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An “Ageing” Operator and Its Use in the Highly Constrained

Topological Optimization of HVAC System Design

Jonathan Wight, Yi Zhang

Department of Civil and Building Engineering, Loughborough University,

Loughborough, Leicestershire,

LE11 3TU, UK

Tel: +44(0)1509-620365

J.A.Wright@lboro.ac.uk

ABSTRACT

The synthesis of novel heating, ventilating, and air-conditioning

(HVAC), system configurations is a mixed-integer, non-linear,

highly constrained, multi-modal, optimization problem, with

many of the constraints being subject to time-varying boundary

conditions on the system operation. It was observed that the

highly constrained nature of the problem resulted in the

dominance of the search by a single topology. This paper,

introduces an new evolutionary algorithm operator that prevents

topology dominance by penalizing solutions that have a dominant

topology.

The operator results in a range of dynamic behavior for the rates

of growth in topology dominance. Similarly, the application of

the ageing penalty can result in the attenuation of topology

dominance, or more severely, the complete removal of a topology

from the search. It was also observed that following the

penalization of a dominant topology, the search was dynamically

re-seeded with both new and previously evaluated topologies. It is

concluded that the operator prevents topology dominance and

increases the exploratory power of the algorithm.

The application of an evolutionary algorithm with ageing to the

synthesis of HVAC system configurations resulted in a novel

design solution having a 15% lower energy use than the best of

conventional system designs.

Categories and Subject Descriptors

I.2.8 [Artificial Intelligence]: Problem Solving, Control

Methods, and Search – heuristic methods.

J.6 [Computer-Aided Engineering]: Computer-aided Design

(CAD).

General Terms: Algorithms, Design.

Keywords: Evolutionary Algorithms, Topological

Optimization, HVAC, System Design.

1. INTRODUCTION

The temperature and humidity of the air in the occupied spaces of

commercial buildings, is maintained by “heating, ventilating and

air conditioning” (HVAC), systems. HVAC systems ventilate

buildings by taking in outside air and mixing it with air that has

been re-circulated from within the building. The ventilation air is

further conditioned by heating, cooling, and humidifying

components. The ventilation air maintains the room temperature

and humidity by being supplied to the room at a condition that

offsets thermal loads on the room.

The air conditioning components can be connected together in a

number of ways to produce a viable system configuration. Over

the last 100 years, a number of recognized configurations have

evolved through the process of heuristic design [2]. The

alternative design solutions have been driven predominantly by

the need to limit the capital cost of the systems. This has led to

system configurations that are able to maintain the temperature

and humidity in more than one “control zone” (a control zone

being a collection of rooms that experience similar heat loads).

However, many of the established “multi-zone” systems have a

higher than necessary energy consumption (a “multi-zone” system

is one that simultaneously conditions more than one control

zone).

Concerns over climate change and associated energy use has

renewed interest in the design of HVAC systems for low energy

use. This paper describe an approach to the automatic synthesis of

HVAC system configurations for minimum energy use by an

evolutionary algorithm. A new algorithm operator which is

designed to maintain the power of the search in exploring

alternative system topologies is also described.

1.1 Problem Characteristics and

Optimization Approach

There are three elements to the design of an HVAC system,

1.

the selection of a component set (the choice of type and

number of components);

2.

given the component set, the design of the feasible

topology;

3.

optimization of the system operation (for several

different operating conditions).

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The selection of a feasible topology is dependent on the

component set and similarly, the optimization of system operation

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on the configuration. Hence, conventionally, the optimization of

such systems is conducted in a iterative manner, the iteration

sequence being first, the selection of a component set, followed

by the optimization of the system topology (with the evaluation of

each trail topology requiring the optimization of the system

operation). However, the need for repeated optimization of the

system topology and operation, results in the iteration being

computational intensive. Hence, the approach adopted here [1,7],

is that all three elements of the design problem are optimized

simultaneously. In the experiments described here, however, the

component set has been fixed so that the optimization is

concerned only with the choice of topology and system operation.

Note that HVAC systems are subject to time varying boundary

conditions (namely, the ambient climate and heat gains arising

inside the building), and that the system operation must be

optimized at each boundary condition. In the work reported here,

the system operation has optimized for 9 boundary conditions that

represent a period during the early morning, mid-day, and

afternoon, in each of three seasons (winter, spring or fall, and

summer).

The optimization problem can be described as a mixed-integer,

non-linear programming problem, having non-linear inequality

and equality constraints. The problem is also highly multi-modal

in that there are many alternative system configurations, each

resulting in a local optimal solution for system operation. The

decision variables are derived from the representation of the

system topology, and the system operation (operation being

optimized for all 9 condition simultaneously). The problem

constraints maintain the feasibility of the topology and the system

operation. Finally, the objective function is formed from the

annual system energy use as estimated from the 9 operating

conditions.

The general approach to the problem formulation and solution has

been described previously [1,7], and is therefore only summarized

here.

1.1.1 Decision Variables and Encoding

There are two sets of decision variables, those relating to system

topology

, and those to the system operation,

decision variable

X , can be defined by:

(

XX

=

. The set of

d

X

c

X

)

cdX,

where:

nc

ncccc

nd

ndddd

RxxX

ZxxX

∈=

∈=

) ,....,(

) ,....,(

,,

,,

Each set of variables has been encoded in a separate chromosome,

the topology variables as an integer chromosome, and the system

operation variables as a real vector chromosome. The system

topology is represented by a modified adjacency matrix (system

graph), that has been designed to limit the duplicate

representation of topologies (although it does not eliminate this

effect completely). The topology chromosome has also been

designed to facilitate the implementation of problem specific

recombination and mutation operators [1,7].

The system operation variables are in two groups, those that

define the air–flow rate through the system and those that define

the thermal capacity (output) of the system components.

1.1.2 Objective Function

Formally, we seek to minimize the annual energy use of the

system

. The “annual energy use" is taken as a weighted

average of the system capacity at each of the 9 operating

conditions, the weights being applied in accordance with the

relative proportion of the prevailing climatic conditions. The

system capacity is a function of the capacity (thermal output), of

the air conditioning components, together with the fan power

required to move air through the duct system.

)(Xf

1.1.3 Constraint Functions and Infeasibility

Function

There are three sets of problem constraints, those concerned with

the viability of the system topology, those with the procedure for

evaluating the system performance and thirdly, those concerned

with the system operation. The topology constraints are concerned

with ensuring that that the system graph is connected in a manner

that enables feasible operation. Each topology constraint is

represented by an equality constraint (a binary constraint with a

value∈(0,1)). The aggregated value of topology constraint

violations

has been taken as the average of all topology

constraint violations (the aggregated value then being in the range

0.0 to 1.0).

)(

dtopXc

The system performance evaluation constraints are concerned

with identifying failure in the evaluation of system performance

(in effect, failure to solve the system algebraic equations). The

system operating constraints are concerned with ensuring that all

components operate within their performance limits and that the

system is able to maintain the desired zone conditions (the zone

constraints being on the amount of outside air entering the zone

and the zone temperature and humidity). The operation

constraints are in the form of both inequality and equality

constraints, although a reasonable engineering tolerance has been

applied to the equality constraints. As in the case of the topology

constraints, the constraints relating to the evaluation of system

performance and to system operation, have been aggregated to

have a value in the range 0.0 to 1.0 (0.0 indicating complete

feasibility).

Note that, although there are three groups of constraints (topology

constraints, performance evaluation constraints, system operation

constraints), the system performance can only be evaluated if a

feasible topology exists. Similarly, the system operation can only

be evaluated if the system performance evaluation has been

completed successfully. The constraint space is therefore

discontinuous; this has been reflected in the formulation of an

infeasibility function. If

aggregated constraint values for the topology, system

performance evaluation, and system operating constraints

respectively, then the system infeasibility

, and are the

, is given by:

)(

dtopXc

)(Xcev

)(Xcop

)(Xi

()

()

⎪

⎩

⎪

⎨

⎧

>+

c

>+

=

else),(45 . 0

0 . 0)( if else),(45

X

. 045. 0

0 . 0)(if ),(1 . 09 . 0

)(XcXc

XcXc

Xi

op

evev

dtopdtop

This gives three bands of infeasibility, infeasible topologies result

in an infeasibility value in the range 0.9 to 1.0, failure to evaluate

the system performance, a value in the range 0.45 to 0.9, and

infeasible system operation in the range >0.0 to 0.45. The choice

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of banding is arbitrary and its impact on behavior of the

optimization has not been studied.

Note also, that the constraints on system performance evaluation

and system operation are a function of all variables (albeit an

implicit function of the topology variables), whereas the topology

constraints are only a function of the topology variables.

1.2 The Evolutionary Algorithm

The general form of the evolutionary algorithm used here has

been described previously [1,7]. In brief, a multi-chromosome

genetic algorithm has been developed to solve the topological

optimization problem, with each chromosome having its own

distinct recombination and mutation operators. Ten chromosomes

are active in the experiments reported here, one integer vector

chromosome representing the system topology (system graph),

and nine real vector chromosomes, one for each of the nine

system operating (boundary) conditions.

Several new recombination and mutation operators have been

developed for use with the topology chromosome, the goal of

these operators being to perform effective recombination and

mutation, while attempting to maintain the feasibility of the

system topology. The operators and their effectiveness have been

described previously [1]. In general, standard recombination and

mutation operators were adopted for use with the real vector

chromosomes, although some problem specific operators have

been developed for use in combination with a “hyper-operation”

(where the specific operator used at a given instance is decided

probabilistically) [7].

The fitness assignment is based on the stochastic ranking

algorithm [5], and selection by a binary tournament. A percentage

of the best solutions are treated as elite individuals and copied

directly to the next generation.

The algorithm control parameters used in the experiments

reported here are given in Table 1. The effect of the algorithm

control parameters on search performance has not been studied in

depth, although preliminary experiments where conducted to

confirm that the choice of parameters resulted in acceptable

performance of the algorithm. In particular, although it is

common to use high crossover rates in topological optimization

[3], a 100% probability of crossover for the topology chromosome

was found to be too disruptive, and resulted in a high probability

of generating infeasible topologies; the same behavior was

observed for the high mutation rates with the result that both the

crossover and mutation rates for the topology chromosome are

lower than used in solution of similar optimization problems. The

ageing control parameter relates to the new algorithm operator.

1.3 Topology Dominance

Previously reported results indicated that the evolutionary

algorithm had great potential to synthesize novel HVAC systems

[1]. An analysis of the performance of the approach was

subsequently extended to compare the energy use of the

synthesized systems against that of two conventional systems [7]

(referred to here as the “benchmark systems”). Many established

HVAC system configurations exist, with the choice of which

system is selected for a particular application depending on the

cost constraints on building construction and operation. In this

respect, the first of the benchmark systems is one with a moderate

capital cost, but potentially higher than necessary operating cost

Table 1. Algorithm Control Parameters

Operation Control Parameter Value

Population Size 1,000

Maximum Generations 10,000

Population

Elite Percentage 2%

Tournament

Selection

Number of Individuals 2

Topology

Chromosome

Probability

50%

Recombination

Operation

Chromosome

Probability

100%

Topology

Chromosome

Probability

2%

Mutation

Operation

Chromosome

Probability

10%

Stochastic Ranking

Probability of

Infeasible Solution

Ranking

45%

ageing

Maximum Evaluations

per Generation

20

(energy use); the second system has the higher capital cost, but

for the example building studied, has the lower operating cost

(energy use). In comparison to the performance of the two

benchmark systems, the evolutionary algorithm was able to

synthesize systems with an energy use in the range of the two

benchmark systems. Although this represents a significant

achievement in that the established systems are the result of over

a century of engineering research and development, a

thermodynamic analysis of the benchmark systems [7], suggested

that they were less than optimal in terms of their energy use.

An analysis of the algorithms behavior in respect to the

exploration of the topologies, indicated that at a single topology

could become dominant at an early stage in the search. Figure 1,

illustrates the life-span of every topology synthesized during a

particular search. Each circle in the figure represents a unique

topology, with the horizontal axis indicating the generation in

which it was first synthesized. The vertical axis indicates the life-

span of the topology in terms of the number of times it appeared

in any generation through the search (the total number of times it

was evaluated). Clearly, the search illustrated in Figure 1 has been

dominated by a single topology that first appeared at

(approximately) the 1200th generation, and was subsequently

evaluated more than 7x106 times throughout the duration of the

search. The dominance was further indicated by the topology

accounting for up to 80% of the individuals in a given generation.

It was also concluded that the reason for the dominance was the

implicit dependence of the system performance evaluation and

operating constraints on the system topology. For instance,

consider a solution which is completely feasible and therefore is

likely to have a reasonably high fitness. If the effect of a mutation

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operation on the topology is to move the position of one of the

components in the system, then it is highly likely that the solution

will become infeasible as the system operation will no longer be

valid (the change in position of the component would require a

change in the capacity of one or more components if the system

operation was to remain valid). In particular, the equality

constraints on the condition of the zone air have a high

probability of becoming infeasible when the system topology is

changed.

Figure 1. Topology Dominance

0 200040006000 800010000

0

1

2

3

4

5

6

7

8x 10

6

Number of evaluations

Generation

Dominant topology

2. AN AGEING OPERATOR

The characteristic dominance of the search by a single topology

(Figure 1), is a clear indication that the exploratory power of the

search is limited. Such dominance can be addressed in several

ways; the population could be “partitioned” such that a number of

different topologies where forced to exist; or the spread of

topologies could be maintained by a sharing function with the

niche count derived from the topology chromosome only.

Partitioning the population is likely to have a limited effect as the

number of viable partitions and therefore alternative topologies is

restricted by the population size. Further, the notion of “distance”

between the different topologies as represented by the system

graph (topology chromosome), is not clearly defined and although

the topology chromosome has been designed to limit the duplicate

representation of alternative topologies by numerically different

chromosomes, the duplication is not guaranteed. As a result, a

new algorithm operator, known as “ageing”, was developed to

prevent the dominance of the search by any one topology.

The ageing concept is that any one topology has a maximum

number of evaluations, after which its fitness is reduced (as in real

life, fitness declines with age). The life-span of the topology is

defined here in terms of the number of times the topology is

evaluated, rather than the number of generations that it has

survived. This allows a topology to survive many generations

provided that it does not dominate the population. The ageing

function is given by:

()

[] 0 . 1 , max)()(

'

ge

nqnXjXj

×−×=

where,

ageing,

, the maximum number of evaluations per generation allowed

for a given topology, and

(number of generations to date).

is the fitness of the individual,

the number of times the topology has been evaluated,

its fitness after

the current generation number

)(X

e n

j

)(

'Xj

q

g

n

The effect of the function is that provided the number of

evaluations to date (

), is less than the maximum allowed by

the current generation (

g

nq×

), the fitness remains unchanged. If

however, the number of evaluations exceeds the maximum

allowed, then the fitness function value is increased by a factor

equal to the number of evaluations in excess of the limit

(

ge

nqn

×−

). Note that in this formulation, the higher the fitness,

the lower the function value (

having a function value of 0.0. Since the best solution has a raw

fitness of 0.0, it is unaffected by the ageing operator, although any

other individual in the population having the same topology

would be subject to ageing. This acts to preserve at least one copy

of the current best topology, while limiting its dominance (the

best individual is preserved since we operate with a rank ordered

population, with the top 2% of individuals being copiedto the new

population).

e n

), with the best solution )(Xj

Figure 2, illustrates the effect of the ageing operator on limiting

topology dominance. As for Figure 1, the circles represent unique

topologies, the horizontal axis the generation in which they

appeared, and the vertical axis, the number of times a given

topology was evaluated over the search period. The dashed

diagonal line represents the ageing limit (in this case being 20

evaluations per generation, or 2% of the population per

generation). Topologies with evaluations above the line have been

subject to ageing, whereas those below survived without being

aged.

Figure 2. Effect of Ageing on Topology Dominance

5

0 2000 400060008000 10000

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 10

Number of evaluations

Generation

Topology C (Optimum)

Topology B

Topology A

The three solid lines show the growth in number of evaluations

for three of the topologies (‘A’, ‘B’ and ‘C’). The point at which

the growth line crosses the ageing limit indicates the generation at

which ageing was applied to the particular topology. The three

example topologies exhibit very different behavior in this respect.

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Topology ‘A’ appeared early in the search and since the number

of duplicate topologies in any given population remained

relatively low, the topology survived many generations before

ageing was applied. Once aged, however, further growth in the

number of evaluations was highly attenuated (on the scale of

Figure 2, it would appear that there is no growth, although the

topology may in fact continue to survive with a few copies in

each subsequent generation).

Topology ‘B’ first appeared in mid-search, and remained

“dormant” for many generations, until it experienced a rapid

growth in the number of evaluations and subsequent application

of ageing. The behavior of topology ‘C’ is somewhat different to

that of topology ‘A’ and ‘B’, in that the rapid growth in number

of evaluations occurred soon after the topology first appeared.

Further, unlike topologies ‘A’ and ‘B’, topology ‘C’ remained

active after ageing, with the number of duplicate topologies

appearing in each subsequent generation being in the order of the

2% of the population (equivalent to the ageing limit).

Figure 2, therefore illustrates that the ageing operator allows a

topology to survive provided that it does not dominate the search,

and that the number of topologies having a significant impact on

the search has been increased (from 1 topology in Figure 1, to

many in Figure 2).

Figure 3. Dynamic Effect of Ageing on the Population

Diversity

6000 6500700075008000

0

100

200

300

400

500

600

700

800

900

1000

Topology count

Generation

All topologies

New topologies

Figure 3, illustrates the dynamic effect of ageing on the

population diversity (the vertical axis being the number of

different topologies in a given generation). Each increase in the

number of topologies corresponds to the application of the ageing

operator to a dominant topology. As the fitness of the dominant

topology is penalized, the diversity of the population grows both

in terms of new and previously evaluated topologies.

A detailed analysis of the effect of ageing on the behavior of the

search, or the effect of the ageing parameter

conducted, however, it can be concluded that, the ageing operator:

, has not been

q

•

prevents the long term dominance of the search by any one

topology;

•

allows topologies that have a low growth in number of

evaluations to survive for many generations, whereas

topologies that have a rapid increase in dominance of the

population are prevented from prolonged dominance;

•

and exhibits a dynamic behavior in the re-seeding of the

population with new and previously searched topologies.

3. EXPERIMENTS

Two experiments have been conducted for the same example

problem, one experiment on the performance of the algorithm

without ageing, and second experiment with ageing.

3.1 Example Problem and Benchmark

Systems

The example HVAC optimization problem is for a two zone

building [7]. That is, the HVAC system is required to

simultaneously conditions two separate spaces in the building.

The level of difficultly associated with this “multi-zone” problem

is significantly greater than for the design of a system that serves

only one zone. The difficulty is associated with designing a

system that can condition zones that may be experiencing

different thermal loads and therefore have different conditioning

needs. For example, it is possible that one zone will require

cooling while a separate zone, conditioned by the same system,

may require heating. In the example developed here, the

operation of the system is optimized for 9 different boundary

(load) conditions, within which 5 different system operating

regimes occur (Table 2). As well as requiring the use of

mechanical heating and cooling, under some boundary conditions,

the system should be able to condition one or more zones through

the use of moderately cool outside air alone (which is referred to

as “free cooling” in Table 2).

Table 2. System Operating Regimes

Zone

East West

Heating Heating

Heating Cooling

Heating Free cooling

Free cooling Free cooling

Cooling Cooling

3.1.1 Problem Dimension

The component set selected for the experiments resulted in 21

discrete variables encoded in the integer topology chromosome

[7]. The number of alternative topologies resulting from the

chromosome structure is , which gives a search space of

5.1x10

duplication in the representation of the topologies and so the

number of unique topologies is <

) !(nO

20 topologies (although it is recognized that there is some

). ) !(nO

Each of the 9 boundary conditions results in 11 continuous

operational variables, which gives a total of 99 continuous

variables, encoded as 9 separate real chromosomes. Note that

separate recombination and mutation operators are applied to each

chromosome.

There are 10 equality constraints on the feasibility of the

topology; 1 equality and 9 inequality constraints on the evaluation