present-notes

\begin{frame}{Outline}
\hfill Percent Complete
\bi
\item Modeling cascading power failures, Ch. 2 \hfill (92\%)
\bi
\item Decision dependent uncertainty 
\item Effective capacity and a priori sampling
\item Multi-stage integer program (MSIP)
\ei
\item Risk model for real-time dispatch, Ch. 4 \hfill (78\%)
\bi
\item Endogenous risk due to line loading
\item Line failure risk function and net injection uncertainty
\item Joint chance constraint (JCC)
\ei
\item Solving design problems, Ch. 3 \hfill (42\%)
\bi
\item Efficient OPA simulation with variance reduction
\item Enhanced solves via accessory information
\item Derivative free optimization (DFO)
\ei
\ei

\end{frame}


\begin{frame}{OPA Simulation Algorithm}
%\begin{algorithm}
%\caption{OPA Cascading Algorithm}\label{opa_alg}
\tiny
\begin{algorithmic}
\Procedure{OPA}{$x, D, \xi, \omega$}
\State Solve (DC OPF) to find base case load shed $z_0$
\State $\xi$ occurs and corresponding changes to the grid are made
\State Stage $s \gets 1$
\While{ Not DONE }
\State Solve (DC OPF) to find power injects and branch flows for adjusted grid
\State $\mathbb{O}_s \gets \emptyset$
\For{$\forall e \in \cE $}
\State $\mathbb{O}_s = 
\left\{ 
\begin{array}{lr}
  \mathbb{O}_s + \left\{ e \right\} & \mbox{w/ prob. } h(y_e), \mbox{ draw } \omega_{es} \\
  \mathbb{O}_s & \mbox{o/w }
\end{array}
\right. $ 
\If{ $\mathbb{O}_s \neq \emptyset$ }
\State Modify Grid with $\mathbb{O}_s$
\State $s\gets s+1$
\Else
\State $s^* \gets s,$ calculate $z_{s^*}$, DONE
\EndIf
\EndFor
\EndWhile
\State \label{done}Load Shed $  \lambda\left(x,D,\xi,\omega\right)  = z_{s^*} - z_0$
\EndProcedure
\end{algorithmic}
%\end{algorithm}
\end{frame}

\begin{frame}{Dispatch Point Data}
Cost of the different dispatch points
\begin{center}
 \begin{tabular}{ |c| c c c c c c|}
\hline
& OPF & CC .01 & CC .05 & JCC .98 & JCC .9 & JCC .8 \\
\hline
\hline
Cost & 565.215 & 590.172 & 572.833& 589.594 & 581.92 &579.135\\
%$r$ & 0.00504327 & 0.0015044 & 0.0169355 & 0.000309812 &.0028877 &0.00466\\  -- shown in picture
\hline
\end{tabular}
\end{center}
Expected number of lines over their threshold
\begin{center}
\begin{tabular}{| r  c |}
\hline
Risk :& Ex[$y_e > U_e$] \\
\hline
\hline
OPF:& 0.3875\\
CC .01:& 0.0225\\
CC .05:& 0.1971 \\
JCC .8:& 0.31 \\
JCC .9:& 0.09 \\
JCC .98:& 0.0125 \\
\hline
\end{tabular}
\end{center}
\end{frame}
\begin{frame}{Piece-wise Square}
\begin{center}
\includegraphics[scale=.96]{jcc/fig-sensesquare}
\end{center}
\end{frame}

\begin{frame}{Algorithm}
\tiny
\begin{algorithmic}
\Procedure{JCC}{d,$\Sigma^m$,$\epsilon$,$\epsilon_g$,L,p}
\State $L \gets \emptyset$  (Set of Lines with potential risk)
\State $S \gets \emptyset$  (Set of Cuts)
\State $r \gets 0$ (Risk)
\BState \emph{solve}:
\State $(\hat{x},\hat{\beta},\hat{y}) \gets $Solve DC Power Flow, \cref{jcc_obj,jcc_cons,jcc_kcl,jcc_slack,jcc_limit,jcc_gen}, with cuts $S$, risk $r\leq\epsilon$
\If {Infeasible} \Return Problem Infeasible 
\EndIf
\State Calculate $\hat{s},\hat{z},\hat{r}$ using $(\hat{x},\hat{\beta},\hat{y})$ and \cref{branch_cov,line_risk}
\If {$\hat{r} \leq \epsilon + tol$} \Return Optimal $(\hat{x},\hat{\beta},\hat{y},\hat{s},\hat{z},\hat{r})$
\EndIf
\For{$\forall e$}
\If {$\hat{z_e} \geq tol$}
    \If {$e \notin L$}
            \State $L \gets \left\{L,e\right\}$
            \State Initialize $s_e,z_e$
            \State $r \gets r + z_e$
    \EndIf            
    \State $S \gets$ line risk cuts \ref{line_risk_cuts} for $z_e,y_e,s_e$ dependent on $\hat{z}_e,\hat{y}_e,\hat{s}_e$
    \State $S \gets$ branch variance cuts \ref{branch_var_cuts} for $s_e,\beta_e$ dependent on $\hat{s}_e,\hat{\beta}_e$
\EndIf
\EndFor
\State \textbf{goto} \emph{solve}
\EndProcedure
\end{algorithmic}
\end{frame}

\begin{frame}{Risk Function}
Risk function is convex, but not analytic
\bi
\item Cutting Planes
\ei
\begin{subequations}
\label{line_risk_cuts}
\begin{align}
z_e \geq g(\hye,\hse) &+ \frac{\partial g}{\partial y_e}(\hye,\hse) \left(y_e - \hye \right) 
+ \frac{\partial g}{\partial s_e}(\hye,\hse) \left(s_e - \hse \right) \\
g(y_e,s_e) &= (a + b y_e)\left[ 1 - \Phi(\alpha_L) \right]  + b s_e \phi(\alpha_L)  \\
 \frac{\partial}{\partial y_e}g(y_e,s_e) &= b\left[ 1 - \Phi(\alpha_L) \right]\\
\frac{\partial}{\partial s_e}g(y_e,s_e) &= b \phi(\alpha_L) 
\end{align}
\end{subequations}

\end{frame}

\begin{frame}{Standard Deviation of Branch Flow}
St. Dv. is second order cone, but only need for small number of lines
\bi
\item Cutting Planes
\ei
\begin{subequations}
\label{branch_var_cuts}
\begin{align}
s_e \geq \fehb &+ \sum_j \pfehb \left( \beta_j - \hat{\beta_j} \right)\\
  \feb &= \sqrt{\pe^2 \sD - 2 \pe \se  + \see }\\
  \pfeb &= \frac{A_{ej} \left( \pe \sD - \se \right)}{\sqrt{\pe^2 \sD - 2 \pe \se  + \see }}
\end{align}
\end{subequations}
\end{frame}

Line risk from second term
\begin{equation}\label{line_risk}
z_e(\mu^y_e,\sigma^y_e) = (a + b \mu^y_e)\left[ 1 - \Phi(\alpha_L) \right]  + b \sigma^y_e \phi(\alpha_L) 
\end{equation}
\begin{tabular}{c l}
$\mu^y_e$ & Mean branch flow \\
$\sigma^y_e$ & Standard deviation of branch flow \\
$\Phi(\cdot)$ & CDF of standard normal distrubtion \\
$\phi(\cdot)$ & PDF of standard normal distrubtion \\
$\alpha_L$ & $\frac{L - \mu^y_e}{\sigma^y_e}$
\end{tabular}



\subsection{Example}
\begin{frame}{Northeast America Blackout 2003}
This grid had load of 28,700 MW before event started \\
At worst, load 5,716 MW, $< 20 \% $ of original \\
Took around 4 to 5 hours\\
Initial events 1 hour apart, near end, within seconds \newline
\\
Effects 
\begin{itemize}
\item 45 million people affected in 8 states, 10 million in Ontario
\item Estimated economic impact of around \$6 billion for US, \$2 billion for Canada
\item Power slowly brought back 1-3 days
\item Some areas lost water pressure 
\item Trains in and out of New York shut down
\item Some industry not back to full production until 7 to 8 days later
\item Loss of Life
\end{itemize}
\end{frame}

\begin{frame}{Northeast America Blackout 2003}
\begin{itemize}
\item System load was less than historic peak
\item Large amount of imported power to Cleveland-Akron Area
\item David-Besse Nuclear Plant in long-term forced outage
\item Capacitor banks in planned maintanence
\item Hot and humid, multiple tree contacts
\item Server outages and communication problems

-Inability for operators to react
\pause
\end{itemize}
\BBR{Fast acting trips take out Northeast Power Grid}
\begin{itemize}
\item Thousands of events happened within last minutes 
\item Dynamic power swings responsible, voltage instability companion  
\end{itemize}
\EBR
\end{frame}

\begin{frame}{Line Capacities}
\begin{itemize}
\item Understand the effects line capacities have on overall cascading process
\item Use this information to aid in Derivative Free Optimization
\end{itemize}
\pause
But How?
\pause
\begin{center}
\textbf{Pictures}, of course!

\includegraphics[scale=.5]{pictures}
\end{center}
\end{frame}


\begin{frame}{Expected Load Shed}
\centering
\begin{itemize}
\item Moving along coordinate axis
\item Adding up to 3x capacity to line
\item Standard error about $\pm 1$
\end{itemize}
\includegraphics[scale=.35]{exls1}
\end{frame}


\begin{frame}{Expected Load Shed}
\centering
\includegraphics[scale=.35]{exls2}
\end{frame}


\begin{frame}{Expected Load Shed}
\centering
\includegraphics[scale=.35]{exls3}
\end{frame}

\begin{frame}{Expected Load Shed}
\centering
\includegraphics[scale=.35]{exls4}
\end{frame}

\begin{frame}{and Other Risk Measures}
\centering
\includegraphics[scale=.35]{riskmeasures}
\end{frame}



\begin{frame}{Derivative Free Optimization}
Bad behavior and secondary information \newline
Clusters of lines which help explain expectation of load shed
\begin{center}
\includegraphics[scale=.3]{clusters}
\end{center}

\end{frame}


\section{Discussion}
\subsection{Collaboration}
\begin{frame}{Graph Connections and Optimization}
\alert{Graph Theory Connections}
Find connections with clusters of lines from this and those from graph properties
\begin{itemize}
\item Possible connection with partition and max flow cuts - Halappanavar
\end{itemize}

\alert{Optimization with Simulation}
Use simulation in framework proposed by Ferris 
\begin{itemize}
\item Economic dispatch such that cascade risk measures stays bounded
\end{itemize}
\begin{subequations}
\begin{align}
\min \hspace{10px} & f(x,y)	\\
\mbox{s.t.} \hspace{10px}& g(y) \le 0	\\
	& y = S(x) \in \cB 
\end{align}
\end{subequations}

\end{frame}

\begin{frame}{N-k Criteria and Chance Constraints}
\alert{Economic Dispatch such that no more than $k$ branches lost }

Ex. Avoid $k > 1$ for large percentage of scenarios
\begin{equation}
P\left\{\text{Line Outage} \le k \right\} \ge 1 - \epsilon
\end{equation}
Binary variable for each scenario on whether it has held
\begin{subequations}
\begin{align}
\sum_i z_{is} &\le	M_s \hat{z_s} + | \xi_s | + k \\
\sum_s \hat{z_s} &\le \epsilon | \cS |
\end{align}
\end{subequations}
%where $M_s = | \cE | - | \xi_s | - k$.  



\alert{Redispatch and Stability Regions}
Chance constraint on system staying in stability region after contingencies. - DeMarco simulation

\begin{itemize}
\item Possible to understand stability regions through simulation?
\item What type of constraints can describe the regions?
\end{itemize}

\end{frame}

\begin{frame}{Derivative Free Optimization and Redispatch}
\alert{Derivative free optimization} used on continuous and smooth functions
\begin{itemize}
\item Cascade process doesn't fit this
\item Want to use for local convergence
\item Still need global method to find promising regions
\end{itemize}
\alert{Stochastic Redispatch} seems to be naturally suited to chance constraints
\begin{itemize}
\item Ensure reserves are capable of covering a percentage of contingencies
\item Unit commitment model which ensures robust reserve allocation
\end{itemize}
\end{frame}


\section{Dynamic Equillibrium}
\subsection{Forces}
\begin{frame}{Forces on System}
Full OPA model tries to capture these forces and find an equillibrium that is seen in the real world \newline \\
Short Term Forces:
\begin{itemize}
\item Random demand fluctuations
\item Blackouts, Cascading Process
\item Dispatch Model
\end{itemize}
Long Term Forces:
\begin{itemize}
\item Economic and political processes
\item System repair
\item Load growth
\item Design and upgrade of system
\end{itemize}
\end{frame}


\subsection{Equillibrium}
\begin{frame}{Equillibrium}
Self-organized into dynamic equillibrium where blackouts of all sizes occur\newline
\\ 
Average frequency of blackouts steady for 30 years

Power tail distribution, exponent around $-1.3\pm.2$
\newline \\
With given forces, OPA model gives consistent results with data

\end{frame}


\subsection{Dispatch Models}
\begin{frame}{Critical points}
\alert{Stay away from critical points!}
\begin{itemize}
\item A critical point is when the system is at a maximum throughput operating point, that is, it is constrained by line limits or generator limits. 
\item Power systems tend towards these spots due to economic pressures. 
\item  Critical points are the cheapest way to provide power if there were no concern of outages or uncertainty.
\end{itemize}
Two types of critical points
\begin{itemize}
\item Inadequate generation capacity - leads to a lot of smaller blackouts
\item Inadequate transmission system capacity - leads to few large blackouts.
\end{itemize}
\end{frame}

\begin{frame}{Dispatch Model}

First solve economic dispatch model subject to power flow constraints to find
\begin{itemize}
\item Least cost dispatch at cost $B^*$.
\end{itemize}
 Find ``reliability frontier" for an increased budget.
\begin{subequations} \label{rel}
\begin{align} 
\max	& \hspace{10px} \lambda r_g(g) + (1-\lambda) r_f(f) \\
\mbox{s.t.}	& \hspace{10px}c(g) \le B^* + \delta	\\
  &	\hspace{10px}\mbox{ power flow constraints} 
\end{align}
\end{subequations}
\begin{itemize}
\item $\lambda$ will be used to express the trade off between lots of small outages or few large outages.
\item   $\delta$ represents the increased budget to find a new operating point.  
\item $c(g)$ is the cost function for the generator dispatch $g$.  
\item As $\delta \rightarrow 0$, (\ref{rel}) will approach the economic dispatch point. 
\item $f$ is power flows on the transmission lines.  
\end{itemize}

\end{frame}

\begin{frame}{Reliability Function}

An example of possible reliability functions may be to maximize a weighted excess capacity for generators and transmission lines.
\begin{align}
r_g(g) &= \sum_i \left[ \alpha_i (g_i^+ - g_i ) + \beta_i (g_i - g_i^-) \right]\\
r_f(f) &= \sum_{ij} \gamma_{ij} \left( \frac{U_{ij} - f_{ij}}{U_{ij}} \right)
\end{align}
\begin{itemize}
\item $\alpha$ and $\beta$ will represent the relative need for up-ramping and down-ramping capabilitie
\begin{itemize}
\item desired geographical location of ramping rates through the relative size of $\alpha_i$ at any node $i$.  
\end{itemize}
\item $\gamma_{ij}$ can be developed using electrical information of the transmission line to ensure that the core network is more stable. 
\begin{itemize}
\item More weight can be point on critical lines to the power grid.
\end{itemize}
\end{itemize}

\end{frame}


\begin{frame}{END}
\end{frame}

\section{Break points}
\begin{frame}{Changing Branch Capacities}
What happens when you change branch capacities? \newline\\
Anything! \newline \\
To start to get a handle on problem, fix random variables \newline \\
Estimate Load Shed Distribution \newline \\
Initial Outages  $\hat{\xi} =\left\{ 400, 146, 9, 299, 119, 321, 389, 393 \right\}$ \newline \\
Demand Scenario $\hat{\theta}$ is from test instance data \newline \\
Step along one branch element's capacity, $x_{284}$
\end{frame}

\subsection{More Capacity Less Load Shed}
\begin{frame}{Effects of Capacity Change on Load Shed}
\includegraphics[scale=.4]{examplegood.pdf}
\end{frame}
\begin{frame}{Why Did This Happen}
As capacity was added to line 284, the Mean Load Shed, V@R and CV@R improved \newline \\
This was not monotonic
\begin{itemize}
\item Capacity has large effect on neighboring lines 
\item as $x_{284}$ increases from low values, neighbor lines start to overload,
	leads to larger load shed
\item as $x_{284} \rightarrow 1.1x_{284,ref}$, no important lines overloaded 
\item as $x_{284}$ increases further, no changes to cascade 
\end{itemize}

\end{frame}
\subsection{More Capacity More Load Shed}
\begin{frame}{More Capacity is not Always Good}
\includegraphics[scale=.4]{examplebad.pdf}
\end{frame}
\begin{frame}{Why Did this Happen}
Why does the load shed go up when the system is adding capacity?
\begin{itemize}
\item discontinuous jump in load shed at $x'_{301} \in [1.04, 1.17]$
\item For $x_{301} < 1.04 $ \newline
	Cascade can outage lines 288, 299, 301 \newline
	As $x_{301}$ approaches 1.04, lines 288, 299 no longer overloaded \newline
	Outage of line 301 has minimal system impacts 
\item For $x_{301} > 1.17 $ \newline
	Line 301 no longer overloaded, area of congestion shifted \newline
	But lines 127, 226 now overloaded and can be outaged \newline
	This weakens system considerably, leading to large scale cascades
\end{itemize}
\end{frame}

\section{Risk Model for Real-Time Dispatch}


\end{document}