Simplified Control Guideline Extraction fromModel Predictive Control Results of Mixed-Mode

Buildings

Peter May-Ostendorp

December 16, 2009

Contents

0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Mixed-Mode Buildings . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Motivation for Model-Predictive Control with Rule Extraction 2

0.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.2.1 Model-Predictive Control Environment . . . . . . . . . . . . . 40.2.2 Principal Component Analysis of Optimal Results . . . . . . . 70.2.3 Spectral Analysis of Optimal Results and Principal Components 90.2.4 General Linear Model Development . . . . . . . . . . . . . . . 9

0.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 100.3.1 Sample Results of MPC for Single-Decision-Variable Optimiza-

tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110.3.2 Model Simplification Through PCA and Binomial GLM . . . . 140.3.3 Time-Lagged Model Development . . . . . . . . . . . . . . . . 180.3.4 Model Development from Summer-Long Data . . . . . . . . . 18

0.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

0.1 Introduction

0.1.1 Mixed-Mode Buildings

In most commercial buildings the United States, facades are sealed, and ventilationis provided mechanically through fans and air handling units; cooling is providedthrough vapor compression refrigeration equipment like chillers or direct expansionair conditioning units. This is, naturally, a very energy-intensive way to condition aspace, and today’s commercial buildings now consume about 25% of their electricitythrough these mechanical processes [15]. Natural ventilation, a passive cooling tech-nique, simply entails providing outside air to the spaces of a building directly throughoperable windows or other facade openings. This can be used both to ventilate (i.e.

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provide fresh outdoor air) and cool the space, providing significant energy savings inthe process.

Mixed-mode (MM) buildings represent a new, hybrid approach to space con-ditioning, blending traditional mechanical cooling and ventilation with low-energynatural ventilation. Spaces are naturally ventilated and cooled whenever possible,but intelligent controls allow mechanical systems to serve as a backup during warmerperiods when outdoor air alone cannot meet the cooling demands of the space. MMbuildings have gained prominence in the past 10 to 15 years, particularly in Europe,and initiatives like the United States Green Building Council’s (USGBC) Leadershipin Energy and Environmental Design (LEED) program are driving higher adoptionin the US as well. Case studies currently indicate that cooling energy savings canrange from 20 to 50% compared against buildings designed to code [14, 6].

As noted above, intelligent control is absolutely crucial in MM buildings, as withany other high performance building. However, given that the building industry’spractical experience with MM buildings is relatively short, control sequences mustcurrently be developed on a project-by-project basis. Control algorithms in today’sMM buildings usually involve a series of simple heuristics and if/then statements,such as “if the outdoor temperature drops below 68◦, open all automated windowsand turn off mechanical cooling.” An example of one such algorithm is provided infigure 1 below, in which various logical comparisons are made against the averagezone temperature of the building to determine whether openings in the facade shouldbe made.

It should be noted that most MM buildings are not fully automated. Occupantsretain a great deal of control in MM buildings, particularly in the control of windowpositions for private offices. This “adaptive” approach can reduce the complexityof the control system and has been shown to improve occupant thermal comfort byaffording them greater control over their surrounding environment [5, 2]. However,introduction of occupant windows can undermine the energy savings of MM build-ings since people cannot be expected to operate their windows in an energy-efficientmanner all of the time. As a result, some MM buildings incorporate informationalsystems, such as notification lights, to signal to occupants when windows should beopened [9].

0.1.2 Motivation for Model-Predictive Control with RuleExtraction

Model-predictive control (MPC) is a control methodology that seeks strategies throughtime that minimize a cost function, based on the predictions of a model. In the

2

Figure 1: A MM control algorithm for the Scottish Parliamentary Building. [3]

3

context of building systems, MPC allows for discovery of near-optimal operationstrategies that minimize the energy use, carbon dioxide emissions, or dollar cost ofthe building or parts of the building. The current research applies MPC to a series ofgeneralized MM building designs to determine whether better control algorithms canbe found than are currently employed today, mainly by optimizing decisions in timeregarding window openings and cooling equipment operation. Although MPC hasbeen applied extensively in the HVAC engineering field in the past years [4, 8, 7], ithas only recently been applied to MM buildings through the optimization of neuralnetwork models specifically trained on two unique buildings [13, 12, 11].

The long-term goals of the current research are the following:

1. Apply MPC to examine the near-optimal control of “typical” MM buildings

2. Develop improved, generalized control guidelines for MM buildings that areinformed by the near-optimal results

3. Validate the improved control guidelines by implementing them in actual MMbuildings

This paper will demonstrate a methodology for achieving the first two of thesegoals. Namely, near-optimal window control patterns from MPC runs will be brieflycovered, and then a combination of diagnostic statistical techniques will be usedto extract a reduced-dimension model from the optimal results that approximatesthe near-optimal control behavior in a less computationally expensive and time-consuming way.

0.2 Methodology

0.2.1 Model-Predictive Control Environment

The MPC problem at hand can be expressed in a very general way as the uncon-strained optimization problem.

Min Z(xt) = Ecost + P

Subject to : xt = 0, 1,

where xt is a vector of binary decisions regarding window positions, Ecost is the costof building operation in real dollars over a planning horizon (determined through

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building energy simulation), and P is a general penalty cost applied to discouragecertain undesirable characteristics in the solutions.

The cost function that results in the above unconstrained optimization problemdoes not lend itself well to traditional pattern search techniques, because it cancontain many local minima. As a result, a meta-heuristic search technique, particleswarm optimization (PSO) has been adopted to quickly search the decision space fornear-optimal solutions. The details of the PSO implementation will not be discussedhere.

In figure 2 below, a block diagram schematic of the overall optimization environ-ment is presented to demonstrate the general solution approach. Building models areread in and modified by the PSO algorithm in MATLAB by manipulating schedulesthat control window openings. The resulting models are evaluated using the U.S.Department of Energy’s EnergyPlus simulation engine. Results are read back intoMATLAB, where the cost function is computed, and the PSO algorithm decides howto proceed to the next decision vector. The algorithm recurses until the cost functionvalues for all particles converges below a predefined tolerance or a maximum numberof iterations is reached.

This process is used to optimize decisions over a 24-hour planning horizon L.For the cases under consideration in this paper, even though the time step of thebuilding energy simulation is one hour, the planning horizon is broken up into 2-hour“building modes” to reduce the size of the decision space and the time to convergence.When the near-optimal decision vector is found for the current planning horizon, theoptimizer proceeds to the next planning horizon, preserving the thermal history ofthe decisions implemented on the previous day(s). This concept is qualitativelyillustrated in figure 3 below.

At the end of this process, the near-optimal result is simply a vector N ∗L hourslong, where N is the length in days of the optimization period and L is the 24-hourplanning horizon. Although multiple decision vectors could be considered in the op-timization (e.g. one decision vector for the window controls on each facade of thebuilding), only one global decision vector controlling the operation of all windows inthe building was used. This simplified the initial analysis and shortened run-times;however, the analysis can and will be expanded to consider many other control vari-ables, including spatial disaggregation of windows and on/off signals for mechanicalHVAC equipment.

5

Figure 2: Block diagram of the MATLAB/EnergyPlus environment

6

Figure 3: Procession of MATLAB/EnergyPlus MPC environment. Previous deci-sions (black) determine the thermal history of the building, thus impacting decisionsunder the current planning horizon (gray).

0.2.2 Principal Component Analysis of Optimal Results

One of the major goals of this research is to extract useful control guidelines from theresults of optimization. With day-long optimizations for the single decision vectortaking nearly one hour using an 8-core server — and this is not by any means themost exhaustive optimization approach possible — it would be extremely difficultto currently justify using real-time MPC to control a live building. Simpler controlguidelines that approximate the optimal results are absolutely necessary to takeresults into practice.

Principal component analysis (PCA) was used as a technique for diagnosing thedominant modes of variation in potential control sensor/predictor variables, bothto identify key variables that are highly correlated to the control behavior and toreduce the dimensionality of the predictor dataset for general linear model develop-ment. Many other texts provide a thorough treatment of PCA, and so only the basicframework will be covered here.

The goal of PCA in the current context is to identify the modes of variation in aset of potential control variables that best capture the variance in that dataset. Sinceall principal components are orthogonal to each other by construction, the observedmodes are uncorrelated features of the data set, and we can impute physical meaningto each. Furthermore, we can examine the eigenvector weights on each potentialcontrol variable to determine which are most responsible for variance. These sorts of

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qualitative observations will be extremely useful in later simplified control algorithmdevelopment.

A variety of control sensor/predictor variables were examined based on knowledgeof the physical heat transfer processes involved as well as considerations for the typesof data that could reasonably be sensed and provided to a building automationsystem. The variables examined include:

1. Outdoor dry bulb temperature

2. Outdoor dew point temperature

3. Wind speed

4. Wind direction

5. Direct solar irradiance

6. Mean zone temperature (for all zones)

7. Zone operative temperature (for all zones)

All told for the sample building under consideration, a total of 27 different variableswere considered.

Time series of the predictor variables over the entire optimization period wereplaced as columns into a matrix. The values were then normalized to remove thecolumn means and standard deviations. The covariance matrix of the resultingmatrix x was then found by

S =1

n− 1x>x. (1)

The covariance matrix was then decomposed to solve the eigen equation

SE = λE, (2)

yielding eigenvalues λ and eigenvectors E. The principal components (PC) of theoriginal data were then obtained as shown in equation 3. The PCs represent acoordinate transformation of the original predictor dataset. Each PC is a projectionof the old data onto one of the new coordinate axes. The new coordinates are alignedwith the modes of greatest variation in the data.

U> = E>x> (3)

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0.2.3 Spectral Analysis of Optimal Results and PrincipalComponents

Building simulation is dominated by boundary conditions that are diurnal in na-ture. Outdoor temperature, sunlight, and building occupancy all exhibit uniformlydiurnal frequencies. As will be shown, the near-optimal window control results fromsimulation studies show a similar diurnal frequency. In order to further diagnose thetemporal characteristics of the underlying sensor/predictor dataset, spectral analysiswas conducted. As with PCA, a detailed development of the theory behind spec-tral analysis is not presented here, only the details relevant to its application in thisresearch.

To obtain the smoothed spectrum of the time series signals (both in original andprincipal component space), the autocorrelation function was first computed. Next,the autocorrelation function was transformed into the frequency domain by applyinga finite Fourier transform with a Parzen smoothing window. Frequencies up to theNyquist frequency (fn = 12 cycles/day) were computed.

0.2.4 General Linear Model Development

Ultimately we would like to know if there are underlying patterns, logic, or rulesgoverning the optimal control sequence found through MPC. We would particularlylike to be able to reduce the dimensionality of the problem through a simplified modelso that a near-optimal control policy could be produced without the need for runningcomputationally expensive MPC optimizations. A general linear model (GLM) wasthe tool of choice used as a first step toward identifying whether it is even feasible toreproduce the near-optimal results in any meaningful way through a simplified model.In particular, the signal we wish to reproduce is binary and, thus, is best describedby the binomial distribution. In the parlance of GLM, the binomially distributedphenomena are linearly modeled through the logit link function. In other words,the regression model developed relates our predictor variable, W (window openingbinary signal), to a linear variable θ(x) described by the logit function

θ(x) = logp(x)

1 − p(x), (4)

where p(x) is the probability of a window opening signal being issued. In physicalterms, p(x) could also be interpreted as the percent of windows open or the percentopen for all windows. The value of p(x) is found through the inverse logit function

9

p(x) =eθ(x)

1 + eθ(x). (5)

In this particular case, the model predictor variables x are, in fact, principal com-ponents of our predictor variable set. This is done to reduce the dimensionality andincrease parsimony of the model.

Finally, we would like to compare the predictive power of the model developedagainst the original near-optimal control signal and against a “random” algorithmthat would effectively provide a 50/50 chance of opening/closing a window at alltimes. The statistical measure employed is the ranked probability skill score (RPSS),which has been used in various climatological contexts to compare model forecastsagainst observations and purely random climatological predictions and is developedin detail in [16]. The RPSS compares the accuracy of model predictions againstchance. It assigns a score that is negative if model results are worse than chance,0 if model results reproduce chance events, and positive if model results are closerto observations than chance events. A score of 1 is possible if the model exactlyreproduces the observed process.

The score is computed by dividing window opening predictions into j categories.In this case, two categories were used to separate probabilities above and below50%. A vector of forecast probabilities, pj, is constructed based on the GLM modelpredictions. Similarly, a vector of observed events, Wj, is constructed from theoptimal results, in whichW1 is 1 for window closings andW2 is 1 for window openings.We then take the cumulative density function of pj and Wj, resulting in Pcdf andWcdf . The RPSS is then computed as follows:

RPS =j∑i=1

(Pcdf −Wcdf )2 (6)

RPSS = 1 − RPSmodelRPSchance

. (7)

0.3 Results and Discussion

Several phases of results are presented, starting with the results of simplified opti-mizations so that the reader can be familiarized with the basic characteristics of theMPC results for the building in question. Next, the combination PCA and modelsimplification is applied to two week-long series of optimal results. Finally, the modelsimplification is applied to a longer, summer-long optimization run to see whetherbetter model skill and robustness is achieved.

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0.3.1 Sample Results of MPC for Single-Decision-VariableOptimization

Single-variable MPC optimizations were conducted for a small (approximately 18,000sf or 1,750 m2), three-story office building located in Boulder, CO. The basic Ener-gyPlus model — including surface geometries, materials, and systems (most impor-tantly its three air handling units and associated direct expansion cooling apparati)— was adapted from the U.S. DOE benchmark commercial building models [1]. Thebuilding contains a total of 11 occupied thermal zones, all but one of which hasaccess to operable windows for natural ventilation purposes. In the MPC environ-ment, there is simply one decision vector, xt, that includes a binary on/off decisionregarding window opening for all operable windows in the building for a given dayof optimization. A rendering of the building is presented in figure 4 below.

Figure 4: A 3D rendering of the EnergyPlus building model.

Optimizations for the above case were conducted with a cost function compris-ing HVAC electricity consumption and a “switching penalty” that slightly penalizessolutions for each additional change in state. Highly “noisy” solutions with erraticstate switching behavior can be a sign of model insensitivity to the decision. Thiscould happen, for example, on weekends, when the building’s cooling setpoints areallowed to rise and the mechanical cooling systems are not needed as much. Openingwindows during this period may have a trivial effect on the energy savings. Simi-larly, noisy solutions are undesirable from a user satisfaction standpoint, because in

11

a real building, occupants might not favor control strategies that resulted in windowpositions changing every hour or two.

Each optimization is compared against so-called “base” and “reference” cases.The base case represents the same building operated as a traditional, sealed-facade,mechanically cooled and ventilated office building. The reference case presents asimplified MM building that is not subject to optimal control, but rather simplyallows occupants to operate their own windows. The mean percent of windows openin the model is determined based on an adaptation of the Humphreys algorithm [10].

Optimizations were conducted for three periods during the summer cooling sea-son in Boulder: early summer (mid-June), peak summer (mid-July) and summer-long (mid-June through late August). Third-generation typical mean year (TMY3)weather data were used for each. Sample results for peak summer are presented infigure 5. The top row of the figure shows hourly ambient temperatures. The middlerow plots the actual control pattern for the week (starting with Monday) in black,with the anticipated window opening behavior of occupants (for the reference case)shown in gray. For the optimal case, any simultaneous mechanical cooling and openwindows (known as concurrent operation) are denoted by a red plus symbol. Finally,cumulative energy savings compared against the base and reference case are shownat the bottom.

The results demonstrate several remarkable qualities. First, the optimal windowopening behavior closely approximates what is known as night flush ventilation, inwhich natural ventilation is used to pre-cool the building’s thermal mass when coolambient temperatures are available. This form of passive thermal storage allows thebuilding to ride out hotter parts of the day with less reliance on HVAC equipmentfor cooling. One can clearly see this pattern at the bottom of the above chart,since energy savings accumulate during midday when windows are not open. Thisis when the building harvests the “cool” stored in the thermal mass. The optimizerhas converged on this pattern of operation without any externally imposed expertknowledge or constraints.

A second very interesting characteristic of the optimal control scheme is how itdiffers dramatically in time from the predicted occupant control. Note that for mostdays of the week, occupants open windows near the peak ambient temperatures.Because the HVAC system has not been notified of the occupant behavior (or theoccupants have not been instructed to keep windows closed during warm weather),this results in wasted cooling energy. The result is that the optimal MM buildingfor this case saves consistently more energy when compared to the reference MMbuilding rather than the base case building.

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10

20

30

Ambient Conditions

Tem

pera

ture

(C)

0

1

Single−Variable Optimization: Optimal Window Openings

Win

dow

Pos

ition

OFREFOFOPT

Cumulative Electricity Savings

Ene

rgy

(kW

h)

0

100

200

300 EBASEEREF

13 Jul 14 Jul 15 Jul 16 Jul 17 Jul 18 Jul 19 Jul

Figure 5: MPC optimization results for a week in mid-July in Boulder, CO.

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0.3.2 Model Simplification Through PCA and Binomial GLM

Model results for both July and June were subject to a series of multivariate andtime series statistical techniques to explore the possibility of reducing the dimension-ality of the potential control variables and to eventually develop a simplified modelthat would approximate the MPC optimization results in real time. The data weredecomposed via singular value decomposition, and the eigenvalues, λ, and eigenvec-tors, E, extracted. A plot of the eigenspectrum for the July predictor variables isshown in figure 6. Note that over 90% of the fractional variance in the predictor setis captured by the first four modes of the PCA, meaning that, as expected, thereis significant redundancy in the predictor variables (the various zone temperaturesalone represent significant redundancy).

0 5 10 15 20 25

0.0

0.2

0.4

0.6

Eigen Spectrum Plot for July Predictors

Modes

Frac

tiona

l Var

ianc

e

Figure 6: The fractional variance captured by the various modes of the PCA of theJuly predictors.

The eigenvectors themselves can be used, to some extent, to determine the weight

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of various possible control variables and to impute some logic to the physical behav-iors that should be driving the optimal control behavior. Figure 7 below plots theweights of the components present in the first three eigenvectors. The first eigen-vector indicates, intuitively, that the most important predictors are the ambienttemperature and the zone temperatures. These features alone account for over 70%of the variance and are, in fact, the most common sensor variables used as inputsto conventional MM control algorithms. The second and third eigenvectors can beinterpreted as other driving environmental factors, including solar gains and wind,due to the heavy weighting on the associated “ambient conditions” variables.

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

First Eigen Vector

Variables

Eig

en V

ecto

r Wei

ght

T_db

T_dp

Win

d S

peed

Win

d D

ir.

Dire

ct S

olar

T_core

T_core_op

T_B_1

T_B_1_op

T_B_2

T_B_2_op

T_B_3

T_B_3_op

T_B_4

T_B_4_op

T_M_1

T_M_1_op

T_M_2

T_M_2_op

T_M_3

T_M_3_op

T_T_Open

T_T_1_op

T_T_2

T_T_2_op

T_T_3

T_T_3_op

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Second Eigen Vector

Variables

Eig

en V

ecto

r Wei

ght

T_db

T_dp

Win

d S

peed

Win

d D

ir.

Dire

ct S

olar

T_core

T_core_op

T_B_1

T_B_1_op

T_B_2

T_B_2_op

T_B_3

T_B_3_op

T_B_4

T_B_4_op

T_M_1

T_M_1_op

T_M_2

T_M_2_op

T_M_3

T_M_3_op

T_T_Open

T_T_1_op

T_T_2

T_T_2_op

T_T_3

T_T_3_op

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Third Eigen Vector

Variables

Eig

en V

ecto

r Wei

ght

T_db

T_dp

Win

d S

peed

Win

d D

ir.

Dire

ct S

olar

T_core

T_core_op

T_B_1

T_B_1_op

T_B_2

T_B_2_op

T_B_3

T_B_3_op

T_B_4

T_B_4_op

T_M_1

T_M_1_op

T_M_2

T_M_2_op

T_M_3

T_M_3_op

T_T_Open

T_T_1_op

T_T_2

T_T_2_op

T_T_3

T_T_3_op

Figure 7: The first three eigenvectors of the July predictor set. For zone temperatureson the right hand side, B refers to the bottom level, M the mid level, and T the toplevel.

If the above results alone were used to select the relevant modes upon whichto base a GLM, one might simply use the first PC, since it accounts for such alarge portion of the variance. However, spectral analysis of the original time seriesalongside the PCs can also illuminate which modes of the PCA contain the samedominant periodicities as the optimal control behavior. Sample results are shownbelow in figure 8 comparing the smoothed spectrum of the optimal time series against

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those of the first and second PCs of predictors. Note that from the standpoint of thespectrum, the second PC actually bears a stronger resemblance to the original controlsignal than the first. This is likely due to the fact that significant low frequencytrends should likely exist in the first PC, which contains heavy weighting of outdoortemperature data. Nevertheless, this analysis would indicate that it is advisable toinclude higher modes in the GLM development.

0 2 4 6 8 10 12

0.00

0.15

Spectrum of Optimal Control Time Series

Frequency [Cyc/Day]

Spe

ctra

l Pow

er

0 2 4 6 8 10 12

020

40

Spectrum of First PC of Predictors

Frequency [Cyc/Day]

Spe

ctra

l Pow

er

0 2 4 6 8 10 12

0.0

1.5

3.0

Spectrum of Second PC of Predictors

Frequency [Cyc/Day]

Spe

ctra

l Pow

er

Figure 8: Smoothed spectra for the original control signal and potential PC predic-tors.

GLMs with a logit link function were fit to the first four PCs of the predictorvariables to predict the probability of a window opening signal. The results for theJune and July weeks are presented in figure 9. The RPSS values are both fairlyneutral for each model (RPSSJune = 0.048, RPSSJuly = 0.066), meaning that,although the models visually follow the pattern of the original behavior fairly well,

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they still do not reproduce it much better than a random number generator. Therecould be two reasons for this. One is that the models were trained on a limited set ofdata only spanning one week each. The second reason, which ostensibly holds for anyMPC optimization results generated through a particle swarm algorithm, is that the“optimal” results are themselves generated through a stochastic process. Throughthe emergent behavior of the PSO algorithm, one assumes that results will convergeon a near optimal result, but the absolute optimum is not guaranteed or known.In this way, the optimal control signal itself contains some noise that the eventualcontrol algorithm or model approximation does not necessarily need to replicate.

0 50 100 150

0.0

0.2

0.4

0.6

0.8

1.0

Model Response and Optimal Results: June

Time

Pro

babi

lity

of W

indo

w O

peni

ng

0 50 100 150

0.0

0.2

0.4

0.6

0.8

1.0

Model Response and Optimal Results: July

Time

Pro

babi

lity

of W

indo

w O

peni

ng

Figure 9: GLM response for June (above) and July (below) shown in red. Optimalcontrol signal shown in black.

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0.3.3 Time-Lagged Model Development

A modified approach was also attempted to develop June and July GLMs. Given thethermal mass present in the building and the fact that our original optimal controlsignal was developed based to some degree on future predictions of energy use, it ispossible that certain variables crucial to the heat transfer processes in the building(e.g. outdoor temperature and solar gains) need to be examined as a lagged dataset.The above GLM development process was inserted in a loop in which a subset ofthe predictor variables — namely the variables pertaining to outdoor environmentalconditions — were lagged by k hours up to a maximum of 12 hours. The PCA/GLMprocess was repeated for each lag and the model with the best RPSS was preserved.These results are presented in figure 10 below. The best lags were 2 and 4 hoursfor June and July, respectively. The time lagging of variables, however, did notsignificantly improve RPSS scores (RPSSJune = 0.072, RPSSJuly = 0.073).

0.3.4 Model Development from Summer-Long Data

In the end, the single largest improvement in model prediction skill was achievedby building GLMs around summer-long optimization results. The PCA/GLM pro-cess was repeated for this longer time series, including consideration of time-laggedvariables. Interestingly, the spectral composition of the leading PCs of the predictorvariables compared more favorably with the original control signal. In particular, abetter diurnal component was observed, shown in figure 11.

The overall model prediction results are shown below. The first five PCs of thepredictor set were used to develop the model (capturing 90% of the variance in thepredictor dataset), and the model resulted in a significantly higher RPSS of 0.195.The improved skill of the model is demonstrated in figure 12, where the first week’sworth of results are presented.

To further test the robustness of the summer-long GLM and to explore whetherthe optimal results tend to exploit the same control logic across different MPC runsand different time periods, the model’s prediction performance was tested for theweek-long periods during June and July. The performance of the summer-long modelwas tested against the week-long segments of the training data spanning June 15 - 21and July 13 - 19. In this way, we see how well a model trained on a summer’s worth ofoptimization results performs on a subset of that data. This test also helps illuminatewhether similar strategies are being exploited by the optimizer during different partsof the summer. As figure 13 below demonstrates, the general summer model was ableto predict window behavior for the two different weeks quite well (RPSSJune = 0.091,RPSSJuly = 0.214). These results provide a very positive indication that it will be

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0 50 100 150

0.0

0.4

0.8

Model Response and Optimal Response: June

Time

Pro

babi

lity

of W

indo

w O

peni

ng

0 50 100 150

0.0

0.4

0.8

Model Response and Optimal Response: July

Time

Pro

babi

lity

of W

indo

w O

peni

ng

Figure 10: Model responses with lagged outdoor environment parameters for Juneand July.

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0 2 4 6 8 10 12

0.0

0.4

0.8

1.2

Spectrum of Optimal Control Time Series

Frequency [Cyc/Day]

Spe

ctra

l Pow

er

0 2 4 6 8 10 12

050

150

Spectrum of First PC of Predictors

Frequency [Cyc/Day]

Spe

ctra

l Pow

er

0 2 4 6 8 10 12

05

1015

2025

Spectrum of Second PC of Predictors

Frequency [Cyc/Day]

Spe

ctra

l Pow

er

Figure 11: Spectral analysis of the optimal control signal and the leading PCs of thepredictor variables for the whole summer.

20

0 50 100 150

0.0

0.4

0.8

Model Response and Optimal Results: June to August

Time

Pro

babi

lity

of H

eatin

g

Figure 12: The first week of predictions for the GLM, trained on the entire summer’sworth of optimal control results.

possible to extract meaningful, simplified control algorithms from future optimizationruns that may prove robust enough to span large portions of the cooling season.

0.4 Conclusion

Through a combination of PCA and a binomial GLM, this paper has demonstratedthat the current simplified MM optimization results do exploit an underlying logicthat can be extracted in a reduced-dimension model. The reduced-dimension GLMis of great practical use since it can reproduce the key features of the original opti-mal control signal without the need for running computationally burdensome MPCoptimizations that can take hours to complete.

The initial results are promising, but there are several important areas that shouldbe studied further. One is to implement the GLM model described above in theMATLAB/EnergyPlus environment as a window controller to compare the energyuse of the building under GLM control to that under MPC. The percent energysavings of the simplified model will ultimately be a better arbiter of model skill thanRPSS or any other statistical measure that one could devise.

The second important step yet to be taken is to further simplify the model to thepoint that we achieve a control algorithm with the best balance between maximumenergy savings and minimal input (i.e. minimal sensors). This may involve furtherdimensional reduction by dropping redundant predictor variables (operative temper-ature, for example) and reformulate the logistic regression model as a decision tree

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0 50 100 150

0.0

0.4

0.8

Model Response and Optimal Response: June Segment

Time

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0 50 100 150

0.0

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Model Response and Optimal Response: July Segment

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Figure 13: Subset of model results for June and July weeks using a model trainedon summer-long data.

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comprised of if/then statements.Lastly, the technique must be expanded beyond the simplified optimization cases

presented here to more complicated cases containing multiple decision variables, po-tentially for different pieces of building equipment.

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