacobian matrices, the computational burden of the SG method is significantly reduced. Two cases, IEEE 14-bus and IEEE 118-bus systems, are used to verify the computation speed and efficiency of the proposed method.

In complex power systems, nonlinear load flow equations have multiple solutions. Under typical load conditions only one solution isf stable and corresponds to a normal operating point, whereas the second solution is not stable and is never realized in practice. However, in future distribution grids with high penetration of distributed generators more stable solutions may appear because of active or reactive power reversal. The systems can operate at different states, and additional control measures may be required to ensure that it remains at the appropriate point. This paper focuses on the analysis of several cases where multiple solution phenomena is observed. A noniterative approach for solving load flow equations based on the Gröbner basis is introduced to overcome the convergence and computational efficiency associated with standard iterative approaches. All the solutions of load flow problems with their existence boundaries are analyzed for a simple 3-bus model. Furthermore, the stability of the solutions is analyzed using a derived aggregated load dynamics model, and suggestions for preventive control are proposed and discussed. The failure of naïve voltage stability criteria is demonstrated and new voltage stability criteria is proposed. Some of the new solutions of load flow equations are proved to be both stable and acceptable to the EN 50610 voltage fluctuation standard.

A non-iterative and exact linearization load flow technique for circuit contingency effects in power systems

It is an important part for the power system operation risk control to identify the critical lines. Fluctuation of wind power and fluctuation of load will have a great influence on power transmission and distribution, which will lead to the change of critical lines. Based on probabilistic load flow (PLF) analysis, the risk theory is introduced to put forward, and probabilistic flow betweenness considering uncertainty of wind power and load is presented for critical line identification, which reflects the influence of power grid structure characteristics, change of system running state and uncertain factors on power transmission. It can identify critical lines more accurately. Simulation calculation on CEPRI-36 bus system integrated with wind farms and IEEE-39 bus system with wind farms verify effectiveness of this method. It is able to search for the critical lines according to system running state in actual application, so as to adopt security control measures for the system.

Power flow analysis (PFA) is one of the most important issues of power system analysis and design. This paper aims at developing and applying a high-speed sampling method to produce large-capacity sample set based on pattern feature vector with layer structure (PFV with LS) of IEEE 14-bus system data analytical methods. In this study, sampling based pattern vectors are constructed in three layers. The first and second layer vectors are very low dimensional, and the sampling of the third layer is in parallel for each local area. Samples generated by this sampling method can be applied to the calculation of probabilistic load flow (PLF) and probabilistic static security assessment. Simulation results show that the proposed method can improve efficiency of PLF analysis and probabilistic static security assessment. The advantages of using PLF based on PFV with LS as the power flow limit can minimize the complicated mathematical equations. Furthermore, the algorithm is very simple and accurate, especially when the system connected to the wind power.

Accurate solution of dc ungrounded rail power system load flow requires detailed modelling of negative path components, resulting in large numbers of circuit nodes and branches. Such modelling complexity incurs heavy computational burden and potential ill numerical conditions when solved by nodal voltage equations. Part I of this study presented a ladder‐circuit model for traction network between substations and a power flow formulation adopted from ac distribution load flow. Part II presents the solution algorithm that harnesses the ladder circuit models to achieve efficient load flow computation and numerical robustness for the entire dc traction system by sequentially solving for power flows of individual substations and wayside ladder circuits. The proposed algorithm requires computational burden linearly proportional to the number of substations, much less than those by nodal voltage equations. The computation results from two transit lines in Taiwan illustrate that the proposed algorithm accomplishes the objectives of computational proficiency, numerical robustness and versatility to train load models and various traction circuit configurations such as branch lines.

The power distribution network connects the high-power transmission network to the end-users and thus plays an important role in the power system. The distribution system must be properly planned and operated in order to efficiently transfer power from the generator to the load end. As a result, load flow is a critical requirement for distribution system planning. This article proposes a fast flexible load flow (FFLF) solution for the radial distribution. The introduction of the radial configuration matrix (RCM) with the direct forward sweep method (DFSM) makes the fast flexible load flow method for radial distribution unique. The direct forward sweep method helps to reduce the number of iterations and calculation steps. Furthermore, the radial configuration matrix and simple bus numbering allow for the flexibility to adapt to any changes in the structure of the radial distribution networks, which is difficult with other methods. The proposed FFLF method can also be applied to the different practical loading conditions without forming a complex voltage equation. The proposed FFLF algorithm is tested on both balanced and unbalanced radial distribution systems and validated using IEEE standard test feeders.