efficient computer simulations of large crowds consisting of hundreds or thousands of individuals require simple models which nevertheless provide an accurate description of reality .one such class of models , so - called cellular automata ( ca ) , has been studied in statistical physics for a long time .  in caspace , time and state variables are discrete which makes them ideally suited for high - performance computer simulations. however , ca modelling differs in several respects from continuum models .these are usually based on coupled differential equations which often can not be treated analytically .one has to solve them numerically and therefore the equations have to be discretized . in general , only space and time variablesbecome discrete whereas the state variable is still continuous .one important point is now that ca are discrete from the beginning and that this discreteness is already taken into account in the definition of the model and its dynamics .this allows to obtain the desired behaviour in a much simpler way . on the other hand ,the numerical solution of ( discretized ) differential equations is only accurate in the limit , .this is different in the ca where and are finite and accurate results can be obtained since the rules ( dynamics ) are designed such that the discreteness is an important part of the model .  in order to achieve complex behaviour in a simple fashion one often resorts to a stochastic description .a realistic situation seldomly can be described completely by a deterministic approach .already minor events can lead to a very different behaviour due to the complexity of the interactions involved . for the problem of pedestrian motionthis becomes evident e.g. in the case of a panic where the behaviour of people seems almost unpredictable .but also for `` normal '' situations a stochastic component in the dynamics can lead to a more accurate description of complex phenomena since it takes into account that we usually do not have full knowledge about the state of the system and its dynamics . hereone has to keep in mind that in general some sort of average over different realizations of the process ( e.g. different sequences of random numbers ) has to be taken . even if there are single realizations which yield unrealistic behaviour the average process will be a good description of the real process .furthermore a stochastic description allows to answer questions like `` what is the probability that the evacuation of this building will take longer than 3 minutes ? '' in a natural way .  in the followingwe will present a detailed description of the model and the basic philosophy of our approach .applications are presented in part ii .during the last decade considerable research has been done on the topic of highway traffic using methods from physics .cellular automata inspired by the pioneering works compose by now an important class of models .most studies have been devoted to one - dimensional systems , where several analytic approaches exist to calculate or approximate the stationary state .on the other hand , pedestrian dynamics has not been studied as extensively as vehicular traffic , especially using a cellular automata approach .one reason is probably its generically two - dimensional nature . in recent years, continuum models have been most successful in modelling pedestrian dynamics .an important example are the _ social force models _ ( see e.g.  and references therein ) . herepedestrians are treated as particles subject to long - ranged forces induced by the social behaviour of the individuals .this leads to ( coupled ) equations of motion similar to newtonian mechanics .there are , however , important differences since , e.g. , in general the third law ( `` actio = reactio '' ) is not fulfilled .in contrast to the social force models our approach is closer in spirit to the general strategy of modelling ( elementary ) forces on a microscopic level by the exchange of mediating particles which are bosons . it is therefore similar to _ active walker models _ used so far mainly to describe trail formation , chemotaxis ( see for a review ) etc . herethe walker leaves a trace by modifying the underground on his path .this modification is real in the sense that it could be measured in principle . for trail formation, vegetation is destroyed by the walker and in chemotaxis he leaves a chemical trace .in contrast , in our model the trace is virtual .its main purpose is to transform effects of long - ranged interactions ( e.g. following people walking some distance ahead ) into a local interaction ( with the `` trace '' ) .this allows for a much more efficient simulation on a computer .cellular automata for pedestrian dynamics have been proposed in .these models can be considered as generalizations of the biham - middleton - levine model for city traffic .most works have focussed on the occurrence of a jamming transition as the density of pedestrians is increased .all models have only nearest - neighbour interactions , except for the generalization proposed in which is used for analyzing evacuation processes on - board passenger ships .the other models use a kind of `` sublattice - dynamics '' which distinguishes between different types of pedestrians according to their preferred walking direction .such an update is not easy to generalize to more complex situations where the walking direction can change . to our knowledgeso far most other collective effects encountered empirically have not been reproduced using these models .another discrete model has been proposed earlier by gipps and marksjs .this model is somewhat closer in spirit to our model than the cellular automata approaches of since the transitions are determined by the occupancies of the neighbouring cells .however , this model can not reproduce all collective effects either . in discretized version of the social force model has been introduced .the repulsive potentials by the pedestrians are stored in a global potential , with pedestrians reacting to the gradients of this global potential .although this model is able to reproduce collective effects it is not flexible enough to treat individual reactions to other pedestrians , and collision - avoidance is not always guaranteed for velocities greater than 1 .first we discuss some general principles we took into account in the development of our model .the implementation of the interactions between the pedestrians uses an idea similar to chemotaxis .the pedestrians leave a virtual trace which then influences the motion of other pedestrians .this allows for a very efficient implementation on a computer since now all interactions are local .the transition probabilities for all pedestrians just depend on the occupation numbers and strength of the virtual trace in his neighbourhood , i.e. we have translated the long - ranged spatial interaction into a local interaction with `` memory '' .the number of interaction terms in other long - ranged models , e.g.the social - force model , grows proportionally to the square of the number of particles whereas in our model it grows only linearly .the idea of a virtual trace can be generalized to a so - called _floor field_. this floor field includes the virtual trace created by the pedestrians as well as a static component which does not change with time .the latter allows to model e.g. preferred areas , walls and other obstacles .the pedestrians then react to both types of floor fields .  to keep the model simple , we want to provide the particles with as little intelligence as possible and to achieve the formation of complex structures and collective effects by means of self - organization .in contrast to older approaches we do not make detailed assumptions about the human behaviour . nevertheless the model is able to reproduce many of the basic phenomena .the key feature to substitute individual intelligence is the floor field .apart from the occupation number each cell carries an additional quantity ( field ) which can be either discrete or continuous . this field can have its own dynamics given by diffusion and decay coefficients .interactions between pedestrians are repulsive for short distances .one likes to keep a minimal distance to others in order to avoid bumping into them . in the simplest version of our modelthis is taken into account through hard - core repulsion which prevents multiple occupation of the cells . for longer distancesthe interaction is often attractive .e.g. when walking in a crowded area it is usually advantageous to follow directly behind the predecessor .large crowds may also be attractive due to curiosity .  with two particle species moving in opposite directions , each with its own floor field ,effects can be observed which are so far only achieved by continuous models : lane formation and oscillation of the direction of flow at doors .we consider this model to be another example for the ability of cellular automata to create complex behaviour out of simple rules and the great applicability of this approach to all kinds of traffic flow problems .in contrast to vehicular traffic the time needed for acceleration and braking is negligible .the velocity distribution of pedestrians is sharply peaked .these facts naturally lead to a model where the pedestrians have a maximal velocity , i.e. only transitions to neighbour cells are allowed .furthermore , a greater , i.e. pedestrians are allowed to move more than just one cell per timestep , would be harder to implement in two dimensions , especially when combined with parallel dynamics , and reduce the computational efficiency .the number of possible target cells increases quadratically with the interaction range .furthermore one has to check whether the path is blocked by other pedestrians .this might even be ambigious for diagonal motion and crossing trajectories .also higher velocity models lead to timescales which are much too small ( see sec . [ sec_def ] ) .the area available for pedestrians is divided into cells of approximately .this is the typical space occupied by a pedestrian in a dense crowd .each cell can either be empty or occupied by exactly one particle ( pedestrian ) .for special situations it might be desirable to use a finer discretization , e.g. such that each pedestrian occupies four cells instead of one .the update is done in parallel for all particles .this introduces a timescale into the dynamics which can roughly be identified with the reaction time . in the deterministic limit , corresponding to the maximal possible walking velocity in our model , a single pedestrian ( not interacting with others ) moves with a velocity of one cell per timestep , i.e.  per timestep .empirically the average velocity of a pedestrian is about  .this gives an estimate for the real time corresponding to one timestep in our model of approximately which is of the order of the reaction time , and thus consistent with our microscopic rules .it also agrees nicely with the time needed to reach the normal walking speed which is about .this corresponds to at least timesteps if the pedestrian can only accelerate by one unit per timestep .therefore in models with large a timestep would correspond to a real time shorter than the smallest relevant timescale .this makes the model more complicated than necessary and reduces the efficiency of simulations .each particle is given a preferred walking direction . from this direction , a _ matrix of preferences _is constructed which contains the probabilities for a move of the particle .the central element describes the probability for the particle not to move at all , the remaining 8 correspond to a move to the neighbouring cells ( see fig .[ fig_prefs ] ) .the probabilities can be related to the velocity and the longitudinal and transversal standard deviations ( see for details ) .so the matrix of preferences contains information about the preferred walking direction and speed . in principle, it can differ from cell to cell depending on the geometry and aim of the pedestrians .in the simplest case the pedestrian is allowed to move in one direction only without fluctuations and in the corresponding matrix of preference only one element is one and all others are zero . in the following it is assumed that a matrix of preferences is given at every timestep for each pedestrian .these can e.g. be obtained from some model for route selection which assigns certain routes to each pedestrian .this ansatz can easily be extended by fixing the direction of preference for each cell separately , e.g. to handle structures inside buildings .then the particles would use the matrix belonging to the cell they occupy at a given step .however , a similar effect can be obtained much simpler by introducing a second floor field ( see sec . [ sec_floor ] ) .  in each update step for each particle a desired move is chosen according to these probabilities .this is done in parallel for all particles .if the target cell is occupied , the particle does not move . if it is not occupied , and no other particle targets the same cell , the move is executed .if more than one particle share the same target cell , one is chosen according to the relative probabilities with which each particle chose their target .this particle moves while its rivals for the same target keep their position ( see fig .[ fig_conflict ] ) .the rules presented up to here are a straightforward generalization of the ca rules used so far for the description of traffic flow .the main difference is that in principle transitions in all directions are possible and each pedestrian might have her own preferred direction of motion characterized by a matrix of preferences . the only interaction between particles taken into account so far is hard - core exclusion .   in order to reproduce certain collective phenomenait is necessary to introduce further longer - ranged interactions . in some continuous models thisis done using the idea of a social force . herewe present a different approach . since we want to keep the model as simple as possible we try to avoid using a long - range interaction explicitly .instead we introduce the concept of a _ floor field _ which is modified by the pedestrians and which in turn modifies the transition probabilities .this allows to take into account interactions between pedestrians and the geometry of the system ( building ) in a unified and simple way without loosing the advantages of local transition rules .the floor field modifies the transition probabilities in such a way that a motion into the direction of larger fields is preferred.  the floor field can be thought of as a second grid of cells underlying the grid of cells occupied by the pedestrians .it can be discrete or continuous . as already explained in sec .[ sec_principles ] we distinguish two types of fields which will be called static and dynamic floor fields , respectively .  the _ dynamic floor field _ is just the virtual trace left by the pedestrians ( see sec .[ sec_principles ] ) .it is modified by the presence of pedestrians and has its own dynamics , i.e. diffusion and decay .usually the dynamic floor field is used to model a ( `` long - ranged '' ) attractive interaction between the particles .each pedestrian leaves a `` trace '' , i.e. the floor field of occupied cells is increased .explicit examples where such an interaction is relevant are given in part ii .the dynamic floor field is also subject to diffusion and decay which leads to a dilution and finally the vanishing of the trace after some time .the _ static floor field _ does not evolve with time and is not changed by the presence of pedestrians .such a field can be used to specify regions of space which are more attractive , e.g. an emergency exit ( see the example in ) or shop windows .this has an effect similar to a position - dependent matrix of preference , but is much easier to realize .the _ transition probability _ in direction ( see fig .[ fig_prefs ] ) now depends on four contributions :  * the matrix of preference which contains the information about the aim and average velocity of the pedestrian .* the value of the dynamic floor field at the target cell .this contribution takes into account the effects of the motion of the other pedestrians .in many applications ( see ) it is attractive to `` follow the crowd '' , i.e. transitions in directions with a large value of the dynamic floor field are preferred . *the value of the static floor field .it allows to model effects of the geometry .e.g. in a corridor it is usually less attractive to walk close to the walls .such an effect can be incorporated in a static floor field which decreases near the walls .* the occupation number of the target cell .a motion in direction is only allowed if the target cell is empty ( ) and forbidden if it is already occupied ( ) .one simple possibility to take into account all contributions ( i)(iv ) is to define the transition probability in direction by  is a normalization factor to ensure where the sum is over the nine possible target cells .there are also slightly more general forms of the transition probabilities which have been studied in .since the total transition probability is proportional to the dynamic floor field it becomes more attractive to follow in the footsteps of other pedestrians .this effect competes with the preferred walking direction specified by and the effects of the geometry encoded in .the relative influence of the contributions ( i)(iii ) is controlled by coupling parameters .these depend on the situation to be studied .consider for example a situation where people want to leave a large room ( see ) .normal circumstances , where everybody is able to see the exit , can be modelled by solely using a static floor field which decreases radially with the distance from the door . since transitions in the direction of larger fieldsare more likely this will automatically guarantee that everybody is walking in the direction of the door . if , however , the exit can not be seen by everybody , e.g. in a smoke - filled room or in the case of failing lights , people will try to follow others hoping that they know the location of the exit . in this case the coupling to the dynamic floor field is much stronger and the static field has a considerable influence only in the vicinity of the door . this example will be studied in more detail in .in contrast to the static floor field the dynamic floor field is changed by the motion of pedestrians .furthermore it is subject to diffusion and decay .its dynamics consists of three steps :  * if a pedestrian leaves a cell the dynamic floor field corresponding to this cell is increased by .the increment is a parameter of the model and can either be discrete or continuous . * to model the diffusion ,a certain amount of the field is distributed among the neighbouring cells . * to model the decay of the field , the field strength is reduced by a decay constant .  in ( a ) the virtual trace left by the motion of the pedestrians is created .( b ) is necessary because pedestrians do not necessarily follow exactly in the footsteps of others .diffusion leads to broadening and dilution of the trace .( c ) implies that the lifetime of the trace is finite and that it will vanish after some time .diffusion and decay of the dynamic field lead to an effective interaction strength between the pedestrians which decays exponentially with the distance .  in have introduced two variants of the floor field , a discrete and a continuous one . in the discrete casethe field strength can be interpreted as the number of bosonic particles ( `` bosons '' ) at the cell . in ( a )the number of bosons is increased by one . in ( b ) bosons can move with probability to neighbouring cells and in ( c ) bosons are removed with probability . in the continuous case the dynamics in ( b ) and ( c ) is described by a diffusion - decay equation where is the diffusion constant and the decay constant .details can be found in .the update rules of the full model including the interaction with the floor fields then have the following structure :  1 .the dynamic floor field is modified according to its diffusion and decay rules ( see sec . [ dynfloor ] ) .2 .  for each pedestrian , the transition probabilities for a move to an unoccupied neighbour cell are determined by the matrix of preferences and the local dynamic and static floor fields , e.g. ( see sec . [ sec_floor ] ) .. 3 .each pedestrian chooses a target cell based on the probabilities of the transition matrix .the conflicts arising by any two or more pedestrians attempting to move to the same target cell are resolved , e.g. using the procedure described in sec .[ sec_rules ] .the pedestrians which are allowed to move execute their step .the pedestrians alter the dynamic floor field of the cell they occupied before the move ( see sec .[ dynfloor ] ) .these rules have to be applied to all pedestrians at the same time ( parallel dynamics ) .this introduces a timescale into the dynamics which corresponds to approximately of real time .this allows e.g. to translate evacuation times measured in computer simulations into real times .we have introduced a stochastic cellular automaton to simulate pedestrian behaviour .we focused here on the general concept .the effects which can be observed with the basic approach will be presented in together with simple applications .the key mechanism is the introduction of a floor field which acts as a substitute for pedestrian intelligence and leads to collective phenomena .this floor field makes it possible to translate spatial long - ranged interactions into non - local interactions in time .the latter can be implemented much more efficiently on a computer .another advantage is an easier treatment of complex geometries . in models with long - range interactions ,e.g. the social - force models , one always has to check explicitly whether pedestrians are separated by walls in which case there should be no interaction between them .the general idea in our model is similar to chemotaxis .however , the pedestrians leave a virtual trace rather than a chemical one .this virtual trace has its own dynamics ( diffusion and decay ) which e.g. restricts the interaction range ( in time ) .it is realized through a dynamical floor field which allows to give the pedestrians only minimal intelligence and to use local interactions . together with the static floor field it offers the possibility to take different effects into account in a unified way , e.g.the social forces between the pedestrians or the geometry of the building .  in partii we will demonstrate that the approach indeed is able to reproduce the known collective effects and self - organization phenomena . therefore the model is a good starting point for realistic applications .the model can also be applied to more complex geometries and various characteristics of a crowd can be simulated without major changes .so it should be possible to study the effects of panic ( see and references therein ) . in show results for simple evacuation simulations .the description of pedestrians using a cellular automaton approach allows for very high simulation speeds .therefore , we have the possibility to extract the complete statistical properties of our model using monte carlo simulations .  finally it should be emphasized that we have presented only the basic ideas of the approach . forrealistic applications modifications might be appropriate , e.g. smaller cell sizes etc .one can also introduce more than just one species of pedestrians ( e.g. two groups moving in opposite directions ) . in this caseeach species interacts with its own floor field . in the simplest casethese fields are independent from each other .

Abstract: we present a 2-dimensional cellular automaton model for the simulation of pedestrian dynamics . the model is extremely efficient and allows simulations of large crowds faster than real time since it includes only nearest - neighbour interactions . nevertheless it is able to reproduce collective effects and self - organization encountered in pedestrian dynamics . this is achieved by introducing a so - called _ floor field _ which mediates the long - range interactions between the pedestrians . this field modifies the transition rates to neighbouring cells . it has its own dynamics ( diffusion and decay ) and can be changed by the motion of the pedestrians . therefore the model uses an idea similar to chemotaxis , but with pedestrians following a virtual rather than a chemical trace .


the problem of solving the power flow equations that describe a power system , i.e. computing the steady state of the grid ( typically the bus voltages ) given the state of power generators and loads , is among the most classical tasks in circuit and power system theory .an analytic solution of the power flow equations is typically not available , given their nonlinear nature .for this reason , notable effort has been devoted to the design of numerical methods to solve systems of power flow equations , to be used both in offline analysis of a grid and in real time monitoring and control of the system ( see for example the review in ) .specific tools have been derived for the approximate solution of such equations , based on some assumptions on the grid parameters .in particular , if the power lines are mostly inductive , equations relating active power flows and bus voltage angles result to be approximately linear , and decoupled from the reactive power flow equations , resulting in the _ dc power flow model _ ( see the review in , and the more recent discussion in ) .we focus here on a specific scenario , which is the power distribution grid .more specifically , we are considering a balanced medium voltage grid which is connected to the power transmission grid in one point ( the distribution substation , or pcc , point of common coupling ) , and which hosts loads and possibly also microgenerators .power distribution grids have recently been the object of an unprecedented interest .its operation has become more challenging since the deployment of distributed microgeneration and the appearance of larger constant - power loads ( electric vehicles in particular ) .these challenges motivated the deployment of ict ( information and communications technology ) in the power distribution grid , in the form of sensing , communication , and control devices , in order to operate the grid more efficiently , safely , reliably , and within the its voltage and power constraints .these applications have been reviewed in , and include real - time feedback control , automatic reconfiguration , and load scheduling . in order to design the control and optimization algorithms for these applications , an analytic ( rather than numerical ) solution of the power flow equationswould be extremely convenient .unfortunately , because in the medium voltage grid the power lines are not purely inductive , power flow equations include both the active and reactive power injection / demands , and both the voltage angles and magnitudes , in an entangled way .the explicit dc power flow model therefore does not apply well .the contribution of this paper is twofold .first , we give sufficient conditions for the existence of a practical solution of the nonlinear power flow equations in power distribution networks ( theorem [ theo : taylor_expansion_complex ] ) .second , we derive a tractable approximate solution to the power flow problem , linear in the complex power injections , providing a bound on the approximation error ( corollary [ cor : approx ] ) .  in the remainder of this section , we review relevant related works . in section [ sec : model ], we present the nonlinear equations that define the power flow problem . in section [ sec : main ] we present our main existence result , together with the linear approximate solution .we illustrate such approximation via simulations in section [ sec : simulations ] , and we compare it with the classical dc power flow model in section [ sec : dcpower ] , where we also show how to incorporate voltage controlled ( pv ) buses .conditions have been derived in order to guarantee the existence of a solution to the power flow equations in the scenario of a grid of nonlinear loads .many results are based on the degree theory . in , for example , exponential model is adopted for the loads , and sufficient conditions for the existence of a solution are derived .these conditions are however quite restrictive , and do not include constant power ( pq ) buses . in , on the other hand , the existence of a solution is proved by exploiting the radial structure of the grid , via an iterative procedure which is closely related to a class of iterative numerical methods specialized for the power distribution networks .the existence of solutions to the power flow equations has been also studied in order to characterize the security region of a grid , i.e. the set of power injections and demands that yield acceptable voltages across the network .these results include , and others where however the decoupling between active and reactive power flows is assumed .other works in which the dc power flow assumption plays a key role are and , both focused on active power flows across the grid . on the other hand , the results in focus on the reactive power flows and on the voltage magnitudes at the buses .  in implicit function theorem is used in order to advocate the existence of a power flow solution , without providing an approximate expression for that .it is worth noticing that the linear approximate model that we are presenting in this paper shares some similarities with the method of _ power distribution factors _ , which allow to express variations in the state ( voltage angles ) as a linear function of active power perturbations .this method is also typically based on the dc power flow assumptions , even if a formulation in rectangular coordinates ( therefore modeling reactive power flows ) has been proposed in .notice that , except for the seminal works on power distribution factors , and the more recent results in , most of the related results consists in algorithms that allow to compute this factors only numerically , from the jacobian of the power flow equations .the approximate power flow solution proposed in this paper has been presented in a preliminary form in , where however no guarantees on the existence of such solution and on the quality of the approximation were given .we are considering a portion of a symmetric and balanced power distribution network , connected to the grid at one point , delivering power to a number of buses , each one hosting loads and possibly also microgenerators .we denote by the set of buses , where the index refers to the pcc .we limit our study to the steady state behavior of the system , when all voltages and currents are sinusoidal signals at the same frequency .each signal can therefore be represented via a complex number whose absolute value corresponds to the signal root - mean - square value , and whose phase corresponds to the phase of the signal with respect to an arbitrary global reference .  in this notation ,the steady state of the network is described by the voltage and by the injected current at each node .we define the vectors , with entries and , respectively .each bus of the network is characterized by a law relating its injected current with its voltage .we model bus as a _ slack node _ , in which a voltage is imposed where are such that and .we model all the other nodes as pq buses , in which the injected complex power ( active and reactive powers ) is imposed and does not depend on the bus voltage .this model describes the steady state of most loads , and also the behavior of microgenerators , that are typically connected to the grid via power inverters .according to the pq model , we have that , at every bus , where is the imposed complex power .a more compact way to write these nonlinear power flow equations is the following .let the vectors be vectors in having , as entries .then we have where is the vector whose entries are the complex conjugates of the entries of and where denotes a diagonal matrix having the entries of the vector as diagonal elements .we model the grid power lines via their nodal admittance matrix , which gives a linear relation between bus voltages and currents , in the form   in the rest of the paper , we assume that the shunt admittances at the buses are negligible . therefore the nodal admittance matrix satisfies where is the vector of all ones . under this assumption, the matrix corresponds to the weighted laplacian of the graph describing the grid , with edge weights equal to the admittance of the corresponding power lines .we will show in a remark in section [ sec : main ] how this assumption can be relaxed , so that the entire analysis can be extended seamlessly to the case in which shunt admittances of the lines are not negligible .considering the same partitioning of the vectors as before , we can partition the admittance matrix accordingly , and rewrite as = \left[\begin{array}{cc } y_{0 0}&y_{0 \lnodes}\\y_{\lnodes 0}&y_{\lnodes \lnodes}\end{array}\right ] \left[\begin{array}{c } v_0\\v_\lnodes\end{array}\right].\ ] ] where is invertible because , if the graph representing the grid is a connected graph , then is the only vector in the null space of  . usingwe then obtain where the impedance matrix is defined as   objective of the power flow analysis is to determine from these equations the voltages and the currents as functions of and , namely in general , because of the nonlinear nature of the loads , we may have no solution or more than one solution for fixed and , as the following simple example shows .  consider the simplest grid made by two nodes , node being the slack bus ( where we let ) , and node being a pq bus . in this casewe have that the following equations have to be satisfied assume that , and that is real .the system of equations can then be solved analytically .in fact it can be found that if there are no solutions .when on the contrary , there are two distinct solutions notice that , if is large , then the solutions exist and , since , the current take the two values therefore , when is large , one solution consists in small currents ( thus small power losses and voltage close to the nominal voltage across all the network ) , while the other consists in larger currents , larger power losses , and larger voltage drops .of course , the system should be controlled so that it works at the first working point .the intuition from this simple example is developed in the next section , where the existence and uniqueness of a practical solution to the power flow equations ( i.e. a solution at which the grid can practically and reliably be operated ) is studied , and an approximate power flow solution ( linear in the power terms ) is proposed .define so that we have   by putting together with and , we get \\ & = f+s_\lnodes+\frac{1}{v_0 ^ 2}\operatorname{diag } ( f+s_\lnodes ) \x ( \bar f+\bar s_\lnodes),\end{aligned}\ ] ] and therefore   we can determine a ball where there exists a unique solution to this equation by applying the banach fixed point theorem . in order to do so , define the function   consider the standard 2-norm on defined as let us then define the following matrix normon  where the notation stands for the -th row of .the following result holds .consider the vector 2-norm on , and the matrix norm defined in . if then there exists a unique solution of the power flow equations and in the form where is such that [ theo : taylor_expansion_complex ]  let and . in order to apply the banach fixed point theorem ,we need to show that , under the hypotheses of the theorem , for a suitable constant .we prove first .observe that , by using lemma [ lem : pnorm ] in the case , we have where we used the fact that .now , using the definition of and lemma [ lemma ] in the appendix ( with and ) we can argue that , if is true , then   we prove now .it is enough to notice that , by applying lemma [ lem : f ] in the appendix ( with , and ) we obtain that where finally , notice that by using we obtain   now from and , by applying the banach fixed point theorem , we can argue that there exists unique solution of equation .then by using and we have that in order to prove it is enough to define .  the norm can be put in direct relation with the induced matrix 2-norm , and with structural properties of the graph that describes the power grid .indeed where is the -th vector of the canonical base .it can be shown that  ] and $ ] , respectively .the resemblance of with the dc power flow model will be investigated in section [ sec : dcpower ] , where the proposed approximate solution is presented in polar coordinates .the grid model in which the matrix is assumed to satisfy is based on the assumption of zero nodal shunt admittances . in the casein which shunt admittances are not negligible , the proposed analysis can be modified accordingly . in this more general casethe matrix is invertible , and for almost all grid parameters of practical interest , the submatrix is also invertible. equation becomes where , and where is a perturbation of the vector and corresponds to the normalized no - load voltage profile of the grid .the reasoning of theorem [ theo : taylor_expansion_complex ] can be repeated by defining , where the diagonal matrix is a perturbation of the identity matrix . in this case, has to satisfy the equation where   condition for the existence of a power flow solution is then replaced by the condition and the approximate power flow solution in corollary [ cor : approx ] becomes   the results of theorem [ theo : taylor_expansion_complex ] and corollary [ cor : approx ] hold also for other vector norms , different from the vector 2-norm , as the following remarks show .consider the standard p - norm on defined as if and by us then define the following matrix norm on  where , as before , the notation denotes the -th row of .we have that theorem [ theo : taylor_expansion_complex ] holds with respect to the vector -norm on and the matrix norm on where are such that .this follows from lemma [ lem : pnorm ] .the sufficient condition becomes as before , the statement of theorem [ theo : taylor_expansion_complex ] allows to derive a bound on the error of the approximate power flow solution proposed in corollary [ cor : approx ] ( this time using hlder s inequality ) , replacing with   the particular case , has an interesting physical interpretation .it can be inferred from that , as power line impedances have positive resistance and reactance , therefore , which , in a radial network , is the length of the path ( in terms of the magnitude of the path impedance ) connecting bus to bus .we denote such length by .then , corresponds in a radial network to the _ breadth of the grid _ , defined as the length of the longest path connecting a bus to bus . on the other hand, corresponds to a metric for the _ total load _ of the grid , defined as the sum of the apparent powers at every node .therefore , in this case , the sufficient condition for the existence of a practical solution becomes and the bound on the approximation error is   in general , theorem [ theo : taylor_expansion_complex ] gives only sufficient conditions for the existence and uniqueness of a solution of the nonlinear power flow equations , and a bound on the error of the approximation . by revisiting the example proposed in section [ sec : model ] we show that both the sufficient condition and the bound can be tight .consider the same example considered in section [ sec : model ] .notice that , regardless of the vector norm that is chosen , the sufficient condition in theorem [ theo : taylor_expansion_complex ] then becomes in the case that we were considering in the example , with real and negative ( corresponding to an active power load ) , it is easy to see that this condition is also a necessary condition for the existence of a solution . in the special case in which the condition is marginally satisfied ( ) ,it is possible to compute both the exact solution and the approximate solution according to , the approximation error is clearly , which marginally satisfies the approximation error bound , i.e. order to illustrate the results presented in the previous section , we considered a symmetric balanced testbed inspired to the ieee 123 test feeder .the details and the data of the adopted testbed are available online , together with the matlab / gnu octave and matpower source code used for the simulations .  in a per - unit notation , where is taken as the reference nominal voltage , and taken as the reference nominal power , we have that both the sufficient condition of theorem [ theo : taylor_expansion_complex ] , and its modified -norm version ( , ) are thus verified . therefore , existence and uniqueness of a practical solution of the power flow equations is guaranteed .  in figure[ fig : voltage ] we reported the true solution of the original nonlinear power flow equations ( obtained numerically ) together with the approximate linear solution proposed in corollary [ cor : approx ] .the average and maximum approximation errors are reported in table [ tbl : errors ] . in figure[ fig : voltage ] we also reported the bounds and on the approximation error ( as a solid and dashed curve , respectively ) .both these bounds are quite conservative in the assessment of the quality of the approximation .however , they guarantee that the power flow equations have a unique solution in a region which is practical for the operation of the power distribution grid ( other uninteresting solutions of the power flow equations may exist , at unacceptable voltage levels ) .  in order to better assess the quality of the approximated solution proposed in corollary [ cor : approx ] , an extensive set of variations of the original testbedhave been considered , including :  * different total grid load and load distribution * shunt capacitors for static voltage support * voltage regulation * different tap position at the pcc .these simulations are available online , together with the source code used to generate them , and a selection of them is included hereafter .  in a first case ,the active and reactive power demand of all loads has been doubled , in order to simulate a uniform overload of the grid .the uniform scaling of the vector of power demands affects in the same way ( i.e. quadratically ) both the approximation bounds and .while the bounds are still conservetive , figure [ fig : voltage_overload ] shows that the approximation error is indeed larger compared to nominal case .  in a second case ,the demand of one single bus ( bus 32 ) has been increased to 2 mw of active power and 1 mvar of reactive power , i.e. 50 times the original demand .figure [ fig : voltage_spotoverload ] shows how the approximation error is affected , similarly to the uniform overload case .however , in this case , the effect of the overload on the approximation bounds and is qualitatively different , as different norms of are involved in the two bounds .this example also shows that it is not possible , for a given grid , to know a - priori which particular -norm yields the tightest bounds .the approximation errors for these two cases are reported in table [ tbl : errors ] , for comparison with the nominal case .@ & & + & avg . & max & avg . &max +  + voltage magnitude [ p.u .] & 0.0041 & 0.0056 & 7.88% & 8.45% + voltage angle [ deg ] & 0.0097 & 0.0178 & 0.43% & 0.66% +  + voltage magnitude [ p.u . ] & 0.0191 & 0.0261 & 16.72% & 17.94% + voltage angle [ deg ] & 0.0999 & 0.1782 & 2.09% & 3.02% +  + voltage magnitude [ p.u . ] & 0.0197 & 0.0373 & 18.99% & 21.59% + voltage angle [ deg ] & 0.0994 & 0.3112 & 2.12% & 4.27% +   the relative error for the voltage magnitude and angle at each bus is computed with respect to the voltage drop and to the angle difference , respectively .in this section , we show how to translate the model proposed in corollary [ cor : approx ] to polar coordinates , i.e. in terms of voltage magnitudes and phases .if we denote by the symbol the vector having as entries the magnitudes of the entries of a complex vector , then from the approximate model proposed in corollary [ cor : approx ] we can obtain if we assume that that , i.e. that the voltage drops are much smaller than the nominal voltage , then from the fact that for , we obtain the approximation gives the opportunity to understand how the proposed model can be useful in a more general context , such as when there are voltage regulated ( pv ) buses in the grid .let be the subset of pv buses , whose voltage magnitude is regulated to , and let be the remaining set of pq buses .let then be divided into the corresponding blocks therefore , approximation for the nodes in becomes this linear relation can be inverted , in order to obtain the following expression for the reactive power injection of the nodes in  plugging this expression in the model proposed in corollary [ cor : approx ] yields an approximate solution of the power flow equations that is a linear function of the active and reactive power of the the pq buses , and of the active power and voltage magnitude set - points of the pv buses .the quality of this approximated model has been verified numerically on the same testbed of section [ sec : simulations ] , by assuming that buses 15 and 51 are voltage regulated to .also this simulation is available in , and the results are reported in figure [ fig : voltage_pvbus ] .     we now consider the phases of the approximate power flow solution , and we show how the proposed approximation can be seen as a generalization of the dc power flow model . if we denote by the vector having as entries the phases of the entries of the approximate solution , we have by using the fact that for , we obtain the intermediate approximated model in the special case in which power lines are assumed to be purely inductive , namely , one obtains equation is known in the literature as dc power flow model ( see and references therein ) which is based exactly on the assumption of small voltage drops and purely inductive lines .equation is typically shown in the following equivalent form where and are the vectors in with entries and , where is the power lines susceptance matrix ( i.e. a real matrix such that ) , and where is the slack voltage ( the voltage magnitude at the pcc ) .these different approximations of the bus voltage angles have been plotted in figure [ fig : angles ] for the same testbed .it is clear how the model proposed in corollary [ cor : approx ] approximates very well the true voltage angles . on the other hand ,the dc power flow model presents a much larger approximation error , due to the inaccurate assumption that power lines are purely inductive .indeed , the model , which differs from the dc power flow model only for the fact that this assumption has not been made , provides a much more accurate approximation .  in light of this analysis , the model in corollary [ cor : approx ]can also be interpreted as a generalization of the dc power flow model to the case of generic power line impedances .in theorem [ theo : taylor_expansion_complex ] we derived a sufficient condition for the existence of a practical solution to the nonlinear power flow equations that describe a power distribution network .this condition is proved to be also necessary in at least one example , and is verified in the ieee testbed that we adopted for numerical validation .different variations of this condition can be obtained by adopting different norms ( and thus different invariant sets for the banach fixed point theorem ) , yielding various physical interpretations .  in the case of a grid of constant power ( pq ) buses and one slack bus, the existence result immediately returns the approximate solution of the power flow equations presented in corollary [ cor : approx ] .such approximation is conveniently linear in the active and reactive power references of the buses , and analytical bounds for its error are given .we showed that , via some manipulations , it is also possible to consider different bus models , and in particular voltage regulated ( pv ) buses .we finally assessed the quality of the proposed model via numerical simulations on a set of variations of the ieee 123 test feeder , showing how it outperforms the classical dc power flow model .the proposed model has the potential of serving as a flexible tool for the design of control , monitoring , and estimation strategies for the power distribution grid .[ [ app : norms ] ]  let be the -norm of a complex vector as defined in , and be the matrix norm defined in .assume that are such that .then [ lem : pnorm ]  we first prove the lemma in one direction .we have that , if , then where we have applied holder s inequality .hence , in case we have that in case we have that   in order to prove the other direction first notice that it is well known that holder s inequality is tight in the sense that , for any fixed complex vector there exists a complex vector with the same dimension such that and . from this fact , if is the index such that , we let ( namely the -th vector of the canonical base ) and be a vector such that and . with this choice it is easy to verify that   assume that we have a vector norm and a matrix norm such that for any , .if we define the function then [ lem : f ]  first observe that notice that and so from this we can argue that   [ lemma ] let such that . then satisfies   to prove the lemma it is enough to substitute in ( [ second ] ) and to verify that the inequality holds . doing this we need to verify that this inequality holds if and only if , which is true since by hypothesis we have that .the authors thank prof .florian drfler for his insightful and useful comments on the first draft of the paper .

Abstract: we consider the problem of deriving an explicit approximate solution of the nonlinear power equations that describe a balanced power distribution network . we give sufficient conditions for the existence of a practical solution to the power flow equations , and we propose an approximation that is linear in the active and reactive power demands of the pq buses . for this approximation , which is valid for generic power line impedances and grid topology , we derive a bound on the approximation error as a function of the grid parameters . we illustrate the quality of the approximation via simulations , we show how it can also model the presence of voltage controlled ( pv ) buses , and we discuss how it generalizes the dc power flow model to lossy networks .   power systems modeling , load flow analysis , power distribution networks , fixed point theorem .


studies in astroparticle physics link astrophysics , cosmology , particle and nuclear physics and involve hundreds of scientific groups linked by regional networks ( like aspera / appec ) and national centers .the exciting progress in these studies will have impact on the knowledge on the structure of microworld and universe in their fundamental relationship and on the basic , still unknown , physical laws of nature ( see e.g. for review ) .virtual institute of astroparticle physics ( via ) was organized with the aim to play the role of an unifying and coordinating structure for astroparticle physics .starting from the january of 2008 the activity of the institute takes place on its website in a form of regular weekly videoconferences with via lectures , covering all the theoretical and experimental activities in astroparticle physics and related topics .the library of records of these lectures , talks and their presentations was accomplished by multi - lingual forum . in 2008 via complex was effectively used for the first time for participation at distance in xi bled workshop . since then via videoconferences became a natural part of bled workshops programs , opening the virtual room of discussions to the world - wide audience .its progress was presented in . herethe current state - of - art of via complex , integrated since the end of 2009 in the structure of apc laboratory , is presented in order to clarify the way in which via discussion of open questions beyond the standard model took place in the framework of xv bled workshop .the structure of via complex is illustrated on fig .[ homevia ] .the home page , presented on this figure , contains the information on via activity and menu , linking to directories ( along the upper line from left to right ) : with general information on via ( about via ) , entrance to via virtual rooms ( rooms ) , the library of records and presentations ( previous ) of via lectures ( previous lectures ) , records of online transmissions of conferences(previous conferences ) , apc seminars ( previous apc seminars ) and apc colloquiums ( previous apc colloquiums ) and courses , calender of the past and future via events ( all events ) and via forum ( forum ) . in the upper right anglethere are links to google search engine ( search in site ) and to contact information ( contacts ) .the announcement of the next via lecture and via online transmission of apc colloquium occupy the main part of the homepage with the record of the most recent via events below . in the announced time of the event ( via lecture or transmitted apc colloquium ) it is sufficient to click on `` to participate '' on the announcement and to enter as guest ( printing your name ) in the corresponding virtual room .the calender links to the program of future via lectures and events .the right column on the via homepage lists the announcements of the regularly up - dated hot news of astroparticle physics .  in 2010special cosmovia tours were undertaken in switzerland ( geneva ) , belgium ( brussels , liege ) and italy ( turin , pisa , bari , lecce ) in order to test stability of via online transmissions from different parts of europe .positive results of these tests have proved the stability of via system and stimulated this practice at xiii bled workshop .these tours involved special equipment , including , in particular , the use of the sensitive audio system konftel 300w .the records of the videoconferences at the xiii bled workshop are available on via site .  in 2011 via facility was used for the tasks of the paris center of cosmological physics ( pccp ) , chaired by g. smoot and for the public programme `` the two infinities '' conveyed by j.l.robert .it has effectively supported participation at distance at meetings of the double chooz collaboration : the experimentalists , being at shift , took part in the collaboration meeting in such a virtual way .the simplicity of via facility for ordinary users was demonstrated at xiv bled workshop .videoconferences at this workshop had no special technical support except for wifi internet connection and ordinary laptops with their internal video and audio equipments .this test has proved the ability to use via facility at any place with at least decent internet connection .of course the quality of records is not as good in this case as with the use of special equipment , but still it is sufficient to support fruitful scientific discussion as can be illustrated by the record of via presentation `` new physics and its experimental probes '' given by john ellis from his office in cern ( see the records in ) .  in 2012 via facility , regularly used for programs of via lectures and transmission of apc colloquiums , has extended its applications to support m.khlopovs talk at distance at astrophysics seminar in moscow , videoconference in pccp , participation at distance in apc - hamburg - oxford network meeting as well as to provide online transmissions from the lectures at science festival 2012 in university paris7 . via communicationhas effectively resolved the problem of referee s attendance at the defence of phd thesis by mariana vargas in apc .the referees made their reports and participated in discussion in the regime of via videoconference .the discussion of questions that were put forward in the interactive via events can be continued and extended on via forum .the forum is intended to cover the topics : beyond the standard model , astroparticle physics , cosmology , gravitational wave experiments , astrophysics , neutrinos .presently activated in english and russian with trivial extension to other languages , the forum represents a first step on the way to multi - lingual character of via complex and its activity .one of the interesting forms of forum activity is the educational work . for the last four yearsm.khlopovs course `` introduction to cosmoparticle physics '' is given in the form of via videoconferences and the records of these lectures and their ppt presentations are put in the corresponding directory of the forum .having attended the via course of lectures in order to be admitted to exam students should put on forum a post with their small thesis .professor s comments and proposed corrections are put in a post reply so that students should continuously present on forum improved versions of work until it is accepted as satisfactory .then they are admitted to pass their exam .the record of videoconference with their oral exam is also put in the corresponding directory of forum .such procedure provides completely transparent way of estimation of students knowledge .first tests of via system , described in , involved various systems of videoconferencing .they included skype , vrvs , evo , webex , marratech and adobe connect . in the result of these teststhe adobe connect system was chosen and properly acquired .its advantages are : relatively easy use for participants , a possibility to make presentation in a video contact between presenter and audience , a possibility to make high quality records and edit them , removing from records occasional and rather rare disturbances of sound or connection , to use a whiteboard facility for discussions , the option to open desktop and to work online with texts in any format . the regular form of via meetings assumes that their time and virtual room are announced in advance .since the access to the virtual room is strictly controlled by administration , the invited participants should enter the room as guests , typing their names , and their entrance and successive ability to use video and audio system is authorized by the host of the meeting .the format of via lectures and discussions is shown on fig .[ ellis ] , illustrating the talk `` opera versus maxwell and einstein '' given by john ellis from cern .the complete record of this talk and is available on via website .the sensational character of the exciting news on superluminal propagation of neutrinos acquired the number of participants , exceeding the allowed upper limit .for the first time the problem of necessity in extension of this limit was put forward and it was resolved by creation of a virtual `` infinity room '' , which can host any reasonable amount of participants .the ppt or pdf file of presentation is uploaded in the system in advance and then demonstrated in the central window .video images of presenter and participants appear in the right window , while in the upper left window the list of all the attendees is given . to protect the quality of sound and record, the participants are required to switch out their microphones during presentation and to use lower left chat window for immediate comments and urgent questions .the chat window can be also used by participants , having no microphone , for questions and comments during discussion .the interactive form of via lectures provides oral discussion , comments and questions during the lecture .participant should use in this case a `` raise hand '' option , so that presenter gets signal to switch our his microphone and let the participant to speak . in the end of presentationthe central window can be used for a whiteboard utility as well as the whole structure of windows can be changed , e.g. by making full screen the window with the images of participants of discussion .regular activity of via as a part of apc includes online transmissions of all the apc colloquiums and of some topical apc seminars , which may be of interest for a wide audience .online transmissions are arranged in the manner , most convenient for presenters , prepared to give their talk in the conference room in a normal way , projecting slides from their laptop on the screen .having uploaded in advance these slides in the via system , via operator , sitting in the conference room , changes them following presenter , directing simultaneously webcam on the presenter and the audience .via sessions of xv bled workshop have developed from the first experience at xi bled workshop and their more regular practice at xii , xiii and xiv bled workshops .they became a regular part of the bled workshop s programme .in the course of xv bled workshop meeting the list of open questions was stipulated , which was proposed for wide discussion with the use of via facility . the list of these questions was put on via forum ( see ) and all the participants of via sessions were invited to address them during via discussions . during the xv bled workshop the test of not only minimal necessary equipment , but either of the use of via facility by ordinary users was undertaken . via sessions were supported by personal laptop with wifi internet connection only , as well as for the first time the members of via team were physically absent in bled and all the videoconferences were directed by m.khlopov at distance from paris .it proved the possibility to provide effective interactive online via videoconferences even in the absence of any special equipment and qualified personnel at place . only laptop with microphone and webcam together with wifi internet connectionwas shown to be sufficient not only for attendance , but also for via presentations and discussions .  in the framework ofthe program of xv bled workshop , p. belli , staying in his office in rome , gave his talk `` dama / libra results and perspectives '' ( fig .[ dm ] ) and took part in the discussion of puzzles of dark matter searches , which provided a brilliant demonstration of the interactivity of via in the way most natural for the non - formal atmosphere of bled workshops ( see ) .in the course of this discussion n.s .manko bortnik and g. bregar , being in bled , have considered possible dark matter candidates that follow from the approach , unifying spins and charges , and maxim khlopov presented from paris the state - of - art of composite dark matter scenario , stipulating the open problems of this solution for the puzzles of direct dark matter searches . via sessionswere finished by the via talk `` the atlas experiment at the lhc : present status and its future '' by a.s romaniouk and discussion of problems of experimental search for new physics at accelerators .  via sessionsprovided participation at distance in bled discussions for m.khlopov ( apc , paris , france ) , p. belli ( rome tor vergata , italy ) , e. soldatov ( cern , switzerland ) , k.belotsky , n.chasnikov and a.mayorov ( mephi , moscow ) , j .-cudell and q.wallemacq ( liege , belgium ) , r.weiner ( marburg , germany ) and many others .current via activity is integrated in the structure of apc laboratory and includes regular weekly videoconferences with via lectures , online transmissions of apc colloquiums and seminars , a solid library of their records and presentations , together with the work of multi - lingual via internet forum .the scientific - educational complex of virtual institute of astroparticle physics can provide regular communications between different groups and scientists , working in different scientific fields and parts of the world , get the first - hand information on the newest scientific results , as well as to support various educational programs at distance .this activity would easily allow finding mutual interest and organizing task forces for different scientific topics of astroparticle physics and related topics .it can help in the elaboration of strategy of experimental particle , nuclear , astrophysical and cosmological studies as well as in proper analysis of experimental data .it can provide young talented people from all over the world to get the highest level education , come in direct interactive contact with the world known scientists and to find their place in the fundamental research . via applications can go far beyond the particular tasks of astroparticle physics and give rise to an interactive system of mass media communications .  via sessionsbecame a natural part of a program of bled workshops , opening the room of discussions of physics beyond the standard model for distant participants from all the world .the experience of via applications at bled workshops plays important role in the development of via facility as an effective tool of science and education online .the initial step of creation of via was supported by aspera .i am grateful to p.binetruy , j.ellis and s.katsanevas for permanent stimulating support , to j.c .hamilton for support in via integration in the structure of apc laboratory , to k.belotsky , a.kirillov and k.shibaev for assistance in educational via program , to a.mayorov , a.romaniouk and e.soldatov for fruitful collaboration , to m.pohl , c. kouvaris , j .- r.cudell , c. giunti , g. cella , g. fogli and f. depaolis for cooperation in the tests of via online transmissions in switzerland , belgium and italy and to d.rouable for help in technical realization and support of via complex .i express my gratitude to n.s .manko bortnik , g.bregar , d. lukman and all organizers of bled workshop for cooperation in the organization of via sessions at xv bled workshop .

Abstract: virtual institute of astroparticle physics ( via ) , integrated in the structure of laboratory of astroparticle physics and cosmology ( apc ) is evolved in a unique multi - functional complex of and , supporting at distance interactive online participation in conferences and meetings , various forms of collaborative scientific work as well as programs of education . the activity of via takes place on its website and includes regular videoconferences with systematic basic courses and lectures on various issues of astroparticle physics , regular online transmission of apc colloquiums , participation at distance in various scientific meetings and conferences , library of their records and presentations , a multilingual forum . via virtual rooms are open for meetings of scientific groups and for individual work of supervisors with their students . the format of a via videoconferences was effectively used in the program of xv bled workshop to provide a world - wide participation at distance in discussion of the open questions of physics beyond the standard model . the via system has demonstrated its high quality and stability even for minimal equipment ( laptop with microphone and webcam and wifi internet connection ) and without any technical assistance at place .   * virtual institute of astroparticle physics - science and education online *  maxim yu . khlopov  _ centre for cosmoparticle physics `` cosmion '' + national research nuclear university `` moscow engineering physics institute '' , 115409 moscow , russia + apc laboratory 10 , rue alice domon et lonie duquet + 75205 paris cedex 13 , france _


we consider two simple problems for setting confidence belts .the first problem is that of the poisson distribution in the presence of background . the second problem is the measurement of a non - negative parameter , with a normally distributed error .although simple problems , there has been considerable work and discussion on them in the last few years . most of the discussion has centered on the poisson case , when the number of observed events is small .this is an important current topic because there are many search experiments involving small numbers of events .these include higgs particle searches , supersymmetry searches , and neutrino oscillation searches such as lsnd and karmen .we will briefly describe some recent attempts to address these problems .the standard method for setting 90% cl bounds has been to select a region with a 5% probability above and a 5% probability below the bounds .about two years ago , gary feldman and bob cousins suggested a method , new to physics analyses , called the unified procedure . for the poisson case , for each possible value of the parameter , they looked at the ratio of the probability of getting the observed number of events for that compared to the maximum probability for all physically allowed s , and picked s with the highest ratio to build a 90% confidence region .this solved two problems with the old procedure .it automatically transitioned from an upper limit to a confidence belt and it always produced confidence sets in the physical region .the unified procedure works well for many problems and is a significant improvement over the symmetric tails procedure .however it has a serious problem if few or no events are observed .if 0 events are seen , there are 0 signal and 0 background events . that there are 0 background events is interesting , but irrelevant to the question of whether signal events are seen . a 90% c.l .limit on should come from % , which sets a limit at .( throughout this article probabilities for a fixed parameter are denoted by a subscript .probability densities and probability masses are denoted with lower case letters and distribution functions by upper case letters . )such a case did occur in the summer of 1998 from initial karmen results .they had 0 events with a background of 2.88 , and using the unified method obtained an upper limit of .it is desirable to have a method set a confidence limit near 2.3 , independent of background , for a 90% c.l ., when no events are seen .we presented a variant of the feldman - cousins unified procedure which corrected this problem and introduced the concept of conditional coverage . for any observation we know that the background is less or equal to the observed number of events , .we suggested the use of a sample space not of all experimental outcomes , but of outcomes with background .thus , if , we considered only measurements in which there are background events , even for 20 events observed in the sample space .this probability was used both to set the ratio used in the unified method and to calculate the coverage probability .this method had the advantage of giving an upper limit of about 2.42 for 0 observed events independently of the background mean .see figure 1 .robert cousins found that this procedure had a problem with its lower limit when applied to a continuous observable .suppose one measures a parameter with a measurement error , where is normal ( 0,1 ) .the r - w procedure eliminated for all . however , if , the probability of measuring is 10% . see figure 2 .it appeared that the r - w upper limit and the f - c lower limit were needed .since the conditional coverage concept is , as yet , unfamiliar , it is desirable that a method should have reasonable conventional coverage as well as conditional coverage .in this section , we derive baysian credible intervals when the expected signal is given a uniform prior distribution .these intervals have several desirable properties . like the unified intervals ,bayesian intervals lie in the physical region and automatically change from confidence bounds to two - sided intervals .they avoid the known pitfalls mentioned above , the problems with low counts in unified method and the large lower boundary of the conditional confidence intervals in the continuous case .finally , and importantly , the bayesian intervals are optimal on their own terms .the intervals derived minimize length among all credible intervals .the price paid for these desirable properties is dependence on the prior and metric .we address the first of these concerns by showing that the frequentist coverage probability of the bayesian intervals is quite close to the bayesian posterior credible level .that is , while the two probabilities are conceptually quite different , they are close numerically . in addition , we show that the bayesian credible intervals have exact conditional ( frequentist ) coverage probability , except for discreteness in the poisson case . with regard to the metric, it is primarily the optimality of our procedures that depends on the metric .see section 7 .we will illustrate with two examples , a poisson case in which the mean is composed of an unknown signal mean and a known background mean and the measurement of a parameter , with a measurement error which is normal ( 0,1 ) .we use a uniform ( improper ) prior of 1 for in both problems .this is a simple problem in which the statistical issues are clear and which approximates the poisson problem for large .we measure , where , and the density function for is for the uniform prior probability , the marginal density of becomes where the standard normal distribution function .we now use bayes theorem .the conditional density of given is  is proper ; the improper ( infinite ) prior has cancelled out .we wish to find an upper limit and a lower limit , dependent on , for which in bayes theory , such intervals are called _credible intervals_. it is desirable to minimize the interval  ] where  ] if .the probability mass function and distribution function for the poisson distribution with mean are : and . for our present problem , where is the fixed ( known ) `` background '' mean .if we let have a prior uniform distribution for , we then have  as derived in our previous paper . the derivation proceeded by expanding and noting that .we again use bayes theorem .then , recalling that for and , we have   to set upper and lower confidence bounds for , we need to solve the equation for and . a 90% c.l .corresponds to setting .the second equality here is obtained by writing and using the reasoning described after equation ( 8) . to get the shortest interval we need to include values of with highest density , _i.e. _ , =[\theta : p_b(\theta|n)\ge c]\ ] ] for some . the conditions ( 10 ) and ( 11 ) may be solved numerically by an iterative procedure .this procedure is described in section * 6 * in a more general context .the bayesian credible intervals just derived have exact posterior coverage probability , by construction . in this section, we show that they also have high frequentist coverage probability .   in the continuous example , , and .fix a and find the points at this fixed where meets the upper and lower bayesian limits .let be that which meets the upper limit , and be that which meets the lower limit ,  ] covers iff , and the conventional unconditional coverage is .this is not exactly .a lower limit on the coverage is shown in the appendix to be but this is a very conservative limit . for ,this lower limit is 0.8182 , while a numerical calculation finds a minimum coverage of about 0.86 .the conventional coverage probability can be improved for a very small increase in limits .we will consider a conservative ad - hoc modification of the upper limit on using the one - sided limit for a higher confidence level , .let .\ ] ] let be the corresponding to .the formula for the conventional coverage is , as above .the undercoverage , derived in the appendix , is very small for this conservative modification . for a 90% c.l ., the conventional coverage is at least .900 everywhere to three significant figures .figure 3 shows a plot of the coverage bands , plotted as vs .the confidence bands are shown in figure 4 and compared with the old r - w upper bound and the feldman - cousins unified procedure lower bound .the conventional frequentist coverage for the bayesian model and for this conservative modification are shown in figure 5 .the bayesian credible intervals also have an exact conditional frequentist property , in terms of the error .note that if is observed , then , necessarily , , since . is just the probability that .let  ] covers iff or , equivalently , . in the appendix, it is shown that if is an independent copy of , then =1-\epsilon\ ] ] and  ] .we first find the value at which is maximum , or , equivalently ,  ] .it is clear from ( [ eq : thetahat ] ) that , and it can be shown that the obvious iteration starting at converges to .this iteration is a special case of the em algorithm .next integrate .let then \theta\ ] ] we use the reasoning described after equation(8 ) to evaluate this integral . we can use either equation ( 19 ) , recognizing that the integral is an incomplete gamma function , or use equation ( 20 ) to find .the limits can then be found by iterations as follows .  1 .first solve the equation for .this is straightforward , since is increasing in .2 .  if , then and .3 .  if , solve the equations    for by a double iteration .in solving these equations , start with , where and is the maximum value of , corresponding to .solve the first two equations for .this is straightforward , since is increasing for and decreasing for .then replace ( respectively , ) by , accordingly as ( respectively , ) .note that the iteration procedure for the poisson example without additional parameters measured is just a special case of this iteration procedure .we have proposed methods for setting credible / confidence intervals for the _ means _ of a poisson variable , observed in the presence of background , and of a normal distribution , when is known to be non - negative . in the process , we have made specific choices for the prior distributions and loss structure .the ( uniform ) prior distributions were chosen for mathematical tractability and agreement between the bayesian credible level and conventional confidence level ; and one of the main findings is that such agreement is possible .the intervals have an optimality property : they minimize the length of credible intervals in the scale for the uniform prior . for both problems , the mean provides a physically meaningful and mathematically tractable metric .our procedures depend on the choice of metric , prior , and loss structure as follows : if the interval for is and the metric were changed , to say , then a valid interval for is .this has the same frequentist coverage as the original interval and is an exact credible interval for the induced prior , .the optimality is lost , however .the transformed interval does not minimize length in the -scale .the situation for our method is similar to that in the cramer - rao limit , equation 13.10 of roe .this very useful bound also depends on the metric .the derivations of the ad hoc modifications in section 5 mix bayesian and frequentist reasoning , and this may seem inelegant .we believe that any inelegance is mitigated by the minor nature of the changes and the resulting good properties of the procedure from both the bayesian and frequentist viewpoints .  in the case of a poisson distribution without auxiliary variables , there is a close connection between our method and : the procedure is mathematically equivalent to an upper credible bound with a uniform prior distribution .read calls this a coincidence .we think that there is a deeper connection .the optimality claimed for in read s work is optimality for testing background only versus background plus a specified signal .our procedures are designed to produce intervals of shortest length , not most powerful tests . that these two criteria can lead to different procedureswas noted in the statistical literature by pratt .this research is supported by the national security agency under mda904 - 99 - 1 - 004 and the national science foundation under phy-9725921 .proof : the result follows from = { \phi[min(x , z ) ] \over  \phi(x ) } = prob[x-\theta\le z|x].\ ] ] in words , the conditional frequentist distribution of is the same as the posterior distribution of .it follows that & = & prob[x - u(x)\le  x-\theta\le min(x , d(x))|x ] \\    & = & prob[\ell(x)\le\theta\le u(x)|x ] = 1-\epsilon ,  \end{aligned}\ ] ] where the last equality follows from the definitions of and .next , if  ] , where and , then =prob[-d(x)\le x-\theta\le a(x)|x]=1-\epsilon\ ] ] and , therefore , , since the bayesian limits were chosen to minimize the length of the interval .proof : recall that and were the values of for a given meeting the upper and lower bayesian limits , so and  ] and next suppose .then , = 1-\left[{1 \over 2 } + { 1 \over  2}\left(1-\epsilon\right)\phi(x_\ell)\right] ] , and .observe that for and for .for if , then >\phi^{-1}(1- { 1 \over 2}\epsilon) ] ; and if , then <\phi^{-1}(1- {  1 \over 2}\epsilon) ] for all .  clearly, iff iff .thus , iff , or equivalently , using ( [ eq : approx ] ) , the conditional probability of the latter event given is approximately where the last equality follows from the definitions of and .  to appreciate ( [ eq : approx2 ] ) , it is instructive to use a different approach .suppose that we knew apriori that .then the mass function of is then to form confidence intervals in this model , we might construct tests that the parameter has a given value and then invert this family of tests .this requires finding and for which for each , where and are functions of and .this implies and is nearly equivalent to \ge 1-\epsilon .\label{eq : cndcnf}\ ] ] except for discreteness , the left side of ( [ eq : cndcnf ] ) is .the bayesian credible intervals implicitly determine values of and , namely  the relation ( [ eq : approx2 ] ) shows that these values nearly satisfy ( [ eq : cndcnf ] ) , except for discreteness .feldman and r.d .cousins , unified approach to the classical statistical analysis of small signals , _ phys .rev . _ * d57 * , 3873 ( 1998 ) .k. eitel and b. zeitnitz , the search for neutrino oscillations with karmen , in neutrino physics and astrophysics , procedings of the xviii international conference on neutrino physics and astrophysics , takayama , japan , edited by y. suzuki and y. totsuka , north holland ( 1999 ) . b.p . roe and m.b .woodroofe , improved probability method for estimating signal in the presence of background , _ phys .* 60 * , 053009 ( 1999 ) .robert d. cousins , application of conditioning to the gaussian - with - boundary problem in the unified approach to confidence intervals , arxiv : physics/0001031 , 15 jan 2000 , submitted to _ phys .dempster , n.m .laird , and d.b .rubin , maximum likelihood from incomplete data via the em algorithm , _ roy .soc . _ * b39 * , 1 ( 1977 ) .roe , probability and statistics in experimental physics , second edition , springer - verlag , new york , to be published in 2001 .read , modified frequentist analysis of search results ( the method ) , _ proceedings of the workshop on confidence limits at cern , january 17 - 18 , 2000 _, to be issued as a cern yellow report . h. wang and m. woodroofe , to appear in _ ann ._ j. pratt , _j. amer . statist ._ , * 56 * 549 - 567,(1961 ) .

Abstract: we propose using a bayes procedure with uniform improper prior to determine credible belts for the mean of a poisson distribution in the presence of background and for the continuous problem of measuring a non - negative quantity with a normally distributed measurement error . within the bayesian framework , these belts are optimal . the credible limits are then examined from a frequentist point of view and found to have good frequentist and conditional frequentist properties .