Explicit modeling of individual devices does produce a very accurate simulation, but it can be very computationally intensive. For some time, there has been a desire to build an aggregate load model that incorporates the essential features of demand response, and in particular the three primary types of demand response DR control signals
- Direct load control
- These are DR control strategies that directly command devices to turn <math>on</math> or <math>off</math> deterministically or probabilistically. The parameter that describes this behavior is <math>\eta</math>. Values of <math>\eta</math> that are positive describe the rate at which devices turn <math>on</math> and values of <math>\eta</math> that are negative describe the rate at which devices turn <math>off</math> per unit of time.
- Thermostat reset control
- These are DR control strategies that adjust the thermostat control band, by increasing the hysteresis or by moving the temperature band. The parameters that describes this behavior is <math>L</math> and <math>\delta</math>, which respectively describe the size of the control band and the rate at which the control moves up in units of <math>L</math> per unit time.
- Duty cycle control
- These are DR control strategies that adjust the duty cycle of the device by adjusting the fractional runtime of the devices. The parameter that describes the nominal duty cycle of a device is <math>\varphi</math>, which is unitless. It is related to the rate at which devices move up <math>r_{on}</math> and down <math>r_{off}</math> the control band <math>L</math> per unit time such that <math>r_{on}=\varphi</math> and <math>r_{off}=1-\varphi</math>.
Demand Response Model
The DR model is based on two state queues of size <math>L</math>, one for those devices in the <math>off</math> regime and one for those in the <math>on</math> regime. The rate at which devices migrate down the <math>off</math> queue toward the lower control band limit <math>0</math> is given by the parameter <math>r_{off}</math>. The rate at which devices migrate up the <math>on</math> queue toward the upper control band limit <math>L</math> is given by the parameter <math>r_{on}</math>.
The duty cycle <math>\varphi</math> is the fraction of the time at a device is on with respect to the total time <math>T</math> it takes for the device to complete a cycle. If all the devices have the same load <math>q</math>, then this is also the fraction of devices that are <math>on</math> at any given time as well as the fraction <math>Q=N_{on}q</math> of the maximum load <math>\hat Q = Nq</math>. Thus, nominally
<math>\varphi = \frac{t_{on}}{T} = \frac{r_{off}}{r_{on}+r_{off}} = \frac{Q}{\hat Q} = \frac{N_{on}}{N}</math>.
We will see that this is true only if all the devices are identical, and there are no devices that are "short cycling", i.e., changing state from <math>on</math> to <math>off</math> or from <math>off</math> to <math>on</math> at any point other than the control band limits <math>0</math> and <math>L</math>.
If there is a non-zero probably <math>\eta</math> that a device turns <math>on</math> arbitrarily, regardless of the temperature <math>x \in (0,L)</math>, then we must consider the fact that <math>r_{off}</math> is effectively shorter than if all devices reached the control band limit <math>0</math> in due course without short-cycling. We call the value <math>\eta</math> the excess demand, in contrast the value <math>\varphi</math> which we call the base demand or natural demand.
Equilibrium Solution
The key to the behavior of a population of <math>N</math> devices is to recognize that any change in the values <math>L</math>, <math>\varphi</math>, or <math>\eta</math> will disturb the distribution of devices at the various temperatures <math>x</math>. The effective value of <math>r_{off}</math> in the case that devices are turned <math>on</math> permaturely (when <math>\eta \ge 0</math>) has been shown to be
<math>\rho(\eta) = (1-\eta)r_{off} + \eta</math>.
The natural distribution of devices is given by the density functions
<math>n_{off}(x) = \frac{N \eta r_{on}}{\rho (r_{on}+\rho) (e^{\eta L / \rho}-1)}e^{\eta x/\rho}</math>
<math>n_{on}(x) = \frac{N \eta}{(r_{on}+\rho) (e^{\eta L / \rho}-1)}e^{\eta x/\rho}</math>
The total number of devices that are <math>on</math> is
<math>N_{on} = N \frac{\rho}{r_{on}+\rho}</math>
Thus, we find that the effective duty-cycle <math>\Phi</math> of a population of such devices when the demand is non-zero
<math>\Phi(\eta) = \frac{\rho(\eta)}{r_{on}+\rho(\eta)}</math>,
which is what is generally called the total demand or just demand. We use the symbol <math>\Phi</math> to distinguish the diversity of the population from the duty cycle of single device.
When <math>\eta</math>, <math>\varphi</math>, and <math>L</math> change sufficiently slowly, the equilibrium solution given here is sufficient and accurate. Otherwise, a dynamic solution must be considered.
Dynamic Solution
When <math>\eta</math>, <math>\varphi</math>, or <math>L</math> change too quickly for the equilibrium solution to be valid, a dynamic model must be used. Unfortunately, a solution to the differential equations used to derive the equilibrium model has not yet been found. Instead a set of finite difference equations must be used, one for cases where <math>\eta > 0 </math> and one for cases where <math>\eta < 0</math>.
When <math>\eta > 0</math> we have
<math> \Delta n_{on}(L,t+\Delta t) = -r_{on} n_{on}(L,t) + \eta n_{off}(L,t) + (1-\eta) r_{off} n_{off}(L,t) </math>
<math> \Delta n_{on}(x,t+\Delta t) = -r_{on} n_{on}(x,t) + \eta n_{off}(x,t) + r_{on} n_{on}(x+\Delta x,t) </math>
<math> \Delta n_{off}(0,t+\Delta t) = -(1-\eta) r_{off} n_{off}(0,t) - \eta n_{off}(0,t) + r_{on} n_{on}(0,t) </math>
<math> \Delta n_{off}(x,t+\Delta t) = -(1-\eta) r_{off} n_{off}(x,t) - \eta n_{off}(x,t) + (1-\eta) r_{off} n_{off}(x-\Delta x,t) </math>
and when <math>\eta < 0</math> we have
<math> \Delta n_{on}(L,t+\Delta t) = -r_{on} (1+\eta) n_{on}(L,t) + \eta n_{on}(L,t) + r_{off} n_{off}(L,t) </math>
<math> \Delta n_{on}(x,t+\Delta t) = -r_{on} (1+\eta) n_{on}(x,t) + \eta n_{on}(x,t) + r_{on} (1+\eta) n_{on}(x+\Delta x,t) </math>
<math> \Delta n_{off}(0,t+\Delta t) = -r_{off} n_{off}(0,t) - \eta n_{on}(0,t) + r_{on} (1+\eta) n_{on}(0,t) </math>
<math> \Delta n_{off}(x,t+\Delta t) = -r_{off} n_{off}(x,t) - \eta n_{on}(x,t) + r_{off} n_{off}(x-\Delta x,t) </math>
Note that to guarantee the stability of the numerical solution, we must have <math>r_{on} + r_{off} \le 1</math>, so that we always have at least
<math>r_{on} = \varphi</math> and <math>r_{off} = 1- \varphi</math>.