Aggregate demand response model

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 $$on$$ 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 $$\eta > 0 $$ and one for cases where $$\eta < 0$$.

When $$\eta > 0$$ 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 $$\eta < 0$$ 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 $$r_{off} = 1- \varphi$$.