Modelling and Reliability Analysis of Multi-source Renewable Energy Systems Using Deterministic and Stochastic Petri Net

In this paper, the energy systems based on renewable resources are studied. The purpose is to make an analysis of the dynamic behavior of the component elements of these systems in order to evaluate the performance related to their reliability and availability. Our approach is based on the use of Deterministic and Stochastic Petri Nets, to establish a model to extract performance with the goal to define subsequently a tolerant control process fault reconfigurable to limit risks unavailability.


INTRODUCTION
With the energy challenges of today, the use of Multi-Source Renewable Energy Systems [1] [2] appears as a potential solution for the production of electricity on a large scale.These systems combine different sources (wind, PV, biomass, geothermal ...) and are characterized among other things by the diversity of elements that constitute the Mechanical elements (blades, actuators, etc.), chemical (battery) electrical (generators, converters) and computer (control and supervision program).These multiples components give these systems a complex character and make them quite difficult to study.The evaluation of their dependability (reliability, availability ...) is a major concern, because of the heterogeneity of their components and especially the demands of the energy market.
The methods of assessing the dependability in these fields (mechanical, electrical engineering, chemistry and computer science, ...) are indeed very different from each other.To date, there is no technique to quantify the reliability and availability of such systems.To overcome this problem, solutions need to be made to ensure continuity of service and optimum reliability of operations provided by these systems.The aim is to minimize the risks of defects or incidents which create a system downtime, thus intruding stocks and energy supply.In this article, we focus on the study of the dynamic behavior of these systems in order to make a subsequent quantitative analysis to evaluate their reliability in a model based on Petri nets.These graphs [3] introduced by Carl Adam Petri in 1962 [4], are powerful in modeling Discrete Event Systems [5].These are defined by discrete state spaces where all state variables take their values in a countable set.The choice of this modeling tool is justified by the discrete nature of the state of system to be studied (functioning and failure).Specifically, we use a category of Petri net which models the stochastic phenomenon, represented by the appearance of failures in the Multi-Source Renewable Energy Systems.These graphs are known as Deterministic and Stochastic Petri Nets (DSPN) [6], [7].
In the literature, to our knowledge, there are very few works that are interested in the qualitative and quantitative analysis of multi-source energy based Petri net systems.In [8], the authors propose a Petri net-based modeling for better energy management of a multi-source power system consisting of a photovoltaic source (PV), a storage battery the classic grid.The authors in a study [9] proposed a stochastic Petri net based modeling for the management of energy to power a server set while taking into account their frequency of operation and the number of servers in idle state.In a study [10] the authors proposed a method to quantify and improve the performance of multi-source energy systems, using a PN graph.The work presented in [11] deals with the energy management of a photovoltaic based power station.This power station includes storage units with batteries for long-term energy supply and ultra-capacitors for fast dynamic power regulation.According to the availability of the primary source, the level of the stored energy and the request from the grid operator, the authors have defined three main operating modes for this system with a particular modeling tool: Petri nets.For each operating mode, an energy management algorithm has been designed in order to calculate the energy dispatching of an adjustable power margin for the storage units.
In this work, the DSPN is used to model the behavior related to the dynamic evolution of these systems to extract, by simulation, the instant of passage from the state of functioning towards to a state of breakdown and vice versa, noting that these moments are the time references that will be used to quantify the reliability and availability of these systems.Knowing that this reliability corresponds to the probability for a system to fulfill a required function during a period from 0 to t 1 [12], and availability is the probability that a repairable system operates in an interval from t 1 to t 2 [12].This paper is organized as follows: after a general introduction describing the context of this work, the DSPN which is used for modeling of multi-source energy systems is presented in Section 2. The model governing the evolution of the functional and dysfunctional behavior of these systems is presented in Section 3. The performance evaluation using the simulation of these systems on quantifying the reliability is the subject of Section 4, this is followed by a subsequent analysis of results simulation before concluding this work.

DETERMINISTIC STOCHASTIC PETRI NET (DSPN)
This section presents the Deterministic and Stochastic Petri Nets (DSPN) that will be used for modeling of hybrid systems to renewable energy sources with the purpose to analyse the reliability of this systems.These graphs are defined as 11-tuple (P, T, I, O, MO, H, N, G, W, d, e), where P is a finite set of places.T is the finite set of transitions partitioned into three subsets: one subset of stochastic transitions T e , where their firing follows stochastic law, a subset of deterministic transitions T d and a subset of T i immediate transitions.The set of arcs is divided into two subsets: a subset classical arcs: I is the set of arcs from the place p to transition t, and Y is the set of outgoing arcs showing from transition t to place p. H corresponds to the subset of inhibitor arcs.N is the weight of arcs on classical and inhibitor arcs.The initial state of DSPN is designated by M o and the amount of tokens in place p i is denoted by M (p i ).In DSPN, the places are represented by circles and the transitions by bars.Immediate transitions are presented by black bars and the deterministic transitions by filled rectangles and the stochastic transitions by empty rectangles.Durations ith firing d are represented in the following way: the exponential transition follows the form with and the immediate transition t, , deterministic transition .The immediate transitions have priority over other transitions.In case of conflict between immediate transitions, the transition that has the highest priority e will be fired first, while the default priorities are equal to one.If two transitions are of the same priority, we will associate each immediate transition with a natural number (weight) represented by W, while its role is to resolve conflicts in a probabilistic law.Inhibitor arcs are represented by arcs which end by a small circle.
The use of inhibitor arcs increases the capacities of modelling and the property of a DSPN.If a place p and a transition t are connected by an Inhibitors arc with a weight N, transition t cannot be fired except if the marking of place p is lower than N.However, if a place p and a transition t are connected by a classical arc with a weight N, the transition t can be fired if the marking of the place p is greater than or equal to N. The main difference is when the firing of a transition t removes tokens in place p when are connected by a classical arc, while no token movement will take place in the case of inhibitor arcs.In addition to the symbols presented before, one will add the last symbol G which is a function of activation of the transitions.

Energy System Mono-source with Wind Turbine
This system consists of a wind turbine, connected to a DC bus through an AC/DC converter.The DC bus is connected into a Load through an inverter (Fig. 1a).The functional and dysfunctional model DSPN of this system is depicted in Fig. (1b).The Interpretation of places and transitions of this DSPN is given in Table 1 and Table 2 respectively.In Fig. (1b), we show that a system containing only wind turbine source is in a wait state (p 1 ) or breakdown state (p 3 ), the AC/DC converter, Inverter and the DC bus are stagnated in their wait states (p 4 , p 10 ), and in the free state (p 7 ), the load will remain disconnected (p 14 ).The firing of transition t 1 will allow tokens of place p 1 to pass towards place p 2 , that means that the wind turbine passes from the wait state towards the operating state, represented by the place p 2 .The AC/DC converter, Inverter and DC bus will also pass them towards the operating condition in an occupied state p 5 , p 11 , p 8 respectively.This passage will influence directly the load which will pass towards the state connected p 13 .If the wind turbine is in the wait state or of breakdown and the DC bus is in a breakdown state or the AC/DC converter is in a breakdown state, the oad will be automatically disconnected.

Energy System Mono-source with Photovoltaic Module
This energy system contains only photovoltaic module (PV) connected directly to the DC bus and in its turn with a Load using an Inverter (Fig. 2a).The functional and dysfunctional model of this system is represented by the DSPN depicted in Fig. (2b).In Fig. (2b), the token in place p modelling the wait state of photovoltaic and which is connected directly to the DC bus, also in the free state (p 5 ), and Inverter which is in the wait state (p 6 ) indicates that the Load is disconnected (p 10 ).The activation of transition t 1 will push the photovoltaic module to function p 1 .This operation will influence the DC bus which will pass from the state of vacuum towards the occupied state (p 4 ) and Inverter towards the functional state (p 7 ).The token which is in place p 10 passes towards the connect state (p 9 ) after the firing of the transition t 14 .If the Photovoltaic module or the DC bus or Inverter is in a breakdown state, like represented in Fig. (2b) by the places p 1 , p 3 and p 8 , the Load will remain disconnected.

Multi-source Renewable Energy System
We present in Fig. (3), the first global view on the Multi-source Renewable Energy System, which consists of four blocks connected to each other.This system consists of four blocks interconnected.The first block includes the different energy sources that deal with the production of electrical energy.The second block, which has the characteristic of having a link with the other blocks, consists of the transmission devices, such as the bus of transmission and inverters.The third block is that of storage which one will find inside the various types of storage of energy such the battery.At the end the block of consumption which gathers the consumers.In DSPN depicted in Fig. (5), before the beginning of the production, the Photovoltaic module, Wind turbine, AC/DC converter and Inverter are in wait state, represented by the presence of a token in places p , p 3, p 6, p 12 respectively.Such that the DC bus is in the free state (p 9 ) and the battery in wait state (p 22 ) .This situation prevents the load to connecting itself.When one or both of the transitions t 5 and t 1 are fired, the wind turbine and the photovoltaic pass towards production state, are represented by p 4 and p 1 .This operation will influence directly the other components of the system; the DC bus which will pass towards the occupied state (p 10 ) after firing one of the transitions (t 13 , t 14 , t 15 , t 16 , t 17 , t 18 , t 19 , t 20 , t 21 , t 22 and t 23 ), Inverter and the AC/DC converter which will also pass towards the functional state (p 13 , p 7 ).This situation will influence directly the state of the Load which will pass from the disconnected state (p 16 ) to the connected state (p 15 ) after firing of the transition t 42.
The functioning of photovoltaic module and wind turbine separately or collectively allows the battery to pass through the load state represented by p 21 in order to store the additional energy.
When the token is in place p 17, the load of the battery reaches its maximum value.The battery stops charging (p 20 ) when switching from photovoltaic module and a wind turbine to the breakdown state (p 2, p 5 ) or in wait state (p , p 3 ).In this case, battery ensures energy supply.

AND PERFORMANCE EVALUATION
In this part, we were interested the performance evaluation of Multi-Source Renewable Energy Systems by using the DSPN.The Reliability and availability are the two performance indicators that will be analyzed by using the simulation software for Petri nets, known under the name TOTAL.This latter gives us the instant time of firing different transitions of DSPN.

Principle
To study the reliability and availability of a component, we are interested only in the moment of the first switching from the functional state of the failed state (firing the transition T1), and the end time of the repair, i.e. switching from the failed state to the functional state, firing the transition T2 (Fig. 6).

Graphic Illustration
The following graph illustrates the passage of a component from a functional state (= 1 binary) to the failed state (=0 binary) and vice-versa.
Remember that the different instants of switching from functional state to the fault state are given by the TOTAL simulation software and vice-versa.

Evaluation of Reliability (R)
In this work, we focus on the first firing of the transition T1 to quantify the reliability for any given time instant (LD).

Evaluation of Availability (A)
In the study of the availability, we are interested in firing transitions of failure and repair throughout the duration of the simulation.
The following diagram shows the different cases, depending on the value LD, in order to calculate the reliability and availability.

Parameters of Simulation
To be able to extract the results in this work using simulation on the Total software, we carried out simulation of the systems during 83220 hours (9,5 years).The parameters chosen for the systems of Figs.(1b, 2b and 5) are represented in the table below:  ▪ Index "i" refers to transition "Ti" ▪ occ.h -1 correspond to occurrence per unit time   In Fig. (7b), the reliability of DC bus is 100% throughout simulation but for the two other components (inverter, photovoltaic), their reliability was stable at 100% until t= 20353,7 h (2,32 years) and t = 67195,4 H (7,67 years), respectively.Then it experienced a reduction because of the firing of the transitions of passage towards the state of breakdown.

Fig. (7b).
Reliability of each components of the Mono-source Energy System, with photovoltaic only (system depicted in Fig. 2a)).
The total reliability which is represented in Fig. (8a) is stable with the value 100%.After the first year, it follows a reduction in its value to arrive to the value of 6% after 9.5 years of operation (83220 h).
On the other hand, the total reliability which is represented in Fig. (8b) is stable with the value 100%.After the second year, it follows a reduction in its value to reach the value of 19, 75%, after 9.5 years of operation (83220 h).For the total reliability which is represented in Fig. (9b), it has been stable but the only difference is in the manner of its reduction which was slower than the preceding one, showing thatits value was 19.75% after 9.5 years of operation (83220 h).
In Fig. (10a), the availability of photovoltaic module, DC Bus and battery is stable with the value of 100% throughout the simulation.This stabilization is due to the absence of failure for this length of time.On the other hand, the availability of the wind turbine remains stable with the value 100% until the first shooting of the transition from failure; then its value caned tops and bottoms.This situation is due to the presence of 4 shootings of the transition from failure to t = (16253 h, 32903,6 h, 48823 h, 68026,6h) and 4 shootings of the transition from repair to t = (16450,9 h, 34051,6 h, 48839,1 h, 69435 h).The value of availability of component AC/DC was stable at 100% until t = 8, 84 years (77470, 9 h) and that is only the first shooting of the transition from failure, then it followed a fall up to the value of 99.5% of the availability with the precision that the time of shooting of the transition from repair was equal to t = 77853,3 h.The inverter has a value of availability fixed at 100% until the first shooting of the transition from failure to t = 2,24 years (19639,4 h) after which it fell up to the minimal value with t = 2.5 years which was around 99.29%.It then increase exponentially by specifying that the moment of shooting of the transition from repair was around t = 19794,7 h.
In Figs.(10b, 11a), and 11b), we present the evolution of the total availability of the systems represented in Figs.
(1a, 2a and 4), respectively, according to time.This availability is calculated by the application of the values of the availability of each component according to its architecture.The total availability of the system depicted in Figure 1a is represented by the graph, enabling us to divide the readings into four parts.The first: beginning until the first year when the system of was reliable having 100%, in the second part: the reading goes down again until the fourth year when its value was around 97.5%.The third part: between the fourth and the fifth year, it is around 96.5%.The fourth share: it remains stable until the end of simulation which is around 95%.The total availability which is represented in Figure 11b system of Figure 2a is stable, having its maximum value (100%) until the second year and a half.Then it undergoes a decline to reach the value of 96-68%.Afterward it follows an increase in an exponential way until the end of simulation.For the total availability which is represented in Figure 10b system of Figure 4, he shape of the graph is the same as that in Figure (11b), with the only difference of a decline in the availability to the value of 99.29%.

Comparison of results
From the values obtained during simulation, we let us can leave there with the following comparison and results: Considering reliability of the three systems which are represented in Figs.(1a, 2a, and 4). the system of 4 is better reliable compared to two others.The system of Fig.

CONCLUSION
In this article, we developed two types of energy systems containing renewable resources (mono-source, multi sources).We began with the qualitative analysis (functional and dysfunctional, of waiting) of each system then calculated their performances (analyses quantitative) over one duration of time of 9.5 years and by precisely calculating the reliability and availability of each component, we measured reliability and total availability of these system containing the method which we developed.In end, we compared the results obtained to draw appropriate conclusions.

CONSENT FOR PUBLICATION
Not applicable.

CONFLICT OF INTEREST
The authors declare no conflict of interest, financial or otherwise.

Fig. (
Fig. (1b).DSPN of a Mono-source energy system with wind turbine.

Fig
Fig. (2b).DSPN of a Mono-source energy system with photovoltaic module.

Fig. ( 4 )
Fig. (4) describes a Multi-source Renewable Energy System to study.It consists of Photovoltaic module, Wind turbine, Inverter, AC/DC converter and a DC bus.In addition to one storage system of energies (a battery), all these components are connected to a Load.

Table 5 .
Interpretation of places of the model DSPN depicted in Fig. (5).

Table 6 .
Interpretation of transitions of the model DSPN depicted in Fig. (state towards the functional state of the photovoltaic T2 stochastic Return of photovoltaic of the functional state towards the wait state T3 stochastic Passage of the photovoltaic towards the breakdown state T4 stochastic Repair of photovoltaic T5 stochastic Passage of the wait state towards the functional state of the wind turbine T6 stochastic Return of the wind turbine of the functional state towards the wait state T7 stochastic Passage of the wind turbine towards the breakdown state T8 stochastic Repair of the wind turbine T9 stochastic Passage of the wait state towards the functional state of the AC/DC converter T10 stochastic Return of the AC/DC converter of the functional state towards the wait state T11 stochastic Passage of the AC/DC towards the state of breakdown DC bus of the state of vacuum towards the occupied state of DC bus T24 stochastic Return of the DC bus of the state occupied towards the vacuum state T25 deterministic Passage of load of the battery towards the stopping of the battery T26 stochastic Repair of the battery T27 stochastic Passage of the battery towards the breakdown state T28 stochastic Passage of the battery towards the load state T29 stochastic Return of the battery towards the wait state T30 stochastic Return of the battery towards the wait state T31 stochastic Passage of the battery towards the discharge state T32 stochastic Passage of the battery towards the breakdown state T33 stochastic Repair of the battery T34 immediat Return of the battery towards the discharge state T35 deterministic Passage of the battery of the state of discharge towards the stopped state T36 immediat Return towards the state of load of the battery T37 stochastic Passage of the value of the load (V) of Vmin towards Vmin<V<Vmax (Table 5) contd..... value of the load (V) of Vmin <V< Vmax towards Vmin T39 stochastic Passage of the value of the load (battery) (V) of Vmin <V<Vmax towards Vmax T40 stochastic Return of the value of the load (battery) (V) of Vmax towards Vmin <V<Vmax of the functional state towards the state of breakdown T45 stochastic Return of Inverter towards the wait state T46 stochastic Passage of Inverter towards the functional state T47 stochastic Passage of DC bus towards the breakdown state T48 stochastic Repair of the DC bus

Fig. (
Fig. (7a) shows that the reliability of DC bus and the AC/DC is 100% throughout simulation contrary to the two other components (wind turbine, inverter).Their reliability reduced because of the firing of the transitions representing the passage towards the state of breakdown to t = 11986, 7 h (1, 37 years) and t= 35232, 1 h (4, 02 years).

Fig. (
Fig. (7a).Reliability of each component for the Mono-source Energy System, with wind turbine only (system depicted in Fig. (1a)).

Fig. (
Fig. (8a).Total Reliability of the Mono-source Energy System with a wind turbine only (system depicted in Fig. (1a)).

Fig. (
Fig. (9a)shows the reliability of the DC bus, photovoltaic module and battery which is 100% throughout simulation because of the absence of firing of the transitions, modelling the passage towards the state of breakdown.On the other hand, for three other components, their reliability was stable at 100% until t = 16253h (1,86years) for wind turbine, T =19639, 4 h (2, 24 years) for the inverter, and t = 77470, 9h (8, 84 years) for AC/DC.It then reduced because of the firing of the transitions which indicate the passage towards the state of breakdown.
Fig. (11a).Total Availability of a Mono-source Energy System (with wind turbine only).
(2a) is better reliable than the system of Fig. (1a).Concerning the availability of the three systems which is represented in Figs.(1a), 2a and in 4).Broadly they have a good availability but the system of Fig. (2a) is a little better than the rest of the two .On the economic plan, the system of Fig. (2a) is the system least expensive compared to the system of Fig. (3).The complexity the system of Fig. (3) is high which makes it difficult to install, contrary to the system of Fig. (2a) which is smaller and simpler to install.

Table 1 . Interpretation of places of the model DSPN depicted in Fig. (1b).
P6Breakdown State of AC/DC converter P7 DC bus is free (vacuum) P8 DC bus is occupied P9 DC bus is in a breakdown state P10 Wait state of Inverter

Table 2 . Interpretation of transitions of the model DSPN depicted in Fig. (1b).
T11 stochastic Return of the DC bus of the state occupied towards the state of vacuum T12 stochastic Passage of DC bus of the state of vacuum towards the occupied state of DC Bus T13 stochastic Passage of Inverter towards the functional state T14 stochastic Return of Inverter towards the rest state T15 stochastic Passage of Inverter of functional state towards the breakdown state T16 stochastic Repair of Inverter T17 immediat Desactivation of the Load T18 immediat Activation of the Load