Optimal and suboptimal control policies for a solar collector system

An optimal control problem to maximize the net energy gathered by a flat-plate solar collector system by controlling the collector fluid flow rate is investigated. The problem is formulated in terms of a distributed parameter system and solved using the method of characteristics. It is shown that if the pump of the collector loop is such that its pumping power is greater than a linear function of the fluid velocity, then the optimal control policy is one in which the fluid flow is instantly switched between zero and maximum rates. Necessary conditions that determine the optimal switching times are derived. Because the resultant switching function of the optimal policy is shown to be decomposable into two parts, one that depends on the state of the system and another that requires a priori knowledge of the solar intensity over the entire period of operation, a suboptimal control policy that can be implemented by an on/off feedback controller with hysteresis is proposed. When this suboptimal policy is compared with the optimal policy, it is shown that on a clear day with sufficient solar insolation to dictate a two-switch optimal policy, the two policies are identical. Under other weather conditions, the feedback suboptimal controller will keep the pump off for a slightly shorter period of time than the time dictated by the optimal control.