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 \gt 0 [/math] and one for cases where [math]\eta \lt 0[/math].

When [math]\eta \gt 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 \lt 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].