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Article Excerpt
CURRENT FEATURES OF HVAC OPTIMIZATION PROBLEMS
Owing to the complex nature of a centralized heating, ventilating, and air-conditioning (HVAC) system, it is becoming more popular to make use of the available simulation programs to handle the associated design and energy management problems based on practical engineering experiences. It is useful to study the performance of HVAC systems under the changing boundary conditions, mainly the climatic conditions and indoor heat gains. The developed simulation model is thus conveniently used as a "what-if" evaluator for the purpose of in-depth study. This would be convenient if only one single problem variable were involved, and typically parametric study or regression analysis would be applied in this case. However, if the problem is related to two or more variables under constrained situations, it would be a challenge to devise a suitable scheme for optimization. In order to save on computational demand and user effort, it is common to rely on personal judgment or intuition to facilitate the search progress. However, reliability of the "optimal" problem variables may be in doubt since the solution may be true only locally, but not globally, if the search landscape is rugged and multimodal.
For detailed study of HVAC systems, a simulation-optimization approach has increasingly been applied. A variety of optimization problems have been formulated, such as equipment sizing by Wright (1986), control strategies by Kintner-Meyer (1994), thermal comfort by Huh (1995), plant scheduling by Taylor (1996), routing and distribution by Fong et al. (2001), supervisory control by Hanby et al. (2002), fault diagnosis by Wang and Wang (2002), and energy management by Fong et al. (2006).
In the context of HVAC optimization problems that are developed using a detailed simulation model, the complex interrelationship and possibly discrete nature of the problem variables means that these problems cannot be solved by the traditional analytical or gradient-based methods. Although there are emerging heuristic optimization methods, like simulated annealing, tabu search, particle swarm optimization, and ant colony optimization, they rely heavily on problem-specific parameters that may not be transferable to another problem. In general, the evolutionary algorithm (EA) is advantageous to both the traditional and the heuristic optimization methods because it is free of derivative information and problem-specific parameters and it is not easily trapped by local optima due to its population-based searching strategy.
In this regard, there have been growing applications of the population-search EA in handling different optimization problems. For instance, Simpson et al. (1994) applied the genetic algorithm (GA), one of the paradigms of EA, to the optimization of pipe networks. Wright (1996) used GA for HVAC optimization studies in the sizing of HVAC equipment, and Huang and Lam (1997) used GA for optimizing controller performance in HVAC systems. Sakamoto et al. (1999) examined the application of GA to optimize the operation schedule for a district heating and cooling plant, and Asiedu et al. (2000) focused on the application of GA for duct system design. Chow et al. (2002) used GA to develop an optimal control scheme for an absorption chiller, and Wright et al. (2002) applied a multi-objective GA to identify the optimal pay-off characteristic between building energy cost and the thermal discomfort of occupants. Angelov et al. (2003) applied an EA to propose novel secondary HVAC systems. Lu et al. (2004, 2005) adopted GA to optimize the plant operation of a centralized HVAC system model.
Although a number of the aforementioned research works were based on GA, the working efficiency was inevitably compromised since the population size was commonly in tens or hundreds, leading to a very large number of simulation function calls and lengthy computational time. For the simulation-optimization approach, the efficiency of the optimization method for the HVAC simulation models is a primary concern since the bottleneck for the process is commonly the simulation run for generating the required evaluation function value from the problem variables. Excepting analytical optimization approaches, the working efficiency of any numerical optimization method is directly associated to the number of evaluation function calls. As a result, a robust optimization method should be able to generate the global or near-optimal solution of HVAC problems with minimum calls to the simulation models since most of the HVAC optimization problems reported in the literature were based on the paradigm of GA. The unique features of GA are its binary representation of problem variables and the emphasis on recombination for continual evolution. Although a number of optimization problems have been effectively handled by GA, the performance and efficiency of another paradigm--evolution strategy--has been seldom reported. Therefore, this paper focuses on a newly developed EA based on the paradigm of evolution strategy.
PARADIGMS OF THE EVOLUTIONARY ALGORITHM
EA is a probabilistic and population-based heuristic algorithm developed from the Darwinian paradigm of evolution, which is often viewed as analogous to optimal exploration and optimization. The essential steps are derived from the fundamental principles of variation and selection of the Darwinian evolution throughout generations. The two major paradigms of EA, genetic algorithm and evolution strategy, are discussed in the following sections.
Genetic Algorithm
The GA was developed by J.H. Holland in the 1960s (Holland 1962, 1967). H.J. Bremermann offered conceptually equivalent procedures also in the 1960s (Bremermann 1962; Bremermann et al. 1965), as did A.S. Fraser in 1950s (Fraser 1957). GA closely follows the paradigm of Darwinian biological evolution, so it has an emphasis on crossover (or recombination) and a probabilistic selection operator. Mutation plays a minor role and is treated as a background operator. In order to let the problem variables simulate a chromosome of the required bits, binary strings are commonly used in GA for representation of problem variables under optimization. A building block principle is called schema theorem, which is used to describe the expected number of instances of a schema that are found in the next epoch of GA when the proportional selection is adopted. Holland (1975) introduced the simple genetic algorithm (SGA) with the typical procedures shown in Figure 1. SGA is the basic form of the GA; different GA variants have been developed from SGA.
[FIGURE 1 OMITTED]
Evolution Strategy
Evolution strategy was developed by I. Rechenberg and H. Schwefel in the 1960s (Rechenberg 1965; Schwefel 1965) and is commonly used in problems with real-valued or discrete variables. The genetic operators of evolution strategy include recombination, mutation, and selection. Evolution strategy emphasizes the equal importance of mutation and recombination. A recombination operator reduces the occurrence of scattered individuals around the search landscape. For the mutation operator, similar to evolutionary programming, the Gaussian realization with self-adapted strategy parameter is used. The framework of evolution strategy is presented in Figure 2. Generally, evolution strategy has a parent population size [mu] and an offspring population size [lambda]. The implementation of evolution strategy can be broadly categorized into the commas strategy ([mu], [lambda]) and the plus strategy ([mu] + [lambda]), and this classification is based on the selection approach to be adopted. For the ([mu], [lambda]) strategy, it is common that [lambda] = k[mu], where [lambda] > [mu] and k [member of] [I.sup.+]. The [lambda] offspring becomes the selection pool, and the offspring individuals are all ranked according to their fitness. Then the best [mu] individuals are deterministically selected to be the parent population for the next epoch. For the ([mu] + [lambda]) strategy, the selection pool is the union of [mu] parents and [lambda] offspring, all the individuals are ranked according to their fitness, and the best [mu] individuals are selected to be the next parent population deterministically.
[FIGURE 2 OMITTED]
Comparison of GA and Evolution Strategies
The similarities and differences between the operators and characteristics of these two paradigms of EA are summarized in Table 1. There are several significant features of evolution strategy and evolutionary programming compared to GA:
Table 1. Comparison of Major Operators of Genetic Algorithm and Evolution Strategy Genetic Evolution Algorithm Strategy Recombination * Core * Important (Crossover) operator operator * Recombination * Recombination probability > 0.5 probability = 1 Mutation * Background * Core operator operator * Low mutation * Commonly probability, using usually stochastic 1/L (where L strategy is chromosome parameter with...
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