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08) 1353–1365

Energy and Buildings 40 (20

Predictive controllers for thermal comfort optimization

and energy savings

Roberto Z. Freire a,*, Gustavo H.C. Oliveira b, Nathan Mendes c

a Pontifical Catholic University of Parana, Rua Imaculada Conceicao, 1155, 80215-901 Curitiba, PR, Brazilb Industrial and Systems Engineering Graduate Program – PPGEPS, Pontifical Catholic University of Parana,

Rua Imaculada Conceicao, 1155, 80215-901 Curitiba, PR, Brazilc Thermal Systems Laboratory – LST, Pontifical Catholic University of Parana, Rua Imaculada Conceicao,

1155, 80215-901 Curitiba, PR, Brazil

Received 21 March 2007; received in revised form 19 December 2007; accepted 25 December 2007

Abstract

The present work is focused on the study of indoor thermal comfort control problem in buildings equipped with HVAC (heating, ventilation and

air conditioning) systems. The occupants’ thermal comfort sensation is addressed here by the well-known comfort index known as PMV (predicted

mean vote) and by a comfort zone defined in a psychrometric chart. In this context, different strategies for the control algorithms are proposed by

using an only-one-actuator system that can be associated to a cooling and/or heating system. The first set of strategies is related to the thermal

comfort optimization and the second one includes energy consumption minimization, while maintaining the indoor thermal comfort criterion at an

adequate level. The methods are based on the model predictive control scheme and simulation results are presented for two case studies. The results

validate the proposed methodology in terms of both thermal comfort and energy savings.

# 2008 Elsevier B.V. All rights reserved.

Keywords: Predictive control; Thermal comfort; Energy saving; HVAC systems

1. Introduction

Thermal comfort has a great influence on the productivity

and satisfaction of indoor building occupants. The majority of

heating, ventilation and air conditioning (HVAC) systems for

thermal comfort are based either on a single temperature

control loop or, in some cases, on a multivariable temperature

and relative humidity control loop. However, as far as thermal

comfort optimization is concerned, other parameters should be

considered in order to provide thermal satisfaction to the

occupants. The interactions between people and the thermal

environment are complex and have been the subject of much

study, therefore, the science of thermal comfort is taking into

account all these considerations [1–5].

As it is discussed further in this paper, thermal comfort in

buildings is a concept which is difficult to define. Over the last

* Corresponding author.

E-mail addresses: roberto.freire@pucpr.br (R.Z. Freire),

gustavo.oliveira@pucpr.br (G.H.C. Oliveira), nathan.mendes@pucpr.br

(N. Mendes).

URL: http://www.pucpr.br/lst

0378-7788/$ – see front matter # 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.enbuild.2007.12.007

decades, a large number of thermal comfort indices have been

established for indoor climate analysis and HVAC control

system design (see [4]) and a quite disseminated one is the

PMV (predicted mean vote), proposed in Ref. [1]. This index

considers environmental variables and individual factors and

the closer to zero the PMV value, the better the occupants’

thermal comfort sensation. Another example of thermal

comfort index is presented in Ref. [6], where a least enthalpy

estimator (LEE) is proposed. Such scheme utilizes the

concept of thermal comfort and an enthalpy-based theory to

provide a suitable setting on the effective temperature line for

the HVAC control system to achieve thermal comfort and

energy saving purposes. The effective temperature (ET*)

indicates the rate at which indoor air temperature can be

increased with decreased relative humidity while maintaining

the same thermal comfort.

Associated to the thermal comfort concept, energy

consumption is another important issue related to HVAC

systems performance. Energy efficiency in buildings is

nowadays an important issue due to the growth of energy

costs, energy consumption and environmental impacts,

especially those related to global warming. According to

R.Z. Freire et al. / Energy and Buildings 40 (2008) 1353–13651354

[7,8], in the United States and other developed countries, about

one-third of all energy use can be attributed to buildings.

Actually, research works on the design of efficient building

are growing fast and, in many countries, projects of comercial

and residential buildings must contain improvements on energy

saving strategies. Existing efforts to improve the energy

efficiency of buildings are discussed in Ref. [9], where a

computational fluid dynamics (CFD) model has been applied to

simulate and evaluate changes on energy demand when

different control strategies and changes in the building material

are used. However, there is a trade-off between energy

consumption and indoor thermal comfort, which relevance

has been progressively attracting the attention of industrial and

academic researches. The aim is to save energy while

maintaining the occupants’ thermal comfort.

In this way, the present paper proposes control strategies for

reducing energy consumption and maintaining acceptable

indoor air conditions related to thermal comfort. Therefore, one

should first include the thermal comfort concept into the control

law.

One idea related to the thermal comfort control approach is

to assume a PMV sensor, so that the PMV is measured and

controlled, i.e., it is the feedback variable of an ordinary closed-

loop structure. In Ref. [10], a fuzzy control law is used and, in

Ref. [11], a PID and a fuzzy controller are proposed and

compared. In Ref. [12], a direct neural network, capable to

predict a comfort level (based on PMV) on specific user by

learning the user’s comfort zone, was used to maintain the

indoor comfort level within the desired range. A different

proposal, but still in the PMV-context, is presented in Refs.

[13,14], where the control algorithm set-point is changed on-

line or off-line as a function of the indoor PMV measurement.

In those works, a fuzzy logic expert system defines set-points

for temperature and air velocity signals of a multivariable

controller. Finally in Ref. [15], a genetic algorithm is used to

find optimal set-point values of an HVAC system in order to

reduce energy consumption, while maintaining thermal

comfort within an acceptable range.

Some works treating thermal comfort as comfort zone are

presented in the sequence. Both [16,17] use a fuzzy control

strategy applied to an HVAC system in order to maintain

acceptable thermal comfort conditions and energy savings. In

Ref. [16], the issue is approached by regulating the airflow of

different zones of a residential building and, in Ref. [17], by

maintaining the zone conditions at preferred set point values

applied to the ventilation system and by using the HVAC plant

when necessary. In Refs. [13,18], based on the fact that, on the

thermal comfort context, it might be enough just setting a

temperature band value instead of having a temperature

regulation control at a precise preset value, comfort zone control

strategies are proposed. Another study related to thermal comfort

and energy saving purposes is presented in Ref. [19] where a

hierarchical control strategy is developed integrating the

classical control techniques with fuzzy logic rules.

In Ref. [20], a first approach using PMV related to the

MBPC (model based predictive control) law [21,22] applied to

an HVAC system, in order to optimize occupants thermal

comfort sensation, is presented. The manipulated variable is the

input for the HVAC device. A system identification procedure

has been performed using linear regression, for a heating

system integrated to a building model, presented in Ref. [23].

The aim is to obtain a model that describes the HVAC system

and building behaviour in a structure that is useful for advanced

control law synthesis such as the MBPC. These models have

been used in Ref. [24] as an introduction to the PMV-based

energy saving non-linear control scheme.

The present paper summarizes and extends the previous

mentioned work line. Here, two strategies to define thermal

comfort for the occupants are addressed. The first strategy

assumed that thermal comfort can be identified by a comfort

zone within a psychrometric chart region. Five control schemes

for improving the thermal comfort by using the two approaches

for measuring thermal comfort are addressed in the following

sections. A characteristic of these schemes is the assumption of

a SIMO (single input, multiple outputs) building system, where

indoor temperature and relative humidity are measured

variables and the single manipulated variable is the power

applied to the HVAC device. All these five schemes use model

based predictive control fundamentals. The first control law

assures that the temperature signal lies within a comfort bound

while minimizes the energy consumption. The second assures

that the temperature signal lies whithin a comfort bound while

optimizing the relative humidity. The third one computes the

optimal value for the input power based on cost function using

temperature and relative humidity optimization. The fourth one

is a PMV-based-predictive control since it calculates the control

signal that optimizes the PMV index in terms of thermal

comfort. Finally, the fifth control algorithm optimizes the

energy consumption and maintains the PMV index inside

acceptable conditions.

This article is organized as follows: in the next section

concepts related to thermal comfort are reviewed, including the

PMV prediction equation. In Section 3 the proposed control

laws are presented and in Section 4 some simulation results are

presented in terms of two case studies: (i) constant values for

metabolic rate and cloth index and (ii) occupants metabolic

rates varying between low and moderate levels. Results are

obtained by using an hourly weather data of a cold week in the

city of Curitiba, Brazil. Finally in Section 5 the conclusions are

addressed.

2. Thermal comfort

Definition and control of indoor conditions for reaching

thermal comfort in buildings are hard to be established. As

thermal satisfaction depends on several parameters, research

works on thermal comfort have been conducted and some

comfort indices have been proposed over the last 50 years. An

example is the thermal comfort index called effective

temperature, which is computed by using the indoor

temperature and relative humidity signals and have been

adopted by ASHRAE [4] for decades. The present work is

focused on the development of control laws based on two ways

to measure thermal comfort. This is discussed as follows.

Fig. 1. ASHRAE comfort zone representation.

R.Z. Freire et al. / Energy and Buildings 40 (2008) 1353–1365 1355

Thermal comfort can be identified by a comfort zone within

a psychrometric chart. The ASHRAE comfort zone is shown in

Fig. 1[25].

On the other hand, a quite disseminated index for evaluating

indoor thermal comfort is the PMV [1], which combines

environmental variables and individual parameters. This index

is based on a theoretical model combined with the results from

experiments with approximately 1300 subjects, and can be

written as a function of four environmental variables

(temperature, T; relative humidity, f; mean radiant temperature,

T rm; air velocity, v) and two individual parameters (metabolic

rate, M; cloth index, Icl), as follows:

PMV ¼ Gðtbs;f; T rm; v;M; IclÞ (1)

3. Model based predictive control laws for thermal

comfort and energy savings

Based on the previous section discussions, the aim of a

thermal comfort based control law is to keep the indoor

hygrothermal conditions inside the comfort zone (see Fig. 1) or

within the PMV-based comfort bounds. Moreover, there is a

trade-off between thermal comfort optimization and energy

consumption. Classical control strategies usually cannot

provide these benefits to a single-zone building. The MBPC

technique makes it possible to minimize both problems when

cost functions and constraints are adequately implemented.

The model based predictive control strategy cannot be

defined as a specific control law, but can be viewed as a strategy

which uses several control methods linked by means of some

common ideas [21,22]. The methodology is related to

controllers that present almost the same structure and

characteristics in common, that is, MPC or MBPC controllers

are defined by the process model, which is obtained for control

purposes, and are characterized by the following four main

steps:

(i) Process modeling: where data from input (manipulated)

Fig. 2. MBPC controllers scheme.

and output (controlled) signals are used to predict the

process behaviour (output prediction) in a future horizon,

defined as prediction horizon (Ny).

(ii) C

ost function definition: where the system closed-loop

performance during the prediction horizon is specified. It is

defined by using the output prediction, the reference signal

and the control effort.

(iii) C

ost function optimization: the cost function is optimized

as a function of the set of future control signals (control

horizon, Nu) to be applied to the process during the

prediction horizon. In this step, constraints in the

manipulated and controlled variable can be added in order

to deal with the system operation constraints, e.g., limits on

the actuators of the HVAC system.

(iv) R

eceding horizon strategy: only the first control signal

computed from the cost function optimization is applied to

the real process and, in the next step time, all the algorithm

is repeated.

MBPC has found place in many industrial applications, as

well as in HVAC control applications, for instance [22,26–28].

The generic scheme for the MBPC controllers is shown in

Fig. 2.

The present work deals with MBPC for thermal comfort.

Two ways for including thermal comfort issues in the control

law are discussed, one based on the psychrometric chart and

another based on the PMV index. These are the main points that

make the controllers presented in this paper different from the

classical MBPC laws. The first approach leads to a linear

representation for the process model. The models are defined by

a MISO (multiple-input/single-output) ARMAX (auto-regres-

sive with exogenous input) representation of a building coupled

to the HVAC system. The manipulated input signal is uðkÞ,which represents the signal applied to a heating/cooling system

and the disturbance inputs are the outdoor temperature

(TEXTðkÞ), outdoor relative humidity (HEXTðkÞ) and total solar

radiation (SEXTðkÞ). Finally, the controlled output signals are

the indoor temperature (yTðkÞ) and indoor relative humidity

(yHðkÞ). The two models are presented as follows:

AiðqÞyiðkÞ ¼ BiðqÞuðkÞ þ BT;iðqÞTEXTðkÞ þ BH;iðqÞHEXTðkÞþ BS;iðqÞSEXTðkÞ þ CiðqÞjiðkÞ (2)

where i ¼ T or H represents the indoor temperature and indoor

relative humidity models. jiðkÞ is a random disturbance signal,

zero mean, variance s. The models parameters are given by the

AiðqÞ, BiðqÞ, BT;iðqÞ, BH;iðqÞ, BS;iðqÞ and CiðqÞ, which have the

;

Fig. 3. Representation of a Wiener structure.

R.Z. Freire et al. / Energy and Buildings 40 (2008) 1353–13651356

following form:

AiðzÞ ¼ 1þ a1q�1 þ a2q�2 þ � � � þ ana q�na ;

BiðzÞ ¼ b0 þ b1q�1 þ b2q�2 þ � � � þ bnbq�nb ;

CiðzÞ ¼ 1þ c1q�1 þ c2q�2 þ � � � þ cnc q�nc

(3)

For the controller internal temperature and relative humidity

predictions (first step of the MBPC scheme) the outdoor climate

is assumed non measurable as it is usually found in several

HVAC facilities. Based on models (2) the canonic and

observable state-space model representation is

xiðk þ 1Þ ¼ AixiðkÞ þ biuðkÞ þ gijiðkÞ;

yiðkÞ ¼ cTi xiðkÞ þ jiðkÞ

(4)

where xi is the system state vector and the matrixes A, b, g and

cT can be written as

Ai ¼

�a1 1 0 . . . 0

�a2 0 1 . . . 0

..

. ...

�an 0 0 . . . 0

0 . . . 0

..

. ...

0 . . . 1

0 . . . 0

2666666666664

3777777777775

; bi ¼

0

..

.

0

b0

..

.

bn

266666664

377777775

(5)

gi ¼

c1 � a1

..

.

cn � an

0

..

.

0

266666664

377777775

; cTi ¼ ½1 0 . . . 0� (6)

By using this model structure, output predictions during the

prediction horizon for the temperature and relative humidity

signals, that is, yTðk þ jjkÞ and yHðk þ jjkÞ for j ¼ 1; . . . ;Ny,

can be made (see for instance [29]).

Moreover, as it is classic in MBPC derivations (see [21,22]),

each set of output predictions can be put in a matrix

representation, i.e.:

yT ¼ HT Duþ ylT; yH ¼ HH Duþ ylH (7)

In the set of Eq. (7), yT and yH are vectors containing the

temperature and relative humidity output predictions. HT and

HH are known as dynamic matrices (constructed using the

model step response parameters). Finally, ylT and ylH are

known as the free output predictions for temperature and

relative humidity models.

The second approach, that is, the PMV one, leads to a non-

linear representation for the process model since its output is

obtained by the non-linear PMV computation. In this case the

model is represented by a series structure made with the linear

model that relates the input signal and the temperature and

relative humidity output signals followed by a non-linear static

operator that transforms these signals in the PMV index. This

kind of structure is known as a non-linear Wiener model, as

shown in Fig. 3.

Following, based on these two approaches, the MBPC

algorithms are proposed. They are further divided into two

groups: (i) the algorithms for promoting better thermal comfort

based on a comfort zone, within the psychrometric chart, where

one of them, while optimizing thermal comfort, reduces energy

consumption of the HVAC system; (ii) the PMV-based

algorithms, which are responsible for maintaining acceptable

indoor thermal comfort conditions, where one of them is also

capable to minimize energy consumption by setting PMV

bounds for the indoor air conditions.

3.1. Algorithms based on a comfort zone in the

psychrometric chart

This section presents thermal comfort optimization MBPC

algorithms.

3.1.1. Algorithm based on setting temperature signal

boundaries and on relative humidity optimization

Here, the psychrometric chart’s comfort zone is used to

define hard bounds on the internal temperature. Therefore, the

control law objective is to find the optimal control signal that

brings the internal relative humidity as close as possible to the

comfort zone (see Section 2). Therefore, the control law is

given by the following optimization problem:

minDuðkjkÞ;Duðkþ1jkÞ;...;DuðkþNu�1jkÞ

Jk

s: to

Duðk þ jjkÞ ¼ 0 8 j ¼ Nu; . . . ;Ny

umin � uðk þ jjkÞ � umax 8 j ¼ 1; . . . ;Nu

yT;min � yTðk þ jjkÞ � yT;max 8 j ¼ 1; . . . ;Ny

(8)

The first constraint set is related to the control horizon, while

the second one is the constraint over the control signal, imposed

by the HVAC device. The interval ½yT;min ; yT;max �, that is, the

constraints over the temperature signal are defined equal to the

comfort zone temperature bounds. Duðk þ jjkÞ is the optimal

control signal variation at time k þ j computed at time k. The

cost function Jk is given by

Jk ¼XNy

j¼1

ðyHðk þ jjkÞ � wHÞ2 þXNu�1

j¼0

l Du2ðk þ jjkÞ (9)

where wH is the relative humidity set-point defined as the mean

relative humidity value in the comfort zone and l is a weighting

R.Z. Freire et al. / Energy and Buildings 40 (2008) 1353–1365 1357

factor. This optimization problem can be rewritten as

minDu

ðyH � wHÞT þ l DuTDu

s: to

ADu � b

HT

�HT

� �Du � yT;max � ylTðk þ jjkÞ

�yT;min þ ylTðk þ jjkÞ

� � (10)

where

A ¼

�1 0 0

1 0 0

�1 �1 0 ...

1 1 0

. . .

266664

377775 ; b ¼

�umin uðk � 1Þumax �uðk � 1Þ�umin uðk � 1Þumax �uðk � 1Þ

..

. ...

2666664

3777775

; Du ¼

DuðkjkÞDuðk þ 1jkÞDuðk þ 2jkÞ

..

.

Duðk þ Nu � 1jkÞ

2666664

3777775

(11)

and

yT;min ¼ ½yT;min yT;min . . . yT;min �T;

yT;max ¼ ½yT;max yT;max . . . yT;max �T;

wH ¼ ½wH wH . . . wH�T(12)

Considering QH ¼ ðHTHHH þ lIÞ and f T

H ¼2ðylH � wHÞTHH, the optimization problem is presented as

minDu

DuTQH Duþ f H Du

s: to

ADu � b

HT

�HT

� �Du � yT;max � ylTðk þ jjkÞ

�yT;min þ ylTðk þ jjkÞ

� � (13)

The optimal solution of (13), given by means of Nu future

values for the control signal variation, is obtained by using a

quadratic programming algorithm. The signal uðkÞ applied to

the process at the time instant k is obtained as:

uðkÞ ¼ DuðkjkÞ þ uðk � 1Þ.

3.1.2. Algorithm based on temperature and relative

humidity optimization

Now, the second control strategy based on the psychrometric

chart is described. The building occupants thermal comfort

sensation is given by defining a trade-off between the internal

temperature and relative humidity errors in relation to pre-

defined set-point values. The set-points for these signals are

defined as the center of a comfort region within a psychrometric

chart. Therefore, the control law is given by the following

optimization problem:

minDuðkjkÞ;Duðkþ1jkÞ;...;DuðkþNu�1jkÞ

Jk

s: to

Duðk þ jjkÞ ¼ 0 8 j ¼ Nu; . . . ;Ny

umin � uðk þ jjkÞ � umax 8 j ¼ 1; . . . ;Nu

(14)

In Eq. (14), the cost function Jk is given by

Jk ¼�XNy

j¼1

ðyTðk þ jjkÞ � wTÞ2 þ rXNy

j¼1

ðyHðk þ jjkÞ � wHÞ2�

þXNu�1

j¼0

lDu2ðk þ jjkÞ

(15)

where r defines the trade-off between the temperature and

relative humidity errors. By stipulating the relative weight

given to these two variables on the thermal comfort. Hence,

the optimization problem is

minDu

ðHT Duþ ylT � wTÞTðHT Duþ ylT � wTÞþ rððHH Duþ ylH � wHÞTðHH Duþ ylH � wHÞÞþ l DuT Du

s: to

ADu � b

(16)

In a matricial representation, it can be rewritten as

minDu

DuTQ Duþ f T Du

s: to

ADu � b

(17)

where Q and f are defined as:

Q ¼ HTTHT þ rðHT

HHHÞ þ lI (18)

f ¼ 2ðylT � wTÞTHT þ 2rðylH � wHÞTHH (19)

As before, problem (14) defines a quadratic programming

problem and the control signal applied to the process is

obtained as: uðkÞ ¼ DuðkjkÞ þ uðk � 1Þ.

3.1.3. Algorithm based on setting temperature signal

boundaries and minimizing energy consumption

In this case, if it is assumed that, to provide thermal comfort,

the temperature signal can vary within a thermal comfort zone.

The objective is to find the optimal value for the HVAC power

signal providing that the temperature signal lies inside limits in

order to maintain acceptable indoor thermal comfort. Through

these limits, it is possible to maintain acceptable thermal

comfort conditions and to reduce energy consumption.

In this problem, the cost function is defined by the energy

consumption. The constraints are equivalent to the ones given

by Problem (8).The last constraint set assures that the

remaining of the indoor temperature lies in an acceptable

Fig. 4. Building model.

R.Z. Freire et al. / Energy and Buildings 40 (2008) 1353–13651358

comfort zone. The control law is presented by the following

optimization problem:

minDuðkjkÞ;uðkþ1jkÞ;...;DuðkþNu�1jkÞ

u2ðkÞs: to

Duðk þ jjkÞ ¼ 0 8 j ¼ Nu; . . . ;Ny

umin � uðk þ jjkÞ � umax 8 j ¼ 1; . . . ;Nu

yT;min � yTðk þ jjkÞ � yT;max 8 j ¼ 1; . . . ;Ny

(20)

Problem (20) can be rewritten as

minDu

Du2ðkjkÞ þ 2uðk � 1jkÞDuðkjkÞs: to

ADu � b

HT

�HT

� �Du � yT;max � ylTðk þ jjkÞ

�yT;min þ ylTðk þ jjkÞ

� � (21)

The optimal solution of Eq. (20), given by means of Nu

future values for the control signal variation, is obtained by

using a quadratic programming algorithm. The signal uðkÞapplied to the process at the time instant k is obtained as:

uðkÞ ¼ DuðkjkÞ þ uðk � 1Þ.

3.2. PMV-based algorithms

In this section, two MBPC algorithms for energy savings and

thermal comfort are proposed. In the first one, the thermal

comfort is addressed by the psychrometric chart and in the

second one it is addressed by the PMV index.

3.2.1. Algorithm based on PMV optimization

Following, a control strategy is described where the building

occupants thermal comfort sensation is given by PMV

calculations. The PMV predictions computed by using the

non-linear model are included in the cost function and as

discussed in Section 2, the closer to zero the PMV value, the

better the thermal sensation. Therefore, the control law is given

by the following optimization problem:

minDuðkjkÞ;...;DuðkþNu�1jkÞ

XNy

j¼1

ðyPMVðk þ jjkÞÞ2 þXNu�1

j¼0

l Du2ðk þ jjkÞ

s: to

Duðk þ jjkÞ ¼ 0 8 j ¼ Nu; . . . ;Ny

umin � uðk þ jjkÞ � umax 8 j ¼ 1; . . . ;Nu

(22)

In this way, the solution of Problem (22) given by means of

Nu future values for the control signal variation is

uðkÞ ¼ DuðkjkÞ þ uðk � 1Þ. Since Problem (22) is a non-linear

optimization problem with non-linear constraints, its solution

can be obtained by using a sequential quadratic programming

algorithm.

3.2.2. Algorithm based on setting PMV signal boundaries

minimizing energy consumption

Following, a control strategy which adopts the building

occupants thermal comfort sensation as boundaries for the

PMV signal is presented. As discussed in Section 3.2.1, this

algorithm is also based on a non-linear structure (the Wiener

structure presented in Fig. 3).

Following ASHRAE standards [4], built environment is

comfortable in terms of thermal comfort when PMV is between

the boundaries�0:5 � PMV � þ0:5. Based on these limits for

PMV, the control law for this case can be presented as follows:

minDuðkjkÞ;Duðkþ1jkÞ;...;DuðkþNu�1jkÞ

u2ðkÞs: to

Duðk þ jjkÞ ¼ 0 8 j ¼ Nu; . . . ;Ny

umin � uðk þ jjkÞ � umax 8 j ¼ 1; . . . ;Nu

yPMV;min � yPMVðk þ jjkÞ � yPMV;max 8 j ¼ 1; . . . ;Ny

(23)

where yPMV;min and yPMV;max represent the boundaries for PMV

associated to the thermal comfort conditions. In this problem,

the cost function is related to the energy consumption, as the

one for Problem (20). The third constraints set assures that the

PMV signal will be between the specified boundaries providing

thermal comfort.

This problem assures, at each sample time, an optimal

control signal, minimizing energy consumption while maintain

the PMV inside the thermal comfort range. Similarly to

Problem (8), the optimal control signal can be obtained by a

sequential quadratic optimization algorithm.

4. Simulation results

In order to evaluate the performance of the previous thermal

comfort MBPC control strategies presented in this paper, a

single-zone building used by the BESTest methodology [30], as

shown in Fig. 4, has been adopted.

In this way, the BESTest single-zone building has been

simulated using PowerDomus [31], which is a whole-building

hygrotermal and energy simulation software based on a lumped

formulation where in the energy balance are considered

sensible and latent conductive heat transfer loads, convective

heat transfer, long- and short-wave radiation, infiltration,

ventilation and loads related to the HVAC system among other

parameters. Therefore, the building model presented in Fig. 4,

implemented in PowerDomus, acts as an actual building for the

identification and control experiments, where the external

climate is captured from a TRY (test reference year) weather

data (external temperature, TEXT; external relative humidity,

Fig. 5. External temperature, relative humidity and total solar radiation for the

simulation period in Curitiba, Brazil.

R.Z. Freire et al. / Energy and Buildings 40 (2008) 1353–1365 1359

HEXT; total solar radiation, SEXT) of the city of Curitiba, Brazil.

By means of least-square based identification procedures [32],

the following equations for the combined HVAC (a heating

system of 5 kW maximum power) and building envelope

model under analysis were obtained [23,33].

ð1� 0:97000q�1ÞyTðkÞ

¼ þð0:08395q�1Þu0ðkÞ þ ð0:02527q�1ÞTEXTðkÞ

þ ð0:20340q�1ÞHEXTðkÞ þ ð�0:07245q�1ÞSEXTðkÞ

þ ð1þ 0:67670q�1ÞjTðkÞ(24)

ð1� 0:96920q�1ÞyHðkÞ

¼ þð�0:002107q�1Þu0ðkÞ þ ð0:0002751q�1ÞTEXTðkÞ

þ ð0:01759q�1ÞHEXTðkÞ þ ð0:003342q�1ÞSEXTðkÞ

þ ð1þ 0:72390q�1ÞjHðkÞ(25)

In Eqs. (24) and (25), the units for temperature is (�C) and

the sampling time is 1 min. In these models u0ðkÞ is given by:

u0ðkÞ ¼ uðkÞ þ uPðkÞ (26)

where uðkÞ is the manipulated signal provided by the control

law and uPðkÞ is a disturbance representing the heating gener-

ated by the building occupants.

Now some closed-loop results are presented by using two

case studies. In all cases, the controller is turned on at the 0-th

hour of the third day (time equal to 48 h—two days of warm-up

simulation period) and the simulation period is 1 week (168 h).

The controller parameters are: Ny ¼ 10 and Nu ¼ 1. It is

assumed here a heating system, therefore, uðkÞ is constrained to

the limits [0,5]. The external climate data adopted during the

closed-loop simulations, i.e., temperature, relative humidity

and total solar radiation, are shown in Fig. 5. For the first case

study, the results are related to the analysis of the properties of

the control schemes presented in Sections 3.1 and 3.2, where

the main objective is to maintain the best thermal comfort

conditions with or without energy consumption reduction. In

the sequence, i.e., in the second case study, only the control

schemes related to thermal comfort and energy savings are

discussed (see Section 3.2) and the effect of changes on the

occupants’ metabolic rate and cloth index is also analyzed.

4.1. Case study I—controller performance analysis

The control strategies presented in Sections 3.1 and 3.2 are

evaluated in terms of temperature, relative humidity, PMV and

energy consumption. In this case, two occupants are considered

inside the building environment, where their clothing index and

metabolic rate are constant equal to 0.66 clo and 1.20 met.

These values are related to usual office clothing and metabolic

rate [4].

Following, in Figs. 6–10 , results for the five control

strategies are presented and discussed. Figs. 6(Solution 1) to

8(Solution 3) are related to the strategies based on the

psychrometric chart comfort zone and Figs. 9(Solution 4) and

10(Solution 5) are related to the ones based on the PMV index.

In Fig. 6, the controller is tuned assuming that the temperature

comfort bound in the psychrometric chart is (22 �C, 26 �C) and

the median value of the comfort bound is 0.5 (or 50%).

Following Problem 8, these parameters set the choice of wH and

[yT;min , yT;max ]. It can be noticed that the performance

requirements are satisfied, the constraints are satisfied and

the relative humidity tends to the set-point.

In Fig. 7(Solution 2), the controller set-points are made

equal to 24 � C and 0.5, i.e., the medium value of the comfort

zone. The trade-off r between the temperature and relative

humidity errors is set as 100, which explains why the

temperature error is lower than the one for the relative

humidity signal. PMV index obtained for this case is better than

the one found in the previous solution since the PMV is very

sensitive to the temperature signal.

In Fig. 8(Solution 3), the results were obtained by a

controller that has, additionally, considered energy con-

sumption minimization. The bounds on the temperature

signal are set as the same as the one presented in the first

solution. Table 1 presents the energy consumption of these

three solutions based on the psychrometric chart. As

expected, it can be seen that the lowest energy consumption

for the controllers based on a comfort zone within the

psychrometric chart is obtained.

Figs. 9 and 10 present the results obtained by using the

controller based on the PMV index. The optimal PMV index is

obtained by the controller presented in Fig. 9(Solution 4). It can

be seen by Table 1 that this is the case where the PMV is closer

to zero.

Fig. 6. Case study I—indoor temperature, indoor relative humidity, PMVand control signal for the algorithm based on setting temperature signal boundaries and on

relative humidity optimization.

R.Z. Freire et al. / Energy and Buildings 40 (2008) 1353–13651360

Fig. 10(Solution 5) presents a controller that includes energy

reduction minimization. The constraints over the PMV index

are set equal to [�0:5; 0:5]. As expected, the energy

consumption is lower than in the previous solution. By

comparing the two energy saving solutions (i.e. Figs. 8 and 10),

Fig. 7. Case study I—indoor temperature, indoor relative humidity, PMV and c

optimization.

it can be noticed that for the latter, the PMV based energy

reduction strategy could take advantage of this, particularly,

individual and external climate condition, and provide an

additional energy saving in relation to the other analogous

controller (Solution 3).

ontrol signal for the algorithm based on temperature and relative humidity

Fig. 8. Case study I—indoor temperature, indoor relative humidity, PMV and control signal for the algorithm based on setting temperature signal boundaries and

minimizing energy consumption.

R.Z. Freire et al. / Energy and Buildings 40 (2008) 1353–1365 1361

4.2. Case study II—varying metabolic rate and

clothing index

In this section, only the controllers based on energy

consumption minimization are considered, which perfor-

mance is analyzed for variations on metabolic rate and clothing

index.

Fig. 9. Case study I—indoor temperature, indoor relative humidity, PM

In Figs. 11 and 12, the results are presented in terms of

indoor temperature, indoor relative humidity, PMV and energy

consumption. Variations on the metabolic rate and clothing

index are presented in Table 2, where during the day (from 7 am

to 10 pm), the occupants’ metabolic rate is equivalent to a

physical activity of a person working in an office, while at night

(from 10 pm to 7 am), it is assumed the occupants are sleeping.

V and control signal for the algorithm based on PMV optimization.

Fig. 10. Case study I—indoor temperature, indoor relative humidity, PMV and control signal for the algorithm based on setting PMV signal boundaries minimizing

energy consumption.

R.Z. Freire et al. / Energy and Buildings 40 (2008) 1353–13651362

By analyzing the results in terms of temperature, it can be

noticed that the controller that uses the PMV as a comfort

parameter (Fig. 12) increases the indoor temperature during

the period when metabolic rate is low in order to provide a

better thermal comfort sensation. When the metabolic rate

becomes higher, the internal temperature decreases propor-

Fig. 11. Case study II—indoor temperature, indoor relative humidity, PMV and con

minimizing energy consumption.

tionally in order to provide adequate thermal sensation to the

occupants. On the other hand, analyzing the results presented

in Fig. 11, the temperature signal does not vary due to

changes on the metabolic rate and/or clothing index, which

affects the indoor thermal comfort. The energy consumption

are 114.81 kWh and 202.26 kWh for the psychrometric

trol signal for the algorithm based on setting temperature signal boundaries and

Fig. 12. Case study II—indoor temperature, indoor relative humidity, PMVand control signal for the algorithm based on setting PMV signal boundaries minimizing

energy consumption.

R.Z. Freire et al. / Energy and Buildings 40 (2008) 1353–1365 1363

chart and for the PMV-based strategy situations. The PMV-

based strategy presents a higher consumption, however, due

to its adaptation feature that can provide a better indoor

thermal comfort.

Fig. 13. Variations on the indoor temperature, indoor relative humi

Fig. 13 shows the situation where no adaptation on the PMV-

based controller is taken into account. This highlights that the

main advantage of this controller is the possibility of

considering a global model for thermal comfort, regardless

dity and PMV when different personal parameters are adopted.

Table 2

Case study II—variations on metabolic rates and clothing index

State Icl (clo) M (met)

I 0.66 1.20

II 0.45 0.80

Table 1

Energy consumption for the 5 days simulation period and quadratic sum of the

PMV for the controllers without energy consumption minimization

Solution Energy consumption (kWh)P

PMV2

1 222.47 1:6378� 103

2 192.72 1:1625� 102

3 110.78 –

4 174.77 10.6897

5 98.67 –

R.Z. Freire et al. / Energy and Buildings 40 (2008) 1353–13651364

the changes of individual parameters and take advantage of

such information when available.

5. Conclusions

The indoor thermal comfort control problem in a single-zone

building equipped with an HVAC system has been analyzed.

Five MBPC algorithms to optimize room air conditions focused

on thermal comfort and/or energy savings by using an only-

one-actuator system associated to heating/cooling equipment

have been presented.

Some of the proposed algorithms were based on defining

thermal comfort by a comfort zone within the psychrometric

chart and some by using the PMV index. They were also

classified between those that consider only thermal comfort

optimization and those that additionally provide energy

savings. All control strategies that have been presented reached

their goals in order to provide better indoor thermal comfort

conditions to the occupants.

The importance given to the temperature set-points in this

work is related to its highly correlation to the thermal comfort

sensation. The PMV-based controllers are well adapted in

situations where at least a rough estimator of the occupants

activity can be obtained.

Simulation results of two case studies, by using the Curitiba

weather data file, have shown that the control algorithms

presented in this paper can simultaneously promote thermal

comfort and energy consumption reduction maintaining 100%

of the time in the comfort zone. Due to the ability of the PMV

controller to adapt to the individual parameters, it has been

shown that it can provide a better global performance in terms

of both thermal comfort control and energy consumption.

Acknowledgments

The authors thank CAPES (Coordenacao de Aper-

feicoamento de Pessoal de Nııvel Superior) of the Secretariat

of Education of Brazil and FINEP (Financiadora de Estudos e

Projetos) and CNPq (Conselho Nacional de Desenvolvimento

Cientııfico e Tecnologico) both of the Secretariat of Science

and Technology of Brazil for support of this work.

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