domain: info math

text:
Afafe Zehrouni
email: afafe.zehrouni@emse.fr

Vincent Augusto
email: augusto@emse.fr

Thierry Garaix
email: garaix@emse.fr

Raksmey Phan
email: raksmey.phan@emse.fr

Xiaolan Xie
email: xie@emse.fr

Sophie Denis

Michel Gentile

Hospital Flood Emergency Management Planning using Markov Models and Discrete-Event Simulation

Keywords: flood, healthcare, emergency management planning, modeling, simulation

Disasters such as major floods affect all part of the globe. Hospital and healthcare structures are critical resources during such event and do not always benefit of emergency preparedness. When hospitals are impacted by the disaster, it puts a strain on the system and a reorganisation of all available hospitals on a given territory is necessary. As part of case study applied to the impact of floods on the Île-De-France region's health system, we present in this paper a simulation model that evaluates healthcare emergency plan by combining the healthcare process with the flood dynamics. The results can be used to elaborate an optimized strategy for evacuation and transfer operations. We provide a case study including several medical specialties and quantify the impact of several flood scenarios on the healthcare system. Highlights • Flood dynamics Markov model • Patient flow discrete-event model • Operational flood emergency management planning tool at the territory scale • Evacuation decision tool for hospital manager

Introduction 1.Context

Like the majority of disasters, natural or man-made, floods can result in significant economic loss and human casualties. Disaster Operations Management (DOM) is a set of actions and decisions aimed at preparing responses to natural or man-made disasters by reallocating resources (e.g. health facilities, transportation). The Federal Emergency Management Agency defines DOM as a cycle of four components: mitigation, preparedness, response and recovery [START_REF]Flood insurance reform act of 2004[END_REF] (Figure 1).

Mitigation activities take place before and after emergencies and aim at reducing the occurrence and the impact of a disaster. Preparedness activities take place before the disaster occurs to prepare the right response. A response is the application of the preparedness plans during the disaster by allocating the necessary resources to protect the community. Recovery includes actions taken to return to a normal situation after a disaster. Operations Research (OR) techniques can be used in several components of the DOM cycle. Lian and Yen [START_REF] Lian | Comparison of risk calculation methods for a culvert[END_REF] compares several risk calculation methods applied to flood mitigation. Wei et al. [START_REF] Wei | Artificial neural network based predictive method for flood disaster[END_REF] use artificial neural network to predict flood occurrence and prepare for the impact. Artificial neural network is also used in [START_REF] Shim | Spatial decision support system for integrated river basin flood control[END_REF] as well as geographical information system to predict the dynamic of water during the flood and to optimize the gate-control strategies. A classified survey of the use of OR techniques for DOM is presented in [START_REF] Altay | OR/MS research in disaster operations management[END_REF] and updated in [START_REF] Galindo | Review of recent developments in OR/MS research in disaster operations management[END_REF].

One of the main challenges in DOM is the lack of resources [START_REF] Hoard | Systems modeling in support of evidence-based disaster planning for rural areas[END_REF], and understanding and managing critical resources is essential for the minimization of the disaster impact. Health facilities represent a key resource and must accommodate the usual patients flow and the flow resulting from a disaster [START_REF] Agca | Optimization-based Logistics Planning and Performance Measurement for Hospital Evacuation and Emergency Management[END_REF]. However, hospitals may be themselves at risk of flood and therefore there is a simultaneous (i) increase of the demand for emergency services and (ii) decrease of their provision. In addition, when a hospital is flooded, its evacuation is more constrained than mass evacuation due to the patients' health conditions and the necessity to relocate them in appropriate facilities.

This research is supported by the Île-de-France Regional Health Agency (Agence Régionale de Santé d' Île-de-France, ARSIF ). The mission of ARSIF is to implement health policy in the region. In particular, ARSIF is working and the anticipation, preparation and management of health crises. A practical study case that is the resilience of the healthcare facilities network of Île-de-France region (Paris, France) in case of a major flood is studied in this paper.

Objective and contributions

The main objective of this study consists in proposing a management tool for hospital evacuation planning in the event of a major flood of nearby rivers. Unlike most of the articles in literature, we integrate in our model (i) the flood dynamics (movement in space and time) instead of considering the impact at a predefined time using a Markov model, and (ii) the macroscopic patient flow model on a long horizon (one year) using a discrete-event model.

Hospital evacuation decisions adapt to the evolution of the flood and the model is used for both preparedness before the event and response during the event. We consider hospital capacities on a macroscopic level: our model does not include detailed evacuation decisions from each building as proposed in several papers [START_REF] Chen | Modeling and simulation of a hospital evacuation before a forecasted flood[END_REF], but it allows to control the flow of patients over the different facilities. The resulting tool is used by the ARSIF for both strategic and operational control of evacuation in the event of a major flood.

This paper is organised as follows: after a short literature review (Section 2), we define the problem settings and present the general approach used in this work in Section 3. Next we present the flood model using Markov chain in Section 4. Section 5 defines the healthcare process model in a normal situation and during the event of flood. Several policies and strategies are implemented and applied to the process. Degradation of the process are measured using several key performance indicators, and the results for a case study are presented Section 6. Conclusions and perspectives are given in Section 7.

Related literature

In the literature, hospital evacuation operations have been approached in different ways: project management, mathematical modelling, simulation models and hybrid models [START_REF] Taaffe | Improving hospital evacuation planning using simulation[END_REF]. In DOM, simulation models aim to evaluate several outcomes such as costs, mortality or focus directly on the building architecture (e.g. exits and staircases) that are used during evacuation [START_REF] Hunt | Simulating hospital evacuation[END_REF].

In [START_REF] Voyer | Hospital evacuation in disasters: uncovering the systemic leverage using system dynamics[END_REF], a simulation model is developed to compare the impact of different resources on the evacuation operations. Results indicate that an increase of transportation resources (number of ambulances or transit rate) has a smaller benefit for evacuation than a change in the available capacity of the safe hospitals. In this study, disaster type is not specified.

The study [START_REF] Yi | Hospital Capacity Planning for Disaster Emergency Management[END_REF] focuses on the analysis of the available capacity in safe hospitals in Florida, and estimates the absorption ability of the region in case of an earthquake. In [START_REF] Kevin | Evaluating the robustness of hospital evacuation plans[END_REF], simulation is used to evaluate the effectiveness of an evacuation plan for one hospital under various scenarios and resources (e.g. patient types, nurses, number of ambulances).

A simulation model is developed in [START_REF] Chen | Modeling and simulation of a hospital evacuation before a forecasted flood[END_REF] to evaluate and improve the evacuation of one hospital in case of a flood considering internal resources (e.g. stretchers) and external resources (e.g. ambulances). In a previous work [START_REF] Zehrouni | Health care emergency plan modeling and simulation in case of major flood[END_REF], a discrete-event simulation model was proposed to evaluate the impact of a flow on a network of hospital for strategic decision aid (decisions related to resource preparation and localisation to ensure feasibility of evacuations).

A systematic review of hospital emergency evacuation in disasters is presented in [START_REF] Yaghoubi | Decision-making on hospital emergency evacuation in disasters and emergencies: Findings from a systematic review[END_REF]. To the best of our knowledge, only one study [START_REF] Shim | Spatial decision support system for integrated river basin flood control[END_REF] considers dynamic responses to the progression of the disaster (open or close gates). Other works focus on actions to be taken when the disaster is over. Studies dedicated to healthcare systems are often focused to a single facility.

The scientific objective of this paper is to fill a gap in the literature by providing a formal method that involves dynamic decisions during the progression of the disaster for a healthcare network using a modelling and simulation approach, taking into account a continuous flow of data related to the flood and hospitals characteristics.

3 Problem settings

Position of the problem

In this paper we intend (i) to mimic the behavior of the healthcare network using data from medico-administrative databases, and (ii) to evaluate the emergency plan of a regional healthcare system and assess the region's resilience in case of flood, on the strategic level (several months before the supposed flood period) or on the operational level (a few days before the supposed start of the flood, taking into account meteorological information).

We define the region's resilience as the ability to treat all scheduled patients and emergency arrivals within the region (i.e. with no transfer to hospitals outside of the region). In other words, the resilience is achieved if the non-flooded hospitals can take into care their patients as well as emergency patients and the patients coming from the flooded hospitals. In the context of very limited capacity, such solution may only be achieved by predefined management rules. For example:

• Discharging patients in order to free up as many resources as possible, before and during the flood.

• Preventive evacuation of high risk hospitals based on geographic location and electric fragility.

• Transfer of flooded hospital patients according to pre-established preferences.

Unlike most other disasters (natural and man-made), these rules are feasible in case of flood because of the alert period given by the weather forecast and water level measurements. However, the effectiveness of the flood management rules is highly variable depending on the flood dynamic (water level and speed) as well as the emergency patients flow.

General approach

To evaluate accurately the preparedness plan and the decisions before and during flood, the proposed approach presented in Figure 2 combines a patient flow model using Petri nets with a dynamic flood model using Markov chains. Data sources are depicted as green boxes, models as blue boxes, and dashboards as brown boxes.

Hospital Data

Capacity Flood risk ...

Patient Data

Hospital LoS Arrival time The patient flow model based on Petri nets is implemented in order to simulate all care pathways in all hospitals at the macroscopic level over a long horizon (at least one year) across the considered territory. The patient flow model replicates the operations of the healthcare network and is fed using real data coming from medico-administrative databases (Hospital Data and Patient Data). Using this model, we apply a disturbance at a certain time (start of a flood) to evaluate the impact on the patient flow.

Territorial data

The flood model is a dynamic short term event (few days to few weeks). When the flood alert starts, healthcare processes are adjusted by including the flood management rules until the end of the flooding event. Such processes are considered as degraded care pathway (evacuation of the hospital and/or patients transfers).

The input data feed for the aforementioned models are: (i) hospital data (capacity per medical speciality and hospital, flood risk evaluation per hospital), (ii) patient data (referred hospital, length of stay, arrival time), (iii) geographical data (maps of counties within the region, flooded zones decomposition, transportation time).

Finally, several Key Performance Indicators (KPI) are computed to evaluate the performance of the management rules applied during the flood event. The main KPI is the evaluation of the autonomy of the region in case of major flood. The exhaustive list of KPI is presented in Section 6. In the following we formally define the flood model in Section 4 and the patient flow model in Section 5.

Formal flood modeling using Markov chain

A key contribution of this paper is to propose a flood model taking into account dynamics of water. We present in this section the set of assumptions we considered regarding the flood dynamic related to its starting time and its duration. Then a new Markov chain modeling flood dynamics is proposed. A comprehensive example describing how to build the model from real data is given in Section 6.

Assumptions on flood characteristics

The region is partitioned into zones. All the locations in a zone are impacted at the same level during each period of the flood. A flood is characterized by a sequence of water level values for each zone during rising and receding water times. Of course the water level values satisfy physical constraints related to geography of the ground and the stream-bed configuration. The speed and the extent of the flood is then characterized through the changes and maximal values in the matrix that defines a water level for each pair of zone and time period.

Obviously the water level is not constant during a time period. The value used can either be obtained by aggregating values over time (average, maximum) or taking a value at a specific time-stamp (beginning, middle or end of the period).

In order to link this geographical model with the healthcare network, we combine the zones coordinates with the geographic location of the facilities. We obtain the list of potentially flooded facilities for each water level. Accuracy strongly depends on the number (i.e. the size) of zones and the number (i.e. the length) of time periods involved in the model. Assumption 1 (Starting time of a flood) The starting time of the flood event is selected randomly to capture the impact of seasonality on the healthcare process as arrival distribution of patients varies depending on the period of the year.

Assumption 2 (Flood types) We consider two phases in a flooding event: (i) flood rising and (ii) flood receding. The length of the first phase determines if the flood is slow or fast. The total duration of the flood and the duration of both phases are selected randomly within realistic ranges extracted from historical data and validated by experts. This is a limitation of our experiments design since several phases of increase and decrease may occur during a flood.

When a flood alert is triggered, the severity of the event remains unknown. We consider that the alert period allows the increase of preparedness to the worst possibilities. Consequently, all hospitals in the territory set a priority for their patients and free, if possible, a predefined percentage of their capacity. The performance of the rules applied during the alert period is reflected through the three following outcomes:

1. Less transferred patients in case of flood.

Markov model

Based on the geographical decomposition into zones and the time decomposition into time periods, we define a Markov chain with discrete index set (time) and discrete state space.

States definition

Let Z be the set of potential flooded zones indexed by z, and S z be the set of possible states for the zones. For each zone we define a finite set of n + 1 states {0, 1, . . . , n} with state 0 corresponding to a non-flooded zone and states 1 . . . n corresponding to water levels. The state values do not necessarily correspond to proportional water levels. These values are selected in order to model different impact level on the zone.

Water levels correspond to a flow value at each measuring station. Flow quantities can be easily sum up following the main river and its contributing rivers. Water levels are preferred, since they are more frequently measured and directly related to the flood severity.

A time horizon H discretized in time periods {1, . . . , t, . . . , H} is considered to model flood dynamics. At each time period, the flood advances in the direction of the water flow (for example, the zones on the right (East) of Figure 7 are impacted first), and the changes of states follow that flow.

Consequently, we consider a Markov chain where each State SR i = {s 1 , .., s |Z | } of |Z | items representing the water levels of all the region zones at a time period. The time period index is discarded from the state definition since it is only used to decide on the phase, flood rising or receding.

An example of flood simulation using our Markov chain is proposed in Figure 3 in five zones. No flood is occurring during periods 1 and 2. Zones 1 and 2 are flooded at level 1 during period 3, and then at level 2 during period 4 whereas zones 3 and 4 are flooded at level 1, etc.

t 1 (0, 0, 0, 0, 0) No flood ↓ t 2 (0, 0, 0, 0, 0) No flood ↓ t 3 (1, 1, 0, 0, 0) Zones 1 and 2 are flooded level 1 ↓ t 4 (2, 2, 1, 1, 0) Zones 1,

Initial State

A flood may start at one or several different zones depending on the number of rivers. We suppose that the flood event always starts at State SR 0 :

SR 0 [z] ≥ 1 if z is a source zone of a river, 0 otherwise. ( 1
)
In the example of Figure 3, the flood starts at t 3 from zones 1 and 2.

Transition Rules

The probability to reach state SR j from state SR i is denoted p ij . A transition rule defines how the level of water is changing between two consecutive time periods. Three types of transitions or transition rules are applied in our model : loop, increase and decrease:

• Loop transition. The same state SR i with a probability p ii is kept for all zones between two time slots. The loop transition periods allow to control the speed of the flood rising and receding.

• Increase and decrease transitions. The first (flood rising) and second (flood receding) phases are symmetrically controlled by the following rules.

In order to keep the flow of water consistent with the flood direction, the water level in any downstream zone z is lower (resp. upper) than the level in the upstream neighboring zone z during the previous period. Equations [START_REF] Altay | OR/MS research in disaster operations management[END_REF][START_REF] Chen | Modeling and simulation of a hospital evacuation before a forecasted flood[END_REF] and (4-5) enforce these constraints for State SR j that follows SR i for the increase and decrease transitions, respectively.

∀z ∈ Z , SR j [z] ≤ SR i [z] ≤ SR j [z] + IN C max (2) ∀z, z ∈ Z , SR j [z] ≤ SR i z (3) ∀z ∈ Z , SR j [z] -DEC max ≤ SR i [z] ≤ SR j [z] (4) ∀z, z ∈ Z , SR j [z] ≥ SR i z (5)
Note that the rules are applied to the normalized water levels of the states definition.

Between two consecutive states a maximal increase IN C max (resp. decrease DEC max ) values are defined. Combined with the length of the time periods, these values limit the water rising and receding speed.

We assume that the flood is rising (resp. receding) in all the zones at the same time. As in the example of Section 6.3, slow floods in Paris area satisfy this statement. However some extensions may consider zones in different phases at the same time. But, consistency with the physical constraints of the water flow should be revised.

The rules (2) to (5) enforce consistency between neighbor zones according to the current episode, rising or receding, by limiting the set of reachable states from State SR i . Of course, the transition probabilities p ij have two values; one for each episode.

When few historical data are available, like in our study case described in Section 6, the first value to set is the stationary state probability p ii , since this is the most common case for slow floods. Then the probabilities of reachable states SR j , j = i are set to the same value p ij = (1 -p ii )/Card({j = i : SR j is a reachable state}).

Monte-Carlo simulation

The proposed Markov chain model is integrated in a DES (Discrete Event Simulation) model in the following way: at each time slot, an event of the DES model is triggered in order to activate the transition probabilities of the Markov models. Depending on the result, the states of the zones are updated, resulting in the triggering of new management rules in the patient flow model described in the next section.

Patient flow model

In this paper the patient flow model is defined on a macroscopic scale and specific to a medical specialty. The complete healthcare network is composed of multiple instances of patient flow models, one for each medical specialty.

Assumptions on patient flow

Data are related to hospitals and patients, as detailed below. In our case study, facilities are all hospitals (but other types of facilities can be considered in our model). The capacities of hospitals have a major role in our model as the available capacity of a non-flooded facility can absorb the flow of patients caused by the flood (emergency and evacuation). Moreover, patients have specific needs depending on their health troubles. For instance, a dialysis patient needs a bed specifically equipped for dialysis. Therefore the capacity of each facility is defined by a multidimensional vector with one value for each specialty.

We consider 3 health states for each patient:

• State 1 (ps 1 ): the patient must be taken care of on the scheduled date, or if they is already hospitalized, their care must continue,

• State 2 (ps 2 ): the hospitalization may be delayed by a week maximum,

• State 3 (ps 3 ): the hospitalization may be delayed by a week or more.

Hospitals and patients data sets are used as input of the discrete-event model. Basically, patients arrive to facilities with specified arrival rate to receive care in a specialty for a given length of stay. The patient's specialty does not change during their stay. Parameters used in the model are summarized in Table 1. Binary indicator that equals 1 if delaying patients of specialty s in states ps 2 and ps 3 during a major flood allows the continuity of care of all patients in the state ps 1 , 0 otherwise.

Assumption 3 (Binary impact on the flood on hospitals) We suppose that if a hospital is flooded, it must be evacuated (all the patients are transferred).

Assumption 4 (Independence of specialities) Specialities are considered separately since we assume no transfer between medical specialities is permitted during the flood. The autonomy of the region in the event of a flood can therefore be evaluated for each speciality separately.

Healthcare management strategies during a flood

When a hospital is flooded, hospitalized patients must be transferred. Also, future scheduled patients and emergency patients are referred to other hospitals during the entire flood period.

During the flood period, disaster management operations (especially transportation) are complex and involve several decision makers. We define two strategies for patients' transfers based on organization and feasibility constraints.

1. Strategy 1: patient transfer is prioritized depending on transportation resource management. Thus, depending on the available capacity in the relevant specialty, a patient will be transferred as first choice to a hospital in the same hospital group, then in the same department (county), then outside the county but within the same region, and finally outside of the region (Figure 4).

2. Strategy 2: we define an isolated territory which does not include any major critical bridges. Depending on the available capacity in the relevant speciality, a patient will be transferred to a hospital in the same isolated territory. The purpose is to prevent helicopter transportation outside of the territory which is costly and should be avoided as much as possible. If all hospitals in the territory are saturated, patients will be transferred outside of the region (Figure 5).

Flooded Hospital

Hospital group County Region Out of region Regardless of the selected strategy, all impacted hospitals remain closed after the end of the flood event for repair purpose. The duration of the repair depends on the cause of the impact (submersion or electrical failure).

Petri Net Model of Patient Flow

Basic notions of Petri nets

An ordinary Petri net (PN) is a 4-tuple R = (P, T, F, M 0 ) where P and T are two disjointed sets of nodes called respectively places and transitions, F ⊆ (P × T ) ∪ (T × P ) is a set of directed arcs, M 0 : P → N is the initial marking of the net.

The set of input (resp. output) transitions of a place p ∈ P is denoted by •p (resp. p•). Similarly the set of input (resp. output) places of a transition t ∈ T is denoted by •t (resp. t

•).

A transition t ∈ T is said to be enabled at M 0 if for all p ∈ •t, M 0 (p) ≥ 1. A transition may fire if it is enabled. The firing of a transition t at marking M removes one token from each of its input places and puts one token to each of its output places.

A source transition is a transition without any input place. A source transition is always enabled. A sink transition is a transition without any output place. When firing a sink transition, all tokens are removed respecting usual rules but no tokens are generated.

A T-timed Petri net is a 5-tuple R = (P, T, F, θ, M 0 ) where θ : T → N assigns to each transition t its transition firing time θ(t). Firing a timed transition t at time d removes immediately one token from each input place but add tokens to its output places only at time d + θ(t). In this paper, we use a T-timed Petri net because we need to take into account duration of hospital stays.

A colored Petri net is a 7-tuple CPN = (P, T, C, A, W + , W -, M 0 ) where C : (P ∪ T ) → Ω, C(p), p ∈ P , is the set of colors associated to a place p (i.e. the set of colors that place p may have), C(t), t ∈ T , is the set of colors associated to a transition t (i.e. the set of ways to fire t), W - p,t : C(t) → N |C(P )| is the pre-condition of a transition in relation to a color which defines for each way of firing t the required combination of tokens of different colors in different places, W + t,p : C(t) → N |C(P )| is the post-condition of a transition in relation to a color which defines for each way of firing t the combination of tokens of different colors added to different places. In this paper we use a colored Petri net in order to take into account the various medical specialties during the hospital stays.

Model description

In order to formally define the patient flow model, a colored T-timed Petri net model for each specialty s ∈ {1, . . . , |S|} is proposed in Figure 6. Patients arrivals are modelled using a source transition t s,1 . Place p s,1 models the decision related to the triggering of a degraded mode for patient care in the event of a flood. If the facility is not impacted, transition t s,2 is fired (transportation time within the same facility) and the patient stays in the facility by firing transition t s,3 . The transition duration is defined by a random variable that depends on the type of stay. When the patient exits the facility (patient discharge), sink transition t s,4 is fired.

In the event of a flood, the degraded mode is triggered. Transitions t s,5 , t s,6 , t s,7 model transfers to other hospitals following a predefined priority. Transition t s,8 is a special case and models the transfer to an hospital out of the region, that penalizes the resilience of the region. In that case, we do not consider the stay of the patient and thus transition t s,3 is skipped. For the Strategy 1, t s,5 models a transfer to a facility from the hospital group, t s,6 models a transfer to a facility from the county, and t s,7 models a transfer to a facility from the region.

Place p s,4 models the available beds for each hospital having the specialty s. Then we have as many ways of firing t s,3 as we have hospitals having beds to take into care patients related to specialty s.

The resulting model is replicated for all medical specialties of the region. Formally the Petri Net of Patient Flow (PNPF) for a medical specialty s is a 7-tuple colored Petri net PN PF ∫ = (P s , T s , C s , A s , W + , W -, θ s , M s,0 ) where:

• P s = {p s,1 , p s,2 , p s,3 , p s,4 } is the set of places.

• T s = {t s,1 , . . . , t s,8 } is the set of transitions.

• C s : (P s ∪ T s ) → Ω, C(p), p ∈ P s , is the set of colors associated to a place p (i.e. the set of colors that place p may have) and C(t), t ∈ T s , is the set of colors associated to a transition t (i.e. the set of ways to fire t).

Here C s (p) = {cp 0 } ∀p ∈ {p s,1 , p s,2 , p s,3 }, C s (p s,4 ) = {cp 1 , . . . , cp |H| }, C s (t) = {ct 0 } ∀t ∈ T -{t s,3 } and C s (t s,3 ) = {ct 1 , . . . , ct |H| }.
Thus cp 0 is the color of patient tokens and cp 1 , . . . , cp |H| are the colors of tokens modelling beds in hospitals, one color for each hospital.

• A s is the set of arcs of the PN.

• W - p,t : C(t) → N |Cs(P )| is the pre-condition of a transition in relation to a color which defines for each way of firing t the required combination of tokens of different colors in different places. Here all transitions have only one color except t s,3 : W - p s,2 ,t s,3 (ct 0 ) = 1 (we need a patient token to start the stay) and W - p s,4 ,t s,3 (ct h ) = (0, . . . , 1, . . . , 0) where 1 is at rank h, ∀h ∈ H (we need a bed in a hospital of the network). • θ s : T s → N is a function returning the firing duration for a transition of the PN.

• M s,0 ∈ N P × N |Cs(Ps)| is the initial marking of the PN. Here place p s,4 has as many tokens of color cp h as hospital h has beds for medical specialty s.

Simulation of the Patient Flow Model

Anylogic Pro 7.1 has been chosen to implement both the patient flow model and the flood dynamic model. The implementation of the Petri net model in a simulation software is straightforward: source and sink transitions are implemented using source and sink modules. Decision places are implemented using decision nodes. Transportation or facility stay (temporized transitions) are implemented using process modules. A virtual event is triggered periodically (following the time periods defined in the Markov chain) during the simulation in order to update the Markov chain states. Global variables describing the flow state are updated. Depending on these values, output of place p s,1 is chosen accordingly to the currently used management rule and the flood state.

Key Performance Indicators

As introduced above, the ARSIF coordinates the evacuation operations and the patients transportation over the region during a disaster. Therefore we choose a set of KPIs that quantify the impact of each scenario and identify the decision makers involved (hospital groups, counties or the entire region).

Following the sequence of evacuation and transfers, Table 2 defines the list of indicators measured in this experiment. In this section, we present some results obtained on the Paris region healthcare network which has been impacted by a flood of the Seine and Marne rivers. We use several flood scenarios obtained by the Markov chain model and real data for different specialties. The usual period of major flooding is from November to April. Consequently, most preparedness plans are designed according to the pattern of patients flow during this period. However, flooding may occur at any time of the year, causing the sub-optimality of some of the emergency management rules (e.g. June 2016 flood in Paris).

Data and Scenarios

Several specialties were treated in the experiments and we choose to present as example the results for general medicine as well as a synthesis of other specialties.

The French national hospitalization database (Programme de Médicalisation des Systèmes d'Information, PMSI) is an exhaustive nationwide medico-administrative database that covers the data of patients in both public and private hospitals. We extract from this database the dates of start and end of stays of patients of all hospitals for every considered specialty.

Data collected from the hospitals in the Paris region are summarized in Table 3. Hospitals provided for each specialty their capacity, the number of patient per month, the mean length of stay. The flood risk is evaluated for each flood scenario by mapping the hospitals locations and the flood coverage. In these experiments, we consider 24 scenarios based on the combination of the two following parameters:

• Flood starting time (month): The flow of patients of each specialty strongly depends on the month of the year. It is important to test the impact of a flood on several periods during the year.

• Transfer strategy: Represents the priority sequence selected for the evacuation of hospitalized patients and transferred activities, as presented in Figures 4 and5.

We ran 1000 replications per scenario. Depending on the specialty, the computing time of one replication is between 5 and 15 seconds. All the results presented below are mean values of the replications.

The patient process Petri net model is calibrated using (i) hospitalisation medico-administrative data for length of stay modelling and (ii) geographical data for transportation duration. For each medical speciality, we fit a lognormal distribution for the length of stay in the hospital (transition t s 3 and an exponential distribution for the inter-arrival duration (transition t s,1 in Figure 6). For each pair of hospitals of the considered region, a transportation duration is modelled using a triangular distribution (the minimum value is the required transportation duration without traffic, the maximum value corresponds to the duration with heavy traffic, and the mod value corresponds to the most observed transportation duration).

Flood characteristics

The flood model proposed in this paper is based on the detailed maps of regional zones. These characteristics can be used in order to evaluate transfer strategies face to potential scenarios.

Zones

Water level and flow are monitored by reference stations. We consider that for each reference station there is a potential flood zone (PFZ), i.e., a zone in our Markov chain model. An example of reference stations and 15 associated zones is shown in Figure 7.

Figure 7: Water flow and the potential flooded zones associated to reference stations.

Water levels

The water levels that define possible states of each zone are computed relatively to a reference flood. Based on the results of a hydraulic model ALPHEE [START_REF] Driee | Hydrolic model alphee[END_REF], the Regional and Interdepartmental Directorate for Environment and Energy designed regional scenarios to estimate the impact of floods in the Paris region.

These scenarios use a set of simplified maps of flooded zones according to the flow of the main rivers in the region and are proportional to the flood of 1910, one of the most devastating floods in the region. Figure 8 shows examples of the impact of the three most important scenarios representing, from left to right, 80%, 100% and 115% of 1910's flood. These scenarios are respectively denoted R0.8, R1 and R1.15 in the rest of the paper. We use these data to set four possible states to each zone, as shown in Table 4. The zone is not flooded

s z = 1
The zone is impacted by the scenario R0.8

s z = 2
The zone is impacted by the scenario R1

s z = 3
The zone is impacted by the scenario R1.15 In this case study, we tested the impact of a slow flood on the territory, as it is the most frequent one for the Paris region.

Flood Markov model validation

The parameters (water levels, time periods and transition probabilities) of the Markov chain model are set according to the flood in Paris (2016) in order to validate our approach.

In the current section, the flood model is run starting from different time periods, in order to measure the convergence of the model when real information is revealed time to time.

Reference water levels are real data collected from [START_REF] Driee | Épisodes de crue de mai-juin 2016 sur le bassin de la seine[END_REF]. These data are simplified through a subset of five zones along the Seine river which correspond to the five measuring stations located after the main rivers confluences.

The eleven water levels considered in the Markov chain model are computed as one tenth of the gap between the maximum expected level (115% of the 1910 flood) and the default level.

Real water levels reached by the flood are given at the top of Figure 10 from 2016-05-28 to 2016-06-12. The corresponding normalized water levels according the 1910 flood are given in the bottom graph of Figure 10. The average error over one hundred independent runs of the forecast at the first day (2016-05-28) of the simulation was 1.77. The error is computed as the average deviation between every real and forecast water level (with normalized units), at every station, every day and for every run. Flood simulation model is validated using historical water levels: for one possible flood dynamic we want to test, we adjust probabilities of the Markov model in order to match the real flood. This way it is possible to test the impact of slight variations of a flood on the hospital network.

Patient flow model validation

There are several ways to validate a simulation model. In [START_REF] Sargent | Verification and validation of simulation models[END_REF], the author summarises and presents 15 of the most commonly used validation techniques. Among these techniques, we selected historical data validation for our problem: since we have historical data, part of the data is used to build the model and the remaining data are used to test whether the model behaves as the system does.

Patient flow model is constructed using 80% of historical data to build distributions used in the model. Then we inject the patients from the other 20% of historical data in our simulation model (without any flood occurring) and we compare the KPI length of stay of patients against the actual values. We used a 90% confidence interval and we verified our average length of stay for each specialty is within the given confidence interval.

Figure 12 presents the validation results for all medical specialties taken into consideration in this case study. Note that the simulation model under estimates the length of stay for neuro-vascular and long term care patients. This phenomenom is caused by the occurrence of extreme points in the historical data for these specialties. After discussing with practitioners, we consider the results given by the model are fair considering these particular cases. Also, for dialysis, length of stay is always equal to one. Apart from these 3 particular cases, simulation results are validated as the historical data sample average always fit in the error interval of the simulation results.

Results

General medicine speciality

In the region, 65 institutions are likely to be affected by a flood, of which 12 are due to flooding and 53 to electrical failure. On average, a flood requires the evacuation of 1,500 hospitalised patients and nearly 10,000 scheduled or urgent activities are displaced during the flood and the repair period (Figure 13). There is an impact of seasonality on the activity of general medicine. The months of May and August are the least critical, while the months of June and November are the most critical. Electric frailty is the main cause of flood impact on health facilities: 80.52% of the evacuated patients and transferred activities are due to an electrical power failure, whereas the remaining 19.48% of evacuation are caused by an actual flooding of the hospital. The next results represent the indicators of the absorption ability of the region depending on the scenarios.

Scenario 1

In Scenario 1, we suppose that the major bridges are functional and can be used for evacuation and transfer operations. Using this sequence of priority, the region remains Figure 13: Number of evacuees and transferred activities in the region autonomous for healthcare (Auto = 1). Moreover, the flood impact is absorbed within the county for evacuations and activity transfers. Table 5 shows that hospital groups absorb most of the evacuees from impacted hospitals. This result has a major advantage as hospital groups may share patients files as well as human and material resources. Transfer of activities are less absorbed by the hospital groups but still remain in the same county. We also observe that seasonality have little impact on the distribution of evacuees and transferred activities. In this scenario, no hospitalized patient (resp. no medical activity) is evacuated (resp. transferred) outside of the region. In general medicine, an average of 13% of patients must be treated immediately (ps 1 ), 30% can be postponed for several days as needed (ps 2 ) and more than 56% can be rescheduled in one to several weeks (ps 3 ). By studying the lists of patients and their states, decision makers may avoid evacuations and transfers outside of the region.

Other specialties

As mentioned previously, several major and critical specialties were studied in our work. Each specialty necessitates detailed modeling of the healthcare process as well as specific management rules in case of a flood. We summarize in Table 7 the results obtained : the averages on all the replications of the number of evacuations and transfers, the probability of autonomy for the two selected strategies, the impact of seasonality and the flexibility of the specialty. For all the studied specialties, regional independence is guaranteed if the major bridges and roads are available (Strategy 1). On the other hand, the independence of the isolated territories is not always guaranteed: in the worst case (neuro vascular unit), the probability of independence of the isolated territory amounts to 78%. This means that in 22% of flood scenarios, patients must be evacuated and activities transferred outside the region. For many specialties, there is no flexibility in the rescheduling due to the states of the patients. However, it is possible to free more capacity in other specialties and avoid the loss of the regional autonomy.

Conclusions and perspectives

We presented in this paper the work realized with the Regional Health Agency that aims to evaluate healthcare emergency plan and assess the regional resilience in case of major flood. We developed a modeling and simulation approach that combines a healthcare process model with a flood model using Markov chain. The model is used both as a preparedness tool and a response tool for healthcare decision makers. The results obtained on real data allow us to validate the model and estimate the impact of several flood scenarios on each specialty. The measured key performance indicators also identify the decisions makers involved (the hospital groups, counties or isolated territories).

In future work, additional evacuation strategies will be tested depending on more than the hospitals capacity and political preferences. Moreover, optimization models should be integrated to the model to solve several disaster operations problems, such as transportation operations and human resources assignment.

A Appendix: Software Development

A visualization interface is integrated to the model using the AnyLogic software and the Java programming 1 . It is intended to be used by medical authorities in order to visualize the flood impact and organize the response accordingly. Several parameters are selected at the start of the simulation:

• The initial date of the simulation -a drop-down menu to choose the start day and month;

• The initial force of the flood -a choice between the intensities of 80, 90, 100, 105 and 115 representing the water levels;

• The force and speed variations of the flood -a choice between "no variation", "low variation" and "high variation".

• The availability of major bridges and roads -this choice defines the evacuation strategy.

To visualize the evolution of the simulation we used the Map tool provided by AnyLogic. The flood evolution is illustrated by colored zones corresponding to the water level (on the map of Figure 14).

When a hospital is damaged, its color changes depending on the cause of damage (active, flooded or power failure). KPI measures are visible through data graphs and updated with the flood evolution.

As mentioned previously, the application can be used as a preparedness tool by varying the parameters and executing replications, and also as a response tool if connected to input data in real time as a digital twin application.

Figure 1 :

 1 Figure 1: Life cycle of Disaster Operation Management.

Figure 2 :

 2 Figure 2: General approach

2 .

 2 Better preparedness to face to the flow of emergency patients caused by the flood.3. Excess capacity for patients transferred from flooded hospitals.

1 Figure 3 :

 13 Figure 3: Example of flood dynamics

Figure 4 :Figure 5 :

 45 Figure 4: Strategy 1 for evacuations and transfers.

Figure 6 :

 6 Figure 6: Generic patient flow model

Figure 8 :

 8 Figure 8: Three regional flood scenarios in Paris region as a percentage of 1910's flood.

Table 4 :

 4 Notation and definition of zone states relatively to 1910's flood. State of the zone z ∈ Z Definition s z = 0

Figure 9

 9 Figure 9 shows an example of the combination on a sub part of the map. The yellow areas represent the flooded area in case of scenario R0.8, the pink in case of R1 and the blue areas in case of R1.15. The hospitals are represented by the orange dots.

Figure 9 :

 9 Figure 9: Example of the mapping of the three scenarios and hospitals locations

Figure 10 :

 10 Figure 10: Water levels at five measuring stations during the 2016 flood in Paris We defined empirically the values of the parameters of the flood of 2016. The stability probability p ii = 0.3, the maximal increment for flood rising or receding is equal to 2. In Figure 11, the forecasts performed at three different days are displayed. The forecast computed on 2016-05-28 is displayed in the top graph, on 2016-06-02 in the middle graph, and on 2016-06-06 in the bottom graph. Actually, the forecast for next days can be refined when new measures are collected and used for a warm start of the Markov chain model. The values displayed are averaged over one hundred runs.

Figure 11 :

 11 Figure 11: Update of forecasts on 05-28, 06-02 and 06-06

Figure 12 :

 12 Figure 12: Validation results

Figure 14 :

 14 Figure 14: Simulation Manager

Table 1 :

 1 List of parameters Parameter Definition S = {1, . . . , s, . . . , |S|} Set of medical specialties H = {1, . . . , h, . . . , |H|} Set of hospitals in the network n s,h Number of beds of specialty s in hospital h F lex s

Table 2 :

 2 Key performance indicators. T ot Number of evacuees for the entire region E HG Number of hospitalized patients evacuated to a hospital in the same group E Dep Number of hospitalized patients evacuated to a hospital in the same county E Reg Number of hospitalized patients evacuated to a hospital within the region E IT Number of hospitalized patients evacuated to a hospital within the isolated territory E Out Number of hospitalized patients evacuated to a hospital outside the region T T ot Number of transferred activities for the entire region T HG Number of transferred activities to a hospital in the same group T dep Number of transferred activities to a hospital in the same county T reg Number of transferred activities to a hospital within the region T IT Number of transferred activities to a hospital within the isolated territory T Out Number of transferred activities to a hospital outside the region Auto Binary indicator that equals 1 if the region remains autonomous, 0 otherwise 6 Case study: Paris region flooding

E

Table 3 :

 3 Hospitals and patients data for the studied specialties.

	Specialty Number	Maximum	Maximum	Total	Patients	Mean
		of hospi-	flooded	electrical	capacity	per year	LoS
		tals	hospitals	failures			(in
							days)
	General	189	12	53	20,601	1,748,569 3.9
	medicine

Table 5 :

 5 Percentage of hospitalized patients evacuated to a hospital in the same hospital group (E HG) (resp. same county (E Dep), same region (E Reg), out of the region) and number of activities transferred in the same hospital group (T HG) (resp. In Scenario 2 the main bridges are not functional and thus evacuation and transfers must stay within isolated territories. Isolated territories absorb 97% of evacuations and transferred activities as shown in Table6. The remaining 3% of patients therefore need outof-region transfers. In this strategy as well, seasonality have little impact on the number of evacuees and transferred activities.Technically, this strategy is very difficult to implement in real life for a logistical reason. Transportation resources are managed by either hospital groups or counties. An isolated territory

	same county (T Dep), same

Table 6 :

 6 Percentage of hospitalized patients evacuated to a hospital in the same territory (E IT) (resp. same county (E Dep), same region (E Reg), out of the region) and number of activities transferred in the same territory (T IT) (resp. same hospital group (T HG), same county (T Dep), same region (T Reg), out of the region) for Scenario 2

	Beginning	Patients evacuations			Transferred activities
	of the flood	E IT E Dep E Reg	Out	T IT T HG T Dep T Reg Out
	January	97.93% 2.07% 0.00% 0.00% 10,657	54	198	0	0
	February	97.66% 2.34% 0.00% 0.00% 11,013	69	222	0	0
	March	97.69% 2.31% 0.00% 0.00% 11,187	57	195	0	0
	April	97.74% 2.26% 0.00% 0.00% 10,325	28	148	0	0
	May	97.78% 2.22% 0.00% 0.00%	8,097	15	102	0	0
	June	97.90% 2.10% 0.00% 0.00% 11,771	39	174	0	0
	July	97.75% 2.25% 0.00% 0.00% 10,073	11	130	0	0
	August	97.81% 2.19% 0.00% 0.00%	6,996	15	100	0	0
	September	97.80% 2.20% 0.00% 0.00% 10,919	25	149	0	0
	October	97.96% 2.04% 0.00% 0.00% 11,316	25	147	0	0
	November	97.86% 2.14% 0.00% 0.00% 10,889	39	181	0	0
	December	97.70% 2.30% 0.00% 0.00% 10,212	40	191	0	0

may contain several counties, and a hospital group may have establishments in several isolated territories. An extra effort is needed to coordinate the cooperation between several decision makers.

Table 7 :

 7 Hospitals and patients data for the studied specialties.

	Specialty	Flexibility	Nb of	Nb of Autonomy Autonomy Season.
			evacuees transfers	Strat1	Strat2	impact
	Surgery	YES	1027	6715	100%	99%	YES
	Obstetrics	NO	500	1661	100%	82%	NO
	Neurosurgery	NO	122	200	100%	98%	YES
	Neuro-vascular	NO	1	41	100%	78%	YES
	Dialysis	NO	324	1338	100%	92%	NO
	Cancerology	YES	86	422	100%	100%	YES
	ICU	NO	490	1222	100%	98%	NO
	Long term care	NO	2558	861	100%	86%	NO

Work done with the participation of students from the Engieering School ISIMA of Clermont-Ferrand: M. Bertoni and D. Selimovic


Luc Damé
email: luc.dame@latmos.ipsl.fr

Lucile Conan

Carlo De Santi

Alessandro Caria

Matteo Buffolo

Matteo Meneghini

Pierre Maso

Clément Dias

Halima Ghorbel

Pierre Gilbert

Mustapha Meftah

Philippe Bove

V Sandana

David Rogers

Ferechteh Teherani

Dynamical properties and performances of ß-Ga 2 O 3 UVC photodetectors of extreme solar blindness

Keywords: Deep ultraviolet photodetectors, β-Ga 2 O 3 oxide, Space observations, Dynamical properties, Herzberg continuum, Nanosatellites. DEVINS--

There is a surge in interest for the ultra wide bandgap (Eg ~ 4.9 eV) semiconductor gallium oxide (Ga 2 O 3 ). A key driver for this boom is that single crystal wide area bulk β-Ga 2 O 3 substrates have become commercially available and that a variety of methods have been shown to give high quality epitaxial growth. Amongst a whole range of potential applications (power/switching electronics, solar transparent electrodes, etc.) extreme solar blindness photodetectors in β-Ga 2 O 3 , the more stable monoclinic phase of Ga 2 O 3 , offers the most exciting perspectives for deep ultraviolet observations below 250 nm. We present an overview of the complete realization process (epitaxy, photolithography, singulation and packaging), and space qualification (to TRL8) of a series of β-Ga 2 O 3 photodetectors We insist on their remarkable solar blindness performance (factor 1000 or more between peak responsivity around 215-220 nm and 250 nm) and dynamical properties (rise and fall times < 5 ms), along with their enhanced responsivity, excellent thermal behaviour and radiation hard properties. 4 of these detectors have been integrated on the INSPIRE-SAT 7 nanosatellite expected to be launched in April 2023 to reach the TRL9 "demonstrated in flight" qualification.

INTRODUCTION

We present the development of disruptive ultra wide band gap semiconductor (≥ 4.9 eV, corresponding to a 253 nm vacuum wavelength) UVC photodetectors on oxide-based ß-Ga 2 O 3 that are intrinsically solar blind, radiation-hard and thermally-robust. We have recently shown that the bandgap of these detectors can be engineered upwards through Al alloying, ß-(Al)Ga 2 O 3 , so as to obtain optical transitions from 253 down to 200 nm [START_REF] Rogers | Sharp/tuneable UVC selectivity and extreme solar blindness in nominally grown undoped Ga 2 O 3 MSM photodetectors by pulsed laser deposition[END_REF]. The detectors developed and space qualified are intended to monitor the Solar Spectral Irradiance (SSI) at 215/220 nm (Herzberg solar continuum: 200-242 nm) with an accuracy better than 0.5% (5 times better than previous measurements in Space [START_REF] Meftah | SOLAR-ISS, a New Solar Spectrum Based on Recent Observations[END_REF]).

In this paper we recall the realization process, properties and performance of these detectors and, in particular, the new evaluation of the dynamical response of these detectors (< 5 ms) largely under evaluated in a previous work [START_REF] Arrateig | Development and Simulated Environment Testing of β-(Al)Ga 2 O 3based Photodetectors for Space-based Observation of the Herzberg Continuum[END_REF] and even recently by a factor 2 [START_REF] Damé | Realization, calibrations and spatialization of ß-(Al)Ga 2 O 3 UVC Photodetectors of Extreme Solar Blindness for Space Observations[END_REF]. 4 of these new ß-Ga 2 O 3 -based photodetectors have been space qualified and integrated in the INSPIRE-7 nanosatellite [START_REF] Meftah | INSPIRE-SAT 7, a Second CubeSat to Measure the Earth'sEnergy Budget and to Probe the Ionosphere[END_REF] to be launched in April 2023. Already at TRL8 (qualified for flight) these detectors could reach the TRL9 level (demonstrated in flight) and, since of their very simple implementation process, they could be integrated in several of the Earth and climate constellations currently developed or planned to observe the UVC spectral band 200-242 nm of particular interest for climate (this solar spectral band, absorbed in the stratosphere, causes oxygen to dissociate and creates the ozone layer).

DETECTORS REALIZATION

We realized, from epitaxy to packaging (cf. Fig. 1), detectors in ß-Ga 2 O 3 sensitive between 200 and 240 nm, centred at 215-220 nm, with a half width at half maximum reduced (of the order of 30 nm) and with excellent rejection of the visible spectrum (a factor 1000 between the peak response at 215/220 nm and 250 nm). Oxide materials have this decisive advantage: a significant rejection of the visible adjustable in wavelength as a function of the oxide chosen and its doping (aluminium in our case [START_REF] Rogers | Sharp/tuneable UVC selectivity and extreme solar blindness in nominally grown undoped Ga 2 O 3 MSM photodetectors by pulsed laser deposition[END_REF]). β-(Al)Ga 2 O 3 "solar blind" detectors also present an excellent signal-to-noise ratio (dark current is very low: ≤ 5 pA). These detectors have also other significant properties: they are radiation hard (long term performances) and performing at ambient temperature (no need for cooling).

These detectors were realized in the framework of the ANR DEVINS program (2018-2022) from the wafers to the final packaged sensors going through device architecture development, epitaxy, photolithography, contacting, probing, singulation and packaging in TO-39 caps with sapphire windows. Process is detailed in previous papers [START_REF] Rogers | Sharp/tuneable UVC selectivity and extreme solar blindness in nominally grown undoped Ga 2 O 3 MSM photodetectors by pulsed laser deposition[END_REF][START_REF] Arrateig | Development and Simulated Environment Testing of β-(Al)Ga 2 O 3based Photodetectors for Space-based Observation of the Herzberg Continuum[END_REF][START_REF] Damé | Realization, calibrations and spatialization of ß-(Al)Ga 2 O 3 UVC Photodetectors of Extreme Solar Blindness for Space Observations[END_REF].

PERFORMANCE EVALUATION

We performed probing tests on hundreds of MSM components (5000 realized) to categorize detectors performances and to select a hundred of them for packaging and space qualification. So far, we achieved up to 6 orders of magnitude between dark and light currents at only -5 V bias voltage (constraint of nanosatellite finally released to -10 V). Dark current is very low (≤ 5 pA) and could not be measured most of the time with our equipment on the good selected components. Dynamics on its side (rise and fall times) has been thoroughly studied and better than expected with this material (less than 5 ms), and much better than reported, erroneously, to a few seconds in a previous work [START_REF] Arrateig | Development and Simulated Environment Testing of β-(Al)Ga 2 O 3based Photodetectors for Space-based Observation of the Herzberg Continuum[END_REF].

Dark current

The dark current was measured on hundreds of β-Ga 2 O 3 Metal-Semiconductor-Metal (MSM) photodetectors and on the packaged detectors. Fig. 2 presents one of the result of its measurement on a packaged detector not selected for flight (low responsivity of 1 mA/W only). We notice an excellent 6 order of magnitude between dark and light currents measured at only -5 V bias voltage. Dark current, in addition, was very low (< 5 ± 1 pA) and below the measurements capabilities of our equipment. 5 to 6 orders of magnitude were typically observed on packaged detectors when connectors are properly fixed (wire bonding) on an extra pad to strengthen the soldering process (see Fig. 4 in reference [START_REF] Damé | Realization, calibrations and spatialization of ß-(Al)Ga 2 O 3 UVC Photodetectors of Extreme Solar Blindness for Space Observations[END_REF]). On the MSM, with probe contacts, only 3 to 4 orders of magnitude between dark and light are commonly observed [START_REF] Rogers | Sharp/tuneable UVC selectivity and extreme solar blindness in nominally grown undoped Ga 2 O 3 MSM photodetectors by pulsed laser deposition[END_REF] because of the poor contacts of limited size.

Spectral response

A test bench set-up was developed at LATMOS to characterize the detectors using a Deuterium lamp and powermeter (cf. Fig. 3). Acquisition is automatized under computer control with a LabVIEW program. Note that when the voltage is increased, e.g. to a bias voltage of 10 V instead of 5 V (a possibility implemented on the nanosatellite INSPIRE-SAT 7), the response is multiplied by a factor 5 to 7 owing the non-linearity of the response between 5 and 10 V (cf. Fig. 5). For example, measured packaged detectors 707-23 raised from 20 mA/W to 120 mA/W and 707-04 from 20 to 100 mA/W at peak value 220 nm.

Figure 5. The spectral response of the photodetectors increases significantly as a function of the bias voltage. Between 5 and 10 V of bias voltage the multiplying factor on the responsivity is 5 to 7 while it is more linear after (factor 2 between 10 and 15 V; factor 3 between 10 and 20 V).

Dynamical response

Dynamical response (rise and fall times) were evaluated recently [START_REF] Damé | Realization, calibrations and spatialization of ß-(Al)Ga 2 O 3 UVC Photodetectors of Extreme Solar Blindness for Space Observations[END_REF] to be less than 20 ms. In fact, and even if further work is still necessary, we established after correcting the power line noise (50 Hz) with a Notch filter (40-60 Hz), that this figure is even better: we were able to achieve reliable operation at 480 Hz and measured a stable response time (rise and fall times) of less than 5 ms (cf. Fig. 6). As soon as bias is applied to the photodetector, the photoresponse was found to start decreasing until a steady-state condition is reached (see Fig. 7). In order to have stable operation of the photodetector during the mission, the bias will be permanently applied when in use in space to avoid the decrease observed at start. In order to investigate the physical process limiting the dynamic performance, the measurements were repeated at different ambient temperatures. As reported in Fig. 8, temperature has a significant impact on the speed and amplitude of the response. This behaviour can be caused by various effects, including the variation in leakage level with temperature and the change in defect capture and emission rate. A full analysis is not trivial and will be carried out in the future.

SPACE QUALIFICATION

An intensive testing of our 100 packaged detectors pre-selected for flight was performed in particular thermal cycling and response tests. These are illustrated in Fig. 9 showing the spectral response and the photocurrent/voltage curves (under -5 V bias) at room temperature and after all thermal cycling experiments (from 230 to 340 K, and under vacuum or synthetic air) at atmospheric pressure. Performance achieved with the 4 flight-selected detectors are even better since the bias voltage available on the nanosatellite is finally 10 V (response > 100 mA/W). Characteristics of our packaged ß-(Al)Ga 2 O 3 detectors are summarized in Table 1.

PARAMETER ACHIEVEMENT

Peak response 215/220 nm ± 5 nm Spectral response @ peak > 100 mA/W @ 10 V bias Spectral range 200-242 nm Rejection ratio (peak/250 nm) > 1 000 Mean noise level < 5 pA

Rise and Fall times ≤ 5 ms

CONCLUSION

UVC optimized detectors in Gallium oxide were realized and tested, and present unique performances of extreme solar blindness (factor 1000 or more between peak of responsivity and 250 nm) and dynamical properties (rise and fall times of less than 5 ms). 4 of them are now integrated in the INSPIRE-SAT 7 nanosatellite [START_REF] Meftah | INSPIRE-SAT 7, a Second CubeSat to Measure the Earth'sEnergy Budget and to Probe the Ionosphere[END_REF] to be launched in April 2023. Upon success, the DEVINS ß-(Al)Ga 2 O 3 photodetectors qualified for flight (TRL 8) will be granted TRL 9 (demonstrated in flight).

Figure 1 .

 1 Figure 1. The 5 main steps in the device architecture development of the DEVINS UVC ß-Ga 2 O 3 detectors (epitaxy, photolithography, contacting, probing, singulation and packaging in TO-39 caps).

Figure 2 .

 2 Figure 2. We observed 6 order of magnitude between dark and light currents measured at only -5 V bias voltage. Dark current is very low (< 5 ± 1 pA) and below the measurements capabilities of our equipment.

Figure 3 .Fig. 4

 34 Figure 3. Characterization test bench developed at LATMOS for the photodetectors DEVINS in ß-(Al)Ga 2 O 3 .

Figure 4 .

 4 Figure 4. Typical spectral responsivity (in A/W) of a photodetector (707-26/16-05, selected for flight on INSPIRE-SAT 7) with a maximum at 215 nm, a response of ~20 mA/W (with a bias voltage of -5 V), an FWHM of ~35 nm and, accordingly, an excellent rejection of about 1000 for wavelengths above 250 nm.

Figure 6 .

 6 Figure 6. Steady-state time dependence of the photoresponse of a photodetector at 215 nm (bias of -5 V applied at start).

Figure 7 .

 7 Figure 7. Time dependence of the photoresponse after bias is applied.

Figure 8 .

 8 Figure 8. Temperature dependence of the photoresponse over time.

Figure 9

 9 Figure 9. a) Spectral response (at -5 V bias) and (b) IV curves (under polychromatic illumination) at room temperature before (black) and after (red) atmospheric pressure and vacuum thermal cycling of one typical (but with low response) DEVINS UVC ß-(Al)Ga 2 O 3 detector.

Figure 10 .

 10 Figure 10. Illustrative view of the DEVINS detectors positions on the nanosatellite INSPIRE-SAT 7 and photo of DEVINS-L integration in the nanosatellite (Credits: INSPIRE-SAT 7).

Figure 11 .

 11 Figure 11. DEVINS-L ß-(Al)Ga 2 O 3 detector mounting design (with focusing lens) for the INSPIRE-Sat 7 nanosatellite.

Table 1 .

 1 DEVINS UVC photodetectors characteristics.

ACKNOWLEDGEMENTS

These Gallium oxide detectors were realized in the framework of the DEVINS program (DEep uV INnovative detector technologies for Space observations) financed by the ANR (Agence Nationale de la Recherche) under contract ANR-18-CE42-0015 (2018-2022).