This study only concerns building optimization methods that support design decisions using a single-criterion formulation, with constraints. The study focuses on methods intended for the early stages of a design process. This means that methods based on detailed building models are not considered.
The design decisions considered in this study are the shape of the building, the amount of insulation used in the building envelope, and the type and relative area of the windows, compared with the area of the external walls. Decisions regarding the design and operation of HVAC systems are not considered.
The performance with respect to energy, indoor environment and economy are considered, but not the performance with respect to the environmental impact. Only non-residential buildings are considered.
Reliability and sensitivity analysis are not considered. Programming specific details, such as choosing programming language, developing graphical user interfaces, including computer-aided design modeling environments, are not considered. Furthermore, de-veloping database management systems for managing the data needed for representing buildings is also not considered.
A mathematical model of decisions
The purpose of this chapter is to provide a description of how optimization can be used for estimating optimum design decisions for systems such as buildings, governed by, for instance, partial or ordinary differential equation. A general description of design decision problems is provided, which aims at simplifying the process of translating such problems into optimization problems. Furthermore, an optimization problem is formulated, which represent a wide range of building design decision problems.
The governing equations for the considered system should ideally be solved analytically, but this is not possible in general. If analytical solutions are unavailable, the governing equations can be solved using numerical methods, which are often implemented in spe-cialized simulation software. It is therefore necessary to consider ways in how to combine simulation software with optimization methods. An interface is described that addresses this issue.
3.1 Building design decision problems
Decisions made during the design process have consequences for the performance of the building. For instance, decisions regarding the shape of the building, the total window area and the amount of insulation material used in the building have consequences for the energy performance of the building, the economy of the building and the quality of the indoor environment.
The design decisions, as well as the performance of the building, can be subjected to requirements. For instance, the above mentioned decisions can be subjected to upper or lower limits, and they can be required to assume certain values. They can furthermore be required to be related to each other in certain ways. The window areas on the north-and south-facing fa¸cades can for instance be required to be equal.
Similar requirements can be applied to the performance of the building. Any measure representing the performance of the building, for instance the annual amount of energy required by the building, or the cost of constructing or operating the building, can be subjected to upper or lower bounds, they can be required to assume certain values, or
they can be required to be related to each other in certain ways.
In addition, the performance can be required to be optimized, meaning that a measure representing the performance of the building can be required to assume its minimum or maximum value. For instance, the decision maker may wish to estimate the set of design decisions that provide the building with the least amount of energy required for heating and cooling, or the building with the smallest construction cost.
Decisions:
The shape of the building The window areas The type of windows The amount of insulation
Consequences:
Energy performance:
U-values
Energy consumption Energy frames Indoor environment:
Daylight utilization Number of hours with overheating Comfort values Economy:
Cost of constructing the building Cost of operating the building
Requirements:
Upper/lower limits Assume certain values
Assume minimum/maximum values
Required relations Assume certain values
Requirements:
Upper/lower limits
Required relations
Figure 3.1: A conceptual illustration of a decision problem.
A conceptual illustration of a decision problem is shown in Figure 3.1. The figure further-more provides examples of decisions, consequences and requirements that are relevant for design decisions for buildings. These concepts are defined in the following.
Decisions refers to the set of variables that the decision maker wishes to determine op-timum values for. It is assumed that the decision maker has full control over them.
Significant decision variables for buildings include (but is not limited to) the amount of insulation used in various building components, the area and type of windows used, the overall shape of the building, expressed in terms of, for instance, the width to length ratio, and the number of floors.
Consequences refers to quantifiable parameters that depend on the decision variables, and that can be used as measures for the performance of the system. Consequences of decisions are therefore also referred to asperformance measures in the following. This concept is similar to the conceptutility function used in operations research.
There are many performance measures that are relevant for buildings, for instance, the energy consumption of the building, the cost of constructing the building, the quality of the indoor environment, structural properties, the environmental impact, etc.
Requirements to decision variables and performance measures can be expressed in many different ways. For instance, they can be required to assume their maximum or minimum value, they can be subjected to upper or lower bounds, they can be required to assume specific values, or they can be required to be related to each other in certain ways. The performance measures can furthermore be required to assume their maximum or minimum value.
The following types of requirements are considered:
Optimality requirements: When an individual performance measure, or a linear com-bination hereof, is required to assume its maximum or minimum value.
Inequality requirements: When an individual decision variable or performance mea-sure, or a linear combination hereof, are subjected to upper or lower bounds.
Equality requirements: When an individual decision variable or performance measure, or a linear combination hereof, is required to assume a specific value.
Feasibility: The decision variables are required to be feasible, meaning that it must be possible to assess the consequences of them. This requirement is relevant when combining simulation software with optimization methods, since it for instance can be used for preventing the optimization algorithm from performing simulations using input that has no physical meaning.
Optimality requirements can in theory also be applied to decision variables, but this possibility does not seem to be of any practical use. Furthermore, the considered inequality and equality requirements involve either decision variables or performance measures, but not both.
Estimating the consequences of design decisions for buildings often involve a large number of constant parameters, such as the location and orientation of the building, climate parameters, prices for construction jobs, and physical properties of building components and materials. The constant parameters are not included in this formulation of decision problems.
3.1.1 An example
Assume that a decision maker is required to estimate the amount of insulation and the window area for two fa¸cades of a building, such that the following requirements are sat-isfied:
1. The construction cost must be as small as possible.
2. The total amount of energy required for maintaining a satisfactory indoor environ-ment must be below a certain level.
3. The utilization of natural light (expressed in terms of the daylight factor) must assume a required level.
4. The window area must be the same for both fa¸cades.
In this case, the decision variables are:
1. The amount of insulation
2. The window area for the first fa¸cade 3. The window area for the second fa¸cade.
The performance measures that are relevant for this decision problem are:
1. The total expenses used for constructing the building
2. The total amount of energy required for maintaining a satisfactory indoor environ-ment
3. The daylight factor.
The first requirement to the building can be interpreted as an optimality requirement, the second one as an inequality requirement, and the third one as an equality requirement.
The last requirement specifies a required relation between two decision variables.
Furthermore, if an optimization algorithm is used for estimating a design decision that solves the decision problem, then the following feasibility requirements ensure that the algorithm never attempts to estimate consequences of decisions that can not be carried out physically:
1. The amount of insulation must be positive
2. The window areas of the two fa¸cades must be positive and below the areas of the corresponding fa¸cades.
The next section concerns the details of how decision problems can be expressed as opti-mization problems.