Mathematical models

In the short term, accurate optimization models are used to define scheduling of the power plants, and network coordination, over tipically one day to one week horizon with one hour or tens of minutes discretization. If the electricity system is a market based one, then generation is separated from transmission (e.g. the so called unbundling) and different problems are solved by different actors, basically generation companies (GenCo), Transmission System Operator (TSO) and Market Operator (MO) but there are also Distribution System Operator (DSO), traders and other possible actors. Alternatively if one speaks about monopolist systems, then many problems are solved in a centralized manner by a single centralized entity since the paradigm is quite different.

In the following table a primer of the most important class of problems is presented. As noted before we distinguish the monopolist systems from those market based.

Monopolist Systems

Single Bus Economic Dispatch (SBED)
 Security of Transmission
Single Bus Unit Commitment (SBUC)
Security Constrained UC (SCUC)

Market based Systems

GenCo: Max Profit ED/UC
MO: Energy Markets clearing
TSO: Balancing Markets clearing
 

At the heart of short term optimization problems there is the (short-term) Unit Commitment (UC). This problem requires to optimally operate a set of hydro and thermal generating units, over a given time horizon in order to satisfy a forecast energy demand at minimum total cost. The generating units are subject to various technical restrictions, depending on their type and characteristics. The UC is typically a large-scale, non-convex complex optimization problem.

Here we list several optimization problems from different entities perspective:

In both monopolist and market based models of course production power plants dynamics have to be modeled in a correct way. In the short term, GenCos must consider these constraints in the most detailed way, here we sketch some of the most important:

    1. quadratic cost curves possibly including some important (interdicted) valve point.
    2. min and maximum stable production.
    3. ramp rates and start up rates, possibly depending on the working points for bigger coal plants.
    4. complex operating dynamics for Combined Cycle Gas Turbine (CCGT) that have several Gas Turbine (GT) coupled with Steam Turbine (ST).
    1. Water-to-Power non linear relationships, for thin basin the bilinear dependency of the basin level, together with the discharge, can be included. This severely complicates the models.
    2. Complex cascade dynamics, including delays in the water flows from one basin to another. These delays can be also of different hours for big cascade and as a results their consideration strongly couples the decision variables along the time dimension.
    3. Additionally a forecast of possible natural inflows must be considered, due to rain or snow melt in some situations.