Building HVAC Systems Control Using Power Shaping Approach

V. Chinde, K. C. Kosaraju, Atul Kelkar, R. Pasumarthy, S. Sarkar, N.M. Singh

Abstract— Heating, Ventilating and Air-conditioning (HVAC) the ways to capture the complex interconnection between control systems play an important role in regulating indoor mutiple zones, is to approximate the heat transfer model air temperature to provide building occupants a comfortable using an electrical (RC) network analogy [10]. Various zone environment. Design of HVAC control system to provide an optimal balance between comfort and energy usage is a chal- modeling approaches have been recently compared in [11]. lenging problem. This paper presents a framework for control Once the complex dynamics is represented as an electrical of building HVAC systems using a methodology based on power network, one can use various tools from network theory (e.g., shaping paradigm that exploits passivity theory. The controller passivity-based methods) to devise interesting and novel design uses Brayton-Moser formulation for the system dynamics control approaches [12]. wherein the mixed potential function is the power function and the power shaping technique is used to synthesize the controller Passivity [13] is an input-output property of physical by assigning a desired power function to the closed loop systems that can be used for analysis and synthesis for dynamics so as to make the equilibrium point asymptotically complex systems. The underlying idea is to render a closed- stable. The methodology is demonstrated using two example loop system passive, by an appropriate feedback and as- HVAC subsystems - a two-zone building system and a heat signing a desired closed loop storage (Lyapunov) function. exchanger system. In the context of (port-) Hamiltonian systems [14], this I.INTRODUCTION control technique is referred to as “energy shaping” where Depleting natural energy resources and increasing costs the objective is to shape the energy (the Hamiltonian) of are forcing all countries to look for technologies that can the open-loop system. Another approach is the notion of improve energy efficiency and not just generation of energy. power shaping, having its roots in the Brayton-Moser (BM) It has been well documented that the costs of improving framework [15] for modeling of topologically complete efficiency are much lower than the cost of generating equal nonlinear electrical networks with sources [16]. Passivity is amount of energy. Nearly 40% [1] of the total energy derived using a power like function, also called the mixed consumption in US is due to commercial and residential potential function, as the storage function and one of the port buildings. Heating, ventilation and air-conditioning (HVAC) variables being the derivative of voltages or currents. In this systems are a major source of energy consumption in build- framework we describe the dynamics in terms of physical ings. Statistics reveal that around 40% [2] of the energy (or measurable) variables, such as voltages and currents in used in commercial buildings is by HVAC systems. This case of electrical networks. Moreover, since the derivatives of makes it necessary to tackle energy related issues, such currents and voltages are used as measured outputs, it helps as thermal storage, in building systems by proper dynamic to speed up the transient response of the system. Finally, analysis and control design. Energy costs can be reduced it overcomes the “dissipation obstacle” [17] encountered in by proper control of buildings thermal storage [3]–[5] and classical energy shaping methods. The methodology can be operating the buildings based on demand response [6]. These used to solve the regulation problem in both finite [16] and control techniques require accurate models which captures infinite dimensional systems [18]–[20]. the thermal dynamics of the building. The models obtained In this paper, we use the power shaping paradigm to design should be such that they are computationally efficient so as controllers for two different HVAC subsystems, namely to provide real time feedback inputs for control purposes, thermal zones and heat exchangers. These representative with conflicting objectives of energy efficiency and user examples were chosen as they demonstrate most of the comfort. The models presented in literature based on finite typical complexities found in building HVAC systems. Al- element methods for heat transfer dynamics in buildings have though the models used are simple, its a good starting point proven to be computationally inefficient [7]. Other prevailing and provide analysis as proof-of-concept and can be easily modeling technique is Model Predictive Control (MPC) [8], extended to include detailed building modeling which can [9]. In most cases, the zone temperatures are controlled serve different tasks. First, the dynamics of these two systems using local controllers to ensure comfort of the occupants is transformed into the BM framework, then the input-output which typically leads to high energy consumption due to pair is identified that satisfies the passivity property. The disparate energy demands from individual zones. One of control objective is then to assign a suitable power function to the closed loop system so as to make the equilibrium point V. Chinde, Atul Kelkar and S. Sarkar are with the Department of Mechan- asymptotically stable. ical Engineering at Iowa State University. K. C. Kosaraju, R.Pasumarthy The organization of the paper is as follows. In Section are with the Department of Electrical and Computer Engineering at IIT Madras. N.M.Singh is with the Electrical Department at Veermata Jijabai II, we discuss power-shaping paradigm given the system Technological Institute (VJTI), India. Email: [email protected] dynamics in the BM form. In Section III, we give BM formulation for a multi-zone building model, and solve the where yPB is given by temperature regulation problem using power shaping ap- y = −G˜(x)T x˙ (5) proach. The heat exchanger example is presented in Section PB III-C followed by conclusions presented in Section IV. which is referred as power balancing (shaping) output [16]. Proposition 2: The power balancing output yPB given in II.POWER SHAPINGAPPROACH (5) is integrable. This section briefly describes the underlying idea of power Proof: From Assumption 2, we have that G˜(x) is integrable, shaping. Poincaré’s Lemma ensures the existence of a function Γ(x): n n A. Brayton-Moser form R → R such that > In power shaping the dynamics of the system are written in Γ˙ = −G˜(x) x˙ (6) gradient form using Brayton-Moser formulation, where the using (5) we conclude the proof. storage function has units of power. The gradient structure To achieve the control objective, we need to find a new in the system is exploited to achieve power shaping outputs. storage function Pd of the closed loop system such that Consider the standard representation of a system in Brayton- ˜ ∗ Moser formulation Qx˙ = ∇xPd and x = arg minxPd (7) The closed loop potential function P is difference of power Q(x)x ˙ = ∇xP (x) + G(x)u (1) d function P˜ and power supplied by the controller. In [23], n m the system state vector x ∈ R and the input vector u ∈ R the power supplied by controller is found by solving PDE’s. n (m ≤ n). P : R → R is a scalar function of the state, which Here, we adopt the procedure without solving PDE using the has the units of power also referred to as mixed potential power balancing outputs of the system which is similar to function since in electrical networks it is the combination given in [24]–[26], where they have used for energy shaping of content and co-content functions and the power transfer for a class of mechanical systems. Also recently in [27] between the capacitor and inductor sub systems [21], Q(x): similar idea is used for systems in the port-Hamiltonian form, n n n n n m R → R × R and G(x): R → R × R . The time using the Hamiltonian as the systems stored energy. By ex- derivative of the mixed potential functional is ploiting the Assumption 2, in Proposition 2 we have proved d that the power balancing output is integrable. Using this the P (x) = ∇xP (x) · x˙ dt desired closed loop potential function Pd is constructed in = (Q(x)x ˙ − G(x)u) · x˙ the following way > > > =x ˙ Q(x)x ˙ − u G(x) x˙ 1 2 Pd = kP˜ + ||Γ(x) + a|| (8) 2 kI This suggests that if P (x) ≥ 0 and Q(x) ≤ 0, the system m m×m (1) is passive with storage function P (x) and port power where k > 0, a ∈ R , kI ∈ R with kI > 0. And variables are input u, output y = −G(x)>x˙. But, in general further a is chosen such that (7) is satisfied, which implies ∗ 2 ∗ P (x) and Q(x) can be indefinite [16]. ∇xPd(x ) = 0 ∇xPd(x ) ≥ 0 (9) Assumption: which upon solving gives 1) For the given system, there exists P˜(x) ≥ 0 and −1 ˜† ∗ ˜ ∗ ∗ Q˜(x) ≤ 0 and a := kkI G (x )∇xP (x ) − Γ(x ) (10) ˜† ˜ Q˜(x)x ˙ = ∇xP˜(x) + G˜(x)u (2) where G represents pseudoinverse of G. Proposition 3: Consider the system (1) satisfying the as- describe the dynamics (1) (procedure for finding such n m sumptions 1 and 2. We define the mapping u : R → R pair is given in [22]). Such P˜ and Q˜ are called 1 admissible pairs for (1). u := v + αG˜>x˙ − k (Γ(x) + a) . (11) k I 2) G˜(x) is Integrable. ˜ The control objective is to stabilize the system at the where α > 0, ∇Γ(x) := −G(x). Then system (1) in equilibrium point (x∗, u∗) satisfying closed loop is passive with storage function Pd (8) satisfying (7), input v and output yPB. Further with v = 0 the ˜ ∗ ˜ ∗ ∗ ∗ ∇xP (x ) + G(x )u = 0 (3) system (1) is stable with Lyapunov function Pd(x) and x Proposition 1: Consider the system (1) in BM form (2) as stable equilibrium point. Furthermore, if yPB = 0 =⇒ lim x(t) → x∗, then x∗ is asymptotically stable. satisfying assumption 1. Then the system is passive with t→∞ T input u, output given by yPB = −G˜(x) x˙ and storage Proof: The time derivative of closed loop potential function function P˜. (8) is ˜ Proof: Time differential of P is given by ˙ ˜˙ T Pd = kP + yPBkI (Γ(x) + a) ˜˙ ˜ T T P = (∇xP ) x˙ ≤ yPB[ku + kI (Γ(x) + a)] T ˜ T T T =x ˙ Qx˙ + u yPB ≤ yPBv − αyPByPB T T ≤ u yPB, (4) ≤ yPBv, where we used equations (4),(5),(11) in arriving at the result. where Ni denotes all resistors connected to the ith capacitor This proves that the closed loop is passive with storage (includes zone and surface capacitances), T∞ is the ambient function Pd (8), input v and output yPB. Further for v = 0 temperature. ui is the heating/cooling generation input to the we have ith zone and Qi is the external heat input due to ambient and is nonzero only for the zone nodes. In order to illustrate the P˙ ≤ −αyT y d PB PB proposed idea of power based modeling and regulation of and at equilibrium building systems, we consider the dynamics of simple case k of a two-zone building separated by a surface [29], where the u∗ = − I (Γ(x∗) + a) . (12) k surface is modeled as a 3R2C network is shown in Figure Finally from (10) and (12) we can show that (x∗, u∗) satisfy (1). The dynamics of the system is given by (3). This concludes the system (1) is asymptotically stable ∗ with Lyapunov function Pd and x as equilibrium point [28]. III.CONTROLOF HVAC SUBSYSTEMS In typical building HVAC systems, we have Air-side and Water-side HVAC subsystems. While air side focusses on delivering conditioned air to the zones, water side is responsible for various heat exchanging operations. In this section we apply the proposed approach on examples of both air-side and water-side subsystems. A. Building Temperature control Fig. 1: Two zones separated by surface and lumped RC The building thermal model of a multi-zone building based network model. on first principles such as energy and mass balance equations will lead to coupled partial differential equations. There are several difficulties associated with such kind of models in T3 − T1 C1T˙1 = + Q1 + u1 (14) terms of prediction and control design purpose. Since the R31 building is an interconnected system with individual zones ˙ T4 − T2 as its subsystems and interactions between these zones can C2T2 = + Q2 + u2 R42 occur due to conduction, convection and radiation. In this ˙ T1 − T3 T4 − T3 paper, we assume that the interaction between different zones C3T3 = + R31 R34 occurs only through conduction and contribution due to ˙ T2 − T4 T3 − T4 convection and radiation is negligible. The supplied air to the C4T4 = + R R zone is modulated at the Variable Air Volume (VAV) boxes 42 34 by changing the flow rate and temperature of air through Here T1, T2 are zone temperatures and T3, T4 are surface dampers and reheat coils. In this section, a different view- temperatures. point to modeling and regulation of temperature in a multi- The above system of equations (14) can be written in the > zone building using power is presented. The advantage of Brayton-Moser form (2) with x = T1,T2,T3,T4 , and using Brayton-Moser framework is that it naturally describes (T − T )2 (T − T )2 (T − T )2 the dynamics of systems in terms of measurable quantities. In P (x) = 3 1 + 4 2 + 3 4 the case of building systems the individual zone temperatures 2R31 2R42 2R34 (T − T )2 (T − T )2 are easily measurable and the controller designed can be used + ∞ 1 + ∞ 2 . (15) to improve the transient and the steady state response. 2R10 2R20 1) Building Zone Model: A building zone model is con- Q(x) = diag[−C1, −C2, −C3, −C4] and > structed [10] by combining lumped parameter models of ther- −1 0 0 0 G(x) = . mal interaction between zones separated by a solid surface 0 −1 0 0 (e.g walls). A lumped parameter model of combined heat flow across a surface is modeled as RC-network, with current It is easily verified P (x), Q(x) and G(x) defined in (15) and voltage being analogous of heat flow and temperature. In satisfy assumption 1 and 2. From Proposition 1, system > this modeling framework, the capacitances are used to model (14) is passive with input u = [u1, u2] , power balancing ˙ ˙ > the total thermal capacity of the wall, and the resistances are output y = [T1, T2] and storage function P (x), further from > used to represent the total resistance that the wall offers to Proposition 2, we have Γ(T ) = [T1,T2] . the flow of heat from one side to other. Control objective: The control objective is to stabilize a ∗ ∗ The thermal dynamics of a multi-zone building are given by: given equilibrium point [T1 ,T2 ] satisfying (3) where

∗ ∗ ∗ X (Tj − Ti) (T∞ − Ti) (T −T ) (T∞−T ) C T˙ = + u + u∗ = − 3 1 + 1 i i i (13) 1 R31 R10 Rij Ri0 j∈ (T ∗−T ∗) (T −T ∗) (16) Ni | {z } u∗ = − 4 2 + ∞ 2 Qi 2 R42 R20 B. Controller design

Proposition 4: Consider the closed loop storage function 20 T1 > deﬁned in (8) with kI = diag(k1, k2) and a = [a1, a2] . Pd T2 deﬁned in (8), takes the form T1ref T2ref k1 k2 15 T∞

2 2 C)

P = kP + (T + a ) + (T + a ) (17) ° d 2 1 1 2 2 2 k ∗ ∗ k ∗ ∗ (a) for a1 = − k u1 −T1 , a2 = − k u2 −T2 , Pd is positive 1 1 ∗ ∗ 10 deﬁnite and has a minimum at [T1 ,T2 ].

(b) further with the state feedback controller (11) Temperature ( α ˙ k1 ∗ k ∗ 5 u1 = − k T1 − k T1 − T1 − k u1 1 (18) α k2 ∗ k ∗ u2 = − T˙2 − T2 − T − u . k k 2 k2 2 0 If the tuning parameters α, k, k1, k2 are nonnegative, 0 0.5 1 1.5 2 2.5 3 3.5 4 ∗ ∗ time (hr) then [T1 ,T2 ] is asymptotically stable equilibrium of the closed loop system with Pd as Lyapunov function. (a) Proof: ∗ We need to choose a such that ∇Pd(x ) = 0 and 2 ∗ ∇ Pd(x ) ≥ 0 at the desired equilibrium. Therefore, proof 20 of (a) directly follows from (10) and (12). The proof of (b) follows from Proposition 3. It can also be proved by taking the time differential of the Lyapunov functional Pd deﬁned 15 C) in (17) as shown below °

P˙d = kP˙ + k1(T1 + a1)T˙1 + k2(T2 + a2)T˙2 (19) ˙ ˙ ˙ ˙ 10 = k(T1u1 + T2u2) + k1(T1 + a1)T1 + k2(T2 + a2)T2 T1

Temperature ( T2 = T˙1(ku1 + k1(T1 + a1)) + T˙2(ku2 + k2(T2 + a2)) T1ref 5 T2ref Using u1 and u2 from (18) the resulting equation (19) T becomes ∞ d 2 2 P ≤ −α(T˙ + T˙ ) ≤ 0. 0 dt d 1 2 0 0.5 1 1.5 2 2.5 3 3.5 4 time (hr) The controller obtained is a PI controller with respective (b) to power balancing outputs. The controller needs model information to compute u∗, but the system attains stability for Fig. 2: Zone temperature for constant ambient temperature error in u∗. The analysis provided uses zone heating/cooling a) Same reference b) Different reference. as input, but the proposed approach can be easily extended to more general model where the zone mass flow rate is the control variable [29]. shows the case where the individual zones are subjected Simulations were conducted on the simple two-zone to time varying sinusoidal ambient temperature (T∞ = model, in order to show the effectiveness of the proposed ap- 5 sin(2πt/T )+5◦C,T = 24hrs [10]) with same and different proach. Different operating conditions are considered, where set points, the controller performs reliably under different the zones temperatures have same and different set points ambient temperatures. and with different outside air temperatures. The parameters used for the simulation can be found in [29]. The objective C. Heat exchanger ∗ is to regulate the zone temperatures such that T1 = T1 , Heat exchangers are one of the most important HVAC ∗ T2 = T2 . Figure 2 shows the case where the individual subsystems which transfer heat from one medium (water/air) zones are subjected to constant ambient temperature with to another (water/air). The effectiveness of heat exchangers same and different set points. The controller effectiveness in strongly influences the thermal performance of building terms of transient and steady state performance is verified systems. To illustrate the proposed approach, we consider a in regulating the zone temperatures to their corresponding water-to-water heat exchanger where heating is accomplished set points. The important note is that there is no overshoot either by geothermal or solar energy. We consider a simple in the time response of states before settling to the target tube-shell water-to-water heat exchanger model given in [21], values, which shows the effectiveness of controller compared and the corresponding schematic is shown in Fig. 4. The inlet to energy based controllers. In order to study the actual and outlet temperatures of cold stream are given by Tci,Tco situation, we consider a time varying ambient. Figure 3 whereas the corresponding temperatures on hot-stream side T1 20 T2 T1ref T2ref 15 T∞ C) °

10 Temperature (

Fig. 4: Heat exchanger model. 0 0 0.5 1 1.5 2 2.5 3 3.5 4 time (hr) deﬁned in (20) is passive with storage function P (x) in (21), (a) > input u = [fc, fh] and output > y = [γc(Tco − Tci)T˙co, γh(Tho − Thi)T˙ho] . (22) 20 Further from Proposition 2, we have

γc 2 Γ1 = (Tco − Tci) 2 (23) 15 γh 2 C)

° Γ = (T − T ) 2 2 ho hi Control objective: T1 The control objective is to stabilize system 10 ∗ ∗ ∗ ∗ T2 (20) at operating point [Tco,Tho, u1, u2] satisfying (3) that T1ref is Temperature ( T2ref G (T ∗ −T ∗ ) G (T ∗ −T ∗ ) T∞ ∗ hc co ho ∗ hc co ho 5 u1 = ∗ u2 = − ∗ (24) γc(Tco−Tci) γh(Tho−Thi) Similar to Proposition 4, in this example, using Proposition 0 3 we can show that system (21) in the closed-loop with 0 0.5 1 1.5 2 2.5 3 3.5 4 feedback controller time (hr) α k k (b) ˙ 1 ∗ ∗ u1 = − γc(Tco − Tci)Tco − Γ1 − Γ1 − u1 k k k1 Fig. 3: Zone temperature for varying ambient temperature a) α ˙ k2 ∗ k ∗ Same reference b) Different reference. u2 = − γc(Tho − Thi)Tho − Γ2 − Γ2 − u2 k k k2 ∗ ∗ is asymptotically stable at equilibrium [Tco,Tho] with Lya- are denoted by Thi, Tho, respectively. The control variables punov function (8) defined with kI = diag(k1, k2) and −1 are the volumetric flow rates denoted by fc,fh; the thermal ∗ ∗ a = −kkI u −Γ(x) . The objective is to achieve a desired capacities for the cold and hot stream are denoted by C ,C ; ∗ ◦ c h outlet temperature of cold stream Tco = 80 C. This gives a and the heat transfer in the system is modeled by a thermal ∗ ∗ ∗ desired equilibrium (Tco,Tho), where Tho is determined by conductance G . The differential equations governing the ∗ hc χ and Tco, the admissible equilibrium set χ is given by heat exchanger system are given by χ = {(Tco,Tho) ∈ S|Ghc(Tco−Tho)+γh(Tho−Thi)fh = 0} CcT˙co = −Ghc(Tco − Tho) + γc(Tco − Tci)fc (20) 2 ChT˙ho = Ghc(Tco − Tho) + γh(Tho − Thi)fh and S = {(Tco,Tho) ∈ R |Tco > Tci}. The parameters values used for the simulation are found in [30]. From Fig. The system of equation (20) can be written in Brayton- 5, it can been seen that the desired outlet temperature of Moser form (2) with x = [T ,T ]> and co ho cold stream is attained and lies on the equilibrium manifold, Ghc 2 which shows the performance of the controller in regulating P (x) = (Tco − Tho) 2 the temperature. Q(x) = diag(−Cc, −Ch) and (21) −γ (T − T ) 0 G(x) = c co ci . IV. CONCLUSIONS 0 −γ (T − T ) h ho hi In this paper, a new paradigm is presented for synthesizing It can easily be verified that P (x), Q(x) and G(x) in (21) HVAC control of building systems using power shaping satisfy Assumptions 1 and 2. From Proposition 1, the system approach that exploits passivity property of the system. [7] B. K. McBee, Computational approaches to improving room heating 160 and cooling for energy efficiency in buildings. PhD thesis, Virginia Polytechnic Institute and State University, 2011. 140 [8] Y. Ma, G. Anderson, and F. Borrelli, “A distributed predictive control approach to building temperature regulation,” in American Control 120 Conference (ACC), 2011, pp. 2089–2094, IEEE, 2011. [9] F. Oldewurtel, A. Parisio, C. N. Jones, D. Gyalistras, M. Gwerder, 100 V. Stauch, B. Lehmann, and M. Morari, “Use of model predictive

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