text:
\begin{document}
\maketitle
\begin{abstract}
Plug-in hybrid electric vehicles (PHEVs) are an environmentally friendly technology that is expected to rapidly penetrate the transportation system. Renewable energy sources such as wind and solar have received considerable attention as clean power options for future generation expansion. However, these sources are intermittent and increase the uncertainty in the ability to generate power.  The deployment of PHEVs in a vehicle-to-grid (V2G) system provide a potential mechanism for reducing the variability of renewable energy sources. For example, PHEV supporting infrastructures like battery exchange stations that provide battery service to PHEV customers could be used as storage devices to stabilize the grid when renewable energy production is fluctuating. In this paper, we study how to best site these stations in terms of how they can support both the transportation system and the power grid. To model this problem  we develop a two-stage stochastic program to optimally locate the stations prior to the realization of battery demands, loads, and generation capacity of renewable power sources. We develop two test cases to study the benefits and the performance of these systems.

\end{abstract}
\vspace{-.2cm}
\section*{Nomenclature}
\vspace{-.2cm}

\addcontentsline{toc}{section}{Nomenclature}
\noindent{\bf Index/Set}
\begin{IEEEdescription}[\IEEEusemathlabelsep\IEEEsetlabelwidth{$V_1,V_2,V_3$}]
\item[$G(N,E)$] power grid with bus set $N$ and link set $E$
\item [$J$] set of traffic routes
\item [$I$] set of exchange stations
\item [$u = m(i)$] the bus, $u$, of station $i$
\item [$\Omega$] set of scenarios $\{\omega_1,\omega_2,\ldots\}$
\end{IEEEdescription}
{\bf Deterministic Data}
\begin{IEEEdescription}[\IEEEusemathlabelsep\IEEEsetlabelwidth{$V_1,V_2,V_3$}]

\item [$f_i$] fixed cost to open an exchange station at $i$
\item [$r_i$] storage cost per battery at station $i$
\item [$U_i$] maximum number of batteries at station $i$
\item [$L_i$] minimum number of batteries at station $i$
\item [$c_{ij}$] cost to use station $i$ from traffic route $j$
\item [$h_j$] cost of unmet demand from traffic route $j$
\item [$a$] power per battery (MW)
\item [$C_{uv}$] line capacity of line $(u,v)$
\item [$g_u$] load shedding cost at bus $u$
\item [$b_{uv}$] reactance of line $(u,v)$
\end{IEEEdescription}
{\bf Stochastic Data (per scenario $\omega$}
\begin{IEEEdescription}[\IEEEusemathlabelsep\IEEEsetlabelwidth{$V_1,V_2,V_3$}]

\item [$d^\omega_j$] demand for batteries from traffic route $j$
\item [$l^\omega_{u}$] load at bus $u$
\item [$G^\omega_{u}$] generation capacity at $u$
\item [$o^\omega_{u}$] generation cost per MW at $u$
\end{IEEEdescription}
{\bf Variables}
\begin{IEEEdescription}[\IEEEusemathlabelsep\IEEEsetlabelwidth{$V_1,V_2,V_3$}]

\item [$x_i$] 0-1 variable siting an exchange station at $i$
\item [$w_i$] number of batteries stored at station $i$
\item [$t^\omega_i$] number of batteries for PHEV demand
\item [$s^\omega_j$] number of batteries discharged to the grid
\item [$q^\omega_i$] number of unserved batteries on route $j$
\item [$y^\omega_{ij}$] number of batteries served at station $i$ from route $j$

\item [$\alpha^\omega_{uv}$] power flow in line $(u,v)$
\item [$\theta^\omega_{u}$] phase angle at bus $u$
\item [$\delta^\omega_{u}$] slack variable for unmet demand at bus $u$
\item [$\beta^\omega_{u}$] power generated at bus $u$
\end{IEEEdescription}

\vspace{-.2cm}

\section{Introduction}

Global warming and dependence on fossil fuels pose great challenges to the nation's energy infrastructures and future energy consumption. Smart grid technologies and renewable generation are often touted as central pieces to address these challenges \cite{09-DOE-smartgrid}. Indeed, there have been numerous recent studies analyzing the cost-benefits of large penetration of renewable generation into the power grid \cite{07-Boy, 07-Flo, 09-Too} and the implementation of vehicle-to-grid (V2G) \cite{01-Kem,09-Nem} systems for PHEV to reduce fossil fuel consumption \cite{07-web-history-hv,10-DOE-Veh,07-Gon-phev}.

Recent studies \cite{08-Mor-phev, 06-Sim-phev,08-DOT} on feasibility, cost-and-benefit, and impact of PHEVs indicate that it is necessary to build supporting PHEV infrastructures (charging stations (see Fig.~\ref{fig:chargingstations} and \cite{08-Mor-phev}) and exchange stations (see Fig.~\ref{fig:exchangestations} and \cite{exchangestation})) if the adoption of hybrid electric vehicles continues to increase at current rates. There are challenges in implementing both systems: variability in renewable generation and increased demand for power. In this paper, we focus on how renewable technologies and V2G exchange stations may be coupled to address these challenges.

The crucial connection between the two systems arises from the observation that exchange stations are essentially large battery banks where PHEV drivers can automatically switch their batteries with fully charged batteries.  The exchange station may choose to charge batteries during low power usage times (off-peak) or when renewable power plants are producing large amounts of power, thereby reducing the demands a V2G system places on the grid.  In addition, the station may discharge batteries on to the grid during periods of low renewable generation thereby reducing the variability of renewable generation. Thus, there is considerable potential that a coupling of the two systems will reduce the challenges that arise when the systems are implemented independently. This paper seeks to study how to site the stations so that they benefit both the grid and the V2G system.
This problem shares many similarities to the facility location problem \cite{95-Dre,04-Sny,04-Eis, 06-Wu,02-Dre}. In the facility location problem,  facilities are opened to serve a set of customer demands, and the objective is to minimize the setup cost of opening facilities and transportation cost for customers to use the facility.

The crucial difference between the problem stated in this paper and traditional facility location is that this problem problem contains two networks, a transportation network and a power grid.

The contributions of this paper are two-fold.  First, we develop the first mathematical model to site exchange stations that accounts for impacts to transportation systems and electric power systems. Second, we investigate variations of this model to determine the necessity of accounting for both systems, the conditions in which proper siting can be beneficial to both systems, and potential benefit s trade-offs  to both systems.
The rest of the paper is outlined as follows.

A two-stage stochastic program is developed for designing the location of exchange stations in a V2G in Section~\ref{sec:model}. In Section~\ref{sec:experiment}, we investigate the effect of exchange stations on power grids with renewable generation and we conclude with Section~\ref{sec:conclusion}.

\begin{figure}
\centering
\psfig{file=PHEV-Figures.pdf,scale=.16,page=1} \hspace{.2cm}
\psfig{file=PHEV-Figures.pdf,scale=.16,page=2} \hspace{.2cm}
\caption{\label{fig:chargingstations} \small Commercial charging stations }
\vspace{-0.2cm}
\end{figure}

\begin{figure}
\centering
\psfig{file=PHEV-Figures.pdf,scale=.16,page=3} \hspace{.2cm}
\psfig{file=PHEV-Figures.pdf,scale=.16,page=4} \hspace{.2cm}
\caption{\label{fig:exchangestations} \small Commercial exchange stations }
\vspace{-0.6cm}
\end{figure}

\vspace{-.2cm}

\section{A Two-stage Stochastic Program for Siting Exchange Stations}\label{sec:model}
In a V2G system, exchange stations can serve PHEVs and provide storage services to reduce the variability introduced by renewable generation sources. Thus, it is strategically important to site  exchange stations in locations that are convenient for PHEV drivers and that can be connected to the grid without requiring additional capacity expansion.
In this section, we introduce a two-stage stochastic program that integrates a battery demand model and a power grid model to optimally site exchange stations in a V2G system.

In our stochastic programming model, a time line is defined over the decision variables and scenarios as follows. The (first-stage) decisions on where to site exchanges stations and how many batteries to store at each station are made prior to knowing the future demand for the batteries (scenarios). The scenarios are drawn from discrete random variables that characterize PHEV battery demands on each traffic route, the load at buses, and production capacity of renewable generators.  After a scenario is realized, recourse (second-stage) decisions are made to distribute batteries to satisfy PHEV demand and discharge power back to the grid. The cost of the recourse decisions is dependent on the first-stage decisions and the random scenario.  The overall objective function combines the first-stage cost and the expected cost of recourse actions over all scenarios.

We now formalize the model of the first stage variables. In the first stage, the location and size of exchange stations are decided. The long term decisions are made without knowing the exact PHEV demands, loads, and generation capacities. The objective function is to minimize the overall cost
\begin{equation}
\min_{x,w}   \sum_{i\in I}{(f_ix_i+r_i w_i)} + E_\Omega[h(x,w,\xi)] \label{eqn:obj1}
\end{equation}
In (\ref{eqn:obj1}), the summation includes $f_ix_i$  and $r_iw_i$ which represents the fixed cost to open exchange station $i$ and the cost to store batteries at $i$. A first-stage  constraint is represented by the following equation:
\begin{equation}
l_i x_i \le w_i \le u_i x_i, \hspace{5pt} \forall i \in I \label{eqn:cap_batteries}.
\end{equation}
This constraint states that each exchange station can only store batteries when it is open and must satisfy a lower and upper bound on the storage capacity.
The expectation term in the objective represents the second-stage recourse cost $h(x,w,\xi) $  of satisfying PHEV demands and meeting demands for power over a set $\Omega$ of scenarios.

A scenario specifies a realization of the random variables.  For each scenario, $\omega$, the second stage objective $h(x,w,\xi)$ is defined as
\begin{equation}
\min \sum_{i\in I, j\in J}{c_{ij}y^\omega_{ij}} +\sum_{j\in J}{h^\omega_j q^\omega_j}+\sum_{u\in N}{o^\omega_{u} \beta^\omega_{u}} + \sum_{u\in N}{g_u \delta^\omega_u} \label{eqn:obj2}.
\end{equation}

The term $c_{ij}y^\omega_{ij}$ represents the cost of PHEV drivers on route $j$ to swap batteries at station $i$. The penalty cost on unmet PHEV battery demand from route $j$ is stated with $h^\omega_jq^\omega_j$. The cost $c_{ij}$ from route $j$ to station $i$ is typically a function of the distance a driver must divert from route $j$ to reach station $i$. For the power grid, the costs include $o^\omega_{u} \beta^\omega_{u}$ for generation and $g_u \delta^\omega_u$ for shedding loads, which are used to measure the performance of the power grid and include the availability of batteries to meet demand for power.

At each station, available batteries $w_i$ can be used to satisfy PHEV demand ($t^\omega_i$) or supply power to the grid ($s^\omega_i$). The following constraint is used
to ensure that no station uses more batteries than are stored
\begin{equation}
s^\omega_i + t^\omega_i \le w_i,  \hspace{5pt} i \in I \label{eqn:control_station}.
\end{equation}

For the transportation system in the PHEV battery model, we model traffic as routes. The traffic density at each route is derived from activities which are based on demographic surveys and activity surveys collected from real households in the study areas.  This approach has been used to simulate metropolitan area traffic and a detailed description of the traffic model is found in \cite{FastTrans}.   For each route $j$, there is a demand $d^\omega_j$ for PHEV batteries. In (\ref{eqn:bal_phev}), the demand can be satisfied by a battery at a station or be unsatisfied with some penalty,
\begin{equation}
 \sum_{i\in I}{y^\omega_{ij}}  + q^\omega_j =  d^\omega_j, \hspace{5pt}.  j \in J \label{eqn:bal_phev}.
\end{equation}
Constraint (\ref{eqn:cap_station}) limits the number batteries used to satisfy PHEV demand to the number of batteries allocated to $t^\omega_i$ at each station.
\begin{equation}
\sum_{j\in J}{y^\omega_{ij}} \le t^\omega_i, \hspace{5pt}  i \in I \label{eqn:cap_station}
\end{equation}

We use the so called linearized DC power flow equations for modeling flow in the power grid. Constraints (\ref{eqn:bal_bus}) and (\ref{eqn:cap_line}) are the standard constraints for capacitated network problem: (\ref{eqn:bal_bus}) is the network flow balance constraint at each node and (\ref{eqn:cap_line}) is the capacity constraint for each line. In (\ref{eqn:bal_bus}), the  extra term $\sum_{i:m(i) = u}{a s^\omega_i}$ accounts for the power supplied by  the batteries at station $i$ which is connected to bus $u$.

In this model, constraint (\ref{eqn:flow_phase}) captures the relationship between power flow on a line and the phase angle difference at either end of the line.

At each generator, the generation capacity is enforced by the constraint (\ref{eqn:cap_generation}). For renewable generators, the capacities may vary between scenarios according to a distribution governing the potential output of the generators. For non-renewable generators, the capacity is fixed throughout the scenarios.

\begin{equation}
\sum_{v: (u,v) \in E}{\alpha^\omega_{uv}} = -l^\omega_u + \delta^\omega_{u} + \beta^\omega_u + \sum_{i:m(i) = u}{a s^\omega_i}, \hspace{5pt}  \forall u\in N \label{eqn:bal_bus}
\end{equation}
\begin{equation}
\alpha^\omega_{uv} = (\theta^\omega_u - \theta^\omega_v)/b_{uv}, \hspace{5pt} \forall (u,v) \in E \label{eqn:flow_phase}
\end{equation}
\begin{equation}
 -C_{uv} \le \alpha^\omega_{uv} \le  C_{uv}, \hspace{5pt} \forall (u,v) \in E \label{eqn:cap_line}
\end{equation}
\begin{equation}
\beta^\omega_{u} \le G^\omega_u, \hspace{5pt} \forall u \in N \label{eqn:cap_generation}
\end{equation}

The equations (\ref{eqn:obj1})--(\ref{eqn:cap_generation}) define a two-stage stochastic program for locating PHEV exchange stations in the V2G system, and we denote this model as {\bf V2G-2STAGES}.

\vspace{-.2cm}

\section{Computational Experiments}\label{sec:experiment}

To evaluate the impact of exchange stations in a V2G system and the power grid, we test {\bf V2G-2STAGES} on a variation of the IEEE RTS-79 \cite{79-RTS} benchmark and a problem derived from power grid and transportation datasets for Miami, Florida. Before describing the details of the two case studies, we first discuss the general settings and intended findings for both studies.
Load shedding and unmet battery demands are two measures of the performance of the V2G system. Although the {\bf V2G-2STAGES} model optimally allocates exchange stations to meet battery demands and stabilize a grid with renewable energy resources, it is important to realize that the primary purpose of exchange stations is to serve PHEVs.  In our case studies, we will investigate the trade-off between these two objectives. Through these studies, we aim to answer the following questions: {\em 1) What is the impact of exchange stations in a V2G system that is connected to a power grid with variable renewable generation, 2) What are the trade-offs between the two objectives, and 3) Is it important to consider both networks for strategic planning of exchange station siting? }

To model renewable generators, we vary renewable penetration from 0 to 1 in increments of 0.1. For example at renewable penetration level 0.3, each generator may become a renewable source with probability 0.3. The PHEV battery demand is derived from total population, vehicle-to-population ratio (0.78), phev-to-vehicle ratio (0.1), and 10\% of PHEVs requring battery exchanges in any given scenario.  The demand is allocated to each traffic route weighted by the route's average utilization which is set to be same among all routes in the studies.
One hundred scenarios are generated for the second stage. The random variables include battery demand from PHEVs, demand for power, and generation capacities of renewables. All random variables are assumed to be independent (though in the future we plan to explore dependencies in renewable generation capacities). For a given renewable generator, its generation capacity can be 0, 0.5, or 1 of its maximum generation capacity. Load at a bus is a uniform random variable between 0.5 and 1.0 of its peak load. PHEVs demand on a traffic route is a uniform random variable between 0.5 and 1.5 of the average battery demand.

 For both case studies, we implemented {\bf V2G-2STAGES} in C++. The resulting mixed integer program is solved by the branch-and-bound algorithm in CPLEX 11.0 with an optimality gap of 0.01. The longest computing time was under a minute on a standard desktop personal PC.

\subsection{Case Study: IEEE RTS-79}
Our first case study considers a synthetic city based on the IEEE RTS-79 benchmark designed to mimic the structure of Los Alamos, New Mexico. In this data set, there are  25 buses  and 38 power lines. Loads are distributed at 24 buses and there are 11  buses which have generation capacity, with up to 6 generators located at a single bus. The maximum load and generation capacity are provided for every bus in the data set. The total generation capacity is 2999 MW, and the total demand is 2880 MW.
We used an 8 by 11 lattice as the transportation network and created the synthetic city (Fig.~\ref{fig:2grids}) by connecting the lattice and power grid. The connection between the transportation network and the power grid was done by assigning the nearest bus to each of the 88 nodes in the transportation network.
\begin{figure}
\centering
\psfig{file=PHEV-Figures.pdf,scale=.18,page=5} \hspace{.25cm}
\caption{\label{fig:2grids} \small Transportation grid and power grid}
\vspace{-0.6cm}
\end{figure}
The population for the synthetic city is set to 344,850 which was derived from the population-load ratio of Los Alamos, New Mexico.
Twenty-eight nodes were randomly selected from the lattice as possible locations for exchange stations. Ten traffic routes were created by randomly selecting ten origin-destination pairs and finding a shortest path for each pair. This synthetic city is a model  of a city with some power consuming industries and a relatively low population.

\begin{figure}
\centering
\psfig{file=PHEV-Figures.pdf,scale=.26,page=11}
\psfig{file=PHEV-Figures.pdf,scale=.26,page=12}
\caption{\label{fig:rts-ge} \small RTS-79 load shedding and unmet battery demands in generation expansion model with renewables from 0 to 100 percent}
\vspace{-0.6cm}
\end{figure}
In order to understand how V2G batteries may maximally improve the performance of a power grid with variable renewable generation, we
removed PHEV demands and reduced {\bf V2G-2STAGES} to a generation expansion model.  In the base case (GE-1 in Fig.~\ref{fig:rts-ge}), no additional generation resources (exchange stations) were added and the grid was evaluated with different renewable penetration levels. With renewable levels lower than 30\%, there is no loading shedding due to renewable generation variability. For higher renewable penetration level, load shedding can reach as high as 33\%. In cases GE-2 to GE-5, we varied the load shedding penalty to stimulate the opening of exchange stations as generation capacity reserve. We ran all cases with 11 different renewable penetration levels from 0 to 100 percent.  When load shedding penalities are increased, there is some reduction in load shedding. Load shedding is reduced to below 10\% with renewable penetration levels less than 80\%. The reduction is mostly caused by opening 25 out of 28 stations. The opening of additional stations does not reduce load shedding due to other limits, e.g., line capacity, of the grid.
\begin{figure}
\centering
\psfig{file=PHEV-Figures.pdf,scale=.26,page=13}
\psfig{file=PHEV-Figures.pdf,scale=.26,page=14}
\caption{\label{fig:rts-v2g1} \small RTS-79: load shedding and unmet battery demands in {\bf V2G-2STAGES} with renewables from 0 to 100 percent}
\vspace{-0.6cm}
\end{figure}

Fig.~\ref{fig:rts-v2g1} shows the simulations of {\bf V2G-2STAGES} for the entire V2G system (including both PHEV battery demands and power grid). First, we used {\bf V2G-2STAGES} to determine that the optimal solution  to satisfy all battery demands  is 6 stations. In simulation V2G-1, we fixed the station locations to this optimal solution and added the power grid.   There was some reduction in load shedding, but there is a trade-off in unsatisfied PHEV demand. In simulation V2G-2 to V2G-5, we relaxed the location of the stations and allowed the total number of opened stations to be 6, 8, 10, and 12 stations respectively. The actual number of opened stations from  these simulations are shown in Fig.~\ref{fig:rts-v2g2}.  The improvement of load shedding is limited as the optimal solutions of GE-3 to GE-5 in Fig.~\ref{fig:rts-ge} show that to have significant reduction in load shedding we require as many as 25 stations.  In V2G-2 to V2G-4 the increase in unmet battery demands is large when compared to V2G-1. The reason for this is that in V2G-1, the locations of stations were fixed to the optimal solution of the PHEV only system. These locations severely limited the discharge capability of the exchange stations due to the associated line capacities. This result demonstrates that in order for the V2G to benefit both PHEV and the power grid, both models must be considered when determining locations to site exchange stations. In V2G-5, a sufficient number of stations are opened such that both load shedding and unmet battery demands are smaller than in V2G-1.

In conclusion, at low renewable penetration levels, the V2G system can reduce load shedding caused by renewable generation variability and meet the demand for PHEV battery exchanges. For higher levels of renewables, the benefit of V2G is not as obvious.  In general, the trade-offs between load shedding and unmet battery demands are quite high. These discrepancies may be caused by the relatively low population to load ratio, i.e. there is not enough PHEV demand and only a small number of exchange stations are required.

\begin{figure}
\centering
\psfig{file=PHEV-Figures.pdf,scale=.26,page=15}
\caption{\label{fig:rts-v2g2} \small RTS-79: allocation of exchange stations in {\bf V2G-2STAGES} with renewables from 0 to 100 percent}
\vspace{-0.6cm}
\end{figure}

\subsection{Case Study: Miami, Florida}
In our second case, we use Los Alamos National Laboratory's data sets on power grid and transportation networks in Miami, Florida. The transmission power system in the greater Miami area consists of more than 200 buses and over 275 power lines. For this case study, we used peak load data and maximum generation capacity.

The overall load is over 6400 MW, and the overall generation and import capacity is greater than 8200 MW.  The second stage problem has 100 scenarios, and in each scenario, loads and generation capacities are independently generated in the same fashion as the first scenario.

In the transportation network, there are over 2500 locations and over 3800 roads. A data set consisting of the gas station locations in the Miami area is used to identify 316 potential locations for exchange stations in the transportation network.

One hundred traffic routes were created by the same method in the last case study.
In 2008, the population of Miami was estimated to be 5,414,712, which yields an estimated 422,348 requests for PHEV batteries based upon the set of ratios described earlier. For each of the 100 scenarios, battery demand of a route is generated randomly and independently with respect to a uniform distribution between 0.5 and 1.5 of the average demand.  Miami represents a city with a high population-to-load ratio when compared to the synthetic city created in the first case study.

\begin{figure}
\centering
\psfig{file=PHEV-Figures.pdf,scale=.26,page=6}
\psfig{file=PHEV-Figures.pdf,scale=.26,page=7}
\caption{\label{fig:miami-ge} \small Miami: load shedding and allocation of exchange stations in generation expansion model with renewables from 0 to 100 percent}
\vspace{-0.6cm}
\end{figure}

Once again, by setting the penalty cost, $h$,  of unmet PHEV demands to zero, we reduce {\bf V2G-2STAGES} to a stochastic generation expansion model. This allows us to understand the maximal benefit V2G could provide to this grid.  We tested this generation expansion model on five different cases GE1-5 in Fig.~\ref{fig:miami-ge}. GE-1 is the base case where no batteries are provided by the exchange stations. In GE-2 to GE-4 we applied different penalty costs on shedding load to investigate the effect of additional exchange stations. With the 11 different renewable penetration levels, the load shedding is not always non-decreasing with respect to the renewable level because of constraints in the power system and the stochastic choice for renewable generation locations. In the results, the location of generators is a key factor in the ability deliver power.  Excess generation capacity at certain locations may not be able to supply loads in the other part of the grid due to capacity constraints.  To elaborate this point, we observed that in GE-1, even without the renewable generation, there was a 2\% load shed even though the overall generation capacity exceeded the overall loads.  As the grid includes more renewables as power sources, the load shedding increases and in the worst case, the load shed reaches 33\%. Increasing the number of exchange station did reduce the load shed. In GE3-GE5, load shedding is reduced to under 5\% for all renewable levels, but the reduction is the result of opening many stations. One interesting case is GE-2 where only 2 to 4 stations are opened and the load shedding is reduced to below 5\% with renewable penetration at 50\% or lower.  For  high  renewable penetration levels, a large number of stations are required to reduce load shed caused by the generation variability.

\begin{figure}
\centering
\psfig{file=PHEV-Figures.pdf,scale=.26,page=8}
\psfig{file=PHEV-Figures.pdf,scale=.26,page=9}
\caption{\label{fig:miami-v2g1} \small Miami: load shedding and unmet battery demands in {\bf V2G-2STAGES} with renewables from 0 to 100 percent}
\vspace{-0.6cm}
\end{figure}

Using similar steps to the synthetic city case, we removed the power grid from {\bf V2G-2STAGES}, to determine the optimal allocation of the stations (106) to serve all the battery demands. Next, we  evaluated the performance of the full V2G system with respect to  this optimal allocation.  First, we applied the optimal allocation to the power grid and the result is V2G-1 shown in Fig.~\ref{fig:miami-v2g1}. Importantly, the load shed is less than the base case GE-1; however,  the unmet battery demands increased even at low renewable levels. In V2G-2, the total number of opened stations is still set to 106, but {\bf V2G-2STAGES} model can freely choose the locations of stations.  The load shed  is slightly better in V2G-2 than V2G-1 and at the same time the unmet battery demands drop to almost zero for low renewable penetration levels. This indicates that it is important to combine both traffic and grid models when determining the sites to open stations. For V2G-3 to V2G-5, we allowed the number of stations to increase by 10\%, 20\%, and 50\% respectively from 106.  With 10\% increase, V2G-3 shows that {\bf V2G-2STAGES} is able to select locations such that both load shed and unmet battery demands are very low for the renewable penetration level less than and equal to 50\%, indicating that a small amount of additional V2G infrastructure construction can significantly help the power grid.

In conclusion, with low to medium renewable penetration levels ($\le 50\%$), the V2G system of the greater Miami area  can reduce the variability of renewable sources while maintaining the  service to PHEVs. The location of stations is important for the performance of a V2G system, and integrating both grid and traffic models is important to the planning process. The performance of the V2G system is much better in the Miami case study than in the synthetic city. The difference appears to be a result of  Miami having high population to load ratio, and this high ratio leads to more battery demand, more options for siting exchange stations and greater oppurtunity to benefit the power grid.
\begin{figure}
\centering
\psfig{file=PHEV-Figures.pdf,scale=.26,page=10}
\caption{\label{fig:miami-v2g2} \small Miami: allocation of exchange stations in {\bf V2G-2STAGES} with renewables from 0 to 100 percent }
\vspace{-0.6cm}
\end{figure}

\vspace{-.2cm}

\section{Conclusion}\label{sec:conclusion}
In this paper, we investigated the effect of a V2G system on reducing load shed caused by generation variability  of renewables. After introducing a two-stage stochastic programming model to site exchange stations, we applied the model to analyze the V2G system through two case studies.

\vspace{-.2cm}


TITLE: Can we test QFT on a curved spacetime?
QUESTION [6 upvotes]: It is possible to extend a quantum field theory to a curved spacetime. But does this lead to predictions that can be tested and measured? Had it been confirmed?
The underlying reason I am asking this is: curved spacetime means emergence of gravity and therefore General Relativity regime. And we know that GR and QFT are incompatible. I realise that in order to include gravity, one should put its Lagrangian in from the very beginning and this, I guess, does not work. But does the current mathematical framework for extending the known field theories to a curved spacetime work?

REPLY [5 votes]: Cosmology and inflation provide a hugely important test of quantum field theory in curved spacetime.  During inflation, there is a scalar field (the inflaton) that is providing the vacuum energy that drives inflation.  This scalar field is subject to the rules of quantum field theory in curved spacetime.  The quantum fluctuation of this scalar field lead to the temperature fluctuations we see in the Cosmic Microwave Background today, and as these fluctuations grow due to gravitational collapse, they produce the structure we see today in galaxies and clusters.  Measuring properties of the CMB and large scale structure is thus a test of quantum field theory in curved spacetime from the inflationary epoch.


TITLE: Net work done in lifting
QUESTION [0 upvotes]: ($g$ is assumed as $10\ \mathrm{m/s^2}$)
We lift a stationary object of mass, $m$, to a height $s$, ensuring it has got a resultant upward acceleration of $2\ \mathrm{ m/s^2}$.  The required force to be applied by us for the process is ($12m$). At $s$, in order to bring it to a stop, we apply a force of ($2m$) in downward direction. And thus it stops and remains at rest at height $s$.
By work energy theorem, net work done is zero as kinetic energy is zero.
Net work done = [Work done by gravity] + [Work done by us] = 0
$$[-(mgs)] + [(12ms) - (2ms)] =0$$
which is mathematically right.
The question is in the very last term.
The downward force of $2\ \mathrm{ m/s}$ does not act for a distance $s$. Why is then the displacement $s$ taken there ?
Or is the equation wrong ?
What, then, is the right equation?

REPLY [1 votes]: First of all, the work done in bringing the mass to a rest at height $s$ is done by gravity, not us. So your equation should be rearranged as follows.
$$[-mgs-m2s]+m12s=0$$
But I think there is a bigger problem with the equation. Gravity has to be able to do enough negative work such that the negative change in kinetic energy gravity does equals the positive change in kinetic energy that you gave the body by applying a force of $m12$ for whatever distance you applied to it. The max possible negative work done by gravity is $-mgs$.
The only way that can happen is if you stop applying that force short of the distance $s$. The greater the distance you apply it, the greater the negative work done by gravity needs to be. If the distance is too close to $s$, the negative work of gravity may not be enough.
Bottom line is the positive net work done by you must equal the negative net work done by gravity, or
$$-mgs (gravity)+mgs (you)=0$$ in order for the body to come to rest.
Hope this helps.


TITLE: A lemma about quasicentral-approximate-unit
QUESTION [1 upvotes]: Here is a lemma about quasicentral-approximate-unit:

Lemma 7.3.1Let $J\triangleleft A$ be a separable ideal. Then there exists a quasi-central approximate unit $\{e_{j}\}\subset J$ such that $e_{j+1}e_{j}=e_{j}$ for all $j\in \mathbb{N}$.
Proof. Since $J$ is separable, it contains a strictly positive element $h$ (i.e., $\phi(h)>0$ for all states $\phi$ on $J$-let $h=\sum\frac{1}{2^{j}}e_{j}$ where $\{e_{j}\}$ is any approximate unit). For each $n$ let $f_{n}\in C_{0}(0, 1]$ be the function which is zero on the intercal (0, $\frac{1}{2^{n}}]$, one on the intercal [$\frac{1}{2^{n-1}}$, 1] and linear in between. Evidently we have $f_{n+1}(h)f_{n}(h)=f_{n}(h)$, for all $n$, and it is also readily seen that $\phi(f_{n}(h))\rightarrow 1$ for all states $\phi$ on $J$. The usual Habh-Banach convexity argument allows us to extract a quasicentral approximate unit from  the convex hull of $\{f_{n}(h)\}$, so we leave the remaining details to you.

My question is
In the end of the proof, how to usethe Hahn-Banach convexity argument to find a quasicentral approximate unit?

REPLY [1 votes]: You have to look at Theorem 1.2.1, which is just a re-statement of I.9.16 in Davidson's book. Davidson's proof is a page long, not counting auxiliary results.
Davidson's proof doesn't state it explicitly, but the condition $e_{j+1}e_j=e_j$ can be achieved by going far enough along the index set when defining $\mathcal F$ in Davidson's proof.


TITLE: Finding for what values of $y$ $f_{X|Y}(x|y)$ is defined
QUESTION [0 upvotes]: Find the conditional density  function $f_{X|Y}(x|y)$.   Do  not  forget  to  indicate  where  the density is non-zero. For what values of $y$ is this conditional density defined?
First, I found that $f_Y(y) = \frac{14y}{3}(1-y^{\frac{3}{2}})$ for $0 \leq y \leq x^2$ and $f_Y(y)=0$ otherwise. Then, I found that
$f_{X|Y}(x|y) = 2x^{-4}y$. Now, I am unsure as to how to find for what values of $y$ this is defined. Would it simply be $0 \leq x \leq 1$ and $0 \leq y \leq x^2$, as given in the question?

REPLY [0 votes]: The support for $f_Y(y)$ must not include $x$ ; it is neither an argument nor a free variable.
Now the domain of $0\leq x\leq 1$ and $0\leq y\leq x^2$ is also $0\leq y\leq 1$ and $\surd y\leq x\leq 1$.
$$\begin{align}f_Y(y) &= \int_{\surd y}^1 14 x^2 y\,\mathbf 1_{0\leq y\leq 1}\,\mathrm d x\\ &=\tfrac {14}3 y(1-y^{3/2})\,\mathbf 1_{0\leq y\leq 1}\\[4ex] f_{X\mid Y}(x\mid y) &= \dfrac{3 x^2}{(1-y^{3/2})}\,\mathbf 1_{0\leq y\leq 1}\mathbf 1_{\surd y\leq x\leq 1}\end{align}$$

$\mathbf 1_{a\leq z\leq b}$ is a indicator function; peicewise equalling $1$ when the index is satisfies, and $0$ otherwise:$$\mathbf 1_{a\leq z\leq b}=\begin{cases}1&:& a\leq z\leq b\\0&:&\textsf{otherwise}\end{cases}$$


TITLE: Does observing life on Earth increase the probability of life elsewhere?
QUESTION [11 upvotes]: Say I have an implausibly large sack of balls. All I know is that the balls are numbered randomly from $1$ to $n$. For all I know, any value of $n$ (a positive integer) is equally likely.
I reach into the sack and choose a ball randomly. The ball says $42$. Does this change at all the probabilities of the values of $n$ used to number the balls where $n \geq 42$?
(Intuitively it might seem like $n$ is a low number in that if $n$ were very very large (say $2^{42}$) it seems implausible we would hit on a very low number from the first ball sampled. On the other hand, if $n$ is a very very large number, $42$ is equally as like as any ball to emerge.)

Another simplified version might be where the balls are either blue or red, but I don't know how many are blue or how many are red. The first ball I choose is blue. Does this increase the probability of observing further blue balls in later samples?
(Again if there were only one blue ball, intuitively it seems unlikely we would choose it on the first sample. On the other hand, if there were only one blue ball, that ball is as equally likely to emerge as any on the first sample.)

It seems to be a question that crops up a lot. Like for example in the argument that well there's life here on Earth so it would be an improbable fluke if there were no life elsewhere. Of course this is a more complex question than just what colour the balls are, but the thrust of this argument seems to be a probabilistic one, like it boils down to the idea that we know there's one blue ball in the tiny sample we've seen, so there must be lots of blue balls in the implausibly large sack to explain that.
I'm not convinced this latter argument makes sense, but on the other hand, I don't know how to reason about the problem or prove one way or the other whether seeing a blue ball early on affects the (relative) probability of the number of blue balls in the population. Hence I'm wondering, for example, if there's some sort of general theorem from probability that talks about this?

REPLY [0 votes]: Earlier I was trying to tackle this from an intuitive perspective so perhaps it's worth posting some ideas (as thinking out loud).
Coming at this from someone who knows little about probability/probability theory, the only way I can see to reason about this problem is to think in an intuitive way about simulations and just counting cases.
Let's take the case of blue and red balls for example. Let us assume we have $X$ balls and that $B$ of those are blue. To keep this as general as possible, we don't know the value for $B$ (other than $B\geq 1$; put another way, we have no idea of what "prior" probability a ball has of being blue) nor for $X$ (other than $X\geq B$).
However, for argument's sake, let's fix some arbitrary large value for $X$. So given $X$, now we can try run a great many simulations and count how many times, for various values of $b \leq X$, we sample a blue ball on the first go. Let's say we run $s$ simulations for each value of $b$ ($s \gg X$), drawing a first ball and keeping track of how many times it's blue.
As $s$ approaches infinity for a given value of $b$, the number of times we will sample blue in the first ball will equal $\frac{b}{X}$. Across all values of $b$, we get will have $sX$ simulations in total. The number of times a blue ball will be sampled first will be $\frac{sX}{2}$.
Okay now we assume that we know that a blue ball was sampled first and we look at our $sX$ simulations to see in what ratio of simulations that happened for various values of $b$.
Thus if we know that a blue ball is sampled first, then $s$ of those cases will occur when $b=X$, $s-1$ cases when $b=X-1$ and more generally, $s-n$ cases when $b=X-n$.
Since we're interested in probabilities, rather than counting cases, we can look at ratios. In total, we can see that in half the cases, a blue ball is sampled first. For a given value of $b$, $\frac{1}{X}$ cases will have been run for that value (either blue or not). The ratio of cases where the first ball sampled was blue will be $\frac{b}{X^2}$. So for example, given that the first ball sampled is blue, that leaves us with $\frac{1}{2}$ of the original cases, of which $\frac{1}{X}$ cases are explained by all balls being blue ($b=X$).
This is not so satisfying though since it always goes back to the value of $X$. But we can try to find a way to find a general conclusion or an "invariant": let's try to see in terms of "cumulative probability" at what point 50% of the cases where blue is drawn first are covered. In other words, we're looking for $\beta$ such that $\Sigma_{b \leq \beta} \frac{b}{X^2} = \frac{1}{4}$. We can consider $\beta$ as something of a "tipping point", meaning that knowing that blue was sampled first, the value of $b$ being below $\beta$ or above $\beta$ are equally likely (where we would expect it, for sure, to be somewhere above $\frac{X}{2}$). In fact, as $X$ approaches $\infty$, the value for $\beta$ converges to $\frac{X}{\sqrt{2}}$.
So for example, if $X=100000$, and we know that a blue ball is drawn first, this tells us that $P(B>70711\mid \text{blue ball drawn first}) \approxeq 0.5$. Also that ratio is fixed as $\frac{X}{\sqrt{2}}$, so given $X=10000$, $P(B>7071\mid \text{blue ball drawn first}) \approxeq 0.5$.

However, this all presumes I guess what one would call a "uniform distribution" over the possible values that $B$ might take. That seems like the most natural assumption to take when no other information is given, but I rather tend to think that it's just as unknown as the value for $B$, and hence really, no information can be gained from knowing that a blue ball is drawn first unless one assumes something quite strong: that any value of $B$ is somehow equally likely. Under that assumption, there's a 50/50 chance that $\sim$70.7\% or more of balls are blue according to the above line of reasoning.
What I have difficulty grappling with now – on a more philosophical level – is just how reasonable it is to assume – in the absence of further information – that any value of $B$ is equally likely. It's tempting to assume this to even start to make progress, but equally it seems to be something we do not know and thus cannot use.
(Comments are very welcome.)


TITLE: lambda calculus: predecessor function
QUESTION [1 upvotes]: Apparently this expression can be used to calculate the predecessor of a given church numeral:
$\renewcommand{\l}{\lambda}$
$\l n.(n\ \l p . (p.2,\ s p.2)\ (\overline{0},\ \overline{0})).1$
$<tuple>.n$ being the projection onto the $n$th element.
Now, applying this to some number, let's say $3$ I get:
$\l n.(n\ \l p . (p.2,\ s\ p.2)\ (\overline{0},\ \overline{0})).1\ \overline{3}$
$\rightarrow (\overline{3}\ ((\overline{0},\ \overline{0}).2,\ s\ (\overline{0},\ \overline{0}).2)).1$
$\rightarrow (\overline{3}\ (\overline{0},\ \overline{1})).1$
I assume I made some mistake here, because this makes no sense, because the expression is not a tuple, so there can be no projection being applied to it. Well, maybe I just need to unfold the expression further...
$\rightarrow ((\l f. \l x. f(f(f\ x)))\ (\overline{0},\ \overline{1})).1$
$\rightarrow ((\l x. (\overline{0},\ \overline{1})((\overline{0},\ \overline{1})((\overline{0},\ \overline{1})\ x)))\ ).1$
No.. this does not look any better... I can't apply $x$ the the right most/ inner most tuple, no any of the tuples to any other tuple... so... what am I missing here?

REPLY [2 votes]: The error is where you go from
$$
\l n.(n\ \l p . (p.2,\ s\ p.2)\ (\overline{0},\ \overline{0})).1\ \overline{3}
$$
to
$$
(\overline{3}\ ((\overline{0},\ \overline{0}).2,\ s\ (\overline{0},\ \overline{0}).2)).1\;.
$$
In lambda calculus, function application is by convention left-associative, so
$$
\l n.(n\ \l p . (p.2,\ s\ p.2)\ (\overline{0},\ \overline{0})).1\ \overline{3}
$$
is
$$
((\overline3\ \l p . (p.2,\ s\ p.2))\ (\overline{0},\ \overline{0})).1
$$


TITLE: mapping low dimensional space to high dimensional space injectively?
QUESTION [1 upvotes]: Is it possible to find an injective mapping that transforms low dimensional space like $\mathbb{R}$ to a subset of $\mathbb{R}^2$ with interior points? For instance, can we find a mapping from $\mathbb{R}$ to a unit disk in $\mathbb{R}^2$ ($x^2+y^2\leq 1$)?

REPLY [0 votes]: I won't answer your second question (to allow you some time to think out what kind of function would be an injection), but one obvious map that does this is the map that takes the real number $x$ and associates it with the point $(x,x)$ $\in$ $\mathbb R^2$. You can easily verify that this function is indeed injective.