Duration load definition

Let \(f = \left\{\begin{array}{ccc} [0,T] & \longrightarrow & \mathbb{R}_+ \\ t & \mapsto & f(t) \end{array}\right.\) , a function representing a load or a production as a function of time.

We define the charge monotone of the function \(f\) as follows:

\[\begin{split}C(f) : = \left\{\begin{array}{ccc} \mathbb{R}_+ & \longrightarrow & [0,T] \\ x & \mapsto & \lambda(\{t|f(t)\geq x\}) \end{array}\right.\end{split}\]

\(\lambda(\{t|f(t)\geq x\}) = \int_{\{t|f(t)\geq x\}}dt\) represents time during which the load exceeded the value \(x\).

We define the charge monotone of the function \(f\) as follows:

\[\begin{split}C(f) : = \left\{\begin{array}{ccc} \mathbb{R}_+ & \longrightarrow & [0,T] \\ x & \mapsto & \lambda(\{t|f(t)\geq x\}) \end{array}\right.\end{split}\]

\(\lambda(\{t|f(t)\geq x\}) = \int_{\{t|f(t)\geq x\}}dt\) represents the time the charge exceeded the value \(x\).

Normalization

Normalized variants are possible.

We can do a “power normalization” by setting:

\[\begin{split}\tilde{f} = \left\{\begin{array}{ccc} [0,T] & \longrightarrow & [0,1] \\ t & \mapsto & \frac{f(t)-f_{min}}{f_{max}-f_{min}} \end{array}\right. \text{ where } \left.\begin{array}{l} f_{min} = \underset{t\in[0,T]}{\text{min }} f(t) \\ f_{max} = \underset{t\in[0,T]}{\text{max }} f(t) \end{array}\right.\end{split}\]

We can do a time normalization” by setting:

\[\begin{split}\tilde{C}(f) = \left\{\begin{array}{ccc} \mathbb{R}_+ & \longrightarrow & [0,1] \\ x & \mapsto & \frac{C(f)(x)-C(f)_{min}}{C(f)_{max}-C(f)_{min}} = \frac{\lambda(\{t|f(t)\geq x\})}{T} \end{array}\right. \text{ where } \left.\begin{array}{l} C(f)_{min} = \underset{x\in\mathbb{R}_+}{\text{min }} C(f)(x) \\ C(f)_{max} = \underset{x\in\mathbb{R}_+}{\text{max }} C(f)(x) \end{array}\right.\end{split}\]

Time and power normalisation corresponds to \(\tilde{C}(\tilde{f})\) function.

Note

In (Poncelet et al., 2017), \(L_{c,b} = \tilde{C}(\tilde{f_{c}})(b)\) where \(c\) is an indice to identify a load type.