text:
\section{Introduction and Motivations}
Recent years have witnessed a considerable interest in the scientific community towards \emph{optimal management} of distributed energy resources. It is anticipated that implementation of the low-carbon agenda and the adoption of
intermittent renewable energy sources will exacerbate issues of demand-supply balance and significantly incentivize the adoption of flexible demand paradigms across the grid.

A recent survey, \cite{weitzel}, has systematically classified a
 significant amount of literature dealing with optimal energy management for Electric Vehicles alone (EVs).
While a fraction of such papers is concerned only with scheduling aspects related to a single EV and the benefits that this could enable for the household, many authors are also concerned with the issue of how solutions implemented at the level of the single household may impact the overall system or how to suitably manage (and suitably distribute) the resulting
 large scale optimisation problem.

An emerging viewpoint, in such respect, is that a hierarchical approach, \cite{bernstein}, may allow to cluster multiple households, EVs or more generally prosumers, into groups which are regarded as a single entity by higher level grid abstraction layers, so that only their aggregate power absorption or generation profiles need to be considered, without mention of lower-level detailed dynamics and schedules.
The wisdom is that this could provide an effective modular approach for tackling the inherent complexity of the optimisation and scheduling task.

To make this proposition effective, however, it is important to characterize the degrees of freedom afforded by an aggregated
fleet of flexible devices in a way that is accurate and still
simple. For instance, \cite{hill}, introduces a heuristic aggregated model of flexible demand, to be used for the solution of Unit Commitment problems.
Similarly, \cite{calvillo}, models the aggregate behaviour of \emph{inelastic} flexible demand through heuristic constraints that limit the shifting ability afforded throughout the day.
Promising results, in this respect, are provided in \cite{ETA-pscc, ETA-tops, ETA-tosg, EAST} where optimal causal scheduling policies (affording maximal flexibility) for heterogeneous fleets and arbitrary time-varying demand signals are provided and their intrinsic capabilities characterized graphically through so called E-P transforms. Equivalent time-domain characterizations were independently explored in \cite{zachary} and further expanded and interpreted in \cite{phtr}.
A related line of investigation, \cite{appino}, proposes the use of controlled invariant sets in order to characterize viable power profiles for aggregated fleets, so as to reduce their model complexity for the purpose of higher level optimisation.

More recently, the challenging task of designing optimal dispatch policies for devices with \emph{partial availability} (as typical of Electric Vehicles which are not always connected to the grid) was considered in \cite{submitted,cdcbatteries}.
In particular, \cite{submitted} proposes a scheduling algorithm for fleets with partial availability and at the same time characterizes the set of aggregated demand profiles that  heterogeneous fleets with partial flexibility induce, when operated without \emph{cross-charging}.
For a fleet with $N$ storage devices and a horizon of $T$ time
intervals,
this replaces a description involving $N \cdot T$ variables and $2 \cdot N \cdot T$ constraints with one that only entails $T$ variables (the aggregated demand signal) and
$2^T$ constraints, thus drastically improving the scalability with respect to $N$, the size of the fleet.
In this paper we propose a way to achieve optimality, in a unit commitment problem, by adopting a simplified formulation with a number of constraints for the fleet that scales linearly in the number $T$ of time-slots considered and is also independent of the size of the fleet $N$. This is remarkable, as it shows that, under suitable conditions, a heterogeneous fleet of any size can effectively be treated as a single storage unit. In addition, we provide the relevant theory in support of our simplification algorithms and compare the different approaches on a large number of randomly generated examples and case studies.

The remainder of this paper is organized as follows: Section \ref{sec:aggregatemodel} presents the aggregate model of the storage fleet. Section \ref{sec:fullavailability} introduces the optimization problem for devices with full availability window. The aggregated model and the associated optimization algorithms for partial available storage devices are elaborated in \ref{sec:partialavailability}. The advantages of proposed aggregated model in computation is addressed in \ref{sec:computation}. Section \ref{sec:twoarea} applies the algorithms to a two-area system, and Section \ref{sec:conclusion} concludes this paper.

\section{An aggregate fleet model for optimisation}\label{sec:aggregatemodel}
We consider a finite discrete time horizon $\mathcal{T} = \{1,2,\ldots,T\}$ and a storage fleet with $N$ batteries, which we denote
$\mathcal{N} = \{1,2, \ldots, N\}$. Each battery $j \in \mathcal{N}$ is initially empty (without loss of generality) and needs to reach a certain target energy level
$\bar{E}_j$ within its availability set $\mathcal{A}_j$.
Moreover, each battery has a maximum rated power $\bar{P}_j$ which is the maximum power it can absorb while charging.
We consider the problem of minimizing generation costs for the task of charging the fleet while additionally meeting some assigned inflexible demand profile
$D^{I}(t)$. We assume that a set of generation units $\mathcal{I}$ is available to this end, each one with minimum and maximum power ratings
$\underline{G}_i$ and $\bar{G}_i$, respectively, $i \in \mathcal{I}$.
In particular the following optimisation problem is of interest:
\begin{subequations}\label{original}
\begin{align}
\min_{\begin{aligned}
	u_j(t)\,\,& j \in \mathcal{N}, t \in \mathcal{T} \\ g_i(t)\,\,& i \in \mathcal{I}, t \in \mathcal{T}
\end{aligned} } &  \sum_{t \in \mathcal{T}} \sum_{i \in \mathcal{I} }   C_i (g_i(t)) \tag{\ref{original}}\\
s.t.\quad \sum_{i \in \mathcal{I}} g_i(t) &= D^{I}(t) + \sum_{j \in \mathcal{N}} u_j(t) \quad \forall \, t \in \mathcal{T} \\
0 \leq &u_j(t) \leq \bar{P}_j \quad \forall \, t \in \mathcal{T}, \; \forall \, j \in \mathcal{N}\\
\underline{G}_i \leq & g_i(t) \leq \bar{G}_i \quad \forall \, t \in \mathcal{T}, \; \forall \, i \in \mathcal{I} \\
\bar{E}_j &\leq \sum_{t \in \mathcal{T}} u_j (t) \quad \forall \, j \in \mathcal{N} \\
u_j(t) &= 0 \quad \forall \, t \notin \mathcal{A}_j
\end{align}
\end{subequations}
where $C_i(\cdot)$ are convex increasing functions on their domain
$[\underline{G}_i, \bar{G}_i]$.
Let us consider the set of possible aggregated demand signals of a fleet of batteries operating without cross-charging and within given availability windows:
\begin{equation*}
\begin{aligned}
\mathcal{D}:= \Big \{ d: \mathcal{T} \rightarrow \mathbb{R}:
\exists \, u_j: \mathcal{T} \rightarrow [0, \bar{P}_j]: d(t) = \sum_{j \in \mathcal{N}} u_j(t), & \\
u_j(t) = 0 \; \forall \, j \in \mathcal{N}, \forall \, t \notin \mathcal{A}_j, \text{ and } \sum_{t \in \mathcal{T}} u_j(t) \geq \bar{E}_j,  &
 \Big \}.
\end{aligned}
\end{equation*}
This polyhedron is described implicitly by using $2NT + N$ inequality constraints and $T+TN$ variables, with (typically) $N \gg T$, (i.e. $T=24$, while $N$ could be in the thousands or even larger). \\
Alternatively, one may seek to describe the set of feasible aggregate power demands without explicit reference to individual battery profiles, viz. by only considering constraints on the aggregate demand. This is possible since the set $\mathcal{D}$ is convex and essentially a projection of a polytope
(of dimension $NT + N$) onto a $T$-dimensional polytope.
It was shown in \cite{submitted} that $\mathcal{D}$ can also be characterized as follows:
\begin{equation*}
\begin{aligned}
&\mathcal{D} = \Big \{ d: \mathcal{T} \rightarrow \mathbb{R}_{\geq 0}: \\
&\sum_{t \in W} d(t) \leq \sum_{j \in \mathcal{N}} \min \{ \textrm{card}
( \mathcal{A}_j \cap W) , \bar{E}_j / \bar{P}_j \} \bar{P}_j, \; \forall \, W \subseteq \mathcal{T} \Big \}.
\end{aligned}
\end{equation*}
As a consequence, the previous optimisation problem can be equivalently recast as:
\begin{subequations}\label{modified}
\begin{align}
\min_{\begin{aligned}
	d (t)\,\,& t \in \mathcal{T} \\ g_i(t)\,\,& i \in \mathcal{I}, t \in \mathcal{T}
	\end{aligned} }&  \sum_{t \in \mathcal{T}} \sum_{i \in \mathcal{I} }   C_i (g_i(t)) \tag{\ref{modified}}\\
s.t.\quad \sum_{i \in \mathcal{I}} g_i(t) &= D^{I}(t) + d(t) \quad \forall \, t \in \mathcal{T}  \\
\underline{G}_i \leq & g_i(t) \leq \bar{G}_i \quad \forall \, t \in \mathcal{T}, \; \forall \, i \in \mathcal{I} \\
\sum_{t \in \mathcal{T}} d(t) & = \sum_{j \in \mathcal{N}} \bar{E}_j \\
\sum_{t \in \mathcal{W}} d(t) & \leq \sum_{j \in \mathcal{N}}
\min \{ \textrm{card} ( \mathcal{A}_j \cap \mathcal{W}), \bar{E}_j / \bar{P}_j \} \bar{P}_j \nonumber \\
& \qquad \qquad \qquad \qquad \qquad \quad  \forall \, \mathcal{W} \subset \mathcal{T}.
\end{align}
\end{subequations}

\section{A simplified model for full availability}\label{sec:fullavailability}

While the previous model does not involve detailed descriptions of individual batteries (in that it only involves aggregated demand profiles) it still requires a potentially large number of inequality constraints, $2^T-1$, in order to capture the flexibility afforded by the fleet. Its use is therefore beneficial only when $2^T < 2NT+T$, viz. for very large fleets or for a relatively low number of considered time intervals.

We perform in Section \ref{sec:computation} a systematic analysis of how
problem (\ref{modified}) and (\ref{original}) compare from the
numerical point of view.
While some of the constraints in
(\ref{modified}) will typically be redundant, their a priori elimination appears to be too computationally demanding as it entails the solution of an LP of comparable computational complexity as the problem itself.
We propose next a way to replace the $2^T$ constraints with $T$ inequality constraints which still afford to compute the true optimal solution (or a very close approximation of it). \\

We consider first the case of full availability, viz. $\mathcal{A}_j =
\mathcal{T}$ for all $j \in \mathcal{N}$. Under such scenario,
let $t_1,t_2,\ldots, t_T$ be such that:
\[  D^{I}(t_1) \leq D^{I}(t_2) \leq D^{I}(t_3) \leq \ldots \leq D^{I} (t_T).  \]
Due to the permutation invariance of the problem and the monotonicity of the cost function we expect that:
\[   d^*(t_1) \geq d^*(t_2) \geq \ldots \geq d^*(t_T). \]
Too see this, assume by contradiction that for some $t_k$ we have
$d^*(t_k)< d^*(t_{k+1})$. Then, $D^{I}(t_k) + d^*(t_k) < D^{I}(t_{k+1}) +
d^*(t_{k+1})$. Hence reducing $d(t_{k+1})$ by a small amount of power $\Delta$ (equal for instance to $(d^*(t_{k+1})- d^*(t_k))/2$ and increasing $d(t_{k})$ by the same amount, yields a strict price reduction.
Let us denote $\tilde{d}$ this new demand profile, which fulfills:
\[ \tilde{d} (t) = \left \{  \begin{array}{rl}
d^*(t) & \textrm{if } t \notin \{ t_k, t_{k+1} \} \\
\frac{d^*(t_k)+d^*(t_{k+1})}{2} &
\textrm{if } t \in \{t_k, t_{k+1} \}
\end{array} \right .
 \]
Feasibility is also preserved since, any constraint involving both $t_k$ and $t_{k+1}$ is unaltered by the power swap. Any constraint
involving only $t_{k+1}$ is also fulfilled since $0 \leq \tilde{d}(t_{k+1}) \leq d^*(t_{k+1})$. Every constraint involving only $t_k$ is also fulfilled since the time $t_k$ can be swapped with time $t_{k+1}$:
\begin{equation*}
\begin{aligned}
&\sum_{t \in W} \tilde{d}(t) = \sum_{t \in W \cup \{t_{k+1} \} \backslash \{t_k \} } \tilde{d}(t) \\ &\leq
\sum_{t \in W \cup \{t_{k+1} \} \backslash \{t_k \} } d^*(t)
 \leq \sum_{j \in \mathcal{N}} \min \{  \textrm{card}(W), \bar{E}_j/ \bar{P}_j   \} \bar{P}_j.
\end{aligned}
\end{equation*}
Hence, by convexity of the set $\mathcal{D}$, this contradicts optimality of $d^*(t)$ for all sufficiently small power swap among $t_k$ and $t_{k+1}$ in the direction of $\tilde{d}$ considered above. \\
Notice that, for any $W$ of cardinality $k$ we have the following:
\[   \sum_{t \in W} d^*(t)  \leq \sum_{i=1}^k d^*(t_i),   \]
hence the same optimal generation cost and solution can be computed by solving:
\begin{subequations}\label{modifiedaggregate}
\begin{align}
\min_{\begin{aligned}
	d (t)\,\,&  t \in \mathcal{T} \\ g_i(t)\,\,& i \in \mathcal{I}, t \in \mathcal{T}
	\end{aligned} } & \sum_{t \in \mathcal{T}} \sum_{i \in \mathcal{I} }   C_i (g_i(t)) \tag{\ref{modifiedaggregate}}\\
s.t.\quad \sum_{i \in \mathcal{I}} g_i(t) & = D^{I}(t) + d(t) \quad \forall \, t \in \mathcal{T}  \\
\underline{G}_i \leq & g_i(t) \leq \bar{G}_i \quad \forall \, t \in \mathcal{T}, \; \forall \, i \in \mathcal{I} \\
\sum_{t \in \mathcal{T}} d(t) & = \sum_{j \in \mathcal{N}} \bar{E}_j \\
\sum_{i=1}^{k} d(t_i) & \leq \sum_{j \in \mathcal{N}}
\min \{ k, \bar{E}_j / \bar{P}_j \} \bar{P}_j \nonumber \\ & \qquad \qquad\qquad\,\, \forall \, k \in \{1,2, \ldots, T\}.
\end{align}
\end{subequations}

\section{Aggregate models for partial availability}\label{sec:partialavailability}
Next we aim to address the more general case of batteries with partial availability, viz.
$\mathcal{A}_j \subseteq \mathcal{T}$ with strict inclusion also allowed.
This is of great practical relevance as it corresponds to batteries which are physically connected to the grid only during a limited amount of time, i.e. as typical of EVs.

Unlike for the case of full availability, the expected ordering of total demand levels (and generation prices) cannot be inferred by looking at inflexible demand alone.
This is because different time intervals typically exhibit different sets of available batteries, and this in turn impacts ability and convenience to charge at any given time slot.
The next Lemma plays an important role to gain an a priori understanding of optimal power profiles.

\comment{
We conjecture that whenever the optimal solution $d^*(t)$ is such that at least one constraint is active, then $\mathcal{W}_{\min}$ will have non-empty intersection with the set of $W$s of active constraints. \\
Another plausible conjecture is that there exist $T$ constraints among the $2^T$ ones, that give rise to the same optimal solution of the considered problem.  \\
The following Lemma could be useful.
\begin{lemma}
	Let $d^*(t)$ be optimal.
Define $\underline{W}$ as follows:
\[ \underline{W} = \arg \min_{t \in \mathcal{T}} D^{I}(t) + d^*(t). \]
Then $\underline{W} \in \mathcal{W}_{\min}$.
\end{lemma}
To see this, consider that for any $t \in \underline{W}$ and any $W \subset \mathcal{T}$ the following holds:
\[   \frac{\sum_{t \in \underline{W}} D^{I}(t) + d^* (t)}{\textrm{card}(\underline{W})} = D^{I}(t) + d^*(t)  \]
\[ \leq \frac{ \sum_{t \in W} D^{I}(t) + d^*(t) }{\textrm{card}(W)}
\leq \frac{ \sum_{t \in W} D^{I}(t) + \sum_{j \in \mathcal{N}} \min \{ \textrm{card}(\mathcal{A}_j \cap W ) , \bar{E}_j/ \bar{P}_j \} \bar{P}_j }{\textrm{card}(W)} = \bar{D} (W) \]

Let us consider, for the sake of simplicity, the case in which
$\mathcal{W}_{\min}$ is a singleton $\{ W_{\min} \}$.
We propose the following iteration:
\begin{enumerate}
	\item $W_1 = W_{\min}$;
	\item For $k=1 \ldots T-1$,
	\[ W_{k+1}  := W_k \, \cup \,   \arg \min_{\theta \notin W_k}
	 \frac{ \sum_{t \in W_k \cup \{ \theta \} } D^{I}(t)  + \sum_{t \in W_k \cup \{ \theta \} } \sum_{j \in \mathcal{N}} \min \{
		\textrm{card} (\mathcal{A}_j \cap (W_k \cup \{\theta \} ), \bar{E}_j / \bar{P}_j   \} \bar{P}_j   }{\textrm{card}(W_k)+1}
	\]
\end{enumerate}
We conjecture that the optimal solution of the previous problem can be found using the following reduced set of constraints:

\[
\min_{\begin{array}{rl}
	d (t)&  t \in \mathcal{T} \\ g_i(t) & i \in \mathcal{I}, t \in \mathcal{T}
	\end{array} }  \sum_{t \in \mathcal{T}} \sum_{i \in \mathcal{I} }   C_i (g_i(t)) \]
\[ \textrm{subject to} \]
\[
\begin{array}{rl}
\sum_{i \in \mathcal{I}} g_i(t) &= D^{I}(t) + d(t) \qquad \forall \, t \in \mathcal{T}  \\
\underline{G}_i \leq g_i(t) \leq & \bar{G}_i \qquad \forall \, t \in \mathcal{T}, \; \forall \, i \in \mathcal{I} \\
\sum_{t \in W_k} d(t) & \leq \sum_{j \in \mathcal{N}}
\min \{ \textrm{card} ( \mathcal{A}_j \cap W_k), \bar{E}_j / \bar{P}_j \} \bar{P}_j \qquad \forall \, k \in \{1,2,\ldots,T\}  \\
\sum_{t \in \mathcal{T}} d(t) & = \sum_{j \in \mathcal{N}} \bar{E}_j
\end{array}
\]	}
\begin{lemma}
	\label{optimalagent}
	Let $u_j^*$ be optimal dispatch policies for problem (\ref{original})
	corresponding to the optimal aggregate demand profile $d^*(t)$ for problem (\ref{modified}), where
	\[  \sum_{j \in \mathcal{N}} u_j^*(t) = d^*(t) \qquad \forall \, t \in \mathcal{T}. \]
	Then, for all $j \in \mathcal{N}$, there exists $d_j >0$ such that:
	\begin{equation}
	\label{lemmaclaim}
	u_j^*(t) = \left \{ \begin{array}{rl} \bar{P}_j & \textrm{if } t \in \mathcal{A}_j \textrm{ and } D^{I}(t) + d^*(t) < d_j \\
	0 & \textrm{if } t \in \mathcal{A}_j \textrm{ and } D^{I}(t) + d^*(t) > d_j.   \end{array} \right .
	\end{equation}
\end{lemma}

We say that $W \subset \mathcal{T}$ corresponds to an active constraint for $d^*(t)$ if:
\[  \sum_{t \in W} d^*(t) = \sum_{j \in \mathcal{N}} \min \{
\textrm{card} (\mathcal{A}_j \cap W), \bar{E}_j / \bar{P}_j   \} \bar{P}_j.   \]
Our aim is to devise an efficient algorithmic procedure to a priori select the support of active constraints in (\ref{modified}) so as to possibly retain the same optimal solution.
The following Lemma provides an important insight on some of the active constraints.

\begin{lemma}
	\label{activeconstraints}
	Let $d^*(t)$ be an optimal aggregated demand profile for problem (\ref{modified}). Consider for any $d > 0$, the following sets:
	\begin{equation}
	\label{thesublevelsets}
	  W_d := \{ t \in \mathcal{T}:  D^{I}(t) + d^* (t) < d  \}.
	  \end{equation}
	Any such set is the support of an active constraint, viz.
	\[   \sum_{t \in W_d}  d^*(t) = \sum_{j \in \mathcal{N} }
	\min \{ \textrm{card} ( \mathcal{A}_j \cap W_d ), \bar{E}_j/ \bar{P}_j \} \bar{P}_j.  \]
\end{lemma}

While Lemma \ref{activeconstraints} identifies an important set of active constraints, it does not provide any indication of how to compute these without solving the optimisation problem (\ref{modified}). In fact, the definition of $W_d$ explicitly requires the optimal aggregate power profile $d^*(t)$.
 The next result gives some important hints on how to look for active constraints by only using a priori available information. \\
To this end define, for each $W \subset \mathcal{T}$, the maximum average total demand as follows:
\begin{equation*}
\begin{aligned}
&\bar{D} (W) := \\   &\frac{ \sum_{t \in W} D^{I}(t)  + \sum_{t \in W} \sum_{j \in \mathcal{N}} \min \{
	\textrm{card} (\mathcal{A}_j \cap W), \bar{E}_j / \bar{P}_j   \} \bar{P}_j   }{\textrm{card}(W) }.
\end{aligned}
\end{equation*}
Notice that this is affected both by the inflexible demand and by the availability windows of individual agents, and is computed assuming that within $W$ and compatibly with individual availability windows, batteries are drawing energy at their maximal rates (until completion of the charging task).

In particular, Lemma \ref{achieved} states that $W_d$ achieves the minimum value of maximum average available demand $\bar{D}(W)$ over sets of its cardinality.
\begin{lemma}
	\label{achieved}
	Let $d^*(t)$ be the optimal aggregated demand for problem (\ref{modified}) and $d \geq 0$ be arbitrary. Define $W_d$
	according to (\ref{thesublevelsets}). Then the following holds:
\begin{equation}
\label{minimumcard}
    \bar{D} (W_d) =  \min_{W \subset \mathcal{T}: \textrm{card} (W)= \textrm{card}(W_d) }   \bar{D} (W).
\end{equation}
Moreover $W_d$ is the unique minimizer in (\ref{minimumcard}).
\end{lemma}

While the $W_d$ sets defined in (\ref{thesublevelsets}) are not the only active constraints that one may find by solving (\ref{modified}), it turns out that they are enough to fully characterized the optimal solution $d^*(t)$, as clarified in the following Lemma \ref{wdenough}.

\begin{lemma}
	\label{wdenough}
	Let $d^*(t)$ be any optimal profile of problem (\ref{modified}) and
	$W_d$ be defined as in (\ref{thesublevelsets}).
	Then the minimum of (\ref{modified}) is equal to the minimum of:
	\begin{subequations}\label{modified2}
	\begin{align}
	\min_{\begin{aligned}
		\tilde{d}(t)\,\,&  t \in \mathcal{T} \\ g_i(t)\,\,& i \in \mathcal{I}, t \in \mathcal{T}
		\end{aligned} } &  \sum_{t \in \mathcal{T}} \sum_{i \in \mathcal{I} }   C_i (g_i(t)) \tag{\ref{modified2}}\\
	s.t. \quad  \sum_{i \in \mathcal{I}} g_i(t) &= D^{I}(t) + \tilde{d}(t) \quad \forall \, t \in \mathcal{T}  \\
	\underline{G}_i \leq & g_i(t) \leq \bar{G}_i \quad \forall \, t \in \mathcal{T}, \; \forall \, i \in \mathcal{I} \\
	\sum_{t \in \mathcal{T}} \tilde{d}(t) & = \sum_{j \in \mathcal{N}} \bar{E}_j \\
	\sum_{t \in W_d} \tilde{d}(t) & \leq \sum_{j \in \mathcal{N}}
	\min \{ \textrm{card} ( \mathcal{A}_j \cap W_d), \bar{E}_j / \bar{P}_j \} \bar{P}_j \nonumber \\ & \qquad \qquad \qquad \qquad\qquad \quad \forall \, d \geq 0 .
	\end{align}
	\end{subequations}
\end{lemma}
It is worth pointing out that as $d$ ranges in $[0,+\infty)$ there are at most $T$ distinct values of $W_d$. This is because the $W_d$ are nested with each other. We may therefore rename them and denote in ascending order as:
\[ W_1 \subsetneq W_2 \subsetneq W_3 \subsetneq \ldots \subsetneq W_Q \]
for some positive integer $Q \leq T$.

Let $d^*(t)$ be any optimal profile of problem (\ref{modified}) and
$W_d$ be defined as in (\ref{thesublevelsets}).
We already remarked that there are at most $T$ distinct values of $W_d$ and that they are nested with one another so that we may rename them and denote in ascending order as:
\[ W_1 \subsetneq W_2 \subsetneq W_3 \subsetneq \ldots \subsetneq W_Q \]
Moreover, it is clear by their definition that the following implication holds:
\[     \frac{\sum_{t \in W_k}  D^{I}(t) + d^*(t)}{\textrm{card} (W_k)}  < \frac{ \sum_{t \in W_{k+1}} D^{I}(t) + d^*(t)  }{\textrm{card}(W_{k+1})}.  \]
As a consequence:
\[    \bar{D} (W_k) < \bar{D} ( W_{k+1}). \]
The discussion so far clearly shows that
the following procedure can be adopted to pre-compute $T$ constraints to be included in a simplified version of (\ref{modified}).
Define $W^*_k$ as follows:
\begin{equation}
\label{cardinality}
W^*_k := \arg \min_{W \subset \mathcal{T}: \textrm{card}(W) = k } \bar{D} (W).
\end{equation}
Notice that the sets $W^*_k$, unlike the $W_d$ previously discussed, can be computed on the basis of a priori available information. Moreover,  by virtue of
Lemma \ref{achieved}, the following inclusion holds:
\begin{equation}
\label{nested}
     \{ W_d: d \geq 0 \} \subset \{ W^*_k: k \in \{1,2,\ldots,T \} \} \subset 2^{\mathcal{T}}.
\end{equation}
Hence, we may replace the constraints of support $W_d$ in (\ref{modified2}) with similar constraints of support
$W^*_k$, without affecting the cost of the optimal solution.
This is stated below and is our main result for this section:
\begin{theorem}
		\label{wstarenough}
		Let $d^*(t)$ be any optimal profile of problem (\ref{modified}) and
		$W^*_k$ be defined as in (\ref{cardinality}).
		Then the minimum of (\ref{modified}) is equal to the minimum of:
		\begin{subequations}\label{modified3}
		\begin{align}
		\min_{\begin{aligned}
			\tilde{d}(t)\,\,&  t \in \mathcal{T} \\ g_i(t)\,\,& i \in \mathcal{I}, t \in \mathcal{T}
			\end{aligned} } & \sum_{t \in \mathcal{T}} \sum_{i \in \mathcal{I} }   C_i (g_i(t)) \tag{\ref{modified3}}\\
			s.t. \quad \sum_{i \in \mathcal{I}} g_i(t) &= D^{I}(t) + \tilde{d}(t) \quad \forall \, t \in \mathcal{T}  \\
		\underline{G}_i \leq & g_i(t) \leq \bar{G}_i \quad \forall \, t \in \mathcal{T}, \; \forall \, i \in \mathcal{I} \\
		\sum_{t \in \mathcal{T}} \tilde{d}(t) & = \sum_{j \in \mathcal{N}} \bar{E}_j \\
		\sum_{t \in W^*_k} \tilde{d}(t) & \leq \sum_{j \in \mathcal{N}}
		\min \{ \textrm{card} ( \mathcal{A}_j \cap W^*_k), \bar{E}_j / \bar{P}_j \} \bar{P}_j \nonumber \\ & \qquad \qquad\qquad\,\, \forall \, k \in \{1,2, \ldots, T\}.
		\end{align}
		\end{subequations}
\end{theorem}

\begin{algorithm}
\DontPrintSemicolon

\textbf{Initialization:}

\nonl $k = T$

\textbf{Update procedure:}

\While{$k > 0$}
{

Find $W^{*}_{k}$ by solving \eqref{cardinality}

Select the inequality corresponding to $W^{*}_{k}$ as an active constraint, decrease the cardinality by
$k = k - 1$
}

\textbf{Final results:} The set of selected active constraints are returned.
\caption{Combinatorial Optimisation Algorithm}
\label{CombinatorialAlgorithm}
\end{algorithm}

The active constraint selection is described in Algorithm \ref{CombinatorialAlgorithm}. One drawback of (\ref{cardinality}) is that, for every assigned value of $k$, it is a combinatorial optimisation problem that entails generation of $\binom{T}{k}$
sets and computation of the corresponding maximum available power.
 While this would typically be faster than a linear programming solver dealing with $2^T$ constraints, it might still be computationally inefficient, and lead to scalability issues.
Hence, we propose in the following two greedy algorithms (of limited computational complexity) for the pre-computation of active constraints, which in simulation have achieved near-optimal performance for large classes of randomly generated problems.

\begin{algorithm}
\DontPrintSemicolon

\textbf{Initialization:}

\nonl $k = T; \hat{W}_T = \mathcal{T}; S = 1$

\textbf{Update procedure:}

\While{$k > 1$}
{

Calculate $\hat{W}_{k-S} := \arg \min_{W \subset \hat{W}_k, |W|=k-S} \bar{D} (W)$

\eIf{ $\hat{W}_{k-S}$ is not a singleton or $\bar{D} (\hat{W}_{k-S}) \geq \bar{D} (\hat{W}_k)$}{$S = S + 1$
}
{

Select the inequality corresponding to $\hat{W}_{k-S}$ as an active constraint, decrease the cardinality by $k = k - S$, and reset $S = 1$

}

}

\textbf{Final results:} The set of selected $\leq T$ active constraints are returned.
\caption{Greedy Algorithm I}
\label{GreedyAlgorithm1}
\end{algorithm}

The greedy algorithm I (in Algorithm \ref{GreedyAlgorithm1}) constructs sets $\hat{W}_k$ in descending order, nested within one another, and starting from the set of cardinality $T$.
It avoids a combinatorial explosion as it seeks to minimize $\bar{D}(W)$ only generating sets which are nested in the previously determined minimal set.
However, it is not guaranteed to determine all of the $W^*_k$ sets and consequently it is also not guaranteed to generate all of the $W_d$ ($d \geq 0$) sets.

In most cases, the greedy algorithm I searches for the minimal $\bar{D}(W)$ over a relatively small number of candidate solutions, i.e., $\binom{k}{k-S}$, when $S$ is small. However, large values of $S$ could slow down the algorithm, due to the unavoidable combinatorial explosion. To further release the computational burden, we propose an alternative greedy algorithm II in Algorithm \ref{GreedyAlgorithm2} for active constraints selection as follows:

\begin{algorithm}
\DontPrintSemicolon

\textbf{Initialization:}

\nonl $k:=T; \hat{W}_T:= \mathcal{T}$

\textbf{Update procedure:}

\While{$k > 1$}
{

Calculate $\hat{W}^{(m)}_{k-1} := \arg \min_{W \subset \hat{W}^{(m)}_k, |W|=k-1} \bar{D} (W)$, $\forall \hat{W}^{(m)}_k \in \hat{W}_k = \{\hat{W}^{(1)}_k, \hat{W}^{(2)}_k, \cdots \hat{W}^{(M)}_k\}$

Let $\hat{W}_{k-1} = \arg \min_{\hat{W}^{(m)}_{k-1} \in \{\hat{W}^{(1)}_{k-1}, \hat{W}^{(2)}_{k-1}, \cdots, \hat{W}^{(M)}_{k-1}\}} \bar{D}(\hat{W}^{(m)}_{k-1})$

Select the inequality corresponding to $\hat{W}_{k-1}$ as an active constraint

Decrease the cardinality by $k = k - 1$

}

\textbf{Final results:} The set of selected active constraints are returned.
\caption{Greedy algorithm II}
\label{GreedyAlgorithm2}
\end{algorithm}

The advantage of this algorithm is that the number of candidates $W$ at the $k$-th iteration is $\binom{k}{k-1} = k$ which avoids the combinatorial explosion and terminates in a number of steps of order $T^2$. However, this fast greedy algorithm might lose optimality for certain examples because the minimizer $\hat{W}_{k-1}$ is restricted to be selected from the subset of $\hat{W}_{k}$ which disables the active constraint selection for $S>1$ in greedy algorithm I.

\section{Advantages in computation}
\label{sec:computation}

This section is devoted to apply the proposed aggregate demand model based optimisation problem and the active constraints selection algorithms to some numerical case studies. All the tested optimisation problems were solved by the YALMIP toolbox and Gurobi optimiser in MATLAB R2019a, on a computer with 2-core 3.50GHz Intel(R) Xeon(R) E5-1650 processor and 32GB RAM.

The computational advantages of using optimisation problem \eqref{modified} and the greedy constraint selection algorithms in a large population of storage devices are evaluated. In particular, the following methods for exploring the optimal dispatch of storage fleets are considered throughout the rest of this section
\begin{itemize}
    \item M1: Each storage unit is modeled as an individual agent and the optimisation problem \eqref{original} is solved;
    \item M2: Aggregate demand model based optimisation by solving problem \eqref{modified} with $2^{T}-1$ inequality constraints on aggregate demand;
    \item M3: Aggregate demand model based optimisation by solving problem \eqref{modified3} with $T$ inequality constraints defined in \eqref{cardinality};
    \item M4: Aggregate demand model based optimisation by solving problem \eqref{modified3} with $\leq T$ constraints chosen by greedy algorithm I.
    \item M5: Aggregate demand model based optimisation by solving problem \eqref{modified3} with active constraints chosen by greedy algorithm II.
\end{itemize}

To compare the performance of these methods, a series of case studies are designed considering various length of full time window and different number of storage devices in a fleet, i.e.,
\begin{equation*}
    T_{1} = 8\, \text{h},\,\, T_{2} = 16\, \text{h},\,\, T_{3} = 24\, \text{h}
\end{equation*}
\begin{equation*}
    N_{1} = 10,\,\, N_{2} = 100,\,\, N_{3} = 1000,\,\, N_{4} = 10 000.
\end{equation*}
The data are sampled every hour. Then, given a time window $\mathcal{T}_{i} = \{1, 2, \cdots, T_{i}\}$ where $i \in \{1, 2, 3\}$, the availability window of each device is uniformly generated as a subset of $\mathcal{T}_{i}$. For each storage, the rated power is set to be $\bar{P}_{j} = 1$ kW  and the target energy $E_{j}$ follows a uniform distribution on the continuous interval $[0,  \bar{P}_{j} \cdot \textrm{card}(\mathcal{A}_{j})]$. Based on these randomly generated examples, the following results are observed:

The computational performances of M1 - M5 for $T_{1} = 8$ h and $T_{3} = 24$ h are shown in Fig.~\ref{DifferentN_FixedT8} and Fig.~\ref{DifferentN_FixedT24}. From the figures, it emerges that the time for solving optimisation problem \eqref{original}  is proportional to the number of agents, so M1 is inefficient for problems with large number of agents. In Fig.~\ref{DifferentN_FixedT8}, when the considered time window is of small length, approaches M2-M4 which are based on the aggregate demand model are effective regardless of the number of devices. In particular, these methods are much faster than M1 for $N_{4} = 10000$. However, the aggregate demand based methods may be exposed to high computational burden if the window is large as shown in Fig.~\ref{DifferentN_FixedT24}, i.e., $T_{3} = 24$. Specifically, M2 can provide the optimal aggregate demand profile, but it is computationally expensive due to $2^{24}-1 \approx 1.6 \cdot 10^7$ constraints. Since the active constraint selection in M3 - M5 is determined by the maximum average demand, a larger number of batteries can also increase the time spent on constraint selection, however, the solution of optimisation \eqref{modified3} is extremely fast and unaffected by the agents' number. Moreover, as described in Section \ref{sec:partialavailability}, combinatorial explosion increases computational burden in searching for the optimal $W^{*}_{k}$ in M3 and possibly $\hat{W}_{k-S}$ in M4, the time on active constraint selection is much higher than that of using the greedy algorithm II in M5.

\begin{figure}[!ht]
\centerline{\includegraphics[width=0.48\textwidth]{Figs/DifferentN_FixedT8.pdf}}
\vspace{-3mm}
\caption{Computational time of $T_{1} = 8$ h for different number of batteries.}
\label{DifferentN_FixedT8}
\end{figure}

\begin{figure}[!ht]
\centerline{\includegraphics[width=0.48\textwidth]{Figs/DifferentN_FixedT24.pdf}}
\vspace{-3mm}
\caption{Computational time of $T_{2} = 24$ h for different number of batteries.}
\label{DifferentN_FixedT24}
\end{figure}

Fig.~\ref{DifferentT_FixedN100} and Fig.~\ref{DifferentT_FixedN10000} presents the computational time of each optimal power dispatch algorithm for $T_{1} - T_{3}$ with $N_{2}$ and $N_{4}$ batteries. The results of M1 verify that the computational complexity for solving optimisation problem \eqref{original} is linear with respect to $N \cdot T$. For the methods M3 and M4 which are based on aggregate demand model, the performance is mainly affected by the length of horizon and example parameters. Particularly, comparing the selection time of M4 at $T_{3} = 24$ h in both figures, we can conclude that less combinatorial explosion is encountered for $N_{4} = 10000$.

Notice that the computational performance of M5 using greedy algorithm II for active constraint selection is the best for problems with large number of batteries and long horizon interval. However, this improvement in computation is achieved at the expense of optimality for some examples. Although the methods M1 - M3 are suffering from the computational challenge, the results are the optimal. To quantify the performance in obtaining the optimal solution using M4 and M5, $10^{4}$ scenarios with $N_{1} = 10$ batteries and $T_{4} = 24$ h time horizon are randomly generated. Finally, the success rate to reach the optimal solution for M4 and M5 are 95.6\% and 93.6\%, respectively (with small optimality gaps in the case of missed optimal solutions).

\begin{figure}[!ht]
\centerline{\includegraphics[width=0.48\textwidth]{Figs/DifferentT_FixedN100.pdf}}
\vspace{-3mm}
\caption{Computational time of $N_{1} = 100$ h for different time horizon.}
\label{DifferentT_FixedN100}
\end{figure}

\begin{figure}[!ht]
\centerline{\includegraphics[width=0.48\textwidth]{Figs/DifferentT_FixedN10000.pdf}}
\vspace{-3mm}
\caption{Computational time of $N_{1} = 10000$ h for different time horizon.}
\label{DifferentT_FixedN10000}
\end{figure}

\section{Two-area system}\label{sec:twoarea}

Next, we consider the application of the aggregate demand model on a two-area system $\mathcal{I} = \{1, 2\}$ over a time horizon $\mathcal{T} = \{1, 2, \cdots, 24\}$ h and an hourly discretization step. Each area is equipped with local dispatchable generators and numerous flexible storage devices. In this case study, to simplify the analysis, we assume that the size of a single generator is significantly smaller than the total installed power output capacity, i.e., the dispatchable power generation at area $i \in \mathcal{I}$ can be approximated by one generator whose power is a continuous variable between $\underline{G}_{i} = 0$ GW and $\bar{G}_{i} = 60$ GW. The generation cost functions are expressed as
\begin{equation*}
    C_{i}(g_{i}(t)) = a_{i}\cdot (g_{i}(t))^{2} + b_{i}\cdot g_{i}(t), \quad \forall i \in \mathcal{I}
\end{equation*}
where
\begin{equation*}
\begin{aligned}
    a_{1} = 1\times & 10^{4}\; \text{\pounds/GW}^{2}\text{h}, \; b_{1} = 1.5\times 10^{4}\; \text{\pounds/GWh},\\
    a_{2} = 2\times & 10^{4}\; \text{\pounds/GW}^{2}\text{h}, \; b_{2} = 1.4\times 10^{4}\; \text{\pounds/GWh}.
\end{aligned}
\end{equation*}
Moreover, the inflexible demands of these two areas $D^{I}_{1}$ (Area 1) and $D^{I}_{2}$ (Area 2) are assumed to be equal to historic data of total power consumption in the U.K. National Grid \cite{websitedata}, as shown in Fig.~\ref{InflexibleDemand}.

\begin{figure}[H]
\vspace{-3mm}
\centerline{\includegraphics[width=0.48\textwidth]{Figs/InflexibleDemand.pdf}}
\vspace{-3mm}
\caption{Inflexible demand profiles at two areas.}
\label{InflexibleDemand}
\end{figure}

A significant penetration of flexible demand is required for charging while minimizing the total generation cost. Specifically, there are $4 \times 10^{6}$ batteries in area 1 and $6 \times 10^{6}$ batteries in area 2, i.e., $\mathcal{N}_{1} = \{1, 2, \cdots, 4 \times 10^{6}\}, \mathcal{N}_{2} = \{1, 2, \cdots, 6 \times 10^{6}\}$. It is assumed that the rated power is equal for the whole population, with $\bar{P}_{j} = 5$ kW, $\forall j \in \mathcal{N}_{1} \cup \mathcal{N}_{2}$. Each device can charge only within a continuous time interval $[t^{s}_{j}, t^{s}_{j} + t^{d}_{j}]$ h, where the starting time $t^{s}_{j}$ and available time duration $t^{d}_{j}$ follow normal distributions, with the following mean and standard deviation:
\begin{equation*}
    \mu_{t^{s}} = 18:00 \,\,\text{h}, \,\, \sigma_{t^{s}} = 1 \,\,\text{h}, \,\, \mu_{t^{d}} = 10 \,\,\text{h}, \,\, \sigma_{t^{d}} = 2 \,\,\text{h}.
\end{equation*}
The energy required $\bar{E}_{j}$ for task completion is uniformly generated on the interval $[0, \bar{P}_{j}\cdot t^{d}]$. Moreover, the power transmission between these two area is bounded by the transmission capacity, i.e., $\abs{p(t)} \leq \bar{p}$.

With the parameters in the two-area system, the optimal power management problem to be solved is formulated as
\begin{subequations}
\begin{align*}
\min_{\begin{aligned}
	u_j(t),\;&g_i(t),\;p(t)  \\ j \in \mathcal{N}_{i},\,\, & i \in \mathcal{I},\,\, t \in \mathcal{T}
\end{aligned} } &  \sum_{t \in \mathcal{T}} \left[  C_{1} (g_{1}(t)) + C_{2} (g_{2}(t)) \right]\\
s.t.\quad  \underline{G}_i \leq & g_i(t) \leq \bar{G}_i \quad \forall \, t \in \mathcal{T}, \; \forall \, i \in \mathcal{I} \\
0 \leq u_j(t) \leq \bar{P}_j,\,\, & \bar{E}_j = \sum_{t \in \mathcal{T}} u_j (t) \quad \forall \, t \in \mathcal{T}, \; \forall \, j \in \mathcal{N}_{1} \cup \mathcal{N}_{2}\\
u_j(t) &= 0 \quad \forall \, t \notin \mathcal{A}_j,\; \forall \, j \in \mathcal{N}_{1} \cup \mathcal{N}_{2}\\
g_{1}(t) + p(t) &= D^{I}_{1}(t) + \sum_{j \in \mathcal{N}_{1}} u_j(t) \quad \forall \, t \in \mathcal{T} \\
g_{2}(t) - p(t) &= D^{I}_{2}(t) + \sum_{j \in \mathcal{N}_{2}} u_j(t) \quad \forall \, t \in \mathcal{T}\\
-\bar{p} \leq & p(t) \leq \bar{p}
\end{align*}
\end{subequations}
which has an equivalent form as following using the aggregate demand model, viz.,
\begin{subequations}
\begin{align*}
\min_{\begin{aligned}
	d_{i}(t),\; g_i(&t),\; p(t)  \\ i \in \mathcal{I},\,\,& t \in \mathcal{T}
	\end{aligned} }& \,\, \sum_{t \in \mathcal{T}} \left[   C_{1} (g_{1}(t)) + C_{2} (g_{2}(t)) \right]\\
s.t.\quad \underline{G}_i \leq & g_i(t) \leq \bar{G}_i \quad \forall \, t \in \mathcal{T}, \; \forall \, i \in \mathcal{I} \\
g_{1}(t) + p(t) &= D^{I}_{1}(t) + d_{1}(t) \quad \forall \, t \in \mathcal{T} \\
g_{2}(t) - p(t) &= D^{I}_{2}(t) + d_{2}(t) \quad \forall \, t \in \mathcal{T}\\
-\bar{p} \leq & p(t) \leq \bar{p}\\
\sum_{t \in \mathcal{T}} d_{i}(t) & = \sum_{j \in \mathcal{N}_{i}} \bar{E}_j \quad \forall i \in \mathcal{I} \\
\sum_{t \in \mathcal{W}} d_{i}(t) & \leq \sum_{j \in \mathcal{N}_{i}}
\min \{ \textrm{card} ( \mathcal{A}_j \cap \mathcal{W}), \bar{E}_j / \bar{P}_j \} \bar{P}_j \nonumber \\
& \qquad \qquad \qquad \quad  \quad  \forall \, \mathcal{W} \subset \mathcal{T}, \; \forall \, i \in \mathcal{I}.
\end{align*}
\end{subequations}

As illustrated in Section \ref{sec:computation}, solving the above two optimisation problems directly is computational inefficient when the number of storage units is large or the time horizon is long. Thus, we adopt two heuristic constraint selection methods for the aggregate demand model to ease the computation burden. Specifically, all the considered approaches to minimize the total generation cost for the two-area power system are summarized as below, each of which is represented by an abbreviation for notational convenience,
\begin{itemize}
    \item MM1: Each storage unit is modeled as an individual agent whose operational behavior is the decision variable.
    \item MM2: The storage units in each area are aggregated as an aggregate demand model, and the power subset of full time windows are imposed in constraints formulation for both areas ($2\times(2^{T}-1)$ constraints).
    \item MM3: The storage units in each area are aggregated as an aggregate demand model, and $2T$ constraints are selected where each area selects its $T$ constraints according to the greedy algorithm II.
    \item MM4: The storage units in each area are aggregated as an aggregate demand model, and the $T$ constraints are selected by applying greedy algorithm II to the merged fleet $\mathcal{N}_1 \cup \mathcal{N}_2$.
\end{itemize}
Particularly, in methods MM3 and MM4 with reduced number of constraints, for the aim of feasibility satisfaction, $d_{i}(t) \leq \bar{d}_{i}(t) = \sum_{j \in \mathcal{N}_{i}: t \in \mathcal{A}_{j}} \bar{P}_{j}$ is also enforced, that is the aggregate demand $d_{i}(t)$ of both areas cannot exceed their maximum power rating.

Next, based on these methods, we discuss the impact of transmission congestion on the marginal cost at each area and on the total generation cost. Fig.~\ref{MarginalPrices} compares the electricity price profiles of the two areas. It can be seen that the marginal prices are different at each area and generally flattened by the flexible demand coordination. The price signals of the two areas converge to the same profile with the increase of power capacity so that the transmission line is no longer congested, i.e. for $\overline{p} = 15$ GW and $\overline{p} = 20$ GW. In particular, since the marginal generation costs at area 2 are higher than those of area 1, the power flow is delivered from area 1 to area 2 which is also verified by Fig.~\ref{PowerflowTwoArea}.

\begin{figure}[H]
\centerline{\includegraphics[width=0.48\textwidth]{Figs/MarginalPrices.pdf}}
\vspace{-3mm}
\caption{The marginal price profiles at two areas.}
\label{MarginalPrices}
\end{figure}

\begin{figure}[H]
\centerline{\includegraphics[width=0.48\textwidth]{Figs/OptimalPowerFlow.pdf}}
\vspace{-3mm}
\caption{The optimal power flow under different transmission capacities.}
\label{PowerflowTwoArea}
\end{figure}

The optimal power flow $p(t)$ between the two areas is shown in Fig.~\ref{PowerflowTwoArea}. Negative values indicate that the power is flowing from area 1 to area 2. When the transmission capacity is no more than $5$ GW, the line is congested over 24 h for energy arbitrage. It is noted that the additional required transmission capacity to avoid congestion is around $15$ GW.

\begin{table}[H]
\centering
\vspace{-5mm}
\caption{Total generation costs (million pounds) under different Transmission Line capacity}
\scalebox{1}{
\begin{tabular}{@{}ccccccc@{}}
\toprule
\multicolumn{1}{c}{$\overline{p}$ (GW)}  & 0 & 5 & 10 & 15 & 20 \\
\midrule
\multicolumn{1}{c|}{MM1} & 771.69  & 718.37 & 698.14 & 696.18 & 696.18 \\
\multicolumn{1}{c|}{MM3} & 771.69 & 718.37 & 698.12 & 696.05 & 696.05  \\
\multicolumn{1}{c|}{MM4} & 771.62 & 718.29 & 698.10 & 696.18 & 696.18  \\
\bottomrule
\end{tabular}}
\label{table:TransmissionLineParameters}
\end{table}

The optimal costs using approaches MM1 - MM4 are collected in Table.~\ref{table:TransmissionLineParameters}. Since the two methods MM1 and MM2 are equivalent, only M1 is displayed in the remaining discussion. From Table.~\ref{table:TransmissionLineParameters}, we can observe that the increase of transmission capacity reduces the operational cost of the two-area system. Moreover, it is interesting to note that MM3 (MM4, respectively) achieve the same optimal cost of
MM1 for values of $\overline{p}$ such that transmission constraints are always active  (respectively never active) over the considered time window. When the transmission line is partially congested during the day, both MM3 and MM4 have small gaps with respect to the optimal cost.

\begin{figure*}[htbp!]
\begin{multicols}{2}
    \includegraphics[width=\linewidth]{Figs/FlexibleDemand_PF5.pdf}\par
    \includegraphics[width=\linewidth]{Figs/FlexibleDemand_PF15.pdf}\par
    \end{multicols}
\begin{multicols}{2}
    \includegraphics[width=\linewidth]{Figs/PowerBalance_PF5.pdf}\par
    \includegraphics[width=\linewidth]{Figs/PowerBalance_PF15.pdf}\par
\end{multicols}
\vspace{-3mm}
\caption{Top row: Aggregate demand profiles when the transmission line capacity is $5$ GW (left) and $15$ GW (right). Bottom row: Power supply and demand sources when the transmission line capacity is $5$ GW (left) and $15$ GW (right).}
\label{FlexibleDemandAndPowerBalance}
\end{figure*}

The flexible demand profiles of the two areas  are compared in Fig.~\ref{FlexibleDemandAndPowerBalance}. Note that the profiles computed by using the selection method MM3 are the same as the optimal solutions obtained by MM1, when the transmission line is congested over 24 h. However, this method may fail to obtain the optimal demand profiles if the power transferred across the transmission line is not  constant in time. In these cases, the cost is lower than the minimum cost of MM1 which indicates that the aggregate demand profiles using MM3 are unfeasible. Concerning the method MM4, which is essentially regarding the two fleets as a single merged fleet, it achieves the optimal generation cost provided that there is no congestion on the transmission line. This ensures that the total demand of the merged single fleet is feasible and optimal, but it does not necessarily imply that the demand profile in each area are feasible. To obtain the optimal aggregate demand for each fleet in the two areas, a dispatching policy of the optimal total demand to individual storage unit in the merged population is needed. Moreover, the power profiles in two areas are also included in Fig.~\ref{FlexibleDemandAndPowerBalance}, showing that the supply and demand are balanced. Compared to the results of $\overline{p} = 5$ GW, high transmission capacity allows more generation from area 1 and diverse flexible aggregate demand.

\section{Conclusion}\label{sec:conclusion}
The paper develops new methods for optimal management of heterogeneous storage fleets subject to availability constraints, by representing their aggregate behaviour through a small number of linear inequality constraints. This can be successfully applied to Unit Commitment problems where the goal is to schedule the fleet's power absorption (or, respectively power release) in order to minimize generation costs.
 The novel formulation involves a number of constraints for the fleet that scales linearly in the length $T$ of the considered time horizon and is independent of the size $N$ of the fleet. This is remarkable, as it shows that, under suitable technical conditions, a heterogeneous fleet of any size can effectively be treated as a single storage unit. The approach is compared with formulations involving a detailed description of the fleet (involving roughly $2 N T$ constraints) or aggregated descriptions involving approximately $2^T$ constraints. Numerical experiments confirm the theoretical results, showing that optimal performance can be achieved with a drastic reduction in the cost of the optimisation time. Since constraints selection is affected by a combinatorial complexity growth in the size $T$ of the considered window, we additionally propose two greedy selection procedures which are shown, through extensive simulations, to perform the correct constraint selection in most considered cases. When the choice of constraints is not optimal, this leads to optimal power schedules which are unfeasible for the original fleet (due to the reduced number of constraints considered), however numerical experiment have shown in all such cases a small optimality gap.
 The method is then tested on a two area system, where each area has its own fleet and ability to schedule it in a coordinated fashion. Transmission capacity constraints are also assumed among the two areas, so that marginal price differentials may be experienced at times when capacity constraints are active.
 While the theoretical analysis cannot be applied to this case, we show how the method yields promising results. The constraints reduction is performed by either considering the two fleets as a single fleet (i.e. ignoring capacity constraints) or, ignoring the existence of a transmission line, by carrying out the reduction of each fleet individually.


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The supplement is made of 90% natural ingredients which have healing abilities and cleansing properties. With this natural solution one is able to maintain healthy and effective internal functioning of body by providing necessary nutrition. It is known as hydroxycitric acid and has profound dual action fat busting abilities.
When you choose Whole Body Research Garcinia Cambogia, you get 30% HCA extract, and potassium without any fillers, binders, or synthetic ingredients.
Next Story → As probably already know extensive experimental research studies have revealed that Hydroxycitric acid hinders fat synthesis and decreases food intake.
Which also stars Katharine McPhee and Jack Davenport, debra plays successful Broadway lyricist Julia Houston in the NBC drama.
As the test subjects lost significantly more weight that those in the control group, this study indicated that Garcinia cambogia may have a potential as a weight loss agent.


What do the recent election results prove about Americans belief in gun control? Let's take a look at some of the game changing votes of gun control advocates versus the candidates endorsed by the NRA.
In Senate races, gun control Senator Mark Udall (D-CO) was defeated by NRA-endorsed Cory Gardner (R) and gun control Senator Kay Hagan (D-NC) was defeated by NRA-endorsed Thom Tillis (R). In Kansas, NRA-endorsed Senator Pat Roberts (R) won. In Georgia, NRA-endorsed Senatorial candidate David Perdue (R) won. In Arkansas, NRA-endorsed Tom Cotton (R) won. And in West Virginia, NRA-endorsed Shelley Moore Capito (R) won, marking the first time that state has sent a Republican Senator to Washington DC in over five decades.
The spotlight was also on the race between NRA-endorsed Senate Minority Leader Mitch McConnell (R-KY) and pro-gun control challenger Alison Grimes (D). McConnell won handily.
NRA-endorsed Senatorial candidate Joni Ernst (R-IA) also won.
This completely ruins the liberals view that 90% Americans are for stricter gun control. Way to go America, way to go.
Would you like to know what to do to protect your family better? Click here to see what lies could hurt you about family self defense.


Sometimes I feel left out and sometimes I just feel annoyed – about Pinot Grigio.
I just don't get Pinot Grigio, I sort of understand why it sells – it usually doesn't taste of anything, so the focus of your evening remains the conversation, the wine stays in the background to lubricate your palate.
I need acidity in a white wine though, so the slight bitterness on the finish of virtually every Pinot Grigio is unpleasant to me.
I have long had a fascination for wines from California and as Napa Valley is the most famous wine region in California I was honoured to be invited to attend the Master Napa Valley Course 2009.
Many of will be aware that I tasted some Virginian wines earlier in the year and was pretty impressed. If you want to read about it again, click here.
As discussed before in these pages, I am often attracted to more unusual wines, the less familiar, the less famous. I suppose that I always live in hope that these will prove to be undiscovered gems whose time has come – and sometimes that is the case.


text:
Q: How do I write my function calling data and column? I want my function to:

*

*change the values in a column so that the ',' is a '.'

*convert that column to a double

I have tried many different forms of this equation but I keep getting the error "Unknown or uninitialised column"
convert_number <- function(data, column, y) {
  a <- str_replace(data$column, ",", ".")
  y <- as.double(data$column)
  return(a)
}

How can I fix it?

A: data$column looks for the column literally called "column" which won't exist. data[[column]] will clear this up for you.

A: This doesn't fix your function, but I think is an approach:
library(dplyr)

df <- tibble(a = c("1,2", "3.4"))

mutate_at(df, "a", ~as.numeric(gsub(",", ".", .)))

>
# A tibble: 2 x 1
      a
  <dbl>
1   1.2
2   3.4

or as a function:
fun <- function(x, col) mutate_at(x, col, ~as.numeric(gsub(",", ".", .)))

df2 <- fun(df, "a")

To change multiple columns simultaneously you would do this:
df2 <- fun(df, c("a", "b", "c"))

redpajama_set_name: RedPajamaStackExchange


text:
Q: validate textbox for date of birth The below code is for validating a textbox for date of birth. The conditions are

*

*Textbox can't be empty

*Textbox date should be in format dd/mm/yyyy

*Textbox date should not be larger than current date. ie; no future date (to show error like -U r not born yet dude)

<asp:TextBox ID="txtDateOfBirth" runat="server" CausesValidation="True" />
          (dd/mm/yyyy e.g. : 12/12/2011)
<asp:CustomValidator runat="server" ID="valDateRange" ControlToValidate="txtDateOfBirth" ErrorMessage="enter valid date" />

But the problem is that, the textbox is inside an ajax wrapper so only client side validations will work.
Anybody here, Plz help me with any hints, suggestions or with working code!!. I will be very great-full because i was working on this since morning

A: You could have a ClientValidationFunction property on your CustomValidator...
<asp:CustomValidator runat="server" ID="valDateRange" ControlToValidate="txtDateOfBirth" ErrorMessage="enter valid date" ClientValidationFunction="validateDate" />

Then create a Javascript function:
function validateDate(sender, e) {

    // Split out the constituent parts (dd/mm/yyyy)
    var dayfield = e.Value.split("/")[0];
    var monthfield = e.Value.split("/")[1];
    var yearfield = e.Value.split("/")[2];

    // Create a new date object based on the separate parts
    var dateValue = new Date(yearfield, monthfield - 1, dayfield)

    // Check that the date object's parts match the split out parts from the original string
    if ((dateValue.getMonth() + 1 != monthfield) || (dateValue.getDate() != dayfield) || (dateValue.getFullYear() != yearfield)) {
        e.IsValid = false;
    }

    // Check for future dates
    if (e.IsValid) {
        e.IsValid = dateValue <= new Date()
    }
}

A: I agree with varangian_12's answer, but be sure to also do some sort of server-side validation for the off-case that your user has Javascript disabled, or disables it to get around your validation
You could do a simple DateTime.TryParse([string value]) and then check to ensure the date occurs in the past
You just need to be sure to handle the edge-case scenarios

A: No need to use valodators, use requiredfieldvalidator for checking if not empty and use jquery date picker to make the field have a date calendar when get focus,
here is an example it is very easy, if u need any help please let me know: http://jqueryui.com/demos/datepicker/

A: You should try this .i will run perfect. you can also used "jquery.validate.js" for make Textbox can't be empty. also in css you have to create a class error
having color red.

   type="text/css" media="all" />

   type="text/javascript"></script>


Maps of the spatial distribution of crop heights can strongly support agriculture in terms of efficiency and yield optimization. Recently published results of experiments of the 3D Spatial Data Processing research group describe an approach to easily extend regular agricultural machines with low-cost sensors for capturing crop heights while the machine is in the field.
Based upon a comparison between crop height values derived from 3D geodata captured with the low-cost approach, and high-end terrestrial laser scanning reference data, minimum RMS and standard deviation values of 0.13 m (6.91% of average crop height), and maximum R² values of 0.79 were achieved. A main conclusion of the study is that the crop height measurements derived from data captured with the introduced setup can provide valuable input for tasks such as biomass estimation.


After seven games played, San Giljan TUM Invest is on the top of the BOV Premier Division League with 21 points gathered, the maximum. Till now they seem above all the other contenders who they all beat. The goal difference is also very good for San Giljan till now. The yesterday game (12-4 against Sliema who was second before the game) showed how determinate San Giljan is to maintain their perfect route.
The second (Neptunes), third (Sliema) and fourth (Sirens) share the fact that their difference between won and lost games is slightly positive. They are also very close to each other with only one point difference between the second and the fourth after the last games of this week (12,12 and 11 points). The second place contest is very tight which can benefit to the leader.
In the bottom of the Premier League standings there is Valetta that collected only one point and Exiles Jetfreight that gathered 4 points (1 draw, 1 win).
It remains 8 games to be played in total and San Giljan seem to be doing clearly well enough to think about the championship title, they have 9 points more than the second and third. As the second place is highly contested, San Giljan can also hope in some defeats from his contenders but if they continue to do as well as they did now which is almost perfect the championship title should be assured before the end of the championship. For the contenders the only way to be able to catch up San Giljan is to make a perfect second part of the season if San Giljan don't change his pace.
Here below you can find the actual ranking of the Premier Division League.


February calendar post at last!!
Why has it taken so long – no idea – beyond printer not working, going back to work after summer holiday, and putting my time into painting.
Anyway – here 'tis. This is my world, my green view – this is what I see if I feel like a spot of daydreaming.
And now that the cicadas have come out in force at last, I wonder if a big shiny insect might have been a better image. Still – they need the trees and trust me – they are all over these.
This time of year is full of spirit, talk of the thin veil, acknowledging ancestors.
Here in the Southern Hemisphere, we are well into spring, and I'm focusing on renewal and finding beauty.
Blossoms are everywhere, scenting the air, delicate pale pink petals on my small apple tree. The weeds are doing their best to compete, bright yellow dandelion flowers shouting from the emerald grass.
This is such an intense time, saturated colour, and the startling speed of new growth.
The flax flowers are almost opening, and when they do, tui will circle the house, then land and feed.
This morning began with rain, the gentle rain that collects and settles on leaves.
The photos for this calendar were all taken last year. I took another photo this morning that was almost identical – a year apart in time, but connected by rain and light.
There's little more to say, except that, in these strange times, remembering the gifts the earth gives us could help us remember the gifts we can give each other and ourselves – gentleness, kindness, patience, encouragement.
At first I thought that was all I had to say. I felt an emptiness. I could feel myself casting around for the right words… there are no "right words". There is only whatever you feel the need to say. There is only the saying of whatever it is you are feeling.
I feel the same rage that rises in so many women around the world. Frustration with the inequalities in our societies, with the sense of entitlement that so many express. There's a part of me that wants to join in the cries of "burn it down!". I don't want to feel helpless, I don't want to feel afraid. I'm drawn to myths of Medusa and Kali and the Morrigan. And yet I feel the witch-wound stirring in my blood, and I wonder if I can be courageous.
All I know is that I have welcomed the crone into my body and soul … and she speaks now, the wise woman, the healer, the mystic.
There's a part of me that needs to step back, and breathe. Standing barefoot on the cool wet ground this morning, I felt calm and quiet, connected to the rain and the light.
And now – the sun has is out. I bask in her warmth and the wet ground is drying. And there were words, after all.


Cedar Rapids, Iowa – Involta, LLC, an industry-leading managed IT, cloud computing and data center services company, was recently named a 2018 Triple Crown Award winner. CRN®, a brand of The Channel Company, has recognized 46 solution providers in North America this year. These outstanding providers reached the qualifying revenue, growth and technical expertise to be named to three of CRN's prestigious solution provider lists, earning them the Triple Crown Award.
Each year, CRN announces lists and rankings to distinguish solution providers who are outshining their peers in the IT Channel. It is a great accomplishment for a solution provider to earn a place on any of these lists.
CRN recognizes the exceptional Channel Partners that place on CRN's Tech Elite 250, Solution Provider 500, and Fast Growth 150 lists in the same calendar year. This exclusive short list of companies earns the CRN Triple Crown Award. Less than fifty companies nationwide accomplished this in 2018, making this an impressive, first-class accomplishment in the IT Channel community.
The 2018 Triple Crown Award winners will be featured in the October issue of CRN and can be viewed online at www.crn.com/triplecrown.
Involta is an award winning national IT service provider and consulting firm. Involta helps organizations plan, manage and execute hybrid IT strategies using a broad range of services including colocation, cloud computing, managed IT, cybersecurity, fiber and network connectivity. Involta has industry-specific services for healthcare, manufacturing, financial and technology that enable compliance and IT transformation initiatives.
Involta maintains partnerships with top tier technology vendors and major public cloud providers such as Cisco, Veeam and Pure Storage, employing a large number of highly certified technical engineers dedicated to building reliable and secure solutions. Through innovative consulting engagements, Involta is able to utilize its unique resources and partnerships to deliver advanced hybrid IT solutions that meet the ever-changing needs of organizations while maintaining the Involta brand promise of Superior Infrastructure, Operational Excellence, and People Who Deliver.


text:
Q: this.getWritableDatabase() causes my Android app to stop I am new to Android SQLite database & while executing this.getWritableDatabase() my app stops. Is there any solution to this problem? My database is about getting students' information and saving their marks in the database.
public class DatabaseHelper extends SQLiteOpenHelper {

    public static final String DATABASE_NAME="Dictionary.db";
    public static final String TABLE_NAME="Dictionary_data";
    public static final String COL_1="ID";
    public static final String COL_2="Name";
    public static final String COL_3="SURNAME";
    public static final String COL_4="MARKS";

    public DatabaseHelper(Context context) {
        super(context,DATABASE_NAME,null,1);
    }

    @Override
    public void onCreate(SQLiteDatabase db) {
        db.execSQL("CREATE TABLE"+ TABLE_NAME+ "(ID INTEGER PRIMARY KEY AUTO INCREMENT,NAME TEXT,SURENAME TEXT,MARKS INTEGER)");
    }

    @Override
    public void onUpgrade(SQLiteDatabase db, int oldVersion, int newVersion) {
        db.execSQL("DROP TABLE IF EXISTS"+TABLE_NAME);
        onCreate(db);
    }

    public boolean insert_data(String name,String surname,String marks) {
        SQLiteDatabase db =  this.getWritableDatabase();
        ContentValues contentValues= new ContentValues();
        contentValues.put(COL_2,name);
        contentValues.put(COL_3,surname);
        contentValues.put(COL_4,marks);
        long result = db.insert(TABLE_NAME,null,contentValues);

        if(result==-1)
            return false;
        else
            return true;
    }
}

And my mainActivity class:
public class MainActivity extends AppCompatActivity {

    DatabaseHelper myDB;
    EditText editName,editSurname,editMark;
    Button btn_add_data;

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);
        myDB = new DatabaseHelper(this.getApplicationContext());

        editName = (EditText)findViewById(R.id.namespace);
        editSurname = (EditText)findViewById(R.id.surnamespace);
        editMark = (EditText)findViewById(R.id.markspace);
        btn_add_data = (Button)findViewById(R.id.add_button);
        addData();
    }

    public void addData(){
        btn_add_data.setOnClickListener(
            new View.OnClickListener() {
                @Override
                public void onClick(View v) {
                    boolean id_inserted = myDB.insert_data(editName.getText().toString(),editSurname.getText().toString(),editMark.getText().toString());

                    if(id_inserted)
                        Toast.makeText(MainActivity.this,"Data inserted",Toast.LENGTH_LONG).show();
                    else
                        Toast.makeText(MainActivity.this,"Data not inserted",Toast.LENGTH_LONG).show();
                }
            }
        );
    }
}

A: there is error in create table
db.execSQL("CREATE TABLE"+ TABLE_NAME+ "(ID INTEGER PRIMARY KEY AUTO INCREMENT,NAME TEXT,SURENAME TEXT,MARKS INTEGER)");

please add space after TABLE
db.execSQL("CREATE TABLE "+ TABLE_NAME+ " (ID INTEGER PRIMARY KEY AUTO INCREMENT,NAME TEXT,SURENAME TEXT,MARKS INTEGER)");

also add log cat for more details,i am not able to comment as it's my new account.


text:
Nihel Landolsi, née le , est une judokate tunisienne.

Carrière
Nihel Landolsi évolue dans la catégorie des moins de 70 kg. Elle est médaillée de bronze aux championnats d'Afrique de judo en 2017 à Antananarivo et en 2018 à Tunis. Elle remporte l'or en 2019 au Cap.

Elle est également médaillée de bronze aux Jeux africains de 2015 à Brazzaville et médaillée d'argent aux Jeux africains de 2019 à Rabat.

Elle est médaillée d'or aux championnats d'Afrique 2021 à Dakar.

Elle participe aux Jeux olympiques de 2020 à Tokyo.

Elle est médaillée d'or aux championnats d'Afrique 2022 à Oran puis médaillée d'argent aux Jeux méditerranéens de 2022.

Notes et références

Liens externes

Judokate tunisienne
Judokate aux Jeux olympiques d'été de 2020
Championne d'Afrique de judo
Naissance en juillet 1995
Lieu de naissance inconnu

redpajama_set_name: RedPajamaWikipedia


text:
Carriera

Club
Pedersen iniziò la carriera con la maglia del Selbak Turn. In seguito, vestì le casacche di molte altre squadre, quali Lillestrøm, Fredrikstad, Göteborg, Brann, Blackburn Rovers e Wimbledon. Chiuse la carriera con il Selbak Turn, nel 2005.

Nazionale
Pedersen giocò 45 incontri per la . Debuttò il 12 settembre 1990, nella partita persa per due a zero contro l'.

Note

Collegamenti esterni

Calciatori della Nazionale norvegese


text:
The 1935 International Lawn Tennis Challenge was the 30th edition of what is now known as the Davis Cup. Due to increased political tensions in Europe, entries for the Europe Zone declined sharply, causing the Qualifying Round system to be scrapped. Only 11 teams would enter the Europe Zone, with 17 teams entering the qualifying rounds; while six would enter the Americas Zone, 4 in North America and 2 in South America. Estonia entered the tournament for the first time.

In the America Inter-Zonal Final the United States received a walkover due to Brazil's absence, while in the Europe Zone final Germany defeated Czechoslovakia. The United States defeated Germany in the Inter-Zonal play-off, but would fall to Great Britain in the Challenge Round. The final was played at the All England Club Centre Court in Wimbledon, London, England on 27–30 July.

America Zone

North & Central America Zone

South America Zone

Americas Inter-Zonal Final
United States vs. Brazil

United States defeated Brazil by walkover.

Europe Zone

Qualifying round

 , ,  and  advance to the 1935 Europe Zone main draw.

Main Draw

Final
Czechoslovakia vs. Germany

Inter-Zonal Zone
United States vs. Germany

Challenge Round
Great Britain vs. United States

See also
 1935 Wightman Cup

Notes

References

External links
Davis Cup official website

Davis Cups by year

International Lawn Tennis Challenge
International Lawn Tennis Challenge
International Lawn Tennis Challenge
International Lawn Tennis Challenge
International Lawn Tennis Challenge


text:
How To Protect Healthcare Facilities
April 21, 2020By MikeBusiness Security Systems
How To Protect Healthcare FacilitiesMike2020-04-21T17:14:41+00:00
Healthcare facilities pose a unique security situation. By nature they hold a lot of risk other organizations don't necessarily have to deal with. Most healthcare facilities are open to the public and provide care for a vast array of people on a daily basis. Many facilities also house expensive and often sought after narcotics or medications. Also, recent changes to healthcare policies are having an effect on everyday operations and their implications may not be fully realized yet. These factors combined with the unpredictability of patient and public behavior make healthcare facilities particularly sensitive environments that require special attention when it comes to security.
As licensed security system integrators our team at Perfect Connections, Inc. understands the changing healthcare climate is producing security risks that need to be addressed. We have been providing comprehensive security system solutions to facilities throughout northern and central New Jersey since 1992. Protecting the integrity of any healthcare facility is crucial to the safety of not only its patients and the general public, but medical staff as well.
Unfortunately, healthcare facilities are no strangers to criminal activity. The 2012 Crime and Security Trends Survey, based on responses from 208 healthcare organizations across the United States, showed that incidents of crime at healthcare facilities are on the rise. In just two years from 2010 to 2012 these incidents rose 37% from a reported 15,000 to more than 20,500. Keep in mind these were reported events. The US Department of Justice estimates that at least half of the crimes committed go unreported. A reason organizations fail to report disturbances is at least partially due to the fact that they don't want to look bad in the public eye and potentially lose patronage.
There are many contributing factors to the rise in criminal incidents at healthcare facilities. For one, many facilities are open to the public 24 hours a day, 7 days a week which can leave them vulnerable. Outside disputes sometimes spill into the healthcare environment causing disturbances and putting staff and patients at risk. Many facilities house pharmaceuticals which can attract substance abusers and drug dealers, increasing the risk of narcotic theft and violence. Another security issue facilities face is when their real estate expands. When healthcare organizations acquire new facilities it can be difficult to create consistency in mitigating risks and instituting security protocols, especially across sprawling campuses.
The Affordable Care Act (ACA also known as "Obamacare") poses a not so obvious concern when it comes to healthcare security. How can a healthcare policy affect security you ask? According to James Barbour, Market Research Analyst on the Kratos PSS Marketing Team, the ACA will undoubtedly have implications on healthcare budgets. He says, "Hospitals will guard their budgets cautiously, and security—traditionally viewed as a cost center—will feel the pinch." Experts also estimate a rise in patient volume due to more people being insured. This has the potential to overwhelm facilities that are already stretched too thin due to budget cuts. The repercussions of the ACA have not been fully realized in general, however security integrators and healthcare professionals should be thinking about ways to effectively and economically manage the risks associated with the healthcare environment.
Often times a healthcare facility will employ physical security guards to help ensure the safety of their staff and patients. However, in the changing economy and in light of policy changes we know security may be faced with spending challenges. Security guards may be on the chopping block when it comes to budget costs, so how can facilities make up for the lack of physical security? Implement comprehensive security systems that include access controls, surveillance, monitoring and notification systems, and lockdown capabilities.
Access control systems can help limit who can gain entry to specific areas of your facility. This would help in sensitive areas, like pharmaceutical storage and data centers. According to Joe Liguori, president of the Board of Directors Security-Net, "healthcare facilities are finding that there is a long-term cost savings by implementing access control for storage and supply closets, for example, eliminating the need to change locks and replace lost or stolen keys." Access control can also help decrease or eliminate theft. By using different methods like card readers, fobs, or sometimes biometrics credentials are harder to duplicate or steal making it easier for medical staff to focus on what is most important, their patients.
In addition to access control, surveillance plays a key role in crime prevention and overall safety. Security cameras allow an organization to have eyes across their entire facility including areas that often go unprotected like parking lots/garages. Having a monitored security system, especially in regards to surveillance, helps alert the proper authorities faster in an emergency. In tandem with a monitoring service, a mass notification system can notify the right personnel in an emergency or dangerous situation as it is occurring. Being notified of an event as it happens would allow for immediate action and hopefully prevent further damage. This is where lockdown capabilities would be beneficial as well. All it would take is the touch of a single button from a facility member, and part or all of a building could be shut down. This is a particularly beneficial feature that can keep a perpetrator out of certain areas in high risk scenarios. Fortunately, lockdown capabilities can sometimes be easily integrated with existing access control features.
While crime statistics can seem daunting, it's important to remember that many, if not most, healthcare facilities are generally considered safe. However, as a healthcare professional or facilities manager it behooves one to consider the risks specific to their organization and how they can be mitigated. As security system integrators, our licensed team at Perfect Connections, Inc. has been providing comprehensive security solutions to healthcare facilities throughout northern and central New Jersey for over 25 years. Just like your patients, we know that no two facilities are exactly the same and that it takes an individualized approach to solve specific needs.
If you live or run a healthcare facility in Central or Northern New Jersey and would like information on any of the topics discussed above, please call 800-369-3962 or simply CLICK HERE.
Image Credit: Image by NEC Corporation of America-Flickr-Creative Commons
← An Update From Perfect Connections
Coronavirus Update from Perfect Connections →

redpajama_set_name: RedPajamaCommonCrawl


text:
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StarKid 30 Day Challenge Plus More Questions
4.8/5.0 (5 votes)
Day 1: Favorite StarKid Production
by: Momma_Hedgehog
Okay so I'm doing the StarKid 30 Day Challenge obviously. Feel free to join me!
I love them all! But if I had to pick it would have to be AVPM because that's where it all started for StarKid and for me as a fan. Plus it has one of my favorite Darren Criss songs 'Not Alone'
Skip to Chapter
Day 2: Favorite Male StarKid
by Momma_Hedgehog
InvisibleSkylarGrey 10 years ago
Yeah!! I like Lauren's tweets about the hurricane.I just like all of her tweets. And one of joes: "Just found a bad asssed motorcycle but I need to find a bad asssed leather jacket to match"
Stuck_In_My_Dreams 10 years ago
I would be Lauren Lopez she just seems to be great person.
__adeline 10 years ago
And I do think you should continue^_^
Oh gawsh... I would like to be any of them really. Probably Jim, since I am very short I would like to experience being tall, just for a day is fine xD
Damn_those_Wizard_Cops 10 years ago
OMR DARREN CRISS TOO! XD
Lauren or Joey, cause their soo freakin awesome! xD
StereoHeart 10 years ago
30. Jim xD or..... DARREN CRISS! :D
And yes keep posting please :)
and Lauren Lopez
ussabrefencer07 10 years ago
Yes, more. :)
Uh, definitely Darren, Holden, the Langs, or AJ. They are the main bit of the creative team/process and I for one would LOVE to be a part of that! :D
I never tryed them cause they dont sell them in Ireland but I suppose..... WHAT THE HELL CANT THEY DO!
Created by Momma_Hedgehog
Momma_Hedgehog
Chasing after the Tweedles, NE, US
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Coming Around (A StarKid Story)
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How I Became A StarKid
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Michel Pastoureau observes that "for medieval society, in effect, the animal that was closest to man is not the bear (despite its outward appearance and its supposed similar method of coupling), even less so the monkey (an abominable figure of the devil), but in fact the pig." The notable resemblance between human and pig, apparent in their shared cunning and omnivorousness, is perhaps at its most disquieting in the similarity between human and porcine anatomy, for this internal similarity is precisely what would be on display at Christian meals. If our Christian eaters had done their homework--and I don't doubt that some of them did--they would have recalled medieval medical treatises, which often included an anagrammatic pun on corpus (body) and porcus (pig). More likely, by remembering the common disparagement of pigs in medieval moral treatises, they would have thought about the pig as a speculum of their own gluttony, but the medical pun might be recalled even in this context: Peter the Chanter's On Virtues and Vices (aka Verbum abbreviatum), for example, points out that "the pig has much in common with humans in its body, as is shown from the arrangement of its internal organs (sicut ex anatomia et divisione ejus patet)" (PL 205:337D-338A). Perhaps first noted as long ago as, of course, Aristotle, such observations have the support of modern science: see Wilson Pond and Harry J. Mersmann's Biology of the Domestic Pig (New York, 2001), which observes that "the digestive similarity and nutrient requirements of the pig and human are remarkably similar."
Claudine Fabre-Vassas The Singular Beast: Jews, Christians, and the Pig (trans. Carol Volk, New York, 1997) is essential. Michel Pastoureau's several discussions of pigs can be found in: "Quel est le roi des animaux?" in Le monde animal et ses représentations au Moyen-Age (XIe-XVe siècles) (Toulouse, 1985); "L'homme et le porc: une histoire symbolique," in Couleurs, images, symboles: études d'histoire et d'anthropologie (Paris: Léopard d'Or, 1989); "Histoire d'une mort infâme: le fils du roi de France tué par un cochon (1131)," Bulletin de la société nationale des antiquaires de France (1992): 174-76; "L'animal et l'historian du Moyen Âge." in L'animal exemplaire au Moyen Âge (Ve - Xve Siècle), ed Jacques Berlioz and Marie Anne Polo de Beaulieu (Rennes, 1999), 13-28; "La chasse au sanglier: histoire d'une dévalorisation (IVe-XIV siècle)," in La chasse au Moyen Age: société, traités, symboles, ed. Agostino Paravicini Bagliani and Baudouin van den Abeele, (Florence, 2000), 7-23; and "Une justice exemplaire: les procès intentés aux animaux (XIIe - XVIe s.)," Cahiers du Léopard d'or 9 (2000), 173-200. He reprints some of the above material in Les animaux célèbres, (Paris: Bonneton, 2001) and Une histoire symbolique du moyen âge occidental,(Paris: Seuil, 2004).
Narcissism Corner: if you're still a bit music-obsessed, listen to most or all of music through your computers, and are a medievalist, perhaps you'd care to join up with me here.
Last FM is so cool, Karl! Thanks for the reference.
If you set up an account, look me up so I can add you to my friend list. Harh! This is fun. (if you're using a PC, I've discovered that media monkey is a superior music player (superior, that is, to itunes or winamp), especially when combined with this. It's relatively simple to connect mediamonkey with lastfm, although, in all honesty, it's even easier with itunes or winamp.
I have a longish post in draft for your/Moore's wolf thing that I hope to post in comments tonight. Hope I can get to it!
Words fail in leafing through the thing.
My goodness do I ever know it. I love the drawings! Kearney doesn't seem to be doing medieval things anymore: a pity, perhaps, since what he has, in spades, is sitzfleisch. Fabre-Vassas, Pastoureau, but above all Milo Kearney (plus Tubach) gave me all that I need for my pigs chapter.
Okay, I just glanced at my notes for it. I can't bear to read too deeply for it, but, jeebus, do I love this: " "As the Middle Ages waned, humor waxed" (5).
Oh hell. Glancing further ahead: "But with the passing of this generation in Rome, along with its chiliastic hopes, the disappointment felt by many Romans must have contributed to the subsequent descent of Rome and its pig imagery into lewdness and obscenity" (140-1).


text:
Q: Trying to add a path to classpath through gradle or some other way I have a gradle/SpringBoot project. I'm trying to run some junit tests that use 3rd party annotations to inject classes. The injected classes are creating a text file (call it 'foo.txt') under $(PROJECT_DIR)/app/build/resources/test. However the same injected class then uses a classLoader.getReource("foo.txt") to try to find the file to try to process it. The classLoader looks like it is searching for foo.text under $(PROJECT_DIR)/build/resources/test and since it can't find it there the 3rd party class throws an exception.
So I somehow either need to make classLoader.getResource() called from the 3rd party class search in $(PROJECT_DIR)/app/build/resources/test for foo.txt or make the 3rd party class create foo.txt in $(PROJECT_DIR)/build/resources/test.
I was thinking the way to resolve this might be to add $(PROJECT_DIR)/app/build/resources/test to the classpath in gradle for the test task. But I don't know how to add a new path to the classpath through gradle. I searched all over google but don't see any examples.
Also, If there is a better solution I'm open to that too, but I have limitations in how I can try to resolve this i.e. assume I can't change the 3rd party app for now, and can't refactor project structure of the project I am creating tests in.
Update:
In case it is helpful here is what the test/s looks like:
@ExtendWith({3rdPartyClassResolver.class})
public class 3rdPartyTest {

  //the actual failing test
  @Test
  @3rdPartyClass(
    storageLocation = "https://storage-location.us/",
    application = "app1",
    version = "latest",
    requestor = "app2")
  public void testing3rdPartyClass(Map<3rdPartyClass, List<Event>> map) {
    System.out.println(); //I don't make it to this point
  }
  //this is what the 3rd party class is trying to do.
  //I'm able to reproduce it using this test. it fails the same way.
  @Test
  public void writeToFile() throws Exception {
    Path pathnio = Paths.get("build/resources/test/foo.txt");
    Files.write(pathnio, Collections.singleton("some text"));
    ClassLoader classLoader = Thread.currentThread().getContextClassLoader();
    URL resource = classLoader.getResource("foo.txt");
    assertNotNull(resource); //this assert fails
  }
}


Market research future published a raw research report on Global Insulation Market that contains the information from 2014 to 2022. Global Insulation market is expected to grow with the CAGR of more than from 2016 to 2022, and with this it is predictable to cross USD Million by 2022.
The global Insulation market has seen a remarkable growth over the past few years and it has been expected that the global market will keep on growing during the forecast period. Insulations are majorly used in the Residential, and non-residential. The growing applications are resulting in the increasing demand for Insulation across the globe. The major factors behind the growth of global Insulation market are rising awareness, technological advancements, rising construction in APAC particularly in China, India, Japan and Indonesia. As per analysis, fiber glass is expected to account the highest growth during the forecast period due to the huge demand from the application industries. The global Insulation market is expected to reach US$ Million at the end of the forecasted period and is expected to grow at CAGR from 2016 to 2022.
APAC will be the dominating market for global Insulation, accounting for US Million and is expected to grow by 2022. This share is attributed to the growing demand from application segment, rising awareness regarding energy consumption and emerging economies.
North America, Europe and RoW are the growing market for Global Insulation Market and are expected to grow at CAGR respectively from 2016 to 2022.
We recognize the key players in the Global Insulation Market as include Huntsman Corporation, Knauf Gips, Owens Corning, CertainTeed Corporation, Johns Manville, Bayer AG, Atlas Roofing Corporation, Dow Building Solutions, Nova Chemicals Corporation, GAF Materials and others. This report includes the product portfolios of the same.
This research report has provides the insights, on various levels of analyses such industry analysis, market share analysis leading market players and their profiles. This report also helps in studying the target segments by providing views on emerging & high-growth segments and market conclusion. Together the market data comprise and discuss with the basic assessments on the competitive scenarios & strategies, of the Global Insulation Market, including the high-growth regions, countries and their political, economic and technological environments. Furthermore the project report also provides the views over the historical market values as well as, pricing and cost analysis of the same.
Global Insulation Market is segmented in to 3 Key dynamics for an easy grasp and enhanced understanding.
North America is the second largest region for the building insulation material market. The North America building insulation material market is driven by various factors such as consumer preference for green buildings and sustained investments in commercial real estate. In North America, U.S. is the largest market for the modular construction. It accounted for the largest market share in 2016. In the past few years, the annual value of construction in U.S. had been increasing continuously. Strong development in commercial and residential structures in the U.S., is projected to fuel the demand for insulation in the construction market in the region.


For me too gvfsd-afp consumes 100% CPU after accessing a Synology NAS, but only after suspending the laptop (non-Mac) for the night and trying again the next day. Trying to access the share again using Nautilus gives a popup saying (translated) "Oops, something went wrong. Unhandled error: error while sending data: broken pipe". Only killing gvfsd-afp helps. I'm using Ubuntu 13.10.