Optimal partitioning of multi-thermal zone buildings for decentralized control
aa r X i v : . [ ee ss . S Y ] J un Optimal partitioning of multi-thermal zonebuildings for decentralized control
Ercan Atam, Eric C. Kerrigan
Abstract —In this paper, we develop an optimization-based sys-tematic approach for the challenging, less studied, and importantproblem of optimal partitioning of multi-thermal zone buildingsfor the decentralized control. The proposed method consistsof (i) construction of a graph-based network to quantitativelycharacterize the thermal interaction level between neighborzones, and (ii) the application of two different approaches foroptimal clustering of the resulting network graph: stochasticoptimization and robust optimization. The proposed method wastested on two case studies: a 5-zone building (a small-scaleexample) which allows one to consider all possible partitions toassess the success rate of the developed method; and a 20-zonebuilding (a large-scale example) for which the developed methodwas used to predict the optimal partitioning of the thermal zones.Compared to the existing literature, our approach provides asystematic and potentially optimal solution for the consideredproblem.
Index Terms —Optimal zone partitioning, multi-zone buildings,decentralized control, mixed-integer linear programming, robustoptimization, stochastic optimization.
I. I
NTRODUCTION
Among building thermal comfort control strategies, modelpredictive control (MPC) and its variants are the most popularones [1]–[4], which can provide energy savings of up to40% compared to an on-off or rule-based controller [5]–[8]. In energy-efficient thermal control of buildings, bothcentralized MPC (C-MPC) and decentralized MPC (D-MPC)can be used. Given a multi-zone building consisting of a largenumber of thermal zones (such as a university building oran airport), it is crucial to have a kind of optimal balancebetween centralization and decentralization [9], which in turnmeans that one needs to find an optimal partitioning ofthermal zones into a set of clusters and control of eachcluster using a separate C-MPC. In optimal partitioning theobjective is to partition the building thermal zones into a set ofclusters to achieve (i) minimum thermal coupling between theclusters, (ii) strong thermal connectivity between members ofeach cluster, (iii) numerically-efficient control of each cluster,(iv) acceptable performance loss compared to C-MPC whenthe D-MPC is applied to clusters.Multi-thermal zone partitioning approaches can be dividedinto two categories: approaches where control design is notan integrated part of the partitioning approach and co-design-based partitioning where control design is an integrated part ofthe partitioning. Although the problem of optimal partitioning
Ercan Atam and Eric C. Kerrigan are with Electrical and Electronic Engi-neering Department of Imperial College London, South Kensington Campus,SW7 2AZ, London, United Kingdom. Kerrigan is also with the Departmentof Aeronautics. E-mails: [email protected],uk, [email protected],uk. of multi-zone buildings for decentralized control is a veryimportant problem to be solved, there have appeared veryfew studies in the literature on this topic, probably due tothe challenges involved.We start the literature review by mentioning three existingworks which use approaches in the first category. In [10], theymodelled the interior wall between two neighbor thermal zoneseither using the 3R2C structure if the wall has a high thermalmass, or 1R structure if the wall has significant openings. Toquantify the level of thermal interaction between a given zoneand a neighbor zone, the middle resistance in 3R2C or thesingle resistance in 1R was set to a nominal value and then toa large value (which basically means full de-coupling). Next,by finding the temperature difference between the two casesand normalizing the difference, they obtained a measure of thethermal interaction. Repeating this for all neighboring zones,they quantified inter-zone thermal interactions of a multi-zonebuilding. Next, by specifying a threshold, they eliminated thecouplings where the interaction degree is below the threshold,which in turn resulted in a “manual" partitioning of thezones into a “non-predetermined" number of clusters wheremedium-to-high thermal interactions exist between membersin each cluster. In [11], the concepts of graph theory-basedstructural and output identifiability [12], [13] together withthe relevant metrics were used to decompose a multi-zonebuilding into identifiable clusters satisfying both structuraland output identifiability. Next, using decentralized uncertainmodels, a two-level hierarchical robust MPC scheme wasdeveloped for control of the whole building system. Althoughthe schemes for both identification and control were regardedas “decentralized", the thermal interactions between a zoneand the neighbor zones were explicitly taken into accountas dynamic interaction signals using the outputs of modelsof these zones. As a result, the term “decentralization" inthis work is not used in the same meaning of our defini-tion which is no use of such a dynamic interaction signal,and hence the work of [11] can be seen more as a kindof distributed identification and control. The last work inthe first category is the work done in [14] for the relatedproblem of contaminant detection and isolation in multi-zonebuildings. They presented two methods. (i) An optimization-based graph clustering method where mixed-integer linearprogramming was used to formulate the optimization problemand the objective was optimal decentralization of the systemto minimize interconnection airflows between clusters. (ii) Aheuristic method based on matrix clustering to decentralizethe system where the developed heuristic is computationallyfaster than the optimization-based graph clustering approach but suboptimal compared to it.As an example of co-design-based thermal zone partitioningstudies we can mention only one study. In [15], an analytically-derived optimality loss factor was used to measure the optimal-ity difference between an unconstrained decentralized controlarchitecture and unconstrained centralized MPC. Next, thedeveloped optimality loss factor was used as a distance metricfor optimal partitioning based on an agglomerative clusteringapproach. The main limitation of [15] is that no constraint wastaken into account in the MPCs.In this paper, we present a new method (which can be putinto the first category of zone partitioning categories men-tioned before) for zone partitioning and our contributions canbe summarized as follows. (i) First, we build on an idea similarto the thermal interaction-quantification idea presented in [10]to determine “interval"-type thermal interaction levels betweenzones, which is more realistic compared to “single"-typethermal interaction levels of [10]. (ii) We formulate the optimalclustering of interval-weighted thermal interaction networkgraph using a stochastic optimization and a robust optimizationframework. (iii) We define a new index for performance ofany control architecture, then using this index we defined an“optimality deterioration" and a “fault propagation" metric toassess the performance of C-MPC/D-MPC architectures. (iv)We demonstrated the effectiveness of the presented approachby testing the developed algorithm on two case studies.The rest of the paper is organized as follows. In Section IIthe details of graph-based optimization formulation (includingboth robust and stochastic frameworks) of the developedpartitioning approach are given. Next, in this section we definea performance index to measure the optimality of any controlmethod for thermal comfort of buildings, and then using thisindex we define the optimality deterioration, fault propagationmetrics and a combined metric to determine the best partitionamong the best partitions. Two case studies are given inSection III to demonstrate the developed approach and theresults are discussed in detail. Finally, the conclusions andsome future research work are given in Section IV.II. F
ORMULATION OF OPTIMAL PARTITIONING APPROACH
A. Building thermal model
The building thermal dynamics models were developedin the BRCM toolbox [16] which is a high-fidelity toolboxwhere the thermal models are developed following a thermalresistance-capacitance network approach. The models of thetoolbox were validated against EnergyPlus [17] (a popularbuilding thermal dynamics and energy simulation platform)and it was observed that the temperature difference betweenthe two software is less than 0.5 ◦ C [16]. The models havethe form given in (1) x ( k + 1) = Ax ( k ) + B u u ( k ) + B w w ( k ) , (1a) T ( k ) = Cx ( k ) . (1b)with u denoting the control input (heating or cooling power), w the external predictable disturbances (ambient air temperature,solar radiation and internal gains inside the building) and T the zone air temperatures. The sampling time was set to 15minutes.Let the multi-zone building be represented as the undirectedconnected graph G , ( V, E ) where V , { , , . . . , N z } denotes the set of indexed N z thermal zones and E denotesthe weighted edges, where the edge weights are intervalsrepresenting the level of thermal interaction between neighborzones. We assume that the building zones can be partitionedinto n clusters, where ≤ n ≤ N z . B. Thermal interaction quantification and the thermal inter-action graph
In this section, we present a formula for calculation of ther-mal interaction intervals between neighboring thermal zones.Let w denote the disturbance vector, which consists of internalgains, ambient temperature and solar radiation data over a rep-resentative year; let T cr , [ T l ( k ) , T u ( k )] be the comfort rangefor occupants (which can be time-dependent). Next, we definea modified comfort range ˜ T cr , [ T l ( k ) − , T u ( k )+1] obtainedfrom allowing 1 ◦ C temperature violation which can happendue to model or disturbance uncertainties during control designand its implementation in real control of buildings. Now wewill construct a simulation input, U ti , as follows: based on ˜ T cr , we construct random zone temperature set-points lying in ˜ T cr (that characterize possible controlled zone temperatures)and then we obtain the corresponding control input vector u which will track these set-points under the influence of w . We take U ti as ( u, w ) where u is random but consistentwith building control inputs encountered during controlledbuilding operation to produce zone temperatures that stay in ˜ T cr . It is very important that U ti includes disturbances andcontrol inputs of controlled building during different seasons,different periods in a day (day time and night time), anddifferent weather conditions. Hence, it is crucial to considera one-year period using a representative year when U ti isconstructed. Now, let T i ( u, w ) be the temperature of the i -th zone obtained from the response of thermal model M when it is simulated with input ( u, w ) and let T no ji ( u, w ) be the corresponding response when the dynamics of the j -thneighbor zone is decoupled from M . The interval includingthe thermal interaction degrees between the neighbor i -th and j -th zone is then given byI ′ ij = min k (cid:12)(cid:12)(cid:12) T i (cid:0) u ( k ) , w ( k ) (cid:1) − T no ji (cid:0) u ( k ) , w ( k ) (cid:1)(cid:12)(cid:12)(cid:12) , (2a) ¯ I ′ ij = max k (cid:12)(cid:12)(cid:12) T i (cid:0) u ( k ) , w ( k ) (cid:1) − T no ji (cid:0) u ( k ) , w ( k ) (cid:1)(cid:12)(cid:12)(cid:12) , (2b)I ij = I ′ ij + I ′ ji , (2c) ¯ I ij = ¯ I ′ ij + ¯ I ′ ji , (2d) I ij =[ I ij , ¯ I ij ] . (2e)Here note that (2c)-(2d) are used to obtain a single averagethermal interaction interval between two neighbor zones. Remark (Design of U ti ) . It is important that the systemis in closed-loop operation (but not necessarily operated byan optimal controller) since we are interested in inter-zone thermal interaction levels to be encountered when buildingis under control. These are the interaction degrees whichdetermine the degradation level of a decentralized controlarchitecture compared to a centralized one.C. Formulation of optimal partitioning problem
The optimization-based optimal partitioning formulation ofmulti-thermal zone buildings consists of formulation of twotypes of constraints and the objective function. We start withthe first set of constraints known as “Cluster formation con-straints" which are compactly given in (3). A subset of theseconstraints were also considered in partitioning formulation ofsome other problems [14], [18]–[21] different from the multi-zone partitioning problem which is considered in this studythe first time to the best of authors’ knowledge. We assumethat the set of thermal zones are partitioned into n clusterswhere we represent a generic cluster by c and the clusterset by C = { , , . . . , n } . Let p i,c denote a binary variableindicating whether in the c -th cluster the i -th thermal zone isincluded ( p i,c = 1 ) or not ( p i,c = 0 ); s i,j,c a binary variableindicating whether in the c -th cluster the edge ( i, j ) is included( s i,j,c = 1 ) or not ( s i,j,c = 0 ); and r i,j a binary variableindicating whether the edge ( i, j ) ∈ E is included in anycluster ( r i,j = 1 ) or not ( r i,j = 0 ). Note that when clusters areformed, a subset of edges may not lie in any of these clusters,but rather they may be “crossing edges" between clusters. n X c =1 p i,c = 1 , i ∈ V, (3a) N z X i =1 p i,c ≥ , c ∈ C , (3b) n X c =1 s i,j,c ≤ , ( i, j ) ∈ E, (3c) s i,j,c ≤ p i,c , ( i, j ) ∈ E, c ∈ C , (3d) s i,j,c ≤ p j,c , ( i, j ) ∈ E, c ∈ C , (3e) p i,c + p j,c ≤ s i,j,c + 1 , ( i, j ) ∈ E, c ∈ C , (3f) r i,j = X c ∈C s i,j,c , ( i, j ) ∈ E, (3g) p i,c ∈ { , } , c ∈ C , i ∈ V. (3h) s i,j,c ∈ { , } ≡ s i,j,c ∈ [0 , , c ∈ C , ( i, j ) ∈ E, (3i) r i,j ∈ { , } ≡ r i,j ∈ [0 , , ( i, j ) ∈ E, (3j)In cluster formation constraints, (3a) indicates that a thermalzone should be included in only one cluster; (3b) indicates thateach cluster should include at least one zone; (3c) indicatesthat an edge can be included in at most one cluster; (3d)-(3e) indicate that for an edge to be included in a cluster bothof its vertices should be included; (3f) indicates that if both i -th and j -th zones are included in cluster c , then the edge ( i, j ) should also be included (the connectivity constraint);(3g) is used to know if the edge ( i, j ) is included in anycluster or not; and finally (3h)-(3j) indicate that the variables p i,c , s i,j,c and r i,j are binary variables, but among which s i,j,c and r i,j , equivalently, can be considered as continuous Fig. 1. An example to illustrate equivalence of disconnected clusters to higher-order connected clusters: (a) the thermal network, (b) disconnected two-clusterpartition, (c) equivalent (in terms of decentralized control) connected three-cluster partition. variables in the interval [0 , with the following reason: case1: p i,c = 0 or p j,c = 0 implies s i,j,c = 0 by (3d)-(3e), case2: p i,c = 1 and p j,c = 1 implies s i,j,c = 1 by (3f), and theequivalence in (3j) follows from (3g). Note that if we want,we can eliminate r i,j since it is a dependent variable, but wewill keep it to make the presentation clear. Remark (Connectivity of clusters) . As illustrated in Figure1, in terms of D-MPC design, a disconnected cluster isequivalent to a higher-order connected cluster, since for adisconnected group in a cluster the associated model andhence the corresponding D-MPC has no coupling with therest. As a result, in multi-zone building partitioning problems,it is enough to consider only connected clusters.
Next, we preset the size and relative size constraints in (4)regarding the obtained clusters. Such constraints are importantfor the management of the computational load of cluster levelcontrol design and/or its sensitivity to any fault.M a ≤ X i ∈ V p i,c ≤ ¯ M a , c ∈ C , (4a) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i ∈ V p i,c a − X i ∈ V p i,c b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M r , c a , c b ∈ C , a = b. (4b)Finally, we express the expression for the objective functionand its minimization in (5). Here, we try to minimize thesum of edge weights (which are thermal interaction levelslying inside intervals) for edges not belonging to any cluster.The decision variables p, s, r are the vectors with entries from p i,c , s i,j,c , r i,j , respectively. J = min r X ( i,j ) ∈ E,µ i,j ∈ I i,j (1 − r i,j ) µ i,j (5) Letting p , [ { p i,c } ] ∈ { , } N z × n , s , [ { s i,j,c } ] ∈ R | E |× n + , r , [ { r i,j } ] ∈ R | E | + , and µ , [ { µ i,j } ] ∈ R | E | + , we can writethe optimization problem consisting of the objective functionminimization given by (5) and constraints given (3a)- (3j) &(4a)-(4b) as in (6). min r (1 − r ) T µ, µ ∈ I ⊆ R | E | + subject to A p A s A p A s A s A r A r | {z } A ∈ Z d × d prs ≤ b, (6) p ∈ { , } N z × n , s ∈ R | E |× n + , r ∈ R | E | + , where I is the thermal interaction interval vector, the di-mensions of A are d = 2( N z + n ) + | E | (3 n + 5) , d = N z n + | E | ( n + 1) , and in A the first block row corresponds to(3a) & (4), the second block row to (3b), the third block rowto (3c)-(3e), the fourth block row to (3f), and the fifth blockrow to (3g).In the next sections, we will present a stochastic optimiza-tion approach and a robust optimization approach to formulatethe uncertainty in the objective function. D. A stochastic optimization approach to optimal clustering
In the stochastic optimization framework, the goal is tominimize the expectation of the cost since the constraints aredeterministic and the cost function is the only part involvinguncertainty. As a result, in (5) we replace all the cost coeffi-cients by their expectation. To that end, let each interval I ij be divided into n d sub-intervals and let ˆ I ij,k represent themean of k -th sub-interval with its probability of occurrence P ij,k , k ∈ { , , · · · , n d } . Defining µ sij , (cid:20)n n d X k =1 P ij,k ˆ I ij,k o(cid:21) , µ s , [ { µ i,j } ] ∈ R | E | + , the stochastic formulation of theoptimization problem becomes min r (1 − r ) T µ s subject to A prs ≤ b, (7) p ∈ { , } N z × n , s ∈ R | E |× n + , r ∈ R | E | + . E. A robust optimization approach to optimal clustering
In this section, we will present the robust optimizationversion of the mixed 0,1 LP problem in (6), which has anuncertain objective function due to the uncertainty in µ . Asa first step, by letting (1 − r ) T µ ≤ z and minimizing z (thenew objective function and new variable), we transform theoriginal objective function into a constraint. Next, defining ¯ µ , ( µ max + µ min ) / and µ mg , µ max − ¯ µ , we can write theinterval uncertainty vector as I = ¯ µ + diag ( ε , · · · , ε | E | ) µ mg , where the entries of the random diagonal matrix satisfy (cid:12)(cid:12) ε i (cid:12)(cid:12) ≤ . Following the lines in [22], [23] for transformation ofa robust MILP into an equivalent deterministic MILP (calledthe “robust counterpart"), we obtain the robust counterpart asin (8): min z z subject to − − (¯ µ + µ mg ) T A p A s A p A s
00 0 A s A r A r | {z } ˜ A ∈ Z ( d × ( d zprs (8) ≤ (cid:18) ( µ mg − ¯ µ ) T b (cid:19)| {z } ˜ b ,z ∈ R + , p ∈ { , } N z × n , s ∈ R | E |× n + , r ∈ R | E | + . F. Definition of performance index and optimality deterriora-tion/fault propagation metrics
The performance of any control architecture will be mea-sured through the following performance index (PI):PI , e − k u utotutot − nr × e − k yv − ave yv − aveyv − ave − nr × e − k yv − max yv − maxyv − max − nr , (9)where the first term takes into account the effect of totalconsumed energy, the second term the effect of average com-fort temperature violation over all zones and the period overwhich the system is controlled, and the final term the effect ofmaximum comfort temperature violation; k u , k y v − ave , k y v − max are weight parameters; u tot − nr , y v − ave − nr , y v − max − nr areparameters used for normalization. Note that PI=1 when u tot = 0 and no comfort temperature violation, which isimpossible in practice. As a result, PI=1 is an ideal bound,but the closer PI is to 1, the better the performance of thatcontrol architecture is. The only constraint on the weights k u , k y v − ave , k y v − max is that PI( , no-fault) ( n = 1 correspondsto C-MPC) should have the largest value compared to all otherMPC architectures with/without faults, since fault-free C-MPCis the best solution.Next, we define two metrics that measure the optimality andsensitivity to fault propagation of a given control architecturecorresponding to a “best n-partition (n ⋆ )". The first metric iscalled the “optimality deterioration metric" (ODM) defined asODM ( n ⋆ , no-fault ) , PI(1, no-fault) − PI ( n ⋆ , no-fault ) PI(1, no-fault) , (10)where PI ( n ⋆ , no-fault ) denotes the performance index of thecontrol architecture corresponding to the best n -partition underno fault. The second metric called “fault propagation metric"(FPM) will be used to quantitatively determine the sensitivity of an MPC architecture to a fault in control/measurementequipment in a zone. This metric is defined asFPM ( n ⋆ , fault ) , PI(1, no-fault) − PI ( n ⋆ , fault ) PI(1, no-fault) . (11)Finally, we need to determine which best n-partition is thebest partition. To that end, we define a weighted-performancemetric (WPM) asWPM ( n ⋆ ) , α (1 − ODM ( n ⋆ , no-fault ))+(1 − α )(1 − FPM ( n ⋆ , fault )) , (12)where ≤ α ≤ is a user-parameter. Then, the best n-partition is the one which has the highest WPM value. Remark (Expected performance) . The paradigm of the pre-sented approach is based on the realistic intuition that thelower the level of thermal interaction between clusters, themore likely higher WPM values for the decentalizedly con-trolled clusters. However, there is no guarantee for this.
III. C
ASE STUDIES
In this section we will consider two case studies to demon-strate the developed optimal partitioning approach. Due tospace limitations, we will present the results only for thestochastic optimization version. The first case study is asmall-scale multi-zone building, which allows one to analyseall possible connected partitions for post-assessment (testingthe decentralized controller corresponding to the associatedpartitioning) and to determine the success rate of the par-titioning method. In both case studies, it was assumed thatcomfort range is [22, 24] ◦ C and that each zone temperaturecan be controlled with a separate heater/cooler. In D-MPCdesigns, we had the following assumptions: for a given cluster,interaction temperatures between zones in that cluster andneighbor zones in other clusters were set to 23 ◦ C (themiddle value in the comfort range); the prediction horizonwas taken as 6 hours; the only constraint was to keep zonetemperatures in the comfort band whenever possible; the costfunction was the weighted sum of total energy consumptionand comfort temperature violation where the weight used forcomfort temperature violation penalization was times theweight used for each control input ( w u i = 1 ). In applying thedeveloped partitioning approach to the case studies, we didnot use any size and relative size constraints on the formedclusters. A. Case study 1
This case study, shown in Figure 2, is a 5-zone officebuilding used during 8:00-18:00. There are significant open-ings between Z1-Z5, Z3-Z5, and Z4-Z5 which are modeledby pure resistors. The resistance between Z4-Z5 is taken as R = 0 . m K/W and the resistances between Z1-Z5 and Z3-Z5 are scaled multiples of R where scales are ratios of volumesof Z1 and Z5 to volume of Z4. The thermal interactionintervals for this case study are: I = [0 . , . ◦ C,I = [0 . , . ◦ C, I = [0 . , . ◦ C, I =[0 . , . ◦ C. The thermal interaction distributions to-gether with their mean values are given in Figure 3, which shows that there is no specific pattern. Optimal n-partitionsdetermined from the developed approach are presented inFigure 4.To determine whether the optimal n-partitions are reallyoptimal we designed D-MPCs for all possible connectedclusters including the optimal n-partitions. For this case studywe considered three scenarios: S1-all zone temperatures arecontrolled, S2-Z1-Z4 controlled, Z5 uncontrolled (to investi-gate the thermal effect of an uncontrolled zone on controlledzones), and S3-there is ±
10% error in zone air temperaturesensor for all zones but one zone at a time (to investigatethe effect of fault propagation). The 5-zone building wascontrolled to keep the zone temperatures in the comfort band.We calculated the average energy per day and per zone( u tot ), average temperature violation ( y v − ave ) and maximumtemperature violation ( y v − max ). The results for S1, S2 andS3 for all possible connected partitions are given in thespecified columns in Table I which also includes the ODM,FPM and WPM when all zones are controlled. The weightparameters were taken as k u = k y v − ave = k y v − max = 1 ; u tot − nr = 100 kWh , y v − ave − nr = 1 ◦ C, y v − max − nr = 3 ◦ C ; α = 0 . . B. Case study 2
The second case study is a 20-zone building with a largenumber of thermally interacting zones. The building and itsthermal interaction graph are shown in Figure 5. For thiscase study we considered two scenarios: S1-all zone tem-peratures are controlled, and S2-to investigate the effect offault propagation, we now assumed a different type of a likelyfault: we assumed that actuators at each zone fail, one at atime, so that no heat/cold is supplied ( u i =0). The results aregiven in Table II. The weight parameters were taken as k u = k y v − ave = k y v − max = 1 ; u tot − nr = 1000 kWh , y v − ave − nr =2 ◦ C, y v − max − nr = 5 ◦ C ; α = 0 . . C. Discussions of results
The computation time for each partitioning in both casestudies is less than one minute using a laptop with the follow-ing hardware specifications: 8GB RAM, Intel(R) Core(TM)i7-8550U CPU @ 1.80GHz 1.99 GHz. CPLEX was used asthe solver in the solution of the mixed-integer linear programs.From Table I of Case Study 1 we observe that, when allzone temperatures are controlled, all control architectures (C-MPC and D-MPC) have performances which are close toeach other on the basis of WPM, even though the buildingin case study 1 is a highly thermally-interacting structure todue large openings between zones. The explanation for suchan observation is the fact that when all zone temperatures arecontrolled in a narrow band, then the thermal interaction is notsignificant, even though the building has a huge potential forthermal interaction. When one of the zones was not controlledin Case Study 1, we observe from Table I that temperatureviolation increases in D-MPC cases since thermal interactionsincrease, but the performance of D-MPCs (based on WPM)are still acceptable. In Table I green rows indicate the bestpartition of each n-partition. When these results are compared
Fig. 2. A five-zone building, its data and thermal interaction network. TABLE IP
OST - EVALUATION RESULTS ( OVER ONE DAY ) FOR ALL CONNECTED PARTITIONS . T
HE COMFORT RANGE FOR CONTROLLED ZONES IS [22, 24] ◦ C DURING tot (kWh) y v-ave ( ◦ C) y v-max ( ◦ C) u tot (kWh) y v-ave ( ◦ C) y v-max ( ◦ C) u tot (kWh) y v-ave ( ◦ C) y v-max ( ◦ C) ODM(%) FPM(%) WPM(%)1 {1,2,3,4,5} 58.9649 0 0 49.0952 0 0 58.7341 0.1608 2.0582 0 57.026 71.4872 {1,2,4,5},{3} 58.1501 0.0084 0.0525 48.2976 0.0072 0.1215 58.7303 0.1594 1.9429 1.762 55.277 71.4812 {1,3,4,5},{2} 58.9643 0.0001 0.0004 49.0945 0.0001 0.0004 58.6829 0.1581 1.8642 0.016 54.007 72.9892 {1,3,5},{2,4} 58.8276 0.0036 0.0382 48.9677 0.0055 0.1101 58.6496 0.1587 1.9275 1.488 54.978 71.7672 {1},{2,3,4,5} 57.0280 0.0276 0.1934 47.1860 0.0293 0.5173 58.6615 0.1645 1.8398 7.012 53.917 69.5363 {1,3,5},{2},{4} 58.8270 0.0037 0.0386 48.9672 0.0055 0.1105 58.6340 0.1602 1.8411 1.504 53.726 72.3853 {1,4,5},{2},{3} 58.1495 0.0084 0.0525 48.2969 0.0072 0.1215 58.6336 0.1606 1.8893 1.764 54.483 71.8763 {1,5},{2,4},{3} 58.0123 0.0121 0.0630 48.1740 0.0126 0.1215 58.6649 0.1607 1.8526 2.325 53.942 71.8663 {1},{2,4,5},{3} 56.2089 0.0361 0.1987 46.4418 0.0359 0.5176 58.6844 0.1666 1.8074 7.203 53.526 69.6363 {1},{2,4},{3,5} 56.8894 0.0313 0.1945 47.0676 0.0346 0.5173 58.5631 0.1672 1.8889 7.258 54.741 69.0003 {1},{2},{3,4,5} 57.0274 0.0276 0.1934 47.1854 0.0293 0.5173 58.5562 0.1678 1.7497 7.014 52.619 70.1844 {1},{2,4},{3},{5} 56.0699 0.0398 0.1998 46.3273 0.0410 0.5176 58.5528 0.1708 1.8716 7.449 54.639 68.9564 {1,5},{2},{3},{4} 58.0118 0.0121 0.0630 48.1734 0.0126 0.1215 58.8271 0.1597 1.7811 2.327 52.860 72.4064 {1},{2},{3,5},{4} 56.8888 0.0313 0.1945 47.0670 0.0346 0.5173 58.6167 0.1687 1.8067 7.260 53.579 69.5814 {1},{2},{3},{4,5} 56.2083 0.0361 0.1987 46.4412 0.0359 0.5176 58.7196 0.1674 1.7254 7.205 52.292 70.2525 {1},{2},{3},{4},{5} 56.0693 0.0398 0.1998 46.3267 0.0410 0.5176 58.3571 0.1711 1.8167 7.450 53.723 69.413Fig. 3. Thermal interaction distributions over one year for the first case study. with the optimal partitioning results in Figure 4, which areresults predicted by our approach, we see that out of five best n-partitions, four of them are predicted correctly. However,since there is only one possible partition when n = 1 , , itwill be more correct to say that out of three best partitions,two of them are predicted correctly by our developed approachwith a success rate of 66.7%. Based on the WPM, the bestcontrol architecture for the first case study is n ⋆ = 2 with thezone partition { } , { , , , } .As regards to the second case study, we see from TableII that the best control architecture predicted by our OLBPapproach based on WPM is C-MPC. However, note that theperformance of D-MPC with n ⋆ = 11 is very close to that ofC-MPC, which again shows that a D-MPC control architecturecan work quite well in control of multi-zone buildings.IV. C ONCLUSIONS
In this paper, we presented an approach for optimal par-titioning of multi-thermal zone buildings for decentralizedcontrol. Both stochastic and robust optimization versions of thedeveloped approach were presented, and the stochastic versionof the algorithm was demonstrated on two case studies. The
Fig. 4. Optimal n-partitions for the first case study. Interval and single numbers on edges represent thermal interaction intervals and their mean, respectively.(a) (b)Fig. 5. (a) A twenty-zone building and (b) its thermal interaction network. The values in parenthesis are mean values. Note that zone 9 has 11 thermalinteractions. TABLE IIP
OST - EVALUATION RESULTS ( OVER ONE DAY ) FOR ALL BEST PARTITIONS . T
HE COMFORT RANGE FOR ALL ZONES IS [22, 24] ◦ C DURING ⋆ u total (kWh) y v-ave ( ◦ C) y v-max ( ◦ C) u total (kWh) y v-ave ( ◦ C) y v-max ( ◦ C) ODM (%) FPM (%) WPM (%)1 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ first case study was a small-scale example so that we were able to obtain all possible partitions and post-asses the performance of the corresponding controllers. From the post-assessment, weobserved that the success rate of our optimal partitioning algo-rithm is . , quite good when the difficulty and complexityof the optimal partitioning problem are considered. Next, wetested the method on a 20-zone building and determined theoptimal partitioning predicted for this building. Moreover, inthis paper we presented new metrics (optimality deteriorationand fault propagation metrics) which can be used in a generaldecentralized control framework.The most important findings of this study can be listed asfollows: (i) the presented partitioning algorithm can be used ef-fectively to determine the best partition and the correspondingdecentralized control architecture for multi-zone buildings; (ii)decentralized control can work very well in office buildings(even for cases where there is a high potential for thermalinteraction between zones) since zone temperatures are, ingeneral, strictly controlled in these buildings during workinghours, and as a result, thermal interactions are not significant(except in the morning during the short period when con-trollers start to move the uncontrolled zone temperatures to thecomfort band); (iii) since decentralized control can work quitewell for office buildings, for this category of buildings onecan use a decentralized control-oriented modeling approachper zone cluster, which will ease the current bottle-neck (theconsiderable effort in control model development [24]) for thewide-spread application of MPC in office buildings; (iv) formulti-zone buildings, it is not always correct that D-MPC willoutperform C-MPC in case of fault propagation: the faultsstay more local in D-MPC compared to C-MPC but since thecontrol models in D-MPC are less accurate (since they arenot able to take all thermal interactions into account) the realperformance of D-MPC depends on a combination of thesetwo effects in faulty scenarios.A future research direction is to consider a co-designapproach where MPC design is explicitly integrated into thepartitioning problem. This approach, although computationallyexpensive, has a huge potential to outperform the approachpresented here. R EFERENCES[1] F. Oldewurtel, A. Parisio, C. N. Jones, D. Gyalistras, M. Gwerder, V.Stauch, B. Lehmann, and M. Morari, “Use of model predictive controland weather forecasts for energy efficient building climate control",
Energy and Buildings , vol. 45, pp. 15-27, February 2012.[2] F. Oldewurtel, C. N. Jones, A. Parisio, and M. Morari, “Stochastic modelpredictive control for building climate control",
IEEE Transactions onControl Systems Technology , vol. 22, no. 3, pp. 1198-1205, 2014.[3] E. Atam, “New Paths Toward Energy-Efficient Buildings: A MultiaspectDiscussion of Advanced Model-Based Control",
IEEE Industrial Elec-tronics Magazine , vol. 10, no. 4, pp. 50-66, 2016.[4] E. Atam, “Investigation of computational speed of Laguerre network-based MPC in thermal control of energy-efficient buildings",
TurkishJournal of Electrical Engineering & Computer Sciences , vol. 25, pp.4369-4380, 2017. [5] S. Privara, J. Siroky, L. Ferkl, and J. Cigler, “Model predictive control ofa building heating system: the first experience",
Energy and Buildings ,vol. 43, no. 2, pp. 564-572, 2011.[6] D. Sturzenegger, D. Gyalistras, M. Gwerder, C. Sagerschnig, M. Morari,and R. Smith, “Model Predictive Control of a Swiss Office Building",
Clima - RHEVA World Congress, June 15-19, Prague, Czech Republic ,pp. 3227-3236, 2013.[7] B. Dong and K. H. Lam, “A real-time model predictive control forbuilding heating and cooling systems based on the occupancy behaviorpattern detection and local weather forecasting",
Building Simulation ,vol. 7, pp. 89-106, 2014.[8] S. C. Bengea, A. D. Kelman, F. Borrelli, R. Taylor, and S. Narayanan,“Implementation of model predictive control for an HVAC system in amid-size commercial building",
HVAC& R Research , vol. 20, no. 1, pp.121-135, 2014.[9] E. Atam, “Decentralized thermal modeling of multi-zone buildings forcontrol applications and investigation of submodeling error propagation",
Energy and Buildings , vol. 16, pp. 384-395, 2016.[10] J. Cai and J. E. Braun, “A practical and scalable inverse modelingapproach for multi-zone buildings", , 2014.[11] C. Agbi, “Scalable and robust designs of model-basedcontrol strategies for energy-efficient buildings", PhDThesis,
Carnegie Mellon University
Proceedings of the 15th International Federationon Automatic Control (IFAC) Symposium on System Identification , Saint-Malo, France, July 6-8 2009, pp. 664-669.[13] J. F. M. Van Doren, P. M. J. Van den Hof, J. D. Jansen, and O.H. Bosgra, “Parameter identification in large-scale models for oil andgas production," in
Proceedings of 18th In- ternational Federation onAutomatic Control (IFAC) World Congress , Milano, Italy, August 28-September 2 2011, pp. 10857-10862.[14] A. Kyriacou, S. Timotheeou, M. P. Michaelides, C. Panayiotou, andM. Polycarpou, “Partitioning of intelligent buildings for distributedcontamination detection and isolation,"
IEEE Transcations on EmergingTopics in Compuational Intelligence , vol. 1, no. 2, pp. 72-86, 2017.[15] V. Chandan and A. Alleyne, “Optimal partitioning for the decentralizedthermal control of builings,"
IEEE Transcations on Control SystemsTechnology , vol. 21, no. 5, pp. 1756-1170, 2013.[16] D. Sturzenegger, D. Gyalistras, V. Semeraro, M. Morari and R. S. Smith,“BRCM Matlab toolbox: model generation for model predictive buildingcontrol",
American Control Conference, June 4-6, Portland, USA , pp.1063-1069, 2014.[17] D. B. Crawley, L. K Lawrie, C. O Pedersen and F. C. Winkelmann.“EnergyPlus: energy simulation program", ASHRAE Journal, vol. 42,pp. 49-56, 2000.[18] M. Boulle, “Compact mathematical formulation for graph partitioning,"
Optiiozation and Engineering , vol.5, pp. 315-333, 2014.[19] T.Shirabe, “A model of contiguity for spatial unit allocation,"
Geograph-ical Analysis , vol. 37, pp. 2-16, 2005.[20] J. Conrad, C. P. Gomes, W. J. van Hoeve, A. Sabharwal and J. Suter,“Connections in networks: hardness of feasibility versus optimality," In
Integration of AI and OR Techniques in Constraint Programming forCombinatorial Optimization Problems, Springer , pp. 16-28, 2007.[21] B. Dilkina and C. P. Gomes, “Solving connected subgraph problems inwildlife conservation," In:
A. Lodi, M. Milano, P. Toth (eds.) CPAIOR,LNCS , vol. 6140, pp. 102-116, 2010.[22] D. Bertsimas and M. Sim, “The price of robustness,"
OperationsResearch , vol. 52, no. 1, pp. 35-53, 2004.[23] Z. Li, R. Ding, C. A. Floudas, “A comparative theoretical and com-putational study on robust counterpart optimization: I. Robust linearoptimization and robust mixed integer linear optimization,",
Ind EngChem Res , vol.50, pp. 10567-10603, 2011.[24] E. Atam and L. Helsen “Control-oriented thermal modeling of multi-zone buildings: Methods and issues",