II.3.2 - Monte Carlo Flow Calculation

2. Read period-averaged flows at dams from the flow archive file
3. Modulate period-averaged dam flows to give daily dam flows
4. Modulate losses in reservoirs
5. Propagate upstream flows to determine daily headwater flows as well as gains and losses from river segments
6. Propagate downstream flows through all river segments using the headwater flows and the segments' gains and losses.

• Regulated headwater segment has a dam, a storage reservoir, and a river source.
• Unregulated headwater segment has a confluence at its downstream end and a river source at its upstream end.
• Loss is a withdrawal (+) or deposit (-) of water to a river segment from an unspecified source. Losses are used to represent irrigation removals and ground water returns to river segments.
• Dams are points that regulate flow, but only dams specified in the flow archive file are considered to be regulation points.
• Confluences are points where two flows upstream of the confluence combine to create the flow downstream of the point.

Fig. 6 Main objects for the Flow submodel

Hydroregulation Models

Flow files for the Monte Carlo runs are obtained from Flow Archive files that are generated from runs of hydroregulation models maintained by two agencies:

• HYDROSIM is run by the Bonneville Power Administration
• HYSSR is run by the U.S. Army Corps of Engineers

The models use information on natural runoff, regional electrical demand and storage capacity of the reservoirs to model the stream flow on a period averaged basis. Models use historical flow records for natural runoff and generate river flows that meet power generation demand in monthly periods. The exceptions to the monthly periods are April and August which are each divided into two periods. In addition, the HYDROSIM model provides elevations of all reservoirs.

Flow Modulation

Flow inputs in the Monte Carlo Mode runs consist of predicted daily flow averaged over monthly or bimonthly intervals at each dam used in CRiSP.1. This input generated from HYDROSIM, or HYSSR flow archive files typically looks like Fig. 7 below. While this record retains most of the annual and seasonal flow variations, actual historic river flows (Fig. 8) exhibit considerable weekly and daily variations that are not replicated by the hydroregulation models used as flow data for CRiSP.1.

The purpose of the modulator is to more accurately simulate real flow patterns encountered by adding variations at finer time-scales consistent with historic flows. These variations include both random and deterministic components.

Fig. 7 Hydroregulation model simulated input - Wells, 1981

Fig. 8 Historic flows at Rocky Reach, 1981

Spectral Analysis of Flow

The CRiSP.1 modulators were developed from the following analysis of flows in the Columbia River system. The goal was to develop a modulator that represented daily and weekly variations in flow and had the same spectral qualities as the flows in the river system as it is now operated.

A spectral analysis of an eleven-year time series (1979-1989) of flows revealed the general trend is a decline in spectral power that is qualitatively similar to a pink noise spectrum¹. In addition, the spectrum has distinct peaks at frequencies of 1/7, 2/7, 3/7 etc., indicating a seven day cycle (Fig. 9).

This spectrum suggest several distinct processes. The weekly component is the result of flow decreasing on weekends when electric power consumptions is less. The pink noise element of the spectrum is probably the result of seasonal and short term correlations in weather patterns that alter the power consumption and unregulated runoff directly.

Fig. 9 Spectrogram: eleven year time series

Modulator Applications

The strategy for using period averaged archive flows to simulate flows with the spectral qualities of the actual ones involves adding flow variations at several points in the system (Fig. 10). These variations are produced by modulators. Since flows start in the headwaters and are summed downstream, flow variation can be added sequentially according to the manner by which they are produced. First, the archive flows are prescribed at all dams. Next, three modulations are applied. Weekly and daily modulations are added at the regulated headwaters to reproduce variations that occur between dams from additions and subtractions of water in the river segments and a loss modulation is added at downstream dams. After modulation, an upstream propagation process is applied to calculate the flows in unregulated headwaters. This forces the total modulation into the unregulated streams. In the case of the weekly modulation this is an artifact since it is induced by hydrosystem operation. The error is not significant though, since the weekly modulation is a small fraction of the total variation.

Fig. 10 Points of flow modulation in system.

Weekly Modulators

The weekly modulation, applied in the regulated headwaters, simulates hydrosystem power generations patterns in which electrical demand decreases on weekends. The modulators, producing lower flows on weekends and higher flows midweek (Fig. 11), are approximated with a three-term Fourier series with fixed amplitude. The equation is

(1) where

• F(t)_{week (j)} = weekly variation in flow for headwater dam j
• G = flow scaling factor in kcfs

  This is set to 12.0 to reproduce the observed weekly variation in flow at Wells Dam for the years 1979 to 1989 excluding 1983 for which flows are missing.

• a_n, b_n = Fourier coefficients

  a₁ = 1, a₂ = 2/3, a₃ = 1/3

  b₁ = 6/7, b₂ = 4/7, b₃ = 2/7

• t = day of the year
• = offset for day of week alignment.

Fig. 11 Weekly shape pattern

Daily Modulators

Daily modulation simulates all variations not associated with the weekly and seasonal variations. A discrete realization of an Ornstein-Uhlenbeck (OU) process (Gardiner 1985) was used to generate the daily variation. The process has two important characteristics: variations are slightly correlated from one day to the next and variances stabilize over time. This is a correlated random walk in which autocorrelation decays in time. The stochastic differential equation for an O-U process is

(2) where

• F_day = daily variation in flow in kcfs at headwater dam
• r = deterministic rate of change of flow per unit of flow (the range is confined such that 0 < r < 1)
• = intensity on the random variations in flow
• w(t) = Gaussian white noise process describing the temporal aspects of the flow variation.

(3)

where the mean and variance of the process are defined

(4) (5) When rt is large enough that exp (-2rt) is negligible, m and V² tend to be constant values and the time series is stationary.

Changing the continuous differential equation into a discrete one with t = 1 reservoir time step, and rearranging gives

(6)

r = 0 gives an unbiased random walk, r = 1 gives a series of uncorrelated normal variates.

For the modulators, a system in stochastic equilibrium is sought such that m = 0. Taking X₀ = y = 0 gives m = 0, and discarding the first 35 iterations yields stable variance for any value of r useful in this context. Modulator parameters selected for the different portions of the system are given in Table 1 and are based on daily flow data for the years 1979 to 1989 at Wells and Lower Granite Dams.

Table 1 Daily modulator parameters for river
River	j	rj
Upper Columbia	13	0.5
Lower Columbia	13	0.5
Snake	7	0.5

Random daily variation is added by a numerical form of an Ornstein-Uhlenbeck (O-U) random process created for each run (Fig. 12).

Fig. 12 O-U shape; r = 0.5, sigma = 13

Monte Carlo Flow Modulator Validation

Using daily flow records for Ice Harbor, Priest Rapids and John Day dams during 1981, monthly and bimonthly (April and August) average daily flows were computed and appended to a CRiSP.1 flow archive from which CRiSP.1 generated modulated flows for these dams. Graphs of observed and model-produced flows for the first 300 days of the year at John Day Dam appear in Fig. 13. The model appears to produce realistic patterns of flow variation that mimic natural flows very well.

At a finer scale, however, note that CRiSP-modulated flows generally exhibit less variability than do observed flows, e.g. compare January and July (Fig. 14). In general, modulated flows are about as variable as observed flows in January, but clearly less variable than observed flows in July. This is also reflected in the variance around the mean flow, given in Table 2. This phenomenon is probably due at least partially to "step-like changes" of flows in July that do not occur in January. There is some variation around the mean due solely to that trend, and this will not be captured in a purely random modulation scheme.

Fig. 13 Flows at John Day Dam, 1981

Fig. 14 January and July flows at John Day Dam, 1981

Flow Loss

The term `loss' represents withdrawals from the system, mainly for irrigation. These withdrawals are positive in CRiSP.1. Negative losses are return flows through ground water.

The loss data in a segment represents the change in flow that occurs between the flow input (calculated from the flow of upstream segments) and the flow output (stored as data in the segment). Where not specified, flow loss is set to zero.

During the upstream propagation operation, new flow loss values are computed for reaches that lie between two dams. A dam is said to have no component of unregulated flow if no unregulated headwater flows into the dam without first flowing through some regulation point.

For each reach r enclosed between a dam and upstream regulation points (Fig. 6), a new flow loss F_L(r) is set by distributing any mass imbalance over all reaches between the dam and/or regulated inflow points in proportion to each reach's maximum allowable flow:

(7)

where

• F _D(r) = flow output at dam immediately below reach r
• F_L(r) = new flow loss at reach r, as adjusted for mass imbalance
• F_M(r) = flow maximum at reach r
• F_M(i) = flow maximum at reach i
• F_R(j) = flow at regulation point j
• n = number of upstream regulated points
• p = number of reaches between dam r and all regulation point

Flow loss is not modified by the upstream propagation in any reach not fully enclosed by regulated headwaters or dams. After appropriate loss values are set, flow loss in every segment is used as input data for unregulated headwater calculations.

Fig. 15 Diagram of reach structure for loss calculation

Reservoir Loss Modulation

At downstream dams, variations in flow from losses due to irrigation and evaporation and additions from surface and subsurface groundwater flows are accounted for with loss modulators. The intensity of this variation is based on the differences in flows observed at adjacent dams as indicated in period averaged hydro-model flows (Fig. 16).

Fig. 16 Inputs at Rocky Reach minus inputs at Wells, 1981

The loss modulation is simulated with a white noise process (Fig. 17). A normal variate random factor is added to modulated flow of all run of the river dam. The equation is

(8) where

• F _{loss (i)} = modulated flow loss at downstream dam i
• _i = the standard deviation of the difference in flows (kcfs) at dam i and i +1 as computed by daily observed flows at all dams over the years 1979-1981.

Table 3 Flow loss modulator parameter for eq (8)
Dam	i (kcfs)		Dam	i (kcfs)
Bonneville	11.0		Little Goose	5.4
The Dalles	4.1		Priest Rapids	4.0
John Day	17.0		Wanapum	5.0
McNary	12.75		Rock Island	2.65
Ice Harbor	2.75		Rocky Reach	3.0
Lower Monumental	2.4		Wells	6.5

Fig. 17 Random factor modulation at Rocky Reach, 1981

Headwater Computation

Once flows are modulated at dams and the losses and gains are calculated, the headwater flows can be calculated with the algorithms described below.

Regulated Headwater

Regulated headwaters are storage reservoir outflows for the Monte Carlo Mode. No losses are considered for storage reservoir flows other than the dam outflow.

Unregulated Headwaters

Each unregulated headwater is examined. If the flow for a given headwater has not yet been computed, then flow for that and all adjacent unregulated headwaters is calculated.

The region of computation for a segment is defined as all segments within the river map subgraph with endpoints consisting of the nearest downstream dam, and the nearest regulation points or headwaters upstream from the dam. An example of a region with several unregulated headwaters is given in Fig. 18.

Fig. 18 Region of regulated FR and unregulated FU rivers

To calculate the unregulated headwater flows, first the total unregulated flow input to dam r (D(1) in Fig. 18) is computed by subtracting the total regulated flow from flow at dam r. The equation is

(9) where

• F_TU(r) = total unregulated flow input to dam r
• p = number of regulated flows in region
• F_D(r) = flow output at dam r
• F_R(j) = flow output at regulation point j

(10) where

• K _i = flow coefficient at unregulated headwater i
• q = number of adjacent unregulated headwaters in region
• F_{U max (i)} = maximum flow at unregulated headwater i or j

(11)

The logic for the unregulated flow calculation is complete except when flow at any unregulated headwater falls below the minimum set in columbia.desc for that headwater, which can be zero. In this case

(12)

and then for each reach r enclosed by dams the new loss F_L(r) is

(13)

where

• F _D(r) = flow output at dam immediately below reach r
• F_L(r) = new flow loss at reach r, as adjusted for mass imbalance
• F_M(r) = flow maximum at reach r or i
• F_R(j) = flow at regulation point j
• F_{U (i)} = flow at unregulated headwater i
• m = number of unregulated headwaters above r (m = 3 in Fig. 18)
• n = number of regulated points adjacent to nearest upstream regulation point (n = 2 in Fig. 18)
• p = number of reaches between dam r and all upstream regulation points (p = 9 in Fig. 18).

Downstream propagation of flow in the Monte Carlo Mode is computed after modulation, flow loss and unregulated headwater flows are computed. Starting at a headwater, flow is propagated by traversing the downstream segments, subtracting loss at each to determine new flow values, and adding flows together at confluences. Thus, flows are assigned at each segment in a downstream recursive descent traversal. The flow for each day is

(14) where

• F_i (t) = flow regulation point i at reservoir time increment t
• F_L(i) = flow loss at reach i
• F_j (t) = flow at regulation point j immediately upstream at reservoir time increment t.

The modulators are combined with archive flows to give daily flows at the dams according to the equation

(15)

where

• F(t)_i = modulated flow at dam i
• F(t) _{arch (i)} = archive flow at dam i
• F(t) _{day (i)} = daily modulated flow in regulated headwater j
• F(t) _{week (i)} = weekly modulated flow in regulated headwater j
• F _{loss (i)} = loss modulated flow in river segment upstream of dam i
• F_min(i) = minimum allowable flow at dam i
• J = number of regulated headwaters upstream of dam i
• I = number of dams upstream of dam i, including dam i

Columbia River Salmon Passage Model CRiSP.1.5 Theory, Calibration & Validation Manual
Copyright © 1996, Columbia Basin Research. All rights reserved.