Data Predictive Control for building energy management

Transcription

1 Data Predtve Control for buldng energy management Ahn Jan, Madhur Behl and Rahul Mangharam Abstrat Desons on how to best optmze energy systems operatons are beomng ever so omplex and onfltng, that model-based predtve ontrol (MPC) algorthms must play an mportant role. However, a key fator prohbtng the wdespread adopton of MPC n buldngs, s the ost, tme, and effort assoated wth learnng frst-prnples based dynamal models of the underlyng physal system. Ths paper ntrodues an alternatve approah for mplementng fnte-tme reedng horzon ontrol usng ontrol-orented data-drven models. We all ths approah Data Predtve Control (DPC). Spefally, by utlzng separaton of varables, two novel algorthms for mplementng DPC usng a sngle regresson tree and wth regresson trees ensembles (random forest) are presented. The data predtve ontroller enables the buldng operator to trade off energy onsumpton aganst thermal omfort wthout havng to learn whte/grey box models of the systems dynams. We present a omprehensve numeral study whh ompares the performane of DPC wth an MPC based energy management strategy, usng a sngle zone buldng model. Our smulatons demonstrate that performane of DPC s omparable to an MPC ontroller, wth only 3.8% addtonal ost n terms of optmal objetve funton and wthn 95% n terms of R sore, thereby makng t an allurng alternatve to MPC, whenever the assoated ost of learnng the model s hgh. I. ITRODUCTIO Control-orented predtve models of an energy system s dynams and energy onsumpton, are needed for understandng and mprovng the overall energy effeny and operatng osts. Wth a reasonably aurate foreast of future weather and buldng operatng ondtons, dynamal models an be used to predt the energy needs of the buldng over a predton horzon, as s the ase wth Model Predtve Control (MPC) []. However, a major hallenge wth MPC s n aurately modelng the dynams of the underlyng physal system. The task s muh more omplated and tme onsumng n ase of a large buldng and often tmes, t an be even more omplex and nvolved than the ontroller desgn tself. After several years of work on usng frst prnples based models for peak power reduton, and energy optmzaton for buldngs, multple authors [], [3] have onluded that the bggest hurdle to mass adopton of ntellgent buldng ontrol s the ost and effort requred to apture aurate dynamal models of the buldngs. The user expertse, tme, and assoated sensor osts requred to develop a model of a sngle buldng s very hgh. Ths s Department of Eletral and Systems Engneerng, Unversty of Pennsylvana, Phladelpha, PA 904, USA {ahnj, rahulm}@seas.upenn.edu Department of Computer Sene, Unversty of Vrgna, Charlottesvlle, VA 903, USA madhur.behl@vrgna.edu Ths work was supported n part by TerraSwarm, one of sx enters of STARnet, a Semondutor Researh Corporaton program sponsored by MARCO and DARPA. beause a buldng modelng doman expert typally uses a software tool to reate the geometry of a buldng from the buldng desgn and equpment layout plans, add detaled nformaton about materal propertes, about equpment and operatonal shedules. There s always a gap between the modeled and the real buldng and the doman expert must then manually tune the model to math the measured data from the buldng [0]. Moreover, the modelng proess also vares from buldng to buldng wth the onstruton and types of nstalled equpment. Another major downsde wth physs-based modelng s that enough data s not easly avalable and guesses for parameter values have to be made, whh also requres expert know how. The alternatve s to use blak-box, or ompletely datadrven modelng approahes, to obtan a realzaton of the system s nput-output behavor. The prmary advantage of usng data-drven methods s that t has the potental to elmnate the tme and effort requred to buld whte and grey box buldng models. Lstenng to real-tme data, from exstng systems and nterfaes, s far heaper than unleashng hoards of on-ste engneers to physally measure and model the buldng. Improved buldng tehnology and better sensng s fundamentally redefnng the opportuntes around smart buldngs. Unpreedented amounts of data from mllons of smart meters and thermostats nstalled n reent years has opened the door for systems engneers and data sentsts to analyze and use the nsghts that data an provde, about the dynams and power onsumpton patterns of these systems. The hallenge now, wth usng data-drven approahes, s to lose the loop for real-tme ontrol and deson makng for both small and large sale buldngs. We address these hallenges by ntrodung an alternatve approah (to greybox MPC) for fnte reedng horzon ontrol of buldng energy systems usng data-drven ontrol orented models. We all ths Data Predtve Control. Whle stll beng model based, DPC nvolves usng salable and nterpretable models for the buldng s dynams. In partular, we utlze modfed regresson trees and regresson trees ensembles to mplement suh ontrol. In our prevous work [7], we developed and evaluated DPC usng mult-output regresson trees as predtve models. In ths paper, we present two new approahes wth sgnfant mprovements. Ths work has the followng ontrbutons: ) We address the lmtatons of our prevous work, and present a data predtve ontrol wth sngle-output regresson trees (DPC-RT) algorthm for fnte reedng horzon ontrol. DPC-RT bypasses the ost and tme prohbtve proess of buldng hgh fdelty models of buldngs that use grey and whte box modelng ap-

2 proahes whle stll beng sutable for reedng horzon ontrol desgn (lke MPC). ) Whle DPC-RT provdes omparable performane to a MPC ontroller, we extend the algorthm to work wth an ensemble of regresson trees. The ensemble data predtve ontrol, (DPC-En) s the frst suh method to brdge the gap between ensemble predtve models (suh as random forests) and reedng horzon ontrol. We present a omprehensve ase study to demonstrate how DPC an aheve omparable performane as MPC but wthout utlzng a dynamal model of the system. We begn wth desrpton of a realst buldng model used for our ase study n Se. II. Se. III defnes the fnte reedng horzon ontrol problem wth MPC framework. Se. IV desrbes the tranng and ontrol algorthms for DPC wth regresson trees and random forests along wth model valdaton. We ompare the performane of DPCRT to MPC and dsuss the hallenges assoated wth DPC n Se. V. We onlude the paper wth a summary of the results and a bref dsusson on the future work n Se. VI. II. MODELIG For testng our algorthms, we use a realst model of the buldng obtaned from the HAMLab ISE tool []. HAMLab s an all-n-one olleton of models and tools sutable for MatLab and/or Smulnk. It provdes a lbrary of realst buldng and equpment models. The model under onsderaton s lnear and uses a state-spae representaton. It aptures the essental dynams governng the zone-level operaton whle onsderng external and nternal thermal dsturbanes. The buldng n queston s a sngle zone buldng, the parameters of whh are dentfed by fndng an equvalent RC network, the shemat of whh s shown n Fg.. The model has 4 states: floor temperature T fl, nternal faade temperature T f, external faade temperature T ef and nternal zone temperature T n suh that x := [T fl, T f, T ef, T n ] T, ontrol nput n the form of heat (n Watts) rejeted/added n the zone Q n suh that u := Q n, and 3 unontrollable nputs or dsturbanes: external temperature T ex, nternal heat gan due to oupany Q o and solar heat radaton Q sl suh that d := [T ex, Q o, Q sl ] T. The mathematal model an be represented as x k+ = Ax k + Bu k + B d d k () k = 0,..., T where A, B and B d are tme-nvarant state spae matres alulated at samplng tme T s = 300 s. The nput u s onstraned between u = 500 W (where negatve values denote oolng) and ū = 000 W and the states x between x = 30 o C and x = 50 o C. ISE also provdes 30 years (97-000) of hourly data for the atmospher dsturbanes, whh s an estmate of dsturbanes usng atual meteorologal data. It s assumed that the dsturbane vetor d s presely known for the purpose of smulatons n Se. V. In our future work, we wll also address the problem of dsturbane onstruton glass Fg. : RC network representaton of the buldng model []. past k future u k movng wndow u k+ u k+ k + Fg. : Fnte-horzon movng wndow of MPC: at tme k, the MPC optmzaton problem s solved for a fnte length wndow of steps and the frst ontrol nput u k s appled; the wndow then reedes one step forward and the proess s repeated at tme k +. unertanty. For more detals on physal modelng and value of dentfed parameters that form A, B and B d, we refer the reader to []. The results n ths paper are presented for ths sngle zone, however, the algorthms desrbed next are easly salable to multple zones. In [], we have suessfully modeled a storey, 70 zone buldng wth our data-drven algorthms. The fous of ths work s on omparng the performane of DPC wth MPC and usng a sngle zone suffes for that omparson as t elmnates any onomtants n the performane omparson. III. MODEL PREDICTIVE COTROL We use a fnte reedng horzon MPC ontroller as a benhmark for omparson. The fnte reedng horzon ontrol (RHC) approah nvolves optmzng a ost funton subjet to the dynams of the system and the onstrants, over a fnte horzon of tme [9]. After an optmal sequene of ontrol nputs are omputed, the frst nput s appled, then at the next step the optmzaton s solved agan as shown n Fg.. MPC for the same sngle zone model has been prevously mplemented n [4]. In ths paper, we use MPC for omparson aganst the DPC algorthm by defnng the followng objetve funton. The objetve of the ontroller

3 (supervsory) s to the energy usage,.e Q n whle mantanng a desred level of thermal omfort. Therefore, at tme step k, we want to determne the optmal sequene of nputs [Q n,k,..., Q n,k+ ] that satsfes Q n,k+j + λ (T n,k+j T ref ) subjet to x k+j = Ax k+j + Bu k+j + B d d k+j Q n Q n,k+j Q n T n T n,k+j T n j =,..., where λ s a tunng parameter and T ref s the referene zone temp that mantans thermal omfort. The parameter λ helps trade-off energy savngs aganst dsomfort. IV. DATA PREDICTIVE COTROL Our goal s to fnd data-drven funtonal models that relates the value of the response varable (.e. zone temperature, n ths ase) to ontrol nputs and dsturbanes. When the data has lots of features, as s the ase n large buldngs, whh nterat n omplated, nonlnear ways, assemblng a sngle global model, suh as lnear or polynomal regresson, an be dffult, and an lead to poor response predtons. An approah to non-lnear regresson s to partton the data spae nto smaller regons, where the nteratons are more manageable. We then partton the parttons agan; ths s alled reursve parttonng, untl fnally we get to hunks of the data spae whh are so tame that we an ft smple models to them. Therefore, the global model has two parts: the reursve partton, and a smple model for eah ell of the partton. Regresson trees s an example of an algorthm whh belongs to the lass of reursve parttonng algorthms. The semnal algorthm for learnng regresson trees s CART as desrbed n [3]. The prmary reason for ths modelng hoe s that regresson trees are hghly nterpretable, by desgn. Interpretablty s a fundamental desrable qualty n any predtve model. Complex predtve models lke neural-networks, support vetor regresson et. go through a long alulaton routne and nvolve too many fators. It s not easy for a human engneer to judge f the operaton/deson s orret or not or how t was generated n the frst plae. Buldng operators are used to operatng a system wth fxed log and rules. They tend to prefer models that are more transparent, where t s lear exatly whh fators were used to make a partular predton. At eah node n a regresson tree a smple, f ths then that, human readable, plan text rule s appled to generate a predton at the leafs, whh anyone an easly understand and nterpret. Makng mahne learnng algorthms more nterpretable s an atve area of researh, one that s essental for norporatng human entr models for buldng energy management. The entral dea behnd DPC s to obtan ontrol-orented models usng mahne learnng, and formulate the ontrol () problem n a way that RHC an stll be appled and the optmzaton problem () an be solved effently. In Se. IV-B, we desrbe our tranng algorthm where we use separaton of varables to ft models on ontrollable and unontrollable varables separately. In Se. IV-C and IV-D, we present two algorthms for mplementaton of DPC, namely DPC-RT whh utlzes a sngle regresson tree, and DPC-En whh uses an ensemble of regresson trees as the underlyng model. A. Tranng Data We smulate the HAMLab model gven by () wth a smple rule based strategy for a perod of 3 months (May 99 - July 99) so that we obtan a rh enough tranng data set. The foreast of dsturbanes T ex, Q o and Q sl are obtaned from the ISE data base. Heat due to oupany Q o s a dsrete varable whh s defned to be 500 W from 8 am to 6 pm and 0 W otherwse. An example of dsturbanes s shown n Fg. 6(a). In order to buld regresson trees we need to tran on tmestamped hstoral data. The followng feature varables are used for tranng the model: ) Weather Data W: Ths nludes measurements of the outsde ar temperature T ex, solar radaton Q sl. Sne we are nterested n predtng the power onsumpton for a fnte horzon, we nlude the weather foreast of the omplete horzon n the tranng features. ) Buldng Data B: The state of the buldng s gven by the zone ar temperature T n. Ths s typally known from temperature sensors n the buldng. We use autoregressve (lagged) terms of T n as features and the future predton(s) of T n as the response varable(s). The heat gan nto a zone due to oupany Q o s also reorded and used for model tranng. 3) Controller Data C: Ths nludes urrent and future ontrol atons Q n. We learn a model f whh predts future zone ar temperature gven urrent and future weather predtons, oupany heat gan, and past zone ar temperature: T n,k+ T n,k+. = f (W k,..., W k+, T n,k,..., T n,k δ, T n,k+ C k,..., C k+ ), (3) where δ s the order of autoregresson, or n shorthand notaton Y = f ( X,..., X n) wth n beng the number of features. ote that T n,k, and C k n (3) are same as 4 th omponent of x k and u k n (), respetvely. It s assumed that perfet foreasts of dsturbanes for the entre length of the horzon s avalable to both MPC and DPC. In realty, the foreasts also have an assoated unertanty but analyzng ts effet of ontrol performane s a future researh dreton and not the fous of ths work. B. Tranng Algorthm Even though t s possble to dretly learn a model based on (3) usng algorthms lke regresson trees, random forests

4 or neural networks, the resultng models are not sutable for optmzaton n () beause of multple reasons. Frst, the gradent s not defned for suh models so we may have to settle wth sub-optmal soluton usng evolutonary algorthms [8]. Seond, beause of autoregresson, t wll lead to state spae exploson. The next step s to modfy the regresson trees and make them sutable for syntheszng the optmal values of the ontrol varables n real-tme. Gven buldng data (X, Y), we an separate the ontrollable (or manpulated) varables X and unontrollable (or non-manpulated / dsturbane) varables X d n the feature set suh that X X d X. Applyng ths separaton of varables, the regresson trees are learned only on the non-manpulated varables or dsturbanes (X d, Y). We obtan a model n the followng form: Y = f tree ( X d,..., X n d ). (4) In the leaf R of the trees, we ft a lnear parametr model whh s a funton only of the ontrollable/manpulated varables: Y R = β T [, X ] T. (5) Any parametr regresson model s vald n the leaves. We hoose lnear models beause they smplfy the optmzaton problem when added as onstrants n (7). We valdate ths assumpton n Se. IV-E. In ths manner, we tran the regresson trees usng only X d, and then n eah leaf we ft a parametr (lnear) model whh s a funton only of X. At run tme, to solve the ontrol problem when only X d s known, we navgate to an approprate leaf R and use the lnear onstrant n the optmzaton problem. In the followng seton, we demonstrate how to defne ths optmzaton problem when we have a separate regresson tree for eah predton step k + to k +. In Se. IV-D we extend ths to an ensemble of regresson trees (random forest) to derease the varane n predtons. C. DPC-RT: DPC wth Regresson Trees To replae dynams onstrants () n (), we buld trees to predt the states of the system,.e. T j s used to predt Y j := T n,k+j where j = {,..., }. Whle eah tree s traned only on X d, n the th leaf of the j th tree we ft a lnear model: Y j R = β T [, X k,..., X k+j ] T. (6) Eq. (6) mples that the predton of zone temperature at tme k + j s an affne ombnaton of ontrol nputs from tme k to k+j. Eah tree ontrbutes to a lnear onstrant n the optmzaton problem. Thus, the DPC-RT ounterpart Algorthm Data Predtve Control wth Regresson Trees DESIG TIME proedure MODEL TRAIIG USIG SEPARATIO OF VARIABLES Set X manpulated features Set X d non-manpulated features Buld predtve trees T j wth (Y j, X d ) usng (4) for all trees T j do for all regons R at the leaves of T j do [, X k,..., X k+j Ft Y j R = β T end for end for end proedure RU TIME proedure PREDICTIVE COTROL whle t < t stop do Determne the leaf and regon R (k) usng X d (k) Obtan the lnear model at R (k) Solve optmzaton n (7) to determne optmal ontrol aton [X (k),..., X (k + )] T Apply the frst nput X (k) end whle end proedure of () beomes Q n,k+j + λ ] T (T n,k+j T ref ) subjet to T n,k+j = β T [, Q n,k,..., Q n,k+j ] T Q n Q n,k+j Q n T n T n,k+j T n j =,...,. We solve ths optmzaton n the same manner as fnte reedng horzon ontrol to determne [Q n,k,..., Q n,k+ ], hoose the frst ontrol nput Q n,k and proeed to the next tme step k +. Our algorthm for DPC wth regresson trees s summarzed n Algo. and a shemat s shown n Fg. 3(a). Durng tranng proess, trees are traned only on unontrollable varables wth lnear models n the leaves whh are a funton only of ontrollable varables. Durng ontrol step, at tme k, the unontrollable features X d (k) are known and thus the leaf R (k) of eah tree s known. The lnear models n R (k) at as onstrants n optmzaton problem. From the ontrol aton [X (k),..., X (k + )] T, the frst nput X (k) s appled to the system. The resultng output Y(k) whh s a feature for the next tme step s fed bak to determne X d (k + ). D. DPC-En: DPC wth Ensemble Models Regresson trees obtan good predtve auray n many domans. However, the smple models used n ther leaves have some lmtatons regardng the knd of funtons they are able to approxmate. The problem wth trees s ther hgh (7)

5 varane and that they an over ft the data. It s the pre to be pad for estmatng a smple, tree-based struture from the data. Often a small hange n the data an result n a dfferent seres of splts. The man reason behnd ths s the herarhal nature of the proess: the effet of an error n the top splt s propagated down to all of the splts below t. Whle prunng and ross valdaton an help redue over fttng, we use ensemble methods for growng more aurate trees. The goal of ensemble methods s to ombne the predtons of several base estmators bult wth a gven learnng algorthm n order to mprove generalzablty and robustness over a sngle estmator. Random forests or tree-baggng are a type of ensemble method whh makes predtons by averagng over the predtons of several ndependent base models. The essental dea s to average many nosy but approxmately unbased trees, and hene redue the varane. Injetng randomness nto the tree onstruton an happen n many ways. The hoe of whh dmensons to use as splt anddates at eah leaf an be randomzed, as well as the hoe of oeffents for random ombnatons of features. Another ommon method for ntrodung randomness s to buld eah tree usng a bootstrapped or sub-sampled data set. In ths way, eah tree n the forest s traned on slghtly dfferent data, whh ntrodues dfferenes between the trees. More expltly, tranng features X p wth p < n and the data tself (n-bag samples) are dfferent for eah tree n the forest. ot all estmators an be mproved by shakng up the data lke ths. However, hghly nonlnear estmators, suh as trees, beneft the most. For a more omprehensve revew we refer the reader to [5]. The man dea here s to replae eah tree n Algo. by a forest ( Y = f forest X d,..., X n ) d whh, agan, s traned only on X d, and then ft a lnear regresson model usng X n every leaf of every tree of every forest. We buld forests for predton steps suh that the forest R j uses a lnear model Y j R = Θ T (8) [, X k,..., X k+j ] T (9) n the leaf R of every tree, where j = {,..., }. Wth a slght abuse of notaton, here (X, Y) orrespond to the nbag samples (n-bag samples orrespond to the data samples on whh the tree was traned) for the trees. Whle the offlne tranng burden n DPC-En s slghtly nreased ompared to DPC-RT, n the ontrol step we explot the better auray, and lower varane propertes of the random forest. If a forest has t number of trees, gven the foreast of dsturbanes, we have t set of lnear oeffents. We smply average out all the oeffents from all the trees to get one lnear model represented by ˆΘ for eah forest. ote that the averagng step an only be done n run-tme beause the leaf of eah tree an be narrowed down only when X d s known. Thus, for forests, we agan have exatly lnear Algorthm Data Predtve Control wth Random Forests RU TIME proedure PREDICTIVE COTROL whle t < t stop do for all forests do Determne the leaves R (k) usng X d (k) Obtan all lnear models at R (k) Average out the lnear oeffents ˆΘ end for Solve optmzaton n (0) to determne optmal ontrol aton [X (k),..., X (k + )] T Apply the frst nput X (k) end whle end proedure equalty onstrants n the optmzaton problem below. Q n,k+j + λ (T n,k+j T ref ) subjet to T n,k+j = ˆΘ T [, Q n,k,..., Q n,k+j ] T Q n Q n,k+j Q n T n T n,k+j T n j =,...,. (0) Algo. summarses the ontrol step. The ensemble data predtve ontrol, (DPC-En) s the frst suh method to brdge the gap between ensemble predtve models (suh as random forests) and reedng horzon ontrol. E. Valdaton For horzon length = 6 and order of autoregresson δ = 6, we run a smulaton on June, 000 usng a random sequene of nputs. Sne all dsturbanes as well as ontrol nputs are known, we an predt the zone temperature T n,k+ to T n,k+6 at every tme k usng both trees and ensembles. We have shown these 6 predtons at dfferent tmes n the smulaton n Fg. 4. At tme k, we make next 6 predtons of zone temperature and ompare t to the atual zone temperature that s obtaned by applyng the same ontrol nput. We repeat ths proess at k to k 6. The normalzed root mean-squared error (RMSE) obtaned wth sngle regresson trees and regresson tree ensembles are shown n top and bottom, respetvely. Both aheve lose to 99% auray, but on an average, predtons wth ensembles are more aurate and have a lower varane. The absolute resdual error n the predtons of T n,k+ for full day s shown n Fg. 5 whh agan shows that random forest are more aurate. The absolute error for ensembles s bounded by ±0.4 o C whle for sngle trees t s ±0.8 o C. V. COMPARATIVE STUDY We now run DPC and MPC ontroller n a losed-loop smulaton wth the HAMLab plant model. The objetve funton and the box onstrants on nput and states are

6 Tree X d Tree T Tree X d X d X d X d X d X d3 X d3 X d4 = βt, Xk X d3 X d4 X d4 = βt, Xk, Xk+ Qn,k+j + λ = βt, Xk,..., Xk+j (Tn,k+j Tref ) subjet to Tn,k+j = βt [, Qn,k,..., Qn,k+j ] separaton of varables allows to buld trees on and lnear models on Xd (k) T Qn Qn,k+j Q n run-tme optmzaton solves for Tn Tn,k+j T n j =,..., (a) DPC-RT: At tme k, the algorthm uses the foreast of dsturbanes Xd (k) to selet lnear models β n the leaves of eah tree. Eah lnear model s added as a onstrant n the optmzaton problem whh alulates optmal sequene [X (k),..., X (k + )]T, of whh the frst one s appled, and Xd (k + ) s alulated to proeed to k +. Xd (k) Forest Forest = Θ T, Xk run-tme optmzaton solves for = Θ T, Xk,..., Xk+j = Θ T, Xk, Xk+ Qn,k+j + λ (Tn,k+j Tref ) subjet to Tn,k+j = Θ T [, Qn,k,..., Qn,k+j ] Qn Qn,k+j Q n separaton of varables allows to buld trees on and lnear models on Forest T Tn Tn,k+j T n j =,..., (b) DPC-En: At tme k, the algorthm uses the foreast of dsturbanes Xd (k) to selet lnear models Θ to Θt n the leaves of eah ensemble. The lnear models n eah ensemble are averaged to alulate a sngle model represented by Θ j whh at as onstrants n the optmzaton problem. Agan, the optmal sequene [X (k),..., X (k + )]T, of whh the frst one s appled, and Xd (k + ) s alulated to proeed to k +. Fg. 3: Run-tme algorthm for data predtve ontrol wth regresson trees and tree ensembles.

7 TABLE I: Quanttatve omparson of R sore, mean value of objetve funton, energy onsumpton and mean devane from the referene temperature. R sore objetve value energy mean devane [ ] (% hange) [ ] [kwh] [ o C] MPC ( ) DPC-En 95.3% 6.0 (3.8%) DPC-RT 83.9% (8.%) numerally, we use Gurob Optmzer [6]. Fg. 4: Auray of predtons wth snapshots n tme: at tme k, we make 6 predtons for tme k to k + 5. The numbers represent mean auray n predtons for these steps for trees (top) and ensembles (bottom). Ths proess s repeated at tme k,..., k 5. resduals Tpred Tat[ o C] tree ensemble 00:00 06:00 :00 8:00 00:00 tme [hh:mm] Fg. 5: Resdual error for the tree T and the forest R that predt T n,k+ shows that the resduals for the forest are more onentrated around 0, whle tree predtons have a muh hgher varane. exatly same n both ases. Only dfferene n the optmzaton problem s due to system dynams. In partular, MPC solves (), DPC-RT solves (7) and DPC-En (0). T ref n the optmzaton problems s set to 8 o C. A. Smulaton Settngs We test the performane of DPC ontroller aganst MPC ontroller on June, 000. The hoe of ths day s arbtrary, and does not fall n the tranng perod. For the HAMLab smulaton model under onsderaton, the hourly weather predtons are only avalable from , however, the hoe of the year does not nfluene the omparson. Heat gan due to oupany Q o s dsrete varable whh s defned to be 500 W from 8 am to 6 pm and 0 W otherwse. The dsturbanes on June, 000 are shown n Fg. 6(a). The samplng tme T s n MPC and model tranng for DPC s 5 mn, whle the weather data s avalable after every h, so these dsturbanes are kept onstant for tme steps. and δ are agan hosen to be 6. For solvng optmzaton B. Results Optmal ontrol strateges for all 3 methods are shown n Fg. 6(b). The hosen referene temperature and external dsturbanes requre both heatng and oolng at some pont durng the day. The soluton obtaned from MPC sets the benhmark that we ompare to. ote that the MPC mplementaton uses the exat knowledge of the plant dynams. Therefore, the assoated ontrol strategy s ndeed the optmal strategy for the plant. In realty the qualty of data, and modelng errors also affet the performane of the MPC ontroller []. Qualtatvely, the DPC-En ontrol nput s not muh dfferent from that of MPC. Untl 8 am, when the dsturbanes are not hangng to a great extent, MPC and DPC-En are very lose. Slghtly after 8 am, when solar heat, external temperature and heat due to oupany all nrease abruptly, we observe that DPC-En ools less and dffers slghtly from the benhmark. Although DPC-RT also mantans the trend n the ontrol nput when the dsturbanes nrease or derease, the ontrol strategy s qute dfferent from that of MPC. Due to hgh varane n the predtons, DPC-RT also results n non-smooth nputs, whh are not good for pratal reasons of swthng onstrants on heatng and oolng equpment n buldngs. Fg. 6() shows the plot for the zone ar temperature. Agan, DPC-RT shows a bgger devaton from the optmal soluton whle DPC-En s near-optmal. Ths s attrbuted to lnear model averagng n ensemble learnng whh mproves the model predton. Ths mprovement omes at the ost of redued nterpretablty. The quanttatve results from the smulatons are tabulated n Tab. I. We are nterested n evaluatng how lose DPC performs n omparson to optmal benhmark set by MPC. The oeffent of determnaton or the R sore shows that 95.3% varablty n MPC has been aounted for by DPC- En, whle DPC-RT aptures only 83.9% varablty. In other words, DPC-En and MPC ontrol nputs are 95.3% lose. Fg. 6(d) shows the umulatve sum of the objetve funton as a funton of tme. At the end of the day, the umulatve sum for DPC-En s 3.8% more than MPC, and for DPC-RT t s 8.% more than MPC. Whle both DPC-En and DPC-RT are near-optmal, the value of the objetve funton valdates that DPC-En s more loser to the optmal soluton. In the onsdered senaro, MPC traks the referene temperature more losely offerng better thermal omfort than DPC-RT or DPC-En by spendng more energy.

8 temperature [ o C] ontrol nput Qn[W] nternal temp. Tn[ o C] um. objetve ost [ ] 0 5 T ex Q o Q sl :00 06:00 :00 8:00 00:00 (a) Hourly weather data T ex and Q sl and nternal dsturbane Q n MPC DPC-En DPC-RT heat [W] 00:00 06:00 :00 8:00 00: (b) Optmal ontrol strateges Q n. MPC DPC-En DPC-RT 00:00 06:00 :00 8:00 00: () Optmal nternal zone ar temperature T n. MPC 0.5 DPC-En DPC-RT 0 00:00 06:00 :00 8:00 00:00 tme [hh:mm] (d) Cumulatve sum of the objetve funton. Fg. 6: Comparson of optmal performane obtaned wth MPC, DPCRT and DPC-En on June, 000. All 3 ontrollers start from the same state of the model. VI. COCLUSIO & DISCUSSIO We present two data predtve ontrol algorthms for fnte reedng horzon ontrol usng regresson trees and regresson trees ensembles. Whle DPC-RT uses a sngle regresson tree for eah step of the ontrol horzon, DPC- En uses a random forest at eah nterval of the ontrol horzon. By separatng the ontrollable and unontrollable features durng tranng, and fttng a lnear model on just the ontrollable features, the optmzaton s redued to a smple onvex program. The ensemble data predtve ontrol, (DPC-En) s the frst suh method to brdge the gap between ensemble predtve models (suh as random forests) and reedng horzon ontrol. We mplement DPC ontroller for energy management on a sngle zone buldng model and ompare ts performane wth a MPC ontroller that uses the aurate dynam model. We demonstrate that DPC an aheve omparable performane to MPC, wth only 3.8% addtonal ost n terms of optmal objetve funton and wthn 95% n terms of R sore, whle avodng the ost and effort assoated wth learnng a grey box/whte box model of the system; and effet whh magnfes at larger sale (hundreds of zones n a large buldng). Our urrent and future work s foused on demonstratng the apablty of DPC on muh larger and realst buldng models, and usng real buldng data and test-beds. We are also addressng the queston of learnng relable data-drven models wth lmted funtonal testng of the plant, and provdng stablty guarantees for DPC for model swthng. DPC has applatons whh go beyond buldngs and energy systems, to ndustral proess ontrol, and ontrollng large rtal nfrastrutures lke water networks, dstrt heatng & oolng. DPC s mmensely valuable n stuatons where frst prnples based modelng ost s extremely hgh. REFERECES [] M. Behl, A. Jan, and R. Mangharam. Data-drven modelng, ontrol and tools for yber-physal energy systems. In 06 ACM/IEEE 7th Internatonal Conferene on Cyber-Physal Systems (ICCPS), pages 0. IEEE, 06. [] M. Behl, T. ghem, and R. Mangharam. Model-q: Unertanty propagaton from sensng to modelng and ontrol n buldngs. In Internatonal Conferene on Cyber Physal Systems. IEEE/ACM, 04. [3] L. Breman, J. Fredman, C. J. Stone, and R. A. Olshen. Classfaton and regresson trees. CRC press, 984. [4] J. Drgoňa and M. Kvasna. Comparson of MPC strateges for buldng ontrol. In Proess Control (PC), 03 Internatonal Conferene on, pages IEEE, 03. [5] J. Fredman, T. Haste, and R. Tbshran. The elements of statstal learnng, volume. Sprnger seres n statsts Sprnger, Berln, 00. [6] I. Gurob Optmzaton. Gurob optmzer referene manual, 05. [7] A. Jan, R. Mangharam, and M. Behl. Data Predtve Control for peak power reduton. In Proeedngs of the 3rd ACM Internatonal Conferene on Systems for Energy-Effent Bult Envronments, pages ACM, 06. [8] A. Kusak, Z. Song, and H. Zheng. Antpatory ontrol of wnd turbnes wth data-drven predtve models. IEEE Transatons on Energy Converson, 4(3): , 009. [9] D. Q. Mayne, J. B. Rawlngs, C. V. Rao, and P. O. Sokaert. Constraned model predtve ontrol: Stablty and optmalty. Automata, 36(6):789 84, 000. [0] J. R. ew, J. Sanyal, M. Bhandar, and S. Shrestha. Autotune e+ buldng energy models. Proeedngs of the 5th atonal SmBuld of IBPSA-USA, pages 3, 0. [] D. Sturzenegger, D. Gyalstras, M. Morar, and R. S. Smth. Model predtve lmate ontrol of a swss offe buldng: Implementaton, results, and ost beneft analyss. IEEE Transatons on Control Systems Tehnology, 4():, 06. [] A. Van Shjndel. Integrated heat, ar and mosture modelng and smulaton n hamlab. In IEA Annex 4 workng meetng, Montreal, May, 005. [3] E. Žáčeková, Z. Váňa, and J. Cgler. Towards the real-lfe mplementaton of MPC for an offe buldng: Identfaton ssues. Appled Energy, 35:53 6, 04.