A Robust evolutionary algorithm for HVAC engineering optimization.

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
               length)            Gaussian
                                  realization

                                  * Mutation
                                  probability = 1

                                  * Usually
                                  mutation
                                  follows
                                  recombination

Selection      * Probabilistic    * Deterministic
               selection of       selection of
               parents            offspring
               for                to be parents
               recombination      of next epoch
               and mutation       (selection
               (selection for     for survival)
               reproduction)

a. For GA, recombination (or crossover) is the core operator and mutation is much less significant. However, mutation is the core operator for evolution strategy.

b. In GA, the recombination and mutation probabilities are commonly less than one. This means that some individuals within the population remain intact without evolution. But for evolution strategy, both probabilities are one. This implies that these two operators would definitely be involved for variation of individuals.

c. Apart from the nature of selection, the approach and sequence of selection in their evolution loops are different. For GA, the parents are selected for the ongoing recombination and mutation processes through the selection operator; selection is therefore used for those variation operators and implemented before the variation process. For evolution strategy, selection is applied on the offspring population to generate the parent population of the next epoch, so selection is implemented after the variation operators. In summary, the approach of "selection for reproduction" is adopted in GA, while "selection for survival" is adopted in evolution strategy.

As a result, although there are similarities between these two paradigms, there are distinguishing features for each paradigm, so their optimization performances would be different. In this paper, emphasis is placed on the EA effectiveness in terms of the number of function calls in the evaluation phase, as this is the rate-determining step in solving many HVAC optimization problems developed by simulation models.

ROBUST EVOLUTIONARY ALGORITHM FOR HVAC OPTIMIZATION PROBLEMS

A robust evolutionary algorithm (REA) has been developed by referring to the paradigm of evolution strategy, which emphasizes both mutation and recombination, as well as selection. In the REA, the key operator is mutation, aimed at continual exploration, while recombination is used for effective exploitation. For the constrained problems, a constraint-handling operator is involved instead of applying the penalty-based approach. The structure of the REA is shown in Figure 3, and the features of its operators are discussed in the following sections.

[FIGURE 3 OMITTED]

Operators of Robust Evolutionary Algorithm

Recombination Operator. Arithmetic recombination was advocated by Schwefel (1981). In this recombination operator, the recombined individual [x.sub.xo] is a ratio between the two parents, [x.sub.p1] and [x.sub.p2], randomly selected from the parent population, as per Equation 1.

[x.sub.xo] = [rx.sub.p1] + (1-r)[x.sub.p2] (1)

where r [member of] U(0,1). In arithmetic recombination, it is obvious that [x.sub.xo] would always be generated within the search region bounded by the two parents; therefore, its effectiveness in global search would be highly related to the fitness of the parents.

Mutation Operator. A new mutation operator called Cauchy deterministic mutation was developed in the REA. This mutation approach makes use of the realization effect of the Cauchy random number [C.sub.j](0,1) and carries out the variation of the individuals via Equation 2. For a real-numbered problem variable [x.sub.mu,j], mutation from the recombined variable [x.sub.xo,j] is as follows:

[x.sub.mu,j] = [x.sub.xo,j] + [[sigma].sub.j][C.sub.j](0,1) for j = 1, ...,[n.sub.var] (2)

where [[sigma].sub.j], the deterministic strategy parameter, is defined as

[[sigma].sub.j] = [[sigma].sub.o,j]exp(-[e-[a.sub.1]]/[[a.sub.2].[e.sub.max]]) (3)

and where

[[sigma].sub.o,j] = [[[UB.sub.j] - [LB.sub.j]]/[a.sub.3]]

e = epoch

[e.sub.max] = epoch of termination

[UB.sub.j] = upper bound of the jth problem variable

[LB.sub.j] = lower bound of the jth problem variable

[n.sub.var] = number of variables in the problem individual

[a.sub.1], [a.sub.2], [a.sub.3] = empirical coefficient of deterministic strategy parameter

[C.sub.j](0,1) = Cauchy random number for the jth variable of individual x

In this mutation operator, the deterministic strategy parameter [sigma] is the variance of the distribution and influences the step length of the mutation. The term strategy parameter is based on the idea of the self-adaptive mutation approach as adopted in the paradigm of evolution strategy and evolutionary programming (Back and Schwefel 1993). Since this strategy parameter is evolved in a deterministic manner as shown in Equation 3, so the strategy parameter is called deterministic in contrast to the stochastic one used in classical evolution strategy or evolutionary programming. Owing to the nature of the Cauchy probability density function, which has a longer tail as compared to the probabability density function of Gaussian, the mutation step length from the Cauchy realization would have greater perturbation in the search throughout the epoch.

Selection Operator. Tournament selection is used to select the new population from offspring in the REA. In tournament selection, pair-wise comparison is conducted for all individuals (i.e., [2n.sub.pop], where [n.sub.pop] is the population of parent or offspring) in the union of parent and offspring populations. For each individual, a preset number of q opponents is randomly chosen within the union for comparison purposes, where

q = min = {[round([n.sub.pop]/2)], 10}. (4)

In each comparison, if the fitness of the individual is better or equal to that of the opponent, it receives a "win." Finally, [n.sub.pop] individuals out of the union are selected by counting the most "wins" and hence become the new population.

Constraint-Handling Operator. The constraint-handling operator is only active for constrained optimization problems. This operator is specially designed not to use the penalty approach for the fitness value commonly found in EA or GA but to handle the fitness value and degree of constraint violation separately. This allows a more reliable optimization search without deforming the fitness landscape. This constraint-handling operator is a rank-based instead of a value-based approach. Since the value-based constraint-handling operator has a dominating effect to those individuals with very large values, this renders a biased selection for such individuals. As a result, the rank-based scheme is adopted for constraint handling.

Elitism. Elitism is a strategy to retain the best individual for the offspring population. In an EA with a high population size (e.g., in the order of hundreds in GA), elitism is not commonly used since a large population already includes a number of potentially optimal individuals. In a small population size, like in the current REA, it may be that the elite would not be selected under the selection operator. In this case, elitism is important to enhance the search success of the continual global search. The elite would not just have the lowest function value for the minimization problem, but the elite should have a minimum or even no total constraint violation for the constrained problem.

EXAMPLE APPLICATIONS IN HVAC PROBLEMS

Problem Developed by Simulation Model--Centralized Solar Water-Heating System

System Design. The design of a centralized solar water-heating system for a tall residential building in Hong Kong (Fong et al. 2007; Chow et al. 2006) is shown in Figure 4. The whole plant simulation model was developed using TRNSYS (SEL 2000), which adopted the Davidon-Fletcher-Powell algorithm (Powell 1964) for solving a set of nonlinear equations from the simulation components, especially the thermal solar collector and hot-water calorifier. The flat plate collector was selected and installed at one single orientation of the building. A total of 840 [m.sup.2] (9042 [ft.sup.2]) of the collectors was applied with a nominal capacity of hot-water calorifier of 36 [m.sup.3] (9510 gallons). The collector fin efficiency factor, the emittance, and the absorptance of the absorber plate were 0.78, 0.03, and 0.947, respectively. In this simulation, the global horizontal irradiation, direct normal irradiation, air dry-bulb temperature, and wind speed were all based on the weather data of the newly developed typical meteorological year for Hong Kong (Chan et al. 2006). The global horizontal and direct normal irradiation were input to the simulation component of the solar radiation processor, which adopted the Reindl model (Reindl et al. 1990) to resolve the beam and diffuse incident solar irradiation on the solar collectors at different surface azimuths and tilt angles. The hourly dry-bulb temperature and wind speed were linked to the component models of the thermal solar collector, pipe, and hot water calorifier so that the convective and radiative heat losses from such equipment could be evaluated due to changing weather conditions. Domestic hot water (DHW) was drawn directly from the hot-water calorifier, and an in-line auxiliary heater was operated whenever the DHW supply temperature was below 60[degrees]C (140[degrees]F) in order to comply with the local regulation (CPPLD 2000). The rated power of the circulation pump was determined by considering the flow rate and the total pressure drop of the entire closed-loop circuitry according to local design practice.

[FIGURE 4 OMITTED]

Problem Formulation. This optimization study was carried out to maximize the energy savings due to the application of the solar water-heating system as compared to the conventional domestic electric heating. The related design variables that contribute to a maximum year-round energy saving are as follows:

[x.sub.1] = tilt angle of solar collectors

[x.sub.2] = surface azimuth of solar collectors

[x.sub.3] = capacity of hot water calorifier

[x.sub.4] = flow rate of circulation pump

According to local design practice, [x.sub.1] and [x.sub.2] are 22[degrees] and due south, respectively. For [x.sub.3] and [x.sub.4], although the corresponding nominal design values are 36 [m.sup.3] (9510 gallons) and 47,000 kg/h (1727 lb/min), it was necessary to study carefully for any compromising effect on retaining useful thermal energy with respect to the daily DHW consumption profile. The objective function of this problem was to maximize the yearly 8760-hour energy savings in electricity consumption, [E.sub.sav], by using the solar heating, [E.sub.sh], against that by the domestic electric heating, [E.sub.eh], as follows:

maximize[E.sub.sav] = [8760.summation over (i = 1)]([E.sub.eh,i]-[E.sub.sh,i]) (5)

Energy consumption of electric heating was determined from the conventional domestic electric water heaters. The electrical energy consumption of solar heating would still be contributed by the circulation pump set for the solar collector array [E.sub.cp] and the auxiliary electric heater [E.sub.ah]. For DHW supply temperatures lower than [T.sub.hw,min], the minimum DHW supply temperature stipulated by the local regulation, the auxiliary heater is energized. So the mathematical expressions of [E.sub.eh] and [E.sub.sh] are as follows:

[E.sub.eh] = [m.sub.hw][C.sub.pw]([T.sub.hw,min]-[T.sub.cw]) (6)

[E.sub.sh] = [E.sub.cp] + [E.sub.ah] (7)

[E.sub.ah] = max{[[m.sub.hw][C.sub.pw]([T.sub.hw,min] - [T.sub.hw,o])], 0} (8)

where

[m.sub.hw] = mass flow rate of domestic hot water, kg/h

[T.sub.hw,o] = hot-water outlet temperature of calorifier, [degrees]C

[T.sub.cw] = make-up potable water temperature, [degrees]C

In Equation 8, the "max" function would return a positive value for [E.sub.ah] only if the first entry is greater than zero. This is because in the case of abundant solar irradiation it would happen that [T.sub.hw,o] is greater than [T.sub.hw,min] and no auxiliary heating is required. Then the first entry of Equation 8 would be negative, and it would return [E.sub.ah] to zero. [T.sub.hw,o] of Equation 8 is a function of the four problem variables, [x.sub.1], [x.sub.2], [x.sub.3], and [x.sub.4], so [T.sub.hw,o] was essential output of the simulation model for determining [E.sub.ah], hence [E.sub.sh] in Equation 7 and finally [E.sub.sav] in Equation 5.

Optimization Results. From Table 2 it can be seen that the optimization result of the REA is better than that based on the design values from local design practice--about 12% better in yearly energy savings. This shows the necessity and effectiveness of involving optimization in system design in order to determine the optimal design parameters. In addition, it can be observed that the REA found a slightly better energy savings at a population of 50 as compared to one of 30. On the other hand, the convergence rate of a population of 30 is as effective as that of one of 50, as shown in Figure 5, so a near optimal solution could be determined in both cases. This demonstrates that the optimal search is still effective at a reduced epoch by the REA. It is important to make sure that the optimization method can search for the optimal design variables effectively within the prescribed span of the epoch. As a result, the REA could guarantee an optimal search in this solar water-heating design problem even at a shorter but reasonable epoch.

[FIGURE 5 OMITTED]

Table 2. Optimization Results of REA for Centralized Solar Water-Heating
Design Problem

               Design        REA              REA
              Practice        at               at
                        [epoch.sub.max]  [epoch.sub.max]
                             = 50             = 30

[E.sub.sav]    1412.1       1584.8           1583.0
Yearly                    (12.2% [up       (12.1% [up
energy                      arrow])          arrow])
saving, GJ

[x.sub.1]        22           23.4            22.3
Optimal tilt
angle of
solar
collectors,
degree

[x.sub.2]         0            6.2            7.7
Optimal
surface
azimuth of
solar
collectors,
degree due
south

[x.sub.3]        36           39.9            40.4
Optimal
capacity of
hot water
calorifier,
[m.sup.3]

[x.sub.4]      47,000       22,566.0        22,501.7
Optimal flow
rate of
circulation
pump, kg/h

Problem Developed by Mathematical Expressions--Duct System Design

Tsal et al. (1988a, 1988b) developed the T-method optimization using the approach of dynamic programming for a duct system design through minimization of the life-cycle cost. This method is still recommended and is fully described in 2005 ASHRAE Handbook--Fundamentals (ASHRAE 2005). An illustrative example of duct design, with the schematic layout, is shown in Figure 6. Duct sections 1 to 6 constitute the return-air subsystem, while duct sections 7 to 19 are the supply-air subsystem. The duct lengths, fittings, accessories, and other relevant design information, as well as the results of the T-method optimization, are comprehensively described in ASHRAE (2005). Asiedu et al. (2000) handled this problem by using the GA to minimize the life-cycle cost. Their study found that the GA could generate duct sizes with less life-cycle cost and better pressure balance as compared to the proposed T-method in ASHRAE (2005).

[FIGURE 6 OMITTED]

Objective Function. The objective function that Asiedu et al. (2000) used to minimize the life-cycle cost of the duct system design originated from ASHRAE (2005), as shown in Equations 9, 11, and 12. Equation 10 was adopted in Asiedu et al.'s (2005) paper for direct benchmarking purposes in the study.

minimizeF = [(E.sub.p] x PWEF) + [19.summation over (i = 1)][E.sub.s, i]] (9)

where [E.sub.p], the first year energy cost ($), is calculated as

[E.sub.p] = [G.summation over (g = 1)][[[Q.sub.fan, g]([E.sub.d] + [E.sub.c, g][[PSI].sub.g]T)[P.sub.g.sup.S]]/[[10.sup.3][[eta].sub.f][[eta].sub.m]]], (10)

where

[E.sub.s] = [S.sub.d][pi]LD (initial cost for round ducts, $) (11)

or

[E.sub.s] = 2[S.sub.d](H + W)L (initial cost for rectangular ducts, $), (12)

and where

F = present worth owning and operating cost (i.e. life-cycle cost), $

PWEF = present worth escalation factor

D = diameter of round duct, m

[E.sub.c,g] = unit electrical energy cost in operation mode g, $/kWh

[E.sub.d] = energy demand cost, $/kWh

G = total number of different operation modes g

g

1 for low flow and non-peak utility rate;

2 for high flow and non-peak utility rate;

3 for high flow and peak utility rate; or

4 for low flow and peak utility rate

H = duct height, m

L = duct length, m

[P.sub.g.sup.s] = maximum subsystem path pressure during operation mode g, Pa

[Q.sub.fan,g] = fan flow rate during operation mode g, [m.sup.3/s]

[S.sub.d] = unit duct cost, $/[m.sup.2]

T = operation time, h/year

W = duct width, m

[[eta].sub.f] = fan total efficiency

[[eta].sub.m] = motor drive efficiency

[[psi].sub.g] = fraction of time system operates in mode g

Constraint Functions. The constraint functions were developed according to the duct design problem. There were altogether 88 constraint functions, as described in Equations 13 through 100. The equality constraints [C.sub.1] and [C.sub.2] were used to limit the pressure imbalance of the largest path, while [C.sub.3] and [C.sub.4] were used to limit that of all paths as follows:

[C.sub.1] = [G.summation over (g = 1)][max l[member of][S.sub.s][[[psi].sub.g](P.sub.g.sup.s)-[P.sub.t,g)] = 0 for supply subsystem (13)

[C.sub.2] = [G.summation over (g = 1)][max l[member of][S.sub.s][[[psi].sub.g](P.sub.g.sup.s)-[P.sub.t,g)] = 0 for return subsystem (14)

[C.sub.3] = [G.summation over (g = 1)][summation over.(l[member of][S.sub.s])][[[psi].sub.g.sup.s](P.sub.g.sup.s)-[P.sub.t,g)] = 0 for supply subsystem (15)

[C.sub.3] = [G.summation over.(g = 1)][summation over.(l[member of][S.sub.s])][[[psi].sub.g](P.sub.g.sup.s)-[P.sub.t,g)] = 0 for return subsystem (16)

where

[P.sub.t,g] = path pressure during operation mode g, Pa

t = duct path index

[S.sub.s] = set of paths in subsystem s (i.e., supply or return subsystem)

The inequality constraints [C.sub.5] through [C.sub.42] for air velocity limits of duct sections 1 to 19 were as follows:

[C.sub.5] to [C.sub.23] = [[V.sub.i] - [V.sub.mode.sup.min]][greater than or equal to] = 0 for i = 1, 2, ..., 19, mode = low or high (17 to 35)

[C.sub.24] to [C.sub.42] = [V.sub.mode.sup.max] - [V.sub.i][greater than or equal to] = 0 for i = 1, 2, ..., 19, mode = low or high (36) to (54)

where V is the airflow velocity in duct (m/s).

The inequality constraints [C.sub.43] through [C.sub.87] for width control from the preceding duct section were as follows:

[C.sub.43] = [W.sub.6]-[W.sub.5][greater than or equal to]0 (55)

[C.sub.87] = [W.sub.9]-[W.sub.7][greater than or equal to]0 (99)

The equality constraint [C.sub.88] for the same width of duct sections 11 and 12 (as required in the duct system design problem) was

[C.sub.88] = [W.sub.11]-[W.sub.12][greater than or equal to]0 (100)

Problem Variables. The variables for this duct design problem included the duct sizes for the return-air and supply-air subsystems. All the problem variables [x.sub.1] to [x.sub.19] were integer I [member of] [I.sup.+] in the unit of centimeter, so every increment would be 1 cm. The feasible range of each problem variable was also assigned with reference to Asiedu et al. (2000). For the return-air subsystem, there were six problem variables with the following corresponding bounds:

[x.sub.1] = diameter of duct section 1 I [member of] [1, 80]

[x.sub.2] = diameter of duct section 2 I [member of] [1, 80]

[x.sub.3] = diameter of duct section 3 I [member of] [1, 80]

[x.sub.4] = width of duct section 4 (constant, = 60 cm) I [member of] [60, 60]

[x.sub.5] = diameter of duct section 5 I [member of] [1, 80]

[x.sub.6] = diameter of duct section 6 I [member of] [1, 80]

For the supply-air subsystem, there were thirteen optimization variables with the following corresponding bounds:

[x.sub.7] = width of duct section 7 1 [member of] [1, 80]

[x.sub.8] = width of duct section 8 I [member of] [1, 80]

[x.sub.9] = width of duct section 9 I [member of] [1, 80]

[x.sub.10] = width of duct section 10 I [member of] [1, 80]

[x.sub.11] = width of duct section 11 I [member of] [1, 80]

[x.sub.12] = width of duct section 12 I [member of] [1, 80]

[x.sub.13] = width of duct section 13 I [member of] [1, 80]

[x.sub.14] = width of duct section 14 I [member of] [1, 80]

[x.sub.15] = width of duct section 15 I [member of] [1, 80]

[x.sub.16] = width of duct section 16 I [member of] [1, 80]

[x.sub.17] = width of duct section 17 I [member of] [1, 80]

[x.sub.18] = height of duct section 18 I [member of] [30, 80]

[x.sub.19] = width of duct section 19 (constant, = 80 cm) 1 [member of] [80, 80]

Optimization Results. The results from the GA of Asiedu et al. (2000) and the REA for the duct system design problem are compared in Table 3. The optimal results were obtained from the best solution among the 10 runs as shown. In the table, it is clear that the REA can determine a better and lower total cost using a significant decrease of population and epoch as compared to the GA. Using the REA, the entire duct system with both the supply and return subsystems could be optimized together, as opposed to using the GA, where the two subsystems were optimized separately in Asiedu et al. (2000). Therefore, in terms of the frequency of function calls for evaluation, there was a tremendous 2600-fold reduction using the REA.

Table 3. Comparison of the Results of the GA of Asiedu et al. (2000) and
the REA for the Duct System Design Problem

                GA of Asiedu et            REA
                  al. (2000)

                Implementation

Population            800                   10
size

Epoch of        3359 for supply            200
termination       subsystem;
                2501 for return
                   subsystem

Tournament              5                   5
size

Number of           4,688,000             1801
function     [800 x (3359 + 2501)]  [10 + (9 x 199)]
calls per
run

Number of              10                 10
runs

                 Major Results

Total cost,          11,618          11,467 (1.3%
$                                   [down arrow])

Material              8575               8531
cost, $

Energy cost,          3043               2936
$

CONCLUSION

Based on the paradigm of evolution strategy, the robust evolutionary algorithm (REA) has been developed to handle HVAC optimization problems effectively and efficiently, particularly those modelled by common plant and energy simulation programs. The REA is formed by the operators of arithmetic recombination, Cauchy deterministic mutation (developed by the authors), and tournament selection. Since the evaluation function call by the simulation program is a time-determining step, it is crucial to acquire a reliable result from a limited number of generations. The optimization effectiveness and efficiency of the REA for HVAC problems has been demonstrated through the application example, either developed by the simulation model or mathematical expressions. The REA is capable of handling different HVAC optimization problems that are featured with multimodal, multidimensional, nonlinear, continuous-discrete, and highly constrained characteristics. The applications of the REA can even extend to other engineering optimization problems of a similar nature.

NOMENCLATURE

[a.sub.1], [a.sub.2], [a.sub.3] = empirical coefficient of deterministic strategy parameter

C(0,1) = Cauchy random number with mean = 0 and variance = 1

[C.sub.pw] = specific heat capacity of water at constant pressure, [kJ.[kg.sup.-1].[K.sup.-1]]

D = diameter of round duct, m

e = epoch

[E.sub.ah] = electricity consumption of centralized auxiliary electric heater, kJ

[E.sub.c,g] = unit electrical energy cost in operation mode g, $/kWh

[E.sub.cp] = electricity consumption of circulation pump set, kJ

[E.sub.d] = energy demand cost, $/kWh

[E.sub.eh] = electricity consumption of conventional electric heating, kJ

[e.sub.max] = epoch of termination

[E.sub.p] = first year energy cost, $

[E.sub.s] = initial cost for round/rectangular ducts, $

[E.sub.sav] = year-round savings in electricity by using solar water heating against electric heating, kJ

[E.sub.sh] = electricity consumption of solar heating system, kJ

F = present worth owning and operating cost (i.e., life-cycle cost) $

f = test function

G = total number of different operation modes g

g = operation mode for duct design optimization problem

H = duct height, m

[I.sup.+] = positive integer set

k = positive integer

L = duct length, m

[LB.sub.j] = lower bound of the jth problem variable

[m.sub.hw] = mass flow rate of domestic hot water, kg.[h.sup.-1]

[n.sub.eval] = number of evaluation function calls

[n.sub.pop] = number of population

[n.sub.var] = number of variables in the problem individual

[P.sub.g.sup.s] = maximum subsystem path pressure during operation mode g, Pa

[P.sub.[iota],g] = path pressure during operation mode g, Pa

PWEF = present worth escalation factor

q = number of opponents in tournament selection

[Q.sub.fan,g] = fan flow rate during operation mode g, [m.sup.3]/s

r = random number [member of] U(0,1)

[S.sub.d] = unit duct cost, $/[m.sup.2]

[S.sub.s] = set of paths in subsystem s (i.e., supply or return subsystem)

T = operation time, h/yr

[T.sub.cw] = make-up potable water temperature, [degrees]C

[T.sub.hw,min] = minimum DHW supply temperature stipulated by the local regulation, [degrees]C

[T.sub.hw,o] = hot water outlet temperature of calorifier, [degrees]C

[UB.sub.j] = upper bound of the jth problem variable

V = airflow velocity in duct, m/s

W = duct width, m

[x.sub.mu] = mutated individual

[x.sub.p] = parent individual

[x.sub.xo] = recombined individual

Greek Symbols

[[eta].sub.f] = fan total efficiency

[[eta].sub.m] = motor drive efficiency

[iota] = duct path index

[lambda] = offspring population size in evolution strategy

[mu] = parent population size in evolution strategy

[sigma] = deterministic strategy parameter

[[sigma].sub.o] = initial deterministic strategy parameter

[[psi].sub.g] = fraction of time system operates in mode g

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APPENDIX A--VERIFICATION OF EFFECTIVENESS OF THE REA

Common HVAC application problems are either constrained or unconstrained. There are generally a number of constraint functions that can be of an equality or inequality, linear or nonlinear nature. It is standard methodology for EA researchers to use popular test functions to conduct empirical studies of the effectiveness of any newly developed EA. In order to verify the effectiveness of the REA, a variety of constrained and unconstrained test functions were used in this study; they are listed in Table A1. In the constrained test functions, linear inequality, nonlinear equality, and nonlinear inequality were all involved. In the unconstrained test functions, both the unimodal and multimodal natures were covered. The characteristics of the objective functions include linear, quadratic, cubic, polynomial, trigonometric, and exponential types. The choice of these test functions was targeted at covering different formats of the optimization problems that are encountered in HVAC engineering applications.

Table A1. Summary of Characteristics and Origins of Test Functions

                                           Type of
Category           Test        No. of      Objective       No. of
                 Function     Variables    Function      Constraints

                 [f.sub.c1]      13        Quadratic       LI: 9
               Floundas & P

                 [f.sub.c2]       5        Quadratic       NI: 6
               Himmelblau 11

                 [f.sub.c3]       2        Cubic           NI: 2
               Floundas & P

Constrained      [f.sub.c4]      10        Quadratic       LI: 3 NI: 5
Functions      Hock & S 113

                 [f.sub.c5]       7        Polynomial      NI: 4
               Hock & S 100

                 [f.sub.c6]       8        Linear          LI: 3 NI: 3
                Hock & S HX

                 [f.sub.c7]       2        Quadratic       NE: 1
                   Maa &
                 Shanblatt

                 [f.sub.u1]      10        Quadratic         --
                   Sphere
                   Model

                 [f.sub.u2]      10        Polynomial        --
               Schwefel 2.22

Unconstrained    [f.sub.u3]      10        Quadratic         --
Unimodal       Schwefel 1.2
Functions

                 [f.sub.u4]      10        Polynomial        --
                 Rosenbrock

                 [f.sub.u5]       2        Trigonometric     --
                   Easom                   and
                                           exponential

                 [f.sub.m1]      10        Trigonometric     --
                 Schwefel 7

Unconstrained    [f.sub.m2]      10        Quadratic and     --
Multimodal       Rastrigin                 trigonometric
Functions

                 [f.sub.m3]      10        Exponential       --
                   Ackley                  and
                                           trigonometric

                 [f.sub.m4]      10        Quadratic and     --
                 Griewank                  trigonometric

Category           Test          Origin
                 Function

                 [f.sub.c1]    Floundas and
               Floundas & P     Pardalos
                                 (1987)

                 [f.sub.c2]    Problem 11,
               Himmelblau 11   Himmelblau
                                 (1972)

                 [f.sub.c3]   Floundas and
               Floundas & P     Pardalos
                                 (1987)

Constrained      [f.sub.c4]    Problem 113,
Functions      Hock & S 113     Hock and
                              Schittkowski
                                 (1981)

                 [f.sub.c5]    Problem 100,
               Hock & S 100     Hock and
                              Schittkowski
                                 (1981)

                 [f.sub.c6]      Heat
                Hock & S HX    Exchanger
                                Design
                              Problem, Hock
                                  And
                              Schittkowski
                                 (1981)

                 [f.sub.c7]     Maa and
                   Maa &       Shanblatt
                 Shanblatt       (1992)

                 [f.sub.u1]    Sphere Model
                   Sphere
                   Model

                 [f.sub.u2]   Problem 2.22,
               Schwefel 2.22    Schwefel
                                 (1995)

Unconstrained    [f.sub.u3]    Problem 1.2,
Unimodal       Schwefel 1.2     Schwefel
Functions                         (1995)

                 [f.sub.u4]    Generalized
                 Rosenbrock    Rosenbrock's
                                Function

                 [f.sub.u5]     Easom's
                   Easom        Function

                 [f.sub.m1]
                 Schwefel 7

Unconstrained    [f.sub.m2]    Generalized
Multimodal       Rastrigin      Rastrign's
Functions                       Function

                 [f.sub.m3]     Ackley's
                   Ackley       Function

                 [f.sub.m4]    Generalized
                 Griewank       Griewank
                                Function

Note: LI = linear inequality, NE = nonlinear equality, NI = nonlinear
inequality.

In this set of the experiments for verification, the population and epoch of the REA were typically 10 and 1000, respectively (except the minimum [e.sub.max] for [f.sub.c6] or [f.sub.u4] was 3000, while that for [f.sub.m1] or [f.sub.m2] was 2000 for general practice). The statistical performance of 50 runs of each test function was recorded. The population of 10 was used rather than the typically larger populations of tens or hundreds for a typical GA, since the primary objective of the experiments was to identify the robust approach with minimum evaluation function calls. Since the appropriate population size and span of epoch in handling real-life engineering optimization problems are generally unknown beforehand, an effective and efficient EA should still give near-optimal or acceptable results at a reduced epoch. Therefore, tests with half of the prescribed epoch were carried out so as to understand the performance of the REA in such stringent conditions. In addition, SGAs with a population of 100 were involved in the tests in order to compare the optimization results from the REA. In this exercise, the population of the SGA is ten times that of the REA.

The mean bests and standard deviations of 50 runs of all the test functions are summarized in Table A2. It was found that the optimization results of the REA were close to the corresponding best-known solutions, showing that the REA was effective for handling a variety of constrained and unconstrained unimodal and unconstrained multimodal test functions. It was also found that the REA was still effective at half of the prescribed epoch. On the other hand, even the population of the SGA was tenfold; its optimization results were not stable and had a high standard deviation in many test functions. This leads to the query about SGA's repeatability and stability in determining the optimal solution; particularly, the number of evaluation function calls is critical, like the simulation model of solar water-heating in this study.

Table A2. Performance of REA and SGA in a Variety of Test Functions

                                        REA                  REA
                 [n.sub.pop]            10                   10
               [epoch.sub.max]        1000 (a)             500 (b)

Test Function    Best-Known          Mean Best             Mean Best
                   Solution      (Std. Deviation)       (Std. Deviation)

                                Constrained Functions

[f.sub.c1]           -15               -14               -14 (1.09)
Floundas & P                    (8.02x[10.sup.-1])

[f.sub.c2]         -30665.5      -30647.53 (52.83)  -30603.78 (112.82)
Himmelblau 11

[f.sub.c3]         -6961.81       -6929.11 (17.61)    -6904.97 (25.89)
Floundas & P

[f.sub.c4]          24.31            25.89             30.51 (14.14)
Hock & S 113                    (9.90x[10.sup.-1])

[f.sub.c5]         680.63            680.85           681.13 (32.40)
Hock & S 100                    (1.34x[10.sup.-1])

[f.sub.c6]         7049.33      7611.68 (638.52)     8275.74 (2015.17)
Hock & S HX

[f.sub.c7]           0.75              0.7487            0.7718
Maa &                           (1.84x[10.sup.-3])  (2.68x[10.sup.-2])
Shanblatt

                        Unconstrained Unimodal Functions

[f.sub.u1]            0         7.48x[10.sup.-5]     1.15x[10.sup.-4]
Sphere                          (2.62x[10.sup.-5])  (3.09x[10.sup.-5])
Model

[f.sub.u2]            0         2.19x[10.sup.-3]     3.00x[10.sup.-3]
Schwefel 2.22                   (4.64x[10.sup.-4])  (5.67x[10.sup.-4])

[f.sub.u3]            0         3.65x[10.sup.-4]      8.49x[10.sup.-1]
Schwefel 1.2                    (2.21x[10.sup.-4])        (1.43)

[f.sub.u4]            0           7.90 (14.81)        45.40 (109.76)
Rosenbrock

[f.sub.u5]           -1             -0.99997             -0.99993
Easom                           (3.08x[10.sup.-5])  (6.48x[10.sup.-5])

                         Unconstrained Multimodal Function

[f.sub.m1]         -4189.83     -3725.95 (181.19)   -3591.05 (216.43)
Schwefel 7

[f.sub.m2]            0            3.23 (1.98)          5.93 (2.51)
Rastrigin

[f.sub.m3]            0         3.48x[10.sup.-3]     4.54x[10.sup.-3]
Ackley                          (6.41x[10.sup.-4])  (8.55x[10.sup.-4])

[f.sub.m4]            0         1.85x[10.sup.-1]     1.71x[10.sup.-1]
Griewank                        (1.17x[10.sup.-1])  (8.79x[10.sup.-2])

                          SGA
                          100
                       1000 (a)

Test Function          Mean Best
                   (Std. Deviation)

[f.sub.c1]           -20.98 (11.03)
Floundas & P

[f.sub.c2]         -30412.3 (149.62)
Himmelblau 11

[f.sub.c3]        -2248.72 (1878.91)
Floundas & P

[f.sub.c4]          284.87 (414.65)
Hock & S 113

[f.sub.c5]        730319.9 (1853933)
Hock & S 100

[f.sub.c6]        8642.85 (4800.55)
Hock & S HX

[f.sub.c7]             0.9646
Maa &             (7.03x[10.sup.-2])
Shanblatt

[f.sub.u1]          1.273 (3.934)
Sphere
Model

[f.sub.u2]        5.45x[10.sup.-2]
Schwefel 2.22    (1.25x[10.sup.-2])

[f.sub.u3]        1968.97 (1039.82)
Schwefel 1.2

[f.sub.u4]         864.55 (948.03)
Rosenbrock

[f.sub.u5]       -0.51979 (0.5097)
Easom

[f.sub.m1]        -2895.08 (890.06)
Schwefel 7

[f.sub.m2]          13.10 (4.70)
Rastrigin

[f.sub.m3]               2.15
Ackley            (8.09x[10.sup.-1])

[f.sub.m4]               1.00
Griewank          (4.17x[10.sup.-1])

Notes: (a.) For [f.sub.c6]/[f.sub.u4] and [f.sub.m1]/[f.sub.m2],
[e.sub.max] were 3000 and 2000, respectively.

(b.) For [f.sub.c6] and [f.sub.u4]/[f.sub.m1]/[f.sub.m2], [e.sub.max]
were 1500 and 1000, respectively.

APPENDIX B--DETAILS OF TEST FUNCTIONSS

Test Function [f.sub.c1]--Floundas and Pardalos' Problem (Floundas and Pardalos 1987)

Objective Function: [f.sub.c1](x) = [5x.sub.1] + [5x.sub.2] + [5x.sub.3] + [5x.sub.4] - 5 [4.summation over (i = 1)] [x.sub.i.sup.2] - [13.summation over (i = 5)] [x.sub.i]

LI Constraint Function: 10 - [2x.sub.1] - [2x.sub.2] - [x.sub.10] - [x.sub.11][greater than or equal to]0(active)

10 - [2x.sub.1] - [2x.sub.3] - [x.sub.10] - [x.sub.12][greater than or equal to]0(active)

10 - [2x.sub.2] - [2x.sub.3] - [x.sub.11] - [x.sub.12][greater than or equal to]0(active)

8[x.sub.1]-[x.sub.10][greater than or equal to]0

8[x.sub.2]-[x.sub.11][greater than or equal to]0

8[x.sub.3]-[x.sub.12][greater than or equal to]0

2[x.sub.4] + [x.sub.5]-[x.sub.10][greater than or equal to]0 (active)

2[x.sub.6] + [x.sub.7]-[x.sub.11][greater than or equal to]0 (active)

2[x.sub.8] + [x.sub.9]-[x.sub.12][greater than or equal to]0 (active)

for 0[less than or equal to][x.sub.i][less than or equal to]1, i = 1, 2, ..., 9

0[less than or equal to][x.sub.i][less than or equal to]100, i = 10, 11, 12

0[less than or equal to][x.sub.13][less than or equal to]1

Optimum (minimum): x * = (1,1,1,1,1,1,1,1,1,3,3,3,1) [f.sub.c1](x *) = -15

Test Function [f.sub.c2]--Himmelblau's Problem 11

Objective Function: [f.sub.c2](x) = 5.3578547[x.sub.3.sup.2] + 0.8356891[x.sub.1][x.sub.5] + 37.293239[x.sub.1] - 40792.141

NI Constraint Function:

92 - 85.334407 - 0.0056858[x.sub.2][x.sub.5] - 0.0006262[x.sub.1][x.sub.4] + 0.0022053[x.sub.3][x.sub.5] [greater than or equal to] 0 (active)

85.334407 + 0.0056858[x.sub.2][x.sub.5] + 0.0006262[x.sub.1][x.sub.4] - 0.0022053[x.sub.3][x.sub.5] [greater than or equal to] 0

110 - 80.51249 - 0.0071317[x.sub.2][x.sub.5] - 0.0029955[x.sub.1][x.sub.2] - 0.0021813[x.sub.3.sup.2] [greater than or equal to] 0

25 - 9.300961 - 0.0047026[x.sub.3][x.sub.5] - 0.0012547[x.sub.1][x.sub.3] - 0.0019085[x.sub.3.sup.4] [greater than or equal to] 0

9.300961 + 0.0047026[x.sub.3][x.sub.5] + 0.0012547[x.sub.1][x.sub.3] + 0.0019085[x.sub.3][x.sub.4] - 20 [greater than or equal to] 0 (active)

for 78 [greater than or equal to] [x.sub.1] [greater than or equal to] 102

33 [greater than or equal to] [x.sub.2] [greater than or equal to] 45

27 [greater than or equal to] [x.sub.i] [greater than or equal to] 45, i = 3, 4, 5

Optimum (minimum): [x*] = (78.0, 33.0, 29.995, 45.0, 36.776)

[f.sub.c2](x*) = -30665.5

Test Function [f.sub.c3]--Floundas and Pardalos' Problem (Floundas and Pardalos 1987)

Objective Function: [f.sub.c3](x) = [([x.sub.1]-10).sup.3]+[([x.sub.2]-20).sup.3]

NI Constraint Function: [([x.sub.1]-5).sup.2] + [([x.sub.2]-5).sup.2]-100[greater than or equal to]0(active)

-[([x.sub.1]-6).sup.2]-[([x.sub.2]-5).sup.2] + 82.81[greater than or equal to]0(active)

for 13[less than or equal to][x.sub.1][less than or equal to]100

0[less than or equal to][x.sub.2][less than or equal to]100

Optimum (minimum): [x*] = (14.095, 0.84296)

[f.sub.c3][x*] = -6961.81381

Test Function [f.sub.c4]--Hock and Schittkowski's Problem 113 (Hock and Schittkowski 1981)

Objective Function: [f.sub.c4](x) = [x.sub.1.sup.2] + [x.sub.2.sup.2] [x.sub.1][x.sub.2] - [14x.sub.1] - [16x.sub.2] + [([x.sub.3]-10).sup.2] + 4[([x.sub.4]-5).sup.2] + [([x.sub.5]-3).sup.2] + 2[([x.sub.6]-1).sup.2] + [5x.sub.7.sup.2] + 7[([x.sub.8]-11).sup.2] + 2[([x.sub.9]-10).sup.2] + [([x.sub.10]-7).sup.2] + 45

LI Constraint Function: 105 - [4x.sub.1] - [5x.sub.2] + [3x.sub.7] - [9x.sub.8][greater than or equal to]0 (active)

-3[([x.sub.1]-2).sup.2] - 4[([x.sub.2]-3).sup.2] - [2x.sub.3.sup.2] + [7x.sub.4] + 120 [greater than or equal to]0 (active)

[-10x.sub.1] + [8x.sub.2] + [17x.sub.7] - [2x.sub.8] [greater than or equal to]0 (active)

NI Constraint Function: [-x.sub.1.sup.2] - -2[([x.sub.2]-2).sup.2] + [2x.sub.1][x.sub.2] - [14x.sub.5] + [6x.sub.6][greater than or equal to]0 (active)

[8x.sub.1] - [2x.sub.2] - [5x.sub.9] + [2x.sub.10] + 12[greater than or equal to]0 (active)

[-5x.sub.1.sup.2] - [8x.sub.2] - [([x.sub.3]-6).sup.2] + [2x.sub.4] + 40 [greater than or equal to]0 (active)

[3x.sub.1] - [6x.sub.2] - 12[([x.sub.9]-8).sup.2] + [7x.sub.10][greater than or equal to]0 (active)

-0.5[([[x.bar].sub.1]-8).sup.2] - -2[([x.sub.2]-4).sup.2] - [-3x.sub.5.sup.2] + [x.sub.6] + 30 [greater than or equal to]0

for -10.0[less than or equal to][x.sub.i][less than or equal to]10.0, i = 1, 2, ..., 10

Optimum (minimum): [x*] = (2.1711996, 2.363683, 8.773926, 5.095984, 0.9906548, 1.430574, 1.321644, 9.828726, 8.280092, 8.375927)

[f.sub.c4](x*) = 24.3062091

Test Function [f.sub.c5]--Hock and Schittkowski's Problem 100 (Hock and Schittkowski 1981)

Objective Function: [f.sub.c5](x) = [([x.sub.1]-10).sup.2] + 5[([x.sub.2]-12).sup.2] + [x.sub.3.sup.4] + 3[([x.sub.4]-11).sup.2] + [10x.sub.5.sup.6] + [7x.sub.6.sup.2] + [x.sub.7.sup.4] - [4x.sub.6][x.sub.7] - [10x.sub.6] - [8x.sub.7]

NI Constraint Function: 127 - [2x.sub.1.sup.2] - [3x.sub.2.sup.4][x.sub.3] - [4x.sub.4.sup.2] - [5x.sub.5][greater than or equal to]0 (active)

282 - [7x.sub.1] - [3x.sub.2] - [10x.sub.3.sup.2] - [x.sub.4] + [x.sub.5][greater than or equal to]0

196 - [23x.sub.1] - [x.sub.2.sup.2] - [6x.sub.6.sup.2] - [8x.sub.7] [greater than or equal to]0

[-4x.sub.1.sup.2] - [x.sub.2.sup.2] - [3x.sub.1][x.sub.2] - [2x.sub.3.sup.2] - [5x.sub.6] + [11x.sub.7][greater than or equal to]0(active)

for -10.0[less than or equal to][x.sub.i][less than or equal to]10.0, i = 1, 2, ..., 7

Optimum (minimum): [x*] = (2.330499, 1.951372, -0.4775414, 4.365726, -0.6244870, 1.038131, 1.594227)

[f.sub.c5](x*) = 680.6300573

Test Function [f.sub.c6]--Hock and Schittkowski's Heat Exchanger Design (Hock and Schittkowski 1981)

Objective Function: [f.sub.c6](x) = [x.sub.1] + [x.sub.2] + [x.sub.3]

LI Constraint Function: 1 - 0.0025([x.sub.4] + [x.sub.6])[greater than or equal to]0(active)

1 - 0.0025([x.sub.5] + [x.sub.7] - [x.sub.4])[greater than or equal to]0(active)

1 - 0.01([x.sub.8] - [x.sub.5])[greater than or equal to]0(active)

NI Constraint Function: [x.sub.1][x.sub.6] - 833.33252[x.sub.4] - 100[x.sub.1] + 83333.333 [greater than or equal to] 0(active)

[x.sub.2][x.sub.7] - 1250[x.sub.5] - [x.sub.2][x.sub.4] + 1250[x.sub.4] [greater than or equal to] 0(active)

[x.sub.3][x.sub.8] - 1250000 - [x.sub.3][x.sub.5] + 12500[x.sub.5] [greater than or equal to] 0(active)

for 100 [less than or equal to] [x.sub.1] [less than or equal to] 10000

1000 [less than or equal to] [x.sub.i] [less than or equal to] 10000, i = 2, 3

10 [less than or equal to] [x.sub.i] [less than or equal to] 1000, i = 4, 5, ..., 8

Optimum (minimum): [x*] = (579.3167, 1359.943, 5110.071, 182.0174, 295.5985, 217.9799, 286.4162, 395.5979)

[f.sub.c6](x*) = 7049.330923

Test Function [f.sub.c7]--Maa and Shanblatt's Problem (Maa and Shanblatt 1992)

Objective Function: [f.sub.c7](x) = [x.sub.1.sup.2] + [([x.sub.2] - 1).sup.2]

NE Constraint Function: [x.sub.2] - [x.sub.1.sup.2] = 0

for -1 [less than or equal to] [x.sub.i] [less than or equal to] 1, i = 1, 2

Optimum (minimum): [x*] = ([+ or -]0.70711, 0.5)

[f.sub.c7](x*) = 0.75000455

Test Function [f.sub.u1]--Sphere Model

Objective Function: [f.sub.u1](x) = [n.summation over (i = 1)][x.sub.i.sup.2]

for -100 = xi = 100 [less than or equal to] [x.sub.i] [less than or equal to] 100

Optimum (minimum): x * = (0, 0, ..., 0)

[f.sub.u1](x *) = 0

Test Function [f.sub.u2]--Schwefel's Problem 2.22

Objective Function: [f.sub.u2](x) = [n.summation over (i = 1)][absolute value of [x.sub.i]] + [n.[PI] over (i = 1)][absolute value of [x.sub.i]]

for -10 [less than or equal to] [x.sub.i] [less than or equal to] 10

Optimum (minimum): x * = (0, 0, ..., 0)

[f.sub.u2](x *) = 0

Test Function [f.sub.u3]--Schwefel's Problem 1.2

Objective Function: [f.sub.u3](x) = [n.summation over (i = 1)][([i.summation over (j = 1)][x.sub.j]).sup.2]

for -100 [less than or equal to] [x.sub.i] [less than or equal to] 100

Optimum (minimum): x * = (0, 0, ..., 0)

[f.sub.u3](x *) = 0

Test Function [f.sub.u4]--Generalized Rosenbrock's Function

Objective Function: [f.sub.u4](x) = [[n-1].summation over (i = 1)][100[[([x.sub.[i + 1]] - [x.sub.i.sup.2])].sup.2] + [([x.sub.i]-1).sup.2]]

for -30 [less than or equal to] [x.sub.i] [less than or equal to] 30

Optimum (minimum): [x*] = (1, 1, ..., 1)

[f.sub.u4](x*) = 0

Test Function [f.sub.u5]--Easom's Function

Objective Function: [f.sub.u5](x) = -cos [x.sub.1] cos [x.sub.2] exp {-[[([x.sub.1] - [pi]).sup.2] + [([x.sub.2]-[pi]).sup.2]]}

for -100 [less than or equal to] [x.sub.i] [less than or equal to] 100

Optimum (minimum): [x*] = ([pi], [pi])

[f.sub.u5](x*) = -1

Test Function [f.sub.m1]--Schwefel's Function 7

Objective Function: [f.sub.m1](x) = [n.summation over (i = 1)][-[x.sub.i](sin[square root of [[absolute value of[x.sub.i]]]])]

for -500 [less than or equal to] [x.sub.i] [less than or equal to] 500

Optimum (minimum): [x*] = (420.9687, 420.9687, ..., 420.9687) [f.sub.m1](x*) = -[n.sub.var].418.9829

Test Function [f.sub.m2]--Generalized Rastrigin's Function

Objective Function: [f.sub.u7](x) = [n.summation over (i = 1)][[x.sub.i.sup.2] - 10cos(2[pi][x.sub.i]) + 10]

for -5.12 [less than or equal to] [x.sub.i] [less than or equal to] 5.12

Optimum (minimum): x * = (0, 0, ..., 0)

[f.sub.m2](x*) = 0

Test Function [f.sub.m3]--Ackley's Function

Objective Function: [f.sub.m3](x) = -20exp(-0.2[square root of [[1/n][n.summation over (i = 1)][x.sub.i.sup.2]]]) - exp[[1/n][n.summation over (i = 1)]cos(2[pi][x.sub.i])] + 20 + e

for -32 [less than or equal to] [x.sub.i] [less than or equal to] 32

Optimum (minimum): [x*] = (0, 0, ..., 0) [f.sub.m3](x*) = 0

Test Function [f.sub.m4]--Generalized Griewank Function

Objective Function: [f.sub.m4](x) = [1/4000][n.summation over.(i = 1)][x.sub.i.sup.2] - [n.[PI].(i = 1)]cos([x.sub.i]/[square root of i]) + 1

for -600 [less than or equal to] [x.sub.i] [less than or equal to] 600

Optimum (minimum): [x*] = (0, 0, ..., 0)

[f.sub.m4](x*) = 0

Received February 13, 2008; accepted June 2, 2008

Kwong Fai Fong, PhD

Member ASHRAE

Victor Ian Hanby, PhD

Tin Tai Chow, PhD

Member ASHRAE

Kwong Fai Fong is a lecturer and Tin Tai Chow is a principal lecturer in the Division of Building Science and Technology, City University of Hong Kong, Hong Kong, China. Victor Ian Hanby is a professor in the Institute of Energy and Sustainable Development, De Montfort University, Leicester, United Kingdom.