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Flood hazards for nuclear power plants
Nuclear Engineering and Design 110 (1988) 213-219 North-Holland, Amsterdam
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FLOOD HAZARDS FOR NUCLEAR POWER PLANTS Ben Chie Y E N * University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
Received 30 June 1988 Flooding hazards for nuclear power plants may be caused by various external geophysical events. In this paper the hydrologic hazards from flash floods, river floods and heavy rain at the plant site are considered. Depending on the mode of analysis, two types of hazard evaluation are identified: (a) design hazard which is the probability of flooding over an expected service period, and (b) operational hazard which deals with real-time forecasting of the probability of flooding of an incoming event. Hazard evaluation techniques using flood frequency analysis can only be used for type (a) design hazard. Evaluation techniques using rainfall-runoff simulation or multi-station correlation can be used for both types of hazard prediction.
1. Introduction According to the sources causing them, flood hazards for nuclear power plants can be divided into two groups. Internal floodings are those caused by malfunction of the power plant internal facilities. External floods are those produced by heavy rain, river flood, failure of dams or levees, high wind induced waves, tsunamis, and other external hydrologic events. The risk of internal flooding is plant-design dependent. It should be evaluated individually according to the details of the design, construction and operational procedures of the specific plant. Therefore, it will not be dealt with in this paper. Nevertheless, internal flooding has been found repeatedly as a threat to the safety of nuclear power plants as evidenced from, e.g., the incidents of Quad Cities Unit 1 in 1972 and Three-Mile Island Unit 1 in 1977 involving condenser circulating water system, and Surry 2 in 1975 and 1977, and East Hatch 1 in 1978 involving service water valves. There are many natural and man-made external causes that would induce floods affecting the safety of a nuclear power plant. It appears that in current practice in reactor safety consideration, flood risks attract far less attention than the risks of earthquake and fire. This is due partly to the following reasons: (a) the probability of occurrence of extremal external floods is low, (b) extremal floods due to external causes usually have a relatively long warning time allowing implementation of emergent safety actions, (c) the false security given by
* Professor of Civil Engineering, now at the University of Virginia.
the use of probable maximum precipitation (PMP), and probable maximum flood (PMF) in the design, and (d) no standardized and comprehensive procedure to quantitatively estimate flood risks. Nevertheless, there are reasons that floods from external causes should not be ignored in safety consideration for nuclear power plants. First, PMF analysis is not without its flaws, and predictions are under high uncertainties. Second, flash floods such as those produced by local heavy storms or failure of nearby dams and levees may not allow enough lead time for warning and action. Third, present knowledge on flood hydrology and risk analysis allows development of improved procedures for flood risk estimation. Fourth, under various circumstances the side effects of floods (not necessarily extremal) may be important in safety considerations, e.g., flood induced erosion at the plant site or interruption of transport in case of a concurrent accident. The potential causes of external floods include the following: (a) Heavy rain or snowfall at the plant site. (b) Flash flood from nearby watershed. (c) River flood. (d) Failure of hydraulic structures. (e) Wind waves. (f) Surges and seiches. (g) Tsunamis. The last three groups, (e), (f) and (g) involve hydrodynamics of water waves in restricted or large water bodies. Procedures dealing with these waves have been developed on the basis of information from the U.S. Army Corps of Engineers' Coastal Engineering Research Center and other sources. Hence they will not be elaborated here.
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Analysis of flood hazards caused by group (d), failure of hydraulic structures such as dams, dikes and levees, involves more hydraulics than hydrology. Recent developments in flood routing and dam-break hydraulics pave the way for improved techniques to evaluate such hazards. However, they will not be dealt with in this paper because of the space and time limitation. It may be presented in the future when opportunity arises. A flood risk analysis for a nuclear power plant consists of identification of flood hazard causes, flooding hazard evaluation, analysis of plant component fragility and vulnerability, a plant and system risk analysis, and a radioactive release risk analysis. In this paper, only the hazard evaluation aspects of the first three cause groups (a), (b) and (c), for river and watershed-drainage-related floods will be discussed from the viewpoints of design and for real-time operation.
2. Definitions of flood hazard and frequency analysis The hazard of a flood can be viewed differently depending on the purpose of the evaluation: (a) It can be expressed as the probability that the magnitude of a characteristic parameter of the flood exceeding a specific threshold value of the parameter, for example, the flood water elevation exceeding a specified stage, or the flood discharge exceeding a specified discharge. (b) It can be expressed as the probability that the action of the flood causes the hydraulic structure or system to perform unsatisfactorily for its intended purposes. Or (c) it can also be expressed in terms of exceeding a threshold value of economic losses or casualties of human lives. The first definition is adopted here. The probability of a flood X exceeding a threshold value x, can be expressed mathematically as
p ( X > x ) = f ~ f ( X: S ) d X
(a)
in which f ( X : S) represents the probability density function of floods, where S denotes a set of parameters. In hydrology, the average return period, T~, (e.g., see Ang and Tang, [1, p. 110]) which is the reciprocal of p ( X > x) on an annual basis is usually used. For a time-invariant (stationary) hydrologic system, once the value of Tr for a given threshold flood magnitude is known, the risk or probability that this threshold, x(Tr), will be exceeding in an n-year period is
P,(X>
X) =
{
1 - /1 \
]]".
gj
(2)
For large T~ approximately
P( X > x ) = 1 - e x p ( - n / T r ) .
(3)
The probability for the Tr-year event to occur less than m times in n years can also be calculated, Yen [15]. Quantitatively the flood at a given location can best be described by the continuous stage or discharge hydrograph. However, often such continuous information is either difficult to obtain or unnecessary from the viewpoint of engineering practice, and knowledge of certain key flood parameters such as the flood volume, duration, peak discharge, maximum stage, and occurrence frequency would suffice. In fact, conventionally floods are often expressed only in terms of the peak discharge and its return period. The concept of comprehensive risk analysis was introduced to flood hazard evaluation only in the last score of years. For decades hydrologists have attempted to define the flood probability density fonction, f ( X : S). When observed data are available and the hydrologic system is stationary, usually a frequency analysis or another stochastic approach is used to determine the probability distribution of X. If data are not available or inadequate, synthetic simulation methods are used. Many different distribution functions have been assumed and examined in the past for floods and precipitation. Some of the distributions are given in Chow [3,4], Kite [6], and Yevjevich [22]. None has proven completely satisfactory. The popular ones include the two-parameter lognormal and extreme type I (Gumbel) distributions and the three-parameter log-Pearson type III (LP3) distribution. The book on finding a desirable distribution function has not been closed. Undoubtedly many more new distributions will be introduced in the future. The set S in eq. (1) should account for all the influential parameters that would affect the probability density function f ( X : S), including the effects of limit data in an n-year period of record, the relative ranking of the event magnitude in the data set, data measurement errors, seasonal and other temporal variations of the hydrologic environment for a nonstationary system, etc. In practice, conventionally only the flood frequency is considered. Even in improved cases, at best only some of the influental factors are accounted for while others are simply ignored. In the standard procedure for hydrologic frequency analysis, such as those recommended by Chow [3] or U.S. Water Resources Council [13], only selected data from the limited n-year record are used. The selected data constitute an annual maximum series or annual exceedance series (or a more extensive partial series).
B. C. Yen / Flood hazardsfor nuclearpower plants The annual exceedance series selects the n largest events from the n-year data without considering the time of their occurrences. The annual maximum series selects from the data the largest event of each year. Year is chosen as the time unit because variations of hydrologic environment from year to year are considerably smaller than the seasonal variations and hence large flood events can be considered to be statistically independent p r o vided the hydrologic system is essentially stationary on a year to year basis (e.g., the watershed is not under urbanization). The events in the data series are ranked and a "plotting position" formula is used to convert the ranking into return period or its reciprocal, the non-exceedance probability. Usually the Weibull formula
Tr = (n + 1 ) / m
(4)
is used for annual maximum series, whereas the California formula
Tr
=
n/m
(5)
is used for annual exceedance series or other partial series, where n is the number of years of record, m is the rank of the event from the largest event, and Tr is the return period in years. Other plotting position formulas can be found, e.g., in Chow [4, p. 8.29]. In this procedure, commonly only the event magnitude is assumed uncertain due to, e.g., measurement errors. But in fact the other dimension, the plotting position, is also subject to uncertainties. Very few studies have been done on this aspect. Moreover, the procedure determines the distribution function in the domain corresponding to the length of the record, whereas the threshold flood magnitude x in eq. (1) is often beyond the domains of data. Therefore, the flood hazard which is evaluated according to eq. (1) as
p(X>x)=~=l1 f~f(g:S)
dX,
(6 /
L being the lower bound of f i X : S), implicity assumes that extrapolation of f i X : S ) beyond the data domain is correct and accurate, which is in fact questionable. The frequency analysis procedure just described considers only the parameters of n and m and it ignores all other influential parameters in the set S in f ( X : S). Thus it provides only an incomplete evaluation of the flood hazard. In reality, it is rather difficult if not impossible to account for all the influential parameters while the integration of eq. (6) is still manageable. Because of the uncertainties on f i X : S), it is desirable to know the reliability of the flood hazard evaluation. The confidence limit or other variance measures are
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often used as such realibility scale. However, the significance of such reliability measures also depends on how thorough the influential factors are accounted for. More researchers are needed on consideration of more influential parameters in S.
3. Hazard of local heavy rainstorm at plant site The flooding hazard of heavy rainstorms at a plant site is a problem of local storm water drainage from the roof, overland and other ground surfaces. Drainage design consists of two stages. The first stage is how much water is to be drained, which is a hydrologic problem. The second stage is how to drain this water which is mostly a hydraulic problem. If properly done the risk due to uncertainties of the second stage is relatively small and insignificant. The rain water concerning plant safety usually comes from local, convective type heavy rainstorms which are characterized by high rain intensity over a relatively short duration, normally less than an hour (There are, of course, exceptions such as the very rare case of rain water column produced by tornadoes which will not be considered here). Traditionally the heavy rain intensity and depth are expressed as a function of the rain duration and frequency (expressed in terms of return period). If gauged point rainfall record is luckily available at the plant site, a frequency analysis (or other types of stochastic analysis) can be preformed on the data and the result is used for the design. However, often gauged record is not available or inadequate at the site. In this case the data of nearby gauged locations are analyzed and the result transferred to the plant site for applications. Typical examples of frequency analysis performed on recorded point rainfall data at gauged stations and the result generalized to ungauged locations are the U.S. National Weather Service's (U.S. NWS) Technical Paper No. 40 [5], Technical Memo No. HYDRO-35 [11] and Atlas No. 2 [10]. However, because of the potential seriousness of the failure of a nuclear power plant, the use of probable maximum precipitation (PMP) is usually recommended for plant design. PMP is defined as the conceivably worst precipitation at a location for specified storm durations. This is the theoretical and hypothetical precipitation produced by the combination of reasonably conceivable worst hydrometeorological conditions occurring concurrently. In other words, this is the top ceiling for the respective durations of all the precipitations at that location corresponding to the current global geophysical situations. Thus, it is the upper limit of the
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rainfall frequency distribution and it carries no occurrence frequency. (In recent years there have been several erroneous attempts to assign a frequency to PMP). The concept of PMP was developed compatible with the concept of nonquantifiable of the worth of human life in hydrologic design. Accordingly, it cannot be subjected to probabilistic risk analysis. Despite this admirable upper-ceiling concept, no satisfactory procedure has been developed to determine PMP accurately. Many methods have been proposed to estimate PMP, Paulhus et al. [8]. They can be grouped into three categories: those by considering the physics of precipitable atmospheric water under the worst possible condition, those by using depth-duration-area relation together with transpose of worst storms, and those based on extrapolation of frequencyfactor relation. In all cases recorded world or regional maximum rainfalls are always used as a guide and adjustment for the estimated results. Inconsistency is a rule rather than exception for the results obtained by using these different methods. A common source of estimated PMP values in the United States is from the U.S. National Weather Service [5,10]. As mentioned previously, in frequency analysis a distribution function is assumed and data are fitted to the assumed distribution to test the goodness of the fit. Subsequently, in application the magnitudes of the events of different frequencies can be estimated from the accepted distribution. Estimates are usually made at low frequencies (large return periods) beyond the length of the record. Therefore, the faithfulness and representativeness of the upper tail (large return period) of the distribution function is of utmost importance. For the popular distributions used in hydrology the upper tail approaches infinity asymptotically. In reality, if the frequency analysis is based on the current global geophysical conditions, not considering global changes in geological time scale, the distribution should be truncated at the upper ceiling, PMP, if it can be accurately determined. The upper ceiling, at the worst, is the total amount of water on earth, which is still finite, although a very large number (1.4 trillion cubic kilometers). Accordingly, any assumed distribution should be adjusted, proportionally if one wishes, in view of the truncation. Very low risk is allowed in nuclear power plant design and hence proper understanding and representation of the tail of the distribution function is important. Once the amount of rain is determined, the runoff on the roof and ground at the plant site can be computed by using an appropriate rainfall-runoff model. These models range from the simple rational method, kinematic wave method to the more sophisticated dy-
namic wave and non-inertia methods. No published existing model has been developed specifically for nuclear power plant sites. However, some urban runoff models (Yen [6]) can be adopted and modified. One must be very careful in selecting an appropriate model. Proper discretization of the overland surface is important. In general, the non-inertia or dynamic wave type models should be used if the backwater from the downstream end of the surface is significant. The design and evaluation of plant site drainage are not without uncertainties. Among the different existing reliability analysis methods applicable to the evaluation of plant site rain-induced flooding hazards and capable of accounting for the input, parameter and model uncertainties and randomness of natural phenomena, the first-order second-moment methods appear most suitable, Ang and Tang [2], Yen et al. [20], and Yen [7]. Examples evaluating the hazards of drainage components can be found, among others, in Yen and Yun [18] and Yen et al. [21].
4. Flash flood hazards
Flash floods from nearby small local watersheds usually is produced by convective type local heavy rainstorms having a relatively short duration of no more than a few hours covering a relatively small area. With such small space and time scales, in real-time operation the lead time of prediction for flash floods is usually short, no more than several hours. This lead time can be lengthened when the rainfall prediction and runoff simulation model are improved, particularly when rain cloud and radar information is utilized. If local flood data are available and the watershed characteristics have not changed over the years, prediction of flash flood hazards for plant design can be done by using the frequency analysis described previously in Section 2, together with consideration of the uncertainty of the parameters. However, normally data on local watershed floods at or near the plant site is not available or inadequate for a frequency analysis. The hydrologic characteristics of the watershed may change because of the construction and development at or near the plant, rendering the past record inappropriate to future conditions. Furthermore, in nuclear power plant design the risk of flash flood generated by the PMP should be investigated. Thus, in most cases flash flood prediction for small watersheds is made through rainfall-runoff simulation rather than by direct application of frequency analysis to recorded data. In the simulation of runoff from rainfall, four con-
B.C. Yen / Flood hazards for nuclearpower plants secutive elements are considered and the stochastic properties of each of these elements contribute to the risk of the flash flood. The four elements are rainfall, abstractions from rainfall (particularly infiltration), generation of overland runoff, and routing of flow in channels. The rainfall may be a specific rain obtained from a standard depth-duration-frequency relationship adjusted for area size reduction or a real rain real-time operation, or it may be the PMP for a critical duration. Usually the rainfall is assumed to be distributed uniformly over its duration and evenly over the entire watershed. In reality there are always spatial and temporal variations of the rainfall. Under some situations the spatially and temporally varying rainfall may p r o duce more severe flash flood than the case of uniform rain. Among the abstractions the most important component for flash flood production is infiltration. Reasonable and realistic estimation of the antecedent soil moisture condition and the temporal and spatial variations of infiltration are important. In the overland runoff generation phase the most important consideration is conservation of flow volume at each time interval. The degree of spatial and temporal discretization also has significant effect on the accuracy of the flood prediction. As to the last element, routing of flow in channels, considerable advances have been made in the past two decades. Various hydraulic and hydrologic routing equations are available to suit the particular situation, Yen [16]. It suffices to note here that usually this element relatively does not impose significantly on the reliability in flash flood estimation for small watersheds. If snowmelt-induced flood is considered, the effect of channel ice jam should also be accounted for. There exist numerous rainfall-runoff simulation models of various degrees of sophistication and detail, including the public-domain models HEC-1, SCS-TR-20 and EPA-SWMM, all developed in U.S. federal agencies. Many lumped-system models combine the elements of abstractions, overland runoff generation and channel routing together as a single transfer component to produce output (flood) from input (rainfall). Typical examples are the rational method and the synthetic unit hydrograph method. The degree of accuracy of these models may contribute significantly to the uncertainty in flood hazard prediction. Melching [7] used the advanced first-order secondmoment method for real-time prediction of single-event flood hazard. Runoff from rainfall is simulated by using the simulation model HEC-1 as well as RORB from Australia. Uncertainties on parameters and models are accounted for.
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By using different values of antecedent soil moisture and rainfall of different frequencies and temporal and spatial distributions as input to the simulation model, different runoff results will be produced. Accordingly flood statistics can be obtained to provide the distribution function f ( X : S ) in eqs, (1) or (6). The reliability of the runoff prediction depends not only on the reliability of the rainfall prediction but also on the reliability of the rainfall-runoff simulation model. The flood hazard can and should be evaluated considering also the parameter and model uncertainties. In hydrologic design with potential loss of lives or extraordinarily large economic losses, often the use of probable maximum flood (PMF) is required. PMF is the theoretical worst flood corresponding to the reasonably conceivable combination of worst precipitation (PMP) and other hydrometeorologic conditions for the watershed being considered. In other words, this is the top ceiling of all the floods at that location corresponding to the existing global condition and it carries no occurrence frequency. Theoretically it serves as the upper limit of the flood frequency distribution, truncating the upper tail of any assumed distribution function if it extends beyond this upper ceiling. If PMF could be determined accurately, it would be very useful in nuclear power plant safety evaluation and design. Unfortunately, no satisfactory procedure has been developed to compute the PMF accurately. Despite this drawback of PMF, it should be used for safety reason as the catchment flash flood for the design of nuclear power plants. Whereas for realtime plant operation the predicted flash flood from real-time rainfall is considered. 5. River flood hazards
Rivers are defined here as waterways draining an area greater than, e.g., 1000 square kilometers or having a flow travel time in the order of one day or longer from its farthest upstream to the location of interest. Thus, in comparison to the small watersheds producing flash floods, the space scale is larger and hence allowing sufficient lead time of prediction for flood warning and appropriate emergency safety actions. In the case of river floods, often river discharge or stage record is available at or near the plant site, the local land use changes (such as urbanization) are relatively small because of the watershed size, and the associated change of watershed characteristics is insignificant. Consequently a frequency analysis (described in Section 2) can be performed on the gauged data and the result transferred to the plant site through interpolation or extrapolation and adjustment, and the result can
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be used for plant design. This, indeed, is the approach used by Wall [14] to estimate the flooding hazard at the Monticello site on the Mississippi River in Minnesota. Wall's work is well known to those dealing with nuclear safety analysis and hence no further detail will be given here. It should only be noted that he considered (a) the assumed distribution (Gumbel) represents the true distribution without correction or truncation (for PMF), (b) the daily peak discharge instead of the actual instantaneous peak discharge is used as the critical flow parameter, and (c) only the randomness of the peak discharge contributes to the flood hazard, uncertainties on model and parameters are not included. In frequency analysis, different distribution functions have been suggested for peak discharges. The popularity of log-Pearson 3 distribution is due primarily to the recommendation of the U.S. Water Resources Council [12,13]. The LP3 distribution is sensitive to the value of the skewness coefficient of the distribution. Unfortunately, field data normally are not accurate enough to provide a reliable estimate of the skewness coefficient. Attempts have been made to improve the estimation by incorporating the regional skewness, U.S. Water Resources Council [13]. However, such remedies have no theoretical basis and their effectiveness has not been verified. If flood data are not available or inadequate, the rainfall-runoff simulation method described in Section 4 for flash floods can also be applied to provide river flood estimation. For rivers the importance of flow routing increases relatively and hence the lumped system approach is undesirable. The selection of proper hydraulic equation for routing, the degree of discretization of the channel reach, and the faithfulness of the parameters to represent the channel characteristics are all significant to the reliability of the estimation. An alternate method is to relate the flow at the site of interest to the flow at upstream stations, preferably a single station sufficiently upstream (for enough lead time) without significant tributaries in between. The correlation between the upstream station and the site of interest can be established either through measured data or by flow routing. Published flow data are usually given in terms of discharge. Actually the measurements are usually made on stages and converted to discharges by using a rating curve. The correlation can be made on either stage or discharge. However, if observed data on stages are available, it would be better to use stage than discharge to avoid the error of the rating curves. Based on a modified form of eq. (1), Yen and Tang [19] suggested a framework to apply the first order second moment analysis to evaluate river flood hazard using
the two-station correlation method. Plate and Ihringer [9] used direct integration of eq. (1) to evaluate the flooding hazard by assuming an exponential form of the density function correlating the stages at two locations in a nonstationary system of a tidal river. For river floods, as in the case of flash floods, the method of frequency analysis can be used for prediction of future floods of specified frequency such as a design flood. It cannot be used for real-time forecast of an incoming flood and prediction of its hazard. Conversely, the rainfall-runoff simulation method, as well as the two-station correlation method, can be used for both prediction of design floods and real-time forecasting. 6. Conclusions
Flooding hazards for nuclear power plants should not be ignored at either the design and construction stage or the operation stage. There are various external geophysical causes that could induce reactor incidents. Among rain related floods, hazards due to heavy rain at the plant site and due to flash floods should also be considered in addition to river floods. There are two types of hydrologic flood hazard prediction: (a) the probability of flooding for an expected service period, and (b) a real-time forecasting of the probability of flooding for an incoming event. In both cases randomness and uncertainties of all the factors such as spatial and temporal variabilities of rainfall and watershed, measurement errors and model accuracy should be accounted for inclusively, not merely the frequency of occurrence of the flood events. Hazard evaluation techniques based on frequency analysis can be used only for type (a) prediction of hazard over a given period. Hazard evaluation techniques based on rainfall-runoff simulation or multi-station correlation can be used for both types of hazard prediction. For type (a) prediction, rainfalls of different frequencies are used as input to the simulation model and the results are analyzed to give flood statistics. For type (b) prediction, a specified rainfall is used as the input. Although different causes of plant flooding have been identified and the concept and general framework of flood hazard prediction have been conceived, much developmental research still need to be accomplished to formulate improved procedures for flood hazard evaluation. The procedure should be adaptable to different levels of available data, or different procedures should be formulated. Most significant factors contributing to the flood hazard should be identified and possible means to reduce the flood hazards should be investigated.
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References [1] A.H.-S. Ang and W.H. Tang, Probability Concepts in Engineering Planning and Design, I. Basic Principles (John Wiley & Sons, New York, 1975). [2] A.H.-S. Ang and W.H. Tang, Probability Concepts in Engineering Planning and Design, II. Decision, Risk, and Reliability (John Wiley & Sons, New York, 1984). [3] V.T. Chow, Frequency analysis of hydrologic data with special application to rainfall intensity, Engineering Experiment Station Bulletin 414, University of Illinois, Urbana (July 1953). [4] V.T. Chow, Handbook of Applied Hydrology (McGraw Hill Book Co., New York, 1964). [5] D.M. Hershfield, Rainfall frequency atlas of the United States for durations from 30 minutes to 24 hours and return periods from 1 to 100 years, Weather Bureau Technical Paper No. 40, U.S. Dept. of Commerce (May 1961). [6] G.W. Kite, Frequency and Risk Analysis in Hydrology (Water Resources Publications, Littleton, Colorado, 1977). [7] C.S. Melching, A reliability analysis on flood event forecasting with uncertainties, Ph.D. thesis, Dept. of Civil Eng., Univ. of Illinois at Urbana-Champaign (1987). [8] J.L.H. Paulhus et al., Manual for estimation of probable maximum precipitation, Opera. Hydrol. Rept. 1, WMO No. 332, Geneva (1973). [9] E.J. Plate and J. Ihringer, Failure probability of flood levees on a Tidal River, in: Stochastic and Risk Analysis in Hydraulic Engineering, ed. B.C. Yen (Water Resources Publications, Littleton, Colorado, 1986), pp. 45-58. [10] U.S. National Weather Service, Precipitation-frequency atlas of the conterminous Western United States (by states), NOAA Atlas 2, 11 vols., U.S. NWS, Silver Spring, Maryland (1973). [11] U.S. National Weather Service, Five to 60-minutes precipitation frequency for the Eastern and Central United States, NOAA Tech Memo NWS HYDRO-35, U.S. NWS, Silver Spring, Maryland (1977). [12] U.S. Water Resources Council, A uniform technique for
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determining flood flow frequencies, Hydrology Committee Bulletin 15, Washington, DC (December 1967). [13] U.S. Water Resources Council, Guidelines for determining flood flow frequency, Hydrology Committee Bulletin 17B, Washington, DC (September 1981). [14] I.B. Wall, Probabilistic assessment of flooding hazard for nuclear power plants, Nuclear Safety, 15 (1974~ No. 4, pp. 399-408. [15] B.C. Yen, Risks in hydrologic design of engineering projects, Jour. Hydraulics Div., ASCE, 96 (1970) No. HY4, pp. 959-966. [16] B.C. yen, Rainfall-runoff process on urban catchmets and its modeling, in: Urban Drainage Modelling, ed. C. Maksimovic and M. Radojkovic (Pergamon Press, Oxford, 1986) pp. 3-26. [17] B.C. Yen, Reliability of hydraulic structures possessing random loading and resistance, in: Engineering Reliability and Risk in Water Resources, ed. L. Duckstein and E. Plate, NATO ASI Series E (Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1987) pp. 95-113. [18] B.C. Yen and B.-H. Jun, Risk consideration in design of storm drains, in: Proceedings, 3rd IAHR/IAWPRC Internat. Conf. Urban Storm Drainage, Vol. 2, ed. P. Balmer, P. Malmqvist and A. Sjoberg (Chalmers University of Technology, Goteborg, Sweden, 1984) pp. 695-704. [19] B.C. Yen and W.H. Tang, Reliability of flood warning, in: Stochastic Processes in Water Resources Engineering (Water Resources Publications, Littleton, Colorado, 1977) pp. 333-347. [20] B.C. Yen, S.T. Cheng and C.S. Melching, First order reliability analysis, in: Stochastic and Risk Analysis in Hydraulic Engineering, ed. B.C. Yen (Water Resources Publications, Littleton, Colorado, 1986) pp. 1-36. [21] B.C. Yen, S.T. Cheng and W.H. Tang, Reliability of hydraulic design of culverts, in: Proceedings, Internat. Conf. Water Resources Development, IAHR APD Second Congress, Vol. 2, (Taipei, May 1980) pp. 991-1001. [22] V. Yevjevich, Probability and Statistics in Hydrology (Water Resources Publications, Littleton, Colorado, 1972).
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FLOOD HAZARDS FOR NUCLEAR POWER PLANTS Ben Chie Y E N * University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
Received 30 June 1988 Flooding hazards for nuclear power plants may be caused by various external geophysical events. In this paper the hydrologic hazards from flash floods, river floods and heavy rain at the plant site are considered. Depending on the mode of analysis, two types of hazard evaluation are identified: (a) design hazard which is the probability of flooding over an expected service period, and (b) operational hazard which deals with real-time forecasting of the probability of flooding of an incoming event. Hazard evaluation techniques using flood frequency analysis can only be used for type (a) design hazard. Evaluation techniques using rainfall-runoff simulation or multi-station correlation can be used for both types of hazard prediction.
1. Introduction According to the sources causing them, flood hazards for nuclear power plants can be divided into two groups. Internal floodings are those caused by malfunction of the power plant internal facilities. External floods are those produced by heavy rain, river flood, failure of dams or levees, high wind induced waves, tsunamis, and other external hydrologic events. The risk of internal flooding is plant-design dependent. It should be evaluated individually according to the details of the design, construction and operational procedures of the specific plant. Therefore, it will not be dealt with in this paper. Nevertheless, internal flooding has been found repeatedly as a threat to the safety of nuclear power plants as evidenced from, e.g., the incidents of Quad Cities Unit 1 in 1972 and Three-Mile Island Unit 1 in 1977 involving condenser circulating water system, and Surry 2 in 1975 and 1977, and East Hatch 1 in 1978 involving service water valves. There are many natural and man-made external causes that would induce floods affecting the safety of a nuclear power plant. It appears that in current practice in reactor safety consideration, flood risks attract far less attention than the risks of earthquake and fire. This is due partly to the following reasons: (a) the probability of occurrence of extremal external floods is low, (b) extremal floods due to external causes usually have a relatively long warning time allowing implementation of emergent safety actions, (c) the false security given by
* Professor of Civil Engineering, now at the University of Virginia.
the use of probable maximum precipitation (PMP), and probable maximum flood (PMF) in the design, and (d) no standardized and comprehensive procedure to quantitatively estimate flood risks. Nevertheless, there are reasons that floods from external causes should not be ignored in safety consideration for nuclear power plants. First, PMF analysis is not without its flaws, and predictions are under high uncertainties. Second, flash floods such as those produced by local heavy storms or failure of nearby dams and levees may not allow enough lead time for warning and action. Third, present knowledge on flood hydrology and risk analysis allows development of improved procedures for flood risk estimation. Fourth, under various circumstances the side effects of floods (not necessarily extremal) may be important in safety considerations, e.g., flood induced erosion at the plant site or interruption of transport in case of a concurrent accident. The potential causes of external floods include the following: (a) Heavy rain or snowfall at the plant site. (b) Flash flood from nearby watershed. (c) River flood. (d) Failure of hydraulic structures. (e) Wind waves. (f) Surges and seiches. (g) Tsunamis. The last three groups, (e), (f) and (g) involve hydrodynamics of water waves in restricted or large water bodies. Procedures dealing with these waves have been developed on the basis of information from the U.S. Army Corps of Engineers' Coastal Engineering Research Center and other sources. Hence they will not be elaborated here.
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Analysis of flood hazards caused by group (d), failure of hydraulic structures such as dams, dikes and levees, involves more hydraulics than hydrology. Recent developments in flood routing and dam-break hydraulics pave the way for improved techniques to evaluate such hazards. However, they will not be dealt with in this paper because of the space and time limitation. It may be presented in the future when opportunity arises. A flood risk analysis for a nuclear power plant consists of identification of flood hazard causes, flooding hazard evaluation, analysis of plant component fragility and vulnerability, a plant and system risk analysis, and a radioactive release risk analysis. In this paper, only the hazard evaluation aspects of the first three cause groups (a), (b) and (c), for river and watershed-drainage-related floods will be discussed from the viewpoints of design and for real-time operation.
2. Definitions of flood hazard and frequency analysis The hazard of a flood can be viewed differently depending on the purpose of the evaluation: (a) It can be expressed as the probability that the magnitude of a characteristic parameter of the flood exceeding a specific threshold value of the parameter, for example, the flood water elevation exceeding a specified stage, or the flood discharge exceeding a specified discharge. (b) It can be expressed as the probability that the action of the flood causes the hydraulic structure or system to perform unsatisfactorily for its intended purposes. Or (c) it can also be expressed in terms of exceeding a threshold value of economic losses or casualties of human lives. The first definition is adopted here. The probability of a flood X exceeding a threshold value x, can be expressed mathematically as
p ( X > x ) = f ~ f ( X: S ) d X
(a)
in which f ( X : S) represents the probability density function of floods, where S denotes a set of parameters. In hydrology, the average return period, T~, (e.g., see Ang and Tang, [1, p. 110]) which is the reciprocal of p ( X > x) on an annual basis is usually used. For a time-invariant (stationary) hydrologic system, once the value of Tr for a given threshold flood magnitude is known, the risk or probability that this threshold, x(Tr), will be exceeding in an n-year period is
P,(X>
X) =
{
1 - /1 \
]]".
gj
(2)
For large T~ approximately
P( X > x ) = 1 - e x p ( - n / T r ) .
(3)
The probability for the Tr-year event to occur less than m times in n years can also be calculated, Yen [15]. Quantitatively the flood at a given location can best be described by the continuous stage or discharge hydrograph. However, often such continuous information is either difficult to obtain or unnecessary from the viewpoint of engineering practice, and knowledge of certain key flood parameters such as the flood volume, duration, peak discharge, maximum stage, and occurrence frequency would suffice. In fact, conventionally floods are often expressed only in terms of the peak discharge and its return period. The concept of comprehensive risk analysis was introduced to flood hazard evaluation only in the last score of years. For decades hydrologists have attempted to define the flood probability density fonction, f ( X : S). When observed data are available and the hydrologic system is stationary, usually a frequency analysis or another stochastic approach is used to determine the probability distribution of X. If data are not available or inadequate, synthetic simulation methods are used. Many different distribution functions have been assumed and examined in the past for floods and precipitation. Some of the distributions are given in Chow [3,4], Kite [6], and Yevjevich [22]. None has proven completely satisfactory. The popular ones include the two-parameter lognormal and extreme type I (Gumbel) distributions and the three-parameter log-Pearson type III (LP3) distribution. The book on finding a desirable distribution function has not been closed. Undoubtedly many more new distributions will be introduced in the future. The set S in eq. (1) should account for all the influential parameters that would affect the probability density function f ( X : S), including the effects of limit data in an n-year period of record, the relative ranking of the event magnitude in the data set, data measurement errors, seasonal and other temporal variations of the hydrologic environment for a nonstationary system, etc. In practice, conventionally only the flood frequency is considered. Even in improved cases, at best only some of the influental factors are accounted for while others are simply ignored. In the standard procedure for hydrologic frequency analysis, such as those recommended by Chow [3] or U.S. Water Resources Council [13], only selected data from the limited n-year record are used. The selected data constitute an annual maximum series or annual exceedance series (or a more extensive partial series).
B. C. Yen / Flood hazardsfor nuclearpower plants The annual exceedance series selects the n largest events from the n-year data without considering the time of their occurrences. The annual maximum series selects from the data the largest event of each year. Year is chosen as the time unit because variations of hydrologic environment from year to year are considerably smaller than the seasonal variations and hence large flood events can be considered to be statistically independent p r o vided the hydrologic system is essentially stationary on a year to year basis (e.g., the watershed is not under urbanization). The events in the data series are ranked and a "plotting position" formula is used to convert the ranking into return period or its reciprocal, the non-exceedance probability. Usually the Weibull formula
Tr = (n + 1 ) / m
(4)
is used for annual maximum series, whereas the California formula
Tr
=
n/m
(5)
is used for annual exceedance series or other partial series, where n is the number of years of record, m is the rank of the event from the largest event, and Tr is the return period in years. Other plotting position formulas can be found, e.g., in Chow [4, p. 8.29]. In this procedure, commonly only the event magnitude is assumed uncertain due to, e.g., measurement errors. But in fact the other dimension, the plotting position, is also subject to uncertainties. Very few studies have been done on this aspect. Moreover, the procedure determines the distribution function in the domain corresponding to the length of the record, whereas the threshold flood magnitude x in eq. (1) is often beyond the domains of data. Therefore, the flood hazard which is evaluated according to eq. (1) as
p(X>x)=~=l1 f~f(g:S)
dX,
(6 /
L being the lower bound of f i X : S), implicity assumes that extrapolation of f i X : S ) beyond the data domain is correct and accurate, which is in fact questionable. The frequency analysis procedure just described considers only the parameters of n and m and it ignores all other influential parameters in the set S in f ( X : S). Thus it provides only an incomplete evaluation of the flood hazard. In reality, it is rather difficult if not impossible to account for all the influential parameters while the integration of eq. (6) is still manageable. Because of the uncertainties on f i X : S), it is desirable to know the reliability of the flood hazard evaluation. The confidence limit or other variance measures are
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often used as such realibility scale. However, the significance of such reliability measures also depends on how thorough the influential factors are accounted for. More researchers are needed on consideration of more influential parameters in S.
3. Hazard of local heavy rainstorm at plant site The flooding hazard of heavy rainstorms at a plant site is a problem of local storm water drainage from the roof, overland and other ground surfaces. Drainage design consists of two stages. The first stage is how much water is to be drained, which is a hydrologic problem. The second stage is how to drain this water which is mostly a hydraulic problem. If properly done the risk due to uncertainties of the second stage is relatively small and insignificant. The rain water concerning plant safety usually comes from local, convective type heavy rainstorms which are characterized by high rain intensity over a relatively short duration, normally less than an hour (There are, of course, exceptions such as the very rare case of rain water column produced by tornadoes which will not be considered here). Traditionally the heavy rain intensity and depth are expressed as a function of the rain duration and frequency (expressed in terms of return period). If gauged point rainfall record is luckily available at the plant site, a frequency analysis (or other types of stochastic analysis) can be preformed on the data and the result is used for the design. However, often gauged record is not available or inadequate at the site. In this case the data of nearby gauged locations are analyzed and the result transferred to the plant site for applications. Typical examples of frequency analysis performed on recorded point rainfall data at gauged stations and the result generalized to ungauged locations are the U.S. National Weather Service's (U.S. NWS) Technical Paper No. 40 [5], Technical Memo No. HYDRO-35 [11] and Atlas No. 2 [10]. However, because of the potential seriousness of the failure of a nuclear power plant, the use of probable maximum precipitation (PMP) is usually recommended for plant design. PMP is defined as the conceivably worst precipitation at a location for specified storm durations. This is the theoretical and hypothetical precipitation produced by the combination of reasonably conceivable worst hydrometeorological conditions occurring concurrently. In other words, this is the top ceiling for the respective durations of all the precipitations at that location corresponding to the current global geophysical situations. Thus, it is the upper limit of the
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B.C. Yen / Flood hazards for nuclear power plants
rainfall frequency distribution and it carries no occurrence frequency. (In recent years there have been several erroneous attempts to assign a frequency to PMP). The concept of PMP was developed compatible with the concept of nonquantifiable of the worth of human life in hydrologic design. Accordingly, it cannot be subjected to probabilistic risk analysis. Despite this admirable upper-ceiling concept, no satisfactory procedure has been developed to determine PMP accurately. Many methods have been proposed to estimate PMP, Paulhus et al. [8]. They can be grouped into three categories: those by considering the physics of precipitable atmospheric water under the worst possible condition, those by using depth-duration-area relation together with transpose of worst storms, and those based on extrapolation of frequencyfactor relation. In all cases recorded world or regional maximum rainfalls are always used as a guide and adjustment for the estimated results. Inconsistency is a rule rather than exception for the results obtained by using these different methods. A common source of estimated PMP values in the United States is from the U.S. National Weather Service [5,10]. As mentioned previously, in frequency analysis a distribution function is assumed and data are fitted to the assumed distribution to test the goodness of the fit. Subsequently, in application the magnitudes of the events of different frequencies can be estimated from the accepted distribution. Estimates are usually made at low frequencies (large return periods) beyond the length of the record. Therefore, the faithfulness and representativeness of the upper tail (large return period) of the distribution function is of utmost importance. For the popular distributions used in hydrology the upper tail approaches infinity asymptotically. In reality, if the frequency analysis is based on the current global geophysical conditions, not considering global changes in geological time scale, the distribution should be truncated at the upper ceiling, PMP, if it can be accurately determined. The upper ceiling, at the worst, is the total amount of water on earth, which is still finite, although a very large number (1.4 trillion cubic kilometers). Accordingly, any assumed distribution should be adjusted, proportionally if one wishes, in view of the truncation. Very low risk is allowed in nuclear power plant design and hence proper understanding and representation of the tail of the distribution function is important. Once the amount of rain is determined, the runoff on the roof and ground at the plant site can be computed by using an appropriate rainfall-runoff model. These models range from the simple rational method, kinematic wave method to the more sophisticated dy-
namic wave and non-inertia methods. No published existing model has been developed specifically for nuclear power plant sites. However, some urban runoff models (Yen [6]) can be adopted and modified. One must be very careful in selecting an appropriate model. Proper discretization of the overland surface is important. In general, the non-inertia or dynamic wave type models should be used if the backwater from the downstream end of the surface is significant. The design and evaluation of plant site drainage are not without uncertainties. Among the different existing reliability analysis methods applicable to the evaluation of plant site rain-induced flooding hazards and capable of accounting for the input, parameter and model uncertainties and randomness of natural phenomena, the first-order second-moment methods appear most suitable, Ang and Tang [2], Yen et al. [20], and Yen [7]. Examples evaluating the hazards of drainage components can be found, among others, in Yen and Yun [18] and Yen et al. [21].
4. Flash flood hazards
Flash floods from nearby small local watersheds usually is produced by convective type local heavy rainstorms having a relatively short duration of no more than a few hours covering a relatively small area. With such small space and time scales, in real-time operation the lead time of prediction for flash floods is usually short, no more than several hours. This lead time can be lengthened when the rainfall prediction and runoff simulation model are improved, particularly when rain cloud and radar information is utilized. If local flood data are available and the watershed characteristics have not changed over the years, prediction of flash flood hazards for plant design can be done by using the frequency analysis described previously in Section 2, together with consideration of the uncertainty of the parameters. However, normally data on local watershed floods at or near the plant site is not available or inadequate for a frequency analysis. The hydrologic characteristics of the watershed may change because of the construction and development at or near the plant, rendering the past record inappropriate to future conditions. Furthermore, in nuclear power plant design the risk of flash flood generated by the PMP should be investigated. Thus, in most cases flash flood prediction for small watersheds is made through rainfall-runoff simulation rather than by direct application of frequency analysis to recorded data. In the simulation of runoff from rainfall, four con-
B.C. Yen / Flood hazards for nuclearpower plants secutive elements are considered and the stochastic properties of each of these elements contribute to the risk of the flash flood. The four elements are rainfall, abstractions from rainfall (particularly infiltration), generation of overland runoff, and routing of flow in channels. The rainfall may be a specific rain obtained from a standard depth-duration-frequency relationship adjusted for area size reduction or a real rain real-time operation, or it may be the PMP for a critical duration. Usually the rainfall is assumed to be distributed uniformly over its duration and evenly over the entire watershed. In reality there are always spatial and temporal variations of the rainfall. Under some situations the spatially and temporally varying rainfall may p r o duce more severe flash flood than the case of uniform rain. Among the abstractions the most important component for flash flood production is infiltration. Reasonable and realistic estimation of the antecedent soil moisture condition and the temporal and spatial variations of infiltration are important. In the overland runoff generation phase the most important consideration is conservation of flow volume at each time interval. The degree of spatial and temporal discretization also has significant effect on the accuracy of the flood prediction. As to the last element, routing of flow in channels, considerable advances have been made in the past two decades. Various hydraulic and hydrologic routing equations are available to suit the particular situation, Yen [16]. It suffices to note here that usually this element relatively does not impose significantly on the reliability in flash flood estimation for small watersheds. If snowmelt-induced flood is considered, the effect of channel ice jam should also be accounted for. There exist numerous rainfall-runoff simulation models of various degrees of sophistication and detail, including the public-domain models HEC-1, SCS-TR-20 and EPA-SWMM, all developed in U.S. federal agencies. Many lumped-system models combine the elements of abstractions, overland runoff generation and channel routing together as a single transfer component to produce output (flood) from input (rainfall). Typical examples are the rational method and the synthetic unit hydrograph method. The degree of accuracy of these models may contribute significantly to the uncertainty in flood hazard prediction. Melching [7] used the advanced first-order secondmoment method for real-time prediction of single-event flood hazard. Runoff from rainfall is simulated by using the simulation model HEC-1 as well as RORB from Australia. Uncertainties on parameters and models are accounted for.
217
By using different values of antecedent soil moisture and rainfall of different frequencies and temporal and spatial distributions as input to the simulation model, different runoff results will be produced. Accordingly flood statistics can be obtained to provide the distribution function f ( X : S ) in eqs, (1) or (6). The reliability of the runoff prediction depends not only on the reliability of the rainfall prediction but also on the reliability of the rainfall-runoff simulation model. The flood hazard can and should be evaluated considering also the parameter and model uncertainties. In hydrologic design with potential loss of lives or extraordinarily large economic losses, often the use of probable maximum flood (PMF) is required. PMF is the theoretical worst flood corresponding to the reasonably conceivable combination of worst precipitation (PMP) and other hydrometeorologic conditions for the watershed being considered. In other words, this is the top ceiling of all the floods at that location corresponding to the existing global condition and it carries no occurrence frequency. Theoretically it serves as the upper limit of the flood frequency distribution, truncating the upper tail of any assumed distribution function if it extends beyond this upper ceiling. If PMF could be determined accurately, it would be very useful in nuclear power plant safety evaluation and design. Unfortunately, no satisfactory procedure has been developed to compute the PMF accurately. Despite this drawback of PMF, it should be used for safety reason as the catchment flash flood for the design of nuclear power plants. Whereas for realtime plant operation the predicted flash flood from real-time rainfall is considered. 5. River flood hazards
Rivers are defined here as waterways draining an area greater than, e.g., 1000 square kilometers or having a flow travel time in the order of one day or longer from its farthest upstream to the location of interest. Thus, in comparison to the small watersheds producing flash floods, the space scale is larger and hence allowing sufficient lead time of prediction for flood warning and appropriate emergency safety actions. In the case of river floods, often river discharge or stage record is available at or near the plant site, the local land use changes (such as urbanization) are relatively small because of the watershed size, and the associated change of watershed characteristics is insignificant. Consequently a frequency analysis (described in Section 2) can be performed on the gauged data and the result transferred to the plant site through interpolation or extrapolation and adjustment, and the result can
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B. C Yen / Flood hazards for nuclear power plants
be used for plant design. This, indeed, is the approach used by Wall [14] to estimate the flooding hazard at the Monticello site on the Mississippi River in Minnesota. Wall's work is well known to those dealing with nuclear safety analysis and hence no further detail will be given here. It should only be noted that he considered (a) the assumed distribution (Gumbel) represents the true distribution without correction or truncation (for PMF), (b) the daily peak discharge instead of the actual instantaneous peak discharge is used as the critical flow parameter, and (c) only the randomness of the peak discharge contributes to the flood hazard, uncertainties on model and parameters are not included. In frequency analysis, different distribution functions have been suggested for peak discharges. The popularity of log-Pearson 3 distribution is due primarily to the recommendation of the U.S. Water Resources Council [12,13]. The LP3 distribution is sensitive to the value of the skewness coefficient of the distribution. Unfortunately, field data normally are not accurate enough to provide a reliable estimate of the skewness coefficient. Attempts have been made to improve the estimation by incorporating the regional skewness, U.S. Water Resources Council [13]. However, such remedies have no theoretical basis and their effectiveness has not been verified. If flood data are not available or inadequate, the rainfall-runoff simulation method described in Section 4 for flash floods can also be applied to provide river flood estimation. For rivers the importance of flow routing increases relatively and hence the lumped system approach is undesirable. The selection of proper hydraulic equation for routing, the degree of discretization of the channel reach, and the faithfulness of the parameters to represent the channel characteristics are all significant to the reliability of the estimation. An alternate method is to relate the flow at the site of interest to the flow at upstream stations, preferably a single station sufficiently upstream (for enough lead time) without significant tributaries in between. The correlation between the upstream station and the site of interest can be established either through measured data or by flow routing. Published flow data are usually given in terms of discharge. Actually the measurements are usually made on stages and converted to discharges by using a rating curve. The correlation can be made on either stage or discharge. However, if observed data on stages are available, it would be better to use stage than discharge to avoid the error of the rating curves. Based on a modified form of eq. (1), Yen and Tang [19] suggested a framework to apply the first order second moment analysis to evaluate river flood hazard using
the two-station correlation method. Plate and Ihringer [9] used direct integration of eq. (1) to evaluate the flooding hazard by assuming an exponential form of the density function correlating the stages at two locations in a nonstationary system of a tidal river. For river floods, as in the case of flash floods, the method of frequency analysis can be used for prediction of future floods of specified frequency such as a design flood. It cannot be used for real-time forecast of an incoming flood and prediction of its hazard. Conversely, the rainfall-runoff simulation method, as well as the two-station correlation method, can be used for both prediction of design floods and real-time forecasting. 6. Conclusions
Flooding hazards for nuclear power plants should not be ignored at either the design and construction stage or the operation stage. There are various external geophysical causes that could induce reactor incidents. Among rain related floods, hazards due to heavy rain at the plant site and due to flash floods should also be considered in addition to river floods. There are two types of hydrologic flood hazard prediction: (a) the probability of flooding for an expected service period, and (b) a real-time forecasting of the probability of flooding for an incoming event. In both cases randomness and uncertainties of all the factors such as spatial and temporal variabilities of rainfall and watershed, measurement errors and model accuracy should be accounted for inclusively, not merely the frequency of occurrence of the flood events. Hazard evaluation techniques based on frequency analysis can be used only for type (a) prediction of hazard over a given period. Hazard evaluation techniques based on rainfall-runoff simulation or multi-station correlation can be used for both types of hazard prediction. For type (a) prediction, rainfalls of different frequencies are used as input to the simulation model and the results are analyzed to give flood statistics. For type (b) prediction, a specified rainfall is used as the input. Although different causes of plant flooding have been identified and the concept and general framework of flood hazard prediction have been conceived, much developmental research still need to be accomplished to formulate improved procedures for flood hazard evaluation. The procedure should be adaptable to different levels of available data, or different procedures should be formulated. Most significant factors contributing to the flood hazard should be identified and possible means to reduce the flood hazards should be investigated.
B. C Yen / Flood hazards for nuclear power plants
References [1] A.H.-S. Ang and W.H. Tang, Probability Concepts in Engineering Planning and Design, I. Basic Principles (John Wiley & Sons, New York, 1975). [2] A.H.-S. Ang and W.H. Tang, Probability Concepts in Engineering Planning and Design, II. Decision, Risk, and Reliability (John Wiley & Sons, New York, 1984). [3] V.T. Chow, Frequency analysis of hydrologic data with special application to rainfall intensity, Engineering Experiment Station Bulletin 414, University of Illinois, Urbana (July 1953). [4] V.T. Chow, Handbook of Applied Hydrology (McGraw Hill Book Co., New York, 1964). [5] D.M. Hershfield, Rainfall frequency atlas of the United States for durations from 30 minutes to 24 hours and return periods from 1 to 100 years, Weather Bureau Technical Paper No. 40, U.S. Dept. of Commerce (May 1961). [6] G.W. Kite, Frequency and Risk Analysis in Hydrology (Water Resources Publications, Littleton, Colorado, 1977). [7] C.S. Melching, A reliability analysis on flood event forecasting with uncertainties, Ph.D. thesis, Dept. of Civil Eng., Univ. of Illinois at Urbana-Champaign (1987). [8] J.L.H. Paulhus et al., Manual for estimation of probable maximum precipitation, Opera. Hydrol. Rept. 1, WMO No. 332, Geneva (1973). [9] E.J. Plate and J. Ihringer, Failure probability of flood levees on a Tidal River, in: Stochastic and Risk Analysis in Hydraulic Engineering, ed. B.C. Yen (Water Resources Publications, Littleton, Colorado, 1986), pp. 45-58. [10] U.S. National Weather Service, Precipitation-frequency atlas of the conterminous Western United States (by states), NOAA Atlas 2, 11 vols., U.S. NWS, Silver Spring, Maryland (1973). [11] U.S. National Weather Service, Five to 60-minutes precipitation frequency for the Eastern and Central United States, NOAA Tech Memo NWS HYDRO-35, U.S. NWS, Silver Spring, Maryland (1977). [12] U.S. Water Resources Council, A uniform technique for
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determining flood flow frequencies, Hydrology Committee Bulletin 15, Washington, DC (December 1967). [13] U.S. Water Resources Council, Guidelines for determining flood flow frequency, Hydrology Committee Bulletin 17B, Washington, DC (September 1981). [14] I.B. Wall, Probabilistic assessment of flooding hazard for nuclear power plants, Nuclear Safety, 15 (1974~ No. 4, pp. 399-408. [15] B.C. Yen, Risks in hydrologic design of engineering projects, Jour. Hydraulics Div., ASCE, 96 (1970) No. HY4, pp. 959-966. [16] B.C. yen, Rainfall-runoff process on urban catchmets and its modeling, in: Urban Drainage Modelling, ed. C. Maksimovic and M. Radojkovic (Pergamon Press, Oxford, 1986) pp. 3-26. [17] B.C. Yen, Reliability of hydraulic structures possessing random loading and resistance, in: Engineering Reliability and Risk in Water Resources, ed. L. Duckstein and E. Plate, NATO ASI Series E (Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1987) pp. 95-113. [18] B.C. Yen and B.-H. Jun, Risk consideration in design of storm drains, in: Proceedings, 3rd IAHR/IAWPRC Internat. Conf. Urban Storm Drainage, Vol. 2, ed. P. Balmer, P. Malmqvist and A. Sjoberg (Chalmers University of Technology, Goteborg, Sweden, 1984) pp. 695-704. [19] B.C. Yen and W.H. Tang, Reliability of flood warning, in: Stochastic Processes in Water Resources Engineering (Water Resources Publications, Littleton, Colorado, 1977) pp. 333-347. [20] B.C. Yen, S.T. Cheng and C.S. Melching, First order reliability analysis, in: Stochastic and Risk Analysis in Hydraulic Engineering, ed. B.C. Yen (Water Resources Publications, Littleton, Colorado, 1986) pp. 1-36. [21] B.C. Yen, S.T. Cheng and W.H. Tang, Reliability of hydraulic design of culverts, in: Proceedings, Internat. Conf. Water Resources Development, IAHR APD Second Congress, Vol. 2, (Taipei, May 1980) pp. 991-1001. [22] V. Yevjevich, Probability and Statistics in Hydrology (Water Resources Publications, Littleton, Colorado, 1972).