I'm working on an energy load scheduler. Basically, I'm trying to find an algorithm that turns on deferrable load (think of a dishwasher or a washing machine) when there is a surplus of solar energy. The idea is to turn the deferrable appliances in the most optimal way from an economic point of view, i.e. paying the least amount on the bill.
I found a paper dealing with the same problem, and I'm using it as my roadmap.
In the paper the problem is formulated as a multistage stochastic program. So they try to minimize the average cost to deal with uncertainty.
To model the problem we define a temporal horizon denoted by $\tau$. We divide time in slots , and then we calculate the value of the decision vector $e_t$ for each time slot, t, that minimizes the total cost during the considered time horizon $c_t(e_t)$.
The feasible set for each slot, values of $e_t$ that respect the constrains, is denoted by $x_t$
So far there is no problem. But, what I don't get is why in the paper the problem is formulated in the following way:
$$\begin{equation*} \begin{aligned} & \underset{e_t\in x_t}{\text{minimize}} & & c(e_t) + \mathbb{E}[ \inf_{e_{t+1}\in x_{t+1}}c(e_{t+1})+...+\mathbb{E}[\inf_{e_{T}\in x_{T}}c(e_{T})]]\\ & \end{aligned} \end{equation*}$$
I would rather come up with something like that: $$\begin{equation*} \begin{aligned} & \underset{e_t\in x_t}{\text{minimize}} & & c(e_t) + \mathbb{E}[ c(e_{t+1})+...+c(e_{T})]\\ & \end{aligned} \end{equation*}$$
Can someone explain to me why the author has formulated the problem that way? And what's wrong with the way I formulated the problem?
Here you can find the paper.
Thank you in advance.
