Publication: HVAC & R Research
Publication Date: 01-JAN-07
Delivery: Immediate Online Access
Author: Jiang, Wei ; Reddy, T. Agami ; Gurian, Patrick
Article Excerpt
A companion paper (Jiang and Reddy 2007) presents a general and computationally efficient methodology for dynamic scheduling and optimal control of complex primary HVAC & R plants using a deterministic engineering optimization approach. The objective of this paper is to complement the previous work by proposing a methodology by which the robustness of the optimal deterministic strategy to various sources of uncertainties can be evaluated against non-optimal but risk averse alternatives within a formal decision analysis framework. This specifically involves performing a sensitivity analysis on the effect of various stochastic factors that impact primary HVAC & R plant optimization, such as the uncertainty in load prediction and the uncertainties associated with various component models of the equipment. This is achieved through Monte Carlo simulations on the deterministic outcome, which allow additional attributes, such as its variability and the probability of insufficient cooling, to be determined along with the minimum operating cost. The entire analysis is then repeated for a specific non-optimal but risk-averse operating strategy. Finally, a formal decision analysis model using linear multi-attribute utility functions is suggested for comparing both these strategies in a framework that explicitly models the risk perception of the plant operator in terms of the three attributes. The methodology is demonstrated using the same illustrative case study as the companion paper.
INTRODUCTION
Incomplete knowledge, as well as noisy or erroneous data, can all be viewed as sources of uncertainty. The existence of uncertainty transforms conventional deterministic process models into stochastic ones, the solution of which indisputably remains challenging yet is of great practical importance. Beginning with the works of Dantzig (1955), Tintner (1955), and Charnes and Cooper (1959), optimization under uncertainty experienced rapid development in both theory and algorithms. There are several approaches for dealing with optimization under uncertainty--sensitivity analysis, stochastic programming, and robust optimization (Mulvey et al. 1995), to name a few. While sensitivity analysis is a reactive approach to controlling uncertainty in that it is used to assess the impact of uncertainty on the optimization result without providing a controlling mechanism, the other two are constructive in that they can yield solutions that are less sensitive to the model data than classical mathematical programming.
The issue of uncertainty, as it relates to the optimization of HVAC & R plants, has been addressed in only a few papers. The effects of many factors affecting the cooling load in a building, as well as several parameters of the dynamic chiller sequencing (DCS) procedure described in the companion paper (Jiang and Reddy 2007), have been systematically explored by Olson (1987), who found that the DCS is fairly insensitive to errors in cooling load prediction. Henze and Krarti (1999) investigated the effect of forecasting uncertainty on the cost savings performance of a predictive optimal controller for thermal energy storage. They concluded that the predictive optimal controller is fairly robust and does not require high levels of accuracy in predicting the cooling loads and the real-time pricing (RTP) electricity rates. Henze et al. (2003) investigated whether thermal storage systems can be controlled effectively in situations where cooling loads, non-cooling loads, non-cooling electrical loads, weather information, and the cost of electricity are uncertain and have to be predicted. The analysis concluded that predictive optimal control strategies were greatly superior in reducing utility cost compared to conventional partial-storage thermal energy storage strategies, even with inaccurate forecasts.
To date, most publications addressing optimization and control of HVAC & R plants were based on sensitivity analysis. Further, most of the HVAC & R literature limits itself to load prediction and electrical rate uncertainties (which can be referred to as external prediction uncertainties), while the other two categories of uncertainties, model-inherent uncertainty and process-inherent uncertainty, are largely ignored. It could well be that most optimization problems of HVAC & R plants do not warrant formulating them as stochastic problems due to the low degree of associated uncertainty (for example, current load forecast algorithms are relatively accurate, and so are the modeling techniques and control devices). However, studies specifically addressing the aspect of stochastic optimization would be helpful in supporting (or disproving) such a contention.
Furthermore, it may not always be clear beforehand whether the optimal strategy is robust or not and what potential advantage the best stochastic strategy could be expected to provide over the best deterministic strategy (Seferlis and Hrymak 1996). So, it is important to first determine the best deterministic operating strategy and then investigate the additional benefit of using stochastic optimization, which for many systems may be small and may not justify the additional modeling, optimization, and hardware costs.
OBJECTIVES
Despite recent advances in computer power and the development of better optimization algorithms, only 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. The companion paper (Jiang and Reddy 2007) presents a computationally efficient deterministic engineering optimization for scheduling and controlling complex primary HVAC & R plants over a specified planning horizon (taken as 12 hours). It involves developing response surface models for different combinations of system configurations and then using them for static optimization at any given hour. This capability is then used in conjunction with the modified Dijkstra's algorithm (Olson 1987) for dynamic scheduling and optimal control under different operating conditions and pricing signals.
Granted, the operating cost of a plant is a critical criterion or attribute for choosing an appropriate operating strategy; however, it is not the only factor. Different operating strategies may result in different uncertainty profiles with different robustness levels. Minimizing the operating cost and minimizing the risk associated with this optimal strategy are two separate issues altogether; how to trade off between these objectives is basically a problem of decision analysis. Furthermore, different people have different risk attitudes and thus are willing to accept different levels of risk. How to combine different risk preferences into engineering results and, more basically, how to define risk in such a setting are issues investigated in this paper. The objectives of this paper are to
* systematically assess the robustness of the identified optimal deterministic operating strategy when subject to various sources of uncertainties; in other words, to ascertain the extent to which the optimal solution is stable and evaluate the relative effect of different sources of uncertainty on this optimal solution (this is called sensitivity analysis);
* compare the uncertainty of the optimal strategy found from our rigorous optimization with those of more conservative non-optimal heuristic strategies;
* model the trade-off between the expected value of the savings, its variability, and the increased probability of being unable to provide sufficient cooling to meet the building load (called loss of load capability) and combine these results into a decision analysis framework involving the individual's attitude toward risk and returns; and
* apply this model to both the optimal and heuristic strategies in order to identify the one with the higher operator utility value.
Neglecting the last three aspects is probably one of the reasons why many of the purely analytical optimization solutions are not widely embraced by plant operators. Finally, this paper illustrates optimization methodology and decision analysis techniques using the same hybrid chiller plant selected in the companion paper for two electric price signals, real-time pricing (RTP), and time-of-use rates (TOU) (Jiang and Reddy 2007).
DECISION ANALYSES
Decision analysis provides a formal conceptual framework involving an "overall paradigm and a set of tools by which a decision maker can construct and analyze a model of a decision situation" (Clemen and Reilly 2001). Such methods have been applied in various fields to choose among alternatives in an optimal fashion, including complex engineering systems, economics, physics, social sciences, medical decision making, and others (Hillier and Lieberman 2001). First of all, a decision analysis approach is warranted only when one is faced with a situation involving conflicting objectives, uncertainty in the payoffs, and more than one appropriate alternative. Only once this is determined does one decide on statistical methods for simulating the uncertainty and modeling the results so as to reflect the individual's risk perception or attitudes (low or high risk).
According to Pistikopoulos (1995), a suitable classification of different types of uncertainties is based on the nature of the source of uncertainty: (1) model-inherent uncertainty, which includes uncertainties of the various component models arising from inaccurate or incomplete data and/or lack of perfect regression fit in the response model;...
NOTE: All illustrations and photos have been removed from this article.
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