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# EMF SR 6: Computation of Electric Power Production Cost with Transmission Constraints

This thesis develops an analytical model for multi-area productioncosting. The advantage of this approach is that it explicitly examinesthe underlying structure of the problems. The major contributions ofour research are as follows. First, we develop the multivariate modelnot just for transportation type of models or electric power networkflows, but also for the direct current power flow model. This overcomesthe objection that power flows are unrealistically modeled by atransportation network. Most of the competing approaches suffer fromthis problem. In fact, with the approach developed here, otherexogenous restrictions could be placed on the system subject to someconditions. Second, this thesis derives the multi-area production costcurve in the general case. This new result gives a simple formula fordetermination of system cost and the gradient of cost with respect totransmission capacities. Third, we give an algorithm for generating thenon-redundant constraints from a Gale-Hoffman type region. TheGale-Hoffman conditions characterize feasibility of flow in a network.This is useful not only in calculating readability, but it turns outthat in order to calculate the system cost we integrate overGale-Hoffman type regions as well. As a result, for many broad classesof networks, enormous computational effort is saved. We also gathertogether some existing and new results on Gale-Hoffman regions and putthem in a unified framework. Fourth, in order to derive the multi-areaproduction cost curves and also to perform the integration of themultivariate Edgeworth series, an asymptotic series used to representprobability densities, we need wedge shape regions (a wedge is theaffine image of an orthant). We give an algorithm for decomposing anypolyhedral set into wedges. Fifth, multivariate integration of thenormal distribution is a problem with importance in many areas andcentral to calculation of the production cost. This thesis gives a newmethod for one-dimensional numerical integration of the trivariatenormal. The best methods previously known were only able to reduce theproblem to a two dimensional numerical integration.