Model Predictive Control of thermal comfort as a benchmark for controller performance

Ion Hazyuk a*, Christian Ghiaus b, David Penhouet c

a INSA-Toulouse, Université de Toulouse, Institut Clément Ader, Toulouse, France b INSA-Lyon, CETHIL, UMR5008, F-69621, Villeurbanne, France c CSTB (Centre Scientifique et Technique du Bâtiment), 84 avenue Jean Jaurès, 77421 Marne-la-Vallée, France

Abstract

Assessing controller performance in normal operation needs reproducible conditions and comparison with the best possible result. Tests in emulation are reproducible. Model Predictive Control (MPC) gives the best possible performance when the future inputs and the model of the process are known. When the benchmark is used for building energy management, the cost function of MPC becomes a linear programming problem with constraints given by the comfort. In emulation, the model of the building used in MPC may be obtained by gray-box parameter identification, using signals which excite all the modes of the complete model. The proposed benchmark was used to test a PID and a scheduled start PID-based energy management system. During the test periods, MPC benchmark always outperformed the PID controllers. It reduced the occupants discomfort by up to 97%, the energy consumption by up to 18%, and the number of on-off cycles of heat pump by up to 78%.

Keywords: thermal control; nonlinear Model Predictive Control (NMPC); energy efficiency; intermittently occupied buildings; optimal heating restart time; multi-source multi-consumer system.

1 Introduction Building thermal behavior is characterized by, generally, great inertia and it is strongly

influenced by the weather and occupation type. Most often, the occupation is intermittent, which implies variable indoor temperature set-point. Since space heating is responsible for over 50% of the total energy consumption in residential and tertiary sectors [1], thermal control has an important impact on energy consumption. Nevertheless, energy savings must not affect the comfort during the occupied periods because the cost of people discomfort is much higher than the operational cost of the building [2].

Several surveys on the current building thermal control strategies have shown that these are, generally, room thermostats or thermostatic valves on radiators [3-5]. In the best case, radiator valves are driven by PID controllers to cope with room overheating. Although these controllers are omnipresent in the field, they are not specifically designed or adjusted to minimize the energy consumption. Furthermore, their feedback loop introduces a lag between the indoor temperature and the set-point, which affects negatively the comfort.

* Corresponding author: mechanical department, 135 avenue de Rangueil, F-31077

Toulouse, France. Tel.: +33 (0) 5 67 04 88 23. E-mail address: ion.hazyuk@insa-toulouse.fr (I. Hazyuk)

Ideally, a building thermal controller should take maximal advantage of the weather and the building inertia in order to provide the comfort with minimal energy consumption. Model Predictive Control (MPC) is regarded as being one of the most suited for thermal control of intermittently occupied buildings. It inherently minimizes criteria that may be related to discomfort and energy; it may include weather forecast, future set-point schedule and constraints in the optimization. In simulation studies, MPC outperformed the other tested controllers in terms of energy consumption and comfort criteria [6-9]. Field tests confirmed the trend obtained in simulations [10-16].

Three different approaches can be used to test an innovative algorithm – simulation, in-situ test and emulation. The advantage of simulation models is to obtain repeatable test conditions which allow testing and comparing different controllers. If the same model is used for simulation and for the MPC, there are no modeling errors in the MPC algorithm and, therefore, the robustness of the controller to modeling errors cannot be assessed. On the other hand, in-situ tests permit to capture model uncertainties which usually are not considered in low-order simulation models. However, beside the fact that in this case testing is much more expensive, it is difficult to have repeatable test conditions in order to make valid comparisons of different controllers. Therefore, it is difficult to quantify the improvements of MPC control structure over the other ones.

Testing in emulation combines the advantages of simulation and in-situ experiments. The principle is to simulate the building and its afferent heating system by a detailed model which was calibrated on existing buildings and to implement the model on a process computer which has electrical signals (0 – 10V) as inputs and outputs. Then, the physical controllers are tested by connecting them through electrical signals to the emulated building running in real time, i.e. one emulation second equals one real-time second. In these conditions, the tested controller will make no difference between a real building and a simulated one. The advantage is that the real controller is tested in reproducible conditions.

This allows comparing different controllers operating in the same conditions but do not answer the question of how a controller behaves in comparison with the best possible performance obtainable in the given situation. This paper suggests that MPC with known evolution of the future inputs and high accuracy of the internal model of the process can be used as a benchmark or a yardstick to assess the performance of any controller. The proposed benchmark uses a new cost function for MPC, recently introduced in [17], which ensures the thermal comfort with minimal energy consumption. The cost function is formulated so that MPC becomes a linear optimization problem solved by Linear Programming (LP). This paper focuses on buildings with hydronic heating systems. Due to the hydronic systems, the control problem is more delicate because the overall model is nonlinear. However, it is possible to represent the building by a Hammerstein model. Thereby, physical knowledge is used for the identification of the nonlinear part of the model and linear least squares identification method for the linear part of the model. Then, the inverse of the identified nonlinear characteristic is used to remove the nonlinearity from the control loop. We have already presented the control algorithm [17] and the identification procedure [18] used in this paper. They were tested by simulating only the building (without its heating system) by using a low order model. This paper presents results by using a real-time emulator of the building and of its heating system.. The elements of our previous work which are necessary in order to explain the findings presented hereafter are resumed in §2-4.

The proposed MPC is embedded in a Building Energy Management System (BEMS), tested and compared in emulation with two classical PID based management systems. For the assessment of the overall performance, the heating system that prepares the hot water is also considered in the tests. The controller adjusts the energy flow from the maximum available

power down to zero. Therefore, the heating system has an important impact on the overall control performance.

2 Control problem formulation

2.1 Requirements in building thermal control Usually, the requirements in building temperature control refer to two aspects: thermal

comfort and energy savings. The comfort requirements are generally imposed as a temperature range, defined by an upper and a lower bound, within which should lie the indoor temperature. This range has different width for occupied and unoccupied periods which changes instantly (Figure 1). During the occupied period (occupancy), this temperature range will be called comfort zone and during the unoccupied period – safety zone.

Since the building has a rather slow dynamics, the heating must be restarted in advance so that the indoor temperature does not remain below the comfort zone at the beginning of the occupancy. If, due to the heating system, the temperature reaches the lower bound of the comfort zone before the occupancy starts, than excessive energy is consumed. Consequently, an optimal heating restart generates a temperature variation that attains the lower bound of the comfort zone just around the beginning of the occupied period (Figure 1).

In order to increase the indoor temperature, the heating system consumes energy. Therefore, it can be admitted that a minimal energy control strategy acts against temperature rise and will tend to keep it at the lower acceptable bound. Thus, since in this paper building cooling is not considered, for comfort requirements it is sufficient to define only the lower bound of the comfort and of the safety zones. The minimal energy control strategy naturally constraints the temperature for the upper bound.

Figure 1. Comfort requirements and possible scenarios for indoor temperature

The economic criterion can be formulated for fixed or variable energy price as:

dttΦtJt

e )()( (1)

where t represents the energy price and tΦ is the heat flux injected into the building. The

criterion eJ is related through the efficiency of the heating system to the energy bill. When

the energy cost is constant, minimizing eJ results in minimizing the energy consumption. In

this paper, only energy consumption minimization is considered ( 1t ). Thus, the second

performance requirement in building thermal control is to minimize the criterion eJ from the

relation (1).

2.2 A cost function for thermal comfort with minimal energy consumption MPC calculates a command sequence which minimizes a cost function over a finite

future time horizon. The performance, which is embedded in the cost function, is predicted by using the system model, future variations of the set-points and, if available, future variations of disturbances. Since the system model is indispensable, there are several MPC algorithms that are built around different model representations. We can find algorithms that use artificial neural networks [19, 20], genetic algorithms [21], fuzzy logic [22, 23] or classical formulations using transfer functions, state space or convolution models [24, 25]. Building models are naturally defined in state-space representations and discrete-time MPC is easier to understand than the continuous one [24]. Therefore, in this paper it is used the discrete-time MPC algorithm based on state-space model.

The goal of ensuring thermal comfort with minimal energy consumption can be mathematically formulated as the minimization of the heat flux integral subject to constraints on the indoor temperature. The indoor temperature should be above the lower bound of the comfort/safety zone. Physical limitations of the heating system also should be considered in the optimization. Therefore, additional constraints are imposed on the heat flux, which should be in the feasible range of the heating system. Considering that the heat flux is the manipulated variable, u , and the indoor temperature is the system output, y , the new MPC

problem formulation is [17]:

y

u

N

i

Niikiky

Niuikutosubject

ikukJminimizeu

1),()(ˆ

1,)(0:

)()(:

min

max

1

(2)

where min is the lower bound of the comfort/safety zones and maxu is the maximal power of

the heating system, �� is the predicted output for the next �� time samplings, whose

calculation is detailed in §2.3 and �� is the future command horizon. The above problem formulation is subject to two antagonist constraints (minimal

constraint on the output and maximal constraint on the input). Therefore, from a mathematical point of view, the solution existence is not always guaranteed. However, it is assumed that the heating system is correctly sized, i.e. the maximal power suffices to maintain the indoor temperature above the lower bound of the comfort zone in the worst-case situation (extreme outdoor temperature). In order to guarantee the solution existence under dynamic constraints, introduced by the inertia of the building, the prediction horizon �� should be longer than the

unoccupied period. Thus, at the end of the occupation period, the optimization problem captures also the beginning of the next occupation period. If it detects that at full power the time needed to recover the building from set-back will be longer than that of the unoccupied period, it decides not to turn off the heating system at all during inoccupation. Consequently, the solution is guaranteed thanks to the assumption of sufficient size of the heating system. Since the command is constrained to be always positive, the cost function (2) is positive-definite.

2.3 Solving the optimization problem In order to find the command sequence that meets the requirements from §2.1, MPC

must solve the optimization problem from the equation (2). Given that the problem has a

linear formulation and supposing a linear system model†, it can be solved by Linear Programming (LP). In order to solve it by LP, the control problem (2) needs to be formulated in the following canonical form:

bMu

uc

:

:

tosubject

nimizemi T

(3)

where u is the vector of variables, which in our case is the command sequence, c , b and M are vectors and matrix of known coefficients.

To formulate the optimization problem (2) under the LP canonical form (3), the estimation of the future output y from (2) should be eliminated. The estimation of the future

output can be obtained from the system model. Considering that the system can be represented by a linear discrete state-space representation:

)()()()(

)()()()1(

21

21

kkukky

kkukk

wDDxC

wBBxAx (4)

where u represents the manipulated input, i.e. the heat flux and w represents the measurable but uncontrollable inputs (disturbances), i.e. outdoor temperature and solar radiation, the

estimation of the future yN outputs can be expressed in matrix form as:

dΨuΨxFy 21)(ˆ k (5)

where the matrices 1ΨF, and 2Ψ are functions of constant parameters of the model (defined

in Appendix A), and the vectors are:

Ty

TTTT

T

y

T

y

Nkkkk

Nkukukuku

Nkykykyky

)1()2()1()(

)1()2()1()(

)(ˆ)3(ˆ)2(ˆ)1(ˆˆ

wwwwd

u

y

(6)

It can be noted in equation (5) that the estimation of the future outputs depends only on

the actual state, )(kx , and actual and future input, )1()( yNkuku , and disturbances,

)1()( yNkk ww . The current state can be estimated by a Kalman filter, while the future

disturbances usually may be obtained from weather forecast. By defining the lower bound of the comfort/safety zone in vectorial form:

TyNkkkk )()3()2()1( minminminminmin y (7)

and replacing in (2) the estimation of the future outputs by the relation (5), the optimization problem in LP canonical form becomes:

† Though the building model is nonlinear, it will be shown in §4 that the nonlinearity can be eliminated

from the control loop and the controller design can rely only on the linear part of the model

min2

max

1 )(

:

:

ydΨxF

c

0

u

Ψ

I

I

uc

k

utosubject

nimizemi T

(8)

where c and I are unitary vector and identity matrix of proper size, respectively. Thus, a control law, which maintains the indoor temperature above the lower limit with minimal energy consumption, is obtained by using LP to solve the optimization problem from the equation (8).

3 Dynamic modeling of the building MPC techniques require a dynamical model of the system in order to estimate the future

output. Although there are nonlinear MPC techniques to deal with nonlinear systems [26], generally, linear and low-order models are preferred. In control applications, black-box and lumped capacity models are encountered more often. However, despite the existence of a solid theoretical background for system identification [27], in practice, model identification for real buildings remains difficult. Generally speaking, persistent inputs with large frequency spectrum are essential for good quality identification of the models [28]. Best candidates are pseudo-random binary sequences (PRBS). In inhabited buildings, PRBS of usable length and amplitudes are practically prohibited by the comfort norms. Additionally, the presence of numerous unmeasurable disturbances may reduce considerably the quality of the identified model. However, if the building is emulated by a highly detailed model, which was experimentally validated, there are no limitations on excitation characteristics.

The low-order model of the building is obtained in two stages. First, a model structure is obtained from a lumped capacity representation of the building via thermal networks and then the parameters of the model are obtained by experimental identification.

3.1 State-space modeling Low-order building models used in control are most often obtained by solving a linear

thermal network representation of the building with lumped parameters [14, 18, 29-36]. In this paper it is considered a single-zone building, whose equivalent linear thermal network is represented in Figure 2. The envelope is represented by a 2R-C network, with its capacity lumped in wC and its conductive resistance divided in two halves, 2wR , and placed on each

side of the capacity. The convective resistances between the envelope and the indoor/outdoor air are represented by ciR and coR , respectively. The thermal capacity of the internal mass is

lumped in aC and the ventilation and infiltrations are modeled by vR . The active elements of

the circuit from Figure 2 represent the outdoor air temperature, solar radiation on the building envelope and the internal heat rate. The internal heat rate is the sum of internal free gains (from occupants, electrical appliances, solar radiation through windows, etc.) and heat from heating terminals. The outdoor air temperature is modeled by an ideal source of variable temperature o while solar radiation and internal gains are modeled by ideal sources of

variable flux, sΦ and gΦ respectively.

Figure 2. Equivalent linear thermal network representation of a low-order thermal model of a building

The model output is the indoor temperature; the considered inputs are the outdoor

temperature, solar radiation and internal heat flux. Since the internal heat flux is a sum of multiple contributors, the heat coming from radiators is considered to be the manipulated input (command), while the free gains are considered to be unmeasurable disturbances because they are uncontrollable and usually unmeasurable. Outdoor temperature and solar radiation are also uncontrollable sources but they can be measured. Therefore, they are considered to be measurable disturbances. The building model relaying on these inputs/output signals is considered being linear across the operating temperature range so the superposition theorem [18] or a Modified Nodal Analysis [37] can be applied to find the state-space representation of the circuit from Figure 2. By applying it and separating the controllable and uncontrollable inputs, the following multi-input single-output (MISO) state-space model is obtained:

wDDxC

wBBxAx

21

21

uy

u

(9)

where:

Tzw ][ x – is the state vector: w is the wall temperature and z is the zone temperature;

zy – is the output of the system;

gΦu – is the command (i.e. the total internal heat flux, which comes mainly from the

radiators but also includes internal gains from occupants, solar radiation through the windows, etc.);

Tso Φ ][w – are the measurable disturbances of the system: o is the outdoor air

temperature and sΦ is the solar radiation on the walls;

avwciw

vwciw

awciw

wwciwwwciwwco

ciwwco

CRRR

RRR

CRR

CRRCRRRR

RRR

2

2

2

12

1

22A – is the state matrix;

aC

10

1B ,

01

)2()2(

1

2

av

wwco

co

wwco

CR

CRR

R

CRRB – are the input matrices;

10C , 01 D , 002 D – are the output and feed-through matrices, respectively.

3.2 Parameter identification The considered building in this paper is a typical one storey detached house (Figure 3),

having a 100 m2 footprint, which is extensively used by the French technical research center for buildings (CSTB) for testing thermal control strategies. It is implemented in Simbad toolbox [24] under Matlab/Simulink environment by a detailed “white-box” model that was experimentally validated by CSTB. The heating system is water-based (see §5.1). Data records of input/output signals for the identification are obtained by simulating this detailed model of the reference building in Simulink.

Figure 3. The blueprint of the reference building

For parameter identification, least squares method was used. Since it estimates the

parameters of a discrete transfer function, the state-space model (9) was transformed into its discrete transfer equivalent:

22

11

232

131

22

11

222

121

22

11

212

111

1

1

1

)(

)(

)(

)(

)(

)(

)(

zaza

zbzb

zaza

zbzb

zaza

zbzb

zΦ

z

zΦ

z

z

z

zH

g

z

s

z

o

z

(10)

Thus, the identified parameters are 321121 ,,, bbaa from the above model representation.

Statistical weather records were used for outdoor temperature and solar radiation. For the internal heat flux was used the signal obtained by switching the inlet water temperature between 20 and 60 °C according to a pseudo-random binary sequence (PRBS), where the switching periods varied between eight hours and three days. Such long periods are motivated

by the very slow dynamics of the building. The input signal corresponding to the solar radiation was obtained by calculating the sum of solar radiance on each side of the envelope, multiplied by its surface [18]. The length of the simulated period was four months using a sampling time of one minute. Weather data records corresponded to Lyon, France, for December – March period.

The detailed model of the building from Figure 3 calculates the temperature in each room. However, the model (9) is defined for a single zone building. Therefore, the output is considered to be the mean temperature of the building, which is computed as the average temperature of the rooms, weighted by the room surfaces. In order to use different input variations for parameter identification and model validation, the data records obtained from simulation were divided in two halves.

To assess the capability of the low-order model to reproduce the output simulated by the detailed building model, the fit criterion was used [27]:

%100)()(

)()(1

ddT

dd

dlT

dl

yyyy

yyyyfit (11)

with ly being the output of the low-order model, dy – the output of the detailed model and

dy – the mean of the detailed model output.

Using the simulation signals over the first sixty days, the following transfer function model was obtained, having a fit of 96% (Figure 4 a):

21

2515

21

2717

21

2313

1

1

1

1

1

1

1

9907.0991.11

10589.110589.1

9907.0991.11

10024.910033.9

9907.0991.11

10283.210284.2

)(

)(

)(

)(

)(

)(

)(

zz

zz

zz

zz

zz

zz

zΦ

z

zΦ

z

z

z

z

g

z

s

z

o

z

H (12)

Though the fit between the outputs for the validation data sequence (the last two months

of the simulation) is a little smaller, 93% (Figure 4 b), the obtained second order model reproduces well enough the dynamics of the mean temperature of the building in order to be used for control purpose. The obtained value of the fit factor is very high which is unrealistic for real life building model identification but normally achievable when the process is a detailed linear model, which is the case in benchmark by emulation of building thermal behavior. Such high value is explained by the fact that the detailed building model can be viewed as a linear RC circuit with some nonlinearities to model the radiance and the convection coefficients, for example. In this case the parameter identification of the linear part is equivalent to the reduction of the order of the model, which works very well for linear models.

Figure 4. Comparison between the mean temperatures of the building obtained by simulating the detailed

white-box model and the low-order model; (a) during parameter identification, (b) during model validation

4 Compensation of system nonlinearity The controllable input of the building model from equation (9) is considered to be the

heat flux gΦ . However, in water-based heating systems, the real manipulated variable is

usually the temperature or the flow into the radiator. A possible solution is to replace the heat flux in equation (9) by the inlet water temperature. The heat flux is related to the inlet water temperature by [18]:

zing hSΦ (13)

where: hS – represents the total conductance coefficient of the radiators, where S is the total surface

and h is the mean convection coefficient,

in – is the inlet water temperature of the radiators.

The total conductance coefficient in equation (13) depends on the difference between

the inlet water temperature and the zone temperature. In Figure 5, the total conductance coefficient is calculated from the relation (13) where the heat flux is calculated from the following heat balance equation of the radiator:

outinwwg cmΦ (14)

with:

wm – water mass flow through all radiators,

wc – specific heat capacity of the water,

out – outlet water temperature of the radiators.

Figure 5. The variation of the total conductance depending on the difference between the inlet water

temperature and zone temperature

This variable coefficient introduces a nonlinearity in the model. Nevertheless, considering that the input of the system is the difference between the inlet water and the zone temperature‡, the system can be approximated by a Hammerstein model. Therefore, the system behavior can be described by a linear dynamics block, )(sH , which is preceded by a variable static gain, hS ,

as it is shown in Figure 6. In this application, physical knowledge is used for the identification of the nonlinear part of the model and least squares identification for the parameters of the linear part.

According to empirical correlations, the variation of the total conductance coefficient may be approximated by an exponential law. Since the heat flux may be calculated using the relation (14) and inlet/outlet water temperature measurements, there is no need to use more advanced algorithms for parameter identification. The coefficients of the static characteristic are estimated by fitting the exponential law on the measured data from Figure 5, where the following correlation is obtained:

2544.085.36 zinhS (15)

Thus, the relation between the heat flux delivered by the radiators and the inlet water temperature is given by the equation (13) where hS is given in the equation (15).

Although there are nonlinear MPC techniques [26, 38], the optimization problem is generally non-convex due to the nonlinearity. This leads to increased computational time and risk to end up in local minimum. A possible solution would be to linearize the system model around the operating point. However, since the water temperature in radiators usually varies on a rather wide range (20 – 60 °C), the control performance may be reduced when the system operates outside the validity range of the linearized model. A more adequate solution is to remove the nonlinearity from the control problem by placing the inverse of the nonlinearity,

1Sh , between the controller and the system [39, 40], as it is shown in Figure 6. Since the

linear part of the model is used in the optimization, the calculated command by MPC, u , is

still the heat flux gΦ . Furthermore, this is in accordance with the control goal – to minimize

the energy consumption. For this input, the system model is linear so the optimization problem is convex.

‡ This may be admitted since the inlet water temperature can be obtained as the difference between the

calculated command and the actual temperature of the zone

Once the command is calculated by MPC, the control system calculates the difference between the inlet water and the zone temperature, , by multiplying the heat flux with the inverse of the nonlinearity (see Figure 6). The inverse of the nonlinearity is obtained by inversing the correlation from equation (15), which gives (see Figure 5):

2544.01 02714.0)()( hS (16)

Then, the inlet water temperature is obtained by adding the zone temperature to the result.

Figure 6. Static nonlinearity compensation in a thermal control loop

The temperature in the radiator inlet being the real controllable input of the system, it is

constrained by its maximal admissible value. Since the command calculated by MPC represents the heat flux, the constraints should be translated into maximal heat flux which corresponds to the maximal water temperature. The minimal heat flux constraint is always zero.

5 Performance assessment of MPC and PID based Building Energy Management Systems

As stated in the Introduction, the heating system should also be considered for the assessment of the overall performance of the thermal controller. Usually, hydronic heating systems are multipurpose systems, i.e. they heat the building but also prepare the domestic hot water (DHW). The presence of renewable and conventional energy sources in the same building turns them into multi-source – multi-consumer systems. Being rather complex, they require for operation a Building Energy Management System (BEMS) which should manipulate all the elements of the heating system. Hereafter it is presented the hydronic heating system, which is used for the test of the proposed thermal controller, followed by a description of the associated BEMS. The proposed MPC thermal controller is embedded in this BEMS, for which are assessed the performances described in §2.1. These performances are assessed also for two conventional controllers, PID and scheduled start PID, embedded in the same BEMS.

5.1 Hydronic system specification The hydronic heating system used for the tests consists of (Figure 7): Three energy sources, which are a solar panel of 20 m2 (1), an air-water heat pump

(YZ 1) and a 2 kW electrical heater (YZ 2). The heat pump supplies water at 65 °C, regardless the outdoor temperature.

Two energy consumers, which are the hot water consumption (8) and building heating system by water radiators (9). Each room contains a radiator but the building is regarded as a single thermal zone.

Two hot water storage tanks. The solar tank (2), of 300 liters, is coupled to the solar panel and the DHW tank (3), of 100 liters, is dedicated exclusively to DHW preparation. The electrical heater is placed at the top of the DHW tank.

A hydraulic decoupling bottle (4) creates a neutral hydraulic point between the primary and secondary circuits in order to avoid any interactive dynamic pressure induced by their pumps [41].

Water in the solar tank is heated only by the solar panel through the heat exchanger (5).

Water in the DHW tank can be heated by the solar tank through direct fluid exchange, by the heat pump through the heat exchanger (7), and by the electrical heater. The hot water for radiators is supplied either by the heat pump either by the solar tank. A 3-way mixing valve adjusts the temperature of the water for the radiators by mixing the hot water coming from the decoupling bottle with colder water from the radiators outlet. The opening ratio of the 3-way mixing valve is driven by a local temperature control loop (TC1), whose temperature reference is given by the indoor temperature controller (TC2). The heat pump cannot supply hot water to radiators and DHW tank simultaneously, since the 3-way valve (ZZ2) can redirect water only in a single direction. On the contrary, the solar tank can supply simultaneously water to the DHW tank and the heating system. The DHW tank can be heated by all three sources simultaneously, while the radiators cannot be supplied by both sources concomitantly. All the components of the heating system have been sized based on a study of annual needs of heat and hot water and all the hydraulic pumps are of constant flow.

Figure 7. Hydronic system of the test building

5.2 BEMS specification The role of the BEMS is to act on each pump and valve of the hydronic system in order

to keep the indoor temperature above the minimal limit and to ensure the availability of DHW, while using less possible energy.

The solar panel circuit is managed by a differential controller with hysteresis, which is triggered by the fluid temperature difference between the solar panel outlet (TI3) and the bottom of the solar tank (TI4). The control of DHW production is also of differential type. When the temperature at the top of the DHW tank (TI7) drops below 55 °C, DHW is reheated by the heat pump if the latter is not used for building heating at that moment. If the heat pump is not available, DHW is reheated by the electrical heater. When the temperature at the top of the DHW tank exceeds 60 °C, the heat pump or the electrical heater are stopped. Every time the water temperature at the top of the solar tank (TI5) gets higher than that from the bottom of the DHW tank (TI6), a differential controller with hysteresis circulates the water between these two tanks. This permits to take more advantage of the stored solar energy.

The indoor temperature is controlled by either the proposed MPC, a PID or a scheduled start PID controller (TC2 in Figure 7). The output of the controller gives the set-point for the local control loop (TC1) on the 3-way mixing valve. Thus, the thermal controller acts on the radiator inlet water temperature by mixing the hot water coming from the decoupling bottle with that from the radiator outlet. Once there is a heat demand from the thermal controller, it is supplied by the solar tank, if the outlet water temperature from its upper heat exchanger (TI8) is higher than the controller output. If not, the hot water is automatically supplied by the heat pump. If at that moment the heat pump is in the middle of DHW preparation, DHW is automatically heated by the electric heater.

The PID thermal controllers were tuned by using the Ziegler-Nichols method [42] and their set-point is the lower bound of the comfort/safety zones. The difference between the PID controllers is that for scheduled start PID the set-point change at the beginning of the occupied period is shifted by two hours in advance. This gives the time to the heating system to recover the building from set-back earlier than with a classical PID. Thus, there can be expected a better comfort, but also a higher energy consumption as compared to the classical PID control.

Since the heat pump is always able to provide water at 65 °C, the upper constraint for the command in MPC corresponds to this temperature, i.e. heat rate that can be delivered at this temperature.

5.3 Definition of the criteria for performance comparison Performance comparison between different control systems is usually done through

some criteria that reflect the investigated performances. The aim of the BEMS is to ensure thermal comfort with minimal energy consumption. Furthermore, when actuators like heat pumps, which are sensitive to frequent on-off cycles, are present in the system, a particular feature of the control system is the aggressiveness of the command. By “aggressiveness” we mean the number of on-off switching. This plays a decisive role for the wear and tear of the actuators.

Hereafter the performance criteria, which are usually used to reflect these performances, are presented.

5.3.1 Criterion for DHW availability

For DHW, the comfort is considered to be ensured if the water temperature at the top of the DHW tank is always higher than 55 °C. This prevents the development of Legionella. Thus, the criterion to assess the discomfort in this case is the number of hours for which the temperature at the top of DHW tank is lower than 55 °C.

5.3.2 Excess-weighted PPD (PPD.h)

Predictive Mean Vote (PMV) and Predicted Percentage of Dissatisfied people (PPD) indices, developed by Fanger [43], are used by numerous national and international

regulations, and especially by the international norm ISO 7730 [44]. While PMV and PPD give information about instantaneous discomfort, the excess-weighted PPD, used in the European norm EN 15251 [45], assesses the thermal discomfort over a time period. Therefore, in this paper the excess-weighted PPD is used.

The calculation of the excess-weighted PPD is as follows. Considering that

%10min PPD corresponds to the lower bound of the comfort zone (category 2 in EN 15121),

the following weighting function is defined for the occupied period of the building:

10,0

10,min

PPDwhen

PPDwhenPPD

PPD

wf

(17)

Then, the distribution of the obtained weighting function is used to calculate the excess-weighted PPD as the sum of the weighting function values multiplied by the time interval over which each particular value was obtained:

n

iwfi itwfhPPD

1

.

(18)

Here n is the number of distinct values in the obtained weighting function and iwft is the time

interval over which the ith value of the weighting function was obtained.

5.3.3 Optimal start

One of the crucial points of a BEMS is its ability to restart the heating at the right moment in order to recover the building from night set-back in due time. The lack of such functionality leads to either discomfort at the beginning of the daytime period, either energy wasting. The procedure for the optimal start test is given by the European norm EN 12098-2 [46]. The test is passed if the indoor temperature intersects the optimal start check window, which is illustrated in Figure 8. This check window foresees a temperature range of 1 °C (0.5 °C below the lower comfort bound and 0.5 °C above it) during 30 minutes (15 minutes before the set-point change in the morning and 15 minutes after). Thus, the controller respects the norm EN 12098-2 if the test is passed across the entire test period during the heating season.

Figure 8. Optimal start test

5.3.4 Criterion for energy consumption

The elements of the heating system that consume energy are the electrical heater, the heat pump and all the hydraulic pumps and vans. Since all of them consume electrical energy, the criterion used for comparison is the total electrical energy consumption during the test period.

As the difference between the tested BEMS is the thermal controller, of particular interest is the energy consumed for building heating:

dtttcmE outinwwheating )()( (19)

where wm is the water mass flow through all radiators, wc is the specific heat capacity of the

water, in is the inlet water temperature of the radiators, and out is the outlet water

temperature of the radiators.

5.3.5 Number of on-off cycles

The hydronic system used in the tests contains a heat pump, which is very sensitive to multiple restart cycles. Therefore, an important characteristic of a BEMS is the number of on-off cycles of this equipment. A smaller number of restart cycles is regarded to be better because it reduces the wear and tear of the actuator.

6 Benchmark with classical controllers The building and the hydronic heating system were implemented in Simbad software

[47], which is a dedicated building simulation toolbox under Matlab/Simulink environment, developed by the CSTB for building emulation and performance assessment of real controllers. The BEMS were implemented in an industrial computer dedicated to rapid prototyping. The communication between these two devices were achieved via input/output modules using 0-10 [V] electric signals. The simulation sampling time of the model was one minute while the controller’s sampling time was five minutes.

Since the tests were going to be done for three different BEMS in real time, it was chosen to run the test for six-day periods, representative for winter and mid-season weather. The choice was motivated by the fact that in winter the outdoor temperature is low but usually with lower amplitude variations across the day; in mid-season it has larger variations so there may be periods when the heating system must be turned on and off during the same day. The tests also were done for two different geographical zones in order to examine the impact of a shift in weather conditions on the comfort and energy savings achievable by the MPC. The geographical zones correspond to Paris (oceanic climate) and Marseille (Mediterranean climate). The outdoor temperature and solar beam/diffuse radiations are shown in Figure 9 and Figure 10, respectively.

The considered lower bound of the comfort zone is 19 °C between 07:00am and 10:00pm, and the lower bound of the safety zone is 16 °C between 10:00pm and 07:00am. These bounds were used as a base for set-point on PID controllers. The response of the system driven by a PID, tuned using Ziegler-Nichols method, oscillates. In order to prevent temperature drop below the lower bound, the actual set-point transmitted to PID is raised by 0.5 °C above the lower bound, which represents the value of the first undershoot.

Figure 9. Outdoor air temperature samples for test periods

Figure 10. Beam and diffuse solar radiation samples for test periods

Indoor mean temperature variations for Paris weather, obtained by all three BEMS, are

presented in Figure 11. It can be noticed that the mean temperature obtained by PID or scheduled start PID controllers is generally higher than that resulted with MPC. Nevertheless, in the case of MPC, the temperature practically does not fall below the lower bound of the comfort/safety zone. This means that PID and scheduled start PID controllers consumed more energy than necessary. As for the comfort, it can be seen that at the beginning of the occupation period PID controller performs the worst. It restarts the heating at the moment of the set-point change thus introducing a lag between the indoor temperature and its set-point. As it can be expected, PID did not pass the optimal start test. The scheduled start PID controller did not pass it either. It can be seen in Figure 12(a) that for the fifth day in winter the temperature reaches its set-point too late, thus compromising the comfort, while for the fourth day in mid season (Figure 12 b), the heating system was restarted too early, thus consuming more energy than necessary. Only in the case of MPC the optimal start test is positive for the entire test period.

Figure 11. Comparisons of the indoor temperature evolution obtained with PID, scheduled start PID and

MPC controllers for Paris weather in (a) winter and (b) mid-season

Figure 12. Optimal start test of PID, scheduled start PID and MPC controllers for Paris weather; (a) fifth

day of winter test period (b) fourth day of mid-season test period

The results and comparison of the test criteria for Paris weather are centralized in Table

1. The first day of the test period was not considered for criteria calculation; it served only for model and controller initialization. For the winter test period, MPC consumed the lowest amount of thermal energy for building heating at the same time offering the best thermal comfort. Though the achieved energy savings are not as substantial, 3.5% as compared to PID and 4.7% as compared to scheduled start PID, the improvement of thermal comfort is by far higher; the excess-weighted PPD was lowered by 97% as compared to PID and by 65% as compared to scheduled start PID. The total electrical energy consumed by MPC based BEMS is insignificantly higher than that of PID but still lower than that of scheduled start PID. The slightly larger consumption may be explained by the fact that in the case of MPC the heat pump was used when the weather was unfavourable, performing at a poorer Coefficient Of Performance (COP), as compared to PID. MPC also drastically reduced the number of on-off cycles of the heat pump. The DHW temperature was virtually always above 55 °C in all three cases, therefore it does not appear in the table.

Analysing Figure 11, a particular point deserves additional attention. As discussed in §2.2 the optimization should always find a solution that ensures the imposed constraints. Nevertheless, despite the given arguments, the output constraints are sometimes violated. This

is actually due to the reduced model inaccuracy. Since the fit is not 100%, it is possible that the optimization underestimates the recovery time from set-back. Thus, when approaching the beginning of the occupation period, if the optimization does not find a solution, the heating system operates at full power. This is materialized by a short period of discomfort at the beginning of the occupation period. This is why in the case of MPC the PPD is not zero, although given the problem formulation it should be.

Table 1. Comparison of the test criteria for Paris weather

Performance criterion

Winter Mid-season

PID

Sch

edul

ed

star

t P

ID

MP

C

MP

C v

s.

PID

MP

C v

s.

sche

dule

d st

art

PID

PID

Sch

edul

ed

star

t P

ID

MP

C

MP

C v

s. P

ID

MP

C v

s.

sche

dule

d st

art

PID

Total energy consumption [kWh]

234 240 236 +0.9%

-1.7% 54 56 46 -15% -18%

Energy consumption for building heating [kWh]

315 319 304 -3.5% -4.7% 87 94 72 -17% -23%

Excess-weighted PPD [h]

168 14.3 5 -97% -65% 70 0 8 -88% –

Optimal start Not OK

Not OK

OK – – Not OK

Not OK

OK – –

On-off cycles of the heat pump

136 144 35 -74% -76% 58 56 34 -41% -40%

The results for the mid-season show the same trend as for the winter season, but with

significantly more energy savings achieved by MPC as compared to PID controllers. A particular phenomenon appears in the excess-weighted PPD for scheduled start PID, where this index is zero. The reason is that in mid-season the outdoor mean temperature is higher than in winter (see Figure 9), so restarting the heating system two hours earlier is always more than enough to recover the building from set-back. Since the excess-weighted PPD is null when the indoor temperature is above the lower comfort bound (equations (17) and (18)), this index resulted to be zero. Although this index is zero for scheduled start PID, the problem is that the heating is restarted too early and thereby more energy than necessary is consumed. This is reflected by an increase of 23% in consumption for building heating and by the fact that scheduled start PID did not pass the optimal start test.

The test results for Marseille weather (thermal control in Figure 13 and performance criteria comparison in Table 2), show the same trend as for Paris weather. MPC lowered the total energy consumption as compared to PID controllers, for both, winter and mid-season weather. The consumption reduction is even more accentuated than for Paris weather. The same phenomenon is obtained with scheduled start PID: restarting the heating in advance does not penalize the excess-weighted PPD, but the energy consumption is higher and the optimal start test is not passed.

Figure 13. Comparisons of the indoor temperature evolution obtained with PID, scheduled start PID and

MPC controllers for Marseille weather in (a) winter and (b) mid-season

Table 2. Comparison of the test criteria for Marseille weather

Performance criterion

Winter Mid-season

PID

Sch

edul

ed

star

t P

ID

MP

C

MP

C

Vs.

PID

MP

C v

s.

sche

dule

d st

art

PID

P

ID

Sch

edul

ed

star

t P

ID

MP

C

MP

C v

s.

PID

MP

C v

s.

sche

dule

d st

art

PID

Total energy consumption [kWh]

138 142 116 -15.9%

-18.3% 22 22 18 -18.2% -18.2%

Energy consumption for building heating [kWh]

214 222 192 -10.3% -13.5% 36 36 30 -16.7% -16.7%

Excess-weighted PPD [h]

95 0 10 -89.5% – 15 0 13 -13.3% –

Optimal start Not OK

Not OK

OK – – OK Not OK

OK – –

On-off cycles of the heat pump

118 123 27 -77.1% -78% 45 42 36 -20% -14.3%

The conclusion that may be drawn after analysing the test results is that MPC

outperforms both PID and scheduled start PID controllers regarding comfort and energy consumption. Assuming that an accurate building model, occupation schedule and weather forecast are available, MPC is able to calculate the command that ensures the comfort with minimal energy consumption. Furthermore, in the case of MPC, the BEMS restarted less frequent the heat pump, which extends the lifetime of this expensive equipment. These characteristics make the MPC a good yardstick for estimating the performance of real controllers in emulation.

7 Conclusions This paper investigates the control performance obtained by a Model Predictive

Controller (MPC) and compares them with those obtained by PID and scheduled start PID controllers. The controllers were embedded in a Building Energy Management System (BEMS) and implemented in a physical prototype while the building and the afferent heating system are emulated in a dedicated software and hardware by highly detailed models.

The used MPC implements a recently introduced cost function, which keeps the indoor temperature above an acceptable limit with minimal energy consumption. The optimization problem formulation is adapted in order to be solved by Linear Programming. MPC requires a model of the system, and usually a low-order model is preferred. The used modeling approach

is to obtain a low-order building model in two steps. First a lumped capacity model structure was obtained from a thermal circuit representation of the building, and then, the values of its parameters were estimated by least squares identification method using input/output data sets obtained by simulation of the detailed building model. During the modeling, a nonlinearity in the static characteristic of the model was revealed by physical analysis. Physical knowledge was used to eliminate the nonlinearity from the control loop by using its inverse.

The ensemble was tested in real time for periods of five days long, in winter and mid-season weather, for oceanic (Paris) and Mediterranean (Marseille) climates. The same tests were done for two classical solutions, where a PID and a scheduled start PID controller were used instead of MPC. Performances with regard to thermal comfort, total energy consumption and on-off cycles number of the heat pump were assessed for all three control systems. Their comparison has shown that MPC reduces thermal discomfort, the number of on-off cycles of the hydraulic pump, and the energy consumption. During the test period, MPC reduced the excess-weighted PPD up to 97% compared to PID and 65% compared to scheduled start PID while still reducing the energy consumption. The consumption was reduced by up to 18% as compared to PID controllers, though this rate varies depending on the weather. Additionally, MPC reduced the number of on-off cycles of the heat pump from 144 to 35 cycles. The optimal start test was passed only by the MPC controller, which means that MPC is able to adapt to the actual meteorological conditions.

Thought it was shown that MPC can significantly improve thermal control of the buildings an unknown related to MPC is the impact of weather and occupancy forecast reliability on the control performance. In this study, the predictions of the disturbances were considered as 100% reliable and the parameter identification was done by using a pseudo-random binary sequence of the command input. This approach is feasible in emulation, when the future disturbances are known and the emulated building (i.e. the detailed simulation model) can be excited with any type of command input. In this context, the MPC is an optimal controller and its performance may be considered as a benchmark or a standard performance with which the other real controllers may be compared in emulation. However, the uncertainty of the prediction of the disturbances and the precision of the identified model can highly influence the performance of MPC when applied to a real building [48].

8 Acknowledgement This work was supported by the French National Research Agency (ANR) through three research projects: 1) MIGRER: Smart Homes and Rational Management of Renewable Energy (contract n° 0604C0055 with ADEME), 2) 4C: Comfort in Hot Climate Without Air-Conditioning (contract n° ANR-08-HABISOL-019-06 with CEA), and 3) AIDE-3D: Fault Detection and Performance Assessment for Energy in Buildings (contract n° ANR-09-HABISOL-0-02 with CEA).

Appendix A Starting from the discrete state space representation of the system model:

)()()()(

)()()()1(

21

21

kkukky

kkukk

wDDxC

wBBxAx (20)

and considering that the feed-through matrices, 1D and 2D are null, the future yN states can

be estimated as:

)1()1()(

)1()1()(

)()(ˆ

)1()()1()()(

)1()1()1()2(ˆ

)()()()1(ˆ

22

2

2

1

11

2

1

1

22112

21

21

y

NN

y

NN

N

y

Nkkk

Nkukuku

kNk

kkkukuk

kkukk

kkukk

yy

yy

y

wBAwBAwBA

BABABA

xAx

wBwBABBAxA

wBBxAx

wBBxAx

(21)

Thus, the estimation of the future yN outputs can be written as:

)1()1()(

)1()1()(

)()(ˆ

)1()()1()()()2(ˆ

)()()()1(ˆ

22

2

2

1

11

2

1

1

22112

21

y

NN

y

NN

N

y

Nkkk

Nkukuku

kNky

kkkukukky

kkukky

yy

yy

y

wBACwBACwBAC

BACBACBAC

xAC

wBCwBACBCBACxAC

wBCBCxAC

(22)

By defining the following vectors:

Ty

TTTT

T

y

T

y

Nkkkk

Nkukukuku

Nkykykyky

)1()2()1()(

)1()2()1()(

)(ˆ)3(ˆ)2(ˆ)1(ˆˆ

wwwwd

u

y

(23)

the estimation of the future yN outputs can be expressed in the following matrix form:

dΨuΨxFy 21)(ˆ k (24)

where:

22

3

2

2

2

1

2222

22

2

2

11

3

1

2

1

1

1112

11

1

13

2

0

00

000

0

00

000

BACBACBACBAC

BCBACBAC

BCBAC

BC

Ψ

BACBACBACBAC

BCBACBAC

BCBAC

BC

Ψ

AC

AC

AC

AC

F

yyy

yyyy

NNN

NNNN

(25)

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