@TECHREPORT\{IMM2000-01041,
    author       = "H. A. Nielsen and H. Madsen and T. S. Nielsen",
    title        = "Load scheduling for decentralized {CHP} plants",
    year         = "2000",
    keywords     = "optimization, dynamic programming",
    number       = "",
    series       = "",
    institution  = "Informatics and Mathematical Modelling, Technical University of Denmark, {DTU}",
    address      = "Richard Petersens Plads, Building 321, {DK-}2800 Kgs. Lyngby",
    type         = "",
    url          = "http://www.imm.dtu.dk/~han/pub/efp98akk.pdf",
    abstract     = "This report considers load scheduling for decentralized combined heat
and power plants where the revenue from selling power to the
transmission company and the fuel cost may be time-varying.  These
plants produce both heat and power with a fixed ratio between these
outputs.  A heat storage facility is used to be able to deviate from
this restriction.

The load scheduling must be performed with only approximate knowledge
about the future.  At present in Denmark this uncertainty is only
associated with the heat demand, but in the future revenues of
produced energy and the fuel costs might also be uncertain and
dependent on time.  It is suggested to use a combination of background
knowledge of the operator and computer tools to solve the scheduling
problem.  More specificly it is suggested that the plant is equipped
with (i) an automatic on-line system for forecasting the heat demand,
(ii) an interactive decision support tool by which optimal schedules
can be found given the forecasts or user-defined modifications of the
forecasts, and (iii) an automatic on-line system for monitoring when
conditions have changed so that rescheduling is appropriate.  In this
report the focus is on methods applicable for items (ii) and (iii).
For item (i).

The approach taken in this report is explicitly to describe how the
total revenue from running the plant depends on the schedule for the
heat and power producing units of the plant.  Hereafter optimization
theory, in this case dynamic programming, is applied to find the
optimal schedule.  To take the uncertainties into account it might be
considered to use stochastic dynamic programming.  However, it is
argued that this is unpractical because the forecasting system will
need to be integrated into the optimization system, whereby a modular
design of the software cannot be obtained.  Furthermore, we believe
that all relevant forecasting methods are far too complicated to allow
for this integration; both uncertainties originating from the
dependence of heat load on climate and from meteorological forecasts
need to be taken into account.  Instead we suggest that the decision
support system allows the operator to investigate the sensitivity of
the optimal schedule to variations in the input.  Furthermore, we
suggest that the system is equipped with the possibility to simulate
realistic realizations of the heat demand based on the actual forecast
and previous forecast errors.  By letting the system find optimal
schedules for each of these realizations the operator can gain some
insight into the importance of the uncertainties.

It is shown that with modern personal computers (e.g. 1 GHz Pentium
 {III}), operating systems (e.g. RedHat Linux 6.0), and compilers (e.g.
{GNU} C 2.91) the calculations can be performed quickly enough to allow
use to be applicable in practice.  One optimal schedule covering one
week can easily be found within 5 to 10 seconds.  When considering
many possible realizations of the future heat demand some techniques
are needed to reduce the amount of {CPU} time required.  The results
indicate that it is possible to find optimal schedules for 100
realizations of heat demand using less than 3 minutes of {CPU} time.
Furthermore, the methods allow for massive use of parallel processing."
}