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(703f) Mathematical Programming Models and Solution Methods for Online Scheduling of Central Heating/Cooling Plants

Risbeck, M., University of Wisconsin--Madison
Maravelias, C. T., University of Wisconsin-Madison
In large commercial buildings or campuses, the heating, ventilation, and air conditioning (HVAC) system is often served by a set of equipment operating in a "central plant." Use of this equipment must be scheduled so as to supply the time-varying heating and cooling demands of the overall system. While finding a feasible schedule is typically not challenging, it is desirable to operate the central plant at the lowest possible cost, which depends on a number of factors. First, utilities (primarily electricity) are often subject to time-varying prices or other rate structures that cause nominal operation (i.e., meeting system demand exactly on time) to be expensive. Second, equipment efficiency can vary highly with load, and so units should be operated at levels where they are most efficient. Third, to provide operational flexibility, thermal energy storage (TES) is often installed to temporally decouple chilled or hot water production and consumption. Fourth, equipment startup and shutdown may be accompanied by unmodeled transient dynamics, which requires that units not be switched rapidly. To address all of these considerations while minimizing operational costs, schedules can be determined by optimization using typical production scheduling methods. This approach avoids the use of heuristics and can lead to significant cost savings. However, a single schedule is not sufficient due to disturbances, and thus a practical implementation must address not offline scheduling but online scheduling.

Determining an optimal schedule requires forecasts for both utility prices (used in the objective function) and system thermal load (used in operational constraints). As the system evolves, these forecasts are updated to reflect measured system behavior, which can invalidate prior schedules. In addition, units may break down or be removed from service for maintenance. Thus, the schedule must be constantly updated to reflect the best available information, which imposes constraints on how much time can be spent determining an individual schedule. To meet these requirements, the scheduling formulation must be balanced so as to account for primary cost contributors (e.g., variable electricity price and equipment efficiency) while not considering too much detail so as to be computationally intractable. Although general production scheduling formulations can describe HVAC central plants, a specifically tailored formulation can provide more flexibility while yielding faster solution times.

In this talk, we present a mixed integer linear programming (MILP) formulation for online scheduling of HVAC central plants. Similar to a resource-task network (Pantelides, 1994) formulation, we use an abstract representation of the central plant in terms of "resources," which are mass or energy flows within the system, and "generators," which are pieces of equipment in the central plant. After adjusting equipment utilization and demand satisfaction constraints to more directly model the central plant, we embed piecewise-linear approximations of generator production/consumption models to accurately account for variable generator efficiency. Then, we employ symmetry-removal reformulations that greatly improve solution times for systems with multiple identical generators. The end result is a compact formulation that can accommodate a wide variety of central plant configurations and that can be optimized online for practically sized central plants.

Using this scheduling formulation, we also present a case study of closed-loop operation for a medium-size central plant consisting of chillers, cooling towers, pumps, and a storage tank. In this study, we discuss the effects of different prediction horizons and also inaccurate demand forecasts. We show that, even for moderate prediction horizons, the scheduling model can be solved to within 0.5% of optimality in 30 s. We also show that operating costs are not significantly impacted by uncertain forecasts due to closed-loop feedback. Finally, we discuss a cascaded approach that separates the long-term storage utilization decisions from the short-term unit commitments by using a simplified surrogate model with a long horizon at the upper level and a fully-detailed model with a short horizon at the lower level. Thus, rapid rescheduling can correct for inaccurate long-term forecasts, and existing central plants can be operated at significantly lower costs.


Pantelides, C. C. (1994). Unified frameworks for optimal process planning and scheduling. In Proceedings on the second conference on foundations of computer aided operations (pp. 253--274).