## Contents

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

These are DR control strategies that directly command devices to turn $on$ or $off$ deterministically or probabilistically. The parameter that describes this behavior is $\eta$. Values of $\eta$ that are positive describe the rate at which devices turn $on$ and values of $\eta$ that are negative describe the rate at which devices turn $off$ 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 $L$ and $\delta$, which respectively describe the size of the control band and the rate at which the control moves up in units of $L$ 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 $\varphi$, which is unitless. It is related to the rate at which devices move up $r_{on}$ and down $r_{off}$ the control band $L$ per unit time such that $r_{on}=\varphi$ and $r_{off}=1-\varphi$.

# Demand Response Model

The DR model is based on two state queues of size $L$, one for those devices in the $off$ regime and one for those in the $on$ regime. The rate at which devices migrate down the $off$ queue toward the lower control band limit $0$ is given by the parameter $r_{off}$. The rate at which devices migrate up the $on$ queue toward the upper control band limit $L$ is given by the parameter $r_{on}$.

The duty cycle $\varphi$ is the fraction of the time at a device is on with respect to the total time $T$ it takes for the device to complete a cycle. If all the devices have the same load $q$, then this is also the fraction of devices that are $on$ at any given time as well as the fraction $Q=N_{on}q$ of the maximum load $\hat Q = Nq$. Thus, nominally

$\varphi = \frac{t_{on}}{T} = \frac{r_{off}}{r_{on}+r_{off}} = \frac{Q}{\hat Q} = \frac{N_{on}}{N}$.

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 $on$ to $off$ or from $off$ to $on$ at any point other than the control band limits $0$ and $L$.

If there is a non-zero probably $\eta$ that a device turns $on$ arbitrarily, regardless of the temperature $x \in (0,L)$, then we must consider the fact that $r_{off}$ is effectively shorter than if all devices reached the control band limit $0$ in due course without short-cycling. We call the value $\eta$ the excess demand, in contrast the value $\varphi$ which we call the base demand or natural demand.

# Equilibrium Solution

The key to the behavior of a population of $N$ devices is to recognize that any change in the values $L$, $\varphi$, or $\eta$ will disturb the distribution of devices at the various temperatures $x$. The effective value of $r_{off}$ in the case that devices are turned $on$ permaturely (when $\eta \ge 0$) has been shown to be

$\rho(\eta) = (1-\eta)r_{off} + \eta$.

The natural distribution of devices is given by the density functions

$n_{off}(x) = \frac{N \eta r_{on}}{\rho (r_{on}+\rho) (e^{\eta L / \rho}-1)}e^{\eta x/\rho}$

$n_{on}(x) = \frac{N \eta}{(r_{on}+\rho) (e^{\eta L / \rho}-1)}e^{\eta x/\rho}$

The total number of devices that are $on$ is

$N_{on} = N \frac{\rho}{r_{on}+\rho}$

Thus, we find that the effective duty-cycle $\Phi$ of a population of such devices when the demand is non-zero

$\Phi(\eta) = \frac{\rho(\eta)}{r_{on}+\rho(\eta)}$,

which is what is generally called the total demand or just demand. We use the symbol $\Phi$ to distinguish the diversity of the population from the duty cycle of single device.

When $\eta$, $\varphi$, and $L$ change sufficiently slowly, the equilibrium solution given here is sufficient and accurate. Otherwise, a dynamic solution must be considered.

# Dynamic Solution

When $\eta$, $\varphi$, or $L$ 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 $\eta \gt 0$ and one for cases where $\eta \lt 0$.

When $\eta \gt 0$ we have

$\Delta n_{on}(L,t+\Delta t) = -r_{on} n_{on}(L,t) + \eta n_{off}(L,t) + (1-\eta) r_{off} n_{off}(L,t)$

$\Delta n_{on}(x,t+\Delta t) = -r_{on} n_{on}(x,t) + \eta n_{off}(x,t) + r_{on} n_{on}(x+\Delta x,t)$

$\Delta n_{off}(0,t+\Delta t) = -(1-\eta) r_{off} n_{off}(0,t) - \eta n_{off}(0,t) + r_{on} n_{on}(0,t)$

$\Delta n_{off}(x,t+\Delta t) = -(1-\eta) r_{off} n_{off}(x,t) - \eta n_{off}(x,t) + (1-\eta) r_{off} n_{off}(x-\Delta x,t)$

and when $\eta \lt 0$ we have

$\Delta n_{on}(L,t+\Delta t) = -r_{on} (1+\eta) n_{on}(L,t) + \eta n_{on}(L,t) + r_{off} n_{off}(L,t)$

$\Delta n_{on}(x,t+\Delta t) = -r_{on} (1+\eta) n_{on}(x,t) + \eta n_{on}(x,t) + r_{on} (1+\eta) n_{on}(x+\Delta x,t)$

$\Delta n_{off}(0,t+\Delta t) = -r_{off} n_{off}(0,t) - \eta n_{on}(0,t) + r_{on} (1+\eta) n_{on}(0,t)$

$\Delta n_{off}(x,t+\Delta t) = -r_{off} n_{off}(x,t) - \eta n_{on}(x,t) + r_{off} n_{off}(x-\Delta x,t)$

Note that to guarantee the stability of the numerical solution, we must have $r_{on} + r_{off} \le 1$, so that we always have at least

$r_{on} = \varphi$ and $r_{off} = 1- \varphi$.