Applied Energy 164 (2016) 69–88
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Applied Energy
journal homepage: www.elsevier .com/locate /apenergy
Developing a whole building cooling energy forecasting modelfor on-line operation optimization using proactive system identification
http://dx.doi.org/10.1016/j.apenergy.2015.12.0020306-2619/� 2015 Elsevier Ltd. All rights reserved.
⇑ Corresponding author. Tel.: +1 215 895 6941; fax: +1 215 895 1364.E-mail addresses: [email protected] (X. Li), [email protected] (J. Wen),
[email protected] (E.-W. Bai).
Xiwang Li a,⇑, Jin Wen a, Er-Wei Bai b
aDepartment of Civil, Architectural and Environmental Engineering, Drexel University, Philadelphia, PA 19104, USAbDepartment of Electrical and Computer Engineering, The University of Iowa, Iowa City, IA 52242, USA
h i g h l i g h t s
� Developed and verified a novel general methodology for building energy forecasting.� Quantitatively evaluated energy system nonlinearity and system response time.� Developed and adapted system identification model for building energy forecasting.� Compared the proposed system identification model against four inverse models.
a r t i c l e i n f o
Article history:Received 19 October 2015Received in revised form 30 November 2015Accepted 1 December 2015
Keywords:Building energy modelingModel based optimizationSystem identificationSystem nonlinearitySystem response timeMonte Carlo simulation
a b s t r a c t
Optimal automatic operation of buildings and their subsystems in responding to signals from a smart gridis essential to reduce energy demand, and to improve the power resilience. In order to achieve such auto-matic operation, high fidelity and computationally efficiency whole building energy forecasting modelsare needed. Currently, data-driven (black box) models and hybrid (grey box) models are commonly usedin model based building control. However, typical black box models often require long training periodand are bounded to building operation conditions during the training period. On the other hand, creatinga grey box model often requires (a) long calculation time due to parameter optimization process; and (b)expert knowledge during the model development process. This paper attempts to quantitatively evaluatethe impacts of two significant system characteristics: system nonlinearity and response time, on theaccuracy of the model developed by a system identification process. A general methodology for buildingenergy forecasting model development is then developed. How to adapt the system identification processbased on these two characteristics is also studied. A set of comparison criteria are then proposed to eval-uate the energy forecasting models generated from the adapted system identification process againstother methods reported in the literature, including Resistance and Capacitance method, Support VectorRegression method, Artificial Neural Networks method, and N4SID subspace algorithm. Two commercialbuildings: a small and a medium commercial building, with varying chiller nonlinearity, are simulatedusing EnergyPlus in lieu of real buildings for model development and evaluation. The results from thisstudy show that the adapted system identification process is capable of significantly improve theperformance of the energy forecasting model, which is more accurate and more extendable under bothof the noise-free and noisy conditions than those models generated by other methods.
� 2015 Elsevier Ltd. All rights reserved.
1. Introduction
Buildings are responsible for over 40% of the primary energyand 70% of the electricity consumption in the U.S. [1] More than25% of the U.S. electricity demand could be dispatchable if build-
ings can respond to the dispatch through advance operation strate-gies and smart grid infrastructure [2]. Recently, model basedpredictive control (MPC) has been proven to be a promising solu-tion for this active operation [3]. As the basis of MPC, high fidelityand computationally efficient building energy forecasting modelsare indispensable. How to develop an accurate, robust, and cost-effective building energy forecasting model is an urgent problemand therefore the objective of this study. The goals of this paperare twofold. One is to propose a system identification methodology
70 X. Li et al. / Applied Energy 164 (2016) 69–88
that is able to adapt based on a building’s characteristics, to gener-ate a whole building cooling energy forecasting model. The otherone is to compare the performance of the proposed methodologywith other modeling methods reported in the literature.
Although there are a large number of studies regarding buildingenergy forecasting using different methods, they all can be catego-rized as white box, black box and the grey box models. All thesethree types of models have their own limitations in applicationto real field building control. For example, black box models, suchas autoregressive exogenous (ARX), Artificial Neural Networks(ANN), Support Vector Machine for Regression (SVR), and N4SIDstate space model have been applied in building energy forecastingand control studies [4–11]. These data-driven models, however,often require long training period and the model extensibility islimited to the training data. In this study, model extensibility isdefined as the forecasting accuracy of a model, when it is subjectto weather and operation conditions that are different from thoseduring the model training period. This is an important model prop-erty because building systems are often nonlinear systems. Amodel that is trained using one range of operating/weather condi-tions often is not usable for a different operating/weather condi-tion. Grey box models, such as Resistance and Capacitance (RC)network and lumped parameters models, are popular models inbuilding control and operation studies [3]. They are widely usedin MPC for buildings such as those to estimate the cooling energyconsumption [12–14], to utilize the building passive thermal massstorage [15–17], or to utilize active thermal storage devices [18,19]and the energy generation systems [20,21] to reduce energy con-sumption or energy cost. Even though different advance parameterdetermination methods have been implemented to identify theparameters of the grey box models [12,13], the parameter determi-nation process is often computational demanding. In [14], theauthors developed a method for parameters and variable selectionusing Singular value decomposition and solving the RC equation infrequency domain. Developing the structure of a gray box model,however, often requires expert knowledge, and the parameterdetermination process is also time consuming. Therefore, whenapplying these modeling approaches in the real field, each of theseapproaches has its own barriers such as training data availability/quality, implementation time, and implementation cost (whenexpert knowledge is required).
In order to solve technique gaps from these methods, somestudies started to combine different methods to improve the modelperformance. Lee and Tong [22] presented a hybrid grey modelwith genetic programming for energy consumption forecasting.Fux et al. [23] combined RC model with Kalman filter to improvethe model accuracy and robustness. Lü et al. [24] developed a com-bined RC and autoregressive-integrated-moving-average (ARIMA)model for heterogeneous building energy forecasting. These meth-ods tried to reduce the efforts in the grey-box modeling, but theinherited limitations from the grey box models are still there. Itis also difficult to develop a general model structure for differentbuildings, and it requires high engineering effort in implementingit into real model predictive controllers. On the other hand, datadriven models have also been combined with Kalman filter[25,26] to improve the data driven model performance by bringingin the real measurements. Similarly, the inherited drawbacks ofdata driven models still cannot be solved there.
As results, a novel generalmethodology for building energy fore-castingmodel development has been proposed and validated in thisstudy to solve the limitations of the existing methods. Differentfrom the above described modeling approaches, which collects sys-tem data in a passive manner, system identification (SID) is a pro-cess of developing or improving a mathematical representation ofa physical system using data that is collected from a designedoperation or experiment, in an active manner. Although system
identification techniques have been widely used in other engineer-ing applications, there are only limited applications in the buildingenergymodeling field. In an earlier study by the authors [27], a sys-tem identificationmethodology, using frequency response functionwith an active system excitation, is proposed and tested for build-ing energy forecasting. The method is demonstrated to be able todevelop accurate and computationally efficient energy forecastingmodel for a small commercial building. However, when the pro-posed SID process is applied to develop an energy forecastingmodelfor a medium commercial building, the model accuracy is not satis-factory. It is suspected that a building system’s nonlinearity andresponse time affect the SID model’s accuracy since frequencyresponse function method is better used for more linear systems[28]. Therefore, this study focuses on investigating such impactsand how to adapt the SID process systematically based on asystem’s nonlinearity and response time. The goal is to develop asystematic SID methodology which can be scaled for buildings withvarying nonlinearity and response time.
This study firstly proposes a method to quantitativelydetermine a system’s nonlinearity and response time, and theirimpacts on the SID model development. Based on such character-istics (nonlinearity and response time), a methodology is thendeveloped to adapt the SID modeling process. A comparison studyis also conducted to evaluate the performance of the adapted SIDmodel, developed based on a building’s nonlinearity and responsetime, against literature-reported RC model, SVR model, ANN modeland N4SID model. Four criteria, namely, energy forecasting accu-racy, calculation speed, extendibility and uncertainty are used forthe model performance comparison. Again, forecasting extendibil-ity concerns the model forecasting accuracy when the weatherand/or operating conditions are different from those during thetraining period. Forecasting uncertainty concerns the model fore-casting performance when training and forecasting data containsnoise. Two commercial building, a small and a medium commer-cial building, with varying chiller nonlinearity, are simulated usingEnergyPlus in lieu of real buildings for model development andcomparison. In the following sections, the methodology for systemcharacteristics test and SID model development is introducedfirstly in Section 2, the EnergyPlus modeling and data generationprocess are discussed in Section 3, the system characteristic testresults and SID model adaptation results are summarized in Sec-tions 4 and 5, and then the comparison study is presented inSection 6.
2. Methodology
In this section, the test method used to determine a system’snonlinearity and response time is first introduced. How to adaptthe SID model development based on the nonlinearity andresponse time are then discussed.
2.1. Building energy system characteristics test method
2.1.1. System nonlinearity testIt is believed that a system’s nonlinearity is one of the most
important characteristics for a system’s model development, espe-cially for nonparametric methods [29]. In this study, a magnitudesquared coherence based method for system nonlinearity test[29] is adopted. This method is based on the cross-spectral densityof the inputs and outputs:
Cxy ¼Sxy�� ��2SxxSyy
ð1Þ
where the magnitude squared coherence (Cxy) estimate the powertransfer between input and output to estimate the causality
X. Li et al. / Applied Energy 164 (2016) 69–88 71
between system input and output. Sxy is the cross power spectraldensity between system inputs (x), such as outdoor air temperature,and system output (y), such as building energy consumption. Sxxand Syy are the auto power spectral density of x and y, respectively.They can be estimated from the Fourier transformation of theauto-correlation of the inputs (Rxx) and outputs (Ryy), and thecross-correlation between inputs and outputs (Rxy). The equationsfor the transformation are presented as Eqs. (2)–(4):
Sxy ¼ 1N
XNs¼1
RxyðsÞe�j2pksl ð2Þ
Sxx ¼ 1N
XNs¼1
RxxðsÞe�j2pksl ð3Þ
Syy ¼ 1N
XNs¼1
RyyðsÞe�j2pksl ð4Þ
where l is sampling window length of the spectral density analysis.N is the number of sampling windows. The purpose of the Fouriertransformation is to convert the signal into frequency domain tocapture the system dynamics. Rxy is calculated in (Eq. (5)), Rxx andRyy are calculated in Eq. (6).
Rxy sð Þ ¼ 1l
Xl�1
i¼1
u ið ÞyT iþ sð Þ ð5Þ
RxxðsÞ ¼ 1l� s
Xl�ss¼1
xðjÞxTðjþ sÞ ð6Þ
During an auto-correlation process, the similarity between anobservation and the same variable with a time lag is analyzed todiscover similar patterns in a signal. Cross-correlation is a measureof similarity between one signal and another signal with lag time.For example, cross-correlation can be used to examine the similar-ity between temperature setpoint and cooling energy consump-tion. The power spectral density describes how the power of asignal is distributed over different frequencies. The cross power
Fig. 1. Data sampling window and excitation injection during a
spectral density can be calculated from the Fourier transformationof the cross-correlation between two signals, and the auto powerspectral density can be calculated from Fourier transformation ofthe auto-correlation of one signal.
In this nonlinearity evaluation process, the analysis time period(e.g. one day) will be divided into multiple moving Welch’s over-lapped l-long (e.g. 6 h) segments. The nonlinearity evaluation pro-cess (Eqs. (2)–(4)) will be conducted in each segment. Theoverlapping portion in this study is chosen as 50%. That meansthe 6-h sampling window will move forward for 3 h at each time.The sampling window and excitation injection are illustrated inFig. 1. The excitation signals will be injected into the system ateach excitation injection time. The injection interval is predeter-mined and will be updated based on the system response time testresults. The details about system response time test will be intro-duced in Section 2.1.2. During the nonlinearity evaluation, the non-linearity index (Cxy), frequency response (Suu and Syu in Fig. 2), andMarkov parameters (G(t) in Fig. 2) will be evaluated at each mea-surement time step for each sampling window. In the initial tryingcase, the time step is 15 min. So there will be 60
15 � 6 ¼ 24 nonlin-earity indexes for each sampling window. As the sampling windowmoving forward, the total number of sampling windows will be243 � 7 ¼ 56. Where 24 is the hours per day, 3 is the sampling win-dowmoving speed, and 7 is the number of days in the initial tryingcases.
At last, the results from each segment (sampling window) willthen be ‘‘time-averaged” to generate an overall nonlinearity index.The reason why overlapping segments window are used here isthat most ‘‘window” functions afford more influence to the dataat the center of the data set than to data at the edges, which rep-resents a loss of information [29]. To mitigate that loss, the individ-ual data sets are commonly overlapped in time. Therefore, the finalnonlinearity indexes will be:
Cxy ið Þ ¼Pn¼N
n¼1Cxy;nðiÞN
ð7Þ
nonlinearity evaluation and system identification process.
Fig. 2. SID model development procedure.
Table 1Variables of system identification model.
Variable Variable name Type
Ebldg Building cooling energy (J) OutputTout Outdoor air temperature (C) InputTzone,stp,i Zone i temperature setpoint (C) InputRin,i Lighting/equipment schedule in zone i (–) InputQdir Direct solar radiation (W/m2) InputQdif Diffuse solar radiation (W/m2) InputVoa Ventilation rate (m3/s) Input
72 X. Li et al. / Applied Energy 164 (2016) 69–88
where Cxy ið Þ is the overall nonlinearity index at the time stepi; Cxy;nðiÞ is the nonlinearity index for the nth sampling window atthe time step i, and N is total number of sampling window.
If Cxy ¼ 1, then the system is a linear system. Supposey tð Þ ¼ h tð ÞxðtÞ is a linear system, the nonlinearity index can be cal-culated in Eq. (8).
Cxy ¼Gxy
�� ��2GxxGyy
¼ Hðf ÞGxxðf Þj j2G2
xxðf Þ Hðf Þj j2¼ Gxxðf Þj j2
G2xxðf Þ
¼ 1 ð8Þ
If 0 < Cxy < 1, then the system is a nonlinear system. The proveof this statement can be found in [29]. And the closer the Cxy is to 1,the more the system behaves like a linear system. Details about thesystem nonlinearity test, including testing signal generation andtesting results are discussed in Section 4.1.
The length of a sampling window needs to be large enough toinclude all of the operational data that is influenced by each systemexcitation signal but is not too large so that the impact of differentexcitation signals can be differentiated. The sampling windowlength is initially chosen as 6 h, but is later updated based on thesystem characteristics as discussed in Section 5.2.
2.1.2. System response time testBesides system nonlinearity, system response time is another
critical factor in determining system identification methodology,especially when determining the excitation plan and samplingwindow for nonlinearity test and system identification. The excita-tion signal generation frequency should be calculated based on thesystem response time (Appendix A), and the signal injection inter-view and sampling window should be larger than the responsetime to allow the system to stabilize.
System response time is a measure of how quickly the systemresponds to an input change. System response time is usually mea-sured by experiments. For example, if an response of a dynamicsystem can be expressed as [29]:
x tð Þ ¼ axðt ¼ 0Þe�t=T ð9Þwhere T is the response time constant, the response time of a sys-tem measurement, x tð Þ, to reach 95% of its final steady state valueafter the input change, is defined as T0:95. a is the response coeffi-cient, which is 0.95 in this study. In this study, the building zonetemperature is chosen as the measurement in this response timeexperiment. How fast a building zone’s temperature stabilizesreflects how fast the building HVAC systems respond and how largea building’s thermal mass is. Since both the HVAC system capacity
and the building thermal mass could affect its zone temperatureresponse time, two tests are performed to evaluate the responsetime respectively. The first one is to change the zone temperaturesetpoint from one stabilized temperature to another value. Thenthe time is measured between the beginning of the setpoint changeand when the zone temperature reaches the 95% of the new set-point. This measured time is the first response time. This responsetime reflects the combined impacts of both building thermal massand the HVAC system capacity. The second test is to switch offthe HVAC system at night, when weather disturbances are minimal,and then measure the time that the zone temperature takes todecrease to a steady state. Here, this steady state is defined as lessthan 0.5% of the state change in 15 min. This second test evaluatesthe impact of building thermal mass on the system’s response time.The detailed testing procedure and results of the system nonlinear-ity and response time tests will be presented in Section 4.2.
2.2. System identification method
Various types of model structure and signal excitation methodsexist in SID methods. Even though building energy systems arenonlinear systems, the nonlinearity is often found to behave in arelative longer time interval. Linear models, when selected prop-erly, could still lead to satisfactory forecasting results [3]. Themodel structure and signal excitation methods chosen here arebased on those reported in [27]. In [27], frequency response func-tion approach is used due to its good performance in capturing sys-tem dynamics in frequency domain and computation efficiency[29].
In our previous study [27], where a small commercial buildingis studied, the selection of input and output variables aresummarized in Table 1. The input variables are chosen based ontheir availability from a typical commercial building. Although
X. Li et al. / Applied Energy 164 (2016) 69–88 73
the chosen solar inputs (Qdir and Qdif) are not commonly availablefrom a building’s control system, they can be obtained from anearby weather station or National Solar Radiation Database [30].In this study, a medium commercial building is the focus. Due tothe increased complicity and nonlinearity, which is discussed inSection 4, three SID models, one for each floor, is used to forecastthe energy consumption. The overall building energy is thesummation of these three SID model output. In each SID model,the inputs and output are similar to those described in Table 1.
Buildings are usually operated within a very narrow range oftemperature setpoints and internal equipment schedules. To pro-vide training data that cover a wider range of operating conditions,the zone temperature setpoint, and equipment operation scheduleare chosen to be modified systematically (excited),
Details about how to generate training data for a SID model, i.e.,data collected when the building is actively excited, is described in[27,31]. Fig. 2 also provides a brief summary. In this figure, U is theinputs during a training period, Y is the outputs data during atraining period, PSD is the power spectral density model for inputs,and CPSD is cross power spectral density model for input andoutput. Suu and Syu are the results of PSD and CPSD, respectively.As illustrated in Fig. 1, the PSD and CPSD will be calculated for eachsampling window. Then, for the entire training period, all of thePSDs and CPSDs calculated from each sampling windows will beaveraged obtain overall PSD and CPSD. Finally, GðzÞ which is thefrequency response function as a transfer function in frequencydomain, is calculated as the ratio of the overall results of CPSDbetween system output and each input (Syu) to the results ofPSD between each input (Suu). GðzÞ will be converted to timedomain transfer function GðtÞ using inverse Fourier functiontransformation. The results of the inverse Fourier functiontransformation are then saved as a set of Markov parameters formodel forecasting. Notice that the Markov parameters are vectorswith a dimension of the length of sampling window length, andfor every measurement time step there is a Markov parameterfor each input variable. Therefore, the SID model proposed anddeveloped in this study is presented as a Markov model. Finally,y is the forecasting output, which is calculated as the convolutionof the Markov parameters and system inputs. The detailedprocedure of Markov parameter calculation has been introducedin Section 2.1.1.
Suggested by an early study [32], sum of sinusoids function isused to generate excited inputs (Eq. (10)).
Usþ1 ¼ Us þffiffiffiffiffiffiffiffi2as
psinðxstT þusÞ ð10Þ
where U is the excitation signal (such as temperature setpoint);ffiffiffiffiffiffi2a
pis a magnitude scale parameter from 0 to 1; T is the sampling
time, and u is the phase lag parameter from 0 to 2p, which do notaffect the signal spectrum; x is periodic frequency parameter from0 to 2p. All these parameters, including
ffiffiffiffiffiffi2a
p; T; u, and x, will be
updated at each time step, s. The detailed process to determineand update the model parameter during the training period isdiscussed in [27] and also briefly summarized in Appendix A.
3. Building description simulation and data generation
3.1. Building description
To evaluate how building size, envelope type, and HVAC sys-tems affect the SID process, two building sizes with different HVACsystems are used in this study as the objects. One is a single storyfive zone small office building (same as that used in [27]). It hasfive conditioned zones and an unconditioned attic. The total floorarea is 510 m2. The window-to-wall ratio is approximately 21.2%.The overall U-factor of its single pane windows is 3.4 W/m2 K with
a solar heat gain factor of 0.36. The overall U-factor of the externalenvelopes is 0.68 W/m2 K. The R-value and solar absorptivity valueof the roof are 5.1 W/m2 K and 0.9, respectively. The HVAC systemsused in this building are constant-air-volume (CAV) air handlingunits (AHUs) with five direct expansion (DX) coils. Each DX coilserves on zone. The nominal COP is 3.07.
The other building is a three-story office building, and each floorhas five conditioned zones. The total floor area is 4982 m2. Thewindow-to-wall ratio is around 33%, and the U-factor of these win-dows is 3.3 W/m2 K with a solar heat gain factor of 0.36. The R-value and solar absorptivity value of the roof are 0.33 W/m2 Kand 0.7, respectively. This building uses multi-zone variable-air-volume (VAV) system with electric reheat. The VAV system isserved by a chiller with 414 kW capacity and a nominal COP of 2.8.
Very detailed physics based building simulation models usingEnergyPlus are used in this study in lieu of real building system[33]. Extensive studies about calibrating and validating these Ener-gyPlus models against real field data are conducted by NationalRenewable Energy Laboratory [34] and Pacific Northwest NationalLaboratory [35]. Therefore these models are used as referencemodels for other research and projects. In the small buildingmodel, the coefficients of performance (COP) of the DX coils ismodeled using quadratic equation [34]. In the medium buildingmodel, the performance of the central chiller system is modeledusing second-order and third-order polynomial equation [34].The location of these two building is selected as in Philadelphia,PA, USA for this study. Typical Meteorological Year (TMY) weatherdata file provided by the DOE is used as the weather input. A vir-tual building cluster emulator [36] is used here to simulate thebuilding operation and apply the introduced system identificationschemes.
3.2. Training and testing condition
The two buildings, as described in Section 3.1, are simulated forthree time periods under Philadelphia TMY data: (1) from August1st to August 7th; (2) from August 1st to August 14th; (3) fromAugust 22nd to August 28th; and (4) from July 1st to September30th. Simulated building operation data during time period 1 isused as training data for the SID model. Simulated data during timeperiod 2, which is longer than time period 1, is used as the trainingdata for other models in the comparison study (described in Sec-tion 6.2.1). Simulated data during time period 3 is used for modeladaptation (described in Section 5.2) and model uncertainty com-parison study (described in Section 6.2.3). Simulated data duringtime period 4 is used as the testing data for all models.
3.3. Data noise
Two sets of simulated data are generated in this study for bothbuildings and for all time periods. One set of data assumes no mea-surement noise and is labeled as noise-free data in the followingsections. The other set of data includes measurement noise, byadding Gaussian distributed random white noise to each measure-ment. More details about the noise generation process will beintroduced in Section 6.
4. Results of system characteristic testing
The results of the system nonlinearity and response time anal-ysis are discussed in this section. These results are later used toadapt the SID model to improve the overall model performance.
74 X. Li et al. / Applied Energy 164 (2016) 69–88
4.1. System nonlinearity test results
Using the system nonlinearity test methods introduced in Sec-tion 2.1.1, the system nonlinearity between each system inputand output is calculated for both of the two buildings. Section 4.1.1describes the observations of the nonlinearity results using theoriginal excitation scheme used in our previous study [27] (Sec-tion 2.4). Based on these observations, the excitation scheme isadjusted for the medium building. The nonlinearity test resultsusing the new excitation scheme are summarized in Section 4.1.2.
4.1.1. Results using original excitation schemeFig. 3 summarizes the nonlinearity test results for both buildings
(a for small and b for medium building) using excitation schemedescribed in Section2.1.1. In Fig. 3, each subplot illustrates the calcu-lated nonlinearity index C (Eq. 1) between the output (building cool-ing energy) and a specific input, as a function of the input frequency.
(a) Small building nonlinearity test
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Tout
Non
linea
rity
Inde
x
0 0.2 0.40
0.2
0.4
0.6
0.8
1Tzo
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Qdif
Normalized Frequency (h-1)
Non
linea
rity
Inde
x
0 0.2 0.40
0.2
0.4
0.6
0.8
1R
Normalized F
0 0.5 10
0.2
0.4
0.6
0.8
1Tout
Non
linea
rity
Inde
x
00
0.2
0.4
0.6
0.8
1T
0 0.5 10
0.2
0.4
0.6
0.8
1x 10-5 Qdif
Normalized Frequency (h-1)
Non
linea
rity
Inde
x
00
0.2
0.4
0.6
0.8
1x 10-5
Normalized
(b) Medium building nonlinearity
Fig. 3. Nonlinearit
Each point on the plot represents an averaged nonlinearity index Cxy
over the samplingwindow (6 h). Comparing the nonlinearity resultsbetween these two buildings (Fig. 3a and b), it is shown that the val-ues of nonlinearity indexes for each input in small building are ingeneral higher than the corresponding ones in medium building.This indicates that the relationships betweensystem inputs andout-put in the medium building are more nonlinear than those in thesmall building. For example, the nonlinearity index is between 0.1and 0.5 for zone temperature setpoint input in the small building,while that for the medium building is less than 0.001 (Fig. 3b).
Another observation is that the nonlinearity index often varieswith the system input frequency. For example, the nonlinearityindex for zone temperature setpoint input is much higher whenthe frequency is lower. This indicates that the relationshipbetween the system output, building energy consumption, andsystem input, zone temperature setpoint, is much more linearwhen the setpoint does not vary too frequently (less than 0.2 for
results (subplot title indicates the input)
0.6 0.8 1
ne,setp
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Qdir
0.6 0.8 1
in
requency (h-1)0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1Voa
Normalized Frequency (h-1)
0.5 1
sep
0 0.5 10
0.2
0.4
0.6
0.8
1x 10-5 Qdir
0.5 1
Rin
Frequency (h-1)0 0.5 1
0
0.2
0.4
0.6
0.8
1x 10-4 Voa
Normalized Frequency (h-1)
test results (subplot title indicates the input)
y test results.
TM
H
X. Li et al. / Applied Energy 164 (2016) 69–88 75
small building and less than 0.1 for medium building). Similarobservation can be found for equipment schedule (Rin).
These observations are important information for SID modeldevelopment, since the SID model used in this study is more accu-rate when representing linear relationships. For example, the zonetemperature setpoint and equipment excitation frequencies needto be selected in a lower frequency region to yield a more linearrelationship between inputs and the output.
4.1.2. Solar radiation delay factorIn addition, the nonlinearity indexes between the solar radia-
tion (direct and diffuse solar radiation) and the system outputare less than 0.001 in both small and medium building cases. Itis well-understood that the impact of solar radiation on a build-ing’s energy consumption is often delayed due to a building’s ther-mal mass storage [37]. Therefore, in order to capture the impactfrom this time delay, a variable correlation test is conducted. In thistest, the correlations between the solar radiations (direct and dif-fuse) and the cooling energy consumption with varying time delay(30 min to 3 h) is studied using the building cluster emulator. Theresults are summarized in Table 2 for small building and in Table 3for medium building.
As illustrated in Table 2, the solar radiation with 1.5-h timedelay has the highest correlation with the cooling energy con-sumption. However, in the medium building case, as shown inTable 3, the solar radiation time delay varies from floor to floor.For the first floor, 1 h time delay yields the highest correlationbetween solar radiation and cooling energy consumption; for thesecond floor, 1.5-h time delay yields the highest correlation; forthe third floor, 2-h time delay yields the highest correlation. There-fore, in the following model adaptation process, the solar radiationtime delay will vary accordingly.
4.1.3. New excitation scheme and resultsThe system nonlinearity test results for medium building indi-
cate that the relationships between the system inputs and outputare fairly nonlinear. In order to use the proposed SID method,
Table 2Small building solar radiation time delay correlation test results.
Time delay (h) Correlation factor between building cooling energyand
Direct solar irradiance Diffuse solar irradiance
0 0.482 0.5170.5 0.507 0.5411 0.513 0.5551.5 0.518 0.5652 0.496 0.5642.5 0.478 0.5583 0.449 0.586
Highest correlations are in bold.
able 3edium building solar radiation time delay correlation test results.
Time delay (h) Correlation factor between first floor coolingenergy and
Correlation factorenergy and
Direct solar irradiance Direct solar irradiance Direct solar irradia
0 0.674 0.632 0.6390.5 0.692 0.646 0.6591 0.702 0.652 0.6721.5 0.701 0.649 0.6772 0.692 0.637 0.6732.5 0.671 0.617 0.6563 0.642 0.586 0.630
ighest correlations are in bold.
which is better to be used for linear systems, the following changesare made for the excitation scheme: (1) for the medium building,different excitation schemes are used for different zones to betteridentify the impacts of inputs from each zone on the buildingenergy consumption; (2) for both small and medium buildings,considering time delay of the solar inputs (introduced in Sec-tion 4.1.2); and (3) for both small and medium buildings, use morelow frequency excitation for zone temperature set point, equip-ment and ventilation flowrate.
Based on the new system excitation plan and system inputs, thesystem nonlinearity for both small and medium building are testedagain. The results of this new test are shown in Fig. 4. Comparedwith the original excitation scheme (Fig. 3a), the nonlinearity forsolar radiation (4th and 5th inputs) in the small building case(Fig. 4a) is improved significant. The improvement of the solarradiation nonlinearity indexes in the medium building test is notas significant as that in the small building case. This means thatthe relationship between the solar radiation and the buildingenergy consumption is more nonlinear in the medium buildingcase than that in the small building. For the medium building(Fig. 4b), the nonlinearity indexes for the zone temperature set-points (2nd to 6th inputs) under the new ‘‘zone-by-zone” excita-tion scheme are all between 0.5 and 0.8 for the four perimeterzones and between 0.2 and 0.8 for the core zone. This means thatthe relationship between individual zone input and the buildingenergy consumption is more linear than that of the lumped zoneinput. The new excitation scheme is later applied in the adaptedSID model development described in Section 5.
After the system nonlinearity tested using the new system exci-tation signals, the distribution of each excitation signals are alsoevaluated, as shown in Fig. 5. In the small building case (Fig. 5a),the frequency distribution of the input signals, except zone set-point (Tzone,stp), are mostly distributed in the lower frequency rangebetween 0 and 0.1. The zone setpoint (Tzone,stp) frequency is dis-tributed between 0.1 and 0.3. They all are in the frequency rangethat system nonlinearity index is closer to 1 (Fig. 4a). Similarresults are found in the medium building case (Fig. 5b), the systeminputs, outdoor air temperature (Tout), solar inputs (Qdir, Qdif), andequipment input (Rin), are all distributed in the frequency rangebetween 0.1 and 0.3. Core zone setpoint (Tcore,stp) is mostly dis-tributed between 0.2 and 0.8. South zone setpoint (Tsouth,stp) ismostly distributed between 0.1 and 0.6. West, north, and east zonesetpoint (Twest,stp, Tnorth,stp and Teast,stp) are distributed between 0.2and 0.7. Checking with the nonlinearity evaluation results inFig. 4b, they are also in the range with nonlinearity index closerto 1. Therefore, in all these input distribution regions, SID modelwill have better performance.
4.2. System response time test results
The system response time is tested using the buildingcluster emulator following the testing procedure introduced in
between second floor cooling Correlation factor between third floor coolingenergy and
nce Direct solar irradiance Direct solar irradiance Direct solar irradiance
0.6 0.603 0.5540.618 0.631 0.5810.628 0.655 0.6030.630 0.670 0.6160.623 0.677 0.6200.604 0.672 0.6130.576 0.657 0.595
(a) Small building nonlinearity test results
(b) Medium building nonlinearity test results
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Tout
Non
linea
rity
Inde
x
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Tzone,setp
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Qdir
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Qdif
Normalized Frequency (h-1) Normalized Frequency (h-1)Normalized Frequency (h-1)
Non
linea
rity
Inde
x
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Rin
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0.4
0.6
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1Voa
0 0.5 10
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0.4
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1Tout
Non
linea
rity
Inde
x
0 0.5 10
0.2
0.4
0.6
0.8
1Tcore,stp
0 0.5 10
0.2
0.4
0.6
0.8
1Tsouth,stp
0 0.5 10
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0.4
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0.8
1Twest
0 0.5 10
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0.4
0.6
0.8
1Tnorth,stp
0 0.5 10
0.2
0.4
0.6
0.8
1Teast,stp
Normalized Frequency (h-1)
Non
linea
rity
Inde
x
0 0.5 10
0.2
0.4
0.6
0.8
1Qdir
Normalized Frequency (h-1)0 0.5 1
0
0.2
0.4
0.6
0.8
1Qdif
Normalized Frequency (h-1)0 0.5 1
0
0.2
0.4
0.6
0.8
1Rin
Normalized Frequency (h-1)0 0.5 1
0
0.2
0.4
0.6
0.8
1Voa
Normalized Frequency (h-1)
Fig. 4. Building nonlinearity test results under new excitation plan.
76 X. Li et al. / Applied Energy 164 (2016) 69–88
Section 2.1.2. As introduced earlier, the system response time istested in two different ways with HVAC system being on (test 1)or off (test 2). Each test is then further divided into three subtests.For Test 1, each subtest indicates a different zone temperaturerange. In Test 2, each subtest indicates a different initial tempera-ture. The results are summarized in Table 4.
The results of test 1 (HVAC-system-on) show that the systemresponse time of the small building is generally less than 10 minfor most of the cases, while those in the medium building caseare generally over 45 min for most of the cases. These test resultsare understandable, considering that the medium building has lar-ger thermal mass. Therefore, it needs more time for the HVAC sys-tem to change the building temperature to a new setpoint in themedium building. This also explains the results of the second testthat it also takes longer time for the zone temperatures of the
medium building to reach to a stable value after the HVAC systemis turned off. Therefore, based on the system response time testresults, the system excitation frequency, excitation injection inter-val, and sampling window length should be adapted accordingly toallow enough time for the building systems to stabilize. The detailsabout the model adaptation based on the system nonlinearity andresponse time test is discussed in Section 5.2.
5. SID model adaptation
In this section, the SID model reported in [27] is adapted for themedium building case based on its system characteristics. Firstly,the original SID ([27], referred to as ‘‘pre-adaptation” hereafter)model is applied for both of the two buildings with varying HVAC
(a) Small building input signal frequency
0 0.2 0.4 0.6 0.80
500
1000
1500
2000Tout
coun
t #
0 0.2 0.4 0.6 0.80
500
1000
1500
2000Tzone,stp
0 0.2 0.4 0.6 0.80
500
1000
1500
2000Qdir
0 0.2 0.4 0.6 0.80
500
1000
1500
2000Qdif
Normalnized Frequency (h-1) Normalnized Frequency (h-1) Normalnized Frequency (h-1)
coun
t #
0 0.2 0.4 0.6 0.80
500
1000
1500
2000Rin
0 0.2 0.4 0.6 0.80
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1000
1500
2000Voa
0 0.2 0.4 0.6 0.80
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2000Tcore,stp
0 0.2 0.4 0.6 0.80
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2000Tsouth,stp
0 0.2 0.4 0.6 0.80
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1000
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2000Twest,stp
0 0.2 0.4 0.6 0.80
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1000
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2000Tnorth,stp
0 0.2 0.4 0.6 0.80
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1000
1500
2000Teast,stp
Normalnized Frequency (h-1)
coun
t #
0 0.2 0.4 0.6 0.80
500
1000
1500
2000Rin
Normalnized Frequency (h-1)0 0.2 0.4 0.6 0.8
0
500
1000
1500
2000Voa
Normalnized Frequency (h-1)
0 0.2 0.4 0.6 0.80
500
1000
1500
2000Tout
coun
t #
0 0.2 0.4 0.6 0.80
500
1000
1500
2000Qdir
NormalnizedFrequency (h-1)0 0.2 0.4 0.6 0.8
0
500
1000
1500
2000Qdif
Normalnized Frequency (h-1)
(b) Medium building input signal frequency
Fig. 5. Building input signal frequency histogram.
Table 4System response time test results.
Setpoint variation Time for 95% temperature change, T0.95 (test 1)
18–22 �C 22–28 �C 18–28 �C(22–18 �C) (28–22 �C) (28–18 �C)
Small 10.0 (9.5) min 8.0 (7.5) min 18.5 (9.0) minMedium 33.5 (49.0) min 69.0 (31.0) min 88.0 (67.5) min
Initial temperature Time for 95% temperature change, T0.95 (test 2)
18 �C 23 �C 28 �C
Small 47 min 39 min 23 minMedium 67 min 48 min 36 min
X. Li et al. / Applied Energy 164 (2016) 69–88 77
systems. In this section, the performance of the pre-adaptation SIDmodel, when used for the small and medium buildings, is summa-rized. Then how to adapt the SID model based on the system char-acteristics identified from Section 4 is discussed. The performanceof the adapted SID models for both buildings is summarized at last.
5.1. Pre-adaptation
5.1.1. Markov parametersAs introduced in Section 2.2, the energy forecasting model is
saved as a set of Markov parameters for each input. The Markov
(a) Small building Markov parameters
(b) Medium building Markov parameters
0 2 4 6
Mar
kov
Para
met
ers
-800
-400
0
400
800Tout
0 2 4 6-800
-400
0
400
800Tzone,stp
0 2 4 6-1
0
1Qdir
Time, h0 2 4 6
Mar
kov
Para
met
ers
-1
0
1Qdif
Time, h0 2 4 6
-1
0
1 Rin
Time, h0 2 4 6
-800
-400
0
400
800 Voa
0 3 4 5 6
Mar
kov
Para
met
ers
-1000
-500
0
500
1000Tout
0 3 4 5 6-1000
-500
0
500
1000Tcore,stp
0 3 4 5 6-1000
-500
0
500
1000Tsouth,stp
0 3 4 5 6-1000
-500
0
500
1000Twest,stp
01 2 1 2 1 2 1 2 1 2 3 4 5 6-1000
-500
0
500
1000Tnorth,stp
Time, h0 3 4 5 6
Mar
kov
Para
met
ers
-1000
-500
0
500
1000Teast,stp
Time, h0 3 4 5 6
-10
-5
0
5
10Qdir
Time, h0 3 4 5 6
-10
-5
0
5
10Qdif
Time, h0 3 4 5 6
-1
-0.5
0
0.5
1Rin
Time, h01 2 1 2 1 2 1 2 1 2 3 4 5 6
-1
-0.5
0
0.5
1Voa
Fig. 6. Building SID model Markov parameters.
78 X. Li et al. / Applied Energy 164 (2016) 69–88
parameters of the SID models for these two buildings are plotted inFig. 6. From Fig. 6, it can be observed that the Markov parametersfor solar radiation (both buildings), equipment schedule (both twobuildings) and ventilation rate (medium building) are much smal-ler than other variables. Therefore, the y-axis ranges for those vari-ables are smaller than other variables.
All Markov parameters of the small building model (Fig. 6a)converge to zero at the end of the sampling window. However, inthe medium building case (Fig. 6b), the Markov parameters for out-door air temperature (1st inputs), temperature setpoints (2nd to6th inputs), and solar radiations (7th and 8th inputs) do not con-verge to zero at the end of the sampling period (6 h). This indicatesthat the 6-h sampling window is not long enough for this mediumbuilding. Therefore, considering the truncating error, the samplingwindow lengths for medium building need to be adaptedaccordingly.
5.1.2. Pre-adaptation SID model energy forecasting resultsIn order to evaluate the forecasting accuracy, three indexes are
employed, namely, Coefficient of Determination (R2), Root MeanSquare Error (RMSE) and Normalized Root Mean Square Error(NRMSE):
R2 ¼Pn
i¼1ðxi � �xÞðxi � �xÞPni¼1ðxi � �xÞ2Pn
i¼1ðxi � �xÞð11Þ
RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn
i¼1ðxi � xiÞ2n
sð12Þ
NRMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn
i¼1ðxi � xiÞ2n
s ,ðxmax � xminÞ ð13Þ
where xi and xi is the true and forecasting value; �x and �x are theaverage of true and forecasting value, respectively.
(a) Small building input signal frequency
(b) Medium building input signal frequency
0 0.2 0.4 0.6 0.8 10
500
1000
1500Tout
coun
t #
0 0.2 0.4 0.6 0.8 10
500
1000
1500Tzone,setp
0 0.2 0.4 0.6 0.8 10
500
1000
1500Qdir
0 0.2 0.4 0.6 0.8 10
500
1000
1500Qdif
Normalnized Frequency (h-1) Normalnized Frequency (h-1) Normalnized Frequency (h-1)
coun
t #
0 0.2 0.4 0.6 0.8 10
500
1000
1500Rin
0 0.2 0.4 0.6 0.8 10
500
1000
1500Voa
0 0.2 0.4 0.6 0.8 10
500
1000
1500Tout
coun
t #
0 0.2 0.4 0.6 0.8 10
500
1000
1500Tcore,stp
0 0.2 0.4 0.6 0.8 10
500
1000
1500Tsouth,stp
0 0.2 0.4 0.6 0.8 10
500
1000
1500Twest,stp
0 0.2 0.4 0.6 0.8 10
500
1000
1500Tnorth,stp
0 0.2 0.4 0.6 0.8 10
500
1000
1500Teast,stp
Normalized Frequency (h-1)
coun
t #
0 0.2 0.4 0.6 0.8 10
500
1000
1500Qdir
Normalized Frequency (h-1)
0 0.2 0.4 0.6 0.8 10
500
1000
1500Qdif
Normalized Frequency (h-1)
0 0.2 0.4 0.6 0.8 10
500
1000
1500
2000Rin
Normalized Frequency (h-1)
0 0.2 0.4 0.6 0.8 10
500
1000
1500
2000Voa
Normalized Frequency (h-1)
Fig. 7. Building input signal frequency histogram in forecasting period.
X. Li et al. / Applied Energy 164 (2016) 69–88 79
In Section 4.1.3, the frequency distribution of system inputs areexamined for the training data. In this section, these frequency dis-tributions are re-examined to see whether the relationshipbetween system output and inputs are also close to linear duringa normal system operation. The input signal distributions duringthe forecasting period are plotted in Fig. 7. In the plot for smallbuilding (Fig. 7a), the input signals are mostly distributed in thefrequency range between 0 and 0.2, except equipment schedule(Rin), and ventilation rate (Voa). Rin distributes more in the fre-quency range between 0.6 and 0.8 and Voa has higher distributiondensity in the following two ranges: 0–0.2 and 0.6–0.8. When com-paring these ranges to their corresponding nonlinearity index dis-tribution (Fig. 3), it is found that all of these distribution rangeswill yield a system nonlinearity index that is also closer to 1.Therefore, the normal input signals in the real building are moredistributed in the range where the system behaves closer to alinear system. In the medium building case (Fig. 7b), the signalsare distributed more in the frequency range of 0–0.4, where the
system nonlinearity index is also higher. Therefore, the SID modeldeveloped under the new excitation scheme matches well to thereal building operation case, which will guarantee the modelperformance.
Again, as described in Section 3.2, training data (August 1st to7th) are used firstly to developed the pre-adaption SID model.The model is then tested in time period 3 (August 22nd to August28th). During a model testing, the model is given an initial condi-tion and then is used to forecast building energy consumption(model output) for the entire testing time period. The detailed test-ing results for pre-adaption SID model are illustrated in Fig. 8.Model forecasting accuracy and speed are summarized in Table 5.It is showed that, for the small building case, the pre-adaption SIDmodel is able to achieve acceptable forecasting accuracy (R2 > 0.95)within less than 1 min calculation time. However, the performanceof the pre-adaption SID model in the medium building case ismuch worse (R2 < 0.8). As a result, adaptation of the SID model toimprove its performance is desired for the medium building case.
(a) Energy forecasting results in the small building case.
(b) Energy forecasting results in the medium building case.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1
2
3
4
5
6
7
8
9
Time, day (0824-0828)
Coo
ling
Ener
gy, k
W
"Measured" value (EnergyPlus),kWForecasted value (SID),kW
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
10
20
30
40
50
60
70
80
90
Time, day (0824-0828)
Coo
ling
Ener
gy, k
W
"Measured" value (EnergyPlus),kWForecasted value (SID),kW
Fig. 8. Building cooling energy forecasting results before adaptation.
Table 5Pre-adaptation SID model performance.
Building Calculation time (s) R2 RMSE (kW) NRMSE (%)
Small 27 0.96 0.48 7Medium 231 0.73 12.15 16
80 X. Li et al. / Applied Energy 164 (2016) 69–88
5.2. SID model adaptation
SID model adaptations, as discussed earlier, are applied in themedium building case to improve the performance of the SIDmodel. The details and results are reported in this section.
5.2.1. System excitation: variable, frequency and injection intervaladaption
In the pre-adaption SID model, the excitation signals for tem-perature setpoint and schedule in each zone are excited together.However, the building energy system in the medium building ismore complex and more nonlinear, as the results shown inFig. 3b, where the nonlinearity indexes are much smaller thanthose for the small building. Therefore, it needs more intense exci-tation to get enough training data for the SID model development.In this medium building case, each zone temperature is excitedindividually. Moreover, the excitation signal frequency is recalcu-lated using Eq. (A1) based on the response time (Table 4). As indi-cated by the system response time test results (Table 4), thesystem response time of the small building is within 30 min inall of the testing cases (except the one of turning off HVAC systemat 18 �C). The response time of the medium building is above
30 min in most of the testing cases. Therefore, the 30-min excita-tion interval in the original system excitation plan for the mediumbuilding is needed to be extended to 60 min to allow enoughresponse time for the tested system.
5.2.2. System identification: sampling window length modificationFrom the Markov parameter plots in Fig. 6, the Markov param-
eters for the medium building do not converge to 0 at the end of 6-h sampling window, but they converge to 0 at around 3th hour forall the inputs of the small building. In order to determine the sam-pling length, a parametric test is conducted for the sampling win-dow length with 4 h, 8 h, and 12 h for the medium building case.The Makarov parameters converge at the end of a 12-h samplinglength. Therefore, the sampling window length is adapted to 3 hfor the small building and to 12 h for the medium building.
5.3. Adapted SID model energy forecasting results
The adapted SID model is developed using the same trainingdata as described in Section 3 and is validated using the same test-ing data as for the pre-adaption SID model. The performance com-parison is tabulated in Table 6. It is clearly showed that theforecasting accuracy of the adapted SID model improved from0.91 to 0.96 for the small building, and from 0.73 to 0.94 for themedium building.
The detailed cooling energy forecasting results from theadapted and pre-adaption SID models are compared in Fig. 9 forthe medium building. Fig. 9a provides the time series comparisonand Fig. 9b summarizes forecasting error for pre-adaption and
Table 6Building SID model performance before and after adaptation.
Model Excitation interval, min Excitation frequency, h�1 Sample length, h R2 RMSE, kW NRMSE, %
Small pre-adaptation 30 1–6 6 0.91 0.66 9Small adapted 30 1–6 3 0.96 0.48 7Medium pre-adaptation 30 1–6 6 0.73 12.15 16Medium adapted 60 1–3 6 0.92 5.92 8
Highest correlations are in bold.
(a) Medium building energy forecasting results
(b) Medium building energy forecasting error analysis
0 05 1 15 2 25 3 35 4 45 50
10
20
30
40
50
60
70
80
90
100
Coo
ling
Ener
gy, k
W
Forecasted value (Adapted SID),kWForecasted value (Pre-adaptation SID),kW"Measured" value (EnergyPlus), kW
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
100
"Measured" Value (EnergyPlus), kW
Fore
scas
ted
Valu
e,kW
Forecasted value (Adapted SID)Forecasted value (Pre-adaptation SID),kWForecasted value="Measured" Value
Fig. 9. Medium building cooling energy forecasting results comparison and error analysis.
X. Li et al. / Applied Energy 164 (2016) 69–88 81
adaption SID models. This figure shows that the pre-adaption SIDmodel (blue1 circle) overestimates the energy consumption duringmost of the forecasting period, while the adapted SID model (blackcircle) produces better forecasting accuracy.
From the model development and adaptation, the proposedmethodology is proven to be able to develop high efficient energyforecasting models. This method is also easy to be implemented inother buildings at other cases. The system excitation for trainingdata collecting can be conducted at unoccupied hours and week-ends. Since the properties of buildings do not change, unless thebuildings are retrofitted. Therefore, the model development pro-cess only needs to be conducted once. This proposed methodologyhas been applied in a real commercial building successfully [38].
6. Building energy forecasting model performance comparison
In this section, the performance of the adapted SID model iscompared with four building energy forecasting methods that have
1 For interpretation of color in Figs. 9 and 10, the reader is referred to the webversion of this article.
been reported in the literature, i.e. RC & Chiller model [13,39], SVR[40], ANN model [41], and state space model (N4SID) [42]. Thesmall commercial building described above is used as the testingsubject. In the following comparison study, all of the models(except adapted SID model) are provided with a 14 day (August1st to August 14th in TMY data) training data. The training datafor the adapted SID model under system active excitation lastsfrom August 1st to August 7th. Testing data generated from July1st to September 30th are used for model performancecomparisons.
Besides the three evaluation indexes used before, forecastingextendibility and uncertainty are added into this comparisonstudy. Details about these two indexes are introduced in Section 1.
6.1. Four other models
6.1.1. RC & Chiller modelIn the comparison study, the RC model developed by Braun and
Chaturvedi [13] and by Wang and Xu [12] (with the Chiller modelbased on [39]) is used. The details about the model developmentcan be found in Appendix B.
82 X. Li et al. / Applied Energy 164 (2016) 69–88
The chiller is modeled as a third-order polynomial equation ofoutdoor temperature and chilled water supply temperature:
Pch ¼ a1 � Qrated � ða2 þ a3T þ a4T2 þ a5T
3 þ b1Q þ b2Q2
þ b3Q3 þ b4TQÞ ð14Þ
T ¼ ðTwb � TchwsÞ= Twb � Tchwsð Þrated ð15ÞQ ¼ Qch=Qch;rated ð16Þwhere Qch;rated (ton) is the rated chiller capacity; Tchws (�F) is chillerwater supply temperature, which is assumed as 35 �F in thisresearch; Twb (�F) is ambient wet bulb temperature; Qch (ton) isthe cooling load.
All of the parameters in the RC & Chiller model are determinedthrough pattern searching based optimization methods in MATLAB[43]. The objective function of this optimization is described as:
JðRs;Cs;psÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN
j¼1 QRC;j � QAct;j
� �2N � 1
sð17Þ
where QRC is the cooling energy forecasted from the RC & Chillermodel and QAct is the one reported by the EnergyPlus model; N isthe total time step of this whole simulation. Rs; Cs, and ps areparameters in the RC & Chiller model.
6.1.2. SVR modelThe SVR model [40] is developed based on support vector
machine classification. SVR approximate the function using:
f xð Þ ¼ x �u xð Þ þ b ð18Þwhere f xð Þ represents the feature spaces which are mapped fromthe input space x. The coefficients x and b can be estimated bysolving an optimization problem:
J ¼ min12kwk2 þ Kðx; xÞ ð19Þ
Subject toyi � f xið Þ 6 ef xið Þ � yi 6 e
�
The first term kwk2 is called regularized term. The second termKðx; xÞ is a function kernel term, which measures the error. In thisstudy, radial basis function kernel (Eq. (20)) was used in LibSVM[44].
K x; x0ð Þ ¼ exp �kx� xk22r2
!ð20Þ
Time, day 0 1 2 3
Coo
ling
Ener
gy F
orec
astin
g, k
W
0
2
4
6
8
10"Measured" value (EnergyPlus),kWForecasted value (Adapted SID),kWForecasted value (RC & Chiller),kWForecasted value (SVR),kWForecasted value (ANN),kWForecasted value (N4SID), kW
Fig. 10. Small building cooling energ
where x and x are real and forecasted energy consumption, respec-tively; r is user-defined parameter.
6.1.3. ANN modelANN is a supervised learning model inspired from biological
neural networks, and is widely used to estimate functions. ANNgenerally consists of a network of notes (neurons) and connections.The connections have different weights that are required to betuned in the training process.
In this study, an ANN model with 10 sublayers is developedusing a Matlab Neural Network Toolbox [45]. The inputs and out-put of the SVR and ANN models are the same as the SID model,described previously.
6.1.4. State space model using N4SIDIn this comparison study, the N4SID model in MATLAB system
identification toolbox [46] is used to develop a 5-order state spacemodel to forecast the building cooling energy consumption. N4SIDis chosen here because it has been used and recommended in theliterature [47,48] for the good performance in building energy fore-casting. The N4SID model is able to identify the system order andmodel parameters (A, B, C and D) for a state space model (Eqs.(21) and (22)):
dxdt
¼ Axþ Bu ð21Þy ¼ Cxþ Du ð22Þwhere x is the state vector, which cantinas all the forecasting statevariables, u is the control vector containing the control variables, yis the measurement vector.
In this comparison study, x is defined as the system inputs: Tout,Qdir, Qdif, and Voa. u is the control variables: Tzone,stp,i, and Rin,i.
6.2. Comparison results
In this section, the performance of the developed SID model iscompared against four other building energy forecasting modelsdescribed above. The forecasting accuracy, speed, extendibilityand uncertainty are compared in the following sections.
6.2.1. Accuracy and speedFirstly, all of the five models developed upon the training data
described above are used to forecast the cooling energy consump-tion during the testing time period (i.e. from August 22 to August
(0822-0828)4 5 6 7
y forecasting case study results.
Table 7Cooling energy forecasting model performance.
Model Linear/nonlinear Excitation Training period Testing period Calculation time (s) R2 RMSE (kW) NRMSE (%)
SID L Yes 0801–0807 0822–0828 21 0.96 0.48 7N4SID L No 0801–0814 0822–0828 2.88 0.89 1.11 15.6RC & Chiller NL No 0801–0814 0822–0828 451 0.87 0.83 11.70SVR NL No 0801–0814 0822–0828 0.02 0.69 1.04 14.59ANN NL No 0801–0814 0822–0828 2.78 0.93 0.68 9.60
Highest correlations are in bold.
Table 8Model extendibility testing condition.
Scenario Testingperiod
Training temperaturerange, �C
Testing temperaturerange, �C
1 0801–0831 20.1–35.6 17.6–31.12 0701–0730 20.1–35.6 12.0–37.33 0901–0930 20.1–35.6 6.7–34.0
X. Li et al. / Applied Energy 164 (2016) 69–88 83
28, TMY). The energy forecasting results from these five models arecompared with the ‘‘measured” cooling energy consumption, i.e.,simulated values from the EnergyPlus model. The time series com-parison is illustrated in Fig. 10. In this figure, the forecasted energyconsumption results from the adapted SID model (purple line) arecloser to the ‘‘measured” value (red line), while the results from theN4SID (cyan line) model are mostly different from the ‘‘measured”value. This figure also shows that the ANN model achieves similaraccuracy as the adapted SID model from the third to the seventhday, when the weather conditions are very similar to the trainingdata. The RC & Chiller model achieves similar results as theadapted SID model at the second day when the weather conditionsare different from the training days which causes the energy con-sumption to decrease suddenly. The forecasting accuracy and cal-culation speed of the five models are further compared in.Table 7. The results show that the adapted SID model achievesthe highest forecasting accuracy, where R2, RMSE and NRMSE are0.96, 0.48 kW and 7%, respectively. Even though the adapted SIDmodel requires longer calculation time than the SVR and ANN
Table 9Cooling energy forecasting model extendibility.
Scenario Forecasting accuracy, R2 F
Adapted SID N4SID RC & Chiller SVR ANN A
1 0.92 0.83 0.77 0.90 0.93 72 0.86 0.67 0.81 0.56 0.79 93 0.89 0.55 0.75 0.88 0.73 7
Highest correlations are in bold.
0 0.5 1 1.5 2 20
2
4
6
8
10
12
14
16
18
20
22
Time, day
Coo
ling
Ene
rgy
Fore
cast
ing,
kW
Fig. 11. Small building cooling energy
models, the calculation time for the adapted SID model for thisseven-day energy forecasting is only 0.004 s with the majority ofthe time spent on model training. Therefore the adapted SID modelspeed is acceptable.
6.2.2. Extendibility: outdoor weather conditionIn order to test and compare the extendibility of all five models,
three testing scenarios are designed using the same training periodas described previously. The testing periods for the three testingscenarios are designed so that Scenario 1 represents the situationwhen weather conditions during the testing period are similar tothose during the training period. Scenario 2 and Scenario 3 usethe months before and after the training period when weather con-ditions are often out of the range of the weather conditions duringthe training period. Table 8 summarizes the temperature duringtraining and testing periods for the three scenarios.
The five models are applied to forecast the cooling energy con-sumption in all these three scenarios. The forecasting result com-parison is summarized in Table 9. For Scenario 1, in which theweather conditions during the testing and training periods are sim-ilar, the ANN model achieves the highest R2 (0.93), followed by theadapted SID model, which achieved a similar R2 (0.92). For thesame scenario, the adapted SID model achieves the lowest NRMSE(7.4%), followed by the SVR and ANN models. The reason for ANNmodel achieving good accuracy in Scenario 1 is that the forecastingconditions in scenario 1 are very similar to the conditions in thetraining periods. Therefore, the model does not extend too much
orecasting accuracy, NRMSE
dapted SID (%) N4SID (%) RC & Chiller (%) SVR (%) ANN (%)
.4 11.6 16.3 9.6 9.8
.6 18.4 13.1 28.2 15.3
.9 33.9 17.8 10.6 18.3
.5 3 3.5 4 4.5 5
(0824-0828)
"Measured" value (EnergyPlus),kWForecasted value (Adapted SID),kWForecasted value (RC & Chiller),kWForecasted value (SVR),kWForecasted value (ANN),kWForecasted value (N4SID),kW
forecasting under DR operation.
84 X. Li et al. / Applied Energy 164 (2016) 69–88
in scenario 1. However, in Scenarios 2 and 3, in which weatherconditions are different in the testing and training periods, theadapted SID model performs much better than other four modelsand achieves the highest R2 and lowest NRMSE. Therefore, theadapted SID model presents much better extendibility thanother four models. Considering that the RC & Chiller model is ahybrid model, which is partially physics-based, it is not surprisingthat it has good extendibility. Yet, as a mostly data drivenmodel, the proposed SID model also demonstrates good extendibil-ity with much less development efforts than physics-based/hybridmodels.
6.2.3. Extendibility: indoor operation conditionIn order to test the model extendibility against different opera-
tion signals, a temperature setting schedule from an existingdemand response study is used here. Similar to the previous stud-ies, the SID model is training using building operation data undersystem excitation, while all the other models are based on the
Table 10Energy forecasting accuracy under DR operation.
R2 RMSE NRMSE (%)
Adapted SID 0.86 1.15 8.1N4SID 0.44 2.62 18.0RC & Chiller 0.81 1.32 9.0SVR 0.73 1.89 13.4ANN 0.57 2.35 17.3
Highest correlations are in bold.
1 2 3-100-80-60-40-20
0204060
1 2 3 4 5 6 7-60
-40
-20
0
20
40
60
Time, daySystem Identification Model, AdaptedSID
Coo
ling
Ener
gy, k
Wh
1 2 3 4 5 6 7-60
-40
-20
0
20
40
60
Time, daySupport Vector Machine, SVR
Coo
ling
Ener
gy, k
Wh
TiSubspac
Coo
ling
Ener
gy, k
Wh
Fig. 12. Boxplots of Monte Carlo daily energy co
building operation data under regular strategies. This comparisonstudy is also conducted in the small commercial building describedin Section 3.1. The temperature setpoints in the training periodsare: 26.7 �C from 0 am to 6 am, 24 �C from 6 am to 6 pm, and26.7 �C from 6 pm to 12 am for RC & Chiller, SVR, ANN andN4SID. Then these five different models are used to forecast theenergy consumption for five weekdays from August 24 to August28. The temperature setpoints in these five days are: 32 �C from12 am to 4 am, 18 �C from 4 am to 6 am, 24 �C from 6 am to6 pm and 32 �C from 6 pm to 12 am. The cooling energy forecastingresults from all these 5 models are plotted in Fig. 11. It roughlyshows that the ‘‘Adapted SID” model and ‘‘RC & Chiller” model havethe better accuracy.
The detailed energy forecasting accuracy statistics are tabulatedin Table 10. The adapted SID and RC & Chiller model achieved thehighest R2 above 0.8, followed by the SVR model of 0.73. For theforecasting error, obviously, adapted SID and RC & Chiller modelhave the lowest RMSE and NRMSE.
1 2 3 4 5 6 7-60
-40
-20
0
20
40
60
-60
-40
-20
0
20
40
60
4 5 6 7
Time, dayResistance-Capacitance and Chiller Model, RC & Chiller
Coo
ling
Ener
gy, k
Wh
Time, dayArtificial Neural Networks, ANN
Coo
ling
Ener
gy, k
Wh
1 2 3 4 5 6 7
me, daye Model, N4SID
nsumption forecasting error (08/22–08/28).
Table 11Energy forecasting accuracy in MC simulation, RMSE (kW h).
RMSE Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7
Adapted SID 6.08 3.88 4.80 4.82 6.51 4.83 10.55N4SID 19.5 21.9 39.9 35.7 38.7 38.9 43.0RC & Chiller 10.98 2.65 36.41 27.65 22.53 8.00 10.89SVR 13.17 7.40 33.25 25.25 25.57 11.15 12.09ANN 7.87 22.12 15.26 3.74 5.58 11.55 13.23
Highest correlations are in bold.
Fig. B1. RC model structure for building energy estimation.
X. Li et al. / Applied Energy 164 (2016) 69–88 85
6.2.4. Impact of measurement noiseA Monte Carlo (MC) simulation is conducted to analyze the
noise impact on model accuracy. During a MC simulation, the fol-lowing process is used:
1. Initialize MC simulation by defining input noise distributions(±5%) and adding the noise to the ‘‘measurement” (outputs fromthe EnergyPlus models).
2. Perform MC: for i = 1 . . .N (in this study, N is chosen as 1000).� Sample noise values from defined distributions (in this
study, 5% Gaussian distributed random white noise is used).� Run each model for energy forecasting.� Calculate MC output (daily energy consumption, kW h) from
each model.3. Analyze the performance of each model.
The Boxplots of the MC simulation output during the testingperiod 3 (August 22nd to August 28th) are shown in Fig. 12. In thisfigure, each box shows the 5–95% percentile of the energy forecast-ing error distribution against the EnergyPlus simulation results, themiddle lines show the mean value of the daily energy consumptionforecasting error (kW h) during the MC simulation. The whiskersrepresent the largest values within 1.5 times of the box range. Out-liers beyond this range (extreme worse forecasting cases) are indi-cated by the small dots above and below the whiskers.
As Fig. 12 shows, the SVR model and ANN models have loweruncertainty, which is reflected, by the narrow boxes. But, theiraccuracies are much worse than that of the adapted SID modeland RC & Chiller model. This means that the adapted SID modeland RC & Chiller model are more sensitive to the input noise, whilethey still can maintain better accuracy than SVR model and ANNmodel.
The energy forecasting accuracy in the MC simulation of eachmodel is summarized in Table 11. The results show that theadapted SID model maintains the lowest RMSE in 6 days (Day 1,3–7), while RC & Chiller model achieves the highest accuracy inDay 2 (seconded by the adapted SID model). By examining theweather condition, the temperature of Day 2 is much lower thanthe temperature of other days, which results in a much lower cool-ing energy consumption. This further shows that the RC & Chillermodel and SID model have better extendibility than other models.
On the other hand, the MC simulation results of N4SID modelhave the highest RMSE in 6 days (Day 1, 3–7) during this 7-daysimulation study, and in Day 2 the RMSE of N4SID simulation
result is slightly better than that of ANN model. As a result, it isconcluded that SID model has the lowest uncertainty and N4SIDmodel has the highest uncertainty in MC simulation study.
7. Conclusion and future work
In this paper, methods to test a building system’s nonlinearityand response time are developed. How to adapt a previouslyreported SID model based on the nonlinearity and response timetesting results is proposed to allow the SID model to be used indifferent types of buildings. Three key SID model parameters, i.e.system excitation signal generation frequency, injection interval,and sampling length, need to be adapted to ensure the model fore-casting accuracy. Simulation studies of a small and a medium com-mercial building show that after such adaptation, the accuracy (R2)of the adapted SID model is maintained above 90% for the smallbuilding case and is improved from 73% to 94% for the mediumbuilding case. The adapted SID model is then compared againstfour other building energy forecasting models that have beenreported in the literature: N4SID model, RC & Chiller model, SVRmodel, and ANN model. Comparison criteria include modelaccuracy, speed, extendibility and impact of measurement noise.The model comparison study results show that the adapted SIDmodel has the capability to achieve higher accuracy and betterextendibility under both of the noise-free and noisy conditions.Future efforts are needed to examine the developed methodologiesfor different types of buildings and HVAC systems, and under realworld conditions.
Acknowledgement
Financial support provided by the U.S. National Science Founda-tion (Award ID: 1239247) is greatly appreciated.
Appendix A
The dominant time constant range tHdom and tLdom makes it possi-ble to place the contents of input signals in the important frequencyrange. Therefore the frequency parameter of the signal generationfunction should be chosen within the range specified in Eq. (A1).
1bst
Hdom
6 xs 6as
tLdomðA1Þ
86 X. Li et al. / Applied Energy 164 (2016) 69–88
where as and bs are user-decisions on high and low frequencycontent. as determines the basis of how much fast the designedclosed-loop speed of response will be relative to open loop, whichis 2 typically. as specifies low frequency information correspondingto the 95% of settling time (as ¼ 3). During a SID process, the systeminputs are systematically varied based on the above-described exci-tation signals. No single criterion is able to comprehensively definethe performance of system excitation signals. Crest factor (CF) isused in this study to determining the parameters in the excitationsignal generation function [29]. Defined as Eq. (A2) CF is the ratioof peak values to the average value of the signal (U), whichmeasures the signal distribution over the input span. Loweringthe CF improves the signal to noise ratio of the excitation signals.
CF2r ¼ maxu2ðtÞ
limN!1 1N
PNt¼1 u2ðtÞ ðA2Þ
Therefore, a constrained minimum crest factor optimizationproblem is created to determine the parameters in excitation sig-nal generation function:
J as;xs;usð Þ ¼ minas ;xs ;us
CFr; s ¼ 1;2; . . . ;N ðA3Þ
Subject to the boundaries for building temperature setpointsand equipment schedule:
10 �C ¼ UminT;s < UT;s < Umax
T;s ¼ 32 �C ðA4Þ0 ¼ Umin
E;s < UE;s < UmaxE;s ¼ 1 ðA5Þ
The more details about the system excitation and SID modeldevelopment can be found in [27,31].
im;2 im;2 im;2 im;2 im;3 im;2
Appendix B
Fig. B1 shows the schematics of the RC model. Rs and Cs net-works represent the building roof(s), external walls, and internalmass. External walls were modeled respectively according to theirorientation: east wall, south wall, west wall, and north wall, due tothe solar effect.
At the first step, the RC model was developed just for the thirdfloor for the simplification. All the external walls, internal masses,and roof are modeled as 3R2C structure. As shown in the schemat-ics, all the Rs and Cs are parameters need to identify, while solar airtemperature (Tsol-air), outdoor air temperature (Tout), solar trans-mitted heat gains (Qtrans), etc. are input (boundary) variables.Therefore the state variables and inputs variables are vectored inEqs. (B1) and (B2):
X ¼ T je;k;i Tim;k;i
h iTðB1Þ
u ¼ Tin;i T jsol�air;i Tout;i Qr1;iQ r2;i
h iTðB2Þ
where j is the index for envelope including east, south, west, north,and roof, k is the index for surface sides including external side (as
1) and internal side (as 2), i is the index for time; T je;k;i is the surface
temperature of the external envelope for given surface j, and time i;Tim;k;i is the average surface temperature of internal mass for giventime i; Tin;i is the average building indoor temperature for given
time i; T jsol�air;i is the solar air temperature envelope j, and time i;
Tout;i is the outside dry bulb air temperature for time i; Tatt;i is theattic temperature for time i, which is unconditioned; Qr;i is the radi-ation heat gains to internal mass, which contains heat gains fromsolar remission and internal equipment radiation; Qcon;i is the con-vection heat gains to internal mass from indoor equipment; QOA;i isthe heat gains from outdoor air, including infiltration andventilation.
The energy balance equation of building can be described by:
CindTin
dt¼X5j¼1
T jei;2 � Tin
R jei;3
þ Tim;2 � Tin
Rim;3þ Tout � Tin
Rwinþ Qcon þ QOA ðB3Þ
where CindTindt is the indoor temperature changing rate multiple by
the thermal capacitance, which equals to the heat gains/loos ofthe building; Cin is the capacitance associated with internal zonemass (air and furnishings); The first term at right hand side is theheat transfer rate from all the external envelops, including east,south, west, north walls and roof; the second term is the heat trans-fer rate from the internal envelopes; the third term is the heattransfer through the window, excluding the solar transmission;Qcon and QOA are the heat convection and outdoor air heat gains.
Besides this whole building heat balance equation, there is oneheat balance equation at each node. For example, the heat balancefor external envelopes can be modeled by Eq. (B4) and (B5):
C jei;1
dT jei;1
dt¼ T j
sol�air � Tin
R jei;1
� T jei;1 � T j
ei;2
Rjei;2
ðB4Þ
C jei;2
dT jei;2
dt¼ T j
ei;1 � T jei;2
Rjei;2
� T jei;2 � Tin
R jei;3
ðB5Þ
Therefore, by rearranging the heat balance equations, a statespace format model can be developed as:
dXdt
¼ AX þ Bu ðB6ÞY ¼ CX þ Du ðB7Þwhere X and u are state vector and input vector, which as describedbefore. A; B; C, and D are parameter matrices which are calculatedby the Rs and Cs. In the case studied in this project, A is a 12� 12matrix, B is a 12� 11 matrix, C is a 1� 12 matrix, and D is a1� 11 matrix. The nonzero elements of these four coefficient matri-ces and vectors are:
A 1;1ð Þ ¼ �1Rsei;1C
sei;1
þ �1Rsei;2C
sei;2
A 1;2ð Þ ¼ 1Rsei;2C
sei;1
A 2;1ð Þ ¼ 1Rsei;2C
sei;2
A 2;2ð Þ ¼ �1Rsei;2C
sei;2
þ �1Rsei;3C
sei;2
A 3;3ð Þ ¼ �1Reei;1C
eei;1
þ �1Reei;2C
eei;2
A 3;4ð Þ ¼ 1Reei;2C
eei;1
A 4;3ð Þ ¼ 1Reei;2C
eei;2
A 4;4ð Þ ¼ �1Reei;2C
eei;2
þ �1Reei;3C
eei;2
A 5;5ð Þ ¼ �1Rnei;1C
nei;1
þ �1Rnei;2C
nei;2
A 5;6ð Þ ¼ 1Rnei;2C
nei;1
A 6;5ð Þ ¼ 1Rnei;2C
nei;2
A 6;6ð Þ ¼ �1Rnei;2C
nei;2
þ �1Rnei;3C
nei;2
A 7;7ð Þ ¼ �1Rwei;1C
wei;1
þ �1Rwei;2C
wei;2
A 7;8ð Þ ¼ 1Rwei;2C
wei;1
A 8;7ð Þ ¼ 1Rwei;2C
wei;2
A 8;8ð Þ ¼ �1Rwei;2C
wei;2
þ �1Rwei;3C
wei;2
A 9;9ð Þ ¼ �1Rrei;1C
rei;1
þ �1Rrei;2C
rei;2
A 9;10ð Þ ¼ 1Rrei;2C
rei;1
A 10;9ð Þ ¼ 1Rrei;2C
rei;2
A 10;10ð Þ ¼ �1Rrei;2C
rei;2
þ �1Rrei;3C
rei;2
A 11;11ð Þ ¼ �1Rim;1Cim;1
þ �1Rim;2Cim;2
A 11;12ð Þ ¼ 1Rim;2Cim;1
A 12;11ð Þ ¼ 1R C
A 12;12ð Þ ¼ �1R C
þ �1R C
ðB8Þ
X. Li et al. / Applied Energy 164 (2016) 69–88 87
B 1;2ð Þ ¼ 1Rsei;1C
sei;1
B 2;1ð Þ ¼ 1Rsei;3C
sei;2
B 3;3ð Þ ¼ 1Reei;1C
eei;1
B 4;1ð Þ ¼ 1Reei;3C
eei;2
B 5;4ð Þ ¼ 1Rnei;1C
nei;1
B 6;1ð Þ ¼ 1Rnei;3C
nei;2
B 7;5ð Þ ¼ 1Rwei;1C
wei;1
B 8;1ð Þ ¼ 1Rwei;3C
wei;2
B 9;6ð Þ ¼ 1Rrei;1C
rei;1
B 10;1ð Þ ¼ 1Rrei;3C
rei;2
B 11;1ð Þ ¼ 1Rim;1Cim;1
B 11;8ð Þ ¼ 1Cim;1
B 12;1ð Þ ¼ 1Rim;3Cim;2
B 12;9ð Þ ¼ 1Cim;2
ðB9Þ
C 1;2ð Þ ¼ As
Rsei;3
C 1;4ð Þ ¼ Ae
Reei;3
C 1;6ð Þ ¼ An
Rnei;3
C 1;8ð Þ ¼ Aw
Rwei;3
C 1;10ð Þ ¼ Ar
Rrei;3
C 1;12ð Þ ¼ Aim
Rim;3
ðB10Þ
D 1;1ð Þ ¼ � As
Rsei;3
þ Ae
Reei;3
þ An
Rnei;3
þ Aw
Rwei;3
þ Ar
Rrei;3
þ Aim
Rim;3þ Awin
Rwin
!
D 1;7ð Þ ¼ Awin
Rwin
ðB11Þ
Finally, the building heat gains can be estimated by:
Qgain ¼ Y þ Qcon þ QOA ðB12ÞTherefore, substituting Eq. (B11) into Eq. (B12):
CindTin
dt¼ Qgain � QHVAC ðB13Þ
The model training process is to determine the Rs and Cs in theA; B; C and Dmatrices. They will be identified by comparing HVACload estimated from simplified Physics model, RC model (Eqs.(B6)–(B13)), with the actual HVAC load from the detailed physicsbased simulation model (EnergyPlus model). Optimization meth-ods will be used to determining the parameters to minimize thedifference between the HVAC load from RC model and EnergyPlusmodel. The objective function for this optimization problem is:
JRC ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN
j¼1 QRC;j � QAct;j
� �2N � 1
s
where QRC and QAct are the HVAC loads (cooling and heating) fromRC model and EnergyPlus model, respectively; N is the total timestep of this whole simulation. The initial guess and boundary ofRs and Cs are determined from the Physics theory. Runge Kuttamethod is used to solve the state space equations Eqs. (B6) and(B7) to calculate the state variables, the estimated building load isthen determined by Eq. (B13). Pattern Searching optimizationmethod is unitized here to update the Rs and Cs to minimize theobjective function.
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