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Distributed Temperature Control via Geothermal Heat Pump Systemsin Energy Efficient BuildingsXuan Zhang, Wenbo Shi, Qinran Hu, Bin Yan, Ali Malkawi and Na LiAbstract— Geothermal Heat Pump (GHP) systems are heating and cooling systems that use the ground as the temperatureexchange medium. GHP systems are becoming more andmore popular in recent years due to their high efficiency.Conventional control schemes of GHP systems are mainlydesigned for buildings with a single thermal zone. For largebuildings with multiple thermal zones, those control schemeseither lose efficiency or become costly to implement requiring alot of real-time measurement, communication and computation.In this paper, we focus on developing energy efficient controlschemes for GHP systems in buildings with multiple zones. Wepresent a thermal dynamic model of a building equipped witha GHP system for floor heating/cooling and formulate the GHPsystem control problem as a resource allocation problem withthe objective to maximize user comfort in different zones andto minimize the building energy consumption. We then proposereal-time distributed algorithms to solve the control problem.Our distributed multi-zone control algorithms are scalable anddo not need to measure or predict any exogenous disturbancessuch as the outdoor temperature and indoor heat gains. Thus,it is easy to implement them in practice. Simulation resultsdemonstrate the effectiveness of the proposed control schemes.I. I NTRODUCTIONAccording to an investigation by the United Nations, buildings are responsible for 40% of energy consumption, 70%of electricity consumption, and result in 30% of greenhousegas emission [1]. Roughly speaking, Heating Ventilation andAir Conditioning (HVAC) systems in buildings account for40% of the energy use [2]. It is therefore necessary to makethem more energy efficient for environmental sustainability.In recent years, Geothermal Heat Pump (GHP) systemsare becoming popular among different HVAC systems, dueto their highly efficient use of energy, i.e., they can usuallydeliver more than 3kWh of heat with 1kWh of electricity [3].GHP systems are heating/cooling systems that use the groundas the temperature exchange medium. In winter, they transferheat from the underground soil/water to buildings for heating,and vice versa in summer for cooling. Conventional controlof the GHP system includes Proportional-Integral-Derivative(PID) control [4] and centralized Model Predictive Control(MPC) [5], [6]. These methods are practically efficient forcases with only a single thermal zone, however, they becomeeither less efficient or costly (due to the centralized operationThis work was supported by NSF ECCS 1548204 and 1608509, NSFCAREER 1553407, Harvard Center for Green Buildings and Cities, andNARI Group Corporation. X. Zhang, W. Shi, B. Yan and A. Malkawi arewith Harvard Center for Green Buildings and Cities, 20 Sumner Road,Cambridge, MA 02138, USA (email: xuan zhang@g.harvard.edu, {wshi,byan, amalkawi}@gsd.harvard.edu). X. Zhang, Q. Hu and N. Li are withthe School of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, MA 02138, USA (email: qinranhu@g.harvard.edu,nali@seas.harvard.edu). Corresponding Author: X. Zhang.with heavy burdens of sensing, communication and computation of MPC) for cases with multiple thermal zones for largebuildings. Since modern buildings are usually large, complexand are with multiple zones, scalable and easy-implementingcontrol schemes are undoubtedly needed for them if equippedwith GHP systems for heating/cooling.This paper aims to develop real-time control schemes forGHP systems in typical multi-zone buildings. Specifically,we aim to design distributed algorithms to guide each controllable component to properly adapt their behavior suchthat system-wide objectives are achieved under given operating conditions. The emergence of distributed/decentralizedcontrol in network systems has been stimulated by smartsensing, communication, computing, and actuation technologies nowadays, e.g., in smart grids [7], [8], smart cities [9],[10], mobile robots [11], and intelligent transportation systems [12]. The advantages of distributed/decentralized control include: good scalability as the network grows; reductionof measurement, communication and computation comparedwith centralized control; privacy preserving. Thus, applyingdistributed/decentralized control to GHP system control andoptimization is becoming an area of active research. Representative work includes, for example, distributed MPC [3],[13]. However, distributed MPC still requires a large amountof sensing, communication and computation. In most cases,it needs good prediction of future disturbances, i.e., outdoortemperature, sunlight, indoor occupancy, etc., which maybe hard to obtain in reality. Different from the work usingMPC, the controllers designed here are based on solvingsteady-state optimization problems via gradient algorithms:they (i) are dynamic feedback controllers that can be implemented without measuring or predicting disturbances, (ii)are scalable with respect to building structures, (iii) satisfythe system operating constraints, and (iv) ensure systemefficiency, reliability and user comfort.The structure of this paper is as follows. In Section II, weprovide the detailed problem setup, including an introductionof the GHP system, a commonly used thermal dynamicmodel of the building network, and the optimization problem formulation. Since the original optimization problem isnonconvex, two different scenarios are considered in whichthe problem can be (approximated and) convexified: (i) thecontrol inputs are only the water flow rates (Section III),and (ii) the control inputs are both water flow rates andthe heat pump supply temperature (Section IV). For bothscenarios, we use a modified primal-dual gradient methodto design real-time distributed/decentralized control schemes.As a result, the thermal dynamics can be driven to equilibriawhich are the optimal solutions of those associated optimization problems. In Section V, two numerical examples areprovided to illustrate the effectiveness of the designed control

The supply temperature is adjusted by regulating the compressor of the heat pump. The return water temperature canbe approximated by the floor temperature which is accurateenough for control design [14]. These facts will be used laterin system modeling as well as control design.Fig. 1: Schematic of a typical GHP system.B. Thermal dynamic model with a GHP systemAccording to the above configuration, we model a givenbuilding as an undirected connected graph (N , E). Here N isthe set of nodes representing zones/rooms, and E N N isthe set of edges. An edge (i, j) E means that zones i and jare neighbors. Let N (i) denote the set of neighboring zonesof zone i. The thermal dynamics for each zone is describedby a reduced Resistance-Capacitance (RC) model [16] (morediscussion on this model is available in Remark 1):X Tj TiTf i TiT o Ti Qi (1)Ci Ṫi RiRijRaf ij N (i)schemes, using a building with four adjacent zones. Finally,conclusions and future work are presented in Section VI.Notation: The positive projection of a function h(y) on avariable x [0, ), (h(y)) x is: h(y)if x 0(h(y)) .x max(0, h(y)) if x 0II. P ROBLEM S ETUPA. GHP system in buildingsThe schematic of a typical GHP system is illustrated inFigure 1, which consists of two hydronic and one refrigerantcircuits, interconnected by two heat exchangers, i.e., anevaporator and a condenser [3], [6]. In the following we takethe heating mode case as an example to explain the workingprocess of the GHP system according to [5], [14], as theheating mode is more commonly used in practice.The underground hydronic circuit contains a mixture ofwater and anti-freeze driven by a small circulating pump, andthe temperature of its underground buried brine-filled side isrelatively constant with a seasonal pattern, i.e., warmer inwinter and cooler in summer than the outdoor temperature.The liquid refrigerant in the refrigerant circuit/the heat pump,first goes into the evaporator to absorb heat from the underground hydronic circuit and is converted to its gaseous state.Then this gaseous refrigerant passes through the compressor,and stops at the condenser in which it is converted to itsliquid state to heat up the water in the distribution hydroniccircuit. Finally this liquid refrigerant passes through theexpansion valve, and stops at the evaporator for the nextcirculation. The distribution hydronic circuit is a grid of indoor under-surface pipes filled with water. Driven by anothersmall circulating pump, this grid of pipes distributes heat toconcrete floors (i.e., floor heating) or hydronic radiators (i.e.,radiator heating) of a building for heating purpose. Here weonly consider floor heating while radiator heating is similarand will be reported in a future paper.In general, the heat pump consumes electrical power totransfer heat to the water in the distribution hydronic circuit.The amount of this heat depends on the flow rate of thewater, the heat pump supply/forward temperature, and thereturn water temperature [6], [15]. Each zone in the buildingis equipped with a Thermal Wax Actuator (TWA) that adjuststhe valve opening of the pipes for regulating the flow rate.where i N , Ci is the thermal capacitance, Ti is the indoortemperature, T o is the outdoor temperature, Ri is the thermalresistance of the wall and window separating zone i andoutside, Rij is the thermal resistance of the wall separatingzones i and j, Tf i is the floor temperature, Raf i is thethermal resistance between the indoor air and the floor, andQi 0 is the heat gain/disturbance from exogenous sources(e.g., user activity, solar radiation and device operation).The thermal dynamics of floors equipped with a GHPsystem is given by a simplified lumped element model [6]:Ti Tf iTwi Tf i , i NRaf iRf wiTf i Twi cw qi (Ts Tf i ), i N Rf wiCf i Ṫf i Cwi Ṫwi(2a)(2b)where Cf i is the thermal capacitance of the floor, Twi isthe temperature of the water in pipes, Rf wi is the thermalresistance between the floor and the water, Cwi is thethermal capacitance of the water, cw is the specific heat ofthe water, qi is the flow rate of the water, and Ts is thesupply temperature of the heat pump. Note that (i) Ts is acommon variable of the whole building [6], and (ii) the termcw qi (Ts Tf i ) stands for the heat transfer from the heatpump to the water in undersurface pipes [15].Proposition 1. When the GHP combined with floor heating/cooling system is off (qi 0), (1)-(2) asymptotically converges to an equilibrium point which is uniquely determinedby disturbances T o , Qi . When the GHP combined with floorheating/cooling system is on, the asymptotic convergenceproperty of (1)-(2) remains and the steady state is uniquelydetermined by disturbances T o , Qi and control inputs qi , Ts .The above proposition can be directly derived by rearranging (1)-(2) in state-space representation, and showing that thesystem matrix is Hurwitz (an alternative way is to construct aquadratic Lyapunov function). So the desiderata is to designqi , Ts only for periods when the GHP combined with floorheating/cooling system is on, more specifically, is to designthe dynamics of qi , Ts to drive (1)-(2) to some desired state.C. The optimization problemIn reality, each zone has a desired temperature which isthe set point determined by users. The control objective

considered in this paper is to regulate the temperature to beclose to the set point in each zone, and to minimize the totalenergy consumption of the GHP system. More specifically,we consider the steady-state optimization problem given by X 1cw qi Ts Zf i ri (Zi Tiset )2 s(3a)minZi ,qi ,Ts ,Zf i2 aTs bi NX Zj ZiT o ZiZf i Zis. t. Qi 0 (3b)RiRijRaf ij N (i)Zi Zf i cw qi (Ts Zf i ) 0Raf i0 qi qimax(3d)Tsmin Ts Tsmax(3e)(3c)where i N in (3b)-(3d), ri and s are positive weightcoefficients, Tiset is the temperature set point, a, b are positivecoefficients in the Coefficient of Performance (COP, i.e., anindicator of the relationship between the produced heat andconsumed electricity by the heat pump [15]), [0, qimax ] isthe range of qi , and [Tsmin , Tsmax ] is the range of Ts . Notethat (i) to avoid confusion between steady-state values andtemperature dynamics, we use Zi , Zf i to denote steady-statetemperature values whereas Ti , Tf i are temperatures in thedynamic model (1)-(2), (ii) T o and Qi are exogenous disturbances, and (iii) we have merged the steady-state equationsfrom (2) to obtain (3c). We assume that problem (3) isfeasible and satisfies Slater’s condition [17]. Moreover, wehave four important remarks. In the objective function (3a), the term relating to the totalenergy/electricity consumption (weighted by s) is given byPPcw qi Ts Zf i : the term i N cw qi Ts Zf i standsi N aTs bfor the total heat exchange between the heat pump and thewater in pipes; aTs b 0 is the COP which has beenapproximated as a linear function of Ts as in [15]; a, b canusually be obtained from the heat pump data sheet [6], [15]. Parameters ri , s are determined by users. If users prefermore comfort, they can increase ri and decrease s, andvice versa. Because of this flexibility, we do not imposeconstraints on the temperature comfortable range. In the heating mode, Ts Zf i (or Tf i ), i hold; in thecooling mode, Ts Zf i (or Tf i ), i hold. This is usuallytrue in practice. For example [6], [15], in the heating mode,Ts 27 C while Tf i 26 C. Once the mode is determined,the sign of Ts Zf i (or Ts Tf i ) is determined. The inequality constraints (3d)-(3e) are in accord withthose in the optimization problem (7) in [6].To conclude, the goal is to design the regulating rule forqi , Ts so that system (1)-(2) can be driven to an equilibriumpoint which is the optimal solution to problem (3).Remark 1. [18] Though system (1) is a 1st-order RC model,using higher order RC models does not affect the formulationof (3) since it is a steady-state optimization problem. Forexample, for the 2nd-order model in [16], [19]X Tij TiT o TiTf i TiCi Ṫi QiRiRijRaf ij N (i)Cij Ṫij Ti TijTj Tij RijRijwhere Tij is the temperature of the wall separating zones iand j, and Cij is the thermal capacitance of the wall, thecorresponding steady-state Equation (3b) is given byX Zj ZiT o ZiZf i Zi Qi 0Ri2RijRaf ij N (i)Z Zwhich results from the steady-state equation Zij i 2 j(Zij is the steady-state temperature of the wall), i.e., thesteady-state equation of the higher order model can bereduced to (3b) by eliminating states of solids in the buildingenvelope. Since our control design procedures proposedlater are based on solving the steady-state optimizationproblem (3), using higher order models will not affect them.III. S CENARIO I: F LOW R ATE C ONTROL O NLYA. Problem reformulationIn this section, we consider the water flow rate qi asthe only control input to each zone, and regard Ts as aknown exogenous signal which satisfies constraint (3e). Suchscenario could happen in practice, for instance, Ts is requiredto track some prescribed curve [3]. In this case, problem (3)can be simplified into X 1 ui set 2ri (Zi Ti ) s(4a)minZi ,ui ,Zf i2 aTs bi NX Zj ZiT o ZiZf i Zis. t. Qi 0 (4b)RiRijRaf ij N (i)Zi Zf i ui 0Raf iui0 qimaxcw (Ts Zf i )(4c)(4d)where i N in (4b)-(4d), ui cw qi (Ts Zf i ) is introducedto replace terms on qi , and constraint (3e) is dropped (underthese actions, problems (3) and (4) are still equivalent). Notethat the GHP system is in either the heating mode or thecooling mode. Once the mode is determined, the signs of uiand Ts Zf i are determined so that (i) ui equals either ui or ui , and (ii) the inequality constraint (4d) can become linearby multiplying cw (Ts Zf i ) on both sides. Thus, problem (4)naturally becomes convex.B. A distributed algorithmOnce the mode is determined, problem (4) can be solvedin either a centralized or distributed/decentralized way. Anycentralized algorithm requires to measure the outdoor temperature T o and the indoor heat gain Qi in every zone (Ts isgiven). Because these exogenous disturbances can fluctuatefrequently and are not easy to obtain, the cost of centralizedalgorithms would be expensive. Next we develop a realtime distributed algorithm that does not need measurementof these exogenous disturbances.Consider the heating mode case (the cooling mode caseis similar). The Lagrangian function of (4) is given by X1uiL ri (Zi Tiset )2 si N 2 aTs b X T o ZiX Zj ZiZf i Zi ζi QiRiRijRaf ii Nj N (i)

X Xi Ni Nλi X Z Zifi ui µ ( ui )i N iRaf imaxµ cw (Ts Zf i ))i (ui qi where ζi , λi , µ i , µi are the Lagrange multipliers/dual variables for constraints (4b)-(4d). Motivated by a modifiedprimal-dual gradient method [20], [21], we design the following algorithm to solve (4): 1 L 1 kZi ri (Tiset Zi ) ζi Żi kZi ZiRiRaf iX 1 X ζjλi (5a)RijRijRaf ij N (i)j N (i) L su̇i kui keui (ui ûi ) kui λi uiaTs b µ (5b)i µi keui (ûi ui )û i k̂eui (ui ûi )(5c) L λ ζiiŻf i kZf i keZf i (Zf i Ẑf i ) kZf i Zf iRaf i max µ cw keZf i (Ẑf i Zf i )(5d)i qi Ẑf i k̂eZf i (Zf i Ẑf i )(5e) To Z L X Zj Zii k ζi ζ̇i kζi ζiRiRijj N (i) Zf i Zi Qi(5f)Raf i L Z Z ifiλ̇i kλi kλi ui(5g) λiRaf i L kµ (ui qimax cw (Ts Zf i )) µ̇ i kµ µ iiµ i µ ii(5h) L µ̇ (5i) kµ ( ui ) i kµ µ iiµi µ iiwhere i N , kZi , kui , keui , k̂eui , kZf i , keZf i , k̂eZf i , kζi , kλi ,kµ , kµ are positive scalars representing the controlleriigains, and we have introduced the auxiliary states ûi , Ẑf i :since the objective function (4a) is not strictly convexin ui , Zf i , a standard primal-dual gradient method [22]could yield large oscillations; after introducing the extradynamics, the transient behavior of the overall system canbe improved (demonstrated in Section V). Note that Ti , Tf ihave their own dynamics given by (1)-(2) and thus can notbe designed, which is why we replace Ti , Tf i with Zi , Zf iinitially, i.e., Zi , Zf i , i N are ancillary state variables.According to [21], [23], it is true that (5) asymptoticallyconverges to an equilibrium point which is the optimalsolution of (4), since the optimization problem is convexand extra dynamics have been included in (5). Now usingui, i N(6)qi cw (Ts Zf i )as the control input to system (1)-(2), we can naturallyobtain a real-time distributed controller to regulate (1)-(2) toa steady state which is the optimal solution to problem (4)((4) is equivalent to (3) under known Ts ).Theorem 1. Given constant/step change/slow-varying T o ,Qi , Ts , the trajectory of system (1)-(2) and (5)-(6) asymptotically converges to an equilibrium point at which Ti , qi , Tf iof the equilibrium point is the optimal solution of (3).This theorem requires T o , Qi to be either constant, stepchange, or slow-varying, which holds in practice as theyvary at a time-scale of minutes. Remark that our controlleroperates in real-time, i.e., at a time-scale of seconds.In Equation (5f), the disturbances T o , Qi appear. Motivated by [21], to make the algorithm implementable withoutmeasuring these terms, we introduce ζ̃i kζζi Ci Ti asiX Ti Zi Tj Zj Ti Zi ζ̃i RiRijj N (i) Ti Zi Tf i Zf i, i N.Raf i(7)Moreover, we substitute ζi kζi (ζ̃i Ci Ti ) into (5a), (5d)to eliminate ζi . Now the proposed control scheme (5a)(5e), (5g)-(5i) and (6)-(7) is completely distributed and canbe implemented as follow. Given Ts , Ci , Ri , Rij , Raf i , ri , s,a, b, qimax , each zone in the building collects Tiset from users,locally measures its indoor temperature Ti and floor temperature Tf i , receives the feedback signals ζj kζj (ζ̃j Cj Tj )and Tj Zj from its neighboring zones, and then uses the information to update Zi , ui , ûi , Zf i , Ẑf i , ζ̃i , λi , µ i , µi , qi .maxHere Ci , Ri , Rij , Raf i , a, b, qiare building parameters,ri , s are parameters specified by users, and Ts is a knownsignal.Remark 2. In reality, due to that Zi , Zj (or Ti , Tj ) ofneighboring zones are often very close to each other andRij is not small, the total heat gain/loss from neighboringzones is (sometimes much) less dominant compared with theheat gain/loss from the outsideP plus Tthe Tindoor heat gain inevery zone. Thus, the term j N (i) jRij i could be ignoredPZ Zin (1) as well as the term j N (i) jRij i in (3)/(4). In thiscase, following the same design procedure, we end up witha completely decentralized control scheme (given by settingRij in (5a) and (7)) that only needs local measurement.IV. S CENARIO II: T HE G ENERAL C ASEA. Problem reformulationIn this section, both qi and Ts are considered as controlinputs to the system. Instead of handling the nonconvexproblem (3) directly, we focus on an approximate version ofthis problem, i.e., rather than minimizing the exact energyconsumption in the objective function, we minimize one ofits upper bound. This approximation can help convexify (3),which will become clear later. The approximate problem is X 1c2 q 2 (Ts Zf i )2minri (Zi Tiset )2 s w i(8a)Zi ,qi ,Ts ,Zf i2 aTs bi Ns. t. (3b)-(3e)(8b)Pc qi Ts Zf i where we have modified the term i N w aTin (3a)s b2 22Pcw qi (Ts Zf i )to be i Nin (8a). The latter term is actually aTs b

an upper bound of the square of the exact energy consumption: using a Cauchy-Schwarz inequality, we haveP2X c2 q 2 (Ts Zf i )2 (i N cw qi Ts Zf i )w i aTs b N ( aTs b)i NPb aTsmax ( i N cw qi Ts Zf i )2 N ( aTs b)2where N is the number of zones in the building. Therefore,rather than minimizing the total consumption directly, theobjective here aims to minimize its upper bound, which issufficient for energy saving purpose.Now we show how (8) can be turned convex. Again, weintroduce variables ui cw qi (Ts Zf i ), i N to get X 1u2iri (Zi Tiset )2 s(9a)minZi ,ui ,Ts ,Zf i2 aTs bi Ns. t. (4b)-(4d) and (3e).P(9b)u2i N iNote that (i) the function E aTis convex in ui , Tss bas its Hessian matrix equals .200. b aTs . 2aui.002 (b aT)s . .2 00.b aTs ···2aui(b aTs )2···P2a2 i N u2i(b aTs )3which is positive semi-definite and (ii) once the mode isdetermined, the signs of ui and Ts Zf i are determinedso that constraint (4d) can become linear by multiplyingcw (Ts Zf i ) on both sides. Next we design a distributedalgorithm to solve the convex optimization problem (9).B. A distributed algorithmSimilar to Section III-B, we consider the heating modecase. The design methodology is motivated by a modifiedprimal-dual gradient method [20], [21]. For simplicity, wedirectly write down the resulting distributed algorithm: 1X 1 1Żi kZi ri (Tiset Zi ) ζi RiRaf iRijj N (i) X ζjλi (10a)RijRaf ij N (i) 2su i u̇i kui λi µ µ(10b)iiaTs bP as 2Xi N ui max Ṫs kTs µqc ν νwii(b aTs )2i N(10c) λ ζ iimaxŻf i kZf i µ qc k(Ẑ Z)weZfififii iRaf i(10d) Ẑf i k̂eZf i (Zf i Ẑf i )(10e) To Z X Zj ZiZf i Ziiζ̇i kζi QiRiRijRaf ij N (i)(10f) Z Z ifi uiRaf i µ̇i kµ (ui qimax cw (Ts Zf i )) µ λ̇i kλiiµ̇ i kµ ( ui ) µ ii ν̇ kν (Ts Tsmax ) ν minν̇ kν (Ts Ts )ν (10g)(10h)i(10i)(10j)(10k)where i N , kZi , kui , kTs , kZf i , keZf i , k̂eZf i , kζi , kλi , kµ ,ikµ , kν , kν are positive scalars representing the controllerigains, and we have introduced the auxiliary state Ẑf i toimprove the performance of the algorithm (since ui and Tsare coupled in E whose Hessian matrix is not always zero,only adding Ẑf i is enough for behavior enhancement). Nowusing (6) and (10c) as the control input to (1)-(2) whereui , Ts , Zf i are determined by (10), we can naturally obtain areal-time distributed controller to regulate (1)-(2) to a steadystate which is the optimal solution to (8).Theorem 2. Given constant/step change/slow-varying T o ,Qi (remark that they vary at a time-scale of minutes),each trajectory of the overall system (1)-(2), (6) and (10)asymptotically converges to an equilibrium point at whichTi , qi , Ts , Tf i of this point is the optimal solution of (8).In Equation (10f), the disturbances T o , Qi appear. Similarto Section III-B, to make the algorithm implementable without measuring these terms, we introduce ζ̃i kζζi Ci Tiiwhose dynamics is given by (7). Moreover, we substituteζi kζi (ζ̃i Ci Ti ) into (10a), (10d) to eliminate ζi . Now theproposed control algorithm (10a)-(10e), (10g)-(10k) and (6)(7) is completely distributed and can be implemented asfollow. Given Ci , Ri , Rij , Raf i , ri , s, a, b, qimax , each zonein the building collects Tiset from users, locally measuresits indoor temperature Ti and floor temperature Tf i , receivesthe feedback signals ζj kζj (ζ̃j Cj Tj ), Tj Zj from itsneighboring zones and Ts from the compressor, and then uses the information to update Zi , ui , Zf i , Ẑf i , ζ̃i , λi , µ i , µi , qi .maxminOn the other hand, given Ts , Ts , the compressormaxreceives the feedback signals ui , µ cw from eachi qizone, updates Ts , ν , ν , and then broadcasts Ts . HereCi , Ri , Rij , Raf i , a, b, qimax , Tsmin , Tsmax are building parameters, and ri , s are specified by users.PT TRemark 3. Similar to Remark 2, the term j N (i) jRij iPZ Zcould be ignored in (1) as well as the term j N (i) jRij iin (8). Then following the same design procedure, we obtainanother distributed control scheme (given by setting Rij in (10a) and (7)) that does not need communication between neighboring zones, i.e., it requires less communication.V. N UMERICAL I NVESTIGATIONSIn this section, we present two numerical examples forscenarios described in Sections III and IV respectively, usinga house with four adjacent zones as illustrated in Figure 1.Only the heating case is presented in the following, while thecooling case is similar under the proposed control schemes.The parameters of the simulations are obtained from [15],[24], [25]: all Ci 20kJ/ C, all Cf i 35kJ/ C, allCwi 25kJ/ C, all Ri 15 C/kW, all Rij 23 C/kW, allRaf i 3 C/kW, all Rf wi 5 C/kW, cw 4.186kJ/kg/ C,

3938024681012141618202224T app3Tset 324.524Time(hour)Supply temperatureT 2appTset 22423.52322.58 10 12 14 16 18 20 22 24T30T202468 10 12 14 16 18 20 22 0Time(hour)Indoor heat gains0.35Temperature(c-degree)-10Zone 224.5T app1Tset 1Time(hour)Zone 32423.52322.522Time(hour)21.50246Zone 424Temperature(c-degree)-5Zone e)Outdoor temperature5T4T app4Tset 423.52322.52221.52120.5208 10 12 14 16 18 20 22 2402468 10 12 14 16 18 20 22 24Time(hour)Time(hour)Fig. 2: Profiles of the exogenous inputs (Q1 Q2 ).Zone 10.032Zone 20.0450.040.028Flow rate(kg/s)Flow rate(kg/s)0.030.0260.0240.022q10.02q app10.01602460.0350.03q2q app20.025q max10.0180.028 10 12 14 16 18 20 22 24q max202468 10 12 14 16 18 20 22 24Time(hour)Time(hour)Zone 30.05Flow rate(kg/s)Flow rate(kg/s)0.0350.040.0350.03q30.025q app30.020.015Zone 40.040.0452460.02q40.015q app40.01q max300.030.0258 10 12 14 16 18 20 22 240.005q max40246Time(hour)8 10 12 14 16 18 20 22 24Time(hour)Fig. 4: Flow rates in Scenario I.q1q2q3q4With extra dynamics0.05Flow me(hour)q noex1q 2noexWithout extra dynamics0.05q noex3q 4noex0.04Flow rate(kg/s)[qimax ] [0.03, 0.04, 0.045, 0.035]kg/s, [Tsmin , Tsmax ] [38, 42] C, a 0.11/ C, b 8.4, all ri 0.5p.u., all kZi 0.025p.u., all kZf i 0.033p.u., kTs 0.05p.u., all kui k̂eZf i kζi kλi kµ kµ kν kν 1p.u., alliikeui 10p.u., all k̂eui 0.1p.u., and all keZf i 2p.u. (p.u.means per unit). The outdoor temperature, indoor heat gains,and supply temperature in Scenario I are shown in Figure 2.The simulation result of the first scenario is illustrated inFigures 3-4, in which we set s 0 before 15h and s 10p.u.thereafter. The curves labelled with “app” indicate the caseof using the decentralized controller given in Remark 2,i.e., a communication free scheme. It can be seen that thedifference between these two cases is not large, i.e., less than1.3 C in temperatures, indicating that the performance ofthe decentralized controller could be acceptable in practice.Before 15h, since there is no consumption reduction purpose,i.e., s 0, the temperature trajectories under (5a)-(5e), (5g)(5i) and (6)-(7) track their set points unless the correspondingwater flow rate saturates (although not shown here, using aPID controller will result in the same temperature trajectoriesduring this period). After 15h, deviations from temperatureset points appear due to the consideration of energy saving(while only using a PID controller can not reduce energyconsumption unless forcing users to change their set points).On the other hand, we compare the flow rate response underthe distributed controller, with and without extra dynamics.The result shown in Figure 5 demonstrates that after introducing those extra dynamics, the system performance hasbeen improved that the oscillations are largely attenuated.In the second scenario, s 1p.u. holds before 13h and s 5p.u. thereafter. We can see that the difference between usingthe distributed controller (10a)-(10e), (10g)-(10k) and (6)(7) and using its simplified version given in Remark 3 isalso not large, as shown in Figures 6-8. The temperaturedeviations with respect to their set points before increasingthe weight coefficient s are smaller than those thereafter sincestarting from 13h, energy saving becomes more importantwhile user comfort becomes less. All these two scenariosinspire us that tuning the weight coefficient s (or equival

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