Optimal Power Dispatch via Multistage Stochastic Programming

The short term cost optimal dispatch of electric power in a generation sys tem under uncertain electricity demand is considered The system comprises thermal and pumped storage hydro units An operation model is developed which represents a multistage mixed integer stochastic program and a conceptual solution method using La grangian relaxation is sketched For xed start up and shut down decisions an e cient algorithm for solving the multistage stochastic program is described and numerical results are reported Introduction Mathematical models for cost optimal power scheduling in hydro thermal systems often combine several di culties such as a large number of mixed integer variables nonlinearities and uncertainty of problem data Typical examples for the latter are uncertain prices in electricity trading the future electric power demand and future in ows into reservoirs of hydro plants Incorporating the uncertainties directly into an optimization model leads to stochastic programming problems In the context of power scheduling such models are developed e g in In the present paper we consider a short term optimizationmodel for the dispatch of electric power in a hydro thermal generation system over a certain time horizon in the presence of uncertain demand The generation system comprises coal red and gas burning thermal and pumped storage hydro units without in ows which is typical for the eastern part of Germany Short and long term energy contracts are regarded and modelled as particular thermal units The operation of such a generation system is very complex because it creates a link between a decision in a given time interval and the future consequences of this decision Even for optimal on line power scheduling future costs created by actual decisions have to be taken into account Since a longer time horizon e g one week is often needed due to the pumping cycle of the hydro storage plants the stochastic nature of the demand cannot be ignored The optimization model thus represents a multistage stochastic program containing mixed integer stochastic decisions which re ect the on o schedules and production levels of the generating units for all time intervals of Humboldt University Berlin Institute of Mathematics Berlin Germany This research is supported by the Schwerpunktprogramm Echtzeit Optimierung gro er Sys teme of the Deutsche Forschungsgemeinschaft DFG the horizon The increase of stages in the stochastic programming model corresponds to a decrease of information on the power demand The stochastic model will be developed and discussed in some detail in section for more information we refer to In section we sketch a conceptual decompo sition method by applying Lagrangian relaxation to the loosely coupled multistage stochastic program and in section an e cient algorithm for solving the stochastic program for xed on o decisions is described and numerical results are reported Stochastic Model The mathematical model represents a mixed integer multistage stochastic program with linear constraints Let T denote the number of hourly or shorter time intervals in the optimization horizon and fd t Tg the stochastic demand process on some probability space A P re ecting the stochasticity of the electric power demand It is assumed that the information on the demand is complete for t and that it decreases with increasing t This is modelled by a ltration of elds A f g A At AT A where At is the eld generated by the random vector d d Let I and J denote the number of thermal and pumped storage hydro units in the system respectively According to the stochasticity of the demand process the decisions for all thermal and hydro units f uti p t i t Tg i I f stj w t j t Tg j J are also stochastic processes being adapted to the ltration of elds The latter condition means that the decisions at time t only depend on the demand vector d d nonanticipativity Here uti f g and p t i denote the on o decision and the production level for the thermal unit i and time interval t respectively and stj w t j are the generation and pumping levels for the pumped storage plant j during time interval t respectively Further let ltj denote the water level in terms of electrical energy in the upper reservoir of plant j at the end of interval t The objective function is given by the expected value of the total fuel and start up costs of the thermal units