text:
\section{}
\section{Discrete state dynamics}
\label{a:stateDynamics}

Consider the linear time-invariant state dynamics \eqref{eq:stateDynamics}: $${\partial x(t,w)/\partial t = ax(t,w)+bu_s+cp\opt(t,w)},$$ ${t\in\set{T}\sys(s)}$, ${s\in\set{N}^{\ts{S}}}$, with ${x_0:=x(0)\in [x_0^{\min},x_0^{\max}]}$ and $p\opt(t,w)$ according to \eqref{eq:defTargetPower}. Standard integration techniques can be used to express the state ${x_s(w):=x(sT^{\ts{S}},w)}$ as
\begin{align}
	x_{s}(w) =&\, fx_{s-1}(w)+gu_s+ h_1p^{\ts{ref}}_{s-1}(\tilde{w})+h_2p^{\ts{ref}}_{s}(\tilde{w}) + \gamma v_s,
	\label{eq:discreteDynamics}
\end{align}
where we have used
\begin{align}
	f &:= e^{aT^{\ts{S}}},\hspace*{21mm} g :=b\int_0^{T^{\ts{S}}}e^{a\tau}d\tau,\\
	h_1 &:=\frac{c}{T^{\ts{S}}}\int_0^{T^{\ts{S}}}e^{a\tau}\tau d\tau,\ \
	h_2 :=\frac{c}{T^{\ts{S}}}\int_0^{T^{\ts{S}}}e^{a\tau}(T^{\ts{S}}-\tau) d\tau,\\
	v_s &:=c\int_0^{T^{\ts{S}}}e^{a\tau} w(sT^{\ts{S}}-\tau)d\tau.
\end{align}
Note that the power reference is determined by the averaged activation $\tilde{w}$, whereas the term $v_s$ depends on the unfiltered activation signal $w(t)$. Because $\tilde{w}$ and $v$ are computed from $w$, we write $x(w)$ only.
The discrete-time state evolution can be written in vector form as
\begin{equation}
	x(w) = Fx_0+Gu+Hp^{\ts{ref}}(\tilde{w})+\gamma K v,
	\label{eq:discreteDynLD}
\end{equation}
with ${F\in\R{N^{\ts{S}}}}$, ${G,K\in\R{N^{\ts{S}}\times N^{\ts{S}}}}$, and ${H\in\R{N^{\ts{S}}\times (N^{\ts{S}}+1)}}$.

By plugging \eqref{eq:defPowerRef} into \eqref{eq:discreteDynLD}, we have
\begin{equation}
	x_s(w) = \sum\limits_{i=1}^{s}\{ Q_{s,i}\tilde{w}_i + \gamma K_{s,i} v_i\} + \kappa_s,

\end{equation}
where we have used
\begin{align}
	Q &:= HR(MQ^{\ts{DA}} A^{\ts{DA}} + Q^{\ts{ID}} A^{\ts{ID}})\\
	\kappa &:= Fx_0+Gu+HR(Mq^{\ts{DA}} +q^{\ts{ID}})
\end{align}
for brevity. Using the definitions of $\tilde{w}$ and $v$, we can write
\begin{equation}
	x_s(w) = \sum\limits_{i=1}^{s}\left\{\int_0^{T^{\ts{S}}}\left(\frac{Q_{s,i}}{T^{\ts{S}}}  + c\gamma e^{a\tau} K_{s,i} \right)w(\tau)d\tau\right\} + \kappa_s.
	\label{eq:xsSum}
\end{equation}
Note that in case of self-dissipation, i.e.\,, ${a<0}$, the integrand is time-dependent, making it difficult to evaluate the integral analytically. To circumvent this, we substitute $e^{a\tau}$ by the constant $e^{a\hat{\tau}}$ which is chosen such that for all ${\tau\in [0,T^{\ts{S}}]}$
\begin{equation}
	\abs{e^{a\tau}-e^{a\hat{\tau}}}\leq \epsilon:=(1-e^{a T^{\ts{S}}})/2.
\end{equation}
Because
\begin{multline}
	\max\limits_{w\in\mathcal{W}}\int_0^{T^{\ts{S}}}c\gamma K_{s,i}(e^{a\tau}-e^{a\hat{\tau}})w(\tau)d\tau\\
	= \int_0^{T^{\ts{S}}}c\gamma K_{s,i}\abs{e^{a\tau}-e^{a\hat{\tau}}} d\tau\leq c\gamma\epsilon T^{\ts{S}} K_{s,i},
\end{multline}
the integral in \eqref{eq:xsSum} is bounded from above:
\begin{align}
& \int_0^{T^{\ts{S}}}\left(\frac{Q_{s,i}}{T^{\ts{S}}}  + c\gamma e^{a\tau} K_{s,i} \right)w(\tau)d\tau \nonumber\\
 \leq& \int_0^{T^{\ts{S}}}\left(\frac{Q_{s,i}}{T^{\ts{S}}}  + c\gamma e^{a\hat{\tau}} K_{s,i} \right)w(\tau)d\tau + c\gamma\epsilon T^{\ts{S}} K_{s,i}\nonumber\\
  \leq&\, \abs{Q_{s,i} + c\gamma e^{a\hat{\tau}}T^{\ts{S}} K_{s,i}} + c\gamma\epsilon T^{\ts{S}} K_{s,i}\ \ \forall w\in\set{W}^{N^{\ts{C}}}.
  \label{eq:aux3}
\end{align}
Bounds on $x_s(w)$ can be found by substituting the summands in \eqref{eq:xsSum} with \eqref{eq:aux3}: For all ${w\in\set{W}^{N^{\ts{C}}}}$ and ${s\in\set{N}^{\ts{S}}}$
\begin{align}
\begin{split}
	x_s(w) &\leq \abs{Q_{s} + c\gamma e^{a\hat{\tau}}T^{\ts{S}} K_{s}}\mathbf{1} + c\gamma\epsilon T^{\ts{S}} K_{s}\mathbf{1} + \kappa_s, \\
	x_s(w) &\geq -\abs{Q_{s} + c\gamma e^{a\hat{\tau}}T^{\ts{S}} K_{s}}\mathbf{1} - c\gamma\epsilon T^{\ts{S}} K_{s}\mathbf{1} + \kappa_s. \label{eq:defXsMaxMin}
\end{split}
\end{align}
Inequalities \eqref{eq:defXsMaxMin} are the robust counterparts of \eqref{eq:xsConstr}.

\section{Conclusion}
\label{s:conclusion}
The provision of accurate and reliable SFR services is important to compensate for imbalances in the power grid. We showed how to implement energy schedules in a way that is compliant with ENTSO-E regulations and allows one to explicitly consider limitations of the power-ramp rates of the system providing SFR. Based on piece-wise affine and continuous power trajectories, we derived a reformulation of the energy and reserve bidding problem as a robust linear program that accurately models the different market timescales and lead times. Our approach is versatile and applicable in various market settings. Computations show how adjusting the power reference via day-ahead and intra-day energy markets significantly increases the amount of available SFR capacity.
\section{Description of flexibility}
\label{s:flexDescription}
The decisions a BRP makes on the different energy and reserve markets are restricted by the requirement that the target power trajectory \eqref{eq:defTargetPower} must be feasible for all realizations of the unknown activation signal and must satisfy all physical constraints of the system.
We consider a system with power, power ramp-rate, and state (e.g.\, energy) constraints.

\subsection{Power constraints}
The power the system can draw from or feed into the power grid is limited. For all ${t\in\set{T}\sys(s)}$, ${s\in\set{N}^{\ts{S}}}$, we require that
\begin{equation}
	\un{p}_s \leq p\opt(t,w)\leq \bar{p}_s,\ \forall w\in\set{W}^{N^{\ts{C}}},
	\label{eq:powerConstrCt}
\end{equation}
where ${\un{p},\bar{p}\in\R{N^{\ts{S}}}}$ denote the piece-wise constant bounds on power. The constraints above are satisfied if and only if for all ${\tilde{w} \in\set{W}^{N^{\ts{S}}}}$ it holds that
\begin{align}
\begin{split}
p^{\ts{ref}}_s(\tilde{w}) + \gamma \leq
\begin{cases}
	\bar{p}_1, & s=0,\\
	\min\{\bar{p}_{s},\,\bar{p}_{s+1}\}, & s=1,\dots,N^{\ts{S}}-1,\\
	\bar{p}_{N^{\ts{S}}}, & s=N^{\ts{S}},\\
\end{cases}\\
p^{\ts{ref}}_s(\tilde{w}) - \gamma \geq
\begin{cases}
	\un{p}_1, & s=0,\\
	\max\{\un{p}_{s},\,\un{p}_{s+1}\}, & s=1,\dots,N^{\ts{S}}-1,\\
	\un{p}_{N^{\ts{S}}}, & s=N^{\ts{S}}.\\
\end{cases}
\end{split}
\end{align}

The robust counterparts of the above constraints are derived according to \cite{Bertsimas2011} by computing the maximum and minimum of $p^{\ts{ref}}(\tilde{w})$ with regard to $\tilde{w}$ according to \eqref{eq:pRefDepW}. An analytic solution exists because each element of $\tilde{w}$ is restricted to $\set{W}$: the power constraints \eqref{eq:powerConstrCt} are satisfied if and only if for all ${\tilde{w} \in\set{W}^{N^{\ts{S}}}}$
\begin{align}
\begin{split}
	p^{\ts{ref}}(\tilde{w}) &\leq \abs{R(MQ^{\ts{DA}} A^{\ts{DA}} + Q^{\ts{ID}} A^{\ts{ID}})}\mathbf{1}+R(Mq^{\ts{DA}}+q^{\ts{ID}}),\\
	p^{\ts{ref}}(\tilde{w}) &\geq -\abs{R(MQ^{\ts{DA}} A^{\ts{DA}} + Q^{\ts{ID}} A^{\ts{ID}})}\mathbf{1}+R(Mq^{\ts{DA}}+q^{\ts{ID}}),
\end{split}
	\label{eq:powerConstrRobust}
\end{align}
where $\mathbf{1}$ denotes a column vector of ones of appropriate size.

\subsection{Power ramp-rate constraints}
\label{ss:rampConstr}
Limits on the rate at which power can vary over time play an important role, in particular for providing ancillary services with high accuracy. For all ${t\in\set{T}\sys(s)}$, ${s\in\set{N}^{\ts{S}}}$, we require that the rate of change of the target power be bounded, i.e.\,,
\begin{equation}
	\un{r}_s \leq \partial p\opt(t,w)/\partial t \leq \bar{r}_s,\ \forall w\in\set{W}^{N^{\ts{C}}},
	\label{eq:rampConstrCt}
\end{equation}
where ${\un{r},\bar{r}\in\R{N^{\ts{S}}}}$ denote the piece-wise constant power ramp-rate limits. The above inequalities are satisfied if and only if for all ${\tilde{w} \in\set{W}^{N^{\ts{S}}}}$ it holds that
\begin{equation}
\begin{split}
	(p^{\ts{ref}}_{s}(\tilde{w})-p^{\ts{ref}}_{s-1}(\tilde{w}))/T^{\ts{S}}+2\gamma/T^{\ts{C}} &\leq \bar{r}_s,\\
	(p^{\ts{ref}}_{s}(\tilde{w})-p^{\ts{ref}}_{s-1}(\tilde{w}))/T^{\ts{S}}-2\gamma/T^{\ts{C}} &\geq \un{r}_s,\ s\in\set{N}^{\ts{S}}.
	\label{eq:rampConstr}
\end{split}
\end{equation}
The above inequalities illustrate that ramp-rate constraints limit the sum of the rate of change of the power reference and the rate of change of activated regulation power. They are equivalent to
\begin{align}
\begin{split}
&\hspace*{4mm}\abs{(R_{s+1}-R_{s})(MQ^{\ts{DA}} A^{\ts{DA}} + Q^{\ts{ID}} A^{\ts{ID}})}\mathbf{1}\\
&\hspace*{2mm}+(R_{s+1}-R_s)(Mq^{\ts{DA}}+q^{\ts{ID}})+2\gamma T^{\ts{S}}/T^{\ts{C}} \leq \bar{r}_s T^{\ts{S}},\\
&-\abs{(R_{s+1}-R_{s})(MQ^{\ts{DA}} A^{\ts{DA}} + Q^{\ts{ID}} A^{\ts{ID}})}\mathbf{1}\\
&\hspace*{2mm}+(R_{s+1}-R_{s})(Mq^{\ts{DA}}+q^{\ts{ID}})-2\gamma T^{\ts{S}}/T^{\ts{C}} \geq \un{r}_s T^{\ts{S}},
\end{split}
\label{eq:rampConstrRobust}
\end{align}
with ${s\in\set{N}^{\ts{S}}}$. The case of maximum negative ramp rates is illustrated in Fig.\,\ref{fig:ramps}, where SFR activation changes from maximum positive activation (${w_{l-1}=1}$) to maximum negative activation (${w_{l}=-1}$) within one control time step $T^{\ts{C}}$. In this case, maximum negative ramp rates are required during the time interval $[(l-1)T^{\ts{C}},lT^{\ts{C}}]$.

\begin{figure}[tb]
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\caption{The target power trajectory $p\opt(t)$ (solid line) requires maximum negative ramp rates if ${w_{l-1}=-w_{l}=1}$.}
\label{fig:ramps}
\end{figure}

\subsection{State constraints}
Many types of flexible energy resources owe their flexibility to an energy buffer with limited capacity. Examples of such systems are pumped hydro-power plants, batteries, heating and cooling systems, capacitors, and flywheels. Here we consider systems whose energy buffer level ${x(t)\in\R{}}$ is governed by the linear time-invariant dynamics
\begin{equation}
	\partial x(t,w)/\partial t=ax(t,w)+bu_s+cp\opt(t,w),
	\label{eq:stateDynamics}
\end{equation}
with ${t\in\set{T}\sys(s)}$, ${s\in\set{N}^{\ts{S}}}$. Exogenous (uncontrollable) inputs, such as the weather conditions in the case of heating systems, or trips in the case of electric-vehicle batteries, are summarized by ${u:=[u_1,\dots,u_{N^{\ts{S}}}]\trans}$ and are assumed to be known. Because the bidding problem has to be solved ahead of time, the initial state $x(0)$ is not known precisely, but is known to lie in the interval ${[x_0^{\min},x_0^{\max}]}$. The dynamics \eqref{eq:stateDynamics} are characterized by the parameters ${\{a,b,c\}}$, where ${a\leq 0}$ can be interpreted as the self-dissipation rate of the energy buffer, and $b,c$ determine the conversion efficiencies of exogenous inputs and electric energy into buffered energy, respectively. Our sign convention implies ${c\geq 0}$. The behavior of an ideal battery, for instance, can be modeled by setting ${a=b=0}$ and ${c=1}$.

We consider the piece-wise constant state constraints
\begin{equation}
	\un{x}_{s}\leq x(t,w) \leq\bar{x}_{s},\ \forall w\in\set{W}^{N^{\ts{C}}},
	\label{eq:stateConstrCt}
\end{equation}
for all ${t\in\set{T}\sys(s)}$, ${s=\set{N}^{\ts{S}}}$, where ${\un{x},\bar{x}\in\R{N^{\ts{S}}}}$ denote the physical limits of the system state. The initial state is assumed to be feasible, i.e.\,, ${\un{x}_1\leq x^{\min}_0\leq x^{\max}_0\leq\bar{x}_1}$.

The exact evolution of the state \eqref{eq:stateDynamics} is unknown because it depends on the uncertain activation ${w\in\set{W}^{N^{\ts{C}}}}$.
For our case of piece-wise affine power references, the state $x(t,w)$ can evolve non-monotonically during a discretization interval $\set{T}\sys(s)$.
To ensure that \eqref{eq:stateConstrCt} holds would require bounding the extreme values of $x(t,w)$ for ${t\in\set{T}\sys(s)}$. This, however, would result in non-convex constraints. Instead, we approximate the right-hand side of \eqref{eq:stateDynamics} by affine functions of time and define for every interval $\set{T}\sys(s)$, ${s\in\set{N}^{\ts{S}}}$, the auxiliary dynamics
\begin{align}
	\dot{y}\ind{s}(\tau,\tilde{w}) :=&\,a\bar{x}_s+bu_s+c(p^{\ts{ref}}_{s-1}(\tilde{w})-\gamma)\nonumber\\
	&+[\min\{0,a\bar{x}_s+bu_s+c(p^{\ts{ref}}_{s}(\tilde{w})-\gamma)\}\label{eq:defAuxDyn1}\\
	&-(a\bar{x}_s+bu_s+c(p^{\ts{ref}}_{s-1}(\tilde{w})-\gamma))] \tau/T^{\ts{S}},\nonumber\\
	\dot{z}\ind{s}(\tau,\tilde{w}) :=&\,a\un{x}_s+bu_s+c(p^{\ts{ref}}_{s-1}(\tilde{w})+\gamma)\nonumber\\
	&+[\max\{0,a\un{x}_s+bu_s+c(p^{\ts{ref}}_{s}(\tilde{w})+\gamma)\}\label{eq:defAuxDyn2}\\
	&-(a\un{x}_s+bu_s+c(p^{\ts{ref}}_{s-1}(\tilde{w})+\gamma))]\tau/T^{\ts{S}},\nonumber
\end{align}
with ${\tau\in [0,T^{\ts{S}}]}$, ${y\ind{1}(0)=x_0^{\min}}$, ${z\ind{1}(0)=x_0^{\max}}$, and ${y\ind{s}(0,\tilde{w})=z\ind{s}(0,\tilde{w})=x((s-1)T^{\ts{S}},\tilde{w})}$. For brevity we use $\dot{y}\ind{s}(\tau,\tilde{w})$ to denote $\partial y\ind{s}(\tau,\tilde{w})/\partial\tau$

\begin{prop}
	If for all ${s\in\set{N}^{\ts{S}}}$ and ${\tilde{w}\in\set{W}^{N^{\ts{S}}}}$
	\begin{equation}
		y\ind{s}(\tau,\tilde{w}) \geq \un{x}_s\ts{ and }z\ind{s}(\tau,\tilde{w}) \leq \bar{x}_s	,\,\tau\in [0,T^{\ts{S}}],
		\label{eq:yzConstrCt}
	\end{equation}
	then the state constraints \eqref{eq:stateConstrCt} are satisfied for all ${t\in\set{T}\sys(s)}$.
\end{prop}
\begin{proof}
	Consider the time interval $\set{T}\sys(s)$ and divide it into ${\kappa\in\set{N}^+}$ subintervals, each of duration ${\delta:=T^{\ts{S}}/\kappa}$. For ${\tau=0}$, the constraints \eqref{eq:yzConstrCt} are equivalent to ${\un{x}_s\leq x((s-1)T^{\ts{S}},w)\leq \bar{x}_s}$, ${\forall\tilde{w}\in\set{W}^{N^{\ts{S}}}}$. By definitions \eqref{eq:defAuxDyn1}--\eqref{eq:defAuxDyn2}, it follows that ${\dot{y}\ind{s}(0,\tilde{w})\leq \dot{x}((s-1)T^{\ts{S}},w)\leq \dot{z}\ind{s}(0,\tilde{w}),\,\forall \tilde{w}\in\set{W}^{N^{\ts{S}}}}$.
Approximate
\begin{align}
	x((s-1)T^{\ts{S}}+\delta,w) &\approx x((s-1)T^{\ts{S}},w) + \delta\dot{x}((s-1)T^{\ts{S}},w),\\
	y\ind{s}(\delta,\tilde{w}) &\approx x((s-1)T^{\ts{S}},w)+\delta\dot{y}\ind{s}(0,\tilde{w}),\\
	z\ind{s}(\delta,\tilde{w}) &\approx x((s-1)T^{\ts{S}},w)+\delta\dot{z}\ind{s}(0,\tilde{w}),
\end{align}
and note that ${y\ind{s}(\delta,\tilde{w})\leq x((s-1)T^{\ts{S}}+\delta,w)\leq z\ind{s}(\delta,\tilde{w})}$, which, via \eqref{eq:yzConstrCt}, implies that ${\un{x}_s\leq x((s-1)T^{\ts{S}}+\delta,w)\leq\bar{x}_s}$. The same line of arguments holds for all subsequent subintervals. Consequently, in the limit case ${\delta\rightarrow 0}$, we have that ${\un{x}_s\leq x(t,w) \leq\bar{x}_s}$ ${\forall t\in\set{T}\sys(s)}$, ${s\in\set{N}^{\ts{S}}}$, ${w\in\set{W}^{N^{\ts{C}}}}$.
\end{proof}

Note that $\dot{y}\ind{s}(\tau,\tilde{w}),\,\dot{z}\ind{s}(\tau,\tilde{w})$ are affine functions of $\tau$ that can be integrated explicitly. In addition, the $\min\{\cdot\}$ and the $\max\{\cdot\}$ in the definitions \eqref{eq:defAuxDyn1}--\eqref{eq:defAuxDyn2} ensure that the minimum value of $y\ind{s}(\tau,\tilde{w})$ and the maximum value of $z\ind{s}(\tau,\tilde{w})$ over ${\tau\in [0,T^{\ts{S}}]}$ are reached at the edges of the interval, i.e.\,, at ${\tau\in\{0,T^{\ts{S}}\}}$. Thus, \eqref{eq:yzConstrCt} holds if for all ${s\in\set{N}^{\ts{S}}}$ and ${\tilde{w}\in\set{W}^{N^{\ts{S}}}}$
\begin{equation}
		y\ind{s}(\tau,\tilde{w})\geq\un{x}_s,\, z\ind{s}(\tau,\tilde{w}) \leq \bar{x}_s,\,\tau\in \{0,T^{\ts{S}}\},
\end{equation}
which is equivalent to requiring that for all ${s\in\set{N}^{\ts{S}}}$,  ${w\in\set{W}^{N^{\ts{C}}}}$,  and ${\tilde{w}\in\set{W}^{N^{\ts{S}}}}$
\begin{align}
	\un{x}_s\leq x((s-1)T^{\ts{S}},w) \leq \bar{x}_s,\label{eq:xsConstr}\\
	y\ind{s}(T^{\ts{S}},\tilde{w})\geq \un{x}_s,\label{eq:ysConstr}\\
	z\ind{s}(T^{\ts{S}},\tilde{w})\leq \bar{x}_s,\label{eq:zsConstr}
\end{align}
Thus, \eqref{eq:xsConstr}-\eqref{eq:zsConstr} are sufficient conditions for the state constraints \eqref{eq:stateConstrCt}.
The robust counterparts of \eqref{eq:xsConstr} are derived in Appendix \ref{a:stateDynamics}. The robust counterparts of \eqref{eq:ysConstr} and \eqref{eq:zsConstr} can be found similarly.

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\caption{Illustration of ${\dot{y}\ind{s}(\tau,\tilde{w})\leq\dot{x}(t)\leq\dot{z}\ind{s}(\tau,\tilde{w})}$ for ${t\in\set{T}\sys(s)}$, $\tau\in [0,T^{\ts{S}}]$.}
\label{fig:stateApprox}
\end{figure}

\subsection{Description of flexibility}
The power, ramp-rate, and state constraints \eqref{eq:powerConstrCt}, \eqref{eq:rampConstrCt}, and \eqref{eq:stateConstrCt}, respectively, are satisfied if the offered SFR capacity $\gamma$ together with the energy trading policies ${(Q^{\ts{DA}},q^{\ts{DA}})}$, ${(Q^{\ts{ID}},q^{\ts{ID}})}$ meet the constraints \eqref{eq:powerConstrRobust}, \eqref{eq:rampConstrRobust}, and the robust counterparts of \eqref{eq:xsConstr}-\eqref{eq:zsConstr}. All these constraints are linear in the decision variables ${\zeta:=\{Q^{\ts{DA}},q^{\ts{DA}},Q^{\ts{ID}},q^{\ts{ID}},\gamma\}}$ and thus define a convex polytope ${\set{P}(\Phi)}$ parameterized by the system constraints parameters ${\Phi:=\{\un{p},\bar{p},\un{r},\bar{r},\un{x},\bar{x},x_0^{\min},x_0^{\max}\}}$. This polytope can serve as a concise and convenient description of the flexibility the system offers with regard to trading energy in the day-ahead and intra-day markets and offering regulation power on the SFR market.

\section{Introduction}
\label{s:introduction}

\IEEEPARstart{B}{alance} Responsible Parties (BRPs) act as intermediaries between electric energy resources, electricity markets, and Independent System Operators (ISOs). BRPs are legal entities representing a single or a group of electric energy systems. They are responsible for trading electric energy on energy markets to satisfy the needs of their balance group and to adjust energy schedules to match actual electricity production and/or consumption as accurately as possible.
If the balance group of a BRP comprises systems whose electric energy generation and/or consumption is flexible, the BRP can offer this flexibility to the ISO in the form of ancillary services that are traded on dedicated flexibility markets \cite{ENTSO-E2009}.
We consider a BRP of a single flexible system that wishes to maximize the economic profit made from trading energy on day-ahead and intra-day energy markets and offering Secondary Frequency Regulation (SFR) services to the ISO. We refer to the problem of making optimal decisions on energy and reserve markets as the \emph{energy and reserve bidding problem}.

Characterizing and exploiting the energetic flexibility of various types of electric energy resources has recently attracted growing interest because the need for flexibility in power grids is expected to increase as less predictable renewable energy resources, such as wind and solar, are connected to the power grid. In particular, the provision of SFR has been studied for heating, ventilation, and air-conditioning systems \cite{Vrettos2014a,Qureshi2016,Hao2015a}, plug-in electric vehicles \cite{Vagropoulos2013,Liu2014}, and systems behaving like a generalized battery \cite{Hao2014b,Sanandaji2013}, among others.

The energy and reserve bidding problem faced by the BRP is to decide on the amount of SFR to offer on the reserve market and to optimally trade on energy markets. These decisions are mutually dependent and constitute a trade-off problem between the revenues from offering reserves and the costs of procuring energy. The problem has recently been formulated as a stochastic optimization problem \cite{Zhang2015a,Lymperopoulos2015} and as a robust optimization problem \cite{Vrettos2014a,Vrettos2016b,Gorecki2017,Warrington2012,Warrington2013}. The robust formulations rely on the concept of affinely adjustable robust control (AARC) policies, which have first been applied to the regulation power context in \cite{Warrington2012,Warrington2013,Bertsimas2013,Zhang2014a}, building on the initial work in \cite{BenTal2004} and the references therein.

The provision of SFR services requires the tracking of an activation signal broadcast by the ISO on the timescale of a few seconds. The ability of a system to accurately follow such signals is limited by the physical properties of the system, in particular by power, power-ramp rate, and energy constraints. While power and energy constraints are considered by most AARC approaches, all works mentioned above neglect power ramp-rate constraints. However, the crucial importance of ramping constraints has been recognized in the context of grid balancing \cite{Borsche2014} and unit commitment \cite{Morales-Espana2014,Parvania2016}, and has lead to the adoption of new performance-based schemes for the remuneration of SFR provision by several ISOs \cite{FERC2011}. But the methods developed in \cite{Borsche2014,Morales-Espana2014,Parvania2016} cannot be directly applied to the energy and reserve bidding problem considered here.

In our formulation of the bidding problem, we adopt the AARC formulation used in \cite{Vrettos2014a,Vrettos2016b,Gorecki2017,Warrington2012,Warrington2013}, and make two main contributions.
First, in contrast to the aforementioned works, we explicitly incorporate constraints on power-ramp rates in our modeling framework to guarantee the feasibility and accurate provision of SFR. Instead of the commonly used piece-wise constant power schedules, we base our formulation on piece-wise affine and continuous power trajectories, which makes it possible to accurately model power-ramp rates on arbitrary timescales. We then derive a reformulation of the AARC optimization problem which is necessary to guarantee the feasibility of the piece-wise affine solutions.
Second, in our problem formulation we include all the different timescales and market lead times involved in the bidding problem. This allows us to study how different energy and reserve market parameters affect the amount of SFR capacity a system can provide.
These two contributions make our approach versatile and adaptable, and pave the way for applying the energy and reserve bidding problem in real-world settings, where accurate modeling of system constraints and market specifications is indispensable.

The energy and reserve bidding problem of a BRP is introduced in Sec.\,\ref{s:problemDescription}. In Sec.\,\ref{s:reqsPowerRef}, we illustrate how piece-wise affine power schedules can account for finite ramp rates. Section \ref{s:flexDescription} provides a robust reformulation of the the energetic flexibility of an energy resource. It is used in Sec.\,\ref{s:scheduling} to formulate and solve the energy and reserve bidding problem. A conclusion is provided in Sec.\,\ref{s:conclusion}.

\section*{Acknowledgment}

\ifCLASSOPTIONcaptionsoff
  \newpage
\fi

\bibliographystyle{ieeetr}

\section{Problem Description}
\label{s:problemDescription}

We consider a BRP that participates in energy and reserve markets and whose balance group comprises a single system. The BRP strives to schedule the electric energy generation and/or consumption of the system so as to maximize the profit made from trading energy and from offering SFR services over a planning horizon of duration $T^{\ts{H}}$. The bidding problem involves different timescales and requires the BRP to take decisions on multiple stages both offline, i.e.\,, prior to the time of delivery, and online, i.e.\,, at delivery.

\subsection{Offline decisions}
\textit{1) Reserve capacity market:} An ISO is responsible for the safe and reliable operation of the transmission grid. In particular, it is accountable for keeping the supply and demand of electric energy balanced at all times. To be able to compensate fluctuating generation and consumption and other unexpected disturbances in a timely manner, it procures ancillary services on dedicated markets ahead of time. Different ancillary services exist, and their specifications vary among ISOs \cite{Swissgrid2017,ENTSO-E2009,FERC_MarketPrimer}. We focus on Secondary Frequency Regulation\footnote{Secondary Frequency Regulation is also referred to as Frequency Restoration Reserves, Spinning Reserves, or Load-Frequency Control.} and adopt the product specifications that apply to the Swiss SFR market, cf.\,\cite{Swissgrid2017}: every week on Tuesday not later than at 1 p.m., a BRP can bid a constant and symmetric capacity ${\gamma\in\R{+}_0}$ for the subsequent week \cite{Swissgrid2017}.\footnote{More precisely, multiple bids consisting of capacity-price pairs can be submitted.} If the bid is accepted, the ISO has the right, but not the obligation, to ask the BRP to deviate from the planned power reference by at most $\pm\gamma$ units of power at any time during the subsequent week. The SFR tendering period is denoted by $T^{\ts{RES}}$ (1 week in Switzerland). We choose the planning horizon to comprise a single SFR tendering period, i.e.\,, ${T^{\ts{H}}=T^{\ts{RES}}=1}$ week. In return for keeping the reserve capacity available over the time $T^{\ts{RES}}$, the BRP receives the capacity reservation payment ${R^{\ts{RES}}:=c^{\ts{RES}}\gamma}$, where $c^{\ts{RES}}$ is the reserve capacity price  \cite{Swissgrid2017}.

\textit{2) Day-ahead energy market:} On the day-ahead energy market, a BRP can trade energy for every time interval of duration $T^{\ts{DA}}$ of the next day. The Swiss day-ahead market runs on an hourly timescale, i.e.\,, ${T^{\ts{DA}}=1}$ h, and closes at 11 a.m. on the day before delivery \cite{Abrell2016}. The outcome of the day-ahead markets is the energy schedule ${e^{\ts{DA}}\in\R{N^{\ts{DA}}}}$ of energy quantities to be produced or consumed during each of the ${N^{\ts{DA}}=7\cdot 24=168}$ time intervals in the planning horizon $T^{\ts{H}}$. The day-ahead energy procurement costs are ${C^{\ts{DA}}:=c^{\ts{DA}\trans} e^{\ts{DA}}}$, where ${c^{\ts{DA}}\in\R{N^{\ts{DA}}}}$ denotes the day-ahead market clearing prices, which are unknown at the time of bidding.

\textit{3) Intra-day energy market:} To make adjustments to the day-ahead energy schedule during the day of delivery, the BRP can trade energy continuously on the intra-day markets for time intervals of durations ${T^{\ts{ID}}}$.
We focus on quarter-hourly energy block products, i.e.\,, ${T^{\ts{ID}}=15}$ min, as traded on the Swiss intra-day market with a lead time of 1 hour \cite{Abrell2016} and denote by $N^{\ts{ID}}$ the number of intra-day time slots in the planning horizon $T^{\ts{H}}$, i.e.\,, ${N^{\ts{ID}}=7\cdot 24\cdot 4=672}$. The energy procurement costs of the intra-day energy schedule ${e^{\ts{ID}}\in\R{N^{\ts{ID}}}}$ are ${C^{\ts{ID}}:=c^{\ts{ID}\trans} e^{\ts{ID}}}$, where ${c^{\ts{ID}}\in\R{N^{\ts{ID}}}}$ denotes the intra-day market prices, which are unknown at the time of bidding.

\subsection{Online decisions}
As will be discussed in detail in Sec.\,\ref{s:reqsPowerRef}, the day-ahead and intra-day energy schedules of a BRP are translated into a continuous-time power reference. The BRP is responsible for ensuring that its balance group follows this power reference in order to comply with all energy contracts concluded on the markets. BRPs are held accountable for any mismatches between their actual and planned energy schedules that arise from unforeseen outages of generation units or inaccuracies of load forecasts. The ISO constantly observes the state of the power grid and, in case of imbalances, takes measures to reestablish the nominal operating state. SFR is activated to bring the grid frequency back to its nominal value and make actual energy profiles match their corresponding schedules \cite{Swissgrid2017}. This is achieved by adjusting the power consumption or generation of all energy resources that have offered SFR according to an activation signal computed and broadcast by the ISO \cite{ENTSO-E2009}.
Power deviations due to SFR activation can result in energy deviations from the energy schedule of a BRP. In Switzerland, this so-called up- and down-regulation energy is measured over each time interval $T^{\ts{ID}}$ separately for positive and negative activation, and is denoted by $e^{\ts{up}}$ and $e^{\ts{dn}}$, respectively. The BRP is remunerated for tracking the activation signal based on the amount of up- and down-regulation energy delivered, i.e.\,, the remuneration is  ${R^{\ts{REG}}:=c^{\ts{up}\trans} e^{\ts{up}} - c^{\ts{dn}\trans} e^{\ts{dn}},}$
where ${c^{\ts{up}},c^{\ts{dn}}\in\R{N^{\ts{ID}}}}$ are the corresponding regulation energy prices \cite{Swissgrid2017}.

\subsection{The energy and reserve bidding problem}
\label{ss:problemDescription}
The energy and reserve bidding problem of the BRP is to decide how much reserve capacity $\gamma$ to offer on the SFR market over the planning horizon $T^{\ts{H}}$ and how to trade day-ahead and intra-day energy $e^{\ts{DA}}$ and $e^{\ts{ID}}$, respectively, such as to maximize the total expected profit. These decisions must be made subject to the constraints that the reference power trajectory associated with the energy schedules $e^{\ts{DA}}$ and $e^{\ts{ID}}$ \textit{i)} meets the energy needs and physical constraints of the systems in the balance group, \textit{ii)} complies with the energy contracts concluded on the energy markets, and \textit{iii)} keeps the SFR power capacity $\gamma$ offered to the ISO available at all times. The more reserve capacity the BRP offers, the more restrictive become the constraints on possible trades in the energy markets. Consequently, offering reserves and trading energy constitutes a trade-off between the reserve reward $R^{\ts{RES}}$, the potential remuneration of regulation energy $R^{\ts{REG}}$, and the energy procurement costs ${C^{\ts{DA}}+C^{\ts{ID}}}$.

\subsection{Different timescales}
The bidding problem takes place on different timescales. Offline decisions are made on the timescales $T^{\ts{RES}}$, $T^{\ts{DA}}$, and $T^{\ts{ID}}$ corresponding to the reserve market, and the day-ahead and the intra-day energy market, respectively. Online activation of SFR occurs at the faster timescale $T^{\ts{C}}$. The dynamics of the system are discretized on an intermediate timescale $T^{\ts{S}}$, which depends on the time constants of the system, the frequency with which set points can be changed, and possible computational challenges of the resulting problem. Commonly, $T^{\ts{S}}$ will lie in the range of 5--15 min. In general, it holds that ${T^{\ts{H}}\geq T^{\ts{RES}}\geq T^{\ts{DA}}\geq T^{\ts{ID}}\geq T^{\ts{S}}\gg T^{\ts{C}}}$. For simplicity, we assume that longer time horizons are integer multiples of shorter ones, and for ${\ast\in\{\ts{RES},\ts{DA},\ts{ID},\ts{S},\ts{C}\}}$ define ${N^{\ast}=T^{\ts{H}}/T^{\ast}}$, ${\set{N}^{\ast}=\{1,\dots,N^{\ast}\}}$, and $\set{T}^{\ast}(k)$ the continuous time interval ${[(k-1)T^{\ast},kT^{\ast})}$.
Different timescales are used in different market regions. All timescales $T^{\ast}$ are used as parameters in our approach making it versatile and applicable to the various market settings. Here we consider the Swiss market setting and use ${T^{\ts{H}}=T^{\ts{RES}}=1}$ week, ${T^{\ts{DA}}=1}$ h, ${T^{\ts{ID}}=15}$ min, ${T^{\ts{S}}=5}$ min, and ${T^{\ts{C}}=1}$ s.

\section{Energy Baseline and Power Reference}
\label{s:reqsPowerRef}

\subsection{Energy baseline}

The day-ahead and intra-day energy trades of a BRP result in the energy market baseline ${e^{\ts{base}}\in\R{N^{\ts{ID}}}}$, which comprises the total energy trades in each interval of duration $T^{\ts{ID}}$ in the planning horizon. The energy baseline is defined as
\begin{equation}
	e^{\ts{base}}_k:=e^{\ts{DA}}_j T^{\ts{ID}}/T^{\ts{DA}}+e^{\ts{ID}}_k,\ j:=\lceil k T^{\ts{ID}}/T^{\ts{DA}}\rceil,\ k\in\set{N}^{\ts{ID}},
\end{equation}
and is written in vector form as ${e^{\ts{base}}=Me^{\ts{DA}}+e^{\ts{ID}}}$ with ${e^{\ts{base}}:=[e^{\ts{base}}_1,\dots,e^{\ts{base}}_{N^{\ts{ID}}}]\trans}$ and ${M\in\R{N^{\ts{ID}}\times N^{\ts{DA}}}}$.

\subsection{Power reference}
It has been common in recent works, e.g.\,\cite{Vrettos2014a,Qureshi2016}, to associate with the energy baseline $e^{\ts{base}}$ the piece-wise \emph{constant} continuous-time power reference
\begin{equation}
p(t):= e^{\ts{base}}_k/T^{\ts{ID}},\ t\in\set{T}\id(k),\ k\in\set{N}^{\ts{ID}},
\label{eq:defPrefConst}
\end{equation}
that the BRP is committed to follow. Whereas the energy content of $p(t)$ over each time interval $\set{T}\id(k)$ matches $e^{\ts{base}}_k$ as required by market contracts, implementing the instantaneous changes of $p(t)$ at the interface of subsequent market time intervals $\set{T}\id(k)$ would require infinite ramping capabilities. Inaccurate tracking of the power reference \eqref{eq:defPrefConst} can cause the grid frequency to deviate from its nominal value when switching between market time intervals, making it necessary to activate frequency reserves \cite{Borsche2014,Morales-Espana2014}.
To alleviate this difficulty, we propose to translate the energy baseline $e^{\ts{base}}$ into a continuous-time piece-wise \emph{affine} \cite{Morales-Espana2014} and continuous power reference $p^{\ts{ref}}(t)$ whose breakpoint intervals are each of duration $T^{\ts{S}}$. The continuous-time power reference $p^{\ts{ref}}(t)$ is fully defined by its breakpoint values ${p^{\ts{ref}}:=[p^{\ts{ref}}_0,\dots,p^{\ts{ref}}_{N^{\ts{S}}}]\trans}$ according to
\begin{equation}
	p^{\ts{ref}}(t):=p^{\ts{ref}}_{s-1}+(p^{\ts{ref}}_{s}-p^{\ts{ref}}_{s-1})(t-(s-1)T^{\ts{S}})/T^{\ts{S}},
	\label{eq:defContPref}
\end{equation}
for ${t\in\set{T}\sys(s)}$, ${s\in\set{N}^{\ts{S}}}$. An example power reference is given in Fig.\,\ref{fig:baseline}.
In accordance with ENTSO-E standards \cite{ENTSO-E2009}, the power reference comprises $N^{\ts{ID}}$ periods during which the power is constant and equal to ${e^{\ts{base}}_k/T^{\ts{ID}}}$, ${k\in\set{N}^{\ts{ID}}}$. These periods of constant power are connected by linear ramps, each of duration $T^{\ts{RP}}$ as illustrated in Fig.\,\ref{fig:baseline}. We assume that $T^{\ts{RP}}$ is an integer multiple of $T^{\ts{S}}$ and that ${0\leq T^{\ts{RP}}\leq T^{\ts{ID}}}$. For the Swiss case of ${T^{\ts{ID}}=15}$ min, ${T^{\ts{RP}}=10}$ min, and ${T^{\ts{S}}=5}$ min, the relation between $e^{\ts{base}}$ and $p^{\ts{ref}}$ is the following:
\begin{align}
p^{\ts{ref}}_s =
\begin{cases}
	e^{\ts{base}}_1/T^{\ts{ID}}, & s=0,\\
	e^{\ts{base}}_k/T^{\ts{ID}}, & s=3k-2,3k-1,\\
	& k\in\set{N}^{\ts{ID}},\\
	(e^{\ts{base}}_k+e^{\ts{base}}_{k+1})/(2T^{\ts{ID}}), & s=3k,\,k\in\set{N}^{\ts{ID}-1},\\
	e^{\ts{base}}_{N^{\ts{ID}}}/T^{\ts{ID}}, & s=N^{\ts{S}}.
\end{cases}
\label{eq:mapEtoPref}
\end{align}
Thus, the power reference can be expressed compactly as
\begin{equation}
	p^{\ts{ref}} = Re^{\ts{base}}=RMe^{\ts{DA}}+Re^{\ts{ID}},
	\label{eq:defPowerRef}
\end{equation}
where we encode \eqref{eq:mapEtoPref} by ${R\in\R{(N^{\ts{S}}+1)\times N^{\ts{ID}}}}$.
Note that \eqref{eq:mapEtoPref} reverts to \eqref{eq:defPrefConst} in the limit case ${T^{\ts{RP}}=0}$. For all ${T^{\ts{RP}}>0}$, our choice  \eqref{eq:defContPref} is continuous and, thus, allows us to model the power-ramp rates.
We make the assumption that a BRP with energy baseline $e^{\ts{base}}$ will use the piece-wise affine continuous-time power trajectory $p^{\ts{ref}}(t)$ \eqref{eq:defContPref} defined by its breakpoint values $p^{\ts{ref}}$ \eqref{eq:defPowerRef} as its power reference.

\begin{figure}[tb]
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\caption{The piece-wise affine continuous-time power reference $p^{\ts{ref}}(t)$ (solid black line) defined by its breakpoint values ${p^{\ts{ref}}:=[p^{\ts{ref}}_0,\dots,p^{\ts{ref}}_{N^{\ts{S}}}]\trans}$ with breakpoint intervals $T^{\ts{S}}$. The energy content (shaded gray area) of $p^{\ts{ref}}(t)$ over the time interval $\set{T}\id(k)$ is $e_k^{\ts{ref}}$. The power ramps of duration $T^{\ts{RP}}$ are located symmetrically at the intersections of market time intervals.}
\label{fig:baseline}
\end{figure}

\subsection{Activation of SFR}
\label{ss:activationOfSFR}
The SFR service is activated by the ISO by sending out an activation signal to all BRPs that offer SFR for the corresponding time interval. The activation signal ${w:=[w_1,\dots,w_{N^{\ts{C}}}]\trans}$ is a discrete-time signal with ${w_l\in\set{W}:=[-1,1]}$, ${l=1,\dots,N^{\ts{C}}}$. The signal is computed by the ISO and broadcast \emph{sequentially} on the timescale $T^{\ts{C}}$ (e.g.\,, ${T^{\ts{C}}=1-5}$ s in Continental Europe \cite{ENTSO-E2009}, ${T^{\ts{C}}=1}$ s in Switzerland \cite{Lymperopoulos2015}, and ${T^{\ts{C}}=4-6}$ s in the USA \cite{FERC2011}). That is, the activation $w_l$ becomes available to the BRP only at time $(l-1)T^{\ts{C}}$. We interpret the discrete signal $w$ as the continuous-time piece-wise affine and continuous activation signal
\begin{equation}
	w(t):= w_{l-1}+(w_l-w_{l-1})(t-(l-1)T^{\ts{C}})/T^{\ts{C}},
\end{equation}
where ${t\in\set{T}\ctrl(l)}$, ${l\in\set{N}^{C}}$. We denote the set of all such functions $w(t)$ by $\mathcal{W}$, which is a closed set of continuous functions.

The target power level $p\opt(t)$ that the BRP is responsible to track continuously is a function of the continuous time $t$ and the activation $w(t)$ and is given by
\begin{equation}
	p\opt(t,w):=p^{\ts{ref}}(t)+\gamma w(t).
	\label{eq:defTargetPower}
\end{equation}
Compared with the power reference $p^{\ts{ref}}(t)$, which is piece-wise affine on the timescale $T^{\ts{S}}$, the activation $w(t)$ can vary at the higher rate $T^{\ts{C}}$. Thus, accurate tracking of the target power level \eqref{eq:defTargetPower} is challenging. The minimum required tracking accuracy of $p\opt(t)$ is specified by the ISOs individually (cf.\,\cite{TestSC} for details on the Swiss requirements). In the USA, ISOs are required to base their regulation compensation payment $R^{\ts{REG}}$ not only on the quantity of regulation provided, but must also take into account how accurately the activation signal has been tracked \cite{FERC2011}. This motivates the use of a continuous power reference according to \eqref{eq:defPowerRef} and the explicit consideration of ramp-rate constraints.

\subsection{Adjustable energy schedules}
\label{ss:adpatingSchedules}

The decision on the amount of SFR to be offered is made once for each reserve tendering period and cannot be changed afterwards. In contrast, energy schedules can be adjusted from day to day or intra-day by trading energy on respective markets. These adjustments can depend on past SFR activation, for example. Let ${\tilde{w}\in\set{W}^{N^{\ts{S}}}}$ be the activation signal $w(t)$ averaged over time intervals of duration $T^{\ts{S}}$, i.e.\,,
\begin{equation}
	\tilde{w}_s:=\frac{1}{T^{\ts{S}}}\int_{(s-1)T^{\ts{S}}}^{T^{\ts{S}}} w(t) dt,\ s\in\set{N}^{\ts{S}}.
\end{equation}
Following the work in \cite{Warrington2012,Warrington2013,Zhang2014,Fabietti2016,Gorecki2017}, we express the adjustable day-ahead and intra-day energy schedules as affine functions of the averaged activation signal:
\begin{align}
\begin{split}
	e^{\ts{DA}}(\tilde{w}) &= Q^{\ts{DA}} A^{\ts{DA}} \tilde{w}+q^{\ts{DA}},\\
	e^{\ts{ID}}(\tilde{w}) &= Q^{\ts{ID}} A^{\ts{ID}} \tilde{w}+q^{\ts{ID}},
	\label{eq:defControllers}
\end{split}
\end{align}
with parameters ${Q^{\ts{DA}}\in\R{N^{\ts{DA}}\times N^{\ts{DA}}}}$, ${q^{\ts{DA}}\in\R{N^{\ts{DA}}}}$, ${Q^{\ts{ID}}\in\R{N^{\ts{ID}}\times N^{\ts{ID}}}}$, and ${q^{\ts{ID}}\in\R{N^{\ts{ID}}}}$. The matrices ${A^{\ts{DA}}\in\R{N^{\ts{DA}}\times N^{\ts{S}}}}$ and ${A^{\ts{ID}}\in\R{N^{\ts{ID}}\times N^{\ts{S}}}}$ are used to average $\tilde{w}$ over time intervals of duration $T^{\ts{DA}}$ and $T^{\ts{ID}}$, respectively. For instance, $A^{\ts{ID}}$ is block-diagonal with each block given by a row-vector of length $T^{\ts{ID}}/T^{\ts{S}}$ whose elements all equal $T^{\ts{S}}/T^{\ts{ID}}$. By plugging \eqref{eq:defControllers} into \eqref{eq:defPowerRef} we can write the power reference as an affine function of the averaged activation:
\begin{equation}
	p^{\ts{ref}}(\tilde{w}) = R(MQ^{\ts{DA}} A^{\ts{DA}} + Q^{\ts{ID}} A^{\ts{ID}})\tilde{w}+R(Mq^{\ts{DA}} + q^{\ts{ID}}).
	\label{eq:pRefDepW}
\end{equation}

The policies \eqref{eq:defControllers} describe how much energy should be traded on the day-ahead and intra-day energy markets depending on the average amount of SFR activated during past time intervals. The policies must be causal, i.e.\,, they can only depend on past activation, and must respect the different lead times of the markets. These requirements can be incorporated by imposing a particular structure onto $Q^{\ts{DA}}$ and $Q^{\ts{ID}}$. Figure \ref{fig:structureQ} illustrates which elements of $Q^{\ts{DA}}$ and $Q^{\ts{ID}}$ can be nonzero. Because the day-ahead market has a lead time of 13 h, the trading decisions for the second day can only depend on the first 11 h of the first day. Similarly, the ${4\cdot 15}$ min lead time of the intra-day market forces the first 5 rows of $Q^{\ts{ID}}$ to be zero. The size of the bidding problem can be reduced by not taking into account all the available historic activation data in \eqref{eq:defControllers} but only considering the $N^{\ts{DA}}_{\ts{lb}}$ and $N^{\ts{ID}}_{\ts{lb}}$ most recent measurements. The policy parameters $N^{\ts{DA}}_{\ts{lb}}$ and $N^{\ts{ID}}_{\ts{lb}}$ determine the width of the blocks in $Q^{\ts{DA}}$ and the number of off-diagonals in the lower-triangular part of $Q^{\ts{ID}}$, respectively, \cfFig.\,\ref{fig:structureQ}.

\begin{figure}[tb]
\setlength{\belowcaptionskip}{-5pt}
\centering
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\begin{pspicture*}(-55,18)(197,-169)

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	\psline[linewidth=0.5pt]{->}(-19,-37)(-2,-37)\rput[r](-20,-37){\scriptsize day 2}
	\psline[linewidth=0.5pt]{->}(-19,-156)(-2,-156)\rput[r](-20,-156){\scriptsize day 7}

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	\rput[bc](69,-104){\scriptsize $N^{\ts{DA}}_{\ts{lb}}$}

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\caption{Structure of $Q^{\ts{DA}}$ (a) and $Q^{\ts{ID}}$ (b) for the case ${N^{\ts{DA}}=168}$ and ${N^{\ts{ID}}=672}$. The shaded grey areas denote the location of potentially nonzero elements. The amount of historic activation data considered by the policies \eqref{eq:defControllers} is governed by the parameters $N^{\ts{DA}}_{\ts{lb}}$ and $N^{\ts{ID}}_{\ts{lb}}$.}
\label{fig:structureQ}
\end{figure}

\section{Solving the Bidding Problem}
\label{s:scheduling}

\subsection{Formulation of the bidding problem}

The energy and reserve bidding problem consists of finding the energy trading policies \eqref{eq:defControllers} and the reserve capacity $\gamma$ that maximize the expected profit while satisfying all system constraints.

The profit
\begin{align}
J(\zeta,w) :=&\ R^{\ts{RES}}(\gamma)+R^{\ts{REG}}(\gamma,w)-C^{\ts{DA}}(Q^{\ts{DA}},q^{\ts{DA}},w)\nonumber\\
&\ -C^{\ts{ID}}(Q^{\ts{ID}},q^{\ts{ID}},w),
\label{eq:defObjective}
\end{align}
is uncertain because of its dependency on the SFR activation signal and the market clearing prices. Thus, we maximize the expectation of \eqref{eq:defObjective} over the uncertain parameters ${\Psi:=\{w,c^{\ts{DA}},c^{\ts{ID}},c^{\ts{RES}},c^{\ts{up}},c^{\ts{dn}}\}}$, and write the energy and reserve bidding problem as the linear program

\begin{equation}
	\max\limits_{\zeta} \mathbf{E}_{\Psi}[ J(\zeta,w)]\ \ts{ s.t. } \zeta\in\set{P}(\Phi).
	\label{eq:schedulingOpti}
\end{equation}
The number of decision variables and constraints of \eqref{eq:schedulingOpti} is determined by various factors, such as the planning horizon, the different market timescales, the system discretization time, and also the trading policies \eqref{eq:defControllers} used.

\subsection{Computational study}
\label{ss:simulationStudy}
The results of the economic bidding problem \eqref{eq:schedulingOpti} do not only depend on the market characteristics and technical constraints of the system, but are also influenced by market prices. An economic analysis of \eqref{eq:schedulingOpti} can be found in \cite{Qureshi2016}. To investigate the maximum available SFR capacity independent of economic considerations, we solve \eqref{eq:schedulingOpti} with objective ${\max_{\zeta}\gamma}$ and investigate the effects that different market time scales, lead times, and trading policies have on the maximum available SFR capacity $\gamma\opt$.
To facilitate the interpretation of the results, we consider an ideal battery modeled according to \eqref{eq:stateDynamics} with parameters ${a=0}$, ${b=c=1}$, and no exogenous input, i.e.\,, ${u=0}$. The physical constraints of the battery are similar to those of a \emph{Tesla Powerwall 2} battery \cite{Powerwall2016}: ${-\un{p}_s=\bar{p}_s=5}$ kW, ${\un{x}_s=0}$ kWh, ${\bar{x}_s=15}$ kWh, ${s=1,\dots,N^{\ts{S}}}$, and $x_0=7.5$ kWh. We use ${T^{\ts{S}}=5}$ min, ${T^{\ts{C}}=1}$ s, ${N^{\ts{DA}}_{\ts{lb}}=13}$ h, and ${T^{\ts{H}}=T^{\ts{RES}}}$.
Table \ref{tab:capResults} provides the market settings and trading policy parameters considered and the results obtained.
The maximum available SFR capacity $\gamma\opt$ and the minimum ramp rates $\bar{r}\opt$ required to offer this service are provided as quantities relative to the rated power $\bar{p}$.

As a reference case, the first setting described in Tab.\,\ref{tab:capResults} considers the case where the power reference cannot be adjusted via day-ahead or intra-day energy trades; it remains fixed regardless of potential SFR activation. This setting results in the least amount of SFR capacity, which, for the simple system at hand, can be computed analytically by dividing the available up- and down energy buffer capacity by the duration of the planning horizon: ${\gamma\opt=}$ 7.5 kWh / (7$\cdot$24) h $\approx$ 0.0446 kW, which corresponds to 0.89\% of $\bar{p}$.
More SFR capacity can be made available by adjusting the power reference via day-ahead energy trades depending on past activation. The available capacity $\gamma\opt$ increases as trading decisions are based on more data on past activation, see settings 2--6, where the value of $N^{\ts{DA}}_{\ts{lb}}$ is gradually increased. Considering more than 24 h of past activation data, i.e.\,, $N^{\ts{DA}}_{\ts{lb}}>24$, does not yield more SFR capacity because the activation that occurred more than 24 h in the past has already been compensated for in the previous day-ahead market.

Whereas day-ahead adjustments of the power reference yield only minor increases of SFR capacity over the reference setting, intra-day adjustments make more than 50\% of the rated power available for SFR, see settings 7--9. Reducing the intra-day market lead time $T^{\ts{ID}}_{ld}$ from 1 h to 1/4 h, however, has only minor effects on $\gamma\opt$, see settings 11--13. Reducing the planning horizon from 7 d to 1 d results in an increase of $\gamma\opt$ from 50.26\% to 51.87\%, see settings 7 and 11.
The results also illustrate the ramp-rate constraints \eqref{eq:rampConstr}. Consider setting 13, for instance: Tracking SFR activation of size ${\gamma\opt=52.63}$ \% of $\bar{p}$ around a constant $p^{\ts{ref}}$ would require ${\bar{r}=2\cdot 52.63/T^{\ts{C}}=105.26}$ (\% of $\bar{p}$)/s, whereas in reality the slightly larger amount ${\bar{r}=105.42}$ (\% of $\bar{p}$)/s is needed.

The results highlight the importance of the intra-day market which operates at faster time scales than the day-ahead market. It allows energy-constrained systems such as batteries to compensate for the regulation energy provided. The introduction of energy markets operating at even shorter time intervals, e.g.\, 5 min, could further increase the amount of available reserves. Additional reserves can be made available by reducing the length of the tendering period $T^{\ts{RES}}$ and by relaxing the requirement that a constant amount of SFR must be offered.

The bidding problem \eqref{eq:schedulingOpti} has been solved with \textsc{IBM ILOG}\textregistered\footnote{\label{fnote}ILOG and CPLEX are trademarks of International Business Machines Corp., registered in many jurisdictions worldwide. Other product and service names might be trademarks of IBM or other companies.} \textsc{CPLEX}\textregistered\footnotemark[3]\, v12.6 on a system featuring an Intel\textregistered\, Xeon\textregistered\, 12-core CPU @2.4 GHz and 20 GB of RAM. The solving times are provided in the rightmost column of Tab.\,\ref{tab:capResults}. For fixed discretization time $T^{\ts{S}}$, the problem size grows quadratically with the planning horizon length $T^{\ts{H}}$. An efficient means to reduce the problem size is to reduce the number of free variables in the energy trading policies \eqref{eq:defControllers} via the parameters $N_{lb}^{\ts{DA}}$ and $N_{lb}^{\ts{ID}}$, cf.\, Section \ref{ss:adpatingSchedules}.

\begin{table}[t]
\setlength{\abovecaptionskip}{-5pt}
\caption{Maximum SFR capacity and minimum required ramp rates for different market  settings and trading policies.}
\label{tab:capResults}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline
\# & ${T^{\ts{RES}}}$ &  $T^{\ts{ID}}_{ld}$ & $N_{lb}^{\ts{DA}}$ & $N_{lb}^{\ts{ID}}$ & $\gamma\opt$ & $\bar{r}\opt$ & Sol. time\\
 & [d] & [h] &  & & [\% $\bar{p}$] & [(\% $\bar{p}$)/s] & \\\hline
1 & 7 & 1 & 0  & 0 & 0.89 & 1.78 & 8 min\\
2 & 7 & 1 & 1  & 0 & 0.93 & 1.87 & 9 min\\
3 & 7 & 1 & 2  & 0 & 0.96 & 1.94 & 12 min\\
4 & 7 & 1 & 6 & 0 & 1.14 & 2.29 & 40 min\\
5 & 7 & 1 & 12 & 0 & 1.55 & 3.13 & 100 min\\
6 & 7 & 1 & $\geq$ 24 & 0 & 4.05 & 8.25 & 270 min\\ \hline
7 & 7 & 1 & 0 & $\geq$ 1 & 50.26 & 100.69 & 30 min\\
8 & 7 & 1/2 & 0 & $\geq$ 1 & 50.34 & 100.84 & 30 min\\
9 & 7 & 1/4 & 0 & $\geq$ 1 & 50.37 & 100.91 & 30 min\\ \hline
10 & 1 & 1 & 0 & 0 & 6.25 & 12.50 & 5 s\\
11 & 1 & 1 & 0 & $\geq$ 1 & 51.87 & 103.90 & 25 s\\
12 & 1 & 1/2 & 0 & $\geq$ 1 & 52.38 & 104.92 & 25 s\\
13 & 1 & 1/4 & 0 & $\geq$ 1 & 52.63 & 105.42 & 25 s\\
\hline
\end{tabular}
\end{center}
\end{table}

redpajama_set_name: RedPajamaArXiv


> > even if they don't have guide info.
plus the DB entries for one channel that was inserted incorrectly.
It should be easy to fix given that info.
> > Without the logs it is hard to tell.
> As described above, I have three tuners.
> different channels, watching different channels live, etc.
what you did at each moment, and what you expected to happen vs.


text:
Total Joint Center
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Home St. Luke's Orthopedic Care News * *
St. Luke's Orthopedic Care Welcomes Region's First Orthopedic Oncologist
Justin Miller, DO, Treats Cancers of the Bones and Musculoskeletal System
After finishing four years of medical school and a five-year orthopedic surgery residency, Justin Miller, DO accepted one more challenge. He completed a two-year fellowship in orthopedic oncology because he wanted to help patients through their cancer journey.
The only orthopedic oncologist in the Lehigh Valley, and one of approximately 200 in the United States, Dr. Miller joined St. Luke's Orthopedic Care to bring this highly specialized level of care to patients in our region.
"A patient may have a devastating cancer diagnosis, but as an orthopedic oncologist, you can help them through every moment and step of the way," he said. "You create a bond with your patient and their families. Hopefully, you're able to save their life or limb."
As an orthopedic oncologist, Dr. Miller treats patients with cancer of the musculoskeletal system, which consists of bones, joints, ligaments, muscles, tendons and nerves. He treats both cancers that began in the musculoskeletal system, and those that spread there from another part of the body (metastasize).
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Dr. Miller joined St. Luke's after completing a two-year fellowship in orthopedic oncology at Morristown Medical Center, working with experts in the subspecialty. He completed an internal medicine and orthopedics residency at Jersey City Medical Center and earned his medical degree at Philadelphia College of Osteopathic Medicine. Dr. Miller chose St. Luke's because several physicians he knew who worked for the Network described the culture as team-oriented and patient-focused. Also, he wanted a position where he could work in the countryside because he enjoyed a rural medicine rotation during his training.
"I'm here to treat the community and help people in pain or discomfort so they can live their best life," he said. "It makes me happy to take my patients' pain away and help them return to normal activities." While he performs surgery as needed, he first treats patients with non-surgical means. He also guides patients through their entire orthopedic healing and rehabilitation process.
During his fellowship, he treated a 15-year-old patient with a tumor in her thigh bone, just above the knee. He removed the tumor and restored the function of her leg.
"We still connect to make sure her recovery is going well," he said. "She made me a nice bracelet with a little bone on it that I still wear it to this day. I was able to save her life and save her limb from needing an amputation with a large reconstruction procedure."
Pennsburg Carpenter Bounces Back after Revision Knee Surgery to Build His Hew Home
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redpajama_set_name: RedPajamaCommonCrawl


text:
Proteins (GRS)
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Chairs Kevin E. Hauser and Tamara C. Bidone
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Protein science is a rapidly evolving area of research due to advances in both experimental and computational technologies that are providing unprecedented high-resolution structures together with new insights into mechanisms and functions. With the purpose of creating an interdisciplinary environment focusing on cutting-edge problems in protein research, the 2019 Gordon Research Seminar (GRS) on Proteins will bring together students and postdocs that study proteins in cellular environments through the lens of cellular, biophysical, chemical, and computational methodologies.
Session topics of the 2019 GRS are Protein-folding (Design, Dynamics, Mechanisms); Protein-stability (Conformational Switching, Allostery, Dynamics of Large Proteins); Protein-signaling (Protein-Protein Interactions, Bioenergetics, Design); Protein-evolution (Biosynthesis Pathways, Degradation, Aggregation). We welcome applications about Protein topics not listed in this preliminary program and new ideas for session topics; final session topics will be available after September 1, 2019, to best reflect the interests of the attendees. A senior scientist will deliver a keynote lecture.
On Saturday night, there will be a informal event (attendance is optional) in which participants are paired, and then re-paired every few minutes, with a complementary partner (e.g. experimentalists with computationalists, and representative of different sub-disciplines in protein research) to teach each other about their science.
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Discussion Leader: Mireia Sola (Max Planck Institute for Biophysical Chemistry, Germany)
Martin Culka (Institute of Organic Chemistry and Biochemistry, Czech Academy of Sciences, Czech Republic)
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Coffee will be served in the poster area from 11:00 am - 11:30 am
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Conference History Proteins


While the world mourned the death of singing legend Aretha Franklin last month, I couldn't help but wonder whether this tremendous loss could have been avoided. The Queen of Soul battled pancreatic cancer, a ruthless killer known for developing without symptoms until it is too advanced to treat.
But the picture of pancreatic cancer is changing. It is my deepest hope that we get the upper hand on pancreatic cancer, that we stop it from taking our legends before they're ready to go. I believe new detection techniques will allow us to do just that.
We often talk about being blindsided by pancreatic cancer. Unlike other cancers, it kills nearly as many people each year who develop the disease, with the majority of people dying within the first eight months of detection. But the truth is, risk factors present warning signs. If we identify people with strong family histories or other risk factors early enough, we can monitor these high-risk people and catch tumors while they are still treatable.
For example, as someone who battled diabetes, Franklin was at increased risk of developing pancreatic cancer. Nearly 10 percent of pancreatic cancers are hereditary. Other risk factors include obesity, smoking and diabetes. I was not her doctor, and I don't know if Franklin had other risk factors. But for patients at Hoag's Anita Erickson Pancreatic Cancer Early Detection Program whose medical histories raise red flags, we offer aggressive surveillance in hopes of finding cancer early enough for treatment.
Our team of geneticists, genetic counselors, gastroenterologists, GI surgeons and a clinical nurse navigator work together to detect early stage pancreatic cancer through labs, imaging and diagnostic testing, as well as genetic counseling and testing.
To participate in the program, patients submit their de-identified information to a registry, shared by similar organizations in the nation. This database will help future clinicians to better understand this complex disease and how to effectively treat and manage it.
By proactively monitoring higher risk patients, we can detect cancer at an early stage, surgically remove the tumors and increase survival rates. Surveillance and early detection is a new frontier for us, but with pancreatic cancer on the rise, it is an issue that cannot be ignored.
Hoag Memorial Hospital Presbyterian is one of the first hospitals in the nation to create a program targeted at finding pancreatic cancer early, when it might still be treatable. Because medicine has evolved to allow us to take genetic information and family history to determine who to test and how to test them, I expect many other hospitals will soon follow suit.
We lost a legend. It is my hope to live in a world in which we can stop pancreatic cancer from claiming another legend – or mother, father, sister or friend. The warning signs are there. Let's pay attention and turn the tables on this insidious killer.
Valentina Dalili-Shoaie, M.D., is a medical geneticist and a physician leader for Anita Erickson Pancreatic Cancer Early Detection Program at Hoag.


Only take the possibility to come up with a potent functional resume. If you should be asking to get a job and also you also wished to earn a fresh look on your own resume which is fine, presentable and clean, then you arrived into the suitable spot. If you are hunting for work being a account executive, then your resume has to prove you have the wisdom and experience required to notify customers and control finances. In addition, inside the event that you usually switch tasks, you may additionally look at a resume that is operational. In the event you have held temporary occupations which are in maintaining the job which you're using for, mention these temporarily. There was an range of explanations for why some one would want to take a project within this enterprise.
Simply see their site and then make and then edit your own Resume once you desire. These types of resumes are perfect for those who've gaps in their occupation foundations or are brand new to a particular business. Your resume should be formatted within an easy, expert method. Your resume must perhaps not mention your personal hobbies, or unless those have an immediate bearing in your own preferred career course. Useful resumes are perfect for people that don't possess plenty of function experience about the job they have been applying for.
Your resume needs to start out using the going. Oahu is the specific same with hints. Functional and traditional resumes are not the only resume types to choose from.
Depending upon your own work history, you may possibly like to check at utilizing a functional resume. You need to use a chronological study if you've got extensive job history that is certainly seeing the job which you're applying for. If you have found an executive-level job and decades of expertise, then you may perhaps possibly not be capable of meet all your capabilities, qualifications, and accomplishments onto a single page. Insteadyou also have to take into consideration your years of knowledge.
You don't need to sign up in order of using their site. The very ideal issue of the website is you might go back into Resumizer no cost Resume Creator on the web to edit your file you've generated with them. The other website for creating your CV is the cvmaker. There are a lot of free on line sites that gives totally free downloadable Demo Vitae and Effectiveness Templates. Educational advice is included with certifications along with abilities that are exceptional. Activity lists aren't likely to force you to stand out by the audience. For seasoned bookkeeping professionals, it truly is encouraged to look in a set of abilities that are essential.
A premier excellent sample bookkeeping resume will comprise each of the crucial phrases and terms that an employer will likely soon be hunting for a number of occasions during the resume. Lots of accountant resume samples you will see online tend not to give the level of depth required to aid one to framework yours. A lot of employees wind up feeling that their existing employers discover that it's extremely hard to supply what they want and move to scan the market.
If you are considering a non existent resume format, it's even more explanation why you need to receive it assessed before you utilize it as a way to employ to jobs. You first ought to pick a particular format for your resume. Both key formats in use today days are chronological and functional format. At Primer publication there are just 7 downloadable templates which you are able to select from and also an additional 12 templates they have added lately. So it will likely be quite easyto select out of their many templates. You will find 20 totally free Resume templates that you will download out of Resumetemplates.in. You also have to pick the layout of your own restart.
Employ our accounting resume samples to find a clearer strategy of this information companies may possibly be trying to find from you. Compatibility screening a section of non-functional tests carried on the software element or the full applications to estimate the compatibility of their app with all the calculating atmosphere. It's very important to notethat in the event you opt for manual testing, it really is a recognized truth that the job will soon be tedious and laborious. Consequently, efficiency testing is the task to ensure that you determine the operation of a software. There are some industry-recognized certificates that might help them grab the most best tasks in the enterprise.
It's mandatory that you focus on your own education and techniques. The real-life surroundings will help discover the flaws from the machine and protect against costly bugs been discovered at another period. It also features a succinct collection of knowledge, and a thorough group of abilities. Function knowledge also needs to be in reverse chronological order. The comprehension of of the most recent systems is vital. On the contrary, it centers upon the relevant skills you need developed which fit the requirements of the job for which you are applying. Additionally, it is advisable to figure out strategies to demonstrate your analytic abilities along with time-tested plan.
Designers can not have the job done alone and no object design staff could function having a bothersome artist. A designer ought to be in a situation to edit fashions as an application evolves through the many decades. Ahead of you begin composing a graphic designer restart, don't neglect to ask yourself exactly what you will need. If you really will need to become a designer that is quick, then you still want to create an endeavor to be one. As an alternative, designers learn how to draw in order that they could express their own thoughts. This manner in which the designer can concentrate along the means by which the elements go collectively in the place of always designing or copying the exact same components. Likewise applications element designers want to encompass the circumstance where this function has been supposed to be used, by an individual, to find the entire capacity.
This short article discussions whether pupils should utilize rev-it. Qualitative investigation comprises a small choice of participants, determined on criteria characterized by the research. You really don't need to become a professional in everything, nevertheless, also you also ought to exhibit a pure fascination by the breadth of your adventure.
The template looks clean, simple, and nominal, is famous for spaciousness and power to accommodate all your firm special articles in a full page website. As an example, it may be very handy to have expert layout because it will give an outstanding impression to any one of your long term organizations. Design your resume and then discover a job Do not let me tell you exactly how important the restart is and also how it can affect you in acquiring the exact job. Or you adore the specific design and you also take pleasure in the method of producing your theme with different pre-made designs and choices. Therefore it is immaterial the best way to show it upon your own design restart. Architecture isn't a ninetofive profession. You're definitely going to write a killer structure resume.
In the event the object isn't reachable, instead of observing every single notification, it may be more straightforward to fix the matter by correcting your structure. If it's possible to do so give attention to visual elements, and emphasize only a handful images. Along with the labour knowledge, the abilities section another significant role to acquire in a restart.
Conversation is one of the countless knowledge designers need to take within their own arsenal. Notifications are frequently compared contrast with delegates. It could look as a very simple remedy to complex problems. As software increase in percentage, retaining a watch out for the most of its components state and data leaks can get dull, and also are usually challenging to take care of and purge. Think about an intensive resume being a expert tool to assist you then feel noticed among the competition employed. The process for choosing a member of staff in massive and smaller corporations isn't likely to shift dramatically or be greatly disrupted. Volunteering, in your business or within your subject of expertise into just another firm, is one way to alleviate the transition back again to full-time occupation.
You must send me a restart if you may browse the whole item. It is fast and simple to make use of. While you can find a lot of ways in which ATI advertises, Bruce expresses that societal media is your overriding way at which the company creates buzz about the things they are able to give. If you would enjoy ideas, take a look at those sample visual hints under control. Exactly the specific same idea pertains to a niche website. Considering for circumstance has been proven practical in so a number of our adventures. Quite simply, the lack of context consideration generally seems to cause scenarios of bugs and usability conditions that might have already been prevented.
The entire winter is almost overspring is right round the corner. You can find a significant lot of places on the web at which it's possible to find at no cost. The above the fold is just one of the primary aspects of your own organization. A restart to get design isn't a laundry list. A depth comprehension of 3 d modeling computer software like AutoCAD is also critical. After having a long amount of involuntary unemployment, an individual could possibly be fearful that his abilities have escalated to the level he won't have the ability to use in a brand new location.
Let's search at each and every part of the electronic mail carefully to identify what is wrong. A superb email may bring about business achievement. Professional email now is a very beneficial device to make your work less challenging and time consuming because it's 1 advantage of employing today's technology.
E mails are not exactly the exact same as business meetings. Additionally, make sure that your mails signify that you have while in the actual world. Hence if you are not too sure seeing how to compose a specialist email, then it is probably advisable to abide by an skilled electronic mail template. As it's been discussed it is critical to write properly framed professional emails.
Together along with maintaining your mails short, following a regular structure additionally allows you to create fast. You are composing a resignation email for the very 1st time and you end up staring at a blank piece of newspaper or even word record. Before registering your email, ensure to feature a final line. As email getting more and more usually used methods for communicating, it's very important this one must learn just how to compose a formal email. It is quite tough to compose a formal email for an expert entity.
E mails are frequently used to get firsthand data at senior school and school essays. Check your electronic mail address more often, and should you get a contact, answer it if possible if required. Therefore it's important that should sending a message address, it ought to have a subject that may discover a person's eye of the recipient. Therefore, not all enterprise enterprise email ought to be written and formatted in the specific way. The absolute most efficient business emails are polite and don't comprise S-Lang or violent phrases. Usually, they're the ones given to you by your organization, usually including yours along with your institution's title. Business emails written to coworkers are commonly straight and have for specific activities to be taken.
Your information should become distinct and clear. Subsequent to the information is routed, the file goes to be downloaded and connected to this message automatically. The most important message is wherever your recipient gets evident about this function of this official emailaddress. Even in case you write the most eloquent, persuasive communication from the Earth, a badly designed touch will make you look to be an newbie.
An filled matter matter line to inform the reader just exactly what the letter is all about. When you have finished that, you are prepared to compose a resignation letter that is appropriate. The sample letters given are supposed to provide a concept of how to write efficiently for a variety of predicaments. Earlier composing, make note of the things you desire to put in your letter. The company letter additionally conveys an impression of their association at various manners. A professional business hint must make a beneficial and profound effect on the minds of the receivers and should carry out a part in raising the disposition of the firm.
Finding the very appropriate words to create is very challenging and you may detect that it's simpler to confer with your boss inperson. Email still remains the basic way of communicating for various documented and company offers. In the official email previously mentioned, the use of this email has been initially clearly stated.
The most typical individual applying electronic mail for company sends and receives more than 100 mails per day, as shown by a written report published from the Radicati Group. In addition, your subject should not be excessively long however temporarily explained. The subject of your email must ready your receiver upon what things to expect in the big body of their mail .
Just take a little time to analyze the example and see whether you may explain to what's wrong by it. The next example demonstrates how to compose an everyday email. The exact first example demonstrates how to compose a formal business email.
Broadly speaking, the advice included on your letter should be prepared in a concise manner, together with the information you would rather communicate clearly stated. Ensure to receive the information you're searching for. When it's crucial to deliver negative info in a contact, it ought to be completed in a confident, constructive way. CC-ing someone is necessary if there's relevant data within the mail or you're connecting them with the genuine recipient. If you have received little numerical info to present, it needs to be described in the writing from your own newspaper. That you don't will need to get stressed in the event that you missed important information. Before mailing a sales letter, it's crucial that you've accumulated as much info on the consumer as feasible.
Variety of Resumes Resume is necessary for every type of job that you just simply apply. Conventional, history-based resumes told companies about sooner times now's executive restart has to indicate strategies to make a change later on. Opportunely, engineering resumes come with a very straightforward format.
If a supplier wishes to acquire hydraulic gear, then it might require the in-house hydraulic professionals to get out more on the topic of the marketplace and establish products and suppliers in giving the perfect price and effectiveness. Find out more about the industry you wish to enter and also the company you would like to utilize for. Companies are competing in the regions of responsiveness along with customization. Next, join with those who work on the firm you're seeking job in.
You'll find lots of companies who promote their staff to dress appropriately as well, however, there are employers that are strict as it pertains to grooming within the right small business attire. They need to see you learn howto speak about your knowledge in a manner that results in value for them and introduces your own accomplishments at the greatest possible lighting. Manufacturing purchasers also make an effort to track down new services and services which may improve their organization's functionality. An fabricating company and planner is someone who handles the organization and get of services and merchandise to get an company.
Each business and profession includes specific essential terms. Computer-integrated production (CIM) in technology is really something of fabricating at which in fact the entire producing procedure has been controlled through personal computer. Besides being the ability to switch functions in a company, you may try out a different industry if you want a change. The aerospace firm is one solid business.
Some engineering specialty it will be potential to receive work in doc pdf entirely free. There are various tasks on the market for engineers. Industrial assembly line work increases creation and permits your employer to rapidly train employees. For an start in your hunt for a coming role, have a look at the Engineering work Research web page. Nonetheless, the tasks are very different from one another. On the flip side, you can also obtain a rewarding task in search to help detect improved ways to extract and refine oil.
In depth comprehension of largescale manufacturing and production is essential. Previous relevant experience is often required or preferred as well. Any proper experience from your telecom business will be a incentive.
You can possibly be attracted to technology as a result of the creative element. Manufacturing engineering is an exceedingly crucial area throughout the world. It's simply a single part of this technology manufacturing industry. Afterward it can be your perfect professional alternative. Industrial technology is really a excellent livelihood industry. As soon as it really isn't the most famous engineering discipline, it is the most diverse between the many expert engineering areas of study. Even the exceptional industrial technology restart has the suitable blend of important info, strength-highlighting along with formatting.
Not many engineers need the programming experience to build designs, nevertheless, you'll need to have an understanding of how models work you realize what kind of issues can be mimicked and what the simulations me-an for the undertaking. Regardless of quantity of work knowledge you need being an engineer, it is very important that you make an eye-grabbing resume. In the end, mechanical engineers need to take on a serious number of team assignments within the course of these livelihood. You'll find a lot of ideas you must think about while creating a profitable resume to find industrial engineer's informative article. It's not necessary that in the event you should be an excellent engineer then you'll undoubtedly get a great job. So, the very first thing in becoming an avowed electrical engineer would be always to learn the exact requirements on your condition. Licensed electric engineers who move for another state will find they will need to earn a permit inside their brand new condition.
Health and safety engineers are incredibly much like industrial engineers. In order to discover methods to minimize waste and increase overall effectiveness, industrial engineers begin by analyzing product criteria and requirements. To learn more on that which it must be an Industrial Engineer, then take a look at our complete Industrial Engineer Job Description. Industrial engineers also take care of the plan and workings of the mills which make products. In addition, lots of industrial engineers pursue a MBA degree in the place of a master's degree once they truly come at the job power. Budding industrial engineers know how to plan, design and execute complex methods for a specific business. Excellent assurance engineers are located in virtually every manufacturing enterprise, and their main task would be to be sure items are always large quality.
Simply see their site and create and edit your Resume if you desire. Resumes in the us must maybe not include some marital, family, or health advice. Your resume will property from the precise same inbox, which often means you must measure your game up. The things which can make people unique or interesting cannot be found within their own resumes. The tips prepared for shifting livelihood are made by placing several challenging spending and work extra time. Along with that, the recruiter will like to be more convinced they will set you at the leading part of the consumer and appear convinced, professional. There are quite amazing recruiters out there who'd like to position you into opportunities that are amazing.
Functional formats are not advised unless there's a really unusual situation. An practical sample or template is the one that may possibly be accommodated to your purposes of people hunting for a career change. You will find 20 completely free Resume templates which you can download out of Resumetemplates.in.
That which you really need to make the journey to could be your role. Mark's current part is undefined. For just about almost any applicant who's using for a specific position which will call for many technical skills and processes, employing these resume format samples 20 17 is likely to work to their own advantage, letting them emphasize it and earn a potential employer along with recruiters realize you're fit to the project. It is fantastic for people that've already been working in precisely the exact very same area or sector, which means to those intending to keep inside the exact same market.
There aren't any two perspectives about the component of resumes for shifting your own career. Additionally, it is recommended to always check the work description to find out if the organization would like a certain document form. It will function as a useful reference for a number of your questions as you start your journeys in becoming an internet programmer. Furthermore, in the situation where the last occupation mention was not the best 1, someone might desire to select the guidance of the practical resume template.
A functional resume is rather beneficial for someone who is seeking to modify monitors, professionally talking, that might want to enter another niche or separate marketplace entirely. It's an extended, tiring grind which often feels thankless as well as also endless. It is worth time to emphasise the terms first because you'll find lots of selections of recruiters.
Should you require some advice, then you can look at a handful job descriptions which could help you decide which keywords to include on your resume. You'll acquire complete wisdom and hands-on experience when acquiring or boosting the skills needed for HR experts as well as other organizational leaders. The skills a chronological restart may not be able highlight are all set to become emphasized in a resume that is practical. Folks are hunting for leadership capabilities, also building which form of lateral sway is an exceptional means to prove that, Cramm clarified. You'll find certain resume formats that exist and you could select from. You have the capability to write an updated restart in number of the steps. Along the precise traces as grammar, there are in fact a excellent offer of opportunities to exude self confidence within your own resume.
The domain is typically the firm or functional area which you want todo it in. The best point of the site is that might go back to Resumizer Free Resume Creator Online to edit your own file that you've created together with them. You'll find lots of free on line sites that delivers free downloadable cv and Resume Templates.
Once you are hired, you wish to match the essential requirements of your job without worrying. This kind of restart writing is well suitable for candidates who would like to move towards another job industry. You will be cranking out applications and will require a method to continue to keep an eye on these. The application form is currently open. The process may be quicker, particularly if it is during BMC.
Applicant monitoring techniques must take a position to read your own resume. These candidate monitoring methods utilize software to scan your resume and choose no matter if you are a excellent fit to your circumstance. It may not be the sole technology they utilize however, its the backbone of every one of them.


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Q: Help to convert this sql query to 'JOIN' syntax I am new to SQL. Can someone help to convert this query, so it uses join syntax?
SELECT groups.name,
       users.firstname,
       users.lastname,
       users.emailaddress,
       groupmembers.userid,
       reminders.groupid,
       reminders.ownerid,
       reminderdetails.recno,
       reminderdetails.cardid,
       reminderdetails.message
FROM   reminderdetails,
       reminders,
       groupmembers,
       users,
       groups
WHERE  assocdate = 'Y'
       AND ( reminder1 != 99
             AND ( Datediff(reminderdate, '2011-09-22') = reminder1 )
              OR ( reminder2 != 99
                   AND Datediff(reminderdate, '2011-09-22') = reminder2 ) )
       AND reminders.cardid = reminderdetails.cardid
       AND groupmembers.groupid = reminders.groupid
       AND groupmembers.sendemail = 'Y'
       AND users.recno = groupmembers.userid
       AND groups.recno = reminders.groupid

A: Try this:
SELECT g.name,
       uu.firstname,
       uu.lastname,
       uu.emailaddress,
       gp.userid,
       rm.groupid,
       rm.ownerid,
       rd.recno,
       rd.cardid,
       rd.message
FROM   reminderdetails rd
JOIN   reminders rm on rm.cardid=rd.cardId
JOIN   groupmembers gp on gp.groupid=rm.groupId and gp.sendemail='Y'
JOIN   users uu on uu.recNo=gp.userId,
JOIN   groups g ON g.recno=rm.groupId
WHERE  assocdate = 'Y'
       AND ( reminder1 != 99
             AND ( Datediff(reminderdate, '2011-09-22') = reminder1 )
              OR ( reminder2 != 99
                   AND Datediff(reminderdate, '2011-09-22') = reminder2 ) )

A: Bit of a guess without knowing your table structure, but I think this seems about right.
Edit: Just seen the other one; I'd generally prefer to not load up the JOIN with too many clauses, as you end up going hunting all over the place when you want to find out what WHERE clauses are being applied. If possible, I think it's better to keep JOINs containing just the JOIN and put the rest into the WHERE.
I'm sure someone else can come along and shoot me down on that one, but that's my take on it :)
SELECT groups.name,
       users.firstname,
       users.lastname,
       users.emailaddress,
       groupmembers.userid,
       reminders.groupid,
       reminders.ownerid,
       reminderdetails.recno,
       reminderdetails.cardid,
       reminderdetails.message
FROM   groups
INNER JOIN groupmembers ON groups.groupid = groupmembers.groupid
INNER JOIN users ON groupmembers.userid = users.userid
INNER JOIN reminders ON users.userid = reminders.ownderid
INNER JOIN reminderdetails ON reminders.cardid = reminderdetails.cardid
WHERE  assocdate = 'Y'
       AND ( reminder1 != 99
             AND ( Datediff(reminderdate, '2011-09-22') = reminder1 )
              OR ( reminder2 != 99
                   AND Datediff(reminderdate, '2011-09-22') = reminder2 ) )
AND groupmembers.sendemail = 'Y'

redpajama_set_name: RedPajamaStackExchange


But not even Hari Seldon could have predicted the birth of the extraordinary creature called The Mule—a mutant intelligence with a power greater than a dozen battle fleets . . . a power that can turn the strongest-willed human into an obedient slave.