Day-to-night heat storage in greenhouses: 1 Optimisation for periodic weather

Abstract

Day-to-night heat storage using water tanks (buffers) is common practice in cold-climate greenhouses, where gas is burned during the day for carbon dioxide enrichment. In this study an optimal control approach is outlined for such a system, based on the idea that the virtual value (shadow price) of the stored heat, its ‘co-state’, could be used to guide the instantaneous control decisions. If this value is high, the system has an incentive to fill the heat storage (buffer), and vice versa if the co-state is low. The optimal co-state trajectory maximises the net income (performance criterion). To illustrate the method, a system description and a parameter-set roughly representative of tomato greenhouses in The Netherlands is used. The results, for daily-periodic weather, show: (1) The optimal co-state is constant (same value night and day), in contrast to the varying set-points and control fluxes. (2) The optimal solution is associated with minimum time on the storage bounds (minimum time of full or empty buffer). (3) The optimal virtual value (co-state) of stored heat is about the same as the actual cost of boiler heat during winter and about zero in summer. (4) The gain from installing a buffer is highest during spring and minimal in winter. (5) The intensive utilisation of the heat buffer in summer and its low utilisation in winter indicate that the justification of the heat storage practice, under the assumed conditions, is more the need for CO2 enrichment in summer than the need for heating in winter.