Duration-differentiated Energy Services with a Continuum of Loads
  As the proportion of total power supplied by renewable sources increases, it gets more costly to use reserve generation to compensate for the variability of renewables like solar and wind. Hence attention has been drawn to exploiting flexibility in demand as a substitute for reserve generation. Flexibility has different attributes. In this paper we consider loads requiring a constant power for a specified duration (within say one day), whose flexibility resides in the fact that power may be delivered at any time so long as the total duration of service equals the load's specified duration. We give conditions under which a variable power supply is adequate to meet these flexible loads, and describe how to allocate the power to the loads. We also characterize the additional power needed when the supply is inadequate. We study the problem of allocating the available power to loads to maximize welfare, and show that the welfare optimum can be sustained as a competitive equilibrium in a forward market in which electricity is sold as service contracts differentiated by the duration of service and power level. We compare this forward market with a spot market in their ability to capture the flexiblity inherent in duration-differentiated loads. 
Ashutosh Nayyar, Matias Negrete-Pincetic, Kameshwar Poolla and Pravin   Varaiya
cs.SY