PROOF OF CONCEPT FOR THE BAYESIAN ANALYSIS OF COMPUTER CODE FOR BUILDING ENERGY MODELLING

Michael Wood, Matthew Eames, and Peter Challenor

College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, UK

ABSTRACT Building energy simulations are computationally expensive processes. One method of reducing the computation time is to create meta-models or emulators, but these simplified models often sacrifice precision for speed. Recently, regression based emulators have proven popular (Jenkins, Patidar, Banfill, & Gibson, 2011), however, this paper presents an alternative approach to emulation based on the Bayesian analysis of computer code output (BACCO). The Bayesian building emulator (BBE) detailed in this paper is based on knowledge of 30 training simulations, which vary the building’s glazing ratio, orientation and insulation thickness. Validation shows that the emulator represents the uncertainty surrounding the untested simulator inputs well.

INTRODUCTION Dynamic models of buildings using an electrical analogy originated in the 1950s (Burnand, 1952), the ideas of which are still being used in today's computational building models. Building models vary in both complexity and accuracy, and are generally used in industry to comply with building regulations and to assist in make decisions about the design of the building (King, 2010). Early computer models were, by necessity, simplified versions of the thermal behaviour of real buildings (Achterbosch, Jong, Meulen, & Verberne, 1985), but as computational power has increased, so has the complexity of the models themselves. Modern simulators such as Energy Plus and IES Virtual Environment are based on physical principles making them credible to engineers and building scientists. However, the added computational power can restrict the number of iterations of a model in an optimisation process. Such complexity is therefore likely to prohibit an exhaustive search of all possible variations of the input parameter space. One method for speeding up the optimisation process is to reduce the number of simulations needed to reach an optimal solution. This can be achieved by using genetic algorithms or other optimal search techniques (Siddharth, Ramakrishna, Geetha, & Sivasubramaniam, 2011). Another method is to

create a meta-model which can complete simulations faster than the simulator. Meta-models can be created by using neural networks (Kumar, Aggarwal, & Sharma, 2013), with regression based meta-models also having been shown to be successful (Jenkins et al., 2011). More recently, supercomputing techniques have been used to speed up the simulation process by brute force (Greenberg et al., 2013). In this paper we present a meta-model (emulator) which was created by using Bayesian analysis (O’Hagan, 2006). The proposed emulator is a statistical model of the building simulator and has several advantages over traditional meta-modelling techniques. Like all meta-modelling techniques, the Bayesian emulator requires a number of training runs to ‘learn’ about the simulator it is modelling. The emulator produced is fast, accurate and has an advantage over regression methods in that it includes a measure of its own uncertainty. However, the creation of the emulator is based on the assumption that the output of the simulator can be modelled as a Gaussian Process (GP). Examples of previous work using Bayesian emulators to model building performance have focussed on particular building elements, such as glazing (Kim, Ahn, Park, & Kim, 2013), cooling systems (Kang, Kim, Ahn, & Park, 2013) or applied to specific complex problems such as computational fluid dynamics (CFD) (Tagade, Jeong, & Choi, 2013). The work detailed in this paper demonstrates a Bayesian emulator which accounts for the effect of several non-related building parameters (i.e. glazed area, insulation thickness and orientation).

METHOD The Bayesian meta-model of a building simulator is created using a four-stage process:

• Run 'training simulations' of the building model to 'learn' about the relations between the input and the output.

• Use the results of the training simulations to train the emulator.

• Validate the emulator by completing additional runs of the building model to compare to the output of the emulator.

• Use the emulator to make inferences regarding the output of the simulator.

To avoid confusion, the original building model is always referred to as the simulator and the newly created model of the simulator is always referred to as the emulator. The building simulator is based on a three-zoned thermal network model of a two-storey building with a roof space, which is glazed to one of the façades (Figure 1). Thermal network models allow any building to be represented by an analogy with an electrical model and then solved as a series of first order differential equations. Such models have been extensively used to solve such thermal modelling problems in the past as the calculations are completed analytically rather than numerically. The system consists of a series of nodes, which are coupled by conduction and convection and the thermal storage is represented by a capacitor (Mathews, Richards, & Lombard, 1994). The heat flow by radiation is represented by a current source on the node where the radiation is incident and the node voltage represents its temperature. Each layer of the construction contributes to the dynamic response of the building and is modelled as an independent element when creating the thermal model. The nodes and boundary conditions of the thermal elements are determined from the diffusivity of the material and time step of the dynamic model (Tabares-Velascoa & Griffith, 2012).

Figure 1: The building model

The building model is shown in Figure 1. Each building story is a heated zoned of dimensions 8 m × 8 m × 2.4 m and has a constant air exchange rate of 0.2 air changes per hour. The roof is pitched at an angle of 30º with no overhang. The roof space is unheated and has a constant air exchange rate of three air changes per hour. For simplicity, at each time step, the heating and cooling loads are calculated for each heating zone to maintain the internal temperature at 21 ºC using an ideal plant with an unlimited capacity. The purpose of this work is to demonstrate the effectiveness of the emulation procedure to represent the simulation of the building model as such this simplification is appropriate. The thermal properties for the materials used for the construction for each surface are shown in Table 2 and the glazing system is shown in Table 1. The building is located in Plymouth and the standard

CIBSE Test reference year is used. No other internal gains are considered. The building model created allows the glazed area, insulation thickness (of the external wall only) and orientation of the building to be varied. The glazed area, insulation thickness and the orientation form the input to the simulator.

CREATING A SIMPLE BAYESIAN EMULATOR The Bayesian Building Emulator (BBE) treats the building simulator as a multi-input function with one output (in this case the annual energy use). The emulator is based on the assumption that we can model the output of the simulator as a Gaussian Process. Representing the output in this way is a pre-requisite of the Bayesian method. This is because complete probability distributions are required to specify uncertainty, and Gaussian Processes are by far the simplest way of achieving this. The emulator outputs are therefore represented as multivariate normal (Gaussian) distributions. To create the BBE, the building simulator is represented by a function, !(!), where ! is the input vector of a particular input configuration and !(!) is the output of the simulator (in this case the annual energy use). If we run a number of training simulations, this gives us a number of values for ! for which we know the output (!(!)). These training inputs (!, where!! ∈ !), each have an associated output (!(!)).

GLAZING LAYER GLASS 1 ARGON CAVITY GLASS 2

Thickness / mm 4 16 4 Thermal Conductivity / Wm-1K-1

1.06 0.016 1.06

Transmittance 0.899 - 0.672 Forward Reflectance 0.08 - 0.188 Backward Reflectance 0.08 - 0.163

External emissivity 0.837 - 0.059 Internal emissivity 0.837 - 0.837

Table 1: Properties of the glazing system used in the building model. The frame makes up 20 % of the

glazing area and has a U-value of 2 Wm-2K-1 and a linear transmittance of 0.08 Wm-1K-1.

SU

RFA

CE

/ M

ATE

RIA

L

THIC

KN

ESS

/ M

M

THER

MA

L C

ON

DU

CTI

VIT

Y

/ WM

-1K

-1

DEN

SITY

/ K

GM

-3

HEA

T C

APA

CIT

Y

/ JK

G-1

K-1

External Wall Cast concrete 220 1.13 2000 1000

Insulation 10-100 0.038 25 1030 Cast concrete 220 1.13 2000 1000

Ground floor London Clay 750 1.41 1900 1000

Brick 220 0.77 1750 1000 Concrete 100 1.13 2000 1000 Insulation 80 0.025 30 1400 Chipboard 25 0.15 800 2093

Carpet 10 0.06 160 2500 Internal Floor/ceiling

Carpet 10 0.06 160 2500 Chipboard 25 0.15 800 2093

Cavity 100 - - - Insulation 10 0.04 15 1300

Plaster Board 13 0.16 600 1000 External Roof

Clay tile 5 0.84 1900 800 Glass fibre 25 0.04 12 840

Roofing felt 5 0.19 960 837 Insulation 180 0.043 12 840

Plaster Board 13 0.16 600 1000

Table 2: Thermal properties of the constructions used in the building model.

The BBE represents the output as a mean function (!(!)) and a covariance function (!(!,!′)) (Kennedy & O’Hagan, 2001). The mean function is the expected value of !(!) and the covariance function provides a measure of the uncertainty around the mean function. The form of both these functions is set prior to running the simulator and are then modified by hyper-parameters1 following the emulator training phase. Note that the posterior mean will always evolve so that it passes exactly through the training set where the emulator uncertainty goes to zero. The mean function takes the form ! ! = ℎ(!)!! where ℎ ! = 1, !!, !!, !! ! and ! is a vector of four hyper-parameters, each associated with a column in ℎ(!). The covariance function is defined as ! !,!′ = !!!!(!,!!), where !(!,!′) is the correlation function. For the purposes of the simple one-dimensional BBE, a Matérn function (with the shape parameter set to 3/2) is used as the correlation function:

! !,!! = ! 1 − 3 !!√!! , where

1 Note that in Bayesian statistics, the term hyper-parameter describes a parameter in the prior distribution rather than a parameter of the computer model itself. It is therefore important to note that these hyper-parameters have no direct association to the physical parameters in the model.

! = ! ! − !! !∆(! − !!) !!

The matrix Δ is a diagonal matrix of the correlation lengths !!, where ∆!!= !!!!!. Each dimension of the input will have an associated !!, which defines the 'roughness' of the output. A low value of ! implies a more rough output than a higher value, which in turn implies the output is smoother for variation in this input dimension. Like the other hyper-parameters of ! and !, values for ! must all be estimated based on the results of the training simulations.

Estimating the Hyper-Parameters For the BBE, ! and ! are determined analytically for a given !, and only ! needs to be changed to estimate the other parameters. The most likely values of ! for a given training set (and hence ! and !) are determined by maximising the log likelihood function (Bastos & O’Hagan, 2009). ln !!∗(!) = !−0.5 ! − ! ln !! − 0.5 ln ! −0.5ln!( !!!!!! ), (2) where ! = 2×ln!(!), ! = ℎ(!!) (note ℎ ! =! 1,! ! in our model), ! is the number of training inputs, ! is the number of elements in ! and !!,! = !(!! , !!). The function !!∗(!) is the posterior distribution function of ! (and hence !) with !!! calculated from !! =(! − ! − 2)!!!! ! ! !!! −!!!! !!!!!! !!!!!!! !(!). (3) The log likelihood function was maximised numerically by calculating ln!(!!∗(!)) for different trial combinations of!!. To do this it was assumed that each element of delta would lie between 0.1 and 20 respectively and would be equally likely to fall between these two values (thereby assuming a uniform prior between these two values). Due to the computational cost of calculating the log likelihood function for each delta, a two-stage search was used. First, a coarse-grid search was used, followed by a fine-grid search around the maximum of the initial search. The resolution of both these grid searches is 10x10x10. The value of ! that yields that largest value for ln!(!!∗(!)) was carried forward (!). Using !, !!was calculated from equation 3 and ! was calculated from ! = !!!!!! !!!!!!!!(!). The Posterior Mean and Covariance Function The posterior mean !∗(!) and covariance function !∗(!,!′) are defined by the following equations (J. Oakley, 2004).

!∗ ! = ℎ(!)!! + ! !,! !!!!(! ! − !!), (4) !∗ !,!! = !!! ! !,!! − ! ! !!!!! !! +(ℎ ! ! − ! ! !!!!!)(!!!!!!)!!(ℎ !′ ! −! !′ !!!!!)! , (5) Detailing the derivation of equations 4 and 5 would take up too much space here, but interested readers will find O'Hagan's tutorial (O’Hagan, 2006) a good general introduction to Bayesian emulation. Further information on how the log likelihood functions and posterior mean and variance functions are linked to Bayes’ Theorem can be found in Gaussian Processes for Machine Learning appendix A1 and A2 by CE Rasmussen and CKI Williams (Rasmussen & Williams, 2006)

Figure 2: An emulator trained from five simulations points (orientation = 0 radians, glazed area = 50%). The Matérn (!! = !

!) correlation function is used to evaluate the mean function and the covariance

function.

Simple One Dimensional Emulator Figure 2 shows the results of a simple one input / one output emulator which models variation in the external wall insulation thickness. The original output of the simulator is shown by the blue circles (training points), with the true range of the simulator output shown by the blue line (solid). The red (close-dotted) line shows the posterior mean function and the two red (long-dotted) lines show the 95% confidence intervals. Table 3 shows the hyper-parameters and the training simulation results associated with this emulator. These values provide the information to be able to evaluate the posterior mean and covariance functions that constitute the Bayesian Emulator.

Table 3: Variables and values for the simple emulator

VARIABLE VALUE

! (scaled between 0 and 1)

[0.0, 0.11, 0.22, 0.33, 0.44, 0.56, 0.67, 0.78, 0.89, 1.0]

!(!) (kWh) (scaled between 0 and 1)

[23,390, 16,153, 13,915, 12,859, 12,249, 11,855, 11,579, 11,376, 11,220, 11,097]T

! 0.95

! 0.149 (note that this value of sigma is based on FD being scaled by dividing it by its maximum)

! [1.12, 0.67]T

Although the building emulator shown in Figure 2 provides good emulation, it can be improved by adding more training simulation points. Figure 3 shows improved accuracy with the addition of five additional training points.

Figure 3: An emulator trained from twelve

simulations points (orientation = 0 radians, glazed area = 50%)

Training a Bayesian emulator therefore requires a balance between number and location of training simulations and emulator accuracy. The more training simulations undertaken, the more accurate the emulator is likely to be, but the more time the emulator will take to create.

Creating a Three Dimensional Bayesian Emulator A three-dimensional emulator Bayesian emulator was created to examine the effect of changing three inputs:

• Percentage glazed area (between 50-90%) on one surface.

• External wall insulation thickness (between 10 – 100 mm)

• Orientation of glazed surface (between -π/2 and +π/2 radians, where 0 radians is due South)

The three input parameters create a three-dimensional input space. In order to train the emulator well, the input configurations need to be chosen carefully. Given that we would like the simulator to be run as few times as possible, it is important that the input space is 'filled' efficiently. When training the one dimensional simulator given in the previous example, providing a 'good fill' of training variables was trivial, since the most efficient fill is achieved by equally spaced inputs. However, for the three dimensional input space a more sophisticated Latin Hypercube design process approach is required. For the Latin Hypercube design process, it has been shown that 10 training simulations per input dimension provides a good balance between minimising the number of training simulations and providing enough information for the emulator (Loeppky, Sacks, Welch, & Welch, 2009). Since the model in question has three input dimensions (glazed area, insulation thickness and orientation), 30 training simulations were used. To generate the input configurations, 100 random, 30-point Latin Hypercubes and the hypercube with the maximum minimum distance between the design points was selected. The input variables are scaled to range between 0 and 1 to simplify both the creation of the Latin Hypercube and the mathematics of the emulator. The simulator was then run for each of the 30 input configurations. Maximising the log likelihood function yielded the hyper-parameters shown in Table 4.

Table 4: Parameters for the 3D emulator

HYPER-PARAMETER

VALUE

! [0.95, 0.12, 0.02, -0.42]T

! [3.42, 19.78, 0.76]

! 0.0073

RESULTS To check that the emulator was performing as expected, it was subjected to a number of validation procedures (Bastos & O’Hagan, 2009).

Individual Standardised Errors The individual standardised errors provide a measure of the accuracy of the emulator. However, they require additional validation simulations to calculate them. The standardised errors (!) are given by,

!! =! !!! !!∗(!!!)

!∗(!!!,!!!), (6)

where !! are the individual standardised errors, !!! are the individual simulator inputs for the validation samples, and ! is the index of each sample (where ! = 1, 2, 3… !′). A set of 15 validation simulations (!′) were used to calculate the individual standardised errors. The validation inputs in !′ were generated using the same Latin Hypercube design process that was used for the original training set. Since the emulator provides both a mean (the expected value) and variance for each input, it is important to note that we do not expect all the simulation points to lie on the mean function, but to be distributed about the mean function according to the emulator’s covariance function. Therefore, if the building emulator is accurate, then on average 95% of the standardised errors would be within 2 standard deviations of the mean. For the validation samples, 93.4% of the errors lay within two standard deviations of the mean function. This result demonstrates that there is a high probability that the emulator is representing the simulator output well by providing a good estimation of its own uncertainty (for a normal distribution, 95% of the residuals would be within two standard deviations of the mean). Calculating the Mahalanobis Distance The Mahalanobis distance (!) is a scalar value that provides an alternative means of measuring the validity of the emulator (Bastos & O’Hagan, 2009). ! is calculated using the simulation and emulation results for the validation sample (!′). ! = ! !! −!∗(!′) ! !∗ !!,!′ !! ! !! −!∗(!′) , (7) For a valid emulator, ! will have a scaled F-distribution with degrees of freedom given by !′ and (! − !), and an expected value of !’. For the validations points considered, the expected value of ! is 15 with a variance of 53.2. The Mahalanobis distance for the validation samples was 15.7, which falls close to the mean of the expected distribution of ! (Figure 4).

Figure 4: The Mahalanobis Distance of the

validation sample (vertical line) in relation to its predicted probability density function

Comparing the Simulator to the Emulator over a Larger Number of Points A 9×9×9 grid of equally spaced input points (!!"#) was simulated to further test the validity of the building emulator. The simulator results were then compared to those of the emulator. Figure 5 shows a histogram of the standardised errors in the emulation over the 729 points, and Figure 6 shows the location of emulation inputs the simulation output is more than two standard deviations from the mean function. Figure 7 and Figure 8 compare the results of the emulator and simulation.

Figure 5: Histogram of the standardised errors

between the emulator and the simulator based on 729 simulations and emulations.

Figure 6 shows that most of the points where the simulator outputs are more than two standard deviations from the mean function are at the ‘edges’ of the input space and represent 5.5% of the input values tested. It is expected that around 5% of emulator outputs would be greater than two standard deviations from the true value (given the probabilistic nature of the emulator). Figures 7 and 8 show how closely the trends in the output of the emulator match those of the simulator.

Figure 6: Configurations of the input points were the true output of the simulator is more than 2 standard deviations from the mean function (as predicted by

the covariance function of the emulator)

DISCUSSION AND CONCLUSION This paper has shown that a Bayesian Emulator can represent the modelled energy use of a simple building. Based on a training set of 30 simulations, the validation diagnostics show that the emulator's mean and variance predictions fit the modelled data well. Further interrogation of the emulator by using a brute-force method showed that 94.5% percent of 729 simulation values were within two standard deviations of the mean function, a finding that in agreement with the uncertainty specified by the emulator. The validation diagnostics show that the emulator is valid, however, some areas of the emulator creation need improvement. Such improvements include an improved method for estimating the hyper-parameters, particularly!!. One of the major advantages of Bayesian emulation is the ability to identify key trends and patterns in the output of the simulator with only a limited number of simulator runs. Figure 7 and Figure 8 show that trends in the output of the emulator closely match those of the simulator. Patterns identified by the building emulator include:

• The orientation of the building is not as important as might have been expected in determining the total heating and cooling load (i.e. there is not much variation observed in the orientation axis).

• An increase in glazed area leads to a linear increase in the total amount of energy used (combined heating and cooling).

• Increasing the amount of insulation used in the building provides a very steep reduction in the amount of energy used from around 10 mm to 40 mm, but between 40 mm and 100 mm the increasing thickness of the insulation appears to lessen in effect.

The Bayesian methodology also allows a variety of analyses, which are usually too computationally expensive to perform with the building simulator. These analyses include applying MC analysis to the emulator mean and MC uncertainty analysis (i.e. assessing the effect of uncertainties in input values on the output) as well as sensitivity analyses (J. E. Oakley & O’Hagan, 2004) and model calibration (Kennedy & O’Hagan, 2001). MC analysis using the emulator rather than the simulator can achieve high levels of accuracy in significantly quicker time than could be achieved with the computationally expensive simulator. For example, using the analysis of 729 simulation points discussed earlier, a MC evaluation of the emulation mean for these points reveals a value of 16,018 kWh for the annual energy use. For the same set of simulation samples, the simulator has a mean of 16,039 kWh, which represents a difference of 0.13% from the output of the emulator. Following the 30 training simulations, the emulator is 5.4×10! times quicker than the simulator in producing this output. The research has demonstrated that a Bayesian Building emulator can be produced with a limited number of training runs, allowing a statistical

analysis of the building to be completed quickly. Further work will include the investigation of the effects of varying other input parameters to the building model, and improvements to the accuracy of the emulator. In addition to this, we will expand the emulator to include dynamic inputs, along with improvements to the ! search function. Dynamic inputs will add significant additional capabilities to the emulator, since we can consider then consider time-varying effects.

NOMENCLATURE !, simulator / emulator input; !(∙), function representing the simulator; !, training data; !′, validation data; !, Mahalanobis distance; !", Monte Carlo; !!, standardised errors;

ACKNOWLEDGEMENTS This work was supported by the Engineering and Physical Sciences Research Council [EPSRC grant number EP/J002380/1].

Figure 7: Comparison between the simulator (left) and the emulator (right) for a fixed glazing ratio of 50%

! !

Figure 8: Comparison between the simulator (left) and the emulator (right) for a fixed orientation of !!! radians

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