callaway.20090116.direct load control

7/28/2019 Callaway.20090116.Direct Load Control

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(2)

( ) ( )1 11

1Nn i i n

i i

t Pm t

+ +=

= (3)

The variable t denotes time and n is the integer-valued time step; h will be defined as the

time elapsed per step (i.e. 1n nh t t+= ). All other variables and parameters (defined in the

following paragraphs) are indexed by i , which takes on a unique value for each TCL in the

population.

Parameters in Equation (1) are as follows: a governs the thermal characteristics of the

thermal mass and is defined as ( )exp /a h CR= [10], where C is the thermal capacitance

(units of kWh/C), and R is the thermal resistance (C/kW). a is the ambient

temperature, P governs the rate of energy transfer to or from the thermal mass (units of

kW). w is a noise process accounting for all heat gain and loss not modeled explicitly, e.g.,

in the case of a building, that which occurs as a result of opening and closing of doors, solar

gains, infiltration, and the operation of other loads. It will be assumed that w is normally

distributed with variance 2h (bulk units of C2). Because this study focuses on cooling

loads, the sign convention in (1) is chosen so that the parameterP is positive for cooling

loads.


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(4)

In practice, time will elapse between when the system operator issues a command and the

moment the PCT receives the signal, and the TCL itself may not immediately respond to


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M&C used the reverse.

Equations (5)-(6) are coupled by boundary conditions at the upper and lower limits of the

thermostat deadband that enforce conservation of probability:

( ) ( ) ( )0 0 1, , , 0a b bf t f t f t

=

(7)


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> PRELIMINARY: PLEASE DO NOT CITE WITHOUT THE AUTHORS PERMISSION ). In addition to conservation of probability, there are absorbing boundaries and

continuity conditions at the limits of the deadband, as well as natural boundary conditions

governing conditions at infinity, as follows:

( ) ( )0 1, , 0b bf t f t + = = (9)

( ) ( )0 0, ,a bf t f t = (10)

( ) ( )1 1, ,a bf t f t + += (11)

( ) ( )1 0, , 0c af t f t = = (12)

Finally, the total probability mass in the system must always be unity, and therefore

( )0 0 1 1 1a b b cf d f f d f d

+

+

+ + + = . (13)

Equations (5)-(13) make up the complete system of Fokker-Planck equations. M&C

approximate the stationary solution by assuming the heating and cooling rates are


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callaway.20090116.direct load control

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