There are models for which the evaluation of the likelihood is infeasible in practice. For these models the Metropolis-Hastings acceptance probability cannot be easily computed. This is the case, for instance, when only departure times from a "G"/"G"/1 queue are observed and inference on the arrival and service distributions are required. Indirect inference is a method to estimate a parameter "&thgr;" in models whose likelihood function does not have an analytical closed form, but from which random samples can be drawn for fixed values of "&thgr;". First an auxiliary model is chosen whose parameter "&bgr;" can be directly estimated. Next, the parameters in the auxiliary model are estimated for the original data, leading to an estimate . The parameter "&bgr;" is also estimated by using several sampled data sets, simulated from the original model for different values of the original parameter "&thgr;". Finally, the parameter "&thgr;" which leads to the best match to is chosen as the indirect inference estimate. We analyse which properties an auxiliary model should have to give satisfactory indirect inference. We look at the situation where the data are summarized in a vector statistic "T", and the auxiliary model is chosen so that inference on "&bgr;" is drawn from "T" only. Under appropriate assumptions the asymptotic covariance matrix of the indirect estimators is proportional to the asymptotic covariance matrix of "T" and componentwise inversely proportional to the square of the derivative, with respect to "&thgr;", of the expected value of "T". We discuss how these results can be used in selecting good estimating functions. We apply our findings to the queuing problem. Copyright 2004 Royal Statistical Society.
Abstrxt -The paper describes a method for calculating margins to voltage instability. The method is based on sensitivity techniques. The measure to voltage instability can be given in terms of MW, MVAr or MVA depending on the components in the increase in load. The extensions to earlier publications of the method are the ability to increase the load and generation simultaneously on several buses without increasing the computation time. A single forward and backward substitution is sufficient to find the required sensitivities when the load is increased. The productions of the plant can be increased either by a specified profile or by using the turbine droop characteristics. A method for identification of contingencies needing an explicit reserve evaluation is briefly described.Kevwords -Reactive power margins, Voltage collapse, Sensitivity methods. INTRODUCTIONVoltage stability problems have got increasing attention the latest years. Many utilities are already experiencing such problems and the tendency is to operate closer to the margins. This call for a more exact evaluation of the actual security level in order to prevent wide spread disturbances and to further increase the utilization of the system. Work is going on within IEEE and CIGRE [I ,2] and the number of publications increase. The methods applied for evaluating and predicting voltage stability range from simulation methods to methods that focus on the existence of the load flow solution. The simulation methods are accurate and can account for all dynamic phenomena. However, they are time consuming when evaluated for many cases [ 11. The methods investigating the static aspects of the load flow solution apply various methods for identification of the point of bifurcation and to estimate the stability conditions of the system. Proximity indicatorsSeveral indicators are used to estimate the proximity to voltage instability. Some of the indicators are based on the linearized load flow equations. Examples are sensitivities, singular value decomposition [3] and eigen values. Since these indicators do not take into account the non-linearities, they are not suited as stand alone indicators of the voltage 92 SM 395-5 PWRS A paper recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the IE&E/PES 1992 Summer Meeting, Seattle, WA, July 12-16, 1992. Manuscript submitted January 22, 1992; made available for printing May 13, 1992. stability margins.A promising method is to apply a MVAr or a MW distance to voltage collapse as a measure of the robustness of the power system. The application of the MVAr measure is to increase the reactive load on one or several buses and find the maximum increase before the system becomes unstable. The MW measure can be used to determine the maximum load that can be supplied on a bus, or to find the maximum transfer between areas before the voltage collapses. Methods appliedThe calculation of the margins are normally based on the static load flow equations. Th...
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