The basic principle underpinning loadshape is that the total energy $E$ consumed by a series of $N$ pulses

$E = N P \Delta t \!$
Eq. 1

where

• $P$ is the power of each pulse
• $\Delta t$ is the duration of each pulse.

At least one of the variables E, N, P, $\Delta t$ must remain unspecified by the user in order for Eq. 1 to not be overspecified. Thus

$N = \frac{E}{P \Delta t} \!$
Eq. 1a

is used to determine the number of pulses,

$\Delta t = \frac{E}{P N} \!$
Eq. 1b

is used to determine the pulse duration, and

$P = \frac{E}{P \Delta t} \!$
Eq. 1c

is used to determine the power of each pulse.

In cases where the standard deviation $\sigma$ is provided, the power of the pulse is adjusted so that

$P_{adjusted} = Normal[P,\sigma]$
Eq. 2

When the analog loadshape power $P$ is given, the loadshape amplitude $y_t$ at the time $t$ shall be computed from schedule value $x_t$ as

$y_t = P x_t \!$.

When the analog loadshape energy $E$ is given, the loadshape amplitude $y_t$ at the time $t$ shall be computed from the schedule values $x_t$ in the interval $t=(0,T($ as

$y_t = \frac{E x_t}{\sum_{t=0}^T x_t \Delta t}$

where

• $\Delta t$ is the time interval over which the schedule value $x_t$ is given.

When neither the energy nor the power are given, the loadshape amplitude $y_t$ at the time $t$ shall be the schedule value $x_t$.

## Pulsed shapes

When a pulse loadshape is specified, the power amplitude $y_t$ at the time $t$ based on the schedule value $x_t$ is given by

$y_t = \frac{E}{N \Delta t} x_t$.

When the energy is not given, the power amplitude may be computed from the power and time interval

$y_t = \frac{P T}{N \Delta t} x_t$.

When both the energy and power are given, the power amplitude may be computed as

$y_t = \frac{P^2 T}{E}x_t$

## Modulated shapes

TODO:  describe how modulated shapes are generated from schedules

## Queued shapes

TODO:  describe how queued shapes are generated from schedules

## Scheduled shaped

TODO:  describe how pulse shapes are generated from schedules