Control-oriented Thermal Building Modelling

Demand side flexibility (DSF) is becoming an important tool for the power system operation. With air-conditioned buildings being key candidates for DSF, this paper presents a control-oriented thermal building modelling approach which is kept in the linear domain and therefore suited for the employment in convex optimization problems. Along with the building model, two DSF indicators are proposed and applied to assess the DSF potential for an office building in Singapore.


I. INTRODUCTION
The integration of demand side flexibility (DSF) into the power system is an important step towards dealing with renewable generation and increased peak demand due to electric mobility. For example, DSF can help to match generation and demand by shifting flexible loads to time periods with high renewable generation [1], thereby avoiding the shedding of renewable generation and decreasing the need for additional energy storage systems [2]. Additionally, DSF can ensure that electric grid constraints, i.e., thermal limits and voltage limits, are maintained throughout the operation [3] or they can support the grid stability by offering reserves [4].
Heating, ventilation and air-conditioning (HVAC) systems are an important candidate for DSF as they account for a large share of the electricity demand in buildings. DSF from HVAC systems has seen increased attention with the advances in model predictive control (MPC) applications for buildings [5]. With MPC, the control problem of the HVAC system is expressed as a numerical optimization problem aimed at minimizing the operation cost, i.e., energy cost, while satisfying the occupant comfort constraints, i.e., acceptable limits for thermal comfort and indoor air quality. The building operator benefits from MPC through cost savings which arise from the ability to consider dynamic electricity tariffs [6], i.e., the electric demand is shifted to hours with low prices.
Appropriate mathematical building models are the key ingredient for MPC and thus for DSF. Specifically, the equations of the building model are constraints in the numerical optimization problem of the MPC. To this end, convex building model formulations are beneficial, because of the high computational efficiency of convex optimization solvers. This paper introduces a control-oriented thermal building model catering specifically to the formulation of MPC problems by keeping all model equation in the linear, i.e., convex, domain. This model expresses the relationship between the electric load of the HVAC systems and the indoor air climate with consideration for interactions of the building with its environment, its occupants and appliances. Compared to a similar modelling approach in [7], the present model proposes more detailed surface and HVAC model formulations which are better suited particularly for tropical climate. Furthermore, two DSF indicators are proposed and the model is employed to evaluate the DSF of an office building in Singapore.

NOMENCLATURE
Let R be the domain of real numbers. Non-bold letters x, X denote scalars R 1×1 , bold lowercase letters x denote vectors R n×1 and bold uppercase letters X denote matrices R n×m . The transpose of a vector or matrix is denoted by () . Symbols for physical properties are aligned with ISO 80000.

II. BUILDING MODEL
The thermal comfort is expressed in terms of the indoor air temperature. Hence, the thermal building model expresses the relationship between the indoor air temperature, the electric load of the HVAC system, the local weather conditions and the building occupancy. The indoor air temperature, i.e. zone temperature, within each zone is assumed to be uniformly distributed. As a starting point, the differential equation of the zone temperature T z of zone z is expressed as: z is the thermal heat capacity of zone z ∈ Z b , which is obtained according to ISO 13790. The symbol Z b is the set of all zones z in building b ∈ B and B is the set of all buildings b. The heat transfer towards zone z is composed of the the convective heat transferQ cnv,int s,z from surfaces s ∈ S z towards zone z, heat transfer towards zone z due to infiltrationQ inf z , heat transfer towards zone z due to occupancy gainsQ occ z and heat transfer towards zone z from the HVAC systemsQ hvac z , where S z is the set of all surfaces adjacent to zone z.

A. Exterior surfaces
Exterior surfaces are modelled as two thermal resistances with a centered heat capacitance between the exterior and zone z. Each surface s is adjacent to exactly one zone z.
1) Heat balance: The heat balance for the exterior side of surface s is expressed as: On the interior side,Q cnd,int s is the conductive heat transfer from the core towards the interior side of the surface,Q cnv,int s is the convective heat transfer from the interior side of surface s towards zone z andQ irr,int s is the incident irradiation reaching surface s through exterior windows adjacent to the same zone z.
The heat balance for the core of surface s is expressed as: Where C thm s is the heat capacity of surface s. If the heat capacity of surface s is neglectable C thm s = 0, e.g., for windows, the term simplifies toQ cnd,ext s =Q cnd,int s .
2) Exterior convection: The exterior convective terṁ Q cnv,ext s is expressed as: Where A s is the surface area of surface s and h cnv,ext is the exterior convective heat transfer coefficient which is given according to ISO 6946 as h cnv,ext = 0.04 m 2 K/W −1 . The symbol T amb is the ambient temperature and T ext s is the temperature at the exterior side of surface s.
3) Exterior irradiation: The exterior irradiation terṁ Q irr,ext s is expressed as: Where α s is the absorption coefficient of surface s assuming a uniform absorption across the spectrum of the incident irradiation. The symbolq irr,ext d is the total incident irradiation onto a surface oriented towards direction d = {N, E, S, W, H}, i.e., vertically facing North N , East E, South S, West W or horizontally facing upwards H, depending on the respective surface's orientation d = d(s).

4) Exterior emissions:
The exterior sky emission terṁ Q ems,sky s describes the radiative heat loss through emission towards the sky. The term is expressed as: In this linear approximation, the symbol h sky s is introduced as the sky heat transfer coefficient of surface s, whereas T sky is the sky temperature. The sky heat transfer coefficient h sky s in turn is defined as: is the direction orientation of the surface. 6) Interior irradiation: The interior irradiation terṁ Q irr,int s is expressed as: Whereq irr,int z is the interior irradiation incident to all surfaces of zone z. The interior radiationq irr,int z is in fact the irritation which has entered zone z by passing through adjacent windows and is assumed to be uniformly distributed to all surfaces. This term is expressed as: Where τ w is the transmission coefficient of window w. The sets W z and S z contain all windows w and surfaces s that are adjacent to zone z.

B. Interior and adiabatic surfaces
Interior surfaces are modelled as two thermal resistances with a centered heat capacitance between the zone z 1 and zone z 2 . In principle, interior surfaces are modelled equivalently to exterior surfaces, where the heat balance for the interior side (eq. (3)) is applied for both sides of the surface. Adiabatic surfaces are modelled as a single thermal resistances between a heat capacitance and zone z. Adiabatic surfaces are modelled equivalently to exterior surfaces which are only the heat balance for the interior side (eq. (3)) is applied. For the sake of brevity, the full equations are omitted here.

C. Infiltration
The heat transfer towards zone z due to infiltrationQ inf z is defined as: Where V z is the volume of zone z, C th,air is the heat capacity of air and n inf z is the infiltration rate.

D. Occupancy gains
Assuming perfect knowledge of the building occupancy schedule, the heat transfer towards zone z due to occupancy gainsQ occ z , i.e., internal gains, is expressed as: Where A z is the area of zone z andq occ z is the specific thermal gain due to occupancy.

E. Heating, ventilation and air-conditioning (HVAC) systems
HVAC systems are distinguished into 1) generic HVAC system, 2) air handling unit (AHU), 3) terminal units (TUs), 4) heating and chiller plant. The generic HVAC system (section II-E1) provides thermal heating / cooling power to each zone z, i.e. it directly adds / removes thermal energy to / from the zone. This generic HVAC system is an auxiliary system type which helps to model 1) simplified HVAC systems in case that a detailed model is not required or 2) HVAC system types for which a detailed model has yet to be implemented. The air handling unit (AHU) (section II-E2) serves supply air, i.e., conditioned outdoor air at a fixed temperature and humidity level, to each zone z. A terminal unit (TU) (section II-E3) serves supply air, i.e., re-conditioned zone air at a fixed temperature, to each zone z. Note that the TU takes in zone air, whereas the AHU draws fresh outdoor air. The heating and cooling demand of the AHU and TUs is provided in form of supply water, i.e., hot and chilled water, by the heating and chiller plant. The presented HVAC system models can in principal be amended for further system types, e.g., hydronic radiators, once appropriate linear models are formulated.
The total heat transfer towards zone z from the HVAC systemsQ hvac y = [T z ] z∈Z , V ahu z z∈Z , P hvac,el (26) The vectors x, u, v, y are the state, control, disturbance and output vectors. The time-discrete form of the thermal building model is obtained by application of zero-order hold discretization, which is omitted here for the sake of brevity. The final discrete-time state space model is expressed as: Where the matrices A, C are the state and output matrix, and B u , D u , B v , D v are the input and feed-through matrices, on the control and disturbance vectors respectively.

III. DEMAND SIDE FLEXIBILITY (DSF) INDICATORS
For evaluating the DSF of a particular building with the presented building model, two DSF indicators are proposed below. As a starting point, the optimal operation which will serve as the baseline for the DSF indicators is expressed as: Where T min z,t , T max z,t andV min z,t are the minimum and maximum air temperature and minimum fresh air supply required at zone z, where () t denotes the time-dependency of the constraints, e.g., to consider a night-setback. The symbols c t and ∆t are the electricity price at time step t and ∆t is the time step length. For the following test case, the electricity price is set as c t = 1. The solution of eq. (28), i.e., P base t = P hvac,el t , is taken as the baseline electric load in the following.
A. Maximum load reduction DSF can be characterized by the ability to defer electric load for a particular time period. To this end, the maximum load reduction for a fixed time period with respect to the baseline electric load is proposed as an indicator for DSF. The maximum load reduction factor r max is determined with the following optimization problem: Where [t 1 , t 2 ] is the desired load reduction time period and β is a small weighting factor to ensure that the optimal solution considers demand minimization for time periods which are not in [t 1 , t 2 ]. The optimization problem eq. (29) is iterated for different time period lengths ∆t = t 2 − t 1 and start times t 1 to obtain an average value for r max ∆t .

B. Price sensitivity
Assuming indirect control through energy prices, the building operator will determine its load schedule by minimizing the energy costs as in eq. (28). Therefore, the sensitivity of the electric load for changes in the price is proposed as an indicator for DSF. The load sensitivity s ∆c,t for a change ∆c in the price c t at time step t is expressed as: Where P hvac,el t is the solution of eq. (28) for the updated c t . The procedure is iterated over t and ∆c to obtain an average value for s ∆c .

IV. RESULTS AND DISCUSSION
The presented building model is applied for one storey of approx. 750 m 2 at the CREATE Tower in Singapore, which is entirely occupied by office space. The test case building is equipped only with an AHU system, i.e., generic HVAC systems or TUs are not considered.
The maximum load reduction for the test case building is depicted in fig. 1. The largest load reduction is observed for a time period of 30 min, while the possible load reduction decreases for longer time periods. This result demonstrates the limited thermal storage capabilities of the building.
The price sensitivity results are shown in fig. 2. The load variation takes positive values, i.e., the load increases, for price reductions and negative values, i.e. the load decreases, for price raises, which is consistent with the expected behavior.

V. CONCLUSION
This paper introduced a control-oriented thermal building modelling approach for assessing the DSF of HVAC systems. Two DSF indicators where proposed and employed along with the building model to evaluate the DSF potential of a office in Singapore. The presented models, DSF indicators and test case are implemented in [8] and available open source 1 .