From GridLAB-D Wiki
Jump to: navigation, search


Figure 1 - The Equivalent Thermal Parameters (ETP) model

The Commercial Module uses a simple Equivalent Thermal Parameters (ETP) model for small single-zone office buildings (Taylor and Pratt 1988), shown in Figure 1, with first-order ordinary differential equations (ODEs):

[math] \begin{align} T_i' & = \frac{1}{C_a} \left [ T_m H_m - T_i \left ( U_a - H_m \right ) + \sum_{end\ uses}Q_x + T_o U_a \right ] \\ T_m' & = \frac{1}{C_m} \left [ H_m \left ( T_i - T_m \right ) + Q_m \right ] \end{align} [/math]

where

  • [math]T_i[/math] = the temperature of the air inside the building
  • [math]T_i'[/math] = dTi/dt
  • [math]T_m[/math] = the temperature of the mass inside the building (for example, furniture, inside walls)
  • [math]T_m'[/math] = dTm/dt
  • [math]T_o[/math] = the ambient temperature outside air
  • [math]U_a[/math] = the UA of the building envelope
  • [math]H_m[/math] = the UA of the mass of the furniture, inside walls, etc.
  • [math]C_m[/math] = the heat capacity of the mass of the furniture inside the walls, etc.
  • [math]C_a[/math] = the heat capacity of the air inside the building
  • [math]Q_i[/math] = the heat rate from internal heat gains of the building (for example, plugs, lights, people)
  • [math]Q_h[/math] = the heat rate from heating, ventilating, and air conditioning unit
  • [math]Q_s[/math] = the heat rate from the sun to air (solar heating through windows, etc.)
  • [math]Q_m[/math] = the heat rate direct to the mass (e.g, solar radiation direct to mass)

The general first order ODEs (C1-C5 defined by inspection above) is

[math] \begin{align} T_i' & = c_1 T_i + c_2 T_m + c_3 \\ T_m' & = c_4 T_i + c_5 T_m + c_6 \end{align} [/math]

with the constants [math]c_1[/math] through [math]c_6[/math] defined as

  • [math]c_1 = - (U_a+U_m)/C_a[/math]
  • [math]c_2 = U_m/C_a[/math]
  • [math]c_3 = (Q_a+U_a T_o)/C_a[/math]
  • [math]c_4 = U_m/C_m[/math]
  • [math]c_5 = - U_m/C_m[/math]
  • [math]c_6 = Q_m/C_m[/math]

The general form of the second-order ODE is [math]p_1 T_i'' + p_2 T_i' + p_3 T_i = p_4[/math]. The solutions to the second-order ODEs for indoor and mass temperatures are

[math] \begin{align} T_i(t) & = k_1 e^{r_1 t} + k_2 e^{r_2 t} + \frac{p_4}{p_3} \\ T_m(t) & = \frac{T_i'(t) - c_1 T_i(t) - c3}{c2} \end{align} [/math]

where:

  • [math]p_1 = 1/c_2[/math]
  • [math]p_2 = -(c_1+c_5)/c_2[/math]
  • [math]p_3 = c_1 c_5 / c_2 - c_4[/math]
  • [math]p_4 = -c_3 c_5 / c_2 + c_6[/math]
  • [math]r_1,r_2[/math] are the roots of the [math]p_1 r^2 + p_2 r + p_3 = 0[/math]
  • [math]k_1 = [ r_1 T_i(0) - r_2 p_4/p_3 - T_i'(0) ] / (r_2 - r1)[/math]
  • [math]k_2 = [T_i'(0) - r_1 k_1] / r_2[/math]
  • [math]t[/math] = the elapsed time
  • [math]T_i(t)[/math] = the temperature of the air inside the building at time [math]t[/math]
  • [math]T_i'(t)[/math] = the rate of temperature change of the air inside the building at time [math]t[/math]

so that

[math] T_i'(0) = c_2 T_m(0) + c_1 T_i(0) - \left ( c_1 + c_2) T_o + c_7 \right ) [/math]

and

[math] T_m(t) = k_1 \frac{r_1 - c_1}{c_2} e^{r_1 t} + k_2 \frac{r_2-c_1}{c_2} e^{r_2 t} + \frac{p_4}{p_3} + \frac{c_6}{c_2} [/math]

with

  • [math]c_7 = Q_a / C_a[/math]

Defaults

All end-use power factor default to 1.0.

The outdoor air defaults to 59 F, relative humidity to 75%, and solar exposures as follows

  • South : 0
  • South-east : 0
  • South-west : 0
  • East : 0
  • West : 0
  • North-east : 0
  • North-west : 0
  • North : 0
  • Horizontal : 0

The default interior air and mass temperatures are set to the default outdoor air temperature.

The control defaults are as follows:

  • Heating setpoint : 70F
  • Cooling setpoint : 75F
  • Auxiliary cut-in: 20F
  • Economizer cut-in: 60F
  • Setpoint deadband : 1F
  • Ventilation fraction : 1 /h
  • Lighting fraction : 0.5 pu

The default occupancy schedule is M-F 8-17h. When occupied, the default occupancy is 0.002 occupants/sf.

Initialization

The default heating capacity is computed by solving the heat flow equation for the peak heating condition, which gives:

[math]Q_{max_{heat}}=UA ( T_{set_{heat}}-T_{design_{heat}} )[/math]

The default cooling capacity is computed by solving the heat flow equation for the peak cooling condition, which gives:

[math] Q_{max_{cool}} = UA ( T_{design_{cool}}-T_{set_{cool}}) + \sum_{windows}{A_{window} Q_{solar} c_{glazing}} + \sum_{loads}{Q_{load}} +Q_{vent} [/math]

where

  • [math]Q_{vent}=0.2402 \times 0.0735(T_{design_{cool}}-T_{set_{cool}}) \times V_{air} \times ACH [/math]

The heating COP is given by [math]COP_{heat} = Dist_{triangle}(1,2)[/math] and the cooling COP is given by [math]COP_{cool} = Dist_{triangle}(3,5)[/math].

Controls

The HVAC system has 6 control modes:

OFF 
In the OFF mode, the HVAC system is completely off. No ventilation and

no heating or cooling of any kind of performed. This mode is engaged whenever [math]T_{off_{heat}} \lt T_{air} \lt T_{off_{cool}}[/math] and [math]occupancy = 0[/math].

VENT 
In the VENT mode, the HVAC system is ventilating the zone at the

minimum_ach rate. This mode is engaged whenever [math]T_{off_{heat}} \lt T_{air} \lt T_{off_{cool}}[/math] and [math]occupancy \gt 0[/math].

HEAT 
In the HEAT mode, the primary heating (COP>1) system is on and the building is

ventilating at the minimum_ach rate only if occupancy is non-zero. This mode is engaged whenever [math]T_{cutin_{aux}} \lt T_{air} \le T_{on_{heat}}[/math].

AUX 
In the AUX mode, the secondary heating (COP=1) system is on and the building is

ventilating at the minimum_ach rate only if occupancy is non-zero. This mode is engaged whenever [math]T_{air} \le T_{cutin_{aux}}[/math].

COOL 
In the COOL mode, the active cooling (COP>1) system is on and the building

ventilating at the minimum_ach rate only if occupancy is non-zero. This mode is engaged whenever [math]T_{air} \ge T_{on_{cool}} and T_{out} \gt T_{cutin_{econ}}[/math].

ECON 
In the ECON mode, the passive cooling (COP=&infty;) system is on and the building

is ventilating using the rate required to cool using outdoor air only. This mode is engaged whenever [math]T_{air} \ge T_{on_{cool}} and T_{out} \le T_{cutin_{econ}}[/math].

Power calculations

Except as noted below, when ventilation is required, [math]Pvent = floor_area (0.1 - 0.01\imath) [/math] VA/sf, and [math]Qvent=0.2402 \times 0.0735 (T_{out}-T_{air}) V_{air} \times ventilation_rate[/math].

HVAC

OFF 
COP = 0, Qactive = Qpassive = 0, Pvent = 0
VENT 
COP = 0, Qactive = 0, Qpassive = Qvent
HEAT 
COP = 1.0 + (COP_{heating}-1) (T_{out} - T_{aux}) / Trange, zone.hvac.heating.capacity + zone.hvac.heating.capacity_perF*(zone.hvac.heating.balance_temperature-Tout), Qpassive = Qvent
AUX 
COP = 1.0, Qactive = COP * heating_capacity, Qpassive = Qvent
COOL 
COP = -1.0 - (zone.hvac.cooling.cop+1)*(Tout-TmaxCool)/(TmaxCool-Tecon), zone.hvac.cooling.capacity - zone.hvac.cooling.capacity_perF*(Tout-zone.hvac.cooling.balance_temperature), Qpassive = Qvent
ECON 
COP = 0, Qactive = 0, Qpassive = Qvent

Lighting

lights.load = lights.fraction*(lights.capacity + (lights.capacity/lights.power_factor)*(sin(arccos(lights.power_factor)))J) (kW)

lights.heatgain = lights.load*lights.heatgain_fraction (kW)

Plugs

plugs.load = plugs.fraction*(plugs.capacity + (plugs.capacity/plugs.power_factor)*(sin(arccos(plugs.power_factor)))J) (kW)

plugs.heatgain = plugs.load * plugs.heatgain_fraction (kW)

Other loads