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21Journal of Siberian Federal University. Engineering & Technologies, 2017, 10(3), 414-425
УДК 621.311.245
Reliability Evaluation Method of the Wind-Diesel Installation with Application of Dynamic Fault Tree
Vladimir A. Tremyasov and Tatyana V. Krivenko*
Siberian Federal University 79 Svobodny, Krasnoyarsk, 660041, Russia
Received 18.05.2016, received in revised form 19.08.2016, accepted 27.12.2016
Renewable energy sources becoming more widespread in decentralized power supply systems. However their functioning depends on the natural resources potential having variable character. This dependence influences reliable work of autonomous power supply system. Reliability analysis of a renewable energy based autonomous generating system is the actual practical task requiring the solution. For the reliability assessment of autonomous wind-diesel installation it is offered to use the method of dynamic fault tree with Markov models for representation of dynamic operators. The offered method is appliedfor reliability calculation of the wind-diesel installation functioning in the autonomous power supply system in the north of Krasnoyarsk region.
Keywords: renewable energy source, reliability, wind-diesel installation, dynamic fault tree, dynamic operators.
Citation: Tremyasov V.A., Krivenko T.V. Reliability evaluation method of the wind-diesel installation with application of dynamic fault tree, J. Sib. Fed. Univ. Eng. technol., 2017, 10(3), 414-425. DOI: 10.17516/1999-494X-2017-10-3-414-425.
Метод оценки надежности ветродизельной установки с применением динамического дерева отказов
В.А. Тремясов, Т.В. Кривенко
Сибирский федеральный университет Россия, 660041, Красноярск, пр. Свободный, 79
Возобновляемые источники энергии становятся все более распространенными в децентрализованных системах электроснабжения. Однако их функционирование зависит от потенциала природных ресурсов, имеющих переменный характер. Эта зависимость влияет на надежную работу автономной системы электроснабжения. Анализ надежности автономных систем генерации на основе возобновляемых источников энергии является актуальной практической задачей, требующей своего решения. Для оценки надежности
© Siberian Federal University. All rights reserved
Corresponding author E-mail address: emf_tva@mail.ru, tanya-1991mir@mail.ru
автономной ветродизельной установки предлагается метод динамического дерева отказов с использованием марковских моделей для представления динамических операторов. Предложенный метод применен для расчета надежности ветродизельной установки, функционирующей в системе автономного электроснабжения на севере Красноярского края.
Ключевые слова: возобновляемые источники энергии, надежность, ветродизельнаяустановка, динамическое дерево отказов, динамические операторы.
The power sources on the basis of renewable energy sources, such as the sun and wind, gain ground in the world to supply decentralized consumers with electricity. However, the work of renewable energy sources (RES) completely depends on the natural resources potential which possess variable character [1]. Therefore it is necessary to apply RES together with traditional power sources for autonomous power supply of decentralized areas. Usually diesel generators (DG) and storage batteries (SB) are used together with RES. These power sources allow smoothing possible fluctuations of RES power.
Instability of RES electricity production, because of stochastic character of the natural energy resources, influences reliability of autonomous electrical power system. Therefore the reliability analysis of generation systems on the basis of RES is an important and actual practical task requiring solution.
The purpose of this work is the development of the dynamic fault tree method for the reliability analysis of autonomous wind-diesel installation (WDI).
Functioning reliability of the power complex containing RES is defined by reliability indicators of separate elements [2].
The work of wind turbine generator (WTG) is defined by the mode of arriving wind flow and depends on its power characteristics. The main power characteristics of the wind flow are wind speed and annual course of average monthly wind speeds [3].
One of the widespread methods of modeling and reliability prediction of the complex system at the stage of its design is the fault tree (FT) analysis. The FT analysis represents the systematic analysis of events which can cause a system fault, including faults of subsystems and elements, which are the primary causes of system faults [4].
The FT method consists in creation of structural diagram (a fault tree) representing a graphic display of cause-and-effect relationships causing certain types of faults. Event symbols and logical operators are the elements subdividing and connecting a large number of events and conditions, necessary for the fault tree creation. Events and conditions are formulated by specialists according to the system engineering design by means of analysis of its behavior (at emergence of different faults and modes) and the records in the form of conjunctions (0) and disjunctions (U).
In most cases the purpose of the FT analysis consists in defining probability of a final event. As the final event is the system fault, such analysis gives the probability value of the system fault. The FT structure allows developing algorithms by means of which it is possible to calculate probability of the system fault [5].
For quantitative assessment of the system reliability indicators by means of the FT analysis, the following methods are applied: method of the minimum sections of faults (MSF), methods of mathematical logic, method based on the fault function (FF) [5, 6]. Basic data in the FT method are failure rate I and repair rate ^ of the system elements.
The quantitative FT assessment allows receiving reliability indicators for the system in general. Usually such indicators are: system forced outage rate q(s) and stream parameter of fault a>(s).
The FT analysis promotes predicting of potential faults and increasing of system reliability on the design stage. Thus, the FT method is a powerful instrument of the reliability analysis of the RES generation system.
In the [7] reliability analysis of autonomous wind-diesel system, the FT method on the basis of MSF allows considering wind turbine fault because of the weather conditions. In this work the classical static fault trees with logical operators AND, OR were used. However the traditional static fault trees cannot capture the behavior of components of complex systems and their interactions such as sequence-dependent of events, spares and dynamic redundancy management, and priorities of failure events.
In order to overcome this difficulty, the concept of dynamic fault tree (DFT) is introduced by adding sequential notion to the traditional fault tree approach. Modeling of dynamic reliability allows overcoming some problems which can arise in the classical fault trees.
DFT is a further development of the traditional FT method by integration of dynamic operators in the fault trees [8]. DFT allows eliminating defects and restrictions of the classical fault trees due to integration of new logical-dynamic operators considering sequence-dependence of events, timing relationships and priorities.
The DFT introduces four basic (dynamic) operators: the priority AND (PAND), the sequence enforcing (SEQ), the spare (SPARE), and the functional dependency (FDEP) [1].
The graphic designation and description of the dynamic operators are presented in Table 1.
Integration of dynamic operators in FT allows considering many features of the elements faults in the reliability model leading to emergence of undesirable events, and also technical and organizational measures for reliability ensuring.
Reliability modeling by means of DFT has drawn attention of many engineers working in the field of reliability and safety of complex systems [8].
However, DFT method has some difficulties with respect to the submission of the above dynamic statement.
References [9-13] proposed methods to solve DFT. In reference [9] shown, through a process known as modularization, that it is possible to identify the independent sub-trees with dynamic operators and to use different a Markov model for each of them. However, the solution of a Markov model is much more time and memory consuming than the solution of a standard fault tree model. As the number of components increases in the system, the number of states and transition rates grows exponentially. Development of a state transition diagram can become very cumber some and a mathematical solution may be infeasible. To reduce state space and minimize the computational time, an improved decomposition scheme where the dynamic sub-tree can be further modularized (if there exist some independent sub-trees in it) is proposed by in [10]. In reference [11] proposed a numerical integration technique for solving dynamic operators. Although this method solves the state-space problem, it cannot be easily applied for repairable systems. In reference [12] proposed a Bayesian network-based method to further reduce the problem of solving DTFs with state-space approach.
In order to overcome limitations of the above-mentioned methods, a Monte-Carlo simulation approach has been attempted in [13] to implement dynamic operators. The Monte-Carlo simulation
Table 1. Dynamic operators
Desi grationiE DFT Operanor Deseniadion
* PAND Thep riorityAND inoEsi.aes sequence oyoeenta)
¡1 i nnspd The sequevce eaforeiyg (considers sequence-dependenceofevents)
1 -rfT SPAA Tho spare tcens-eers -f-ha ethte spare)
1 W-n-T FDEP The functional dependency (sTnsiderst-euence of evnvts Dnd timinn relafianyh-ps)
0,4 kV
Fig. 1. Autonomous wind-diesel installation: W- wind turbine generator; Gj - working diesel generator; G2 - reserve diesel generator; I - inverter; A - storage battery; X, Z, Yj, Y2 - circuit breaker; Pn - load
based reliability approach,due toitsinherent capability in simulating the actud process and random behavior of the system, can eliminate uncertainty in reliability modeling.
In the autonomous power systems with RES a small number of basic elements is usually used. Therefore for representation of dynamic operators it is possible to apply Markov models.
As an example of the DFT method application for the reliability analysis of autonomous power system, we will consider the scheme of wind-diesel installation (WDI) (Fig. 1), functioning in the northern regions of Krasnoyarsk region taking into account the wind characteristics [3].
WDI includes basic components: the 225 kW wind power unit, the DG system (two diesel generators, 125 kW each) and storage battery with accumulated energy of 100 kWh.
The wind-diesel complex includes: two diesel generators G1 and G2, the working one and the reserve one. If the working DG G1 has a fault, then the reserve DG G2 is automatically started, and the power supply is recovered. It is often supposed that the reserve unit is ready to work every time there is a need of it. For the model to be true to life, we will make an assumption that the reserve unit can sometimes be not ready for work. So, the fault of DG G2 can happen in attempt of turning it on because of the fault of DG G1, and also during inaction of G2. Transition of the reserve generator to the
Fault of all WTG-DG-SB system
Fig. 2. Dynamicfaulttree forreliabilityanalysisof autonomous wind-dieselinstallation
fault condition will have the fault density equa^^wherea = 0,1. After repairofDG itisplaced in operation. Reservation of DG increases reliability of the autonomous power system.
Use of storage battery as a part of the power complex is necessary for smoothing of possible fluctuations of WTG power connected with variable character of wind speed.
Let's consider fault events of WDI elements, represented in the Fig. 1. The DFT final event is the "Fault of all WTG-DG-SB system". The corresponding DFT is given in the Fig. 2.
In the reliability analysis of the generation systems, containing WTG, it is necessary to consider wind conditions influencing their functioning. Creation of DFT for WDI considers the design wind speed vmin < v < vmax at which WTG generates power from zero to rated, and the off-design wind speed vmax < v; v < vmin at which WTG does not generate electric power.
When modeling of the design wind speed in DFT the dynamic operators PAND and SPARE are used. The dynamic operator PAND (events at the exit A, B, C) models faults of switches in operation at damage of the corresponding accessions which will lead to the blackout of 0,4 kV collecting bar.
The operator SPARE (event at the exit E) is applied to modeling of DG system fault. So in case of damage of WTG the working DG can fault, and the attempt to start the reserve DG will not be successful.
Influence of the off-design wind speed on the wind turbine functioning in DFT (Fig. 2) is shown by means of a logical sign "prohibition" (hexagon). Application in DFT of the logical sign "prohibition" allows considering event which happens with some certain probability [5]. In the Fig. 2 the entrance event V, placed under the sign "prohibition", is the simple WTG because of wind conditions (lack of wind, a hurricane, etc.). The conditional event Q, located sideways from the logical sign, is a restrictive condition which represents a probabilistic weight factor. The restrictive condition is characterized by the value of conditional probability of the off-design wind speed q. The value of the off-design wind speed probability is defined with the help of histogram of wind speed distribution in WDI location. The histogram of wind speed distribution characterizes repeatability of wind speeds for the studied period [3]. It shows how long the wind had a certain speed during the considered period.
Also when modeling of the off-design wind speed in DFT, the dynamic operator SEQ (event at the exit F) is used, which models faults of other system power sources (DG, SB) in the well-defined ordering. There can be no other sequence of faults of these components.
Thus, in DFT of the WTG-DG-SB system three dynamic operators (PAND, SEQ, SPARE) are used, the representation of which can be executed by means of Markov models [8]. It is necessary to construct a Markov state diagram for each dynamic operator.
Figures 3-5 show the state space diagrams for the DFT dynamic operators described above. Failure and recovery densities are used as the parameters of these models. The shaded states in the diagrams show the system failure. In each state the figures 1 or 0 indicate the work or the fault of the system elements respectively.
On the basis of the constructed diagrams it is possible to make the system of differential equations, the solution of which allows receiving necessary indicators of reliability: probabilities of k conditions of Pk(t) installation and KG availability factor. In our case from the received system of differential equations for each state space diagram it is necessary to define KG availability factor for its further use in the DFT analysis.
For calculation of reliability assessment at the initial stage, with rather short time interval, it is necessary to use a non-stationary availability factor. However in most cases it is enough to define a
c)
Fig. 3. Markov models for dynamic operators a) PAND(A>', b) PAND(B>',c) PAND(C>\ 1 - down state of systems element, 0 - upitatt; A s aailure rate; (n - repaér rate
Fig. 4. Markov modelfor dynamic operator OPiat®: aP-failure rate for reservediesel
A
Fig. 5. Markov model for dynamic operator SEO'F>
stationary availability factor - probability that the recovered objectwill be operable in the randomly chosen timepoint during a steady-state process (t
For each condition k it is possible to write downthefollowingdifferential equation
where r e A means thne cummin. is coeducted under all sucli stater i whicltbeleng ta a E(k) - is a set of stdtes from whcch tlir; dcrnct transttéon to come state k is postdble; efitl t- a set of stntea Cn which tha direct treooitiod feom thiss ttstr k isr poecèble; ob- r- proOnbiïity 0s" the tyrtem st^y io u state in t ttaneijoiiii^t;; Ay - dntensitid oi Iransitéo- footio (fie atfte t So ahe r^^^te t.
ir ilts attte gtanh eoadoins n ihéiferitrtt atates, tloitn o deffeoenp deiferonaial oeudtion; can be made as a cetuli. lor definition rf tht avoiïsbitiiy inotor tt is necesrary to writedown n equations and one additional equationof theform
According to(l)and (2)wewill recerve the system of differential equations for definition of the availability factor of the dynamic operator PAND (A)
P'e(d = - We(OlS^, + I>ePi U) ,
(1)
iee(e) ieE(e)
n
i>,=i.
(2)
(3)
To solvo the differential equations system and to find the requiredreliability indieator,a computer pro gram i n theMathcadl5 envronmenl was used.
As ;3i re suit of si olution of ihe equation- oy^t;^mt we re c oive values of states probabilities for the dynamic operaUor talA—Gn nnd the stationfry ovailabililg lactor usi ng a to rmula
Kea«=±pt. (4)
1=1
Avfilabilily factor for this dy iri^iE^malc o^uahas is <F) = 0>d9 •
In the eame wny lleS's end the avaHriliho foDtor foe other DFT 0y oamio operators: KtJ47®1" b) = 0,99?;
K=m®(q I9 ,3^ 9 ^ = 0,9^ K*e=• = ,, 9 8t
Knownnf ttetn acallaf ihty factor olc thr Oynamic KWj, oii ef he ^icl define aforcedoubagf
rate b, tlie equation
sl - IK (5f
Thun, aqe forced rater forDFT dynamic opfrctars oee res^ctively enual: qPANDlA) = 0,01;
^(EO = r(i; qooNhc> q^E© ho,or; d^ = 0,02.
Tht faeced o^0;^itoe rate of dynamic operators q™ is an indiurtnr of an output event of the correspondingoperatoranditcanbeusedforthefurtherDFTanalysis.
aho MSl1 l^l nacj^tec^i0 co apphrd for ouantieative enskcsment of DaT gtnen oaove. This method aUowe findme suel sysertn oaeabilhy inerteaOoso ¡as ihe system forced outage rate q(s) and the stream puoameter af fautt «»(is),
As i^ia^^ of DFT analysis, we -wiLll recerve MSF: A, B, C, WEC, WEI, FV1 Aa it was stotad above, t8e output event of the dynamic operator IA,B,C,E,F) characterized by the receiaed forced oufrga eere q^e Ior the conrespanding ooeiotor. 1he foreed outqge rate foee element rODFT ps equal
^Trf d P ] • (6)
T + ^o
wtereO,, - failure rote of, syeoem efeme nt, 1/yenfi op - repalt t^ci^ts otj system element, 1/year. Theutrenm paoame^e:r<ef faullt Potj elemenr ip defermined by a formula
= l(l --qj. (7)
The probability olf efistence of - MSF is; determlnedbyformulas: ^t abs encd ioi /'MSF of she Id dynemic operator
r^nii (8)
c=i
at tnolue iOd hoi MSF of the h dyeamic operaton
m n
n q... (9)
k=1 j=1
Thestreamparameteroffaultfor i MSF isdetermined byformulas:
- 4ai -
ait absenc e in i MSF of th<s k dynamic operator
*
a
v=l
atinclusion in i MSF of the k dynamic operator
*D O * DO
a,
Ot £0^. VH)
0=0
Let's determme tlv^ itream rarameter of 0aujt for the output events A, B, C c^flli^ dynamic operator PblM) ky f-pmulas
a> PAND(A) = a (l - qw )QX o.e.;
„^(^(-q^^q (12)
where ßov. - a conditional probabilityof switches fault iX,7JPtT) on short circuit cloaring in connection, Qxo.j. =Qn o.j. mf . =0,5.1(r3+0;2-lO"3 .
"Tili; hiream fjaaamvler of lraaillt foo ahe output caents E ov the Cynnmic operator SPARE by tormytas
a
VAWjy^ (S_qm )qoG2 (13)
woeae QG2 0z - a cyndiüynnl probaaüiay ov Ga frit at its; ib Itbiset^ <a°j2 oZ= = • l^-2 ± 00 • 1(T2.
With the off-design wmd tperd, it os ni^ttistseiave'" to define the WTG ifcjercDstd oulage oate in DFd( because oC uhe weather conditions by equaaion
- =--=+(— I1" ^^ ^'l (14)
Aqy + fjy
wheae >2,^0= a^/^re, - vmergecce dnesity of hhe off-desigo ^Iiid speed( 1/yrar; o-e = yTN - recovery density of the design wind speed, 1/year; T0 and TN - the weather periods with off-design and design wind speed nesp e ctively[3].
The srream parameter of fault of theWTG connected with wind speed is determined lby a foimula
cov = Xy (l - qv). (15)
As the entrance event V of the logical sign "prohibition" in FT at the exit gives an event which happennwith some certain prababühy, itisnecessany tomcreashmdicatorh f„ and mv by the conditional probability q.
TOen tWe fauOtssobibility of WTG tiathnseohthe weather conditions
qr =rv ' r (16)
T he expesSed nhmberotWTG faults in a unit of time
a* =av ■ r (17)
Ldt's detemmine the probpbihty of existence of MSF FF with the oif-design wind speed by a formula
q * _ '„SEQiF) (io\
<ano - • U8)
Let's determinethefailure rateforMSF FV by aformula
* * SEO (F) SEO( F) *
afk = q* < O=F>+q as(19)
w here <aSE^<F> - tlie stream para meter of fault for the output event F of the dynamic operator SEQ which is determined by a formula
®SEe(F> = <ZGI<1G2<1A + <<+G2<Gl<A + <<+A <G1<G 2- (20)
Ontte baeie fh tht fornodorghdenobove, it is possible to determsneWDI eeliabdityindkraiers in bg eqyatianu
Ns A * * < * < i < * < *
q^^K-1* + qODO)+q*Fv'; *+ d*do)+<SFV, (21)
1=1 1=1
where Ns -MSFnumberinWDIwith thedesignwindspeed.
The reliability indicators for elements WDI are presented in Table 2 [14, 15]. The necessary meteorological data for accounting of influence of the wind conditions on the work of WDI in DFT and the designvalues ofWTGreliability indicatorsareprovidedin theTable3.
Thd results of the asse ssment oe WTG reliability foe powet supply of the northern regions of Krasnoyarsk region aee given in theTable 4.
Table2.Reliabilitn indicators fortheWDIelements
Code elements Elem ent° WDI ReSiialrility indiratars
X, 1/year j 1/year a, 1/year
W Wind turbine geneeator Vestas V20 COt kW b,l 3,05902 3,6H0-3 1,09
Gi Working dietel grneratar (2±0,5)-10-2 1,0102 1,9710-4 1,9910-2
g2 Reserve dieee1 generator (2±o,5>ia-3 C,bl02 1,9790-5 i 0)>9b 0-3
I Inve rier (5=±i)-iab llO3 0,9-500"5 4,9910-2
A Storage battery (2±1)10-3 1103 1,9910-6 1,9910-3
X, Yi, Y2,Z Circ uitbreakee ((±0,9>Cb-3 9103 1 ,9990-7 09)99 0-3
Table 3. Meteorologicaldata received for the northern regions of Krasnoyarsk region, and thecharacieristicof WTG faults
q T ± o TN XV, 1/year Pv, t /year qr r* a v, 1/yeai ® *, 1/year
0,05 0,42 0,58 2,38 1,72 0,57 0,0285 1,023 0,051
TabletyResults ofWDI reliability analysis
MSF «eO 1/y ear q(s) ty(s), 1/year
А 0,01 5,d 84- 10-4
B 0,01 п^аю-6
С 0,01 4,4810-5 3,0610-6 1,D0410-3
WEA 7,1d3eo-ls 9,34 10-8
WEI 3,573-18-10 4,ev 10-6
FV 5,7- 1П-4 1Д210-3
Conclusions
1. The method based on dynamic fault tree for the reliability assessment of wind-diesel installationproposed.
2. The metliod allows consider failures oo a wind-diesel installation caused by weather conditions in its model. Sequencisdependence of events and priorities of failure; events can be tsken into account due to applicatkm of dynamical ope rators.
3. Reliability of wind-diesel installation is determined by its: forced outage rate q(s) and stream parameter of fault m(s), this two indicators can be used to choose optimal structure of WDI.
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