commands.tex

\newcommand{\plotscale}{0.8}
\newcommand{\plotscaleOne}{0.45}
\newcommand\plotsizeAppendix{0.89}
\newcommand\gapsizeAppendix{\vspace{-0.7cm}}
\newcommand\trimLeft{0}
\newcommand\trimBottom{0}
\newcommand\trimRight{0}
\newcommand\trimTop{8}

\newcommand{\forbidden}{\ensuremath{T}\xspace} % for tabu
\newcommand{\hittingSet}{\ensuremath{H}\xspace} % for tabu
\newcommand{\coloredgraph}{\ensuremath{\mathcal{G}}\xspace} % for tabu

\def\includeplot#1{
    % Solve the relative path problem by recursively included eps files from the gnuplot's tex files
    \let\igx\includegraphics
    %\def\includegraphics##1{\igx{plots/##1-eps-converted-to.pdf}}
    \def\includegraphics##1{\igx{windfarmplacement/plots/##1}}
%   \resizebox{\columnwidth}{!}{\input{plots/#1}}
\input{windfarmplacement/plots/#1}
}

% Notation / Abbreviations.
\newcommand{\leftish}{\ensuremath{\ell}\xspace}
\newcommand{\rightish}{\ensuremath{r}\xspace}
\newcommand{\eg}{e.\,g.\xspace}
\newcommand{\ie}{i.\,e.\xspace}
% \newcommand{\wlog}{w.\,l.\,o.\,g.\xspace}
\newcommand{\Wlog}{W.\,l.\,o.\,g.\xspace}
\newcommand{\wrt}{with respect to\xspace}
\newcommand{\cf}{c.\,f.\xspace}
\newcommandx{\MAX}[1]{\overline{#1}\xspace}
\newcommandx{\MIN}[1]{\underline{#1}\xspace}

% Operations.
\DeclareMathOperator{\mergeop}{merge}
\DeclareMathOperator{\linkop}{link}
\DeclareMathOperator{\shiftop}{shift}
\DeclareMathOperator{\extendop}{extend}
\DeclareMathOperator{\convertop}{convert}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\boundop}{bound}
% \DeclareMathOperator*{\argmin}{arg\,min}
% \DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmax}{\arg\max}
\DeclareMathOperator*{\argmin}{\arg\min}

% General stuff and hacks.
\DeclareRobustCommand{\ubar}[1]{\ensuremath{\underaccent{\bar}{#1}}}
\newcommand{\obar}[1]{\ensuremath{\bar{#1}}}
\newcommand*{\minx}{\mathmakebox[\widthof{$\max$}][r]{\min}}
\newcommand{\mytilde}{\ensuremath{\operatorname{\hspace{2pt}\tilde{\protect\phantom{\protect\raisebox{1.0pt}{c}}}}\hspace{-9pt}}}
\newcommand{\inverse}{{\hspace{0.5pt}\protect\raisebox{0.5pt}{\protect\scalebox{0.6}[1.0]{$\scriptstyle-$}\hspace{-0.6pt}$\scriptscriptstyle1$}\hspace{-1pt}}}
\newcommandx{\transpose}[1]{\ensuremath{#1^\intercal}\xspace}
%Matrices
\newcommand{\adjacencyMatrix}{\ensuremath{\mathbf{A}}\xspace}%A
\newcommand{\adjacencyMatrixVector}{\ensuremath{\mathbf{a}}\xspace}%A
\newcommandx{\adjacencyMatrixEntry}[2]{\ensuremath{\mathbf{\adjacencyMatrix_{#1,#2}}}\xspace}
% 
\newcommand{\admittanceMatrix}{\ensuremath{\mathbf{Y}}\xspace}%Y
\newcommand{\admittanceMatrixElement}{\ensuremath{\mathbf{Y_{\edge}}}\xspace}%y
\newcommand{\admittanceMatrixNode}{\ensuremath{\mathbf{Y_n}}\xspace}%y
\newcommandx{\admittanceMatrixEntry}[2]{\ensuremath{
\mathbf{\admittanceMatrix_{#1,#2}}}\xspace}
% 
\newcommand{\polyhedron}{\ensuremath{P}\xspace}%P
\newcommand{\righthandsidevector}{\ensuremath{\realpower}\xspace}
\newcommandx{\frighthandsidevector}[1]{\ensuremath{\vv{\righthandsidevector_
{#1}}}\xspace}
\newcommand{\constraintmatrix}{\ensuremath{\mathbf{A}}\xspace}%A
\newcommandx{\constraintmatrixRowElement}[1]{\ensuremath{\mathbf{A}_{#1}}\xspace}%A
\newcommand{\treeAdmittanceProduct}{\ensuremath{\triangle_n}\xspace}%
\newcommand{\cycleMatrix}{\ensuremath{\mathbf{B}}\xspace}
\newcommand{\incidenceMatrix}{\ensuremath{\mathbf{I}}\xspace}
\newcommandx{\incidenceMatrixElement}[1]{\ensuremath{\vv{i_{#1}}}\xspace}
\newcommandx{\incidenceMatrixEntry}[2]{\ensuremath{\mathbf{\incidenceMatrix_{#1,#2}}}\xspace}
\newcommand{\cutsetMatrix}{\ensuremath{\mathbf{Q}}\xspace}
\newcommand{\laplaceMatrix}{\ensuremath{\mathbf{L}}\xspace}
\newcommandx{\laplaceMatrixEntry}[2]{\ensuremath{\mathbf{\ell_{#1,#2}}}\xspace}
\newcommand{\diagonalMatrix}{\ensuremath{\mathbf{D}}\xspace}

\newcommand{\rank}{\ensuremath{\mathrm{rk}}\xspace}
\newcommandx{\rankx}[1]{\ensuremath{\rank(#1)}\xspace}
% 
\newcommand{\nullity}{\ensuremath{n}\xspace}
\newcommandx{\nullityx}[1]{\ensuremath{\nullity(#1)}\xspace}
% 
\newcommand{\dimension}{\ensuremath{\mathrm{dim}}\xspace}
\newcommandx{\dimensionx}[1]{\ensuremath{\dimension(#1)}\xspace}
% 
\newcommand{\image}{\ensuremath{\mathrm{img}}\xspace}
\newcommandx{\fimage}[1]{\ensuremath{\image\left(#1\right)}\xspace}
% 
\newcommand{\kernel}{\ensuremath{\mathrm{ker}}\xspace}
\newcommandx{\fkernel}[1]{\ensuremath{\kernel\left(#1\right)}\xspace}
% 
\newcommand{\distance}{\ensuremath{d}\xspace}
\newcommand{\algo}{\ensuremath{\mathcal{A}}\xspace}
\newcommand{\solutionmap}{\ensuremath{\mathtt{SolutionMap}}\xspace}
\newcommand{\instancemap}{\ensuremath{\mathtt{InstanceMap}}\xspace}
\newcommand{\reversesolutionmap}{\ensuremath{\mathtt{ReverseSolutionMap}}\xspace}
\newcommandx{\fmagnitude}[1]{\ensuremath{\lvert{#1}\rvert}\xspace}
% Graph.
\usepackage{esvect}
\newcommand{\network}{\ensuremath{\mathcal{N}}\xspace}
\newcommand{\graph}{\ensuremath{G}\xspace}
\newcommand{\dualgraph}{\ensuremath{\graph^\star}\xspace}
\newcommand{\forest}{\ensuremath{\mathcal{T}}\xspace}
\newcommand{\tree}{\ensuremath{T}\xspace}
\newcommand{\treeSet}{\ensuremath{\mathcal{T}}\xspace}
\newcommand{\embedding}{\ensuremath{\mathcal{E}}\xspace}
\newcommand{\points}{\ensuremath{\mathcal{P}}\xspace}
\newcommand{\branches}{\ensuremath{B}\xspace}
\newcommand{\minor}{\ensuremath{H}\xspace}
\newcommand{\subgraph}{\ensuremath{H}\xspace}
\newcommand{\connectedComponent}{\ensuremath{C}\xspace}
\newcommand{\face}{\ensuremath{\cycle}\xspace}
% \newcommand{\blockpath}{\ensuremath{T_B}}
\newcommand{\blocks}{\ensuremath{B}\xspace}
\newcommand{\block}{\ensuremath{\beta}\xspace}%b
% \newcommand{\paralleledgemap}{\ensuremath{\mathrm{para}}\xspace}
\newcommand{\paralleledgeMap}{\ensuremath{\mathrm{\foneToOneMap{para}}}\xspace}
\newcommand{\bilinearmap}{\ensuremath{\mathrm{\foneToOneMap{bilin}}}\xspace}
% \newcommand{\bilinearmap}{\ensuremath{\mathrm{bilin}}\xspace}

% Vertices
\newcommand{\vertices}{\ensuremath{V}\xspace}
\newcommand{\factsbus}{\ensuremath{\vertices_F}\xspace}
\newcommandx{\ffactsbus}[1]{\ensuremath{\glssymbol{factsbus}^{c=#1}}\xspace}
\newcommand{\factsbranch}{\ensuremath{\edges_F}\xspace}
\newcommandx{\ffactsbranch}[1]{\ensuremath{\glssymbol{factsbranch}^{c=#1}}\xspace}
\newcommand{\generators}{\ensuremath{\vertices_G}\xspace}
\newcommand{\consumers}{\ensuremath{\vertices_D}\xspace}
\newcommand{\terminals}{\ensuremath{T}\xspace}
\newcommand{\sourceTerminals}{\ensuremath{\terminals_{\source}}\xspace}
\newcommand{\sinkTerminals}{\ensuremath{\terminals_{\sink}}\xspace}
\newcommand{\source}{\ensuremath{s}\xspace}
\newcommand{\sink}{\ensuremath{t}\xspace}
\newcommand{\target}{\ensuremath{t}\xspace}
\newcommand{\vertex}{\ensuremath{v}\xspace}
\newcommand{\vertexa}{\ensuremath{u}\xspace}
\newcommand{\vertexb}{\ensuremath{v}\xspace}
\newcommand{\vertexc}{\ensuremath{w}\xspace}
\newcommand{\vertexStar}{\ensuremath{c}\xspace}

% Circuit
\newcommand{\circuit}{\ensuremath{i}\xspace}
\newcommand{\cost}{\ensuremath{\gamma}\xspace}
\newcommandx{\fcost}[1]{\ensuremath{\cost_{#1}}\xspace}
\newcommand{\setPiecwisefct}{\ensuremath{F}\xspace}
\newcommandx{\fsetPiecwisefct}[1]{\ensuremath{\setPiecwisefct_{#1}}\xspace}
\newcommand{\loss}{\ensuremath{\ell}\xspace}
\newcommandx{\floss}[1]{\ensuremath{\loss_{#1}}\xspace}
\newcommand{\gencost}{\ensuremath{c_g}\xspace}
\newcommand{\losscost}{\ensuremath{c_\ell}\xspace}
\newcommand{\totalgenlosscost}{\ensuremath{c_\lambda}\xspace}

% Edges
\newcommandx{\makeUndirected}[1]{\ensuremath{\overleftrightsmallarrow{#1}}\xspace}
% 
\newcommand{\edges}{\ensuremath{E}\xspace}
\newcommand{\factsedge}{\ensuremath{\edges_F}\xspace}
\newcommandx{\foneToOneMap}[1]{\ensuremath{\mathrm{\mu}_{#1}}\xspace}
\newcommand{\oneToOneMap}{\ensuremath{\mathrm{\foneToOneMap{dual}}}\xspace}
\newcommand{\embeddingMap}{\ensuremath{\mathrm{\foneToOneMap{\glssymbol{embedding}}}}\xspace}
\newcommand{\undirectededges}{\ensuremath{\makeUndirected{\edges}}\xspace}
\newcommand{\undirectededge}{\ensuremath{\makeUndirected{e}}\xspace}
\newcommandx{\undirected}[2]{\ensuremath{\makeUndirected{(#1,#2)}}\xspace}
\newcommand{\edge}{\ensuremath{e}\xspace}
\newcommand{\dualedge}{\ensuremath{\edge^\star}\xspace}
\newcommand{\edgea}{\ensuremath{\edge_1}\xspace}
\newcommand{\edgeb}{\ensuremath{\edge_2}\xspace}
\newcommand{\cuts}{\ensuremath{H}\xspace}
\newcommand{\apath}{\ensuremath{\pi}\xspace}
\newcommand{\edgeCost}{\ensuremath{a}\xspace}

% Flow
\newcommand{\flow}{\ensuremath{f}\xspace}
\newcommand{\flowa}{\ensuremath{\flow_1}\xspace}
\newcommand{\flowb}{\ensuremath{\flow_2}\xspace}
\newcommand{\netflow}{\ensuremath{\flow_{\mathrm{net}}}\xspace}

%%%%%%%%%%%%% Others %%%%%%%%%%%%% 
\newcommand{\partition}{\ensuremath{P}\xspace}
\newcommand{\neighbor}{\ensuremath{N}\xspace}
\newcommand{\inneighbor}{\ensuremath{N^+}\xspace}
\newcommand{\outneighbor}{\ensuremath{N^-}\xspace}
\newcommand{\partitiona}{\ensuremath{\glssymbol{partition}_1}\xspace}
\newcommand{\partitionb}{\ensuremath{\glssymbol{partition}_2}\xspace}
\newcommand{\boolsubstation}{\ensuremath{z}\xspace}
\newcommand{\cycle}{\ensuremath{c}\xspace}
\newcommandx{\fcycle}[2]{\ensuremath{\cycle(#1,#2)}\xspace}
\newcommand{\cycles}{\ensuremath{C}\xspace}
\newcommand{\cutset}{\ensuremath{K}\xspace}
\newcommand{\cutsets}{\ensuremath{\mathcal{\cutset}}\xspace}
% 
\newcommand{\labels}{\ensuremath{L}\xspace}
\newcommand{\paretolabels}{\ensuremath{D}\xspace}
\newcommand{\lab}{\ensuremath{\mathrm{\ell}}\xspace}
\newcommand{\labnew}{\ensuremath{\lab_\mathrm{new}}\xspace}
\newcommand{\col}{\ensuremath{c}\xspace}
\newcommand{\colorlabels}{\ensuremath{C}\xspace}
\newcommand{\orderedlist}[1]{\ensuremath{\langle #1\rangle}\xspace} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Power Grid
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Network tuple
\newcommand{\acnetworktuple}{
    \ensuremath{\glssymbol{network} = (}
        \ensuremath{\glssymbol{graph}=(\glssymbol{vertices},\glssymbol{edges}),}
        \glssymbol{generators},
        \glssymbol{consumers},
        \ensuremath{\glssymbol{capacity},}
        \glssymbol{resistance},
        \glssymbol{reactance},
        \glssymbol{susceptance},
        \glssymbol{conductance},
        \glssymbol{voltageangledifferencemin},
        \glssymbol{voltageangledifferencemax},
        \glssymbol{voltagemin},
        \glssymbol{voltagemax},
        \glssymbol{realpowergenerationmin},
        \glssymbol{realpowergenerationmax},
        \glssymbol{reactivepowergenerationmin},
        \glssymbol{reactivepowergenerationmax},
        \glssymbol{realpowerdemandmin},
        \glssymbol{realpowerdemandmax},
        \glssymbol{reactivepowerdemandmin},
        \glssymbol{reactivepowerdemandmax}
    \ensuremath{)}
    \xspace
}
\newcommand{\dcnetworktuple}{
    \ensuremath{\glssymbol{network} = (}
        \glssymbol{graph}, 
        \glssymbol{generators}, 
        \glssymbol{consumers}, 
        \ensuremath{\glssymbol{capacity},}
        \glssymbol{susceptance},
        \glssymbol{realpowergenerationmin}, 
        \glssymbol{realpowergenerationmax}, 
        \glssymbol{realpowerdemandmin},
        \glssymbol{realpowerdemandmax}
    \ensuremath{)}
    \xspace
}
\newcommand{\flownetworktuple}{
    \ensuremath{\glssymbol{network} = (}
        \glssymbol{graph}, 
        \source, 
        \sink, 
        \ensuremath{\glssymbol{capacity},}
        \glssymbol{realpowergenerationmin}, 
        \glssymbol{realpowergenerationmax}, 
        \glssymbol{realpowerdemandmin},
        \glssymbol{realpowerdemandmax}
    \ensuremath{)}
    \xspace
}
\newcommand{\rectangulardissection}{\ensuremath{\mathcal{R}}\xspace}
% Resistance
\newcommand{\resistance}{\ensuremath{r}\xspace}
\newcommand{\resistances}{\ensuremath{R}\xspace}
% Resistivity
\newcommand{\resistivity}{\ensuremath{\rho}\xspace}
% Reactances
\newcommand{\reactance}{\ensuremath{x}\xspace}
\newcommand{\reactances}{\ensuremath{X}\xspace}
% Susceptance
\newcommand{\susceptance}{\ensuremath{b}\xspace}
\newcommand{\linechargingsusceptance}{\ensuremath{b_{\mathrm{ch}}}\xspace}
\newcommand{\susceptances}{\ensuremath{B}\xspace}
% Conductance
\newcommand{\conductance}{\ensuremath{g}\xspace}
\newcommand{\conductances}{\ensuremath{G}\xspace}
% Conductivity
\newcommand{\conductivity}{\ensuremath{\sigma}\xspace}
% Impedance
\newcommand{\impedance}{\ensuremath{z}\xspace}
\newcommand{\impedances}{\ensuremath{Z}\xspace}
\newcommand{\impedancebase}{\ensuremath{\impedances_{\mathrm{base}}}\xspace}
% Admittance
\newcommand{\admittance}{\ensuremath{y}\xspace}
\newcommand{\linechargingadmittance}{\ensuremath{y_{\mathrm{ch}}}\xspace}
\newcommand{\admittances}{\ensuremath{Y}\xspace}
\newcommand{\admittancebase}{\ensuremath{\admittances_{\mathrm{base}}}\xspace}
% Transformer
\newcommand{\tapratio}{\ensuremath{\tau}\xspace}
% Frequency
\usepackage{fge}
\newcommand{\frequency}{\ensuremath{\fges}\xspace}%\tilde{f}
\newcommand{\angularfrequency}{\ensuremath{\omega}\xspace}%\tilde{f}
% Capacity
\DeclareMathOperator{\capacity}{\ensuremath{cap}\xspace}
\DeclareMathOperator{\capacitybf}{\ensuremath{\mathbfsf{cap}}\xspace}
\DeclareMathOperator{\mincapacity}{\ensuremath{\MIN{\glssymbol{capacity}}}\xspace}
\newcommand{\NOMcapacity}{\ensuremath{\glssymbol{capacity}}\xspace}
\newcommandx{\fcapacity}[3]{\ensuremath{\glssymbol{capacity}_{#1}(#2,#3)}\xspace}
\newcommandx{\fmincapacity}[2]{\ensuremath{\MIN{\capacity}_{#1}(#2)}\xspace}
\newcommand{\capacityTightness}{\ensuremath{\xi}\xspace}
\newcommand{\substationCapacityVariance}{\ensuremath{\delta}\xspace}
\newcommand{\shapeAspectRatio}{\ensuremath{\beta}\xspace}
\newcommand{\kNearestNeighbor}{\ensuremath{k}\xspace}
\newcommand{\edgeInclusionValue}{\ensuremath{\epsilon}\xspace}
\newcommand{\thermalConductivityCapacity}{\ensuremath{\tau}\xspace}
% Voltages
\newcommand{\voltage}{\ensuremath{v}\xspace}
\newcommand{\voltages}{\ensuremath{V}\xspace}
\newcommand{\voltagemin}{\ensuremath{\MIN{\voltage}}\xspace}
\newcommand{\voltagemax}{\ensuremath{\MAX{\voltage}}\xspace}
\newcommand{\vmagnitude}{\ensuremath{|\voltage|}\xspace}
\newcommand{\voltagebase}{\ensuremath{\voltages_{\mathrm{base}}}\xspace}
\newcommand{\voltagerms}{\ensuremath{\voltages_{\mathrm{RMS}}}\xspace}
% Current
\newcommand{\current}{\ensuremath{i}\xspace}
\newcommand{\currents}{\ensuremath{I}\xspace}
\newcommand{\currentmax}{\ensuremath{\MAX{\current}}\xspace}
\newcommand{\currentbase}{\ensuremath{\currents_{\mathrm{base}}}\xspace}
\newcommand{\currentrms}{\ensuremath{\currents_{\mathrm{RMS}}}\xspace}
% Voltage and current angle
\newcommand{\aangle}{\ensuremath{\theta}\xspace}
\newcommand{\vangle}{\ensuremath{\aangle^\voltage}\xspace}
\newcommand{\vanglemin}{\ensuremath{\MIN{\vangle}}\xspace}
\newcommand{\vanglemax}{\ensuremath{\MAX{\vangle}}\xspace}
\newcommand{\iangle}{\ensuremath{\aangle^\current}\xspace}
\newcommand{\vangleshift}{\ensuremath{\aangle^\voltage_\mathrm{shift}}\xspace}
\newcommand{\anglea}{\ensuremath{\vangle(\vertexa)}\xspace}
\newcommand{\angleb}{\ensuremath{\vangle(\vertexb)}\xspace}
\newcommand{\deltaangle}{\ensuremath{\Delta\vangle}\xspace}
\newcommand{\deltavangle}{\ensuremath{\Delta\vangle}\xspace}
\newcommand{\deltavanglemin}{\ensuremath{\MIN{\Delta\vangle}}\xspace}
\newcommand{\deltavanglemax}{\ensuremath{\MAX{\Delta\vangle}}\xspace}
\newcommand{\angledifference}{\ensuremath{\Delta}\xspace}
% Complex power
\newcommand{\complexpower}{\ensuremath{s}\xspace}
\newcommand{\complexpowers}{\ensuremath{S}\xspace}
\newcommand{\complexpowermax}{\ensuremath{\MAX{\complexpower}}\xspace}
\newcommand{\complexpowerbase}{\ensuremath{\complexpowers_{\mathrm{base}}}\xspace}
% Real (active) power
\newcommand{\supply}{\ensuremath{p}\xspace}
\newcommand{\realpower}{\ensuremath{p}\xspace}
\newcommand{\realpowers}{\ensuremath{P}\xspace}
\newcommand{\realpowersmin}{\ensuremath{\MIN{\realpowers}}\xspace}
\newcommand{\realpowermin}{\ensuremath{\MIN{\realpower}}\xspace}
\newcommand{\realpowergenerationmin}{\ensuremath{
\MIN{\realpower_g}}\xspace}
\newcommand{\realpowerdemandmin}{\ensuremath{
\MIN{\realpower_\demand}}\xspace}
\newcommand{\realpowerdemand}{\ensuremath{\realpower_\demand}\xspace}
\newcommand{\realpowersmax}{\ensuremath{\realpowers_{\max}}\xspace}
\newcommand{\realpowermax}{\ensuremath{\MAX{\realpower}}\xspace}
\newcommand{\realpowergenerationmax}{\ensuremath{\MAX{\realpower_{g}}}\xspace}
\newcommand{\realpowergeneration}{\ensuremath{\realpower_{g}}\xspace}
\newcommand{\realpowerdemandmax}{\ensuremath{\MAX{\realpower_{\demand}}}\xspace}
\newcommand{\realpowerbase}{\ensuremath{\realpowers_{\mathrm{base}}}\xspace}
% Reactive power
\newcommand{\reactivepower}{\ensuremath{q}\xspace}
\newcommand{\reactivepowers}{\ensuremath{Q}\xspace}
\newcommand{\reactivepowermin}{\ensuremath{\MIN{\reactivepower}}\xspace}
\newcommand{\reactivepowergenerationmin}{\ensuremath{
\MIN{\reactivepower_{g}}}\xspace}
\newcommand{\reactivepowerdemandmin}{\ensuremath{
\MIN{\reactivepower_{\demand}}}\xspace}
\newcommand{\reactivepowerdemand}{\ensuremath{\reactivepower_{\demand}}\xspace}
\newcommand{\reactivepowersmin}{\ensuremath{\MIN{\reactivepowers}}\xspace}
\newcommand{\reactivepowermax}{\ensuremath{\MAX{\reactivepower}}\xspace}
\newcommand{\reactivepowergeneration}{\ensuremath{
\reactivepower_{g}}\xspace}
\newcommand{\reactivepowergenerationmax}{\ensuremath{
\MAX{\reactivepower_{g}}}\xspace}
\newcommand{\reactivepowerdemandmax}{\ensuremath{
\MAX{\reactivepower_{\demand}}}\xspace}
\newcommand{\reactivepowersmax}{\ensuremath{\MAX{\reactivepowers}}\xspace}
\newcommand{\reactivepowerbase}{\ensuremath{\reactivepowers_{
\mathrm{base}}}\xspace}
% 
\newcommand{\excess}{\ensuremath{x}\xspace}
\newcommand{\excessSet}{\ensuremath{X}\xspace}
\newcommand{\minexcess}{\ensuremath{\MIN{\excess}}\xspace}
\newcommand{\maxexcess}{\ensuremath{\MAX{\excess}}\xspace}
% Demand
\newcommand{\demand}{\ensuremath{d}\xspace}
\newcommand{\mindemand}{\ensuremath{\MIN{\demand}}\xspace}
\newcommand{\maxdemand}{\ensuremath{\MAX{\demand}}\xspace}
% \newcommandx{\fdeltaangle}[3]{\ensuremath{\deltaangle_{#1}(#2,#3)}\xspace}
\newcommand{\paths}{\ensuremath{\Pi}\xspace}
\newcommand{\pathu}{\ensuremath{\pi}\xspace}
\newcommandx{\fpath}[3]{\ensuremath{\glssymbol{path}_{#1}(#2,#3)}\xspace}

\newcommand{\deltawye}{\ensuremath{\Delta\text{-}Y}\xspace}
\newcommand{\wyedelta}{\ensuremath{Y\text{-}\Delta}\xspace}
\newcommand{\timestamp}{\ensuremath{t}\xspace}
\newcommand{\timestamps}{\ensuremath{T}\xspace}
% Switch
\newcommand{\switch}{\ensuremath{z}\xspace}
\newcommand{\switched}{\ensuremath{S}\xspace}
% P-Q-V-model
\newcommand{\pqv}{\ensuremath{PQV}\xspace}
\newcommand{\iv}{\ensuremath{IV}\xspace}
% Buse Type codes
\newcommand{\vtbus}{\ensuremath{\mathsf{\voltages\vangle}}\xspace}
\newcommand{\qtbus}{\ensuremath{\mathsf{\reactivepowers\vangle}}\xspace}
\newcommand{\pqbus}{\ensuremath{\mathsf{\realpowers\reactivepowers}}\xspace}
\newcommand{\pvbus}{\ensuremath{\mathsf{\realpowers\voltages}}\xspace}
\newcommand{\cvbus}{\ensuremath{\mathsf{C\voltages}}\xspace}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Centrality %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\sbc}{\ensuremath{\mathsf{\dtp B}}\xspace}
\newcommand{\sbcbf}{\ensuremath{\mathbfsf{\dtpbf B}}\xspace}
\newcommand{\SBC}{\textsc{\DTP Betweenness Centrality}\xspace}

\newcommand{\scu}{\ensuremath{\mathsf{S}}\xspace}
\newcommand{\scubf}{\ensuremath{\mathbfsf{S}}\xspace}
\newcommand{\SCu}{\textsc{Switching Centrality}\xspace}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Functions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\DeclareMathOperator{\len}{\ensuremath{len}\xspace}
\DeclareMathOperator{\costsubstation}{\ensuremath{\totalcost_\mathrm{sub}}\xspace} %csub
\newcommand{\totalcost}{\ensuremath{c}\xspace} 
\DeclareMathOperator{\flowcost}{\ensuremath{cost}\xspace}
\DeclareMathOperator{\degree}{\ensuremath{deg}\xspace} % degree of a vertex

\newcommandx{\deriv}[3]{%
    \frac{\diff^{#1}#2}{\diff #3^{#1}}}%
\newcommandx{\pderiv}[3]{%
    \frac{\partial^{#1}#2}{\partial #3^{#1}}%
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Problems %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \usepackage{lmodern}
\usepackage[T1]{fontenc}
\DeclareMathAlphabet{\mathbfsf}{\encodingdefault}{\sfdefault}{bx}{n}
\newcommand{\tens}[1]{\mathbfsf{#1}}
%
\newcommand{\problems}{\ensuremath{\mathcal{P}}\xspace}
%
% 
\newcommand{\edp}{EDP\xspace}   
\newcommand{\EDP}{\textsc{Economic Dispatch Problem}\xspace} 
% Alternating Current
\newcommand{\ac}{\ensuremath{\mathsf{AC}}\xspace}
\newcommand{\acbf}{\ensuremath{\mathbfsf{AC}}\xspace}
\newcommand{\ACu}{\textsc{Alternating Current}\xspace}
% 
% AC Optimal Power Flow
\newcommand{\acopf}{\ensuremath{\mathsf{\ac\opf}}\xspace}
\newcommand{\acopfbf}{\ensuremath{\mathbfsf{ACOPF}}\xspace}
\newcommand{\ACOPF}{\textsc{\ACu\ \OPF}\xspace}
% 
% AC Optimal Transmission Switching
\newcommand{\acots}{\ensuremath{\mathsf{\ac\ots}}\xspace}
\newcommand{\acotsbf}{\ensuremath{\mathbfsf{\acbf\otsbf}}\xspace}
%
% Alternating Direction Method of Multipliers
\newcommand{\admm}{\ensuremath{\mathrm{ADMM}}\xspace}
\newcommand{\admmbf}{\ensuremath{\mathbfsf{ADMM}}\xspace}
\newcommand{\ADMM}{alternating direction method of multipliers\xspace}
% 
% Capacitated Minimum Spanning Tree (CMST)
\newcommand{\cmst}{\ensuremath{\mathsf{CMST}}\xspace}
\newcommand{\cmstbf}{\ensuremath{\mathbfsf{CMST}}\xspace}
\newcommand{\CMST}{\textsc{Capacitated Minimum Spanning Tree}\xspace}
% 
% Combinatorial Optimization Problem
\newcommand{\cop}{\ensuremath{\mathsf{COP}}\xspace}
\newcommand{\copbf}{\ensuremath{\mathbfsf{COP}}\xspace}
\newcommand{\COP}{\textsc{Combinatorial Optimization Problem}\xspace}
% 
% Direct Current
\newcommand{\dc}{\ensuremath{\mathsf{DC}}\xspace}
\newcommand{\dcbf}{\ensuremath{\mathbfsf{DC}}\xspace}
\newcommand{\DC}{\textsc{Direct Current}\xspace}
% 
% DC Maximum Power Flow
\newcommand{\dcmpf}{\ensuremath{\dc\mathsf{MPF}}\xspace}
\newcommand{\dcmpfbf}{\ensuremath{\dcbf\tens{MPF}}\xspace}
\newcommand{\DCMPF}{\DC\textsc{Maximum Power Flow}\xspace}
% 
% DC Optimal Transmission Switching
\newcommand{\dcots}{\ensuremath{\mathsf{\dc\ots}}\xspace}
\newcommand{\dcotsbf}{\ensuremath{\mathbfsf{\dcbf\otsbf}}\xspace}
% 
% DC Optimal Power Flow (DCOPF)
\newcommand{\dcopf}{\ensuremath{\mathsf{\dc\opf}}\xspace}
\newcommand{\dcopfbf}{\ensuremath{\mathbfsf{DCOPF}}\xspace}
\newcommand{\DCOPF}{\textsc{\DC\ \OPF}\xspace}
% 
\newcommand{\dfs}{\ensuremath{\mathsf{DFS}}\xspace}
% 
% Dominating Theta Path (DTP)
\newcommand{\dtp}{\ensuremath{\mathsf{DTP}}\xspace}
\newcommand{\dtpbf}{\ensuremath{\mathbfsf{DTP}}\xspace}
\newcommand{\DTP}{\textsc{Dominating Theta Path}\xspace}
% 
% Flexible AC Transmission System
\newcommand{\facts}{FACTS\xspace}
\newcommand{\FACTS}{Flexible~\ac\ Transmission Systems\xspace}
% 
% Feasibility Problem
\newcommand{\feas}{\ensuremath{\mathsf{FEAS}}\xspace}
\newcommand{\feasbf}{\ensuremath{\mathbfsf{FEAS}}\xspace}
\newcommand{\FEAS}{\textsc{Feasibility Problem}\xspace}
% 
% Concerning flows
\newcommand{\flowvalue}{\ensuremath{F}\xspace}
\newcommand{\flowvaluebf}{\ensuremath{\mathbfsf{F}}\xspace}
% 
% Kirchhoff's Current Law
\newcommand{\kcl}{KCL\xspace}
\newcommand{\KCL}{Kirchhoff's Current Law\xspace}
% 
% Kirchhoff's Voltage Law
\newcommand{\kvl}{KVL\xspace}
\newcommand{\KVL}{Kirchhoff's Voltage Law\xspace}
% 
% Line Outage Distribution Factor
\newcommand{\lodf}{\ensuremath{\mathrm{LODF}}\xspace}
\newcommand{\lodfbf}{\ensuremath{\mathbfsf{LODF}}\xspace}
\newcommand{\LODF}{line outage distribution factor\xspace}
% 
% Linear Programming
\newcommand{\lp}{\ensuremath{\mathsf{LP}}\xspace}
\newcommand{\lpbf}{\ensuremath{\mathbfsf{LP}}\xspace}
\newcommand{\LP}{\textsc{Linear Programming}\xspace}
% 
% Maximum Spanning Tree
\newcommand{\maxst}{\ensuremath{\mathsf{MaxST}}\xspace}
\newcommand{\maxstbf}{\ensuremath{\mathbfsf{MaxST}}\xspace}
\newcommand{\MAXST}{\textsc{Maximum Spanning Tree}\xspace}
%
% Multicommodity Capacitated Network Design (MCND)
\newcommand{\mcnd}{\ensuremath{\mathsf{MCND}}\xspace}
\newcommand{\mcndbf}{\ensuremath{\mathbfsf{MCND}}\xspace}
\newcommand{\MCND}{\textsc{Multicommodity Capacitated Network Design}\xspace}
%
% Minimum Cost Flow
\newcommand{\mcf}{\ensuremath{\mathsf{MCF}}\xspace}
\newcommand{\mcfbf}{\ensuremath{\mathbfsf{MCF}}\xspace}
\newcommand{\MCF}{\textsc{Minimum Cost Flow}\xspace} 
%
% Minimum Cost Flow Problem
\newcommand{\mcfp}{\ensuremath{\mathsf{MCFP}}\xspace}
\newcommand{\mcfpbf}{\ensuremath{\mathbfsf{MCFP}}\xspace}
\newcommand{\MCFP}{\textsc{Minimum Cost Flow Problem}\xspace} 
% 
% Maximum Flow
\newcommand{\mf}{\ensuremath{\mathsf{MF}}\xspace}
\newcommand{\mfbf}{\ensuremath{\mathbfsf{MF}}\xspace}
\newcommand{\MF}{\text{maximum flow}\xspace} 
% 
% Maximum Flow Problem
\newcommand{\mfp}{\ensuremath{\mathsf{MFP}}\xspace}
\newcommand{\mfpbf}{\ensuremath{\mathbfsf{MFP}}\xspace}
\newcommand{\MFP}{\textsc{Maximum Flow Problem}\xspace} 
% 
% Minimum Cut
\newcommand{\mc}{\ensuremath{\mathsf{MC}}\xspace}
\newcommand{\mcbf}{\ensuremath{\mathbfsf{MC}}\xspace}
\newcommand{\MC}{\text{minimum cut}\xspace} 
% 
% Minimum Cut Problem
\newcommand{\mcp}{\ensuremath{\mathsf{MCP}}\xspace}
\newcommand{\mcpbf}{\ensuremath{\mathbfsf{MCP}}\xspace}
\newcommand{\MCP}{\textsc{Minimum Cut Problem}\xspace} 
% 
% Mixed-integer Linear Program
\newcommand{\milp}{\ensuremath{\mathsf{MILP}}\xspace}
\newcommand{\milpbf}{\ensuremath{\mathbfsf{MILP}}\xspace}
\newcommand{\MILP}{\textsc{Mixed-integer Linear Program}\xspace}
% 
% Mixed-integer NonLinear Programming
\newcommand{\minlp}{\ensuremath{\mathsf{MINLP}}\xspace}
\newcommand{\minlpbf}{\ensuremath{\mathbfsf{MINLP}}\xspace}
\newcommand{\MINLP}{\textsc{Mixed-integer NonLinear Programming}\xspace}
% 
% Maximum Power Flow
\newcommand{\mpf}{\ensuremath{\mathsf{MPF}}\xspace}
\newcommand{\mpfbf}{\ensuremath{\tens{MPF}}\xspace}
\newcommand{\MPF}{\text{maximum power flow}\xspace}
% 
% Maximum Power Flow
\newcommand{\mpfp}{\ensuremath{\mathsf{MPFP}}\xspace}
\newcommand{\mpfpbf}{\ensuremath{\tens{MPFP}}\xspace}
\newcommand{\MPFP}{\textsc{Maximum Power Flow Problem}\xspace}
% 
% Maximum Spanning Tree
\newcommand{\mst}{\ensuremath{\mathsf{MST}}\xspace}
\newcommand{\mstbf}{\ensuremath{\mathbfsf{MST}}\xspace}
\newcommand{\MST}{\textsc{Minimum Spanning Tree}\xspace}
% 
% Maximum Transmission Switching Flow
\newcommand{\mtsf}{\ensuremath{\mathsf{MTSF}}\xspace}
\newcommand{\mtsfbf}{\ensuremath{\mathbfsf{MTSF}}\xspace}
\newcommand{\MTSF}{\textsc{Maximum Transmission Switching Flow}\xspace}
% 
% Maximum Transmission Switching Flow Problem
\newcommand{\mtsfp}{\ensuremath{\mathsf{MTSFP}}\xspace}
\newcommand{\mtsfpbf}{\ensuremath{\mathbfsf{MTSFP}}\xspace}
\newcommand{\MTSFP}{\textsc{Maximum Transmission Switching Flow Problem}\xspace}
%
% Maximum Transmission Switching Flow - Decision Problem
\newcommand{\kmtsf}{\ensuremath{k}-\gls{mtsf}}
\newcommand{\kmtsfbf}{\ensuremath{k}-\mtsfbf}
\newcommand{\kMTSF}{\ensuremath{k}-\acrlong{mtsf}}
% 
\newcommand{\kmtsfp}{\ensuremath{k}-\gls{mtsfp}}
\newcommand{\kmtsfpbf}{\ensuremath{k}-\mtsfpbf}
\newcommand{\kMTSFP}{\ensuremath{k}-\acrlong{mtsfp}}
% 
% Minimum Number of Switches
\newcommand{\mns}{\ensuremath{\mathsf{MNS}}\xspace}
\newcommand{\mnsbf}{\ensuremath{\mathbfsf{MNS}}\xspace}
\newcommand{\MNS}{\textsc{Minimum Number of Switches}\xspace}
% 
% Minimum Number of Switches
\newcommand{\mnsp}{\ensuremath{\mathsf{MNSP}}\xspace}
\newcommand{\mnspbf}{\ensuremath{\mathbfsf{MNSP}}\xspace}
\newcommand{\MNSP}{\textsc{Minimum Number of Switches Problem}\xspace}
% 
% Optimal Transmission Switching (OTS)
\newcommand{\ots}{\ensuremath{\mathsf{OTS}}\xspace}
\newcommand{\otsbf}{\ensuremath{\mathbfsf{OTS}}\xspace}
\newcommand{\OTS}{\textsc{Optimal Transmission Switching}\xspace}
% 
% Optimal Transmission Switching Flow Problem (OTSP)
\newcommand{\otsp}{\ensuremath{\mathsf{OTSP}}\xspace}
\newcommand{\otspbf}{\ensuremath{\mathbfsf{OTSP}}\xspace}
\newcommand{\OTSP}{\textsc{Optimal Transmission Switching Problem}\xspace}
% 
% Maximum FACTS Flow
\newcommand{\mff}{\ensuremath{\mathsf{MFF}}\xspace}
\newcommand{\mffbf}{\ensuremath{\mathbfsf{MFF}}\xspace}
\newcommand{\MFF}{\textsc{Maximum~\facts\ Flow}\xspace}
% 
% Maximum FACTS Flow Problem
\newcommand{\mffp}{\ensuremath{\mathsf{MFFP}}\xspace}
\newcommand{\mffpbf}{\ensuremath{\mathbfsf{MFFP}}\xspace}
\newcommand{\MFFP}{\textsc{Maximum~\facts Flow Problem}\xspace}
%
% Minimum Number of FACTS
\newcommand{\mnf}{\ensuremath{\mathsf{MNF}}\xspace}
\newcommand{\mnfbf}{\ensuremath{\mathbfsf{MNF}}\xspace}
\newcommand{\MNF}{\textsc{Minimum Number of~\facts}\xspace}
%
% Minimum Number of FACTS Problem
\newcommand{\mnfp}{\ensuremath{\mathsf{MNFP}}\xspace}
\newcommand{\mnfpbf}{\ensuremath{\mathbfsf{MNFP}}\xspace}
\newcommand{\MNFP}{\textsc{Minimum Number of~\facts Problem}\xspace}
%
% Optimal FACTS Flow
\newcommand{\off}{\ensuremath{\mathsf{OFF}}\xspace}
\newcommand{\offbf}{\ensuremath{\mathbfsf{OFF}}\xspace}
\newcommand{\OFF}{\textsc{Optimal~\facts Flow}\xspace}
% 
% Optimal FACTS Flow Problem
\newcommand{\offp}{\ensuremath{\mathsf{OFFP}}\xspace}
\newcommand{\offpbf}{\ensuremath{\mathbfsf{OFFP}}\xspace}
\newcommand{\OFFP}{\textsc{Optimal~\facts Flow Problem}\xspace}
% 
% Optimal Power Flow (OPF)
\newcommand{\opf}{\ensuremath{\mathsf{OPF}}\xspace}
\newcommand{\opfbf}{\ensuremath{\mathbfsf{OPF}}\xspace}
\newcommand{\OPF}{\textsc{Optimal Power Flow}\xspace}
% 
% Optimal Power Flow (OPF)
\newcommand{\opfp}{\ensuremath{\mathsf{OPFP}}\xspace}
\newcommand{\opfpbf}{\ensuremath{\mathbfsf{OPFP}}\xspace}
\newcommand{\OPFP}{\textsc{Optimal Power Flow Problem}\xspace}
% 
% Power Flow
\newcommand{\pf}{\ensuremath{\mathsf{PF}}\xspace}
\newcommand{\pfbf}{\ensuremath{\tens{PF}}\xspace}
\newcommand{\PFu}{\textsc{Power Flow}\xspace}
% 
% Power Flow Problem (PFP)
\newcommand{\pfp}{\ensuremath{\mathsf{PFP}}\xspace}
\newcommand{\pfpbf}{\ensuremath{\tens{PFP}}\xspace}
\newcommand{\PFPu}{\textsc{Power Flow Problem}\xspace}
% 
% Power Transfer Distribution Factors
\newcommand{\ptdf}{\ensuremath{\mathrm{PTDF}}\xspace}
\newcommand{\ptdfbf}{\ensuremath{\mathbfsf{PTDF}}\xspace}
\newcommand{\PTDF}{power transfer distribution factors\xspace}
% 
% Quadratic Program
\newcommand{\qp}{\ensuremath{\mathrm{QP}}\xspace}
\newcommand{\qpbf}{\ensuremath{\mathbfsf{QP}}\xspace}
\newcommand{\QPu}{Quadratic Programming\xspace}
% 
% Restoration Order Problem
\newcommand{\rop}{\ensuremath{\mathsf{ROP}}\xspace}
\newcommand{\ropbf}{\ensuremath{\mathbfsf{ROP}}\xspace}
\newcommand{\ROP}{\textsc{Restoration Order Problem}\xspace}
% 
% Second Order Cone Programming
\newcommand{\sfp}{\ensuremath{\mathsf{SFP}}\xspace}
\newcommand{\sfpbf}{\ensuremath{\mathbfsf{SFP}}\xspace}
\newcommand{\SFP}{\textsc{Simultaneous Flow Problem}\xspace}
% 
% Second Order Cone Programming
\newcommand{\socp}{\ensuremath{\mathsf{SOCP}}\xspace}
\newcommand{\socpbf}{\ensuremath{\mathbfsf{SOCP}}\xspace}
\newcommand{\SOCP}{\textsc{Second Order Cone Programming}\xspace}
% 
% Shortest Path (SP)
\newcommand{\spp}{\ensuremath{\mathsf{SP}}\xspace}
\newcommand{\sppbf}{\ensuremath{\mathbfsf{SP}}\xspace}
\newcommand{\SPPu}{\textsc{Shortest Path}\xspace}
% 
% Shortest Path Problem
\newcommand{\sppp}{\ensuremath{\mathsf{SPP}}\xspace}
\newcommand{\spppbf}{\ensuremath{\mathbfsf{SPP}}\xspace}
\newcommand{\SPPPu}{\textsc{Shortest Path Problem}\xspace}
% 
% Shortest Path (SP)
\newcommand{\lpp}{\ensuremath{\mathsf{LP}}\xspace}
\newcommand{\lppbf}{\ensuremath{\mathbfsf{LP}}\xspace}
\newcommand{\LPPu}{\textsc{Longest Path}\xspace}
% 
% Shortest Path Problem
\newcommand{\lppp}{\ensuremath{\mathsf{LPP}}\xspace}
\newcommand{\lpppbf}{\ensuremath{\mathbfsf{LPP}}\xspace}
\newcommand{\LPPPu}{\textsc{Longest Path Problem}\xspace}
% 
% Voltage Normalized Lossless Real Power Flow (VNLP)
\newcommand{\sinac}{\ensuremath{\mathsf{VNLP}}\xspace}
\newcommand{\sinacbf}{\ensuremath{\mathbfsf{VNLP}}\xspace}
\newcommand{\SINAC}{\textsc{Voltage Normalized Lossless Real Power Flow}\xspace}
% 
% Voltage Normalized Lossless Real Power Flow Problem (VNLPP)
\newcommand{\sinacp}{\ensuremath{\mathsf{VNLPP}}\xspace}
\newcommand{\sinacpbf}{\ensuremath{\mathbfsf{VNLPP}}\xspace}
\newcommand{\SINACP}{\textsc{Voltage Normalized Lossless Real Power Flow
Problem}\xspace}
% 
% s-t-Rainbow Path
\newcommand{\strp}{\ensuremath{\text{$\source$-$\sink$-}\mathrm{RP}}\xspace}
\newcommand{\strpbf}{\ensuremath{\source-\sink-\mathbfsf{RP}}\xspace}
\newcommand{\STRP}{\textsc{Rainbow $\source$-$\sink$-Path}\xspace}
% 
% Subset Sum Problem
\newcommand{\ssp}{\ensuremath{\mathsf{SSP}}\xspace}
\newcommand{\sspbf}{\ensuremath{\mathbfsf{SSP}}\xspace}
\newcommand{\SSP}{\textsc{Subset Sum Problem}\xspace}
% 
% Transmission Network Expansion Planning
\newcommand{\tnep}{\ensuremath{\mathsf{TNEP}}\xspace}
\newcommand{\tnepbf}{\ensuremath{\mathbfsf{TNEP}}\xspace}
\newcommand{\TNEP}{\textsc{Transmission Network Expansion Planning}\xspace}
% 
% Transmission Network Expansion Planning Problem
\newcommand{\tnepp}{\ensuremath{\mathsf{TNEPP}}\xspace}
\newcommand{\tneppbf}{\ensuremath{\mathbfsf{TNEPP}}\xspace}
\newcommand{\TNEPP}{\textsc{Transmission Network Expansion Planning
Problem}\xspace}
% 
% Transmission System Operators
\newcommand{\tso}{TSO\xspace}
\newcommand{\TSO}{Transmission System Operators\xspace}
% 
\newcommand{\twoapprox}{\ensuremath{\mathsf{2-approx}}\xspace}
\newcommand{\twoapproxbf}{\ensuremath{\mathbfsf{2-approx}}\xspace}
\newcommand{\TWOAPPROX}{\textsc{2-Approximation}\xspace}
% 
% Unit Commitment Problem
\newcommand{\ucp}{\ensuremath{\mathsf{UCP}}\xspace}
\newcommand{\ucpbf}{\ensuremath{\mathbfsf{UCP}}\xspace}
\newcommand{\UCP}{\textsc{Unit Commitment Problem}\xspace}
% 
\newcommand{\vc}{\ensuremath{\mathsf{VC}}\xspace}
\newcommand{\vcp}{\ensuremath{\mathsf{VCP}}\xspace}
\newcommand{\vcpbf}{\ensuremath{\mathbfsf{VCP}}\xspace}
\newcommand{\VCP}{\textsc{Vertex Cover Problem}\xspace}
% 
\newcommand{\fvs}{\ensuremath{\mathsf{FVS}}\xspace}
\newcommand{\fvsp}{\ensuremath{\mathsf{FVSP}}\xspace}
\newcommand{\fvspbf}{\ensuremath{\mathbfsf{FVSP}}\xspace}
\newcommand{\FVSP}{\textsc{Feedback Vertex Set Problem}\xspace}
% 
\newcommand{\partitionp}{\ensuremath{\mathsf{PP}}\xspace}
\newcommand{\partitionpbf}{\ensuremath{\mathbfsf{PP}}\xspace}
\newcommand{\PartitionP}{\textsc{Partition Problem}\xspace}
% 
% Circuit Problem
\newcommand{\wcp}{\ensuremath{\mathsf{CP}}\xspace}
\newcommand{\wcpbf}{\ensuremath{\mathbfsf{CP}}\xspace}
\newcommand{\WCP}{\textsc{Circuit Problem}\xspace}
% 
% Substation Problem
\newcommand{\wsp}{\ensuremath{\mathsf{SP}}\xspace}
\newcommand{\wspbf}{\ensuremath{\mathbfsf{SP}}\xspace}
\newcommand{\WSP}{\textsc{Substation Problem}\xspace}
% 
% Full Farm Problem
\newcommand{\wffp}{\ensuremath{\mathsf{FFP}}\xspace}
\newcommand{\wffpbf}{\ensuremath{\mathbfsf{FFP}}\xspace}
\newcommand{\WFFP}{\textsc{Full Farm Problem}\xspace}
% 
% flexible Full Farm Problem
\newcommand{\wfffp}{\ensuremath{\mathsf{fFFP}}\xspace}
\newcommand{\wfffpbf}{\ensuremath{\mathbfsf{fFFP}}\xspace}
\newcommand{\WfFFP}{\textsc{flexible Full Farm Problem}\xspace}
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% Others 
\newcommand{\tmp}{\mathrm{tmp}}
\newcommand{\twodots}{\mathinner {\ldotp \ldotp}}
\newcommand{\parent}{\ensuremath{\mathrm{parent}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Sets %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Naturals
\newcommand{\naturals}{\ensuremath{\mathbb{N}}\xspace}
% Rationals
\newcommand{\rationals}{\ensuremath{\mathbb{Q}}\xspace}
\newcommand{\posrationals}{\ensuremath{\rationals_{\ge 0}}\xspace}
% Reals
\newcommand{\reals}{\ensuremath{\mathbb{R}}\xspace}
\newcommand{\strictposreals}{\ensuremath{\reals_{>0}}\xspace}
\newcommand{\posreals}{\ensuremath{\reals_{\ge0}}\xspace}
\newcommand{\negreals}{\ensuremath{\reals_{\le0}}\xspace}
% Integer
\newcommand{\integers}{\ensuremath{\mathbb{Z}}\xspace}
% Interval
\newcommand{\interval}{\ensuremath{I}\xspace}
\newcommand{\intervals}{\ensuremath{\mathcal{I}}\xspace}
% Complex
\newcommand{\complexes}{\ensuremath{\mathbb{C}}\xspace}
\newcommand{\imgPart}{\ensuremath{\textbf{j}}\xspace}
% 
\newcommand{\determinant}{\ensuremath{\mathrm{det}}\xspace}
\newcommandx{\fdeterminant}[1]{\ensuremath{\determinant\left(#1\right)}\xspace}
\newcommandx{\imaginary}[1]{\ensuremath{\mathcal{I}\!\!m\!\left
(#1\right)}\xspace}
\newcommandx{\imaginarySmall}[1]{\ensuremath{\mathcal{I}\!\!m\left
(#1\right)}\xspace}
\newcommandx{\real}[1]{\ensuremath{\mathcal{R}\!e\!\left(#1\right)}\xspace}
\newcommandx{\conjugate}[1]{\ensuremath{{#1}^\star}\xspace}
\newcommandx{\realSmall}[1]{\ensuremath{\mathcal{R}\!e\left(#1\right)}\xspace}
\newcommandx{\fsin}[1]{\ensuremath{\sin\!\big(#1\big)}\xspace}
\newcommandx{\fcos}[1]{\ensuremath{\cos\!\big(#1\big)}\xspace}

% Complexity
\newcommand{\NPbf}{\ensuremath{\mathbfsf{NP}}}
\newcommand{\bigO}{\ensuremath{\mathcal{O}}}
\newcommand{\smallO}{\ensuremath{\mathcal{o}}}
\newcommand{\logO}{\ensuremath{\mathcal{\widetilde{O}}}}
\newcommand{\bigT}{\ensuremath{\Theta}}
\newcommand{\opt}{\ensuremath{\mathrm{OPT}}}
\newcommand{\optbf}{\ensuremath{\mathbfsf{OPT}}}

% Mathematical norms
\newcommandx{\norm}[2]{\ensuremath{\left\lVert#1\right\rVert_{#2}}}
\newcommand{\bnorm}[1]{\ensuremath{\norm{#1}{\glssymbol{susceptance}}}}
\newcommand{\leinsnorm}[1]{\ensuremath{\norm{#1}{\ell 1}}}

% Mathematical operators
\newcommand{\ringsum}{\ensuremath{\oplus}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%% Simulated Annealing %%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Turbines
% \newcommand{\turbines}{\ensuremath{\vertices_{T}}}
\newcommand{\turbines}{\ensuremath{\generators}\xspace}
\newcommand{\turbine}{\ensuremath{t}\xspace}
% Substations
% \newcommand{\substations}{\ensuremath{\vertices_{S}}}
\newcommand{\substations}{\ensuremath{\consumers}\xspace}
\newcommand{\substation}{\ensuremath{j}\xspace}
% Possible substations for fFFP
\newcommand{\possiblesubstations}{\ensuremath{\vertices_{L}}\xspace}
\newcommand{\locations}{\ensuremath{L}\xspace}


\newcommand{\temperature}{\ensuremath{T}}
\newcommand{\energy}{\ensuremath{\totalcost}\xspace} %usually E for energy, but E is already used for the set of edges 
%
%%%%%%%%%%%%% Solutions %%%%%%%%%%%%% 
\newcommand{\solutions}{\ensuremath{S}\xspace}
\newcommand{\solution}{\ensuremath{s}\xspace}
\newcommand{\solutiona}{\ensuremath{\glssymbol{solution}_1}\xspace}
\newcommand{\solutionb}{\ensuremath{\glssymbol{solution}_2}\xspace}
% Best Solution
\newcommand{\bestsolution}{\ensuremath{\overline{s}}\xspace}
\newcommand{\bestsolutiona}{\ensuremath{\glssymbol{bestsolution}_1}\xspace}
\newcommand{\bestsolutionb}{\ensuremath{\glssymbol{bestsolution}_2}\xspace}
%
%%%%%%%%%%%%% Probability %%%%%%%%%%%%% 
\newcommand{\probability}{\ensuremath{\mathcal{W}}\xspace}
%
%%%%%%%%%%%%% Representation %%%%%%%%%%%%% 
\newcommand{\representations}{\ensuremath{R}\xspace}
\DeclareMathOperator{\potential}{\ensuremath{\theta}\xspace}
%
%%%%%%%%%%%%% Activity %%%%%%%%%%%%% 
\newcommand{\activity}{\ensuremath{\mu}\xspace}
\newcommand{\activitythreshold}{\ensuremath{\overline{\mu}}\xspace}
%
%%%%%%%%%%%%% Instances %%%%%%%%%%%%% 
\newcommand{\instances}{\ensuremath{\mathcal{I}}\xspace}
\newcommand{\instance}{\ensuremath{I}\xspace}
\newcommand{\instancea}{\ensuremath{\glssymbol{instance}_1}\xspace}
\newcommand{\instanceb}{\ensuremath{\glssymbol{instance}_2}\xspace}
\newcommand{\instancec}{\ensuremath{\glssymbol{instance}_k}\xspace}
%
%%%%%%%%%%%%% Cable Types %%%%%%%%%%%%% 
\newcommand{\cabletypes}{\ensuremath{K}\xspace}
\newcommand{\cabletype}{\ensuremath{\kappa}\xspace}
\newcommand{\cabletypea}{\ensuremath{\glssymbol{cabletype}^{(1)}}\xspace}
\newcommand{\cabletypeb}{\ensuremath{\glssymbol{cabletype}^{(2)}}\xspace}
\newcommand{\cabletypec}{\ensuremath{\glssymbol{cabletype}^{(3)}}\xspace}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Test Cases %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{xifthen}%
\newcommand{\ifequals}[3]{\ifthenelse{\equal{#1}{#2}}{#3}{}}%
\newcommand{\case}[2]{#1 #2} % Dummy, so \renewcommand has something to overwrite...
\newenvironment{switchcase}[1]{\renewcommand{\case}{\ifequals{#1}}}{}%
% 
% Example: Pick case by ID
\newcommand{\nestacase}[2]{%
  \begin{switchcase}{#1}%
    \case{3}{\texttt{nesta\_case3\_lmbd}}%
    \case{4}{\texttt{nesta\_case4\_gs}}%
    \case{5}{\texttt{nesta\_case5\_pjm}}%
    \case{6c}{\texttt{nesta\_case6\_c}}%
    \case{6ww}{\texttt{nesta\_case6\_ww}}%
    \case{9}{\texttt{nesta\_case9\_wscc}}%
    \case{14}{\texttt{nesta\_case14\_ieee}}%
    \case{24}{\texttt{nesta\_case24\_ieee\_rts}}%
    \case{29}{\texttt{nesta\_case29\_edin}}%
    \case{30as}{\texttt{nesta\_case30\_as}}%
    \case{30fsr}{\texttt{nesta\_case30\_fsr}}%
    \case{30ieee}{\texttt{nesta\_case30\_ieee}}%
    \case{39}{\texttt{nesta\_case39\_epri}}%
    \case{57}{\texttt{nesta\_case57\_ieee}}%
    \case{73}{\texttt{nesta\_case73\_ieee\_rts}}%
    \case{89}{\texttt{nesta\_case89\_pegase}}%
    \case{118}{\texttt{nesta\_case118\_ieee}}%
    \case{162}{\texttt{nesta\_case162\_ieee\_dtc}}%
    \case{189}{\texttt{nesta\_case189\_edin}}%
    \case{300}{\texttt{nesta\_case300\_ieee}}%
    \case{2383}{\texttt{nesta\_case2383wp\_mp}}%
    \case{2736}{\texttt{nesta\_case2736sp\_mp}}%
    \case{2737}{\texttt{nesta\_case2737sop\_mp}}%
    \case{2746wop}{\texttt{nesta\_case2746wop\_mp}}%
    \case{2746wp}{\texttt{nesta\_case2746wp\_mp}}%
    \case{3012}{\texttt{nesta\_case3012wp\_mp}}%
    \case{3120}{\texttt{nesta\_case3120sp\_mp}}%
    \case{3375}{\texttt{nesta\_case3375wp\_mp}}%
    #2%
  \end{switchcase}%
}
% 
\newcommand{\tablecase}[1]{%
  % \footnotesize
  \begin{switchcase}{#1}%
    % \case{nestacase3lmbd}{\texttt{case3\_lmbd}}%
    % \case{nestacase4gs}{\texttt{case4\_gs}}%
    % \case{nestacase5pjm}{\texttt{case5\_pjm}}%
    % \case{nestacase6c}{\texttt{case6\_c}}%
    % \case{nestacase6ww}{\texttt{case6\_ww}}%
    % \case{nestacase9wscc}{\texttt{case9\_wscc}}%
    % \case{nestacase14ieee}{\texttt{case14\_ieee}}%
    % \case{nestacase24ieeerts}{\texttt{case24\_ieee\_rts}}%
    % \case{nestacase29edin}{\texttt{case29\_edin}}%
    % \case{nestacase30as}{\texttt{case30\_as}}%
    % \case{nestacase30fsr}{\texttt{case30\_fsr}}%
    % \case{nestacase30ieee}{\texttt{case30\_ieee}}%
    % \case{nestacase39epri}{\texttt{case39\_epri}}%
    % \case{nestacase57ieee}{\texttt{case57\_ieee}}%
    % \case{nestacase73ieeerts}{\texttt{case73\_ieee\_rts}}%
    % \case{nestacase89pegase}{\texttt{case89\_pegase}}%
    % \case{nestacase118ieee}{\texttt{case118\_ieee}}%
    % \case{nestacase162ieeedtc}{\texttt{case162\_ieee\_dtc}}%
    % \case{nestacase189edin}{\texttt{case189\_edin}}%
    % \case{nestacase300ieee}{\texttt{case300\_ieee}}%
    % \case{nestacase2383wpmp}{\texttt{case2383wp\_mp}}%
    % \case{nestacase2736spmp}{\texttt{case2736sp\_mp}}%
    % \case{nestacase2737sopmp}{\texttt{case2737sop\_mp}}%
    % \case{nestacase2746wopmp}{\texttt{case2746wop\_mp}}%
    % \case{nestacase2746wpmp}{\texttt{case2746wp\_mp}}%
    % \case{nestacase3012wpmp}{\texttt{case3012wp\_mp}}%
    % \case{nestacase3120spmp}{\texttt{case3120sp\_mp}}%
    % \case{nestacase3375wpmp}{\texttt{case3375wp\_mp}}%
    % 
    % \case{nestacase3lmbd}{\texttt{3\_lmbd}}%
    % \case{nestacase4gs}{\texttt{4\_gs}}%
    % \case{nestacase5pjm}{\texttt{5\_pjm}}%
    % \case{nestacase6c}{\texttt{6\_c}}%
    % \case{nestacase6ww}{\texttt{6\_ww}}%
    % \case{nestacase9wscc}{\texttt{9\_wscc}}%
    % \case{nestacase14ieee}{\texttt{14\_ieee}}%
    % \case{nestacase24ieeerts}{\texttt{24\_ieee\_rts}}%
    % \case{nestacase29edin}{\texttt{29\_edin}}%
    % \case{nestacase30as}{\texttt{30\_as}}%
    % \case{nestacase30fsr}{\texttt{30\_fsr}}%
    % \case{nestacase30ieee}{\texttt{30\_ieee}}%
    % \case{nestacase39epri}{\texttt{39\_epri}}%
    % \case{nestacase57ieee}{\texttt{57\_ieee}}%
    % \case{nestacase73ieeerts}{\texttt{73\_ieee\_rts}}%
    % \case{nestacase89pegase}{\texttt{89\_pegase}}%
    % \case{nestacase118ieee}{\texttt{118\_ieee}}%
    % \case{nestacase162ieeedtc}{\texttt{162\_ieee\_dtc}}%
    % \case{nestacase189edin}{\texttt{189\_edin}}%
    % \case{nestacase300ieee}{\texttt{300\_ieee}}%
    % \case{nestacase2383wpmp}{\texttt{2383wp\_mp}}%
    % \case{nestacase2736spmp}{\texttt{2736sp\_mp}}%
    % \case{nestacase2737sopmp}{\texttt{2737sop\_mp}}%
    % \case{nestacase2746wopmp}{\texttt{2746wop\_mp}}%
    % \case{nestacase2746wpmp}{\texttt{2746wp\_mp}}%
    % \case{nestacase3012wpmp}{\texttt{3012wp\_mp}}%
    % \case{nestacase3120spmp}{\texttt{3120sp\_mp}}%
    % \case{nestacase3375wpmp}{\texttt{3375wp\_mp}}%
% 
    \case{nestacase3lmbd}{\texttt{\hphantom{000}3\_lmbd}}%
    \case{nestacase0003lmbd}{\texttt{\hphantom{000}3\_lmbd}}%
    \case{nestacase4gs}{\texttt{\hphantom{000}4\_gs}}%
    \case{nestacase0004gs}{\texttt{\hphantom{000}4\_gs}}%
    \case{nestacase5pjm}{\texttt{\hphantom{000}5\_pjm}}%
    \case{nestacase0005pjm}{\texttt{\hphantom{000}5\_pjm}}%
    \case{nestacase6c}{\texttt{\hphantom{000}6\_c}}%
    \case{nestacase0006c}{\texttt{\hphantom{000}6\_c}}%
    \case{nestacase6ww}{\texttt{\hphantom{000}6\_ww}}%
    \case{nestacase0006ww}{\texttt{\hphantom{000}6\_ww}}%
    \case{nestacase9wscc}{\texttt{\hphantom{000}9\_wscc}}%
    \case{nestacase0009wscc}{\texttt{\hphantom{000}9\_wscc}}%
    \case{nestacase14ieee}{\texttt{\hphantom{00}14\_ieee}}%
    \case{nestacase0014ieee}{\texttt{\hphantom{00}14\_ieee}}%
    \case{nestacase24ieeerts}{\texttt{\hphantom{00}24\_ieee\_rts}}%
    \case{nestacase0024ieeerts}{\texttt{\hphantom{00}24\_ieee\_rts}}%
    \case{nestacase29edin}{\texttt{\hphantom{00}29\_edin}}%
    \case{nestacase0029edin}{\texttt{\hphantom{00}29\_edin}}%
    \case{nestacase30as}{\texttt{\hphantom{00}30\_as}}%
    \case{nestacase0030as}{\texttt{\hphantom{00}30\_as}}%
    \case{nestacase30fsr}{\texttt{\hphantom{00}30\_fsr}}%
    \case{nestacase0030fsr}{\texttt{\hphantom{00}30\_fsr}}%
    \case{nestacase30ieee}{\texttt{\hphantom{00}30\_ieee}}%
    \case{nestacase0030ieee}{\texttt{\hphantom{00}30\_ieee}}%
    \case{nestacase39epri}{\texttt{\hphantom{00}39\_epri}}%
    \case{nestacase0039epri}{\texttt{\hphantom{00}39\_epri}}%
    \case{nestacase57ieee}{\texttt{\hphantom{00}57\_ieee}}%
    \case{nestacase0057ieee}{\texttt{\hphantom{00}57\_ieee}}%
    \case{nestacase73ieeerts}{\texttt{\hphantom{00}73\_ieee\_rts}}%
    \case{nestacase0073ieeerts}{\texttt{\hphantom{00}73\_ieee\_rts}}%
    \case{nestacase89pegase}{\texttt{\hphantom{00}89\_pegase}}%
    \case{nestacase0089pegase}{\texttt{\hphantom{00}89\_pegase}}%
    \case{nestacase118ieee}{\texttt{\hphantom{0}118\_ieee}}%
    \case{nestacase0118ieee}{\texttt{\hphantom{0}118\_ieee}}%
    \case{nestacase162ieeedtc}{\texttt{\hphantom{0}162\_ieee\_dtc}}%
    \case{nestacase0162ieeedtc}{\texttt{\hphantom{0}162\_ieee\_dtc}}%
    \case{nestacase189edin}{\texttt{\hphantom{0}189\_edin}}%
    \case{nestacase0189edin}{\texttt{\hphantom{0}189\_edin}}%
    \case{nestacase300ieee}{\texttt{\hphantom{0}300\_ieee}}%
    \case{nestacase0300ieee}{\texttt{\hphantom{0}300\_ieee}}%
    \case{nestacase2383wpmp}{\texttt{2383wp\_mp}}%
    \case{nestacase2736spmp}{\texttt{2736sp\_mp}}%
    \case{nestacase2737sopmp}{\texttt{2737sop\_mp}}%
    \case{nestacase2746wopmp}{\texttt{2746wop\_mp}}%
    \case{nestacase2746wpmp}{\texttt{2746wp\_mp}}%
    \case{nestacase3012wpmp}{\texttt{3012wp\_mp}}%
    \case{nestacase3120spmp}{\texttt{3120sp\_mp}}%
    \case{nestacase3375wpmp}{\texttt{3375wp\_mp}}%
    % #2%
  \end{switchcase}%
}