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

*Occasional Paper*

Author**Robert L. Earle**

Published by

Stanford University, 1996

The production cost in operating an
electric power system is the cost of generation to meet the customer
load or demand. Production costing models are used in analysis of
electric power systems to estimate this cost for various purposes such
as evaluating long-term investments in generating capacity, contracts
for sales, purchases, or trades of power. A multi-area production
costing model includes the effects of transmission constraints in
calculating costs. Including transmission constraints in production
costing models is important because the electric power industry is
interconnected and trades or sales of power amongst systems can lower
costs. Moreover, the ongoing deregulation of the power industry and
growth of a more open market for power make the need to explicitly
account for the effects of transmission on costs more vital.

This thesis develops an analytical model for multi-area production
costing. The advantage of this approach is that it explicitly examines
the underlying structure of the problems. The major contributions of
our research are as follows. First, we develop the multivariate model
not just for transportation type of models or electric power network
flows, but also for the direct current power flow model. This overcomes
the objection that power flows are unrealistically modeled by a
transportation network. Most of the competing approaches suffer from
this problem. In fact, with the approach developed here, other
exogenous restrictions could be placed on the system subject to some
conditions. Second, this thesis derives the multi-area production cost
curve in the general case. This new result gives a simple formula for
determination of system cost and the gradient of cost with respect to
transmission capacities. Third, we give an algorithm for generating the
non-redundant constraints from a Gale-Hoffman type region. The
Gale-Hoffman conditions characterize feasibility of flow in a network.
This is useful not only in calculating readability, but it turns out
that in order to calculate the system cost we integrate over
Gale-Hoffman type regions as well. As a result, for many broad classes
of networks, enormous computational effort is saved. We also gather
together some existing and new results on Gale-Hoffman regions and put
them in a unified framework. Fourth, in order to derive the multi-area
production cost curves and also to perform the integration of the
multivariate Edgeworth series, an asymptotic series used to represent
probability densities, we need wedge shape regions (a wedge is the
affine image of an orthant). We give an algorithm for decomposing any
polyhedral set into wedges. Fifth, multivariate integration of the
normal distribution is a problem with importance in many areas and
central to calculation of the production cost. This thesis gives a new
method for one-dimensional numerical integration of the trivariate
normal. The best methods previously known were only able to reduce the
problem to a two dimensional numerical integration.