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Description
This paper is the first of a two-part sequence that proposes a general methodology for dynamic scheduling and optimal control of complex primary HVAC & R plants, which combines engineering analyses within a practical decision analysis framework by modeling risk attitudes of the operator. The methodology involves a computationally efficient, deterministic engineering optimization phase for scheduling and controlling primary systems over the planning horizon, followed by a systematic and comprehensive stochastic sensitivity and decision analysis phase, where various sources of uncertainties are evaluated along with alternative non-optimal but risk-averse operating strategies. This paper describes the deterministic component of the analysis methodology, which essentially involves the development of response surface models for different combinations of system configurations to be used for static optimization and then using them in conjunction with the modified Dijkstra's algorithm for dynamic scheduling and optimal control under different operating conditions and pricing signals. The proposed methodology is illustrated for a semi-real hybrid cooling plant operated under two different pricing schemes: real-time pricing and time-of-use with electricity demand. We feel that the general methodology framework proposed sacrifices very little in accuracy while being much more efficient computationally than the more complicated optimization methods proposed in the general literature. Moreover, this approach is suitable for online implementation, and it is also general enough to be relevant to other energy systems.
INTRODUCTION
A literature review on supervisory optimal control applied to the operation of complex HVAC & R systems including cooling plants and BCHP plants has been done by Jiang (2005). There are several studies that have adopted mixed integer linear programming (MILP) techniques to this general problem (for example, Dotzauer [1997, 2003] and Yokoyama et al. [2002]). However, the structural design problem has often been treated by considering only a single-period operation (Papoulias and Grossmann 1983) or a multi-period one with a small number of periods (Horii et. al. 1987). Some approaches based on meta-heuristics, such as simulated annealing (SA) and genetic algorithms (GA), have also been proposed (for example, Sakamoto et al. [1999], Curti et al. [2000], and Yin and Wong [2001]). However, these approaches are said to have limitations in the determination of values of search parameters, the judgment of optimality, and the requirement of extensive computation times (Yokoyama et al. 2002).
Several levels of optimal control schemes have been proposed for existing cooling plant operation. These can be grouped broadly as follows:
1. Cookbook solutions, which are simple rules and guidelines for operators to follow (Hydeman 2002).
2. Heuristic control schemes, widely used in the current building control profession, that are developed based on local optimization, system model simplification, estimation, and experience. The 2003 ASHRAE Handbook--HVAC Applications (ASHRAE 2003) describes in detail such control heuristics for operating HVAC systems and components. Further, control heuristics could be used as a starting point in an optimization scheme. In addition, a heuristic type of suboptimal control is often desirable for online implementation purposes.
3. Rigorous optimization algorithms that follow the strict definition of optimal control by proposing optimization algorithms to minimize the objective function (which is often the cost). The approach proposed in this paper falls in this category.
Because of the variety of energy sources used in complex HVAC & R systems, the interdependency between sources, and the variation of technical and economic conditions with time, e.g., change of load, deterioration of equipment, change of fuel and electricity prices, etc., the planning of plant day-to-day operation and evaluation of alternative performance options is not simple. Much of the difficulty is mainly due to the following reasons:
* The objective functions and models are usually nonlinear functions that may contain both discrete (for example, equipment on/off status) and continuous variables--locating the global optimum is not guaranteed. Further, one may have to deal with multiple objectives, which make the problem even more complicated.
* The possible number of independent or decision variables for the problem is large, with a large set of diverse constraints, therefore presenting the engineer with the difficult, if not impossible, task of determining the best operating strategy. Further, if the problem is a multi-period dynamic problem (i.e., involving several stages), with the number of binary scheduling variables increasing with the number of periods, the conventional solution algorithm, which combines the branch and bound method with the simplex method, may require computation times that are not practical (Yokoyama et al. 2002).
Despite recent advances in computer power and the development of better optimization algorithms, only a few are used in industry. What is more remarkable is that most complex HVAC & R plants are still scheduled by humans in a heuristic manner without the aid of computer supporting tools. One possible reason for this often voiced by professionals is the lack of consideration of how to combine pure engineering solutions with individual risk attitudes of how system operators weigh risk over predicted outcome. It is, in essence, this aspect that is addressed by this research.
OBJECTIVE AND SCOPE
The primary objective of this paper is to propose a general and computationally efficient methodology for minimizing the operating costs, including both energy costs and demand costs, of complex HVAC & R plants over the planning horizon, which is taken as 12 hours. The operating cost would include electricity usage cost, gas usage cost, and equipment start-up cost. A true optimization would require the simultaneous optimization of all cost components under the pre-specified thermal load and well-defined performance characteristics and maintenance costs of equipment. An even finer level of analysis would be to consider the reliability associated with different equipment, since the large equipment could be of different vintage and level of degradation.
Utility costs differ with different pricing signals, resulting in different formulations of the optimization cost function, which, in turn, may require different optimization techniques. Only two cases are considered: (a) real-time pricing, which has no demand charge involving energy (electricity and gas) over a certain time period along with start-up cost, and (b) TOU (time of use) with demand, which is more complex since the cost function includes gas and electricity cost (energy cost + demand cost) and start-up cost. The optimization function should explicitly consider start-up cost caused both by additional energy consumption and increased demand. In order to minimize the demand charge, equipment must be operated so that situations that cause large spikes in power consumption (due to having to accelerate components such as fans and motors up to their design speeds) do not occur during periods of peak power.
EQUIPMENT MODELS
Selecting a performance model is an important and essential first step in optimizing the operation of any engineering system. The main components in a cooling plant include chillers, cooling towers, fans, and pumps.
Gordon-Ng (GN) Chiller Model
The semi-empirical GN chiller model (Gordon and Ng 2000) predicts the dependence of chiller COP (defined as the ratio of chiller thermal cooling capacity divided by the electrical power consumed by the compressor) with certain key (and easily measurable) parameters such as the fluid (water or refrigerant) return temperature from the condenser, fluid temperature leaving the evaporator (or the chilled-water supply temperature to the building), and the thermal cooling capacity of the evaporator. A detailed evaluation consisting of over 50 chillers of all types (one-stage, two-stage centrifugal with inlet-guide and VSD, screw, scroll, and reciprocating) and sizes has been conducted by Jiang and Reddy (2003). It was found that the fundamental GN formulation for all types of vapor compression chillers is excellent in terms of its predictive ability.
The following equation is the GN fundamental model for vapor compression chillers:
([1/COP] + 1)[[T.sub.cho]/[T.sub.cdi]] - 1 = [a.sub.1][[T.sub.cho]/[Q.sub.ch]] + [a.sub.2][([T.sub.cdi] - [T.sub.cho])/[[T.sub.cdi][Q.sub.ch]]] + [a.sub.3][[(1/COP + 1)[Q.sub.ch]]/[T.sub.cdi]] (1)
where [a.sub.1], [a.sub.2], and [a.sub.3] are regression coefficients, [T.sub.cho] is chilled-water outlet temperature (K), [T.sub.cdi] is condenser water inlet temperature (K), and [Q.sub.ch] is chiller load (kW).
GN models for single-stage absorption chillers are also valid and have been shown by Jiang and Reddy (2003) to be accurate (with coefficient of variation [CV] about 6%-8%) for steam and hot water two-stage absorption systems.
([[[T.sub.gni] - [T.sub.cdi]]/[[T.sub.gni]COP]] - [[[T.sub.gni] - [T.sub.cho]]/[T.sub.cho]])[Q.sub.ch] = [b.sub.0] + [b.sub.1][[T.sub.cdi]/[T.sub.gni]] (2)
where [b.sub.0] and [b.sub.1] are regression coefficients and [T.sub.gni] is generator inlet temperature (K).
Effectiveness Cooling Tower Model
The effectiveness-NTU model concept, originally proposed for sensible heat exchangers, was modified by Braun (1988) and Braun et al. (1989) to model performance of cooling towers by utilizing the assumption of a linearized air saturation enthalpy. The following general correlation for NTU in terms of the flow rates is used with estimates of the coefficients c and n identified from measurements at different air flow rates [dot.m.sub.a], (with water flow rate [dot.m.sub.w] being, in most cases, constant):
NTU = c([dot.m.sub.w]/[dot.m.sub.a])[.sup.1 + n] (3)
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