# Comparative analysis of optimization approaches to design building envelope for residential buildings.

INTRODUCTION

In order to reduce building energy consumption most effectively, heating and cooling loads due to the building envelope must be addressed early in the design process. Several design parameters can have an effect on these loads, including the shape of the building, wall and roof construction, foundation type, insulation levels, window type and area, thermal mass, and shading. All of these parameters interact and affect the energy performance of the building. Traditionally, this type of analysis has been done with parametric runs using a building simulation engine such as DOE-2 (Winkelmann, 1993) or EnergyPlus (Crawley, 2000). However, varying one parameter while leaving others building envelope features constant can potentially miss important interactive effects, and full combinatory parametric studies are usually infeasible. A better solution is to couple an optimization algorithm to a simulation engine in order to find a minimum for a given cost function including life-cycle cost, annual operating costs, and annual energy use (Wright, 2002; Caldas and Norford, 2003; and Ouarghi and Krarti, 2006).

The objective of this paper is to compare three different optimization techniques to assess their robustness and efficiency for application in building envelope optimization. Robustness is a measure of the algorithm's ability to minimize the cost function, while efficiency is a measure of its speed which is defined in this study as the number of simulations required to reach the minimum cost level. The three methods investigated in this paper include the sequential search used in the Building Energy Optimization or BEopt tool (Andersen, et al. 2004), genetic algorithms or GAs (Goldberg, 1989 and Haupt and Haupt, 2004), and particle swarm optimization or PSO (Wetter, 2006). Each of these methods does not require the calculation of differentials for the cost function, but instead uses discrete values of the cost function to determine the parameter values of the next iteration (i.e. direct search).

DESCRIPTION OF OPTIMIZATION APPROACHES

One approach to classify optimization techniques is by the nature of the problem search space--continuous or discrete. The character of the parameters affecting building envelope optimization lends itself to discrete optimization. A few parameters, such as aspect ratio, orientation, and window area could be considered continuous, but almost all other parameters have a limited number of discrete options. For example, there are a finite number of available wall types for a realistic construction situation. It would be possible to optimize on continuous R-value, but the chance that the optimum solution would correspond to an existing wall type is very small. The same is true of parameters such as window type, foundation type, roof type, and shape.

Continuous optimization methods include the Nelder-Mead simplex method, Hooke-Jeeves method, and various gradient-based approaches (Nash and Sofer, 1996). Because of the discrete nature of the envelope optimization problem, these continuous techniques were not investigated. Discrete optimization methods include global techniques such as genetic algorithms, simulated annealing, tabu search, and particle swarm, as well as direct search techniques such as the sequential search used in BEopt (NREL, 2007). For this study, genetic algorithms were compared to the sequential search, and the particle swarm method was used to validate results.

Sequential Search

The sequential search technique used in BEopt is a direct search method that identifies the building option that will best decrease the cost function after each successive iteration (Christensen et al., 2005). It begins by simulating a user-defined reference building. It then runs a simulation for each potential option one at a time. The most cost-effective option is chosen and used in the building description for the next point along the path. There are a number of discrete options in different categories such as azimuth, aspect ratio, wall type, and ceiling insulation. The most cost-effective option is defined as the one that gives the largest reduction in annual costs for the smallest reduction in source energy use. Annual costs are a combination of mortgage costs (which increase as more expensive energy-efficient options are included) and utility costs. The process is repeated, ultimately defining a path from the reference building to the minimum cost point, and then to a zero net energy building.

Without modifications, this simple algorithm would not reliably find the correct least-cost path, due to the problem of interactive effects between different options. Three special cases have been identified--invest/divest, large steps, and positive interactions (Andersen et al., 2006). The invest/divest case is a result of negative interactive effects. In this case, BEopt removes options which may result in a more cost-optimal point. For example, a highly efficient HVAC system may have been selected as the most cost-effective option at an early point in the process. Later in the search process, however, the improvement of the building envelope may cause the efficient HVAC option to not be cost-optimal, so it is removed from the building design. The large steps case is another example of negative interaction among options. There may be a large energy-saving option that is available at a current point, but is less cost-effective than another option that does not save as much energy. The latter option is chosen, and then the most cost-effective option is again chosen at that second point, which results in a third point. However, it is possible that the original large energy-saving option available at the first point is more cost-optimal than the third point. In order to solve this problem, BEopt keeps track of points from previous iterations and compares them to the current point. If a previous point is more cost-optimal, it replaces the current point. A positive interaction occurs if two options are more cost-effective when present together than they would be if considered separately. An example could be the presence of both large south-facing windows and thermal mass for passive solar heating. BEopt will only find these positive interactions if one of the options is first selected individually. This inability to always identify synergistic options is a potential deficiency with the sequential search method.

Genetic Algorithms

Genetic algorithms (GAs) use the evolutionary concept of natural selection to converge on an optimal solution over many generations GAs (Goldberg, 1989 and Haupt and Haupt, 2004). They differ from traditional optimization methods in a number of areas. First, rather than working with one potential solution at a time, the technique works with a set of solutions called a population. This ensures a global approach to the optimization and helps the GA avoid getting stuck in local minima, which can be a problem with other methods. Second, the GA works with encodings of the parameters, not the parameters themselves. Parameters are traditionally encoded as binary strings, although other encoding options can be used. Finally, GAs use probabilistic methods for determining the parameter values in each successive iteration, rather than deterministic rules. This means that each time a GA is run, the path toward convergence is different, and the end result may be different as well.

Each individual in the population represents a different solution to the problem. Every option for each parameter has a corresponding binary representation, and the parameters are concatenated to form the complete binary string. A new generation is formed at the end of each iteration, consisting of a new population, and this process is repeated until satisfactory convergence criteria are reached, or the maximum number of generations is reached. The algorithm uses only three operators to produce a new population for the next generation -- selection, crossover, and mutation.

There are a number of different ways to handle selection. One method is to rank the population in ascending order by fitness value (after the cost function is evaluated for each individual), and assign probabilities for selection based on each individual's rank. This is called rank weighting. A virtual roulette wheel is spun (by generating a random number between 0 and 1) to determine the members in the new population selected for reproduction.

Once the population for reproduction is selected, the individuals are paired off and "mated" using the crossover process. A crosspoint is selected at random for each pairing, and two new individuals are created by joining the first part of the first string with the second part of the second string, and vice versa. Mutation is the last step in the formation of the population for the next generation, and involves flipping a bit at random in the population from a 0 to a 1 or vice versa mutation is intended to prevent the GA from converging prematurely and helps to maintain a global search. The mutation rate is set at the beginning of the algorithm. Finally, this mutated population becomes the population of the next generation, and the process is repeated until convergence is reached.

Particle Swarm Optimization

Particle swarm optimization (PSO) was chosen as the third optimization method because it is the simplest technique to implement that can deal with discrete options. PSO shares many similarities with genetic algorithms (Kennedy and Eberhart, 1995). Like GAs, the technique works with a set of solutions called a population. Each potential solution is called a particle. Instead of using evolutionary methods, however, the PSO is based on the social behavior of bird flocks or fish schools. Each particle is characterized by a velocity with which it explores the cost function. The velocity and position of each particle are updated after each successive iteration of the algorithm. The particle velocity and position are governed by equations (1) and (2):

[v.sup.new] = [v.sup.old] + [c.sub.1][r.sub.1]([p.sup.localbest] - [p.sup.old]) + [c.sub.2][r.sub.2]([p.sup.globalbest] - [p.sup.old]) (1)

[p.sup.new] = [p.sup.old] + [v.sup.new] (2)

where:

v = particle velocity

P = particle position

[r.sub.1],[r.sub.2] = independent uniform random numbers between 0 and 1

[c.sub.1] = cognitive acceleration constant

[c.sub.2] = social acceleration constant

[P.sup.localbest] = best local solution (best particle in current population)

[P.sup.globalbest] = best global solution (best particle so far in all generations)

The two acceleration constants are usually numbers between 0 and 4. The particle swarm optimization has become popular for the same reasons as the GA, in that it is easy to implement with relatively few parameters to adjust.

EVALUATION METHODOLOGY

In order to test the different optimization techniques, and validate them against each other, three test cases were carried out--these consisted of "small", "medium", and "large" optimizations, described in more details later on in the following section. The accuracy and the efficiency of genetic algorithms are compare to the sequential search and the particle swarm method.

The sequential search technique was tested using BEopt, a software tool available from the National Renewable Energy Laboratory (NREL, 2007). The particle swarm method was implemented using GenOpt (Wetter, 2006). GenOpt is a generic optimization program that can be used to minimize an objective function evaluated by an external simulation program. The genetic algorithm was programmed in MATLAB.

Building Features

The basic features of the residential building used throughout the comparative analysis are shown in Table 1. It consists of a typical detached single-family home commonly used in the Building America Program (Hendron, 2004 and 2006). The economic parameters used in the comparative analysis are shown in Table 2. All the parameters that are not optimized had the fixed values shown in Table 3. The lifetime for all options was set to 20 years.

DISCUSSION OF RESULTS

The optimization results for the small and large test cases are described in details below for the sequential search, genetic algorithm, and particle swarm optimization methods. The results for the medium test case are summarized at the end of this section (refer to Figure 4). The cost function that is minimized is the annual cost of the mortgage plus utilities for the building. The annual mortgage cost consists of the additional cost of building components relative to the reference building, divided by the mortgage period. The costs of buildings components such as wall insulation and window glazing are obtained from RS Means (2007).

Small Optimization

The small optimization test case investigated four parameters, each with four discrete options. The four parameters and associated options are listed in Table 4. The characteristics of the reference building used in the sequential search optimization are in bold.

A full enumeration of all potential cases would require 256 (i.e. [4.sup.4]) simulations. The characteristics of the optimum point found using the sequential search method (BEopt) are shown in Table 5.

In an effort to assess the performance of the genetic algorithm, several cases were run with different parameter values for the GA optimization. The two parameters that affect the GA performance are population size and mutation rate. Since the GA is guided by probabilistic rules, a number of trials were run for different combinations of algorithm parameters, and the results of the trials were averaged in order to identify basic trends.

A summary of the results from the GA runs for the small optimization case are displayed in Table 6. The normalized simulations column is the simulations required by the GA divided by the simulations required by the sequential search method. The percent difference column is the difference between the minimum point found by the GA and the absolute minimum point.

A few general trends are evident from the results. One of the findings is that the number of simulations required increases as the number of mutations increases. This finding makes sense since the objective of mutations is to avoid premature convergence. The number of simulations also increases as the population size increases, since the search begins from a greater number of points. Figure 1 shows the relationship between mutation rate, population size, and simulations required. Figure 2 shows that a linear relationship can be established between the mutation rate, population size, and number of simulations required. Table 7 summarizes the performance of the three methods. The results shown in Table 7 indicate that for the small optimization case, the sequential search technique slightly outperforms the genetic algorithm in robustness, and the efficiency is comparable. This could be due to the fact that the GA starts out by searching from a population of points instead of a single point, so for such a small optimization it loses whatever advantage it may have.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Large Optimization

The large optimization test case looked at all of the available building envelope options in BEopt as outlined in Table 8 (Christensen et al., 2005). A full enumeration of all cases would require 20,275,200 simulations. The characteristics of the optimum point found using the sequential search method are shown in Table 9. A summary of the results from the GA runs for the large optimization case are displayed in Table 10.

Figure 3 shows robustness plotted against efficiency for all of the different cases. At first, the robustness improves in a linear fashion as the number of simuations increases and then levels out at a relatively low percent difference. If an acceptable robustness is within 0.5% of the minimum, the genetic algorithm achieves this threshold with roughly 60% of the simulations required by the sequential search method (used by the BEopt tool).

[FIGURE 3 OMITTED]

There is a cluster of four points around the 60% efficiency mark, corresponding to the higher mutation rates for a population size of 32, and the lower mutation rates for a population size of 48. The performance is comparable.

Table 11 summarizes the performance of the three optimization methods. The results indicate that the GA method outperforms in terms of efficiency the other two optimization methods when the search domain is large.

Comparative Summary

A summary of the genetic algorithm performance relative to the sequential search technique is shown in Figure 4 for small, medium, and large optimization tests.

[FIGURE 4 OMITTED]

Within a robustness of 1%, the GA method is generally more efficient than the sequential search method and save more than 50% of simulation efforts especially for medium and large optimization cases.

SUMMARY AND CONCLUSION

The performance in terms of accuracy and efficiency of the three optimization approaches was compared for various sets of building envelope parameters. The GA method was found to be more efficient than the sequential search and particle swarm optimization when several (more than 10) parameters are considered in the optimization. The advantage of GA method is especially valuable when the cost function becomes more expensive to evaluate (such as using more comprehensive and time consuming simulation (such as using Energy-Plus instead of DOE-2). The efficiency of GAs increases as the size of the search space increases.

A promising area of application of genetic algorithms that could be investigated is the multi-objective optimization Muti-objective optimization looks at more than one cost function, and is useful for illustrating tradeoffs between different cost functions such annual source energy use and annual cost (mortgage plus utilities).

ACKNOWLEDGMENT

The financial support of ICAST to carry this project is acknowledged.

REFERENCES

Andersen, R., C. Christensen, G. Barker, S. Horowitz, A. Courtney, T. Gilver, and K. Tupper, 2004, "Analysis of Systems Strategies Targeting Near-Term Building America Energy-Performance Goals for New Single-Family Homes." NREL/TP-550-36920. November 2004.

Caldas, L. G. and L. K. Norford, 2003, "Genetic Algorithms for Optimization of Building Envelopes and the Design and Control of HVAC Systems." Journal of Solar Energy Engineering 125, 343-3 51.

Caldas, L. G. and L. K. Norford, 2002, "A Design Optimization Tool Based on a Genetic Algorithm." Automation in Construction 11, 173-184.

Crawley, D.B., Lawrie, L.K., Pedersen C.O., Winkelmann, F.C., 2000, "Energy Plus: energy Simulation Program", ASHRAE Journal, Vol. 42, No.4, 49-56.

Christensen, C., S. Horowitz, T. Gilver, A. Courtney, and G. Barker, 2005, "BEopt: Software for Identifying Optimal Building Designs on the Path to Zero Net Energy", Proceedings of ISES 2005 Solar World Congress, Orlando, FL.

Goldberg, David. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, 1989.

Haupt, R. and S. E. Haupt., 2004, Practical Genetic Algorithms, 2nd Edition. JohnWiley and Sons, Inc.

Hendron, R., 2007, "Building America Research Benchmark Definition, Updated December 15, 2006." NREL Technical Report, NREL/TP-550-40968, Golden, CO.

Hendron, R., et. al., 2004, "Building America Performance Analysis Procedures, Revision 1." NREL/TP-550-35567.

Kennedy, J. and R.C. Eberhart, 1995, "Particle Swarm Optimization", Proceedings of IEEE International Conference on Neural Netwroks, Piscataway, NJ, 1942-1948.

MATLAB, Version 7.1.0.124 (R14) Service Pack 3.

Nash, S.G. and A. Sofer, 1996, Linear and Nonlinear Programming, McGraw-Hill, New York, NY.

National Renewable Energy Laboratory, 2007, BEopt. Version 0.8, 2007, NREL, Golden, CO.

Ouarghi, R. and M. Krarti., 2006, "Building Shape Optimization Using Neural Network and Genetic Algorithm Approach." ASHRAE Transactions 112, 484-491.

R.S. Means Building Construction Cost Data, 65th Annual Edition. 2007.

Wetter, M., 2006, "GenOpt Generic Optimization Program User Manual, Version 2.0.0.", Lawrence Berkeley National Laboratory Report, LBNL-54199, Berkeley, CA.

Wright, J., 2002, "Optimization of Building Thermal Design and Control by Multi-Criterion Genetic Algorithm." Energy and Buildings 34, 959-972.

Winkelmann, F.C., B.E. Birsdall, W.F. Bull, K.L. Ellington, A.E. Erdem, J.J. Hirsh, and S. Gates, 1993, "DOE-2 Supplement, Version 2.1E," Technical Report LBL-34947, Lawrence Berkeley National Laboratory, Berkeley, CA.

Daniel Tuhus-Dubrow

Associate Member ASHRAE

Moncef Krarti, PhD, PE

Member ASHRAE

Daniel Tuhus-Dubrow is a graduate student and Moncef Krarti is a professor and associate chair of the Civil, Environmental, and Architectural Engineering Department at the University of Colorado, Boulder, CO.

In order to reduce building energy consumption most effectively, heating and cooling loads due to the building envelope must be addressed early in the design process. Several design parameters can have an effect on these loads, including the shape of the building, wall and roof construction, foundation type, insulation levels, window type and area, thermal mass, and shading. All of these parameters interact and affect the energy performance of the building. Traditionally, this type of analysis has been done with parametric runs using a building simulation engine such as DOE-2 (Winkelmann, 1993) or EnergyPlus (Crawley, 2000). However, varying one parameter while leaving others building envelope features constant can potentially miss important interactive effects, and full combinatory parametric studies are usually infeasible. A better solution is to couple an optimization algorithm to a simulation engine in order to find a minimum for a given cost function including life-cycle cost, annual operating costs, and annual energy use (Wright, 2002; Caldas and Norford, 2003; and Ouarghi and Krarti, 2006).

The objective of this paper is to compare three different optimization techniques to assess their robustness and efficiency for application in building envelope optimization. Robustness is a measure of the algorithm's ability to minimize the cost function, while efficiency is a measure of its speed which is defined in this study as the number of simulations required to reach the minimum cost level. The three methods investigated in this paper include the sequential search used in the Building Energy Optimization or BEopt tool (Andersen, et al. 2004), genetic algorithms or GAs (Goldberg, 1989 and Haupt and Haupt, 2004), and particle swarm optimization or PSO (Wetter, 2006). Each of these methods does not require the calculation of differentials for the cost function, but instead uses discrete values of the cost function to determine the parameter values of the next iteration (i.e. direct search).

DESCRIPTION OF OPTIMIZATION APPROACHES

One approach to classify optimization techniques is by the nature of the problem search space--continuous or discrete. The character of the parameters affecting building envelope optimization lends itself to discrete optimization. A few parameters, such as aspect ratio, orientation, and window area could be considered continuous, but almost all other parameters have a limited number of discrete options. For example, there are a finite number of available wall types for a realistic construction situation. It would be possible to optimize on continuous R-value, but the chance that the optimum solution would correspond to an existing wall type is very small. The same is true of parameters such as window type, foundation type, roof type, and shape.

Continuous optimization methods include the Nelder-Mead simplex method, Hooke-Jeeves method, and various gradient-based approaches (Nash and Sofer, 1996). Because of the discrete nature of the envelope optimization problem, these continuous techniques were not investigated. Discrete optimization methods include global techniques such as genetic algorithms, simulated annealing, tabu search, and particle swarm, as well as direct search techniques such as the sequential search used in BEopt (NREL, 2007). For this study, genetic algorithms were compared to the sequential search, and the particle swarm method was used to validate results.

Sequential Search

The sequential search technique used in BEopt is a direct search method that identifies the building option that will best decrease the cost function after each successive iteration (Christensen et al., 2005). It begins by simulating a user-defined reference building. It then runs a simulation for each potential option one at a time. The most cost-effective option is chosen and used in the building description for the next point along the path. There are a number of discrete options in different categories such as azimuth, aspect ratio, wall type, and ceiling insulation. The most cost-effective option is defined as the one that gives the largest reduction in annual costs for the smallest reduction in source energy use. Annual costs are a combination of mortgage costs (which increase as more expensive energy-efficient options are included) and utility costs. The process is repeated, ultimately defining a path from the reference building to the minimum cost point, and then to a zero net energy building.

Without modifications, this simple algorithm would not reliably find the correct least-cost path, due to the problem of interactive effects between different options. Three special cases have been identified--invest/divest, large steps, and positive interactions (Andersen et al., 2006). The invest/divest case is a result of negative interactive effects. In this case, BEopt removes options which may result in a more cost-optimal point. For example, a highly efficient HVAC system may have been selected as the most cost-effective option at an early point in the process. Later in the search process, however, the improvement of the building envelope may cause the efficient HVAC option to not be cost-optimal, so it is removed from the building design. The large steps case is another example of negative interaction among options. There may be a large energy-saving option that is available at a current point, but is less cost-effective than another option that does not save as much energy. The latter option is chosen, and then the most cost-effective option is again chosen at that second point, which results in a third point. However, it is possible that the original large energy-saving option available at the first point is more cost-optimal than the third point. In order to solve this problem, BEopt keeps track of points from previous iterations and compares them to the current point. If a previous point is more cost-optimal, it replaces the current point. A positive interaction occurs if two options are more cost-effective when present together than they would be if considered separately. An example could be the presence of both large south-facing windows and thermal mass for passive solar heating. BEopt will only find these positive interactions if one of the options is first selected individually. This inability to always identify synergistic options is a potential deficiency with the sequential search method.

Genetic Algorithms

Genetic algorithms (GAs) use the evolutionary concept of natural selection to converge on an optimal solution over many generations GAs (Goldberg, 1989 and Haupt and Haupt, 2004). They differ from traditional optimization methods in a number of areas. First, rather than working with one potential solution at a time, the technique works with a set of solutions called a population. This ensures a global approach to the optimization and helps the GA avoid getting stuck in local minima, which can be a problem with other methods. Second, the GA works with encodings of the parameters, not the parameters themselves. Parameters are traditionally encoded as binary strings, although other encoding options can be used. Finally, GAs use probabilistic methods for determining the parameter values in each successive iteration, rather than deterministic rules. This means that each time a GA is run, the path toward convergence is different, and the end result may be different as well.

Each individual in the population represents a different solution to the problem. Every option for each parameter has a corresponding binary representation, and the parameters are concatenated to form the complete binary string. A new generation is formed at the end of each iteration, consisting of a new population, and this process is repeated until satisfactory convergence criteria are reached, or the maximum number of generations is reached. The algorithm uses only three operators to produce a new population for the next generation -- selection, crossover, and mutation.

There are a number of different ways to handle selection. One method is to rank the population in ascending order by fitness value (after the cost function is evaluated for each individual), and assign probabilities for selection based on each individual's rank. This is called rank weighting. A virtual roulette wheel is spun (by generating a random number between 0 and 1) to determine the members in the new population selected for reproduction.

Once the population for reproduction is selected, the individuals are paired off and "mated" using the crossover process. A crosspoint is selected at random for each pairing, and two new individuals are created by joining the first part of the first string with the second part of the second string, and vice versa. Mutation is the last step in the formation of the population for the next generation, and involves flipping a bit at random in the population from a 0 to a 1 or vice versa mutation is intended to prevent the GA from converging prematurely and helps to maintain a global search. The mutation rate is set at the beginning of the algorithm. Finally, this mutated population becomes the population of the next generation, and the process is repeated until convergence is reached.

Particle Swarm Optimization

Particle swarm optimization (PSO) was chosen as the third optimization method because it is the simplest technique to implement that can deal with discrete options. PSO shares many similarities with genetic algorithms (Kennedy and Eberhart, 1995). Like GAs, the technique works with a set of solutions called a population. Each potential solution is called a particle. Instead of using evolutionary methods, however, the PSO is based on the social behavior of bird flocks or fish schools. Each particle is characterized by a velocity with which it explores the cost function. The velocity and position of each particle are updated after each successive iteration of the algorithm. The particle velocity and position are governed by equations (1) and (2):

[v.sup.new] = [v.sup.old] + [c.sub.1][r.sub.1]([p.sup.localbest] - [p.sup.old]) + [c.sub.2][r.sub.2]([p.sup.globalbest] - [p.sup.old]) (1)

[p.sup.new] = [p.sup.old] + [v.sup.new] (2)

where:

v = particle velocity

P = particle position

[r.sub.1],[r.sub.2] = independent uniform random numbers between 0 and 1

[c.sub.1] = cognitive acceleration constant

[c.sub.2] = social acceleration constant

[P.sup.localbest] = best local solution (best particle in current population)

[P.sup.globalbest] = best global solution (best particle so far in all generations)

The two acceleration constants are usually numbers between 0 and 4. The particle swarm optimization has become popular for the same reasons as the GA, in that it is easy to implement with relatively few parameters to adjust.

EVALUATION METHODOLOGY

In order to test the different optimization techniques, and validate them against each other, three test cases were carried out--these consisted of "small", "medium", and "large" optimizations, described in more details later on in the following section. The accuracy and the efficiency of genetic algorithms are compare to the sequential search and the particle swarm method.

The sequential search technique was tested using BEopt, a software tool available from the National Renewable Energy Laboratory (NREL, 2007). The particle swarm method was implemented using GenOpt (Wetter, 2006). GenOpt is a generic optimization program that can be used to minimize an objective function evaluated by an external simulation program. The genetic algorithm was programmed in MATLAB.

Building Features

The basic features of the residential building used throughout the comparative analysis are shown in Table 1. It consists of a typical detached single-family home commonly used in the Building America Program (Hendron, 2004 and 2006). The economic parameters used in the comparative analysis are shown in Table 2. All the parameters that are not optimized had the fixed values shown in Table 3. The lifetime for all options was set to 20 years.

Table 1. Building Characteristics Parameter Value Location Boulder, CO Floor area 1800 [ft.sup.2] Number of floors 2 Number of bedrooms 3 Number of bathrooms 2 Wall height 8 ft Garage None Roof Flat Table 2. Economic Parameters Parameter Value Electric rate: Marginal 0.08 / kWh Electric rate: Fixed $8 / month Natural gas rate: Marginal $0.80 / therm Natural gas rate: Fixed $8 / month Project analysis period 20 years Elec. Source/Site ratio 3.0 Gas Source/Site ratio 1.0 Table 3. Fixed Options Considered in the Comparative Analysis Category Option Neighbors No neighbors Misc Electric Loads Default Heating set-point 71[degrees]F Cooling set-point 76[degrees]F Thermal mass 1/2" ceiling drywall Infiltration Typical Slab Uninsulated Window areas 20.0% F20 B40 L20 R20 Window type Double clear Air conditioner SEER 10 Furnace AFUE 80% Ducts Typical

DISCUSSION OF RESULTS

The optimization results for the small and large test cases are described in details below for the sequential search, genetic algorithm, and particle swarm optimization methods. The results for the medium test case are summarized at the end of this section (refer to Figure 4). The cost function that is minimized is the annual cost of the mortgage plus utilities for the building. The annual mortgage cost consists of the additional cost of building components relative to the reference building, divided by the mortgage period. The costs of buildings components such as wall insulation and window glazing are obtained from RS Means (2007).

Small Optimization

The small optimization test case investigated four parameters, each with four discrete options. The four parameters and associated options are listed in Table 4. The characteristics of the reference building used in the sequential search optimization are in bold.

Table 4. Options Investigated for Small Optimization Azimuth Aspect Ratio Ceiling Walls 0 0.75 R-30 R-11 batts, 2x4, 16" oc 90 1.00 R-40 R-13 batts, 2x4, 16" oc 180 1.33 R-50 R-15 batts, 2x4, 16" oc 270 1.50 R-60 R-19 batts, 2x6, 24" oc Note: R-1 = 1.0 hr*[ft.sup.2]*[degrees]F/Btu = 0.1761 [m.sup.2]* [degrees]/W.

A full enumeration of all potential cases would require 256 (i.e. [4.sup.4]) simulations. The characteristics of the optimum point found using the sequential search method (BEopt) are shown in Table 5.

Table 5. Sequential Search results for Small Optimization Number of simulations 38 Minimum cost point $1453/year Source energy use 159 MMBtu/yr Azimuth 180 Aspect ratio 1.0 Walls R19 batts, 2x6, 24" oc Ceiling R-30 Note: 1MMBtu = 1.0556 GJ.

In an effort to assess the performance of the genetic algorithm, several cases were run with different parameter values for the GA optimization. The two parameters that affect the GA performance are population size and mutation rate. Since the GA is guided by probabilistic rules, a number of trials were run for different combinations of algorithm parameters, and the results of the trials were averaged in order to identify basic trends.

A summary of the results from the GA runs for the small optimization case are displayed in Table 6. The normalized simulations column is the simulations required by the GA divided by the simulations required by the sequential search method. The percent difference column is the difference between the minimum point found by the GA and the absolute minimum point.

Table 6. Results for GA Small Optimization Population Mutation Simulations Normalized Minimum Percent Size Rate (%) Required Simulations Reached Difference (%) ($/yr) 8 0.5 21.9 57.6% 1468.4 1.06% 8 1.0 25.0 65.8% 1462.1 0.63% 8 1.5 31.6 83.2% 1463.0 0.69% 8 2.0 32.5 85.5% 1460.3 0.50% 12 0.5 37.5 98.7% 1458.0 0.34% 12 1.0 39.2 103.2% 1456.5 0.24% 12 1.5 40.8 107.4% 1467.7 1.01% 12 2.0 48.7 128.2% 1455.0 0.14% 16 0.5 48.4 127.4% 1454.0 0.07% 16 1.0 46.2 121.6% 1456.5 0.24% 16 1.5 54.0 142.1% 1458.5 0.38% 16 2.0 56.3 148.2% 1457.0 0.28% 20 0.5 61.3 161.3% 1454.0 0.07% 20 1.0 57.3 150.8% 1454.0 0.07% 20 1.5 62.5 164.5% 1454.5 0.10% 20 2.0 66.8 175.8% 1454.0 0.07%

A few general trends are evident from the results. One of the findings is that the number of simulations required increases as the number of mutations increases. This finding makes sense since the objective of mutations is to avoid premature convergence. The number of simulations also increases as the population size increases, since the search begins from a greater number of points. Figure 1 shows the relationship between mutation rate, population size, and simulations required. Figure 2 shows that a linear relationship can be established between the mutation rate, population size, and number of simulations required. Table 7 summarizes the performance of the three methods. The results shown in Table 7 indicate that for the small optimization case, the sequential search technique slightly outperforms the genetic algorithm in robustness, and the efficiency is comparable. This could be due to the fact that the GA starts out by searching from a population of points instead of a single point, so for such a small optimization it loses whatever advantage it may have.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Table 7. Performance Summary of the Three Optimization Techniques for Small Optimization Sequential Genetic Particle Search Algorithm Swarm Simulations required 38 45.6 122 Minimum cost reached $1453 $1458 $1453 % difference from optimum 0.0% 0.37% 0.0% % simulations of sequential search - 120% 321%

Large Optimization

The large optimization test case looked at all of the available building envelope options in BEopt as outlined in Table 8 (Christensen et al., 2005). A full enumeration of all cases would require 20,275,200 simulations. The characteristics of the optimum point found using the sequential search method are shown in Table 9. A summary of the results from the GA runs for the large optimization case are displayed in Table 10.

Table 8. List of Possible Values for Shape Parameters Considered in the Optimization Azimuth Aspect Walls Ceiling Thermal Infiltration Foundation Ratio Mass 0 0.67 Rll R30 1/2" Typical Uninsulated Batt, fiberglass ceiling 2x4, drywall 16"oc 22.5 0.75 R13 R40 5/8" Tight 2ft R5 Batt, fiberglass ceiling perimeter 2x4, drywall 16"oc 45 1.00 R15 R50 2x1/2" Tighter 4ft R5 Batt, fiberglass ceiling perimeter 2x4, drywall 16"oc 67.5 133 R19 R60 2x5/8" Tightest 2ft R10 Batt, fiberglass ceiling perimeter 2x6, drywall 24"oc 90 2.5 R21 4ft R10 Batt, perimeter 2x6, 24"oc 112.5 R11 15ft R10 Batt, perimeter 2x4, 16"oc + 1" foam 135 R13 Batt, 2x4, 16"oc + 1" foam 157.5 R19 Batt, 2x6, 24"oc + 1" foam 180 R21 Batt, 2x6, 24"oc + 1" foam 202.5 R19 Batt, 2x6, 24"oc + 2" foam 225 247.5 270 292.5 315 337.5 Azimuth Window Window Area Type 0 20% Double F25 B25 Clear L25 R25 22.5 20% Low-e F20 B40 low SHGC L20 R20 arg 45 18% Low-e F25 B25 std SHGC L25 B25 arg 67.5 18% Low-e F20 B40 high L20 B20 SHGC arg 90 16% Low-e v. F25 B25 high L25 B25 arg 112.5 16% 3-pane, F20 B40 1 HM L20 B20 135 4-pane, 2 HM Kr 157.5 Low-e low SHGC 180 Low-e std SHGC 202.5 Low-e high SHGC 225 Low-e v. high 247.5 270 292.5 315 337.5 Note: 1 in. = 2.53 cm; R-1 = 1.0 hr*[ft.sup.2]*[degrees]F/Btu = 0.1761 [m.sup.2]*[degrees]C/W. Table 9. Sequential Search Results for Large Optimization Number of simulations 1107 Minimum cost point $1186/year Source energy use 98 MMBtu/year Azimuth 180 Aspect ratio 1 Walls R19 batts, 2x6, 24" oc + 2" foam Ceiling R-30 Thermal mass 1/" ceiling drywall Infiltration Tighter Foundation 4ft R10 perimeter, R5 gap Window areas 16.0% F20 B40 L20 R20 Window type Low-e high SHGC arg Table 10. Results for GA Large Optimization Population Mutation Simulations Normalized Minimum Percent Size Rate (%) Required Simulations Reached Difference (%) ($/yr) 16 0.25 119.6 10.80% 1218.4 2.73% 16 0.5 176 15.90% 1214.5 2.40% 16 0.75 273.4 24.70% 1203.7 1.49% 16 1.0 336 30.35% 1203.4 1.47% 32 0.25 360.9 32.60% 1203.5 1.48% 32 0.5 433.2 39.13% 1196.1 0.85% 32 0.75 619.1 55.93% 1189.4 0.29% 32 1.0 652.7 58.96% 1188.1 0.18% 48 0.25 623.2 56.30% 1190.4 0.37% 48 0.5 681.8 61.59% 1190.2 0.35% 48 0.75 799.2 72.20% 1189.5 0.30% 48 1.0 981.5 88.66% 1188.5 0.21% 64 0.25 798 72.09% 1188.4 0.20% 64 0.5 960.8 86.79% 1189.5 0.30% 64 0.75 1151.5 104.02% 1188.7 0.23% 64 1.0 1420.8 128.35% 1188.7 0.23%

Figure 3 shows robustness plotted against efficiency for all of the different cases. At first, the robustness improves in a linear fashion as the number of simuations increases and then levels out at a relatively low percent difference. If an acceptable robustness is within 0.5% of the minimum, the genetic algorithm achieves this threshold with roughly 60% of the simulations required by the sequential search method (used by the BEopt tool).

[FIGURE 3 OMITTED]

There is a cluster of four points around the 60% efficiency mark, corresponding to the higher mutation rates for a population size of 32, and the lower mutation rates for a population size of 48. The performance is comparable.

Table 11 summarizes the performance of the three optimization methods. The results indicate that the GA method outperforms in terms of efficiency the other two optimization methods when the search domain is large.

Table 11. Performance Summary of the Three Optimization Techniques for Large Optimization Sequential Genetic Particle Search Algorithm Swarm Simulations required 1107 649 1545 Minimum cost reached $1186 $1196 $1186 % difference from optimum 0.0% 0.82% 0.0% % simulations of sequential search - 58.6% 140%

Comparative Summary

A summary of the genetic algorithm performance relative to the sequential search technique is shown in Figure 4 for small, medium, and large optimization tests.

[FIGURE 4 OMITTED]

Within a robustness of 1%, the GA method is generally more efficient than the sequential search method and save more than 50% of simulation efforts especially for medium and large optimization cases.

SUMMARY AND CONCLUSION

The performance in terms of accuracy and efficiency of the three optimization approaches was compared for various sets of building envelope parameters. The GA method was found to be more efficient than the sequential search and particle swarm optimization when several (more than 10) parameters are considered in the optimization. The advantage of GA method is especially valuable when the cost function becomes more expensive to evaluate (such as using more comprehensive and time consuming simulation (such as using Energy-Plus instead of DOE-2). The efficiency of GAs increases as the size of the search space increases.

A promising area of application of genetic algorithms that could be investigated is the multi-objective optimization Muti-objective optimization looks at more than one cost function, and is useful for illustrating tradeoffs between different cost functions such annual source energy use and annual cost (mortgage plus utilities).

ACKNOWLEDGMENT

The financial support of ICAST to carry this project is acknowledged.

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Daniel Tuhus-Dubrow

Associate Member ASHRAE

Moncef Krarti, PhD, PE

Member ASHRAE

Daniel Tuhus-Dubrow is a graduate student and Moncef Krarti is a professor and associate chair of the Civil, Environmental, and Architectural Engineering Department at the University of Colorado, Boulder, CO.

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Author: | Tuhus-Dubrow, Daniel; Krarti, Moncef |
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Publication: | ASHRAE Transactions |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jul 1, 2009 |

Words: | 4307 |

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