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 SHAPING APPROACH (5) is integrable. This section briefly describes the underlying idea of power Proof: From Assumption 2, we have that G~(x) is integrable, shaping. Poincar´e’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_ = rxPd and x = arg minxPd (7) The closed loop potential function P is difference of power Q(x)_x = rxP (x) + G(x)u (1) d function P~ and power supplied by the controller. In [23], n m the system state vector x 2 R and the input vector u 2 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) = rxP (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~ + jjΓ(x) + ajj (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 2 R , kI 2 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]. rxPd(x ) = 0 rxPd(x ) ≥ 0 (9) Assumption: which upon solving gives 1) For the given system, there exists P~(x) ≥ 0 and −1 ~y ∗ ~ ∗ ∗ Q~(x) ≤ 0 and a := kkI G (x )rxP (x ) − Γ(x ) (10) ~y ~ Q~(x)_x = rxP~(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, rΓ(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.