Development of room temperature and relative humidity linear parametric models for an open office using BMS data

Energy and Buildings 42 (2010) 348–356

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Energy and Buildings journal homepage: www.elsevier.com/locate/enbuild

Development of room temperature and relative humidity linear parametric models for an open office using BMS data G. Mustafaraj a,*, J. Chen a, G. Lowry b a b

School of Engineering and Design, Brunel University, Kingston Lane, Uxbridge, Middlesex, UB8 3PH, United Kingdom Engineering, Science and The Built Environment, London South Bank University, 103 Borough Road, London SE1 OAA, United Kingdom

A R T I C L E I N F O

A B S T R A C T

Article history: Received 20 September 2008 Received in revised form 23 September 2009 Accepted 4 October 2009

This study investigates Box–Jenkins (BJ), autoregressive with external inputs (ARX), autoregressive moving average with external inputs (ARMAX) and output error (OE) models to identify the thermal behaviour of an office positioned in a modern commercial building in London. These models can all be potentially used for improving the performance of the thermal environment control system. External and internal climate data, recorded over the summer, autumn and winter seasons, were used to build and validate the models. The paper demonstrates the potential of using linear parametric models to predict room temperature and relative humidity for different time scales (30 min or 2 h ahead). The prediction performance is evaluated using the criteria of goodness of fit, coefficient of determination, mean absolute error and mean squared error between predicted model output and real measurements. The results demonstrate that all models provide reasonably good predictions but the BJ model outperforms the ARMAX and ARX models. ß 2009 Elsevier B.V. All rights reserved.

Keywords: Black box linear parametric models Building management system Room temperature and relative humidity prediction

1. Introduction Increased energy costs and strict environmental standards mean that intelligent control over buildings’ energy consumption and Heating, Ventilating and Air Conditioning (HVAC) systems is required. To enable intelligent control, a model is needed and the system model is dependent on internal changes. A number of techniques are available for researchers to model the thermal response of buildings. One such technique is physical modelling and another is the black-box model. Physical modelling involves detailed study of the relationships between all parameters which affect the thermal system. Due to the complex nature of thermal systems and the large number of parameters involved, physical modelling is difficult (sometimes even impossible). The black-box modelling approach, on the other hand, uses ‘‘data mining’’ methods to establish relationships from all observed data. Therefore, system identification modelling with a black-box approach can establish models with very low development costs. No knowledge of the system’s internal structure is needed in these models [1,2]. There are different types of black-box mathematical models. However, in this work we are interested only in black-box linear parametric models because the equations of these models are

* Corresponding author at: Via Rigoni 7, Post Code 27058, Voghera (PV), Italy. E-mail address: [email protected] (G. Mustafaraj). 0378-7788/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2009.10.001

simple and their parameters (see Eq. (8), Section 3) can be interpreted with the physical model of the system. Consequently they will enable future work to identify grey-box models (i.e. based partially on physical knowledge and partially on empiricism [3]). Attempts at building grey-box models can be seen in Norle´n [4] and Jimenez et al. [5]. Another advantage of linear models is that they are easier to use in control schemes than non-linear models. Black-box linear parametric structures such as autoregressive ARMAX and ARX models have been used by many authors to predict thermal behaviour of different office buildings and to control HVAC plants. Norle´n [4] developed a method based on an autoregressive ARMAX model to describe the dynamics of the heat flows in a test cell, while Loveday and Craggs [6,7] used Box–Jenkins time series analysis to describe the thermal behaviour of a building influenced by a number of variables, including external temperature variation, ventilation rate fluctuations and occupancy pattern variation. Past research works [3,8–11] have built models to predict room temperature for different office buildings, with data related to outside weather conditions and internal climate being collected for certain periods (i.e. 36 days, 7 days, 6 days, one year, and three weeks, respectively), and utilizing different sampling rates (i.e. 1, 5 and 15 min). The data collected have been used with OE, BJ and autoregressive ARX and ARMAX models to build models for room temperature prediction. The ARMAX models were used by MacArthur et al. [12] to predict at each hour the next day what the building’s cooling load and ambient temperature profile would

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be. The developed profile was updated on-line to predict 24 h load for the next day based on historical data related to the previous days. The results demonstrate that all models provide reasonably good predictions and some models can be used to characterize dynamic behaviour and to detect faults in air conditioning systems. For example, Yui and Wang [13] explored the use of ARMAX models for modelling and parameter identification of air conditioning processes. In contrast [14–16] used ARX and ARMAX models and the Kalman filter recursive identification method for model parameters identification. These implemented the criteria of residual (i.e. between measurements and predicted model output) and parameter identification methods for fault detection in the air-handling unit of a building’s HVAC system. Finally, ARX and ARMAX models were used in the past by Norle´n [4] and Jimenez et al. [5] to estimate the thermal parameters of physical linear model related to heat exchange through the wall in a test cell, while Pakanen and Karjalainen [17] used an ARMAX model for modelling static heat flows between neighbouring rooms. Despite the contributions of these works, there are still modelling issues that have not been analysed and these are addressed in this paper. Firstly, the research presented in this paper is related to developing models for one real office whereas previous researchers have applied these models mainly to experimental rooms and HVAC plants in which experimental conditions can be managed. Secondly, past research on thermal model development has been related mainly to linear parametric ARX and ARMAX models [3,8–11,18–21], with few research papers [11,18] dealing with BJ and OE models. In contrast, this study undertakes an overall analysis for a period of nine months of room temperature and relative humidity model development, including ARX, ARMAX, BJ and OE models. Thirdly, in this research, predictions of different time scales are investigated, i.e. predictions of room temperature and relative humidity at 6, 12 and 24 steps ahead (30 min, 1 and 2 h) are produced, while in the past most papers mainly dealt with model simulation [3,8–11,19,20]. In this work, the criteria of goodness of fit, mean absolute error, mean squared error and coefficient of determination for various values of step ahead predictions and model simulation are given particular importance. Fourthly, this research uses linear models to predict relative humidity for long periods (nine months) whereas past researches, such as Boaventura Cunha et al. [9] and Lu and Viljanen [22] built linear ARX model and non-linear autoregressive with external input NARX model based on short periods of data collection for 6 and 30 days respectively. In the past, apart from Lu and Viljanen [22] who used a NARX model to predict relative humidity, no research papers have used blackbox linear parametric models to predict relative humidity for different step ahead prediction. In contrast this research adopted black-box linear parametric models for this purpose. Fifthly, the results show that the models obtained in this research are relatively simple (i.e. the model orders are low and therefore there are not many model parameters, e.g. room temperature prediction model orders are na = 1, 2 and nb = 1) and robust (i.e. model parameter variations from one weekday to another are small and hence one universal model can be used for many weekdays). In this research, models are developed using data collected over long periods, whilst in past research works models were developed using a limited period of data collection. For example, Boaventura Cunha et al. [9] used 6 days, Loveday and Crags [6,7] recorded two weeks of hourly data, Lowry and Lee [11] three weeks, Thomas and Mohseni [18] 26 days and nights, Lu and Viljanen [22] 30 days and Moreno et al. [3] 36 days. Models developed and validated using a limited range of data are not reliable for predicting room temperature and relative humidity with high accuracy outside

349

the range of data used for their development and validation. Although models were developed using one year of data in Frausto et al. [10] and Patil et al. [19], the ARX and ARMAX models developed have higher orders (i.e. na = 4 and nb = 5). Finally, unlike many past research works where non-linear neural networks models were developed for room temperature prediction, this research emphasizes linear models because they have a number of advantages: (i) they are simple (low number of model parameters); (ii) they are much easier to deal with due to the potential of connecting them with physical models of the system. On the contrary with non-linear neural networks it is impossible to relate model parameters with the system’s physical parameters; (iii) a disadvantage of non-linear networks is that their parameters (i.e. weights) vary after each trial (i.e. training the network on the same weekday many times), whereas this does not happen with linear models; (iv) linear models are easier to use in control schemes for HVAC plants. The research reported in this paper looks at dry bulb room temperature and relative humidity prediction for the open-plan office positioned on the second floor of Portman House in London throughout the summer, autumn and winter seasons. The main objective is to explore the potential of developing models for room temperature and relative humidity from the data provided by the existing building management system (BMS) and, thus, avoid additional instrumentation. Having collected and identified model inputs and outputs, this research shows that linear parametric ARX, ARMAX, BJ and OE models can be developed using widely available packages such as System Identification Toolbox (SIT) for linear parametric models in Matlab [23,24]. Prediction of room temperature and relative humidity by black-box linear parametric models can be utilized in the control strategy of the building’s temperature and relative humidity. Using a model for prediction is common in many control applications. Where one wants to predict the indoor climate a certain number of steps are required, at least 15–30 min ahead to compute the control signal [18]. This enables the automatic detection of deterioration and failure of HVAC plants, or important changes in the way buildings are operated, including load/ demand management, which increases occupants’ comfort and at the same time reduces the electrical energy consumption of HVAC systems [3,11,18]. The paper is organized as follows: Section 2 describes Portman House and the process of data collection by the BMS. Section 3 introduces modelling techniques for black-box linear parametric models. Section 4 discusses validation criteria, for linear parametric ARX, ARMAX, BJ and OE models, including mean squared error, mean absolute error, goodness of fit and coefficient of determination for different step ahead prediction of room temperature and relative humidity. Section 5 presents model results and discussion. Section 6 addresses conclusion and future work. 2. Portman House description and the process of data collection Portman House is located in the centre of London. An Invensys BMS is installed in the building for operating and monitoring plant/ building services. The room’s temperature and relative humidity in zones 1 and 2 (see Fig. 1) were taken for model training (development) and validation. The dimensions of the room are 210 m2  3.0 m height. The two sensors that measure room temperature and relative humidity are positioned in zones 1 and 2. To identify the parameters of the model describing room temperature and relative humidity of a real building a time series of relevant data was collected. In this research, the only sampling

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G. Mustafaraj et al. / Energy and Buildings 42 (2010) 348–356

Fig. 1. Layout of zones 1 and 2.

interval provided by the BMS installed in Portman House is 5 min. The data collected consisted of:  room temperature for zones 1 and 2 on the second floor, respectively T1 and T2 in degrees Celsius (8C);  relative humidity for zones 1 and 2 on the second floor, respectively H1 and H2 in %;  outside temperature TO in 8C;  outside relative humidity, HO in %;  supply air flow-rate for zones 1 and 2 on the second floor, respectively R1 and R2 in m3/s (supply air flow-rate is related to air coming from the Air Handling Units (AHUs) in zones 1 and 2 (AHU1 and AHU2) positioned on the roof and flowing through the Fan Coil Units (FCUs) positioned in zones 1 and 2 respectively);  supply air temperature for zones 1 and 2, respectively TA1 and TA2 in 8C (supply air temperature is related to air coming from the AHUs in zones 1 and 2 (AHU1 and AHU2) positioned on the roof and flowing through the FCUs positioned in zones 1 and 2 respectively);  supply air relative humidity for zones 1 and 2, respectively HA1 and HA2 in % (supply air relative humidity is related to air coming from the AHUs in zones 1 and 2 (AHU1 and AHU2) positioned on the roof and flowing through the FCUs positioned in zones 1 and 2 respectively);  chilled water temperature TC in 8C (chilled water that flows inside the FCUs positioned in zones 1 and 2 comes from the chillers positioned on the roof);  hot water temperature TH in 8C (hot water that flows inside FCUs positioned in zones 1 and 2 comes from the boilers). The data from Portman House were collected for a period of nine months in the summer, autumn and winter seasons of 2005–2006 through the existing sensors of the BMS. The primary assumption of the model development was that the internal temperature variation is directly influenced by variations of external temperature and the internal air coming from the FCUs. Occupancy, computers, printers and electrical equipment cause additional internal heat gain, but based on the assumption that the zone’s room temperature is very well controlled, their impact is strongly correlated with the internal energy exchanged between the incoming air that flows inside the room from the FCUs, and the circulating water (flows inside FCUs and AHUs). As such, these effects were indirectly included in the model.

The room temperature and relative humidity can be slightly different at different positions in zones 1 and 2. Ideally, a number of sensors in different positions should be installed and data collected for modelling purposes. Unlike some other published research where an experimental building environment set-up is investigated, this study is based on data collection from a real office with a BMS already in operation. The room temperature and relative humidity sensors are positioned at a height of 1.7 m (see Fig. 1) from the floor. No other sensors can be added and the existing sensors’ position cannot be changed. It is acceptable for this study to use limited existing sensors because (1) the temperature and humidity variations in different zones are small and (2) some investigations of good sensor locations were carried out during BMS installation. The BMS in this study only provides dry bulb room temperature (room temperature) and relative humidity. 3. Model development with linear parametric models The procedure to determine a proper linear parametric mathematical model from observed input–output data involves three basic elements: (1) measurement of input–output data, (2) selection of model structure and estimation of parameters and (3) validation of selected model on fresh data set. After the data have been analysed using the SIT, a large collection of models with different orders and structures are obtained. In this research, to choose the model that provides good room temperature and relative humidity prediction, goodness of fit, coefficient of determination, mean absolute error, and mean squared error between model output and real measurements were used (i.e. outlined in Section 5) (see [23–25]). Room temperature Tr (i.e. model output, which for zones 1 and 2 is Tr1 and Tr2) and relative humidity Hr (i.e. model output, which for zones 1 and 2 is Hr1 and Hr2), are modelled by general BJ, OE, ARX and ARMAX models. The OE model which predicts y(t) (it can be either Tr(t) or Hr(t)) is given as follows [24]:

yðtÞ ¼

B1 ðqÞ B2 ðqÞ B3 ðqÞ Rðt  nk1 Þ þ T A ðt  nk2 Þ þ T C ðt  nk3 Þ F 1 ðqÞ F 2 ðqÞ F 3 ðqÞ B4 ðqÞ B5 ðqÞ B6 ðqÞ T O ðt  nk4 Þ þ HA ðt  nk5 Þ þ HO ðt  nk6 Þ þ F 4 ðqÞ F 5 ðqÞ F 6 ðqÞ B7 ðqÞ T H ðt  nk7 Þ þ eðtÞ (1) þ F 7 ðqÞ

G. Mustafaraj et al. / Energy and Buildings 42 (2010) 348–356

B1 ðqÞ ¼ b1;1 þ b1;2 q1 þ B3 ðqÞ ¼ b3;1 þ b3;2 q1 þ B5 ðqÞ ¼ b5;1 þ b5;2 q1 þ B7 ðqÞ ¼ b7;1 þ b7;2 q1 þ F 1 ðqÞ ¼ 1 þ F 3 ðqÞ ¼ 1 þ F 5 ðqÞ ¼ 1 þ F 7 ðqÞ ¼ 1 þ

f 1;1 q1 þ f 3;1 q1 þ f 5;1 q1 þ f 7;1 q1 þ

   

   

þ þ þ þ

þ b1;nb1 qnb1 1 ; þ b3;nb3 qnb3 1 ; þ b5;nb5 qnb5 1 ; þ b7;nb7 qnb7 1 f 1;n f 1 qn f 1 ; f 3;n f 3 qn f 3 ; f 5;n f 5 qn f 5 ; f 7;n f 7 qn f 7

B2 ðqÞ ¼ b2;1 þ b2;2 q1 þ    þ b2;nb2 qnb2 1 B4 ðqÞ ¼ b4;1 þ b4;2 q1 þ    þ b4;nb4 qnb4 1 B6 ðqÞ ¼ b6;1 þ b6;2 q1 þ    þ b6;nb6 qnb6 1

yðtÞ ¼

B1 ðqÞ B2 ðqÞ B3 ðqÞ Rðt  nk1 Þ þ T A ðt  nk2 Þ þ T C ðt  nk3 Þ F 1 ðqÞ F 2 ðqÞ F 3 ðqÞ B4 ðqÞ B5 ðqÞ B6 ðqÞ HO ðt  nk6 Þ T O ðt  nk4 Þ þ HA ðt  nk5 Þ þ þ F 4 ðqÞ F 5 ðqÞ FðqÞ B7 ðqÞ CðqÞ eðtÞ (4) T H ðt  nk7 Þ þ þ F 7 ðqÞ DðqÞ

CðqÞ ¼ 1 þ c1 q1 þ c2 q2 þ    þ cnc qnc ; 1

¼ 1 þ d1 q

2

þ d2 q

nd

þ    þ dnd q

DðqÞ (5)

where nc and nd are the order associated with the disturbance models C(q) and D(q), and e(t) is treated as a white noise. The structure of the ARMAX model is

B1 ðqÞ B2 ðqÞ B3 ðqÞ Rðt  nk1 Þ þ T A ðt  nk2 Þ þ T C ðt  nk3 Þ yðtÞ ¼ AðqÞ AðqÞ AðqÞ B4 ðqÞ B5 ðqÞ B6 ðqÞ T O ðt  nk4 Þ þ T H ðt  nk5 Þ þ HO ðt  nk6 Þ þ AðqÞ AðqÞ AðqÞ B7 ðqÞ CðqÞ T H ðt  nk7 Þ þ eðtÞ (6) þ AðqÞ AðqÞ AðqÞ ¼ 1 þ a1 q1 þ a2 q2 þ    þ ana qna

(2)

F 2 ðqÞ ¼ 1 þ f 2;1 q1 þ    þ f 2;n f 2 qn f 2 F 4 ðqÞ ¼ 1 þ f 4;1 q1 þ    þ f 4;n f 4 qn f 4 F 6 ðqÞ ¼ 1 þ f 6;1 q1 þ    þ f 6;n f 6 qn f 6

where t denotes time instant, B1/F1, B2/F2, B3/F3, B4/F4, B5/F5, B6/ F6 and B7/F7 denote the transfer functions with known structure and unknown model parameters of the system; nb1, nb2, nb3, nb4, nb5, nb6, nb7 and nf1, nf2, nf3, nf4 nf5, nf6, nf7 are the orders associated with the inputs (R, TA, TC, HA, HO, TH and TO, respectively); nk1, nk2, nk3, nk4, nk5, nk6, nk7 are the delays from the inputs to the output Tr or Hr (i.e. the number of sampled data before a change in input will affect the output) and the delay operator q works on signals in the following way: qdx(t) = x(t  d). With this model structure, unbiased model parameter estimation can only be obtained when an additive output error is a white noise with zero mean. The assumption of zero mean white noise cannot be met in many applications because there are many unknown disturbances or noises that affect the system and its measurements. An improvement is to use BJ models, where there is a separate description of the input– output relationship between the dynamic properties of the system B/F and the disturbance noise model properties described as C/D in the following equation [24]:

(7)

where na is the order associated with the polynomials A(q). The structure of the ARX model is a particular form of ARMAX model, where the polynomial C(q) = 1 in Eq. (6). Consequently, comparing the ARX model with the ARMAX model, the white noise e(t) can be described as a moving average of white noise e(t). The difference between the ARMAX and BJ models is that the noise and input are subjected to the same dynamics in the

351

(3)

ARMAX case [24]. Finally, the adjustable parameters to be determined are

u ¼ ½a1 ; . . . ; ana ; b1;1 ; . . . ; b7;nb7 ; c1 ; . . . ; cnc ; d1 ; . . . ; dnd ; f 1;1 ; . . . ; f 7;n f 7 

T

(8)

In Section 5, the orders and delays of the BJ, OE, ARX and ARMAX models are presented as BJ [nb nc nd nf nk], OE [nb nf nk], ARX [na nb nk] and ARMAX [na nb nc nk] respectively. In this study, model orders and input delays are based on the estimation of several ARX models with a range of orders and delays and compare the performance of these models. Consequently, in SIT, model orders and input delays are chosen to correspond to the best ARX model performance and these orders are used as an initial guess for further modelling (i.e. ARMAX, BJ and OE models) [23,24]. The values of na, nb, nc, nd, nf and nk were each varied over the range 1– 10, but for room temperature prediction no advantage in terms of goodness of fit, coefficient of determination, mean absolute error, and mean squared error was achieved beyond the first and second orders (na = 1, 2 and nb = nc = nd = nf = 1) and for input delays nk = 3, 8. Differently from room temperature prediction, for relative humidity, good results in terms of goodness of fit, coefficient of determination, mean absolute error and mean squared error are obtained for higher orders (na = 2, 7, nb = 2, 4, 6 and nc = nd = nf = 1, 2) and input delays nk = 1. The physical basis for the models (1), (4) and (6) is that room temperature and relative humidity at time t are assumed to be linear functions of current and past outside temperatures, outside relative humidity, air flow-rate, air temperature, air humidity, and cooling and heating load released by chilled and hot water respectively. 4. Model training and validation criteria A training set is a group of matched input and output data used for building models, usually by suitable adaption of the parameters in linear parametric models. The ‘‘Prediction Error Method’’ has been used throughout the training stage to calculate the parameter values of linear parametric models [24]. For validation of linear models for different step ahead prediction, in order to compare the prediction accuracy for different models, the following performance measures were calculated for all identified linear parametric models [3,4,10,18,23–25]:  Goodness of fit (G): 0

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi PN 2 B C i¼1 ðyr;i  yi Þ C G¼B @1  rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A  100 PN  P N 1 i¼1 yi  N i¼1 yi ¼ 1; . . . ; N

for i

(9)

G. Mustafaraj et al. / Energy and Buildings 42 (2010) 348–356

352

 Mean squared error (MSE): MSE ¼

1 N

N X

½yi  yr;i 2

(10)

i¼1

 Mean absolute error (MAE): MAE ¼

1 N

N  X

 y  y  i r;i

(11)

i¼1

where N is the number of measured input–output pairs, and y and yr are the measured and predicted model output corresponding to room temperature and relative humidity.  Coefficient of determination r2, obtained through linear regression of the predicted results on the measured data. The linear regression is the measure of correlation between the observed and predicted output.

5. Modelling results and discussion The data collected were analysed by dividing them into weekdays (Monday–Friday) and weekends (the latter are not presented because the HVAC plants are switched off on Saturday and Sunday). The models were trained using data recorded from various weekdays in summer, autumn and winter seasons and then the best model was defined by applying the criteria of model validation (see Section 4). Retuning of ARX, ARMAX, BJ models was carried throughout the three seasons to maintain the following models measures of performance (see Frausto et al. [10], who used this criteria): (1) high values for goodness of fit or coefficient of determination, (2) reduction in the variability of goodness of fit or coefficient of determination from one weekday to another (i.e. to keep close values of goodness of fit or coefficient of determination from one weekday to another) within the same season, and (3) when the performance of the model had fallen below a predefined threshold value, where the predefined goodness of fit and coefficient of determination were set respectively at 40% (or r2 = 0.71) and 35% (or r2 = 0.68) for room temperature and relative humidity. There are some small variations in model parameters from one weekday to another. To obtain the universal model for the whole season, averages of model parameters for different weekdays are used in the final model. Furthermore, as for previous research works by Thomas and Mohseni [18] and Lu and Viljanen [22] where different step ahead predictions were used to determine model output, this section presents the data analysis and model development for room temperature and relative humidity with results for ARX, ARMAX, BJ and OE model structures presented for different step ahead predictions: (i.e. the number of Table 1 Validation criteria between predicted and measured room temperature, summer season. Prediction

Validation criteria

Step z

G

6 12 24 Simulation

Prediction using BJ [1 1 1 1 3] 80–85 0.96–0.98 0.011–0.021 74–81 0.93–0.96 0.017–0.022 68–76 0.91–0.95 0.023–0.032 55–60 0.79–0.85 0.042–0.061

r2

Simulation using OE [1 1 3] 57–63 0.82–0.9

6 12 24 Simulation

MSE

0.035–0.055

Prediction using ARMAX [1 1 1 3] 77–81 0.95–0.96 0.014–0.024 71–76 0.92–0.95 0.018–0.031 64–72 0.88–0.92 0.03–0.05 54–58 0.77–0.8 0.045–0.065

MAE

Table 2 Validation criteria between predicted and measured relative humidity, summer season. Prediction

Validation criteria

Step z

G

6 12 24 Simulation

Prediction using BJ [2 2 2 2 1] 68–73 0.91–0.96 55–66 0.86–0.92 44–52 0.72–0.77 40–46 0.71–0.75

6 12 24 Simulation

r2

MSE

MAE

1.18–1.65 1.75–2.3 2.55–3.2 3.1–4.2

0.51–0.65 0.68–0.82 0.87–1.35 1.32–1.55

Prediction using ARMAX [2 2 2 1] 64–70 0.91–0.93 1.35–1.95 52–60 0.81–0.86 2.5–2.85 40–45 0.71–0.74 3.25–3.5 35–42 0.68–0.71 3.45–5.2

0.62–0.71 0.76–0.87 1.25–1.6 1.58–1.8

successive step ahead predictions are called ‘‘z’’) z = 6, 12 and 24 and model simulation. The analysis of the results for linear models, revealed that their performance in predicting room temperature and relative humidity varied for different step ahead z. According to Ljung [23] and Moreno et al. [3] once the data have been collected, the first two thirds or so of the data recorded were used in order to estimate the model coefficients u (see Eq. (8)) and the remaining one third of data were used for model validation. In this research, 1365 sampled data (114 h, Monday 01:20–Friday 19:00) were used for model development throughout all the weekdays of the summer, autumn and winter seasons, while 683 sampled data related to the following weekdays (57 h, Monday 01:20–Wednesday 11:30) were taken for model validation, respectively for z = 6 (30 min), z = 12 (1 h) and, z = 24 (2 h) step ahead prediction and model simulation (corresponding to step ahead prediction of 57 h). Tables 1–6 show the respective performance of ARMAX, ARX, BJ and OE models (i.e. the range of G, MAE, MSE and r2) resulting from the regression of predicted on measured room temperature and relative humidity, throughout each season. Different models were found to be most appropriate for different periods of each season. The values obtained for G, r2, MAE and MSE are close to each other for different weeks of the summer, autumn and winter seasons and between zones 1 and 2. To avoid repetition, only the results of model validation related to zone 1 are presented in Tables 1–6. The goodness of fit between prediction (i.e. for 1 h step ahead) and measurements are presented in graphs for one weekday’s model validation related to each season. The OE model can be used in SIT only for model simulation [23,24]. Consequently, in Tables 1–6 the results related to this model are not shown for z = 6, 12 and 24 prediction. Analyzing the results obtained with BJ and OE models for model simulation, the BJ model provides slightly better results in terms of G, MAE, MSE and r2. Except for the summer season, the OE model gives slightly Table 3 Validation criteria between predicted and measured room temperature, autumn season. Prediction

Validation criteria

Step z

G

0.19–0.21

6 12 24 Simulation

Prediction using BJ [1 1 1 1 3] 80–85 0.95–0.98 0.008–0.01 74–77 0.92–0.95 0.013–0.024 65–70 0.88–0.91 0.03–0.05 55–60 0.8–0.87 0.035–0.055

0.072–0.092 0.093–0.116 0.12–0.17 0.195–0.22

0.08–0.093 0.095–0.12 0.11–0.19 0.185–0.23

6 12 24 Simulation

Prediction using ARMAX [1 1 1 3] 79–83 0.94–0.97 0.008–0.013 68–77 0.92–0.94 0.015–0.03 58–64 0.83–0.91 0.04–0.06 50–55 0.75–0.78 0.045–0.063

0.081–0.093 0.095–0.119 0.13–0.19 0.21–0.235

0.07–0.09 0.09–0.13 0.12–0.17 0.191–0.22

r2

MSE

MAE

G. Mustafaraj et al. / Energy and Buildings 42 (2010) 348–356 Table 4 Validation criteria between predicted and measured relative humidity, autumn season. Prediction

Validation criteria

Step z

G

6 12 24 Simulation

Prediction using BJ [4 1 1 1 1] 68–79 0.92–0.96 65–74 0.9–0.93 57–65 0.8–0.87 51–55 0.77–0.81

0.85–1.5 1.25–1.8 1.78–2.25 2.3–3.15

0 6 12 24 Simulation

Prediction using ARMAX [7 4 1 1] 73–77 0.93–0.95 0.91–1.2 61–69 0.89–0.91 1.4–2.45 54–60 0.78–0.84 2.5–2.85 48–52 0.74–0.79 2.95–3.5

r2

MSE

Validation criteria

Step z

G

0.42–0.65 0.55–0.75 0.73–0.8 0.85–1.2

6 12 24 Simulation

Prediction using BJ [6 1 1 1 1] 74–77 0.93–0.95 64–71 0.87–0.92 59–64 0.83–0.89 50–58 0.73–0.79

0.45–0.51 0.62–0.78 0.78–0.85 0.87–1.3

6 12 24 Simulation

Prediction

Validation criteria

Step z

G

6 12 24 Simulation

Prediction using BJ [1 1 1 1 8] 72–78 0.92–0.96 0.015–0.025 62–69 0.85–0.9 0.024–0.035 51–59 0.75–0.83 0.04–0.05 48–55 0.71–0.74 0.044–0.057

0.084–0.1 0.11–0.14 0.16–0.2 0.23–0.26

6 12 24 Simulation

Prediction using ARMAX [2 1 1 3] 68–72 0.9–0.94 0.018–0.027 59–66 0.84–0.9 0.026–0.036 50–57 0.8–0.85 0.045–0.065 45–51 0.7–0.73 0.05–0.07

0.086–0.11 0.12–0.15 0.19–0.21 0.2–0.27

MSE

Table 6 Validation criteria between predicted and measured relative humidity, end of autumn and winter season. Prediction

MAE

Table 5 Validation criteria between predicted and measured room temperature, end of autumn and winter season.

r2

353

MAE

better results than the BJ model (see Table 1). Because ARX and ARMAX models have similar performance results throughout the analysis for the three seasons only the ARMAX model is presented (see Tables 1–6 and Figs. 2–7). The analysis for the summer season includes weekdays between 13 June and 09 September 2005, where six inputs (TO, HO, HA1, R1, TA1 and TC) were sufficient to obtain good results throughout the season for the prediction of room temperature and relative humidity. Tables 1 and 2 show the best-fit models, including first and second orders (na = nb = nc = nd = nf = 1, 2), input delays one and three (nk = 1 and 3), elements for each of BJ, OE, ARX and ARMAX models, and the resultant degree of G, MAE, MSE and r2 for each. Examples of model validation for weekdays 04–06 July 2005 are presented in graphs 2–3 for z = 12. The autumn season analysis includes weekdays between 12 September and 11 November 2005

r2

MSE

MAE

0.92–1.2 1.4–1.85 1.82–2.15 2.25–2.8

0.46–0.53 0.57–0.68 0.72–0.78 0.8–1.25

Prediction using ARMAX [2 6 1 1] 72–75 0.92–0.95 0.95–1.15 61–66 0.81–0.87 1.35–2.45 52–59 0.74–0.8 2.55–2.9 45–54 0.71–0.76 2.8–3.6

0.48–0.55 0.65–0.76 0.78–0.87 0.89–1.44

(beginning and middle of autumn), where seven inputs (TO, HO, HA1, R1, TA1, TC and TH) were sufficient to obtain good results throughout for the prediction of room temperature and relative humidity. For this season, the models with orders na = 1 and 7, nb = 1 and 4, nc = nd = nf = 1 and input delays nk = 1 and 3 were chosen to predict Tr1 and Hr1 respectively (see Tables 3 and 4 and Figs. 4 and 5). Finally, for the end of the autumn and winter seasons weekdays between 14 November 2005 and 24 March 2006 were included, where the inputs, TO, HO, HA1, R1, TA1 and TH were sufficient to build models related to Tr1 and Hr1. The ARX, ARMAX and BJ models with orders na = 2, nb = 1 and 6, nc = nd = nf = 1 and input delays nk = 1, 3 and 8 perform better for the prediction of Tr1 and Hr1 throughout these periods (see Tables 5 and 6 and Figs. 6 and 7). Analysis of the results throughout the summer, autumn and winter seasons can be summarised as follows:  For room temperature and relative humidity prediction the results obtained with the BJ model are better than those obtained with ARMAX and ARX models (see Tables 1–6 in terms of G, MAE, MSE and r2).  The model orders na and nb related to room temperature prediction are lower (na = 1, 2 and nb = 1) than those required to predict relative humidity (na = 2, 7 and nb = 2, 4, 6). Consequently, the models’ structure for relative humidity prediction is more complicated and as a result it has more parameters compared to that for room temperature.  A Comparison of the results obtained for z = 6, 12 and 24 and model simulation in terms of G, MAE, MSE and r2 reveals they deteriorate as the number of step ahead predictions z increase (see Tables 1–6). However, the results for model simulation are

Fig. 2. Goodness of fit between predicted and measured room temperature, summer season.

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Fig. 3. Goodness of fit between predicted and measured relative humidity, summer season.

Fig. 4. Goodness of fit between predicted and measured room temperature, autumn season.

still good for the prediction of room temperature and relative humidity.  The results related to room temperature predictions are better than those obtained for relative humidity (see Tables 1–6).  The MAE of relative humidity throughout the analysis is less than the corresponding MSE (see Tables 2, 4 and 6), meaning that errors from some areas are much greater than 1. As a result, these

areas are becoming unpredictable (i.e. as a consequence, the network could not capture the entire information for that point). Relative humidity may be influenced by more factors than room temperature especially in this study where some important information, such as occupancy, solar gains and other effects from computers and printers are not directly incorporated into the models.

Fig. 5. Goodness of fit between predicted and measured relative humidity, autumn season.

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Fig. 6. Goodness of fit between predicted and measured room temperature, end of autumn and winter season.

Fig. 7. Goodness of fit between predicted and measured relative humidity, end of autumn and winter season.

Furthermore, comparing the results obtained in this research for room temperature prediction by ARX, ARMAX, BJ and OE models with those obtained from past works (i.e. that utilize the same models), can be summarized as follows: (1) Models’ performance: in this research the range of values for goodness of fit and coefficient of determination are G = 50–85 and r2 = 0.71–0.98, while in Moreno et al. [3] the coefficient of determination varied between r2 = 0.8861–0.9457, in Patil et al. [19] r2 = 0.65–0.973, and Frausto et al. [10] obtained G = 55.8– 89.1 and r2 = 0.845–0.985. In this research, the range obtained for mean absolute error MAE = 0.07–0.27 and mean squared error is MSE = 0.008–0.057, while Thomas and Mohseni [18] obtained values for MAE = 0.11–0.336 and MSE = 0.05982– 0.232. In addition, for room temperature prediction utilizing non-linear NARX models Lu and Viljanen [22] found MAE = 0.0481–0.1461 and MSE = 0.0062–0.0533. In conclusion, in this research the results for room temperature prediction related to MAE, MSE, G and r2 are close to those obtained in the past. (2) Apart from Thomas and Mohseni [18] and Lu and Viljanen [22] who used different step ahead prediction for room temperature and relative humidity prediction, past research works only presented results related to model simulation. Consequently, in this research analyzing the results for different step ahead

prediction offered the opportunity to choose a number z of step ahead prediction that was more suitable to our requirements. (3) In this research, ventilation air flow-rate and temperature, heating and chilled water temperatures are included as inputs, while in the past only Thomas and Mohseni [18] used ventilation flow-rate and internal heating load. Differently from Thomas and Mohseni [18] who varied the above values experimentally between fixed values, in this research the values of ventilation air flow-rate and temperature, heating and chilled water temperatures were varied by the BMS. (4) For relative humidity prediction the range of values for mean absolute error and mean squared error are respectively MAE = 0.42–1.31 and MSE = 0.85–2.85 for 6 (30 min) and 12 (1 h) step ahead prediction, while for Lu and Viljanen [22], 1 (15 min), 2 (30 min), 3 (45 min) and 4 (1 h) step ahead prediction and sampling interval of 15 min, the range of values are MAE = 0.2568–0.4445 and MSE = 0.7331–1.9242. In both researches, the values for MAE and MSE are bigger than those obtained for room temperature, but in Lu and Viljanen [22], the range of values for MAE and MSE are lower than those obtained in this research. This is because the time scale for obtaining the results in Lu and Viljanen [22] was much less (i.e. 30 days) than the present research (i.e. nine months), and it is reasonable that over such a long period the prediction of relative humidity could deteriorate because various unknown factors that cannot

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be included in the models can affect the thermal behaviour of the building.

could be extended to other types of buildings, including hospitals, supermarkets, airports and schools.

6. Conclusion and future work

Acknowledgments

This study has investigated the prediction of room temperature and relative humidity in one office using black-box linear ARX, ARMAX, BJ and OE models. The models were trained and validated throughout the weekdays of the summer, autumn and winter seasons and the validation results in terms of the range of G, MSE, MAE and r2 between model output and real measurements are presented. There are many important parameters relevant to room thermal behaviour models. However, this paper emphasises the study of temperature and relative humidity predictions because these two parameters are most important for thermal behaviour performance and control. The results illustrate that the BJ model (BJ [1 1 1 1 3]) is more suitable for predicting room temperature, using one week’s data model development throughout the summer and autumn seasons, while BJ [1 1 1 1 8] is more suitable for the winter season. The ARMAX [1 1 1 3] and ARX [1 1 3] models also give good results for the summer and autumn seasons, while ARMAX [2 1 1 3] and ARX [2 1 3] models give good results for the end of the autumn and winter seasons, but overall, for room temperature prediction, the BJ model gives slightly better results than ARX and ARMAX models (see Tables 1, 3 and 5). It is found that for relative humidity higher model orders are required than model for room temperature prediction. In particular, for the summer season the BJ [2 2 2 2 1], ARMAX [2 2 2 1] and ARX [2 2 1] models are chosen as they give good prediction quality. Throughout autumn, the BJ [4 1 1 1 1], ARMAX [7 4 1 1] and ARX [7 4 1] models give good results for the prediction of relative humidity. Throughout the end of the autumn and winter seasons the BJ [6 1 1 1 1], ARX [2 6 1] and ARMAX [2 6 1 1] models are chosen to predict relative humidity. Finally, as for room temperature prediction, for relative humidity, the BJ model provides better results than ARX and ARMAX models (see Tables 2, 4 and 6). Overall, the linear models give good results for the prediction of room temperature and relative humidity. The results show that the BJ model provides better prediction results than ARX and ARMAX models. However, it maynot be concluded that the BJ model is better than the ARX and ARMAX models. The main difference between these three models is the assumption of noise model. For the system investigated, the BJ model’s way of handling noise is more suitable than those of ARX and ARMAX models. Moreover, the results for room temperature are better than those for relative humidity. The models obtained in this study can potentially be applied to control. The PID control is favourable on the assumption that the system model parameters do not change much, and when there is any change in the thermal behaviour of the building (e.g. more equipments are added to the building or there are changes related to office distribution), this causes a change in the thermal model parameters, and the response of PID control will deteriorate. Because the ARX, ARMAX and BJ models can be adapted by changing their parameters (based on immediate past records and actual performance), they can be integrated with PI and PID controller to ensure that the controllers are adaptable to any changes in the controlled environment [16]. In conclusion, these models could be used for the on-line control of HVAC systems and

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