A Simple Model of Flexible Electricity Demand – Why you should think twice before you talk about elasticities of flexible demand.
I consider a very simple model for flexible electricity demand and show that it does not make sense to talk about the elasticity of demand in this model.
- The consumer has to decide electricicty consumption for 24 hours.
- The consumer must at least consumer \(a\) kWh of electricity.
- The comsumer can in any hour at most charge \(b\)
These condition could for example describe the charging of an electric vehicle – a certain amount of energy has to be used and for every hour there is a limit on how much you can charge the battery.
The result is that if decisions are taken as described by the model – where the user simply cost minimize – then
- The elasticity of demand is not well-defined.
So how should we measure the flexibility of demand?
If demand can be concentrated in fewer hours, consumption can be shifted toward the cheapest hours of the day. Flexibility therefore reflects the ability to reallocate electricity consumption across time, rather than changing total energy use. Given this meaning of flexibility one quantitative measure of flexibility is can be given as
\[\phi = 1 – \frac{m}{24}\]
where \(m\) is the number of hours required to satisfy energy demand. The number of hours is measured as
\[
m = \left\lceil \frac{a}{b} \right\rceil
\]
where \(a\) is total energy consumption and the hourly capacity constraint be \(b\).
The measure has the properties that
- \(\phi = 0\): no flexibility (consumption spread across all hours)
- \(\phi \rightarrow 1\): high flexibility (consumption concentrated in few hours)
1. Model setup
Consider a consumer who allocates electricity consumption across 24 hours.
Let \(q_t\) denote consumption in hour \(t\) and \(p_t\) the corresponding price.
The consumer minimizes total cost:
\[
\min_{q_1,\dots,q_{24}} \sum_{t=1}^{24} p_t q_t
\]
subject to
\[
\sum_{t=1}^{24} q_t \ge a
\]
\[
0 \le q_t \le b \quad \forall t
\]
where:
- \(a\) = minimum daily energy requirement
- \(b\) = maximum hourly consumption (capacity constraint)
This formulation represents a simple model of flexible electricity demand where
the consumer chooses the timing of consumption given day-ahead prices.
2. Solution
Sort prices in ascending order:
\[
p_{(1)} \le p_{(2)} \le \dots \le p_{(24)}
\]
Define
\[
m = \left\lceil \frac{a}{b} \right\rceil
\]
The optimal allocation is
\[
q^*_{(t)} =
\begin{cases}
b & t < m \\ a-(m-1)b & t = m \\ 0 & t > m
\end{cases}
\]
Thus the consumer fills the cheapest hours first until the daily energy
requirement is satisfied.
The resulting demand function is therefore
\[
q_t = f(p_1,\dots,p_{24})
\]
which depends on the ranking of prices rather than their absolute level.
3. Total Energy Demand
Total daily consumption is
\[
Q = \sum_{t=1}^{24} q_t
\]
From the optimization problem we obtain
\[
Q = a
\]
provided prices are non-negative.
Thus total energy demand is fixed by the energy requirement.
4. Energy Price Elasticity
The standard price elasticity of demand is
\[
\epsilon = \frac{\partial Q}{\partial P}\frac{P}{Q}
\]
Since
\[
Q = a
\]
we obtain
\[
\frac{\partial Q}{\partial P} = 0
\]
and therefore
\[
\epsilon = 0
\]
Total energy demand is perfectly inelastic in this model.
5. Hourly Demand Response
Hourly demand is instead determined by
\[
q_t = f(p_1,\dots,p_{24})
\]
Demand shifts toward cheaper hours. When the price of an hour rises
relative to others, consumption may move to another hour.
However, the function is piecewise constant because allocation changes only
when the ranking of prices changes.
Hence:
- Local derivatives are typically zero.
- Discrete jumps occur when price rankings change.