DETECTION OF CHILLER ENERGY EFFICIENCY FAULTS USING EXPECTATION MAXIMIZATION

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1 Proceedngs of the ASME 2015 Internatonal Desgn Engneerng Techncal Conferences & Computers and Informaton n Engneerng Conference IDETC/CIE 2015 August 2-August 5, 2015, Boston, USA DETC DETECTION OF CHILLER ENERGY EFFICIENCY FAULTS USING EXPECTATION MAXIMIZATION R Lly Hu Mechancal Engneerng Unversty of Calforna, Berkeley Berkeley, Calforna Emal: lhu@berkeley.edu Jessca Granderson Buldng Technology and Urban Systems Lawrence Berkeley Natonal Laboratory Berkeley, Calforna, Emal: jgranderson@lbl.gov Alce Agogno Mechancal Engneerng Unversty of Calforna, Berkeley Berkeley, Calforna Emal: agogo@berkeley.edu ABSTRACT To detect degradaton n energy effcency of a chller n a chller plant, a multvarate Gaussan mxture model s appled. Ths classfcaton technque was selected to take advantage of an expected correlaton between measurable state varables and equpment and operaton specfcatons and system control targets. The hdden varable s the faultness of the chller and can take on one of three possble states. The fve observed varables correspond to sensor measurements that are typcally avalable and montored n commercally avalable chller plants. The fault detecton algorthm s traned on smulated data for the Molecular Foundry at the Lawrence Berkeley Natonal Laboratory and tested on measured sensor data. The results show that detecton of severe faults and no faults are relatvely accurate, whle detecton of moderate faults s sometmes mstaken for severe faults. The computaton needs are moderate enough for deployment and contnuous energy montorng. Future research outlnes the next steps n regards to senstvty analyses wth alternate probablty densty functons. (FDD). Therefore, an enormous opportunty for energy savngs s n montorng of energy use and contnuous fault analyss. The medan whole buldng energy savngs s 16% and just 13 of the most common faults n U.S. commercal buldngs n 2009 were beleved to have caused over $3.3 bllon of wasted energy [3]. For contnuous fault analyss, manual FDD can be tedous. To automate the process of fault detecton, a statstcal learnng technque s appled. The specfc energy applcaton under nvestgaton s the detecton of effcency degradaton of chllers n a central chller plant. These plants are used n the commercal and ndustral buldngs sectors. A central chller plant s a large faclty that provdes centralzed chlled water to cool large buldngs and mult-buldng campuses. Wthn the chller plant, the chllers are the refrgeraton machnes that provde the chlled water. Chllers are typcally ndustral-grade machnes and are desgned to last more than 20 years. Over such tme scales, performance degradaton and the resultng costs of energy can buld up, hence our focus on ths applcaton. INTRODUCTION Commercal buldngs consume 19% of US prmary energy [1]. Of ths, an estmated 15% to 30% of energy used n commercal buldngs s wasted by poorly mantaned, degraded, and mproperly controlled equpment [2]. Much of ths waste can be prevented wth automated fault detecton and dagnostcs Address all correspondence to ths author. BACKGROUND Fault detecton and dagnostcs have been successfully appled n a range of engneerng domans ncludng manufacturng, n general, and n the aerospace and automotve ndustres. For an overvew of the broad categores of FDD approaches, readers are referred to a revew of FDD n general [4] [5] [6] and a revew of FDD as appled to buldng systems [2] [7]. 1 Copyrght c 2015 by ASME

2 These works frst classfy methods and then compare, and contrast the three FDD categores: qualtatve model-based, quanttatve model-based, and process hstory-based methods. Revew papers are also avalable for surveyng specfc categores of FDD, such as supervsory methods, model-based technques, and trends n applcatons of model-based technques [8]. In addton to revews of FDD methods, the lterature also ncludes proposed FDD algorthms appled to buldng systems [9] [10] [11]. The FDD approaches proposed n the lterature ncludes rule-based methods, expert systems, physcal models, black box models, etc. What makes chllers an nterestng problem s that they operate over a range of effcences dependng on the requred load. Therefore, a good FDD method needs to consder the nfluences of varables that could mpact operatng condtons. Ths paper nvestgates a statstcal learnng approach to account for these varatons. Much of the research on chller applcatons focuses on nternal chller components, for example, ol levels and refrgerant leaks wthn the chller. Ths paper, on the other hand, analyzes the chller n the context of sensor data avalable at the chller plant level. The beneft of ths approach s that newer chllers already have FDD for nternal components bult n by the manufacturer, but system level FDD s rarely used n current busnessas-usual operaton and mantenance of commercal buldngs. Furthermore, exstng FDD bult nto equpment, or people themselves, can detect hard faults falures that occur abruptly and ether cause the system to stop functonng or to fal n meetng comfort condtons. On the other hand, soft faults that degrade performance but allow contnued operaton of the system are more dffcult to detect and dagnose, and can contnually waste energy [12]. One example of such a fault s degradaton n chller effcency. For these reasons, ths paper focuses on a method of detectng soft faults. METHODOLOGY The process used n ths project to detect a soft fault s as follows: 1. Select a fault of nterest. 2. Hypothesze whch sensor data would be useful to detect ths fault. 3. Smulate data under faulty and non-faulty condtons. Preferably, data from a range of faulty and non-faulty operatons would be acqured from measured sensor ponts, possbly from mantenance data or by ntroducng faults to a physcal system. However, t s mportant to recognze that a wde range of operatonal fault data may be hard to acqure wth any statstcal accuracy and smulated fault data may be needed. 4. Investgate whch sensor data s most useful n detectng ths fault and select those sensor ponts for ncluson n the fault detecton algorthm. 5. Chose or develop a mathematcal model for fault detecton. 6. Defne the most crtcal varables n the mathematcal model and select a method to calculate the unknown parameters, e.g., tran for the parameters. 7. Tran the model on a smulated data set. 8. Test the model on smulated data, dfferent from the tranng set. 9. Test the model on measured operatonal data. 10. Analyze the results. 11. Iterate. These steps are appled to a chller effcency applcaton and descrbed sequentally n the next chapters and sectons. Fault Selecton To pck a chller plant fault to focus on, a lterature revew of scentfc and engneerng practce and consultatons wth chller plant staff and buldng researchers resulted n the complaton of a lst of approxmately 50 canddate faults. These faults were then ranked accordng to the mpact on energy, ease of detecton, and frequency of occurrence usng nput from researchers at LBNL and feedback from operators of a smlar plant at the U.S. Washngton Navy Yard. Any faults that could already be detected by the chllers own control system were trmmed from the lst. Addtonal consderatons n rankng the faults were the avalablty of data and the ablty to generate faulty data through manpulatng varables n a physcs-based model of the chller plant. As a result of these nvestgatons, the fault selected for focus was degradaton of chller effcency across the range of coolng loads delvered. Ths fault occurs when the effcency of the chller s lower than expected, gven the current operatng condtons of the chller. Chllers have effcency curves that vary wth coolng load even under deal condtons. These effcency curves are commonly drawn on a plot of kw/ton vs. ton. The chller effcency fault s partcularly nterestng because the effcency of an deal chller naturally depends on the operatng condtons, whch nclude enterng and leavng chlled water temperatures, condenser water temperatures, and the loadng on the chller. Ths property can be dffcult to capture n smpler fault detecton methods such as thresholds on a metrc such as power or kw/ton. Thus, ths fault s a good canddate for a statstcal learnng fault detecton method. VARIABLE SELECTION An ntal canddate lst of 30 varables correspondng to sensor ponts was selected. Then, ths lst was reduced to the most sgnfcant fve varables through exploratory data analyss. 2 Copyrght c 2015 by ASME

3 By plottng the canddate varables, some of them were found to be almost constant or unchangng even when the effcency of the chller decreases. Some of these varables have the same pattern and dstrbuton regardless of the effcency of the chller. Thus, some of the varables were ndependent of the effcency of the chller and were poor canddates as varables to detect neffcency. The choce of the fve varables that were selected s consstent wth physcal ntuton of the effcency degradaton. Four of the varables are assocated wth Chller 1: power consumpton, nstantaneous coeffcent of performance (COP), temperature of the enterng condenser water, and the temperature of the extng condenser water. All of these values are physcally related to the effcency of the chller. The last varable corresponds to the extng condenser water temperature of the other chllers. Ideally workng chllers should have an extng condenser water temperature that s close to the desgn specfcaton, and both Chller 2 and Chller 3 are the same model. The extng condenser water temperature would be expected to be hgher n the case of a faulty chller, snce more electrcal energy s needed to supply the necessary coolng, and hence more heat s generated that results n a larger ncrease n the condenser water temperature. The extng condenser water temperature of the faulty chller can be compared aganst the correspondng temperatures at the other chllers, whch are not faulty. To ntroduce dependence on hstorcal data, the fnal varable lst ncluded the fve sensor ponts at tme t, and at tme t 1. TESTING FACILITY The specfc chller plant used as a case study s the chller plant at the Molecular Foundry at the Lawrence Berkeley Natonal Laboratory. The Molecular Foundry s a 6-floor nanoscence research faclty that s served by a chller plant that has three chllers operatonal for 24 hours a day, seven days a week. A smplfed schematc of the chller plant at the Molecular Foundry s shown n Fgure 1. Chller 1 was selected as the locaton of our nvestgatons of a possble chller effcency fault due to the greater avalablty of sensor data assocated wth t wth whch to test the algorthm. Data for many of the sensor ponts assocated wth the other chllers was unavalable. TRAINING DATA To tran the fault detecton model, data for normal operatons and for operatons under faulty condtons were generated usng a physcs-based model of the plant wrtten n Modelca. It was undesrable to ntroduce faults n the real plant to collect measured data because ntroducng faults would nterfere wth normal operatons of the faclty and damage the expensve equpment. The physcs-based model was created and calbrated by a research partner usng the Modelca modelng language wthn FIGURE 1. SCHEMATIC IMAGE OF THE CONDENSER WATER LOOP OF THE CASE STUDY FACILITY the Dymola modelng and smulaton envronment. The tranng dataset ncludes 24 hours of 10-mnute data of the chller plant under nomnal condtons, under condtons wth a moderate energy effcency fault n Chller 1, and under condtons wth a severe energy effcency fault n Chller 1. Because the fault detecton focuses on the performance of Chller 1, data was only consdered for when Chller 1 was operatng. Ths s because when Chller 1 was not operatng.e., not turned on one cannot deduce nformaton about ts operaton nor ts operatonal effcency. The samplng frequency of the tranng dataset was selected to be 10 mnutes wth a range from March 11 to March 21 to match the data from sensor measurements at the plant. The model nputs nclude the setponts of the plant and program default standard yearly weather profle for San Francsco. The physcs-based program smulates the plant and solves for all thermodynamc states for the whole-plant. The chller effcency faults were ntroduced by modfyng Chller 1 s reference coeffcent of performance (COP). The COP for coolng systems s an effcency metrc for chllers defned as the rato between the coolng output of the chller and the electrcal nput to the chller, as shown n Equaton 1. COP coolng = Q C W where Q C s the rate of heat removal and W s the electrcal nput. The nomnal reference COP for Chller 1 s Reference COPs for a moderate energy effcency fault was chosen as 3.0 and the COP under a severe effcency fault was 1.0. The COPs chosen for the fault condtons are below the mnmum effcency COP dctated by ASHRAE Standard (1) 3 Copyrght c 2015 by ASME

4 MEASURED SENSOR DATA Data from sensors at the Molecular Foundry was collected from July 23, 2012 to Aprl 1, 2013 when contnuous data was avalable durng a season when the chllers were operatng. Fault condtons were not recorded and thus t s not known when faults occurred durng ths tme perod. The only data pont not avalable from the lst of sensors of nterest was Chller 1 COP and thus t was estmated from the followng data ponts avalable: Chller 1 enterng chlled water temperature, Chller 1 extng chlled water temperature, Chller 1 mass flow rate. The resultng dataset covered 750 tmestamps. To concde wth the nstantaneous COP n the tranng dataset, the nstantaneous COP was also calculated from the sensor data usng Equaton 2: COP CH1,t = ṁch1,chw,t(cwt CH1,enterng,t CWT CH1,leavng,t ) w CH1 (2) where all varables are n reference to the chller: ṁ CH1,CHW,t s the mass flow rate of the chlled water, CWT CH1,enterng,t s the temperature of the enterng chlled water, CWT CH1,leavng,t s the temperature of the extng chlled water, and w CH1 s the power consumed by the chller. Fault Detecton Algorthm The fault detecton algorthm used s based on a mxture model, whch can be used to cluster data nto one of the mxture component clusters. In ths case, mxture components represent the faultness of the chller. A mxture model s chosen over estmaton of a sngle model parameter because mxture models have not been used n the lterature for ths specfc fault detecton problem. It would be nterestng to frame fault detecton as a mxture model, by vewng fault states as dfferent mxture components. One strength of ths framework s that each fault state can have a unque probablty dstrbuton of possble observatons from sensors. The parameters of the mxture model can then be learned usng standard technques. Ths model also gves probablstc assgnments of faultness and can be nterpreted as probabltes. In contrast, sngle model parameter estmaton may not be suffcent due to the range of condtons seen even under normal operatons. These condtons nclude start-up and shut-down of chllers, and transents when respondng to changes n coolng demand, or due to control sequences, or behavour of other equpment that s connected n the chller plant. A mxture model of dfferent probablty dstrbutons s more flexble than a sngle dstrbuton. A mxture model s a parametrc probablty densty functon represented as a weghted sum of component (or subpopulaton) probablty denstes. In ths case, the mxture components are modeled as multvarate Gaussan emsson probabltes because the observed varables are expected to be centered on a mean; ths mean could be, for example, the chller manufacturers equpment specfcaton, the as-desgned chller plant specfcaton, or the operaton control setponts. Furthermore, these varables are lkely correlated because of ther physcal meanng. That s, underlyng physcal laws must be obeyed. However, because the system s complex and non-deal, frst prncples cannot be used drectly. However, covarances can capture some of the dependences between varables. To explore how well the data s modelled by a normal dstrbuton, the followng fgures graph hstograms and normal probablty plots of the tranng data. The hstograms compare the data to a normal probablty curve generated from the mean and standard devaton of the data, as seen n Fgures 2, 3, and 4. A normal probablty plot s a type of Q-Q plot (quartle-quartle plot) that plots the quartles of the data aganst the normal dstrbuton, as seen n Fgures 5, 6, and 7. The straght lne represents the normal dstrbuton and departures of the data from the lne ndcate departures from normalty. From these graphs, the tranng data can be seen to somewhat approxmate a normal, wth larger devatons from normalty at the extremes of the data. A Gaussan mxture model assumes that the data s generated from a mxture of a fnte number of Gaussan dstrbutons wth unknown parameters. Ths can be expressed as: p(x µ,σ) = π N(x µ,σ ) (3) = p(z = 1 π )p(x Z = 1, µ,σ ) (4) where the π are the mxng weghts and are constraned to sum to one, x s the data, and Z s the component label. N s an abbrevaton for the Gaussan functon. HIDDEN VARIABLE The mxture model components are the energy effcency fault states of the chller: no fault, moderate fault, and severe fault. The observed varables are the sensor data. Usng ths model, sensor data can be classfed nto one of the fault states of the chller to detect the faultness of the chller. In ths applcaton, the fault state Z represents the chller effcency. The effcency of a chller can take on any value on a contnuous range from 0 to a theoretcal maxmum based on thermodynamcs. For smplcty, the effcency of the chller has been dscretzed. Hence, Z becomes a multnomal random varable and the components Z are dsplayed n Table 1. OBSERVED VARIABLES The observed varables are the fve sensor measurements descrbed n the varable selecton process. To create a dependency 4 Copyrght c 2015 by ASME

5 FIGURE 2. HISTOGRAMS OF THE NO-FAULT DATA COM- PARED TO GAUSSIAN DISTRIBUTIONS FIGURE 5. NORMAL PROBABILITY PLOTS OF THE NO- FAULT DATA FIGURE 3. HISTOGRAMS OF THE MODERATE FAULT DATA COMPARED TO NORMAL DISTRIBUTIONS FIGURE 6. NORMAL PROBABILITY PLOTS OF THE MODER- ATE FAULT DATA FIGURE 4. HISTOGRAMS OF THE SEVERE FAULT DATA COM- PARED TO NORMAL DISTRIBUTIONS FIGURE 7. NORMAL PROBABILITY PLOTS OF THE SEVERE FAULT DATA 5 Copyrght c 2015 by ASME

6 TABLE 1. meanng Colloqual label Components of the hdden varable Z and ther physcal Physcal nterpretaton No energy effcency fault COP = 5.42 Moderate energy effcency fault COP = 3 Severe energy effcency fault COP = 1 on prevous sensor measurements n tme, the observed varables ncluded measurements at both tme t and tme t 1. Thus, there are a total of ten varables: fve sensor measurements at tme t and fve sensor measurements at tme t 1. Fgure 8 shows the graphcal model of the Gaussan mxture model and the nterpretaton to ths applcaton. To solve for the unknown parameters, the Expectaton Maxmzaton (EM) algorthm [13] was chosen. Ths method s an teratve algorthm used to fnd maxmum lkelhood estmates of parameters n probablstc models. It s desrable to maxmze the lkelhood because the lkelhood s a measure for how well the data fts a gven model. The lkelhood s a functon of the parameters of the statstcal model, and therefore we want to fnd parameters that maxmze the lkelhood. The lkelhood for the Guassan mxture model s: l(θ D) = n log(x n θ) = n log π N(x n µ,σ ) (5) The goal of the maxmzaton step of the EM algorthm s to fnd the model parameters that maxmze the log lkelhood. By settng the partal dervatves of the log lkelhood to zero, we obtan the followng formulas for the model parameters. Ths s the maxmzaton step. µ (t+1) = n τn (t) x n n τn (t) (6) Σ (t+1) = n τn (t) (x n µ (t+1) )(x n µ (t+1) n τ (t) n ) T (7) π (t+1) = 1 N n τ (t) n (8) FIGURE 8. CATION GRAPHICAL MODEL AS APPLIED TO THIS APPLI- Where τ s defned as the condtonal probablty that the th component of Z s equal to one. τ p(z = 1 x,θ) = p(x Z = 1,θ )p(z π ) p(x θ) (9) Solvng for the Model Parameters Usng the Expectaton Maxmzaton Algorthm An algorthm s used to estmate the unknown parameters of the mxture model from data of the chller plant when operatng under faulty condtons and under normal condtons. The estmated parameters are the mxture component weghts π and the means µ and varances Σ of the mult-varable normal dstrbutons as shown n Equaton 4. The estmates for these parameters are then used n a condtonal estmate of the fault state. The determnaton of the fault state s analogous to detecton of the fault. Then, for each teraton of the algorthm, we have τn (t) = π(t) j π (t) j N(x n µ (t) N(x n µ (t),σ (t) ) j,σ (t) j ) (10) The expectaton step of the EM algorthm calculates the expected complete log lkelhood, whch n ths case s the calculaton of the posteror probablty τ n. The fault detecton algorthm was mplemented n Matlab. 6 Copyrght c 2015 by ASME

7 RESULTS AND DISCUSSION MODEL TRAINING To tran the model parameters, the Expectaton Maxmzaton algorthm was run for 5 trals of 100 teratons each. From among the trals, the parameters that resulted n the hghest lkelhood were selected. The ntal guess for the mxng proportons was π = 1/3 so that the ntal guess for each of the three states s equal. Ths s because the mxng proportons π wegh the mxture components so equal π wegh each component equally. The ntal guess for the mean of the varables was randomly generated. A good ntal guess was requred for the algorthm to run wthout the covarance matrx becomng non-sngular or the log lkelhood becomng nfnte. Each run of the algorthm took a few mnutes to complete on a 1.6 GHz Intel Core 5 processer wth 4 GB 1333 MHz DDR3 memory. Fgure 9 plots the tranng data wth the fault detecton parameters as calculated by the EM algorthm. The blue damonds are the means of the Gaussan mxture components the means of the fault state clusters as found by the EM algorthm. The black ovals trace the frst standard devaton of the covarances of the Gaussan mxture components the covarances of the fault state clusters. In each plot, two varables are plotted because a two-dmensonal space R 2 s easer to vsualze than ten dmensonal space R 10. The mxture model s actually R 10 Gaussan because the fault detecton model conssts of ten varables. Thus, the means are actually ponts n R 10, and the one standard devaton covarances are manfolds n R 10. Ideally, the mean should fall wthn the mddle of a cluster of data that share the same state; lkewse, the one standard devaton ovals should follow the shape of the cluster of data that share the same state. In Fgure 9, data wth the same fault state s ndcated by the same color. Because the tranng data was smulated, we know the value of the underlyng fault state. In practce, the fault state of the tranng data s often unknown and t s desrable for a fault detecton algorthm to automatcally dentfy data that corresponds to each fault state. The detecton model appears to be better at detectng the severe and no-fault states than the moderate fault state. Ths s expected because extremes are easer to detect than shades of grey. From Fgure 9, ths better detecton can be seen from the locaton of the mean closer to the cluster center and the smaller covarance oval. The covarance s largest for the moderate fault state and the oval representng one standard devaton sometmes encloses data ponts from the severe fault state. When data ponts wth dfferent states overlap, whch data pont belongs to whch state s not as dstngushable. Fgure 9 also shows that for some varables, the EM algorthm found the clusters of data assocated wth the dfferent fault states better than for other combnatons of varables. When the varables are plotted aganst one another n Fgure 9, one can see that some varables are more strongly correlated than others and form dstnct clusters for each of the fault states. The varables FIGURE 9. MEAN AND CONVARIANCES FOR THE GAUSSIAN MIXTURE MODEL PLOTTED AGAINST THE TRAINING DATA that have dstnct clusters for each of the fault states are more valuable for fault detecton. Ths fault detecton model takes ths nto account wth a larger covarance between the varables n the covarance matrx of the Gaussan mxture component. Smlarly, varables that are less correlated and less clustered have a smaller value n the covarance matrx. Some of the poor clusterng may be because some data from dfferent fault states are close to one another and may even overlap. For example, the dstrbuton of data ponts for the condenser 7 Copyrght c 2015 by ASME

8 leavng temperature s smlar between the fault states, whch results n a large overlap between the resultng mxture components. Secondly, the large data range between the fault states also affects the detecton model for some varables. For example, the Chller 1 power for the no-fault state s almost constant whle the power for the severe fault state ranges across an order of magntude. Furthermore, some varables are hghly lnear, as opposed to formng dstnct clusters. The temperature varables n partcular are hghly lnearly dependent on the other varables. Because there are equal amounts of data ponts for each of the fault states, one would expect that the weghtng coeffcents π would each be equal to 1/3. However, the calculated π = {0.53, 0.20, 0.27} for {no fault, moderate fault, severe fault} ndcates that the algorthm favors classfcaton of data ponts nto the no-fault state more than classfcaton nto the other fault states. Ths may be desrable because t favors decreasng the number of false postves. When data belongs to a fault state, the algorthm expects that the Gaussan mxture component wll be evaluated to a hgh probablty; that s, the N(n µ,σ ) term n Equaton 4 wll be close to one. Agan, ths may be desrable because a fault s rased only when the data fts well nto the faulty cluster. On the other hand, ths could be an ndcaton of over-fttng the model. MODEL TESTING ON SIMULATED DATA For each of the data ponts n the generated data set, the probablty of the data pont belongng n each fault state was calculated. Then, the data pont was classfed nto the fault state wth the hghest probablty. The runtme on the same laptop was a few seconds. Thus, the computaton needs of ths approach are practcal and relatvely nexpensve. Fgure 10 shows how the algorthm classfed each of the data ponts wthn the tranng set. Because the data s smulated, we know the underlyng fault state. The green bar represents correct classfcatons whle the red bars represent ncorrect classfcatons. The proposed approach s best at dentfyng the no-fault state and the severe fault state. Alas, a large percentage of moderate fault data s msclassfed as severe faults. The accuracy of the fault state classfcaton can be mproved through mproved nterpretaton of the calculated fault state probabltes. Ths can be acheved by tunng the assgnment rules and assocated thresholds for assgnment of fault states. For example, nstead of classfyng a data pont to the fault state wth the hghest probablty, a data pont may be classfed nto a moderate or severe fault state only f the probablty of t beng n one of the two fault states s greater than a tunable threshold, such as > 50%. These choces allow for tunng of the fault detecton algorthm to dfferent levels of certanty as desred, whch greatly mproves the flexblty of ths approach. FIGURE 10. CLASSIFICATION OF THE TRAINING DATA AS CLASSIFIED BY THE ALGORITHM MODEL TESTING ON MEASURED DATA Measurements collected from the Molecular Foundry are plotted as hstograms for each sensor pont along wth the tranng data for each of the fault states. The results are show n Fgure 11. Tranng data from each of the fault states s plotted ndvdually n a dfferent color. Ths s to vsually compare the dstrbuton of the measured data wth the dstrbuton of each of the fault states. Then, the measured data s plotted as scatter plots wth the mean and covarance of the mxture components ndcated. The results, shown n Fgure 12, have the same varables on each axs as Fgure 9. From Fgure 11, t can be observed that the measured data the grey hstogram s clustered more closely to the data from the no-fault tranng data the green hstogram than the data from the other fault states. Thus, one hypotheszes that the Molecular Foundry chller experences mostly fault-free operaton, and that the fault detecton algorthm should classfy most data measurements as belongng to the no-fault state. Intutvely, ths eght year old chller plant s relatvely new and well nstrumented and montored and one would not expect chller degradaton faults. From Fgure 12, data for some varables seem to better ft nto the mxture components than other varables, smlar to what was observed from the smulated data. The measured data that corresponds well to the dstrbuton of no-fault tranng data nclude: the Chller 2 condenser leavng temperature, Chller 1 enterng condenser temperature, and Chller 1 power. There seems to be an offset between the sensor data and the model mxtures, perhaps due to an offset n the physcs based model used to generate the tranng data. Because most of the sensor data seem to le wthn and closest to the no-fault mxture component, t seems lkely that there are no faults occurrng at the Molecular Foundry durng ths tme. One factor that nfluences the ft of the real sensor data to 8 Copyrght c 2015 by ASME

9 FIGURE 11. HISTOGRAMS OF THE MEASURED DATA COM- PARED TO SIMULATED TRAINING DATA the mxture components found from the tranng data s model msmatch between the real chller plant and the physcal model of the chller plant. One reason for ths msmatch s that data used to calbrate the physcs-based model does not nclude data durng faulty operatons, and thus the tranng data s an estmate of how the plant wll behave based on physcal laws. Thus, there s uncertanty n the tranng data for faulty cases. DEPLOYMENT The varables requred by ths fault detecton approach are commonly sensed n chllers and tracked by chller plant operators. Some systems ntegraton may be requred to extract ths data from the chller plant equpment and ntegraton s not always straghtforward. Exportaton of trend logs from buldng FIGURE 12. REAL DADTA PLOTTED AGAINST THE MIXTURE MODEL COMPONENTS automaton systems s vable, but establshng contnuous data acquston adds complexty. Ths s not unque to fault detecton as unfortunately, gettng data to external applcatons s stll costly and tme consumng n most buldngs projects. Furthermore, the computaton needs of ths approach are not very large. Ths fault detecton approach can be mplemented n real tme to classfy ncomng data nto fault states, or run perodcally or on demand to process hstorcal data. Once faults are 9 Copyrght c 2015 by ASME

10 detected, the nformaton can be relayed to the chller plant operators through a graphcal user nterface. When faults are rased, the operator can close the loop by nvestgatng and remedyng the fault. CONCLUSIONS The multvarate Gaussan mxture model most easly detects the no fault and the severe energy effcency fault n the chller, whch s understandable gven that extremes are the easest to detect. The moderate fault state s sometmes detected as a severe fault. Ths s lkely due to the large spread of data from the moderate fault state, whch overlaps wth data ponts n the severe fault state. Ths rases the queston of what s useful from a decson-analytc prognostcs standpont. What degree of moderate faults would be useful to detect for early warnng of a problem that would beneft by mantenance and nterventon? For example, nstead of three fault states, there could be smaller dscretzatons on fault levels defned for prognostc value. Then, the algorthm could calculate more detaled probablty dstrbutons of fault condtons for each gven measurement to understand the uncertanty n the fault assessment. Ths fault detecton algorthm could be traned usng a few days of data that cover a range of fault condtons. Also, the tranng process s able to dstngush between the dfferent fault states, whch means that one does not need to know whch tranng data ponts correspond to fault condtons, as long as the tranng data covers a range of faulty and non-faulty operatons. The run tme of the algorthm to calculate the fault state of measured data s on the order of a few seconds on a laptop, so that the computaton needs of ths approach are wthn practcal means. Lastly, the fault detecton mathematcal model and tranng procedure s flexble so that t can be appled to detect other types of faults wth other varables as nput. The current proposed fault detecton approach can be mproved to more accurately detect faults wth less false postves and false negatves. For example, further exploratory data analyss may fnd parametrc probablty dstrbutons that descrbe the mxture components better than the Gaussan dstrbuton. Some prelmnary analyss suggests that for the chller energy effcency fault, the exponental dstrbuton may be a better choce than the Gaussan dstrbuton. For other faults, other probablty dstrbutons may be more approprate. To mprove the fault detecton model, dfferent probablty dstrbutons can be explored whch may ft the data better than the normal dstrbuton. The normal probablty plots and hstorgrams show that the data devates from the normal dstrbuton, partcularly at the tals. As a result, more accurate fault detecton classfcaton may be acheved wth a better fttng probablty dstrbuton. To further mprove the model, dfferent choces of observed varables, and transformaton of varables can be consdered. Probabltes could also be condtoned on prevous data or on one another. The scope of the detecton problem could also be expanded to consder combnatons of chllers becomng faulty, other faults, and dagnostcs after fault detecton. To tran the algorthm, nstead of the Expectaton Maxmzaton algorthm, other approaches may be used, such as a Gbbs sampler. Ths may help address the ssue of overfttng of the fault detecton model to the tranng data. Overfttng occurs when the fault detecton model too closely descrbes the tranng data. The ssue of overfttng s an area for further exploraton n fault detecton, partcularly because one of the goals of fault detecton s to dentfy devatons from normal operaton. Because the result of probablstc fault detecton s a probablty dstrbuton, further work can be done to nvestgate the nterpretaton of these probablty dstrbutons. These nvestgatons could nclude, for example, when should a fault be rased, how to assgn severty to the fault rased, and how to balance the certanty of a fault occurrng for the last queston could be through the use of expectaton values of the cost of the fault and the certanty of the fault, whch would also requre the choce of a utlty functon to value the cost of the fault. ACKNOWLEDGMENT The authors would lke to thank Prof. Mchael Jordan for model advce, Lpng Wang for creatng the Modelca Model and for the metered data, Rongxn Yn for background nformaton, and Xupeng Feng, Marco Bonvn, Therry Noudu for help wth Modelca and acqurng the data, and Professor Auslander for hs advce on chller dagnostcs and control. Ths work was supported by the Assstant Secretary for Energy Effcency and Renewable Energy, Buldng Technologes Offce, of the U.S. Department of Energy under Contract No. DE-AC02-05CH The authors would also lke to acknowledge the Envronmental Securty Technology Certfcaton Program program and the U.S. Department of Defense. REFERENCES [1] Admnstraton, U. E. I., Annual Energy Outlook U.S. Energy Informaton Admnstraton. [2] Katpamula, S., and Brambley, M. R., Revew artcle: Methods for fault detecton, dagnostcs, and prognostcs for buldng systemsa revew, part. HVACI&R Research, 11(1), pp [3] Mlls, E., Buldng commssonng: a golden opportunty for reducng energy costs and greenhouse gas emssons n the unted states. Energy Effcency, 4(2), pp [4] Venkatasubramanan, V., Rengaswamy, R., Yn, K., and Kavur, S. N., A revew of process fault detecton 10 Copyrght c 2015 by ASME

11 and dagnoss: Part : Quanttatve model-based methods. Computers I& Chemcal Engneerng, 27(3), pp [5] Venkatasubramanan, V., Rengaswamy, R., and Kavur, S. N., A revew of process fault detecton and dagnoss: Part : Qualtatve models and search strateges. Computers I& Chemcal Engneerng, 27(3), pp [6] Venkatasubramanan, V., Rengaswamy, R., Kavur, S. N., and Yn, K., A revew of process fault detecton and dagnoss: Part : Process hstory based methods. Computers I& Chemcal Engneerng, 27(3), pp [7] Katpamula, S., and Brambley, M. R., Revew artcle: Methods for fault detecton, dagnostcs, and prognostcs for buldng systemsa revew, part. HVACI&R Research, 11(2), pp [8] Isermann, R., Model-based fault-detecton and dagnoss status and applcatons. Annual Revews n Control, 29(1), pp [9] Schen, J., Bushby, S. T., Castro, N. S., and House, J. M., A rule-based fault detecton method for ar handlng unts. Energy and Buldngs, 38(12), pp [10] Cu, J., and Wang, S., A model-based onlne fault detecton and dagnoss strategy for centrfugal chller systems. Internatonal Journal of Thermal Scences, 44(10), pp [11] Lang, J., and Du, R., Model-based fault detecton and dagnoss of {HVAC} systems usng support vector machne method. Internatonal Journal of Refrgeraton, 30(6), pp [12] Comstock, M. C., Braun, J. E., and Groll, E. A., The senstvty of chller performance to common faults. HVACI&R Research, 7(3), pp [13] Bshop, C., Pattern Recognton and Machne Learnng. Sprnger. 11 Copyrght c 2015 by ASME