text:
\section{Introduction}
Energy storage devices such as batteries are key resources in  future energy systems due to their flexibility and fast response speed, and their convenient installations as either large-scale bulk units or as distributed resources.
Most real-time energy storage operations are optimized using predictive control, with applications such as economic dispatch~\cite{korpas2006operation}, frequency control~\cite{li2016connecting}, voltage control~\cite{zhang2015optimal}, renewable integration~\cite{khalid2010model}, energy arbitrage~\cite{krishnamurthy2018energy}, peak shaving~\cite{shi2018using}, electric vehicle charging~\cite{xu2016hierarchical}, or a combination of several aforementioned applications~\cite{megel2014scheduling}. These predictive control strategies solve a multi-period optimization problem over a look-ahead horizon at each control step, obtaining the control and state profile over the entire horizon but only applies the first control result, the problem is then updated with a new horizon and state information for the next control step.

The challenge of using look-ahead control in practice is trading off optimality with computational tractability, as a longer look-ahead horizon incorporates more future information and thus improves solution optimality, but increases the computational challenge significantly. For example, real-time economic dispatches in power systems are typically solved over a single period or with a look-ahead horizon less than one hour~\cite{epri_2016}. However, power system operations have strong daily patterns due to load and weather variations, such as charging storage from solar power during the day and discharge during the night. Thus, being able to incorporate a look-ahead horizon over one day or even longer is crucial for the future power system, but solving such problems over the scale of a realistic power system is extremely computationally challenging~\cite{zhao2018multi}, especially binary variables must be introduced in certain application to prevent simultaneous charging and discharging, making the problem non-convex~\cite{castillo2013profit}. In addition, future uncertainties in power systems are often modeled with scenarios~\cite{wang2011wind}, and modeling uncertainties from multiple sources can easily lead to hundreds of scenarios that makes it almost impossible to solve look-ahead economic dispatch with conventional optimization solvers. While methods such as stochastic dual dynamic programming~\cite{papavasiliou2018application} reduce the solution complexity by introducing inter-temporal and scenario decomposition, the computation is still difficult and requires significant memory usage. On the other hand, the optimal control problem must be solved within a reasonable timescale to fully utilize the fast response speed of energy storage devices. For example, a battery ramps from zero to full discharge power within milliseconds~\cite{gao2002dynamic} thus, a scheme that takes seconds or even minutes to update the control decision is not appropriate for controlling batteries.

Solving storage control from the dual problem is more effective than dealing with the primal problem directly since the storage has only a single state variable with upper and lower bounds. Cruise et a
l.~\cite{cruise2014optimal} concluded the storage control problem can be solved using a search algorithm based on the binding conditions on the state-of-charge, and Hashimi et al~\cite{hashmi2017optimal} has developed an algorithm for energy storage price arbitrage with quadratic time complexity, based on solving the dual problem.
Comparably, the technical contributions of this paper and the main advantages of the proposed algorithm is summarized as follows:
\begin{enumerate}
    \item We show that energy storage control with a generalized time-varying objective functions can be solved in worst-case linear time complexity and constant space complexity, with respect to the look-ahead horizon.

    \item We conclude the optimal control condition for energy storage without having to go through the full look-ahead horizon, i.e., the current control is optimal with respect to any future realizations that may not be included in the current look-ahead window.

    \item We derive a sufficient condition for the occurrence of simultaneous charging and discharging, and provide upper and lower bounds for the prime and dual results under such non-convex conditions.
\end{enumerate}

The rest of this paper is organized as follows: Section~II formulates the problem; Section~III presents main analytical results and the algorithm; Section~IV demonstrates numerical results; and Section~V concludes the paper.

\section{Formulation and Preliminaries}

\subsection{Problem Formulation}
We consider a time period $t\in \mathcal{T}=\{1,\dotsc, T\}$ where $t=1$ is the current control step and $t=2$ to $T$ is the look-ahead horizon.  The optimal control profile $p^*_t$ is a minimizer to the following multi-period optimization problem
\begin{subequations}\label{mso}
\begin{gather}
    p^*_t \in \arg \min_{p_t}\; \sum_{t=1}^T O_t(p_t) + C_T(e_T) \label{mso:obj}\\
    \text{s.t.}\nonumber\\
    p_t = p^+_t - p^-_t \label{mso:c1}\\
    p^+_t = 0 \text{ or } p^-_t = 0 \;\forall t\in \mathcal{T} \label{mso:c0}\\
    e_t - e_{t-1} =   -p_t^+/\eta + p_t^-\eta: \theta_{t-1} \label{mso:c2}
    \end{gather}
    \begin{equation}\label{mso:c3}
    \begin{aligned}
    p^+_t &\geq 0 : \underline{\mu}^+_t\,, \\
    p^-_t &\geq 0 : \underline{\mu}^-_t\,, \\
    e_t &\geq 0 : \underline{\nu}_t\,,
    \end{aligned}
    \quad
    \begin{aligned}
     p^+_t &\leq P : \overline{\mu}^+_t  \\
     p^-_t &\leq P : \overline{\mu}^-_t \\
     e_t &\leq E : \overline{\nu}_t
    \end{aligned}
    \end{equation}

\end{subequations}
where
\begin{enumerate}
    \item $O_t(\cdot)|\mathbb{R}\to\mathbb{R}$ is a scalar time-varying convex objective function. Its derivative is denoted as $o_t = \dot{O}_t$.
    \item $C_T(\cdot)|\mathbb{R}\to\mathbb{R}$ is the terminal cost function of the end state of charge $e_T$. $C_T$ is also convex and its derivative is denoted as $c_T = \dot{C}_T$.  Note that $C_T$ can also be used to model the operation beyond $T$ via dynamic programming~\cite{bertsekas2005dynamic}.
    \item $p_t$ is the control decision variable and it is the energy dispatched from the storage during the time period $t$.
    \item $p^+_t$ is the positive (discharge) component of $p_t$.
    \item $p^-_t$ is the negative (charge) component of $p_t$.
    \item $e_t$ is the state of charge (SoC) at the end of time period $t$, subjects to an initial value of $e_0$.
    \item $\eta \in (0,1]$ is the storage charge and discharge efficiency.
    \item $P \in \mathbb{R}^+$ is the maximum energy that can be charged or discharged into the storage during a single period.
    \item $E \in \mathbb{R}^+$ is the maximum energy that can be stored in the storage.
    \item $p^*_t$ is a set of minimizers to the optimization problem.
    \item $\theta_{t}$ is the Lagrangian multiplier associated with the SoC dynamic, its physical meaning is the marginal value of SoC at the end of time $t$ over the future operation $[t+1, T]$.
    \item $\underline{\mu}^+_t$, $\overline{\mu}^+_t$, $\underline{\mu}^-_t$, $\overline{\mu}^-_t$, $\underline{\nu}_t$, $\overline{\nu}_t$ are positive dual variables associated with inequality constraints.
\end{enumerate}
The objective function \eqref{mso:obj} minimizes the total operating cost over the period $[1, T]$. Constraint \eqref{mso:c1} divides the control $p_t$ into a positive component $p^+_t$ and a negative component $p^-_t$ in order to model the efficiency difference during charge and discharge in the SoC evolution constraint. \eqref{mso:c0} is the non-simultaneous charging and discharging constraint that enforces the storage to only charge or discharge at any given time point. \eqref{mso:c2} models the SoC evolution subjects to efficiencies. Power and energy ratings are modeled in \eqref{mso:c3}.

\begin{figure*}[!t]
    \centering
    \subfloat[]{
        \includegraphics[trim={5mm 0 10mm 3mm},clip, width = .95\columnwidth]{th1_xP.eps}
        \label{fig:th1a}
    }
    \subfloat[]{
        \includegraphics[trim={5mm 0 10mm 3mm},clip, width = .95\columnwidth]{th1_xE.eps}
        \label{fig:th1b}
    }
    \\
    \subfloat[]{
        \includegraphics[trim={5mm 0 10mm 3mm},clip, width = .95\columnwidth]{th1_yP.eps}
        \label{fig:th1c}
    }
    \subfloat[]{
        \includegraphics[trim={5mm 0 10mm 3mm},clip, width = .95\columnwidth]{th1_yE.eps}
        \label{fig:th1d}
    }
    \\
    \subfloat[]{
        \includegraphics[trim={5mm 0 10mm 3mm},clip, width = .95\columnwidth]{th1_zP.eps}
        \label{fig:th1e}
    }
    \subfloat[]{
        \includegraphics[trim={5mm 0 10mm 3mm},clip, width = .95\columnwidth]{th1_zE.eps}
        \label{fig:th1f}
    }
    \caption{\footnotesize Illustration of conditions listed in Theorem~\ref{th1} with three guesses of the Lagrangian value: $x$, $y$, $z\in\mathbb{R}$, $(x>y>z)$ to the policy $\pi$ and the control simulation results are plotted respectively (a): $p\up{\pi}_t(x)$; (b): $\sigma_t(x)$; (c): $p\up{\pi}_t(y)$; (d): $\sigma_t(y)$; (e): $p\up{\pi}_t(z)$; (f): $\sigma_t(z)$. All trials use the same $O_t$ and storage setting, the power limit is between -1 to 1 and the SoC limit is between 0 to 4, which are plotted with red dashes in the figures. As shown in (a) and (b), $\sigma_t(x)$ reached the upper SoC bound at $t=15$, indicating $x\geq\theta_0$ and the rest of the simulation is plotted in dots indicating it is not required. In (c) and (d),  $\sigma_t(y)$ reached the end $t=20$ without hitting either bound, thus the equality relationship between $y$ and $\theta_0$ can be concluded using $c_T(\sigma_T(y))$. In (e) and (f), $\sigma_t(z)$ hit the lower SoC limit at $t=9$ indicating $z\leq \theta_0$, and the remaining simulation is again plotted with dots.}
    \label{fig:th1}
\end{figure*}

\subsection{Karush-Kuhn-Tucker conditions}

The results in this paper are primarily based on the use of the Karush-Kuhn-Tucker (KKT) conditions~\cite{boyd2004convex}, which are listed below for \eqref{mso} ( the non-simultaneous charging and discharging constraint \eqref{mso:c0} is non-convex and is excluded from the KKT condition below, treatment of this constraint will be discussed later):
\begin{subequations}\label{mfv}
\begin{align}
    o_t(p^*_t) + \overline{\mu}^+_t - \underline{\mu}^+_t + \theta_{t-1}/\eta &= 0 : p^+_t \label{mfv_eq2}\\
    -o_t(p^*_t) + \overline{\mu}^-_t - \underline{\mu}^-_t - \theta_{t-1}\eta &= 0 : p^-_t\label{mfv_eq3}\\
    \theta_{t-1} + \overline{\nu}_t - \underline{\nu}_t - \theta_{t} &= 0 : e_t \label{mfv_eq1}\\
    \theta_{T} + c_T(e^*_T) &= 0 : e_T \label{mfv_eq4}
\end{align}
\end{subequations}
and the complimentary slackness conditions associated with the inequality dual variables:
\begin{equation}\label{csc}
\begin{aligned}
\underline{\mu}^+_t [p^*_t]^+ &= 0 \,, \\
\underline{\mu}^-_t [-p^*_t]^+ &= 0 \,, \\
\underline{\nu}_t e^*_t &= 0\,,
\end{aligned}
\quad
\begin{aligned}
\overline{\mu}^+_t (P-[p^*_t]^+) &= 0 \,, \\
\overline{\mu}^-_t (P-[-p^*_t]^+) &= 0 \,, \\
\overline{\nu}_t (E-e^*_t) &= 0\,,
\end{aligned}
\end{equation}
and $\underline{\mu}^+_t$, $\overline{\mu}^+_t$, $\underline{\mu}^-_t$, $\overline{\mu}^-_t$, $\underline{\nu}_t$, $\overline{\nu}_t \geq 0$.
Note that we replaced the use of $p^+_t$ and $p^-_t$ with $p_t$ since $\partial p^+_t = \partial p_t$, $\partial p^-_t = -\partial p_t$, $p^+_t = [p_t]^+$ and $p^-_t = [-p_t]^+$, where $[x]^+ = \max\{0,x\}$ is the positive value function.

\section{Main Results}

We start by relaxing constraint~\eqref{mso:c0} so that the rest of the problem is convex. Thus we establish a closed-form connection between the primal and dual problem in Proposition~\ref{pro:pi}. We then present Theorem~\ref{th1} on identifying the equality relationship between $\theta_0$ and any real number $x\in\mathbb{R}$ using numerical simulation, and develop a binary search algorithm that finds the dual result $\theta_0$ and , thus, the primal result $p^*_1$. Then we discuss  how we can bound the result when it is necessary to incorporate the non-convex non-simultaneous charging and discharging constraint~\eqref{mso:c0} using the proposed algorithm.

\subsection{Optimal Control Policy}
We define the policy $\pi$ that calculates a storage control decision $p\up{\pi}_t(x)$ for time $t$ from an input $x\in\mathbb{R}$ as
\begin{gather}
    p^+_t(x) = \Big[\varphi_t(-x/\eta)\Big]^P_0,\; p^-_t(x) = \Big[-\varphi_t(-x\eta)\Big]^P_0 \\
    p^{\pi}_t(x) = p^+_t(x) - p^-_t(x)\label{es:policy}
\end{gather}
where $[x]^y_z = \max\{\min\{x,y\},z\}$ saturates $x$ between $y$ and $z$ ($z\leq y$),
and $\varphi_t(x) : \mathbb{R}\to \mathbb{R}$ is the inverse function of $o_t$ (derivative of $O_t$) as
\begin{align}
    \varphi_t(x) &= \sup\{y\in\mathbb{R} | o_t(y) \leq x\}\,.
\end{align}
Note that $\varphi_t$ is an alternative definition of the inverse function to $o_t$ while compatible with a piecewise linear $O_t$.

The following proposition states that we can obtain the optimal control $p^*_t$ by using the Lagrangian multiplier $\theta_{t-1}$ as input to policy $\pi$:
\begin{proposition}\label{pro:pi
     Policy $\pi$ is a minimizer to problem \eqref{mso} when using the Lagrangian $\theta_{t-1}$ as the input, i.e.,  $p^*_t = p^{\pi}_t(\theta_{t-1})$\,.
\end{proposition}
\begin{proof}
We start by rewriting the KKT condition associated with $p^+_t$ as
\begin{subequations}
\begin{align}
o_t(p^*_t)+\theta_{t-1}/\eta &< 0 \text{ only if } p^*_t = P \\
o_t(p^*_t)+\theta_{t-1}/\eta &> 0 \text{ only if } p^*_t = 0 \\
o_t(p^*_t)+\theta_{t-1}/\eta &= 0 \text{ if } 0\leq \varphi_t(-\theta_{t-1}/\eta)\leq P
\end{align}
\end{subequations}
where we substitute the complementary slackness condition into \eqref{mfv_eq2} that replaces $\overline{\mu}^+_t$ and $\underline{\mu}^+_t$. It is now trivial to see that we can calculate $p^*_t$ as $\varphi_t(-\theta_{t-1}/\eta)$ and limiting the result between 0 and $P$, hence
\begin{align}
p^+_t = \max\{0, \min\{P, \varphi_t(-\theta_{t-1}/\eta)\}\}\,.
\end{align}
We repeat the similar process for $p^-_t$ with \eqref{mfv_eq3}, and use $p_t = p^+_t-p^-_t$ which gives us the result in Proposition~\ref{pro:pi}.
\end{proof}

The following corollary supplements that with $\theta_{0}$ we can obtain $p^*_1$ as well as a series of consecutive optimal control decisions by recording the accumulated sum of the control results $\sigma_t$ defined as
\begin{align}\label{eq:th_soc}
    \sigma_t(x) =& \sigma_{t-1}(x) - [p^{\pi}_t(x)]^+/\eta - [p^{\pi}_t(x)]^-\eta\,,
\end{align}
with the initial value $\sigma_0(x) = e_0$, where $[x]^+ = \max\{0,x\}$ is the positive value function, and $[x]^- = \min\{0,x\}$ is the negative value function. $\sigma_t(x)$ emulates the SoC evolution but using the control result $p\up{\pi}_t(x)$ which may not be optimal. Another difference is that $\sigma_t(x)$ is not limited between $[0,E]$, instead, whether any $\sigma_t(x)$ falls above $E$ or below $0$ is an indicator on the optimality of $p\up{\pi}_t(x)$, as defined by the following corollary:
\begin{corollary}\label{cor:pi}
$p^*_t=p\up{\pi}_t(\theta_0)$ if $0 \leq \sigma_{\tau}(\theta_0) \leq E$ $\forall$ $\tau\in[1,t]$.
\end{corollary}
Corollary~\ref{cor:pi} means that we can maintain optimal control by using $\theta_0$ as the input to \eqref{es:policy} for control steps beyond $t=1$ if all previous $\sigma_t(\theta_0)$ are within the SoC constraint. Corollary~\ref{cor:pi} is based on Proposition~\ref{pro:pi} and the KKT condition associated with $e_t$ in \eqref{mfv_eq1} that the $\theta_t$ value will not change if both $\overline{\nu}_t$ and $\underline{\nu}_t$ are zeros, indicating $0\leq e^*_t \leq E$ and $\sigma_t(x) = e^*_t$. This corollary is thus proved.

\subsection{Main Theorem on Finding Lagrangian Dual}

\begin{theorem}\label{th1
Given $x\in\mathbb{R}$, its equality relationship with respect to $\theta_0$ can be determined as
\begin{enumerate}
    \item  \textbf{If} a) $\sigma_t(x)$ reaches upper bound first, i.e.,  $\exists \tau\in\mathcal{T} \text{ s.t. } \sigma_{\tau}(x) > E \text{ and }
            0\leq\sigma_{\gamma}(x) \leq E$ $\forall \gamma \in [1,\tau)$;
     \textbf{or} b) $\sigma_t(x)$ reached neither bound and $x > -c_T(\sigma_T(x))$;
    \textbf{then} $x\geq \theta_0$;
    \item  \textbf{If} a) $\sigma_t(x)$ reaches lower bound first, i.e.,  $\exists \tau\in\mathcal{T} \text{ s.t. } \sigma_{\tau}(x) < 0 \text{ and }
        0\leq\sigma_{\gamma}(x) \leq E$ $\forall \gamma \in [1,\tau)$;
        \textbf{or} b) $\sigma_t(x)$ reached neither bound and $x < -c_T(\sigma_T(x))$;
    \textbf{then} $x\leq \theta_0$;
    \item \textbf{If} $\sigma_t(x)$ reached neither bound and $x = -c_T(\sigma_T(x))$, \textbf{then} $x = \theta_0$.
\end{enumerate}
\end{theorem}
Proof of this theorem is deferred to Appendix. The intuition is that the Lagrangian dual is the price of the stored energy, its value does not change despite the change with the SoC evolution, except reaching either the upper or lower SoC bound. The SoC series driven by the optimal dual value should never exceed the SoC bounds as the dual value itself reflects the constrained storage capacity. If the SoC exceed the upper SoC bound, it means SoC value is over estimated as the storage does not have enough capacity to store the excessive energy, hence we picked an $x$ that is higher than the optimal Lagrangian dual value. Vice versa, if the SoC exceed the lower bound, meaning we under estimated the dual value.

\subsection{Solution Algorithm}\label{app:alg}
We design a binary search algorithm that finds $\theta_0$ according to Theorem~\ref{th1}, thus  we find $p^*_1=p\up{\pi}_1(\theta_0)$ (Proposition~\ref{pro:pi}) as well as some consecutive optimal control actions (Corollary~\ref{cor:pi}) without needing to explicitly solve Problem~\eqref{mso}. The algorithm requires a preset search accuracy $\epsilon$ and is described as follows:
\begin{enumerate}
    \item Initialize a search range $L$ and $R$ with which we are confident that $L \leq \theta_0 \leq R$;
    \item Set $x$ to $(L+R)/2$. If $R-L < \epsilon$, return $x$ as the optimal Lagrangian dual value and $p\up{\pi}_t(x)$ as the optimal storage control up to time step $t$;
    \item Run the following iterative simulation
    \begin{enumerate}
        \item Set $1\to t$ and $e_0\to \sigma_0(x)$;
        \item Calculate $p\up{\pi}_t(x)$ using Eq.~\eqref{es:policy};
        \item Calculate $\sigma_t(x)$ using Eq.~\eqref{eq:th_soc};
        \item If $\sigma_t(x) \geq E$, set $x \to R$, go to Step 2);
        \item If $\sigma_t(x) \leq 0$, set $x \to L$, go to Step 2);
        \item If $t<T$, set $t+1 \to t$ and go to Step b);
        \item If $x \geq -c_T(\sigma_T(x))$, set $x \to R$;
        \item If $x < -c_T(\sigma_T(x))$, set $x \to L$;
    \end{enumerate}
    \item Go to Step 2).
\end{enumerate}
An example of a confident search range is that we can assume stored energy always has a positive value and choose $L = 0$ and $R = \max\{o_t(p)/\eta|t\in\mathcal{T}, p\in [-P, P]\}$.

This algorithm achieves the following complexity results:
\begin{enumerate}
    \item \emph{Constant space complexity:} The algorithm achieves $O(1)$ space complexity with respect to the search range and the look-ahead duration $T$, because the equality relationship between $x$ and $\theta_0$ can be identified using only the current simulation result $\sigma_t(x)$ so that previous simulation results are not required to be stored.
    \item \emph{Worst-case linear run-time complexity:} The algorithm achieves a worst-case $O(n)$ complexity with respect to the look-ahead horizon $T$ since the worst-case scenario is to simulate all operations steps from $t=1$ to $T$ during each search, but may terminate before reaching $T$ as stated in step 3-d and 3-e. It also achieves $O(\log n)$ time complexity with respect to the search range for using a binary search algorithm.
\end{enumerate}

\subsection{Non-simultaneous Charging and Discharging}

By far we have concluded the optimal storage control when relaxing the non-simultaneous charging and discharging constraint~\eqref{mso:c1}. A sufficient condition for relaxing this constraint without sacrificing result optimality is illustrated in the following proposition:
\begin{proposition
     A sufficient condition for simultaneous charging and discharging to happen is the Lagrangian dual being negative, i.e., if $\theta_{t-1} \geq 0$ then $p^+_tp^-_t = 0$.
\end{proposition}
\begin{proof}
From \eqref{es:policy}, it is trivial to see that for simultaneous charging and discharge to happen, both terms in \eqref{es:policy} must take non-zero values, hence
\begin{gather}
    \varphi_t(-x\eta) < 0 < \varphi_t(-x/\eta)
\end{gather}
and since $\varphi_t$ is a monotonic increasing function and $0 < \eta \leq 1$, it follows
\begin{align}
    -x\eta < -x/\eta
\end{align}
hence $x$ must be less than zero.
\end{proof}
Recall that $\theta_t$ is the value of the stored energy, hence $\theta_t$ being negative indicates the stored energy has a negative value, i.e., we have an intention to store as less energy as possible. Thus, when \eqref{mso:c1}, the storage can charging and discharging at the same time and use round-trip efficiency loss to consume excessive energy even when the storage is full and has no more storage space. This intuition is useful when deciding whether simultaneous charging and discharging should be considered when formulating the problem, for example, this constraint should be considered in price arbitrage for markets with frequent negative prices.

A common method for enforcing non-simultaneous charging and discharging is to add auxiliary binary variables $\bm{v}=\{v_t\}$ such that the storage can only charge or discharge at one time, as
\begin{align}\label{mso:c0b}
    p^+_t <= Pv_t,\; p^-_t <= P(1-v_t),\; v_t \in \{0,1\}
\end{align}
making \eqref{mso} a mixed-integer programming problem, the Lagrangian dual can thus be calculated given a set of fixed $\bm{v}$.

We assume set $\mathcal{V}$ as the set of all reasonable charging status results, in which $v_t$ may be either 0 or 1 during all periods when simultaneous charging and discharge occur, i.e., when \eqref{mso:c0} must be enforced. It is worth noting that the optimal result for $v\bm{v}$ must be in $\mathcal{V}$. We denote $\theta_0(\bm{v})$ as the resulting Lagrangian dual associated with the charging status $\bm{v}$, then the following proposition stands:
\begin{proposition
\label{pro:dual_b}
     For all $\bm{v}\in\mathcal{V}$, $\underline{\theta}_0 \leq \theta_0(\bm{v})\leq \overline{\theta}_0$, where $\underline{\theta}_0$ is calculated using Algorithm 1 by replacing \eqref{es:policy} with
     \begin{subequations}
     \begin{align}
             p^{\pi-}_t(x) = p^+_t(x)\mathbf{1}_{[p^-_t(x) == 0]} - p^-_t(x)\label{es:policy2}
     \end{align}
     and $\underline{\theta}_0$ is calculated using Algorithm 1 by replacing \eqref{es:policy} with
     \begin{align}
             p^{\pi+}_t(x) = p^+_t(x) - p^-_t(x)\mathbf{1}_{[p^+_t(x) == 0]}\label{es:policy3}
     \end{align}
     \end{subequations}
     where $\mathbf{1}_{[x]} = \{\text{ 0 if $x$ is true and 1 otherwise} \}$ is the indicator function.
\end{proposition}
\begin{proof}
First note that compared to \eqref{es:policy}, \eqref{es:policy2} enforces the storage to charge whenever charging and discharging components are both non-zero. Thus when using \eqref{es:policy2} to simulate the battery operation in Algorithm 1, the resulting SoC must always be lower than using any $\bm{v}\in\mathcal{V}$. Thus according to Theorem~\ref{th1}, the resulting  dual $\theta_0$ must be no smaller than any dual $ \theta_0(\bm{v})$ using charging status $\bm{v}\in\mathcal{V}$. Vice versa, when using \eqref{es:policy3}, the battery prefer discharging over charging, resulting in a lower bound for $\theta_0$.  Note that in \eqref{es:policy2} and \eqref{es:policy3}, $p^{\pi}_t(x)$ is still a monotonic decreasing function to $x$ since $p^+_t(x) \geq 0$ and $p^-_t(x) \geq 0$ for all $x$, $p^+_t(x)$ is an decreasing function and $p^-_t(x)$ is an increasing function. Hence the SoC series is still monotonic increasing with respect to $x$, and the convergence optimality of Algorithm 1 will not be effected.
\end{proof}
And the result on the dual binding can be extended to bind the primal control results:
\begin{proposition
\label{pro:prim_b}
   Let $\underline{p}_t = p^{\pi-}_1(\underline{\theta}_{0})$  and $\overline{p}_t = p^{\pi+}_t(\overline{\theta}_{0})$, and $p^*_t$ is the optimal control, then $\underline{p}_t \leq p^*_t \leq \overline{p}_t$ for all $t\in[1,\;\tau]$.
\end{proposition}
\begin{proof}
First given the dual $\theta_0$, the optimal control $p^*_1$ is either $p^+_t(\theta_0)$ or $-p^-_t(\theta_0)$ as in \eqref{es:policy}, since the battery either charges or discharges. Then according to Proposition~\ref{pro:dual_b}, we know the dual must be within $[\underline{\theta}, \overline{\theta}]$. Then we have
\begin{align}
    \underline{p}_t \leq -p^-_t(\theta_0) \leq p^+_t(\theta_0) \leq \overline{p}_t
\end{align}
which proved this Proposition.
\end{proof}

Hence Proposition~\ref{pro:dual_b} and Proposition~\ref{pro:prim_b} provide bounds for the primal and dual result of the non-convex battery control problem using the proposed algorithm that has worst-case linear time complexity.

\section{Numerical Simulation}

We use randomly generated data sets to compare the proposed algorithm to solving Problem~\ref{mso} with different objectives using Gurobi~\cite{gurobi} (model generated using CVX~\cite{cvx}). All simulations are performed in Matlab~\cite{matlab} on a 2.3 GHz machine with 16GB memory.The storage parameter is set as $P = 1$~p.u., $E = 4$~p.u., $e_0 = 2$~p.u., $\eta = 0.92$, and the terminal cost function is set to $C_T(e_T)= (E-e_T)^2/2$. The accuracy of the search algorithm is set to $\epsilon = 10^{-3}$.

\subsection{Piece-wise linear objectives}

\begin{table}[!t]
    \centering

    \caption{Piecewise-linear simulation results.}
    \begin{tabular}{l r r r r r r}
        \hline
        \hline\Tstrut
          & \multicolumn{3}{c}{ CVX+Gurobi }  & \multicolumn{3}{c}{ Proposed} \\
          Trials & $\theta_0$ & $p^*_1$ & cpu [ms] & $\theta_0$ & $p^*_1$ & cpu [ms]
        \\
        \hline\Bstrut\Tstrut
        & \multicolumn{6}{c}{ $T=10$, $J=100$ } \\
        \hline\Bstrut\Tstrut
        1 & 17.53 & 0.6 & 408.8 & 17.53 & 0.6 & 0.1 \\
        2 & 16.82 & 0.0 & 278.1 & 16.82 & 0.0 & 0.1  \\
        3 & 14.59 & 0.3 & 274.5 & 14.59 & 0.3 & 0.1   \\
        4 & 14.21 & 0.1 & 281.7 & 14.21 & 0.1 & 0.1  \\
        5 & 7.27 & -0.5 & 283.7 & 7.27 & -0.5 & 0.1  \\
                \hline\Bstrut\Tstrut
        & \multicolumn{6}{c}{ $T=10$, $J=1000$ } \\
        \hline\Bstrut\Tstrut
        6 & 18.67 & 0.0 & 1851.5 & 18.66 & 0.0 & 0.1 \\
        7 & 20.28 & 0.8 & 1791.5 & 20.28 & 0.8 & 0.2 \\
        8 & 19.50 & 0.0 & 1788.2 & 19.50 & 0.0 & 0.1   \\
        9 & 17.44 & -0.7 & 1868.4 & 17.44 & -0.7 & 0.2  \\
        10 & 19.00 & 0.0 & 1868.7 & 19.00 & 0.0 & 0.2  \\
                \hline\Bstrut\Tstrut
        & \multicolumn{6}{c}{ $T=100$, $J=1000$ } \\
        \hline\Bstrut\Tstrut
        11 & 19.09 & 0.6 & 18497.7 & 19.09 & 0.6 & 0.1 \\
        12 & 19.95 & 0.8 & 19108.2 & 19.95 & 0.8 & 0.2  \\
        13 & 19.44 & 0.0 & 18786.1 & 19.44 & 0.0 & 0.1   \\
        14 & 18.91 & -1.0 & 19263.2 & 18.91 & -1.0 & 0.2  \\
        15 & 19.55 & 0.0 & 19080.4 & 19.55 & 0.0 & 0.2 \\
         \hline
    \end{tabular}
    \label{tab:sim4}
\end{table}

\begin{figure}[!htb]
    \centering
    \includegraphics[trim={10mm 0 10mm 3mm},clip, width = .95\columnwidth]{pwl_example.eps}
    \caption{Three examples of generated piecewise linear cost curves plotted as derivative of the objective function $o_t$.}
    \label{fig:pwl_example}
\end{figure}

In this section the proposed algorithm is compared with Gurobi using piecewise linear objective function inspired by the supply curves in power system economic dispatches~\cite{kirschen2018fundamentals}. The derivative of the objective function $o_t$ is written as
\begin{align}
    o_t(p) = c_j \text{ if } q_{j-1} \leq p < q_j
\end{align}
where $j\in [1,J]$ is the piecewise segment index, $J$ is the number of segments, $c_j$ is the marginal cost (derivative) of the system when $p$ is between quantities $q_{j-1}$ and $q_j$, and the objective is convex if $c_i \leq c_j$ and $q_i \leq q_j$ for all $i<j$, $i,j\in [1,J]$. Some examples of the generated cost curve are plotted in Fig.~\ref{fig:pwl_example}.
\begin{figure}[!t]
    \centering
    \subfloat[]{
        \includegraphics[trim={10mm 0 10mm 3mm},clip, width = .95\columnwidth]{comp_opt.eps}
        \label{fig:comp1}
    }
    \\
    \subfloat[]{
        \includegraphics[trim={10mm 0 10mm 3mm},clip, width = .95\columnwidth]{comp_alg.eps}
        \label{fig:comp2}
    }
    \caption{\footnotesize Average computation time for solving look-ahead economic dispatch with 5000 cost segments per five minutes using (a) Gurobi and (b) the proposed algorithm. Note that (a) is plotted in minutes while (b) is plotted in milliseconds. In (a), when the look-ahead horizon is beyond 6 hours, the problem takes more than five minutes to solve, which is infeasible for real-time economic dispatch that must be finished within five minutes.}
    \label{fig:comp}
\end{figure}
Similar to the quadratic results, we test the proposed algorithm and Gurobi using different settings and the results are demonstrated in Table~\ref{tab:sim4}, where trials 1--5 have 10 time steps and 100 cost segments $T=10$, $J=100$, trials 6--10 have 10 time steps and 1,000 cost segments $T=10$, $J=1000$, trials 11--15 have 100 time steps and 1,000 cost segments $T=100$, $J=1000$. The result shows the proposed algorithm obtains the same results in all trials compared to Gurobi, while being hundreds or even thousands of times faster.
In particular, in trials 11-15 Gurobi needs around 18 seconds to complete the computation, while the proposed algorithm finishes below 1ms.

In Fig.~\ref{fig:comp}, we further test the computation speed of both methods in solving  look-ahead economic dispatches over the size of realistic power systems with 5000 cost segments per five minute dispatch interval. The result shows that the computation time of Gurobi increases significantly with respect to the look-ahead horizon, and in particular at the 6 hour look-ahead, the problem takes more than 5 minutes to solve which is not feasible since the economic dispatch must be calculated within 5 minutes. In contrast, with our proposed algorithm, the average solution speed is below 0.5 milliseconds for look-ahead horizons up to 8 hours, providing a speed-up up to 100,000 times.

\subsection{Negative Lagrangian dual example}

\begin{table}[!t]
    \centering

    \caption{Negative Lagrangian dual results.}
    \begin{tabular}{l r r r r r r}
        \hline
        \hline\Tstrut
          & \multicolumn{3}{c}{ Dual results }  & \multicolumn{3}{c}{ Control results } \\
          Trials & $\underline{\theta}_0$ & ${\theta}^*_0$ & $\overline{\theta}_0$  & $\underline{p}_1$ & $p^*_1$ & $\overline{p}_1$
        \\
        \hline\Bstrut\Tstrut

        1 & -10.7 & -10.6 & -10.2 & -0.14 & 0.10 & 0.11\\
        2 & -12.9 & -12.5 & -11.8 & 0.49 & 0.53 & 0.56\\
        2 & -8.5 & -8.2 & -8.2 & -0.17 & -0.17 & -0.16\\
        4 & -4.4 & -4.4 & -4.0 & -1.00 & -1.00 & -1.00 \\
        5 & -15.0 & -14.6 & -14.5 & 0.89 & 0.91 & 0.97 \\

        \hline
    \end{tabular}
    \label{tab:sim1}
\end{table}

We consider the following quadratic objective function
\begin{align}
    O_t = \frac{\alpha_t}{2} (\beta_t - p)^2
\end{align}
where $\alpha_t$ are randomly generated between $[0, 10]$, and $\beta_t$ between $[-10,0]$. Recall that negative sign is for charging the battery, hence this is a generation tracking problem where the storage wishes to absorb as much energy as possible. This will result in negative dual prices for the storage and simultaneous charging and discharging will occur if not constrained. Table~\ref{tab:sim1} shows the simulation result for five trails including upper and lower bounds on the dual and control, where ${\theta}^*_0$ and $p^*_1$ is calculated using Gurobi with mixed-integer quadratic programming under default settings, solving \eqref{mso} using the integer constraint \eqref{mso:c0b} for enforcing constraint \eqref{mso:c0}. In all test trails, the primal and dual result fall within the calculated range. In terms of computation speed, the proposed method all solves less than 1 millisecond, while the benchmark method using Gurobi may need up to several minutes to solve the problem depending on the problem size (thousands of steps), due to solving a mixed-integer quadratic programming problem.

\section{Conclusion}
This paper proposed a novel algorithm for solving look-ahead control for energy storage. The numerical results illustrate that the algorithm provides computation speed in milliseconds for controlling a single energy storage device over an extended planning period. In future research, we plan on expanding this method to controlling multiple energy storage devices subject to network constraints. Moreover, using the generalized terminal state function we plan on incorporating this algorithm into scenario-based stochastic programming or dynamic programming. In addition, our results connects the optimal control with the Lagrangian multiplier associated with the state-of-charge constraint, which we will further explore to provide key insights into designing future electricity pricing and distributed control schemes.

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


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text:
ASTRA 1N Roars into Space on Board Ariane 5
Dépèche transmise le 7 août 2011 par Business Wire
LUXEMBOURG & KOUROU, French Guiana--(BUSINESS WIRE)--SES S.A. (Paris:SESG)(LuxX:SESG) announces that its new ASTRA 1N satellite has been successfully launched on board an Ariane 5 rocket from Kourou, French Guiana, on August 6th at 19:52 pm local time (0:52 am Central European Summer Time on August 7th).
"ASTRA 1N will greatly enhance the flexibility of SES' ASTRA satellite fleet at two key SES orbital slots over Europe and we would like to thank our longstanding partners Astrium and Arianespace for this mission success."
ASTRA 1N is set to deliver direct-to-home (DTH) services, notably digital and high definition (HD) television in Europe. The satellite will first provide interim replacement capacity at 28.2° East, and will subsequently be moved to 19.2° East, from where it will mainly serve the German, French and Spanish markets.
ASTRA 1N was built by Astrium on the Eurostar E3000 platform and is equipped with 52 transponders in the Ku frequency band. The satellite had a launch mass of 5,325 kg and is the fourth Eurostar satellite in the SES fleet. ASTRA 1N will be brought into commercial service within the next weeks after extensive in-orbit testing.
"The successful launch of ASTRA 1N is another important milestone in our ambitious launch manifest for 2011, which will see four additional SES launches before year-end," said Romain Bausch, President and CEO of SES. "ASTRA 1N will greatly enhance the flexibility of SES' ASTRA satellite fleet at two key SES orbital slots over Europe and we would like to thank our longstanding partners Astrium and Arianespace for this mission success."
Pictures of the launch will be made available on www.arianespace.com and via AFP, Reuters, EPA, AP, Kyodo News, in Europe via AFP, ANP, ANSA, APA, Belga, DPA, EFE, Keystone, Lusa, Pressebild, and in Japan via JIJI press.
The next SES launch is scheduled for early September, when an Ariane ECA booster will orbit the SES-2 spacecraft from the European Space Port in Kourou.
SES (Paris:SESG)(LuxX:SESG) wholly owns the market-leading satellite operators SES ASTRA and SES WORLD SKIES, and participations in Ciel in Canada and QuetzSat in Mexico as well as a strategic participation in satellite infrastructure start-up O3b Networks. SES provides outstanding satellite communications solutions via a global fleet of 45 satellites. For further information: www.ses.com

redpajama_set_name: RedPajamaCommonCrawl


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Context: Radiologists as a group face unique occupational health hazards among which musculoskeletal injuries, chronic eye strain, and others are yet to receive adequate attention. Constant mental strain due to demanding turnaround times and work pressures may lead to burnout and depression. These combine to decrease overall work satisfaction and productivity. Aims: To study the prevalence of various health issues faced by radiologists in India and to assess whether specific demographic and occupational factors are associated with an increased risk. Settings and Design: Cross-sectional observational study conducted as a voluntary anonymous electronic survey. Methods and Materials: A 36-item survey was sent to radiologists through email and social media. All respondents who completed survey were included in the study. Questions regarding workload, repetitive stress injuries, eye strain, burnout, and so on were asked. Statistical Analysis Used: Chi-square test was used to test significance of correlation (P < 0.05). Results: In all, 383 radiologists completed the survey. A high prevalence of repetitive stress injuries, chronic eye strain, depression, and burnout was found. Significant correlation was found between repetitive stress injuries and burnout. Pre-Conception and Pre-Natal Diagnostic Techniques Act (PCPNDT) related issues and work overload were the most common causes of high stress levels. Radiologists whose practices followed ergonomic design showed significantly less prevalence of neck pain. Conclusion: Radiologists in India have a high prevalence of repetitive stress injuries, chronic eye strain, and burnout, along with unique mental stressors such as PCPNDT-related issues.
Radiologists as a group face unique occupational health hazards. Radiology has a singular work environment not seen in other medical specialties. In addition to working long hours on a computer, many radiologists spend ample time performing ultrasound (USG) requiring constant use of a single upper limb resulting in what is called as the Transducer user syndrome. Interventional radiologists working in angiography suites are also exposed to unique musculoskeletal problems. Besides physical issues, the constant mental strain due to the demanding turnaround times and constant work pressures takes its toll as well. These factors contribute to an increased incidence of repetitive musculoskeletal stress injuries, chronic eye strain, and mental health issues such as burnout, stress, depression, and lack of sleep.
Radiation as an occupational health hazard has received adequate attention with multiple national and international guidelines available. However, these nonradiation hazards have probably not yet received the attention they deserve. These can adversely impact the physical and mental health of the radiologist, decrease productivity, increase medical errors, and lead to early burnout. A few previous studies have been published regarding ergonomics and musculoskeletal problems in radiology as well as mental health issues faced by radiologists. However, no previously published study has looked at the overall physical and mental health of radiologists as a whole, as the two are related and often go hand in hand. Also, such a study has not been performed among the Indian radiology community to the best of our knowledge.
A recent editorial in this journal also highlighted the importance of safe ergonomic practices among radiologists to improve their health and productivity. In going with the same theme, we conducted a cross-sectional survey among practising radiologists and radiology residents in India to study the prevalence of various physical and mental health issues faced by them.
The study was designed according to the STROBE checklist for observational studies. A 36-item anonymous voluntary electronic survey was created online using Google Forms (Google Inc., Mountain View, California, USA), including four questions on demographics and practice patterns. This survey was sent to practising radiologists and radiology residents through email and using the social media platform of WhatsApp (2018; WhatsApp Inc., Menlo Park, California, USA). Responses were collected for a 10 day period starting from 4 April 2018 to 14 April 2018. All respondents who completed the survey were included within the study.
Questions regarding film-based versus workstation reporting, volume of work in USG and other modalities, musculoskeletal injuries, varicose veins, chronic eye strain, needle prick injuries, and contraction of tuberculosis were asked. Two questions were asked on ergonomics, whether the respondent was aware of ergonomics in radiology and whether any ergonomic steps had been taken in the respondent's practice to reduce physical injuries. Questions regarding depression, patient abuse, personal satisfaction, recreational activities, exercise, sleep, burnout, and levels and the causes of stress were also included to assess various mental health issues. The responses were statistically analyzed to evaluate for correlation between patient demographics, the presence of various physical and mental health issues, and other issues. Chi-square test was performed to assess the level of significance of correlation. Statistical analysis was performed using Statistical Package for the Social Sciences (SPSS) software version 20 (IBM Inc., Armonk, New York, USA).
A total of 383 radiologists responded to the survey, of which 67% (258/383) were males and 32% (125/383) females. The radiologists surveyed included a broad mix of radiology trainees along with junior and senior practicing radiologists [Figure 1], with 50% having a work experience of 6 to 20 years. The majority of the respondents (96%; 369/383) practiced purely diagnostic radiology or predominantly diagnostic radiology with few interventional procedures. The working patterns of the radiologists surveyed (predominantly workstation-based vs. predominantly film-based vs. USG-only) are as summarized in [Figure 2]. Most radiologists (74%; 284/383) reported working for 40-70 hours/week [Figure 3]. 15% (59/383) respondents practised tele-radiology, with 81% (48 of 59) practising it for less than 3 hours per day.
As many as 301 of 383 (78%) radiologists reported symptoms consistent with repetitive stress injuries at some point in time after starting their radiology career with the most common being neck pain (52%). The other symptoms are as summarized in [Table 1]. More than one-third (37%; 143/383) of the radiologists did not take any treatment for their repetitive stress injuries [Figure 4]. Neck pain was observed to be significantly more common among female radiologists (P= 0.02). The rest of the repetitive stress injuries did not demonstrate any gender predilection. The prevalence of most repetitive stress injuries predictably increased with years of experience, workload, and number of hours of work per week.
Significant correlation was found between increasing number of USG examinations performed per day and the prevalence of neck pain, low back pain, and shoulder pain (P < 0.05). No significant correlation was found between the number of USG examinations performed and elbow pain, wrist pain, tenosynovitis, and carpal tunnel syndrome.
About 6% of radiologists (23/383) developed varicose veins after joining radiology. 57% of the radiologists (219/383) surveyed said they suffered from symptoms of chronic eye strain and computer vision syndrome (CVS), with the incidence of chronic eye strain significantly increasing with increasing years of experience in radiology (P < 0.05) [Table 2].
32% (124/383) of the respondents had suffered needle prick injuries during their careers in radiology, while one-fourth of all radiologists surveyed (99/383) were not vaccinated against hepatitis B. 9% (35/383) of the respondents had developed tuberculosis after starting their careers in radiology. A majority (57%; 217/383) of radiologists felt their reporting rooms were not adequately aerated or ventilated. However, no significant correlation was found between radiologists acquiring tuberculosis and lack of adequate room ventilation.
45% (173/383) radiologists were not aware of the science of ergonomics, and a majority of respondents (57%; 213/383) either did not know whether ergonomic steps had been taken in their department or reported that no ergonomic steps had been taken. Radiologists in whose practices or departments ergonomic steps had been taken reported a statistically significant lower prevalence of neck pain (P = 0.014).
35% (134/383) of radiologists surveyed reported having an excess workload, and 37% (141/383) reported feeling depressed or having suffered from symptoms of depression at some point in their careers in radiology. The proportion was slightly higher among residents, with 40% saying they suffered from depression; however, the difference was not statistically significant. 46% (178/383) of radiologists reported having faced either physical or verbal abuse from patients in the past. A majority (66%; 255/383) of radiologists reported obtaining either excellent or good personal satisfaction from their work [Figure 5]. However, a high number (40%; 153/383) also felt they were not achieving personal growth in radiology in their current workplace. Only 42% (163/383) of the radiologists said they get adequate time to pursue their hobbies and for recreational activities, and 15% (58/383) reported getting less than 6 hours of sleep every day [Table 3].
The prevalence of burnout among radiologists was 54% (208/383) in our study. A significant correlation was found between the presence of physical repetitive stress injuries and prevalence of burnout (P < 0.05). Significant correlation was also found between radiologist burnout and past patient abuse, depression, lack of time for hobbies and recreation, less than 6 hours of sleep, and lack of personal work satisfaction (P < 0.05). A significant correlation was also found between the prevalence of burnout and radiologists who reported lack of proficiency in management skills (P < 0.05).
On a scale of 1 to 10 to measure levels of stress with 1 being the least and 10 being the maximum possible level of stress, 53% (205/383) of the radiologists reported a stress level of 7 to 10 [Figure 6]. Generally, those radiologists reporting burnout, depression, patient abuse, lack of personal work satisfaction, and improper workplace relationships had higher levels of stress. Among the causes of stress mentioned, the most common cause of stress among radiologists was Pre-Conception and Pre-natal Diagnostic Techniques Act (PCPNDT) related issues (53%; 203/383) and work overload (50%; 193/383). Inadequate income was also cited as a source of stress by 36% (139/383) respondents, with other causes of stress being administrative responsibilities (29%; 110/383) and improper workplace atmosphere and interpersonal relationships (27%; 103/383) [Table 4].
27% (105/383) of the radiologists surveyed felt lack of proficiency in their communication skills, and 41% (159/383) radiologists did not feel proficient in their management skills.
Despite the high prevalence of repetitive stress injuries, depression, and burnout among radiologists in India, a majority (58%; 223/383) did not actively exercise, meditate, or play sports [Table 3]. Radiologists who regularly either exercised or meditated or played sports had a statistically significant low prevalence of burnout as compared to those who did not (P = 0.01).
The incidence of musculoskeletal injuries among Indian radiologists reported in our study was much higher than those reported in western literature, with some injuries having almost double the incidence. The prevalence of musculoskeletal injuries among radiologists in our study versus another 2018 study from the United States in JACR is as follows – neck pain: 52% in our study versus 25% in the JACR study, low back pain: 45% in our study versus 32% in the JACR study, and 78% prevalence of at least one upper limb joint repetitive stress injury in our study versus 16% with an upper limb stress injury in the JACR study.
A significant correlation was found in our study between increasing USG workload and prevalence of neck pain, low back pain, and shoulder pain – a condition known as Transducer user syndrome. The incidence matched with those of studies on sonologists published elsewhere (65%–78%). Radiologists perform a lot of visual (convergence and accommodation) exercise during work, leading to ciliary muscle fatigue and sometimes ciliary muscle spasm. This leads to digital eye strain and CVS. Symptoms of CVS include headache, blurred vision, either dry eyes or excessive watering from eyes, eye strain, and diplopia.
Various studies indicate a prevalence of CVS ranging from 68% to 76% among computer professionals in India and other South Asian countries., Our study also showed a prevalence of 57% for eye strain among radiologists.
Ergonomics (from ergo: work and nomo: natural law) is the science of designing workplaces, products and systems so that they fit the people who use them. Use of ergonomic chairs, ergonomic workstations, and imparting ergonomic training have been reported to cause up to a 70%–80% of improvement in repetitive stress symptoms. In our study, radiologists in whose practices or departments ergonomic steps had been taken reported a statistically significant lower prevalence of neck pain (P = 0.014).
As per World Health Organization estimates, the lifetime prevalence of mental illnesses in the general population is an astonishingly high number of 18%–36%., Approximately 13%–20% of physicians are known to suffer from depression., In our study, 37% of radiologists reported feeling depressed at some point in their careers.
Burnout is increasingly being recognized as a serious mental health condition. It has been defined as a triad of emotional exhaustion, depersonalization, and reduced personal accomplishment., Residency in particular is known to be a period of one's career where one is prone to develop a psychiatric illness. As per previous studies, one-third of residents suffer from depression and up to 75% residents experience burnout., In our study, 40% of residents reported feeling depressed at some point in their radiology career, and 55% reported burnout. Importantly, a 1996 report in Lancet found burnout to be significantly higher in radiologists as compared to nonradiologists (which included surgeons, oncologists, and gastroenterologists), indicating the need to focus more on this issue within the radiology community.
Limitations of our study include the use of a voluntary subjective electronic survey to collect responses, with its inherent biases. Standardized questionnaires to test for burnout are available, the most commonly used being the Maslach Burnout Inventory. Our study does not test for burnout using Maslach Inventory questionnaire, and a subjective question regarding whether the responder has experienced burnout at any time during his or her career in radiology had been asked instead. Further more detailed studies on radiologists in India should be carried out using standardized psychological questionnaires to investigate this serious issue further.
Radiologists in India show a higher prevalence of repetitive stress injuries, chronic eye strain, and burnout as compared to their western counterparts. Indian radiologists are also faced with unique mental stressors with the most common being PCPNDT-related issues, and overall report less work satisfaction when compared with the West. Significant correlation exists between the presence of repetitive stress injuries and burnout among radiologists.
Schoenfeld A, Goverman J, Weiss DM, Meizner I. Transducer user syndrome: an occupational hazard of the ultrasonographer. Eur J Ultrasound 1999;10:41-5.
Benjamin JL, Meisinger QC. Ergonomics in the development and prevention of musculoskeletal injury in interventional radiologists. Tech Vasc Interv Radiol 2018;21:16-20.
Setness PA. Is it real or is it memorex? Discerning whether job-related stress or mental illness is causing physician impairment. Postgrad Med 2003;113:7-9.
Halpenny D, O'Driscoll D, Torreggiani WC. Ocular health among radiologists in the age of PACS: Is it time for our profession to open its eyes to this issue in light of existing European legislation? Br J Radiol 2012;85:e1309-11.
Mohan C. Ergonomics in radiology-Time to revisit. IJRI 2018;28:271-2.
Atwal SS, Prasad A, Deepak D, Agarwal K. Health issues among radiologists: Toll they pay to their profession. J Clin Diagn Res 2017;11:TM01-2.
Parikh JR, Bender C, Bluth E. Musculoskeletal injuries affecting radiologists according to the 2017 ACR human resources commission workforce survey. J Am Coll Radiol 2018;15:803-8.
Murphey S. Work related musculoskeletal disorders in sonography. J Diagnostic Med Sonogr 2017;33:354-69.
Ranasinghe P, Wathurapatha WS, Perera YS, Lamabadusuriya DA, Kulatunga S, Jayawardana N, et al. Computer vision syndrome among computer office workers in a developing country: An evaluation of prevalence and risk factors. BMC Res Notes 2016;9:150.
Thomas J, K V, Sankar N, Jacob GP. Prevalence of ocular symptoms among computer professionals in a university setting in South India. J Evol Med Dent Sci 2014;3:14777-86.
Boiselle PM, Levine D, Horwich PJ, Barbaras L, Siegal D, Shillue K, et al. Repetitive stress symptoms in radiology: Prevalence and response to ergonomic interventions. J Am Coll Radiol 2008;5:919-23.
Kessler RC, Aguilar-Gaxiola S, Alonso J, Chatterji S, Lee S, Ormel J, et al. The global burden of mental disorders: An update from the WHO World Mental Health (WMH) Surveys. Epidemiol Psichiatr Soc 2009;18:23-33.
Brown SD, Goske MJ, Johnson CM. Beyond substance abuse: Stress, burnout, and depression as causes of physician impairment and disruptive behavior. J Am Coll Radiol 2009;6:479-85.
Bender A, Farvolden P. Depression and the workplace: A progress report. Curr Psychiatry Rep 2008;10:73-9.
Center C, Davis M, Detre T, Ford DE, Hansbrough W, Hendin H, et al. Confronting depression and suicide in physicians: A consensus statement. JAMA 2003;289:3161-6.
Bianchi R, Schonfeld IS, Laurent E. Is it time to consider the "Burnout Syndrome" a distinct illness? Front Public Heal 2015;3:158.
Levey RE. Sources of stress for residents and recommendations for programs to assist them. Acad Med J Assoc Am Med Coll 2001;76:142-50.
Thomas NK. Resident burnout. JAMA 2004;292:2880-9.
Ramirez AJ, Graham J, Richards MA, Cull A, Gregory WM. Mental health of hospital consultants: The effects of stress and satisfaction at work. Lancet 1996;347:724-8.
Guenette JP, Smith SE. Burnout: Prevalence and associated factors among radiology residents in New England with comparison against United States resident physicians in other specialties. Am J Roentgenol 2017;209:136-41.
Graham J, Ramirez AJ, Field S, Richards MA. Job stress and satisfaction among clinical radiologists. Clin Radiol 2000;55:182-5.
Lim RCH, Pinto C. Work stress, satisfaction and burnout in New Zealand radiologists: Comparison of public hospital and private practice in New Zealand. J Med Imaging Radiat Oncol 2009;53:194-9.
Zafar HM, Lewis RS, Sunshine JH. Satisfaction of radiologists in the United States: A comparison between 2003 and 1995. Radiology 2007;244:223-31.
Maslach C, Jackson SE. The measurement of experienced burnout. J Organ Behav 1981;2:99-113.


So flattering! Chic high heels to catch the eye! Perfect for upgrading your denims. Shaped 'stack' heels. Pointed toe. Topstitched. In a gorgeous faux suede.
Upper 100% textile. Lining 100% polyurethane. Inner sole 100% polyurethane. Sole 100% synthetic.


Data analysis is the extraction of knowledge and insights from complex data. This course introduces the concepts, issues, and techniques of data analysis. Topics include data cleaning and preparation, feature selection, association rules, classification, clustering, evaluation and validation. Tools implemented in R and Python will be used to explore data sets using these techniques.
This course provides a broad introduction to machine learning and statistical pattern recognition including both supervised and unsupervised learning from a computational perspective. Topics include generative/discriminative learning, parametric/non-parametric learning, neural networks, support vector machines, clustering, dimensionality reduction, and kernel methods. Additional topics as time allows.
It's been a long time since I was in school, and even then my knowledge of AI was pretty eclectic. What belongs in an undergraduate AI course that is not covered by the above two courses?
I took both the AI and (basic) ML undergrad courses at Princeton and currently teach an AI elective at the HS level for some very bright students. I've seen some good online material from Berkeley (CS188) and MIT (6.034), and of course Stanford's ML lecture series is amazing. Russell & Norvig's textbook is basically the bible for these sorts of courses.
From what I've seen, AI undergraduate courses have changed a lot in the past decade to focus on ML, since much of the recent breakthroughs and publicity are in those areas. AI is a very broad and historically rich field beyond just Data Analysis or ML. I would say a comprehensive AI course would start from Alan Turing's classic (and often misused) Turing Test and ground itself in an ongoing definition of rationality as effective problem-solving.
Constraint-Satisfaction Problems (ARC-3, backtracking search) and/or Boolean SAT (DPLL, WalkSAT).
Some courses might go from these topics into Bayes Nets and/or Hidden Markov Models, though I think this leans math heavy since its all about probability.
Usually by this point AI courses have become more or less completely focused on ML, often starting with k-means clustering, basic classification/regression using perceptrons or decision trees, ensemble learning/boosting, reinforcement learning (usually in the framework of Markov Decision Processes), and maybe deeper neural networks.
EDIT: A follow-up; my course has now changed its third unit topic from CSPs/SAT to Markov Decision Processes (Value Iteration and Policy Iteration) and Reinforcement Learning. Its more exciting, leads into ML, and flows out the Adversarial Games nicely because of the shared concept of finding a "policy", especially if Expectimax is covered.
Not the answer you're looking for? Browse other questions tagged curriculum-design undergraduate artificial-intelligence or ask your own question.


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Buck Leerman
Leerman's best days may be behind him, but his nights are still top-notch.
Leerman is a balls-to-the-wall, take-no-shit throwback to the rugged days of early space-frontier exploration with an affection for Old Blue that goes beyond casual indulgence.
His appearance may not be much to talk about (and he might just have more hair on his knuckles than his dome), but he packs more debauchery into a 24-hour period than the average Bear does in a month.
Pushing 60, he's by far the oldest Bear City pilot – but certainly not the most taciturn. He's been grounded countless times for flying while intoxicated, yet somehow, is always able to make it back home in one piece - usually towing more ore than any other pilot.
His predilection for prostitutes is legendary and he will often use the comms channels to regale Bear City with his latest debaucherous undertakings - often involving two or more of the Pleasure Corps working women. His bombastic lasciviousness rub some people the wrong way, most notably the mystique filled Tess, who is often the subject of Leerman's x-rated anecdotal fantasies.
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Q: Javascript (jQuery) performance measurement and best practices (not load time) I'll say right off the bat that this question is NOT about load times; I know about YSlow, the Firebug profiler, and whatever best practices and tools googlage reveals about page component load times.
I am asking what good profiling tools or libraries or add-ons there are for measuring actual execution of Javascript (specifically jQuery), insofar as improving actual user experience goes.  For example, measuring the time from click to visible result on-screen, or helping to determine why a jQuery-based hover effect has slow responsiveness.
We are noticing that when the page/DOM grows relatively large (say, 70kb to 150kb worth of HTML, excluding external CSS, JS and images), and/or has very deep nesting (14-25 levels from <body> to deepest tag), jQuery events fire more slowly, or the whole JS user experience gets sluggish.
I also have googled and learned about best practices for selectors (e.g. selecting by id is much faster than selecting with classes), I will be implementing these practices.  However, once all jQuery is fully loaded, and all events hooked, we still need to improve the actual event firing and execution.
I have implemented some event delegation already, and I do get the sense that having fewer hooked DOM elements makes things slightly better, but the overall experience still needs improvement with large pages.  I should mention that, since the site is AJAX heavy (lots is loaded via AJAX as opposed to initial HTTP hit), we are making heavy use of livequery instead of the plain jQuery event hooks.  I should also mention that we are slightly more focused on IE(7+) performance, but also require good Firefox performance.
As I develop and make changes, I figure I need a way to measure speeds pre- and post-change, so I can have concrete numbers on whether or not a change improves anything.
Any tips, tools, libs, blog posts, URLs?

A: JSLitmus looks like something I might try out.

A: By using Firebug you can go into console tab, then click the firebug-icon in the upper left corner (called "Firebug options"). Then click on the option "Profile Javascript", after that just do what you want to measure, for example clicking on a javascript button, draging a Jquery dragable object or what ever you want that runs a bit of javascript code. When you have executed the javascript you want to measure go back to the same menu option and once again click "Profile javascript" (so that it gets unchecked). Now the console tab will be filled up with all the actions you previously ran aswell as the time it took to execute each method etc (and total time aswell).

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Q: bootstrab nav tab li items in a row i have those nav-tab items with bootstrap
| tab1  |  tab2  |  tab3  |

but when my website becomes responsive they to like this
| tab 1 | tab2 |
| tab 3 |

I have the following code

Recent Events

Drink Time

Food Time

Is there any way to always keep them in the same line and even if they become small there is no problem at all.


Through my paintings, I have met new friends, this culture, nature and landscapes, which expands my aesthetic view in a more broad way.
I work with memory and feelings that marks my being, living with myself in the "here and now". Hearing echos from my background culture that can be amplified in a new enviroment.
I approached China with all my love. With my eyes I have had beautiful deep visions of this reality, of its culture and society, of its economic production, that I filter with my personal aesthetic interpretation and research that I call " Rude beauty".
As my project was to paint about the complexity of human being, I have worked on this topic through my poethic language about the contemporary world, the "Rude Beauty". As common point of view for every place, I am often upset by simple scenes that collides with my eyes and my sensibility, inableing me to recognise beauty, a "Rude Beauty" in all imperfections of our times. A Kind of blue love song , kind of effort to rapresent the "Rude Beauty" of the working class in opposition with the capital global massmedia pseudo culture in which just the superficiality of fame and richness allows for beauty. That brings to the attention the forgotten face, a corner of the real life. The artist as witness of its historic time, as social protagonist, as cultural interpreter.
Looking for a painting quality, simpleness, to reach a rapresentation that can be told as honest.
Colors, mixed media, canvases, are my tools with synthesis of symbols, ideas, forms, related in our time in the most brode way to create a new art piece, which is a new image I gift to the world.
I want to paint what I'm living, refuge myself in Art to avoid death, I feed myself with colors and forms, I amplify my voice through image. Painting is an interior travel, with origin and arrive point in the world. Most than ever today, 2014 a.c., Paint is meaningfull, most than ever today Paint is itself and show to all others Visual-comunicative media of beeing still always the most human. I agree with Picasso about Style. There are artists that have a style that's always the same, maybe for there own individual characters, that brings them to express like this. But for me to have a style seems as I should repeat myself, my creative experience, although that the style in art is recently mostly interesting for art market. My creative process follows various streams, inspiration, to have a vision and try to represent it most directly as possible, through the best pictoric medium or to paint for the pleasure of painting, in colors world, in their abstraction. The serie of still life born from the reading of the writings of Picasso, in which he explains that the represented object doesn't count much, doesn't make a big difference, because what is important is how with painting that object become represented. Taking the still life as classic, humble, subject of the Art history, I will be able maybe one day to represent my epoch in a distinguish manner and then maybe through this that seems little things, several aesthetic perspective will reveal themselves.
The next Text was originally wrote in french and i leaved it as it is.
Ouvrez-moi les yeux et je vous direz ce que je vois! A travers les votre, ce principe émotif de toute communication non verbale…le langage séparé de l´inflation imaginaire, cette pure fonction du désir enthousiaste : la peinture est un outil de fluidité qui rend possible ces choses la !…Une mise en évidence du ciel intérieur, une captation sonore de la nuit que nous portons dans le regard, une mise en garde de l´ombre a travers la lumière et vice-versa…On l´aura beau répété mais il demeure toujours aussi vrai que le plus important n´est pas la chose regardée, mais le regard que l´on pose sur cette chose…Giulia Spugnoli regarde toujours dans les yeux de celui qui regarde !
Laissons voir ce qui se cache sous les apparences ! Mettons ces fameuses apparences sur plan bi dimensionnel : primo verticalement…Deuxio horizontalement…Tercio vice- versa !
Off : Un faste néo-baroque éclairé au néon, un univers symbolique typiquement Berlinois, profondément Florentin, avec ces capsules élliptiques que forment a maintes reprises les cafetières italiennes, des tournoiements Soufis d´objets triviaux et d´éléments translucides…, Iron and Wine, du métal fondu dans le vin, ce même vin qui tache les feuilles sèches de l´angoisse perpétuel…un joyeux bordel sensible a la corde de la spiritualité mais qui casse en mille morceaux les violons de l´émotion. Les figures ne sont que spectres, des archétypes de souffrances individuelles et changeantes qui se déplacent d´un visage à un autre, aussi vite que le bonheur… comme une constellation d´étoiles qui dessinent une croix de mort dans la fracture de la nuit…Un constat désespérant de l´humanité qui peine à combattre son propre germe autodestructeur : les fantômes : ces âmes aussi solitaires que ces objets fatalement inertes ! Une certaine nostalgie ?
On/off : Ainsi, j´ai jeté un coup d´œil… je me suis arrêté, car il me fut impossible de faire autrement… j´ai pris le temps de voir…depuis je regarde…Regarder c´est payer sa dette envers tout ce mérite de l´être vraiment et faire la promesse de toujours laisser filtrer, quelque soit les circonstances, la beauté de la vie… La Grande Belleza ! dit-on chez mon amie.
Das Stillleben wie eine formale ästhetische Erforschung, welches die von menschlicher Arbeit geschaffenen Werke darstellt, die ihrerseits auf eine Abwesenheit-Anwesenheit weisen, auf einen fantasmagorischen, nunmehr verlorenen Gebrauchswert. Wie eine Reflexion über die Ware aus jüngerer Zeit, die stets wichtiger ist als die Menschen, die sie erzeugen. Wie eine Kritik am sozialen und kulturellen Stillstand. Die Objekte sind eine statische, tote Komposition, verweisen jedoch auf ein kreatives Leben, das sie hervorgebracht hat und dessen Fantasmen, Schatten letztendlich wie ein Echo des Lebens wirken.
Shendi Veli, Expert in European cultural politics, Interview with Giulia Spugnoli , as a research in the field of Cultural policies, politics and governance.
Giulia is a painter, performer and art-therapist, born in Florence and currently living and working in Berlin.
Despite her everlasting interest in figurative only at the university Giulia start investing her time in getting an artistic background: she studied and graduated at the Academy of the Fine Arts in Florence with the maximum of the votes.
According to her opinion this was an important step not only for acquiring some technical knowledge around her passion but also for constituting the step that made her pass from a simple hobby to the idea of "working through art".
Her work is fundamentally abstract-figurative painting. Her main topics are either related to socio-political issues or entirely abstract representations. She uses different techniques of drawing (oil, watercolours, collage, recycle etc.) and she normally works in her room.
Giulia is now living in Berlin, in the neighbourhood of Neukolln. She decided to move in Berlin five years ago because she evaluated that here is easier, compared to Florence and Italy in general, to find spaces to produce art and people interested to make projects and connections.
She dedicates almost 20-25 hours a week to painting, moreover she define herself as a quick painter, so her artistic production is quite copious.
About her earnings they are very intermittent: "when you sell a painting you can earn a lot of money in just one day, when you don't you are without income for long periods".
Considering the long term Giulia says that living as a painter , is hard, that's why she started working with projects of art-therapy for children and through public funded programs she was able to use her art in a more social and also profitable way.
The strategy that she uses is to write a project based on art activities, and targeted to children mostly with background migration and apply for funds to the Kulturamt or other neighbourhood based public agency Quartiersmanagement (many of them works on the cultural development of the area), the maximum of funds that she recived for a project is 2500 € . So this work can be complementary, but still insufficient to guarantee her life on the long term.
The other way in which Giulia organizes her work, is the more typical arrangement of exhibitions. She said that in Berlin it is not easy for an emerging artist to find space in the official art galleries. As well she often find galleries very elitist places where arts and people never get contaminated with each other. However many cultural spaces (cafè, bars, associations) are available at hosting art exhibitions in Berlin. She tries to find spaces that would host her work for free and in collaboration with the place she organizes the promotion and the logistic of the event. This is a way to show her work and become more known in the sector. Normally places don't ask for a percentage on her peddling, but sometimes that may happen also.
The kind of institutions or agencies that have been important to her career are basically private associations and cultural public institutions. Between the first one she named the Cafè Tasso where she organized her exhibition and that was important for finding more contact in the field. Among the institutions, one of her most important work-project was with the Alice Salomon Hochschule.
The way Giulia gives back to the community through her work is very important for her. Firstly through her art – therapy projects she got to know all the children of her neighbourhood and to create a feedback with them and their family about which are the needs of the community.
Secondly Giulia went in the occupied school of Olauer Strasse and creates with some of her artists collegues and friends a collective mural painting with refugees. She says that in this places there are no organized activities so many people have nothing to do the whole day and do not have contacts with people outside the school, and she thought that art it is a good way to create for them connection in the city they moved in.
Giulia has also been involved in a feminist art project called Femme Fraktale, all the earnings of this project were devolved to Non profit Associations that supports womens based in Berlin, as Women in Exile.
The cultural policies that made a difference in her career are those who allow many associations to spread, get funds and to get involved in arts. Moreover, has been for her important the public funding of social projects based on artistic activities.
What she thinks that is missing in the cultural policies is a particular attention to promote art made by women.
Giulia says that despite the appearance, in the most famous museums the work of female artist is still under the 10% of the total art hosted. She whishes that more gender-oriented cultural policies will compensate many years (centuries) in which cultural production has been considered only a male activity, she thinks that since culture is an important part of society a better equilibrium between genders would bring benefits to the whole and would be a way to challenge the "main stream culture".
The program www.goldrausch-kuenstlerinnen.de/en/ is according to her a very interesting cultural program, but still too limited in the number of artists it promotes.
Se i quadri potessero parlare cosa direbbero?
Giulia ci accompagna, attraverso le sue pitture, dentro storie e luoghi inattesi, in un lasciarsi trasportare dall' immaginazione a guardare in modo diverso e profondo.
Ogni quadro una storia nascosta da un dettaglio, un'atmosfera, un colore: ogni storia un mondo.
Si avverte, nei suoi dipinti, un richiamo a certe avanguardie figurative dell' inizio del novecento, a quella ricerca artistica fatta di spazio- luce-movimento, determinate nel fornire al teatro di Bertolt Brecht e al suo allestimento scenico nuovi moduli espressivi.
I "Due amici che parlano al Sandman", i commensali di "Underground Szene in Berlin" sono gli attori del teatro espressionistico tedesco del 1° dopoguerra o del Cabaret Voltaire di Zurigo, dove i dadaisti presentavano i loro provocatori spettacoli?
"Warshauerstr. Berlin"- "Boxhagenerstr. " sono le scenografie dove si muovono quei personaggi che, insieme ai commensali del "Progetto collettivo contro la guerra", alla donna di "Disoccupazione e isolamento sociale", anche se attori, appartengono a quel teatro civile che Brecht ha pensato essere fatto per " decostruire le illusioni del pubblico".
Le opere di Giulia non si offrono a una facile lettura, ma tutte trasmettono immediatamente vitalità, passione e generosità. Tratti distintivi della sua personalità artistica.
Del resto ci avverte ancora Brecht, in un suo provocatorio aforisma, riferibile anche alle arti figurative, che "Se la gente vuol vedere solo cose che può capire, non dovrebbe andare a teatro; dovrebbe andare in bagno".