International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Reliability-based calibration of partial safety factors for wave energy converters Ambühl, Simon; Kramer, Morten; Sørensen, John Dalsgaard Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Reliability-based Calibration of Partial Safety Factors for Wave Energy Converters Simon Ambühl PhD student, Dept. of Civil Engineering, Aalborg University, Aalborg, Denmark Morten Kramer Associate Professor, Dept. of Civil Engineering, Aalborg University, Aalborg, Denmark John Dalsgaard Sørensen Professor, Dept. of Civil Engineering, Aalborg University, Aalborg, Denmark ABSTRACT: Wave energy converters (WECs), which harvest energy from the waves and transfer them to electricity, are a new technology, where structural standards need to be developed. An important step towards standardization is the calibration of partial safety factors. A methodology for calibration of partial safety factors for design of welded details for wave energy converter applications is presented in this paper using probabilistic methods. The paper presents an example with focus on the Wavestar device. SN curves and Rainflow counting are used to model fatigue without considering inspections. The influence of inspections is modelled using a fracture mechanics approach, which is calibrated by the SN curve approach. Furthermore, the paper assesses the influence of the inspection quality. The results show that with multiple inspections during the lifetime of the device and by applying a good inspection quality, the safety factor can be significantly reduced.  1. INTRODUCTION Wave energy converters (WECs) harvest kinetic and potential energy from the waves and transfer this to electricity. The idea of using wave energy as a renewable power source goes back to the nineteen seventies when oil crises increased the price of petroleum. Nowadays, many different types of WECs exist and some of them have reached the prototype level. The main reason for the slow development process are, on one hand, the high LCOE (levelized cost of energy) compared with other renewable electricity sources like solar or wind and, on the other hand, the large variety of working principles, which, additionally, make the overall technology development costs large.  For offshore applications, bolted and welded structural elements, which interconnect two structural components, are critical due to corrosion and cyclic loading. For WECs where there are many moving parts and the overall structure is exposed to cyclic loading due to wind and waves, and therefore fatigue failure is of importance. Calibration of partial safety factors can be performed using probabilistic methods where aleatory and epistemic uncertainties are accounted for. The partial safety factors are then used in structural standards to account for these uncertainties. So far, no structural standards for WEC applications exist and the safety factors cannot be directly overtaken from nearby industries like the oil and gas industry or offshore wind turbines since the required reliability levels might not be the same and different control systems which define the amount of harvested energy as well as the loads on the structure are applied. Furthermore, the wind can be assumed to be the dominating load for wind turbines whereas the wave load is assumed to be dominating for WECs. Therefore, partial safety factors for WEC 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 structures need to be calibrated using load characteristics from WECs.  In this paper, partial safety factors for welded details of an existing WEC application are calibrated using the Wavestar technology where a prototype exists at the North Sea coast of Denmark. 2. PROBABILISTIC RELIABILITY ASSESSMENT Probabilistic reliability methods make it possible to include uncertainties related to the limited available data, uncertainties in measurements, physical uncertainties given by Mother Nature (like e.g. inter-annual variation of extreme environmental conditions) as well as modeling uncertainties due to approximations and simplifications as stochastic variables. Considering these different types of uncertainties, the probability of failure, FP , of a certain failure mode and a certain detail can be assessed. Instead of FP , the so-called reliability index,  , is often used:  1 ( )FP X     (1) where X  represents the considered stochastic variables and   the standardized normal distribution. Both the probability of failure as well as the reliability index can be estimated from FORM/SORM (First/Second Order Reliability Methods) or simulation techniques (see e.g. Madsen et al. (2006) and Lemaire et al. (2009)). As a reference time frame, one year is often considered for reliability consideration. Annual probabilities of failures, FP , at time t given survival up to time t as well as annual reliability indices,  , can be determined from the cumulative values FP  and  , respectively:  ( ) ( )( ) (1 ( ))F FFFP t P t tP t P t t      (2) where t t  , t   1 year and ( )FP t  is equal to the failure probability during the time interval  0, t . When a fatigue critical detail is considered, the annual failure probability of total structural collapse ,F COLP is estimated by: , |F COL COL FAT FP P P   (3) where FP  is the annual probability of a certain structural detail and |COL FATP  is equal to the probability of structural collapse of the whole device given fatigue failure of the detail. The probability |COL FATP  accounts for the importance of the considered detail. 3. ACCEPTABLE RELIABILITY LEVELS Acceptable structural reliability levels are dependent on the consequences in case of failure. Consequences may be formulated in monetary units but also in danger of human lives or size of pollution. Table 1 shows the target annual reliability level dependent on the relative measure of safety measures and the consequences in case of failure.  Table 1: Target annual reliability index, Δβ, for structural collapse dependent on the consequences in case of failure and the relative costs of safety measures. Data taken from JCSS (2001). Relative costs of safety measures Consequences of failure Minor Medium Large Large 3.1 3.3 3.7 Normal 3.7 4.2 4.4 Small 4.2 4.4 4.7 For WECs, in case of structural failure, danger of human lives can be assumed to be negligible due to the fact that they are unmanned during operation. Additionally, there is no danger of large pollution in case of failure of WEC devices. The relative cost of safety measured is considered to be Normal to Large. Therefore, the target annual target reliability levels of structural components for WEC applications could considered acceptable if between 3.1 and 3.7, 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 which corresponds to a target annual probability of failure equal to 10-4 and 10-3, respectively. 4. FATIGUE MODELING Fatigue is a time-dependent process driven by cyclic loading, which is important in WEC structures. Two different fatigue models are used in this paper and presented in the following. The SN approach is based on an empirical relationship between a certain stress cycle and the number of cycles leading to failure. The other fatigue model is based on fracture mechanics where crack evolution – the physical reason for fatigue failure – is modelled. As a safety factor, the Fatigue Design Factor  FDF  is considered in this paper. It is the ratio between the design lifetime in years (TFAT) and the expected lifetime of the device (TL): FATLTFDF T   (4) When using a linear SN curve, the FDF  value is directly related to the partial safety factors for fatigue, f , and fatigue strength, m :  mf mFDF      (5) where m  is equal to the slope of the linear SN curve. 4.1. SN Approach This approach is commonly used for design of offshore steel structures. For the SN approach, a linear SN curve is considered. The basic relation between the number of cycles N leading to failure with a certain stress amplitude S  is equal to: mN K S     (6) where K  and m are curve parameters available in standards (see e.g. DNV-RP-C203 (2012)). It is assumed that the stress range /Q z   can be obtained on basis of the load effect Q  (e.g. Normal force) and a design parameter z  (e.g. cross-section area). Furthermore, it is assumed that the total number of stress ranges for a given fatigue critical detail can be grouped into intervals of stress amplitudes such that the number of stress ranges in group i  is in  per year. The number of cycles in  for a given load effect range iQ  is obtained by Rainflow counting, see ASTM Standard E 1049-1085 (1995). The code-based design equation for obtaining the design parameter z, using the Palmgren-Miner rule and a linear SN curve, is written as: , 0,1 ( , ) 0FAT ijk mijk S i ji j kT nG s P H TK    (7) where , 0,( , )S i jP H T is the probability of occurrence of a certain wave state given by the significant wave height ,S iH  and the mean zero-crossing wave period 0, jT , /ijk ijks Q z   the stress range of bin k given ,S iH  and 0, jT . In order to estimate the probability of fatigue failure given a certain design value z , a limit state equation, which is similar to the design equation (see Eq. 7), is considered: , 0,( ) ( , ) 0ijk mijk S i ji j ktng t s P H TK    (8) where t  indicates the time ( 0 Lt T  ), which is equal or smaller than the expected lifetime LT ,   is a stochastic variable representing the uncertainty related to the Palmgren-Miner rule for linear damage accumulation, /ijk SCF M ijks X X Q z  represents the stress range of a wave state given by the significant wave height ,S iH , the mean zero-crossing wave period 0, jT  and a certain stress bin k. The model uncertainty MX models the uncertainties related with the scatter diagram as well as the estimation of wave loads given a certain wave state. The stochastic variable SCFX models the uncertainty related with stress concentrations and the stress estimation given a certain wave load. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 4.2. Fracture Mechanics Approach The SN approach does not enable modeling of the impact of inspections/repairs on fatigue failures. Therefore, a more detailed fatigue model is needed. A fracture mechanics approach can model the crack evolution caused by load cycles and makes it possible to assess the impact of inspections. The basis for crack evolution modeling is the Paris Law, Paris and Erdogan (1963):  ( ) mda C K adN      (9) where a  is the crack size, N  the number of cycles, /da dN  the crack growth rate, K  the stress intensity factor and the parameters C  and m  are parameters to be found from experiments or by comparing the SN and fracture mechanics approach. For a one-dimensional fracture mechanics model, the stress intensity factor can be calculated from the equivalent stress range,eS , and the geometry function Y , which is dimensionless and accounts for different crack forms: ( ) e SCF MK a Y S aX X     (10) where SCFX  represents uncertainties related to the stress concentration and MX  is related to the uncertainties in load modeling. The limit state equation for a fatigue consideration based on fracture mechanics is based on a time-dependent view onto the fatigue crack ( )a t : ( ) ( )cg t a a t     (11) where ca  represents the critical crack depth leading to failure (e.g. thickness of considered plate). Inspections can be implemented by the use of so-called Probability of Detection (POD) curves, which depend on the considered inspection technique as well as the crack size. This curve represents the probability that a certain crack size is detected during an inspection.  When considering visual inspection the POD  curve can be formulated dependent on the smallest detectable crack length  : ( ) 1 exp cPOD c        (12) where c  is the crack length on the material surface. The crack length c  can be approximated from the crack depth a . In general, the models have the following function: 2a afc T       (13) where /a T  is the relative crack depth. Different functions for the crack depth-length ratio (a/2c) dependent on the geometry, and the loading conditions can be found in BS 7910 (2005). In this paper, a constant ratio between a  and c is assumed: 52ac    (14) 5. EXAMPLE – WAVESTAR DEVICE In this example the Wavestar concept is considered. This technology uses point absorbers (floaters), which are moved when a wave is passing. The moving floater drives a hydraulic piston, which impels a turbine and generator. A first prototype exists near Hanstholm (DK) at the North Sea coast of Denmark. Figure 1 shows a picture of the prototype.  Fatigue failure is important for this kind of device due to the fact that during storm conditions the floaters can be moved out of the water and limit extreme loads onto the structure. The next step in the technology development process of the Wavestar device is a device (called C6) consisting of 20 floaters with a floater diameter of 6 meters. The floaters are connected to a platform, which contains all necessary mechanical and electrical devices to transfer hydraulic pressure fluctuations to electricity. This Wavestar prototype device 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 consists of two piles, which connect the platform and floaters to the ground. Figure 2 shows a sketch of the planned C6 Wavestar including numeration of the floaters.  Figure 1: Wavestar prototype at Hanstholm (DK) with two floaters (one in ‘storm protection mode’ and one in ‘production mode’).  Figure 2: Top view of C6 Wavestar device with 20 floaters (numeration included). The considered location has a water depth of roughly 30 meters. Table 2 and Figure 3 respectively show the considered wave-states as well as the probabilities of occurrence of different directions. Wave-state dependent incoming wave direction distributions are not considered. The floaters are out of the water when HS is smaller than 0.5 m because the power production is too small at these wave states. Furthermore, the floaters are taken out of the water when HS is larger than 3.5 m to protect the structure from large wave loads. For this device, a new Power Take Off (PTO) system, which is called Discrete Displacement Cylinder (DDC) and described in Hansen et al. (2013), is developed. This new PTO may lead to different load characteristics compared with the prototype in Hanstholm and needs further investigation. Furthermore, experience at the first prototype in Hanstholm showed that the welded structures at the gyro bearing, which connects the hydraulic cylinder and the platform, showed unexpected fatigue cracks. A calibration of safety factors for this detail is performed here. Figure 4 shows a picture and the location of the critical welded detail. Table 2: Scatter diagram with the probabilities of occurrence of the different wave states. Grey: storm protection mode (no electricity production).  Mean wave period T0,2 (s) 3.0 - 4.0 4.0 - 5.0  5.0 - 6.0 Significant wave height Hm0 (m) < 0.5 10.7 3.1 - 0.5 - 1.0 19.4 12.2 - 1.0 - 1.5 6.9 17.2 - 1.5 - 2.0 - 11.1 5.9 2.0 - 2.5 - 3.2 4.1 2.5 - 3.0 - - 3.4 3.0 - 3.5 - - 2.7   Figure 3: Considered probabilities of occurrences of different incoming wave directions.   Figure 4: Considered welded detail, where fatigue fractures at the prototype in Hanstholm occurred.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6 5.1. Stochastic Model Table 3 and Table 4 respectively show the stochastic models for the SN approach and the fracture mechanics approach. The linear SN curve proposed by DNV-RP-C203 (2012) (‘F’-detail, free corrosion) is considered here. Table 3: Stochastic model for SN approach. LN: LogNormal; N: Normal; D: Deterministic; Var.: Variable; Exp.: Expected; Dist.: Distribution; Std. Standard; Char.: Characteristic. Var. Dist. Exp. value Std. value Char. value Source   LN 1 0.3 1 JCSS (2001) SCFX LN 1 0.0/0.05/0.1/0.15/0.2 1 JCSS (2001) MX LN 1 0.1 1  log( )K  N 11.778 0.2 11.378 DNV-RP-C203 (2010) m  D 3 - 3 DNV-RP-C203 (2010)  Table 4: Stochastic model for fracture mechanics approach. LN: LogNormal; N: Normal; D: Deterministic; Var.: Variable; Exp.: Expected; Dist.: Distribution; Std. Standard. Var. Dist. Exp. value Std. value Source 0a  LN 0.2mm 0.132mm JCSS (2001) ln( )C  N C 0.77 Sørensen and Ersdal (2008) m  D m   0a D T   Y  D 1 0.1 Sørensen and Ersdal (2008) T  D 25mm   5.2. Results When designing a structural detail which is implemented several times on a device, the design sensitivity due to different loads as well as reliability levels given a certain design need to be assessed. The results shown in Table 5 indicate the change in design when focusing on the loads at different floaters as presented in Figure 2. For a certain floater, the same rainflow/load spectrum for the deterministic design as well as the reliability assessment is considered. It is assumed that failure of the welded structure leads to complete collapse of the structure (| 1COL FATP ). In general it can be said that the load influences due to array effects are small due to the fact that the design parameter z differs 3.5% or less for the different floaters.  Table 5: Fatigue Design Factors (FDF) and resulting partial safety factorsf m   for two different annual target reliability levels Δβ. The design parameter z is normalized by z0 (=design parameter of floater 20). COVSCF=0.1 Floater No. 3.1   3.7   FDF  f m  0/z z FDF f m  0/z z  1 2.41 1.34 0.988 4.07 1.60 0.989 2 0.982 0.982 3 0.979 0.979 4 0.977 0.977 5 0.975 0.975 6 0.978 0.978 7 0.983 0.983 8 0.985 0.985 9 0.986 0.986 10 0.987 0.988 11 0.988 0.988 12 0.968 0.968 13 0.969 0.969 14 0.974 0.974 15 0.977 0.977 16 0.980 0.980 17 0.984 0.984 18 0.985 0.985 19 0.988 0.987 20 1.000 1.000 Table 6 shows the results for FDF and partial safety factors with | 1COL FATP   when using the same design for all floaters. This table indicates the sensitivity of the FDF and partial safety factor values given the deterministic design used for floater number 20. The partial safety factor varies between 1.30 and 1.34 for a target annual reliability index of 3.1 and varies between 1.54 and 1.60 for requesting a target annual reliability index of 3.7.   12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7 Table 6: Fatigue Design Factors (FDF) and resulting partial safety factorsf m   for two different annual target reliability levels Δβ when taking for all floaters the same design parameter, but different rainflow spectrums. The design parameter z is normalized by z0 (=design parameter of floater 20). COVSCF=0.1. Floater No. 3.1   3.7   FDF f m  0/z z FDF f m  0/z z  1 2.33 1.33 1.000 3.90 1.57 1.000 2 2.27 1.31 3.82 1.56 3 2.25 1.31 3.79 1.56 4 2.25 1.31 3.74 1.55 5 2.22 1.30 3.76 1.55 6 2.26 1.31 3.80 1.56 7 2.30 1.32 3.82 1.56 8 2.30 1.32 3.86 1.57 9 2.29 1.32 3.85 1.57 10 2.33 1.33 3.91 1.58 11 2.33 1.33 3.90 1.57 12 2.18 1.30 3.63 1.54 13 2.21 1.30 3.70 1.55 14 2.23 1.31 3.72 1.55 15 2.25 1.31 3.74 1.55 16 2.27 1.31 3.78 1.56 17 2.31 1.32 3.87 1.57 18 2.3 1.32 3.86 1.57 19 2.32 1.32 3.91 1.58 20 2.41 1.34 4.07 1.60 The necessary partial safety factor depends among others on the level of uncertainty related with the stress calculations. Simple stress estimation methods (e.g. parametric equations) will have larger uncertainties related with stress concentrations. However, a FEM analysis will increase the needed time for stress calculations and the complexity, but it will decrease the uncertainties related with stress concentrations. Table 7 shows the required FDF and partial safety factors for minimal annual reliability indices of 3.1 and 3.7 dependent on the uncertainty about stress concentrations. The required partial safety factors increase from 1.28 to 1.49 for 3.1   and from 1.49 to 1.86 for 3.7  when increasing COVSCF from 0.00 to 0.20. Table 7: FDF and partial safety factor values dependent on the SCF (stress concentration factor) uncertainty for floater 20(see Figure 2) for different minimal annual reliability indices. COVSCF 3.1   3.7   FDF  f m   FDF  f m   0.00 2.1 1.28 3.31 1.49 0.05 2.19 1.30 3.47 1.51 0.10 2.41 1.34 4.07 1.60 0.15 2.8 1.41 4.95 1.70 0.20 3.33 1.49 6.44 1.86 So far, it is assumed that when the welded detail fails, the whole device fails. But not all details are critical and its criticalness needs to be considered when calibrating partial safety factors. The criticalness of the detail is considered here by the probability  |COL FATP  of total collapse given fatigue failure of the considered detail. Table 8 shows the required FDF and partial safety factor values for different |COL FATP values. The safety factors can be decreased when the overall consequences in case of failure are small. Table 8: FDF and partial safety factors dependent on the probability of failure of the device given failure of the considered detail. COVSCF=0.1 and no inspections considered. |COL FATP  3.1   3.7   FDF  f m   FDF  f m   1.00 2.41 1.34 4.07 1.60 0.50 2.01 1.26 3.45 1.51 0.10 1.04 1.01 2.35 1.33 0.01 1.00 1.00 1.03 1.01 Normally, critical details are inspected in order to guarantee a certain safety level. Furthermore, performing inspections lead to lower FDFs and partial safety factors for the design compared with a design where no inspections are performed. Table 9 (Δβ=3.1) and Table 10 (Δβ=3.7) show the required FDF values and partial safety factors for different number of inspections performed during the lifetime of the device as well as show different minimal detectable crack sizes λ, which depend on the skills of the person performing the inspection as well as the accessibility of the detail. It is assumed here that| 1COL FATP  . Significant reductions of safety factors can be obtained when a good inspection quality is 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8 used (small λ value) and multiple inspections during a lifetime are performed. Table 9: FDF values and partial safety factors dependent on the number of inspections performed during its lifetime as well as the minimal detectable crack size λ requiring an annual reliability index of 3.1.  No. of insp. λ=5mm λ=10mm λ=50mm FDF  f m  FDF  f m  FDF  f m  No insp. 2.41 1.34 2.41 1.34 2.41 1.34 1 insp. 1.60 1.17 1.8 1.22 2.10 1.28 3 insp. 1.00 1.00 1.00 1.00 1.90 1.24 4 insp. 1.00 1.00 1.00 1.00 1.80 1.22 6 insp. 1.00 1.00 1.00 1.00 1.30 1.09 9 insp. 1.00 1.00 1.00 1.00 1.00 1.00 Table 10: FDF values and partial safety factors dependent on the number of inspections performed during its lifetime requiring an annual reliability index of 3.7. No. insp. λ=5mm λ=10mm λ=50mm FDF  f m  FDF  f m  FDF  f m  No insp. 4.07 1.60 4.07 1.60 4.07 1.60 1 insp. 2.40 1.34 2.70 1.39 3.00 1.44 3 insp. 1.30 1.09 2.00 1.26 2.80 1.41 4 insp. 1.10 1.03 1.70 1.19 2.70 1.39 6 insp. 1.00 1.00 1.10 1.03 2.30 1.32 9 insp. 1.00 1.00 1.00 1.00 2.10 1.28 6. CONCULSIONS This paper describes reliability-based calibration of partial safety factors using the Wavestar working principle as a case study. The annual reliability level for WEC applications is assumed to correspond to a maximum annual probability of failure between 10-4 and 10-3.  The results that did not consider inspections indicate that for a maximum annual probability of failure equal to 10-3, the required partial safety factor is equal to 1.34 for fatigue driven details, whereas for a maximum annual probability of failure equal to 10-4, the partial safety factor is increased to 1.60. Further, the effect of inspections is modelled using an exponential POD curve. The results show that significant reductions are possible in the partial safety factor for fatigue driven details when using good inspection qualities and multiple inspections during the expected lifetime. The results in the report can be considered as representative, but more examples should be considered before implementation in standards. 7. ACKNOWLEDGEMENT The authors wish to thank the financial support from the ForskEL - Energinet.dk research program (Contract 12155, ‘Digital Hydraulic Power Take Off for Wave Energy) which made this work possible. 8. REFERENCES ASTM Standard E 1049-1085 (1995). ”ASTM Standard for Cycle Counting Fatigue Analysis” American Society for Testing and Materials (ASTM). BS 7910 (2005). ”Guide to Methods for Assessing the Acceptability of Flaws in Metallic Structures” British Standard (BS). DNV-RP-C203 (2012). ”Fatigue Design of Offshore Steel Structures” DNV. Hansen, R.H.; Kramer, M.M.; Vidal, E. (2013). “Discrete Displacement Hydraulic Power Take-Off System for the Wavestar Wave Energy Converter” Energies, 6, 4001-4044. Lemaire, M. Chateauneuf, Mitteau, J.-C. (2009). “Structural Reliability“, ISTE Ltd., London. JCSS (2001). “Probabilistic Model Code“, Joint Committee on Structural Safety (JCSS). Madsen, H.O, Krenk, S., Lind, N.C. (2006). “Methods for Structural Safety“, Dover Publications Inc., New York. Paris, P.C., Erdogan, F. (1963). “A Critical Analysis of Crack Propagation Laws“ Journal of Basic Engineering, 85, 528-533. Sørensen, J.D., Ersdal, G. (2008). ”Safety and Inspection Planning of Older Installations” Journal of Risk and Reliability, 222(3), 403-417. 

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