FREQUENCY ANALYSES
The results of a SIM, SIMD, or SALT simulation are viewed from the perspective of frequency, probability, percentage-of-time, and reliability metrics associated with stream flows, reservoir storage, water supply diversions, hydroelectric energy production, and salinity concentrations. These metrics for estimating and communicating likelihood are covered in various chapters of the Reference, Users, Salinity, and Daily Manuals. This present Daily
Manual Chapter 6 summarizes those features that are covered elsewhere in the WRAP manuals
as well as introducing additional features that are not covered elsewhere. The new topics introduced in this chapter include flood frequency analysis and the HEC-SSP.
Overview Summary of WRAP Reliability and Frequency Analyses
Table 6.1 lists the various parts of the WRAP manuals that cover reliability and frequency analysis capabilities. The basic reliability and frequency analysis methodologies of the WRAP program TABLES presented in the Reference and Users Manuals were originally developed for application to the results of a monthly SIM simulation but are also applicable to the results of a daily SIMD simulation. The 6REL record activating a reliability analysis for daily diversion requirements is equivalent to the 2REL record which applies to monthly diversion requirements. Likewise, the 6FRE, 6FRQ, and 6RES records that develop frequency analysis tables for daily simulation results are analogous to the 2FRE, 2FRQ, and 2RES records that develop frequency analysis tables for monthly simulation results. Appendix C of this Daily
Manual describes Type 6 input records for TABLES, which consists of 6FRE, 6FRQ, and 6RES
records and time series records, along with types 5 and 7 records. TABLES type 2 records along with other record types are covered in Chapter 4 of the Users Manual.
TABLES types 2 and 6 records deal with monthly and daily, respectively, simulation
results. 2RES and 6RES records build three tables summarizing reservoir storage contents that include storage and draw-down frequency relationships. 2REL and 6REL records build diversion and hydropower reliability tables. 2FRE, 2FRQ, 6FRE, and 6FRQ records, combined with the DATA record, build frequency tables for any of the simulation results variables.
Short-term conditional reliability modeling (CRM) is described in Chapter 7 of the
Reference Manual. TABLES 2REL, 2FRE, 2FRQ, and 2RES records are applied in short-term
CRM as well as conventional long-term reliability and frequency analyses. The resulting 2REL, 2FRE, 2FRQ, and 2RES record tables include CRM parameters in the headings but otherwise have the same appearance as long-term analysis. The quantities in the tables are interpreted differently as explained in the Reference Manual. A SIMD CRM simulation can be performed with a daily computational time step, but the results are aggregated to monthly for reliability and frequency analyses. Thus, 6REL, 6FRE, 6FRQ, and 6RES records are not applicable for CRM.
Salinity simulation capabilities are documented by the Salinity Manual. Frequency analyses of salt loads and concentrations of stream flow and reservoir storage are performed with 8FRE and 8FRQ records, which are analogous to 2FRE and 2FRQ records. The frequency analysis computations are the same regardless of the variable from the SIM or SALT simulation results dataset that is being analyzed. The 8REL record is analogous to the 2REL record.
Simulation of reservoir operations for flood control is covered in the preceding Chapter 5. Flood frequency analysis capabilities are outlined later in Chapter 6.
Frequency analyses of SIM, SIMD, and SALT simulation results can also be performed with the HEC-SSP Statistical Software Package available from the Hydrologic Engineering Center (2010). As discussed later in this chapter, HEC-SSP provides both flexible graphics and comprehensive frequency analysis computational capabilities.
Table 6.1
Outline of Sections of Manuals Dealing with Frequency and Reliability Analyses _____________________________________________________________________
Reference Manual
Chapter 2 Overview
Measures of Water Availability and Reliability
(Volume and Period Reliability, Shortage Metrics, Frequency Analyses) Chapter 5 Organization and Analysis of Simulation Results
DATA Record Transformation of Simulation Results Data Reliability and Frequency Tables
Water Supply and Hydropower Reliability Flow and Storage Frequency Analyses
Reservoir Contents, Drawdown Duration, and Storage Reliability Chapter 6 Additional Auxiliary Modeling Features
Yield versus Reliability Relationships Including Field Yield Chapter 7 Short-Term Conditional Reliability Modeling
Users Manual
Chapter 4 Program TABLES
DATA Record – Data Transformation for Time Series and Frequency Tables 2REL Record – Diversion or Hydropower Reliability Summary
2FRE Record – Flow-Frequency or Storage-Frequency Relationships 2FRQ Record – Frequency for Specified Flow or Storage
2RES Record – Reservoir Content, Draw-Down Duration, and Storage Reliability 5CRM, 5CR1, 5CR2 Records – Conditional Reliability Modeling
Salinity Manual
8FRE, 8FRQ, and 8REL records
Daily Manual
Chapter 6 Frequency Analyses
Chapter 7 Examples Illustrating Daily Simulation Capabilities Appendix C Instructions for Preparing TABLES Input Records
6REL Record − Water Supply Diversion or Hydropower Reliability 6FRE Record − Flow or Storage Frequency Relationships
6FRQ Record − Frequency for Specified Flow or Storage 6RES Record − Reservoir Storage and Drawdown Frequency 7FFA Record − Flood Frequency Analysis
Reliability Metrics for Water Supply and Hydroelectric Power
Reliability tables for meeting water supply diversion or hydroelectric energy generation requirements are developed with the monthly 2REL record or daily 6REL record. A reliability table contains both volume and period reliabilities. Period reliabilities are developed for supplying specified percentages of the SIMD daily demand (6REL record) and SIM or SIMD monthly and annual demands (2REL and 6REL records). Examples of 2REL and 6REL diversion reliability tables are provided in Tables 7.13 and 7.14 of Chapter 7.
Volume reliability is the percentage of the total target demand amount that is actually supplied. For water supply diversions, the amounts are volumes. For hydroelectric power, the amounts are kilowatt-hours of energy generated. Volume reliability (RV) is the ratio of volume
of water supplied or the energy produced (v) to the target (V), converted to a percentage.
(100%) V
v
RV = (6.1)
Equivalently, for water supply, RV is the mean actual diversion rate as a percentage of the mean
target diversion rate. For hydropower, RV is the mean actual rate of energy production as a
percentage of the mean target energy production rate.
Period reliability is the percentage of the total number of periods of the simulation during which the specified demand target is either fully supplied or at least a specified percentage of the target is supplied. The various variations of period reliability (RP) are computed by TABLES
from the results of a SIM or SIMD simulation as: P
n
R = (100%)
N (6.2)
where n denotes the number of periods during the simulation for which the specified percentage of the demand target is met, and N is the total number of periods considered. The 2REL record develops a reliability table with periods defined alternatively in terms of both months and years. The 6REL record develops a reliability table from SIMD simulation results with periods defined alternatively in terms of both days (sub-monthly time step) as well as months and years. The 2REL and 6REL records allow N and n to be defined on a monthly basis optionally either considering all months or only months with non-zero demand targets.
Whereas volume reliabilities are based on total diversion volumes or hydroelectric energy amounts, period reliabilities are based on percent-of-time. Period reliability RP is an expression
of the percentage of time that the full demand target or a specified portion of the demand target can be supplied. Equivalently, RP represents the likelihood or probability of the target being met
in any randomly selected day, month, or year.
Volume and period reliabilities are the standard metrics adopted in most applications of WRAP. However, 2REL and 6REL record field 7 adds a shortage summary table to the standard volume and period reliability table. The metrics tabulated in the shortage summary table include the maximum shortage, vulnerability, resiliency, average severity, shortage index, average number of failures per sequence, and maximum number of consecutive shortages. These diversion shortage metrics are defined in Chapter 5 of the Reference Manual.
Frequency Analyses
The 2FRE, 2FRQ, 6FRE, and 6FRQ records are used to perform frequency analyses of
SIM and SIMD simulation results. These TABLES records allow frequency analyses to be
performed alternatively based on either relative frequency or the normal or log-normal probability distributions. The default relative frequency counting approach is typically adopted for most applications. However, as discussed in Reference Manual Chapter 5, the alternative of applying a probability distribution function may offer improvements in the accuracy of frequency estimates under appropriate circumstances.
Relative Frequency
Relative frequency is expressed by Eq. 6.3 where n is the number of days (6FRE, 6FRQ) or months (2FRE, 2FRQ) during the simulation that a particular flow or storage amount is equaled or exceeded, and N is the total number of days or months considered.
n
Exceedance Frequency = (100%)
N (6.3)
Normal and Log-Normal Probability Distributions
Alternatively, the TABLES frequency analysis records provide options to apply the normal (Eq. 6.4) or log-normal (Eq. 6.5) probability distribution to the simulation results variables generated by SIMD or SIM.
X = X + z S (6.4)
log X = log X + z Slog X (6.5)
X and S are the mean and standard deviation of the data from the SIM output file. log X and Slog X
are the mean and standard deviation of the logarithms of these data. The log-normal distribution consists of the normal distribution applied to the logarithms of X, with Eq. 6.4 expressed as Eq. 6.5 with z still derived from the normal probability distribution. The frequency factor (z) is derived from a normal probability table.
The random variable X in Equations 6.4 and 6.5 is the naturalized flows, regulated flows, unappropriated flows, instream flow shortages, reservoir storage volumes, or other variables from the SIM/SIMD simulation results or variations thereof reflecting adjustments made within
TABLES. The mean and standard deviation computed from these data using Equations 6.6 and
6.7 are the parameters used to model frequency relationship as the log-normal or normal probability distributions. For the log-normal distribution, the mean and standard deviation of the logarithms of the data from the SIM output file are computed and entered into Equation 6.5.
n i i=1 1 X = X n ∑ (6.6)
(
)
2 0.5 n i i=1 1 S = X - X n-1 ∑ (6.7)In the routines activated by the 2FRE or 6FRE record that apply the probability distribution options, exceedance probabilities are assigned to the random variable X by fitting the normal or log- normal distribution in the standard manner consistent with traditional statistical methods. The frequency factor z for specified exceedance frequencies tabulated in Table 6.2 are derived from the normal probability table and are built into TABLES. All of the frequencies listed in Table 6.2 are included if the results of the 2FRE/6FRE frequency computations are tabulated in columnar format. With the standard row format, several of the frequencies are omitted to reduce the width of the table.
Table 6.2
Frequency Factors for the Normal Probability Distribution
Exceedance Factor z in Exceedance Factor z in Frequency Eqs. 6.4 & 6.5 Frequency Eqs. 6.4 & 6.5
99.9% −3.09023 0.1% 3.09023 99.5% −2.57583 0.5% 2.57583 99% −2.32637 1% 2.32635 98% −2.05377 2% 2.05375 95% −1.64487 5% 1.64485 90% −1.28156 10% 1.28156 80% −0.84162 20% 0.84162 75% −0.67450 25% 0.67450 70% −0.52440 30% 0.52440 60% −0.25335 40% 0.25335 50% 0.00000
Format Options for Frequency Tables
The 2FRE and 2FRQ record frequency tables based on monthly quantities may be based on considering all months or alternatively be developed for a specified month of the year such as May or August. Likewise, the 6FRE and 6FRQ record frequency analyses of daily quantitities may be based on considering all days of the year or alternatively be developed for only those days falling within a specified month of the year such as May or August. Optionally, frequency analysis may be applied to moving averages or moving accumulative totals of the data computed for a user-specified number of months or days. The data may be adjusted by a multiplier factor, which may be a unit conversion factor or serve other purposes. The data may be added or subtracted from a constant. For example, the time series data for a frequency analysis of reservoir draw-downs is developed by subtracting water surface elevations from the user specified top of conservation pool elevation or storage volume may be subtracted from capacity. Likewise, deviations of regulated flows from a specified flow amount may be analyzed.
Tables 7.7, 7.8, 7.9, 7.10, 7.11, and 7.12 in Example 7.1 of the next chapter were developed in row format with 2FRE (monthly data) and 6FRE (daily data) records. 6FRE tables in column versus row format are compared in Table 7.28 of Example 7.4. The 2FRQ/6FRQ records also develop frequency table for the same types of variables, but in a different format illustrated by examples in Chapter 5 of the Reference Manual.
Frequency Analysis Variables
In applying a 2FRE or 2FRQ record, TABLES reads monthly simulation results from the
SIM or SIMD output OUT file. With a 6FRE or 6FRQ record, TABLES reads sub-monthly
(daily) simulation results from the SIMD sub-monthly output SUB file. The TABLES DATA record reads all of the simulation results quantities from either a monthly OUT or daily SUB file and creates datasets to be analyzed with 2FRE, 6FRE, 2FRQ, 6FRQ, and time series records.
Exceedance frequency tables may be created with TABLES 2FRE, 6FRE, 2FRQ, and 6FRQ records for the following variables without using a DATA record:
• naturalized flow, regulated flow, unappropriated flow, instream flow shortage, and reservoir storage volume for specified control points
• instream flow shortages and reservoir storage volume for specified water rights • reservoir storage volume and water surface elevation for specified reservoirs • other variables such as reservoir draw-downs computed from these variables
Many applications of 2FRE/6FRE and 2FRQ/6FRQ frequency analysis capabilities deal with stream flows or reservoir storage without needing a DATA record. However, a DATA record allows 2FRE/6FRE/2FRQ/6FRQ frequency analyses to be performed for any of the variables in a SIM OUT or SIMD SUB output file or other daily, monthly, or annual time series developed by the DATA record from the simulation results. The2FRE/6FRE and 2FRQ/6FRQ record routines can be applied to any data series created with a DATA record.
The TABLES DATA record is introduced in Chapter 5 of the Reference Manual and further explained in Chapter 4 of the Users Manual. The sole purpose of the DATA record is to transform SIM or SIMD simulation results to other time series variables of interest to be accessed as input by TABLES frequency analysis record and/or time series record routines. Although the DATA record has broad general applicability, environmental instream flow studies based on daily SIMD simulation results provide a key motivation for its design. The DATA record is illustrated in Examples 7.3 and 7.4 of Chapter 7. The DATA and 6FRE records are used in Example 7.4 to develop frequency metrics for the annual series of the minimum 7-day volume of naturalized flows. Example 7.3 of Chapter 7 applies the DATA and 6FRE records as well as the 7FFA record to perform frequency analyses of maximum annual reservoir storage contents.
The 35 time series record identifiers listed in the second column of Table 6.3 represent 31 time series variables read directly from the SIM/SIMD output OUT file or SIMD sub-monthly output SUB file and four other variables computed by combining simulation result variables. The variables are defined in Chapter 5 of the Reference Manual. The variables are associated with either control points, water rights, or reservoirs. Several of the water right variables are also aggregated by control point or reservoir in the SIM/SIMD simulation results.
The DATA record converts SIM or SIMD simulation results to other related datasets that can be read by the TABLES time series and frequency records. All of the simulation results time series variables from the SIM output OUT or SIMD SUB output files listed in Table 6.3 can be read with a DATA record and manipulated to create other datasets consisting of daily or monthly quantities or annual totals, minima, or maxima covering a specified number of time periods
defining a season of the year. Moving averages or totals may be adopted. The new dataset can be read and organized by time series records just like the original variables listed in Table 6.3. Time series record identifiers are listed in column 2 of Table 6.3. The DATA record also allows 6FRE, 2FRE, 6FRQ, and 2FRQ record frequency analyses to be performed for all of the variables listed in Table 6.3 and other derived datasets.
Table 6.3
SIM and SIMD Simulation Results Variables
Variable from Time Simulation Results for Specified
Simulation Results Series ID Control Point Water Right Reservoir
naturalized stream flow NAT control point
regulated stream flow REG control point
unappropriated stream flow UNA control point
channel loss CLO control point
channel loss credit CLC control point
return flow returned this control pt RFR control point
upstream reservoir release URR control point
control point inflow CPI control point
reservoir storage STO control point water right reservoir
evaporation-precipitation volume EVA control point water right reservoir
stream flow depletion DEP control point water right
diversion target TAR control point water right
diversion shortage SHT control point water right
diversion DIV control point water right
hydroelectric energy target TAR water right
hydroelectric energy shortage SHT water right
firm energy produced DIV water right
instream flow target IFT control point water right
instream flow shortage IFS control point water right
flow switch record volume FSV water right
flow switch record count FSC water right
return flow for water right RFL water right
available stream flow ASF water right
increase in available flow XAV water right
secondary reservoir release ROR water right
hydropower shortage HPS reservoir
electric energy generated HPE reservoir
stream flow depletions RID reservoir
inflows from reservoirs RIR reservoir
releases useable for power RAH reservoir
releases not accessible RNA reservoir
adjusted evaporation-precip depth EPD reservoir
evaporation-precipitation depth EVR reservoir
water surface elevation WSE reservoir
reservoir storage capacity RSC reservoir
HEC-SSP Statistical Software Package
The HEC-SSP Statistical Software Package software and user manual (Hydrologic Engineering Center 2010) are available from the Hydrologic Engineering Center (HEC) of the U.S. Army Corps of Engineers (http://www.hec.usace.army.mil/). The public domain software and manual can be downloaded from the HEC website free-of-charge. HEC-SSP performs frequency analyses with the results presented both in tables and graphs. HEC-SSP is comprised of the following components.
• A general frequency analysis component applies the log-Pearson type III, Pearson type III, log-normal, and normal probability distribution functions to perform frequency analyses for annual series of peak flows or other variables. The Weibull formula is also applied.
• A Bulletin 17B flood flow frequency component applies the log-Pearson type III probability distribution to annual series of peak stream flows following standard procedures outlined in Bulletin 17B of the Interagency Advisory Committee on Water Data entitled Guidelines for
Determining Flood Flow Frequency.
• A volume frequency analysis component is designed for analyzing daily stream flow or stage data. Annual series of minimum or maximum volumes each year during durations of one or more days are analyzed based on an input daily dataset.
• A duration analysis component shows the percent-of-time that a hydrologic variable exceeds specified values.
• A coincident frequency analysis component develops an exceedance frequency relationship for a variable as a function of two other variables.
• A curve combination analysis component combines frequency curves from multiple sources into one frequency curve.
Most of the computational capabilities of HEC-SSP applicable to WRAP are also found in the WRAP program TABLES. However, HEC-SSP provides convenient features for plotting