Surface-water quality and flow Modeling Interest Group

Real-Time Control of the Salt Front in a Complex, Tidally Affected River Basin

¹ OptiQuest Technologies, Greenville, SC
² U.S. Geological Survey, Columbia, SC
³ Riverside High School, Greer, SC

Please direct correspondence to:
Edwin A. Roehl  OptiQuest Technologies, LLC  214 Pelham Davis Circle  Greenville, SC 29615  Internet: info@oqt.com  Phone: (864) 987-0717  FAX: (864) 234-7521	or	Paul A. Conrads  USGS  Stephenson Center, Suite 129  720 Gracern Road  Columbia, SC 29210-7651  Internet: pconrads@usgs.gov  Phone: (803) 750-6140  FAX: (803) 750-6181

This version of the article has all of the figures inline. A version with all of the figures converted to thumbnails, with links to the larger images, is also available; the download time for the thumbnail version will be shorter, but the thumbnail figures may be less convenient for viewing and printing.

Citation:
Roehl, E.A. and Conrads, P.A., 2000, Real-Time Control of the Salt Front in a Complex, Tidally Affected River Basin in Proceedings of the ANNIE 2000 Conference, November 5-8, 2000, St. Louis, MO.

Table of Contents

Abstract
Introduction
Data Preparation
Modeling the Gain
Prediction and Control
Discussion and Conclusions
References

Abstract

Introduction

The U.S. Geological Survey (USGS) participated in comparing artificial neural network (ANN's) models to deterministic finite-difference models of the Cooper River, a complex estuarine system shown in Figure 1 (Conrads and Roehl, 1999). Both models were developed from three years of real-time measurements of water level (WL), dissolved-oxygen concentration, water temperature, and specific conductivity (SC, used to compute salinity) that had been collected by a network of gauging stations. The models predicted the river's hydrodynamic, mass transport, and water-quality behaviors. The ANN's were found to be significantly more accurate and quickly developed. Their compactness and fast execution allowed their use in a prototype control system that was used to investigate regulating wastewater discharges according to the river's assimilative capacity (Roehl and Conrads, 1999).

Figure 1. The Cooper and Wando River, SC.

Subsequently, ANN models were configured to spatially interpolate between gauging stations to predict WL and SC at arbitrary locations within the gauging network (Fig. 2). Of particular interest was predicting the "salt front" location, which is normally in the vicinity of station 02172053 (hereafter "s" replaces the station prefix 021720). This result pointed to a solution for protecting a valuable freshwater reservoir from saltwater migration. A canal located 10 km above s50 connects the west branch of the Cooper River to the reservoir.

Figure 2. ANN model's spatially interpolated SC and WL for one tidal cycle six hours apart.

The location of the salt front depends on tidal conditions and the volume of freshwater flowing downstream from a hydroelectric dam on Lake Moultrie. The WL's power spectrum revealed a strong 28-day lunar cycle. Low freshwater flows coinciding with high tidal levels allows seawater to migrate towards the canal. Figure 3 shows that the SC at s50 peaks a few tidal cycles after peaking at s53 due to upstream migration of the salt front.

Figure 3. Detail of actual SC at s53 and s50 (30 and 45 km upstream of s710 respectively).

Government regulations require minimum total weekly flows from the dam meet water supply demands, however, the timing of the releases is at the discretion of hydroelectric dam operators. The power utility has implemented a number of "action alert thresholds", which when exceeded, require that flows be increased above minimum levels. The SC threshold at s50, which is nearest to the canal, is 1,500 micro-siemens per centimeter (µs/cm), however, it rarely comes into play because thresholds for stations further downstream are exceeded some 20 to 30 times per year. The optimum user of power generation from the dam is to meet peak power demands. When water is released to control the salt front it significantly undermines the commercial operation of the dam.

Below is an alternative to the above regulatory control approach. It fully utilizes the data from the gauging stations to optimize the control of the freshwater releases, bringing significant benefit to the utility. The scheme uses ANN-based models to predict and control the location of the salt front in real-time.

Data Preparation

Modeling the Gain

Figure 4. Detail of actual and filtered WL at s011 and s710.

Separating the signal involves the construction of a function that correlates the WL at the dam to the WL at the river's mouth. For the function's input, the WL at s710 was selected over WL's from upriver stations because it is the least likely to be influenced by the dam releases. Linear correlation (Press et al, 1993) can relate the time series of one variable to that of another. It can also relate a current measurement of a variable to past measurements of the same variable. Successive shifting of one time series relative to another (or to itself) by a delay _d, while computing the correlation for each shift, provides information indicating when the correlation reaches maximum, minimum, and zero values. The first delay at which the correlation equals (or nears) zero is the delay _z at which the times series are "decorrelated".

The bivariate function that was used to correlate the WL's at s011 and s710 is derived from an approach applied to univariate chaotic time series (Abarbanel, 1996). Equation 1 suggests that at time t, a predicted value x_p for variable x can be computed from previous, actual measurements x_a by a function F. The "local dimension" d_L specifies the number of measurements required to optimally predict x_p. Note that x_p(t) = x_a(t) at _d = 0, and the predictive accuracy of F declines towards zero as _d approaches _z.

x_p(t) = F[x_a(t-_d),x_a(t-(_d+_z)),,,x_a(t-(_d+k_z)),,,x_a(t-(_d+(d_L-1)_z))], 0<_d<_z    (1)

The bivariate form of Equation 1 is given by Equation 2, which relates y and x, the WL's at s011 and s710 respectively. Predictions y_p are computed from actual measurements x_a by the function F₁. Comparing Equations 1 and 2, _d is replaced by the delay '_d at which y is maximally correlated to x, while d_L and _z remain characteristics of only x. The residual y_r given by Equation 3 is the difference between the actual and predicted measurements, y_a and y_p.

y_p(t) = F₁[x_a(t-'_d),x_a(t-('_d+_z)),,,x_a(t-('_d+k_z)),,,x_a(t-('_d+(d_L-1)_z))]; '_d>0    (2)

y_r(t) = y_a(t)-y_p(t)    (3)

Within the accuracy limits of F₁ and y_a, y_r contains information about the behavior of y that is unrelated to the tidal effects seen at the river's mouth. This information includes but is not limited to freshwater releases from the dam, therefore, y_r can be used to estimate the gain that relates changes in the WL at the dam to the SC at s50, denoted by y and y₂ respectively. Equation 4 was used to compute predictions y_2p. It was expected that the gain would also depend on tidal conditions, so that F₂ includes information from both y_r and y_p. Note that F₂ uses different delays and local dimensions in relating y_r and y_p to y₂.

y_2p(t) = F₂{[y_p(t-'_pd),y_p(t-('_pd+_pz)),,,y_p(t-('_pd+m_pz)),,,y_p(t-('_pd+(d_pL-1)_pz))],    
      [y_r(t-'_rd),y_r(t-('_rd+_rz)),,y_r(t-('_rd+n_rz)),,,y_r(t-('_rd+(d_rL-1)_rz))]}; '_pd,'_rd>0    (4)

The application of Equation 2 was as follows. '_d and _z were determined to be 10 and 30 hours respectively. F₁ was synthesized from time series of x_a and y_a by a feed-forward ANN. The ANN was trained using back-propagation and conjugate gradient methods. A d_L8 was determined experimentally by adding and removing inputs at delays spaced by _z and tracking the predictive performance of F₁. It was also determined that up to two inputs with delays less than d_Lz could be omitted without significantly degrading F₁. A plot of the predictions made by F₁ and the actual WL are shown in Figure 5.

Figure 5. Detail of actual and predicted filtered WL at s011.

The application of Equation 4 was as follows. The peak correlation '_pd between y_p and y₂ occurred at 0 hours, indicating that the SC at s50 and the WL at s011 move in phase. A high pass filter with a lower limit of 28 days was applied to y_p to remove apparent annual periodicity, whereupon _pz was calculated to be 30 hours (the same as _z). The delay '_rd of the peak correlation between y_r and y_2a was 47 hours, indicating a transport delay of about two days between dam releases and a subsequent effect on the SC. _rz was computed to be 69 hours. A second ANN was developed to synthesize F₂. The rationale for using a single ANN, which combined inputs for both y_r and y_p, was that they were decorrelated by means of their derivation (also verified by correlation analysis). Local dimensions d_rL6 and d_pL8 were estimated as described above for d_L. A plot of the prediction made by F₂ and the actual SC is shown in Figure 6.

Figure 6. Detail of actual and predicted filtered SC at s50.

The gain of F₂ is better understood by viewing the function's response surface under different tidal conditions. Figure 7 shows that the gain's sensitivity to the first two residual inputs y_r(t-'_rd) and y_r(t-('_rd+_rz)), indicated by the difference between the surface's high and low points, varies from 100 to 1,000 µs/cm at low and high tidal levels. Note that F₂ was developed from hourly data which filtering effectively averaged over two tidal cycles. Thus, the 1,000 µs/cm gain corresponds to an unfiltered range of about 1,900 µs/cm (see Fig. 3). Therefore, the 1,500 µs/cm threshold at s50 is well within the interpolative range of F₂.

Figure 7. Predicted SC at s50 on dam releases during low and high tidal levels.

Prediction and Control

Effective control requires predicting ahead of time if salt front migration will pose a problem to allow time for corrective action. A model that predicts SC at s50 47 hours into the future (equal to '_rd) is needed. F₂ is unsuitable because it includes '_pd=0; however, another model F₃, identical to F₂ but with '_pd= '_rd, was synthesized to show feasibility. The predictions of F₃, shown in Figure 8, were poorer than those of F₂, but it was able to predict some of the events corresponding to the largest values of y_2a. F₃ could be easily improved by using additional gauging station variables.

Figure 8. Actual and predicted filtered SC at s50.

Figure 9 shows an idealized model-based control scheme for controlling the salt front, which has been borrowed from the field of industrial process control. Three types of variables are indicated. The variable to be controlled (CV) is the output of the model. Variables that are manipulated (MV) to take corrective action are inputs to the model. Additional inputs are the disturbance variables (DV) that describe the state of the process but cannot be manipulated. Finding values for the MV's so that an undesirable outcome can be avoided requires an optimization program. As DV's change with time, the optimization program searches for MV values that avoid letting the model predict an undesirable outcome. The optimization program adheres to "constraints" that limit the allowable values of the MV's. If it fails to obtain an acceptable outcome within a specified number of iterations, the program returns values that minimize the deviation from a desirable outcome.

Figure 9. Idealized controller uses an ANN-based process model with an optimization.

For this application, F₂ is the process model, the CV is y_2p, the MV is y_r(t-'_rd), and the DV's are the remaining values specified by Equation 4. A minor problem is that because '_pd=0 and t=+47 hours into the future, the DV y_p(t-'_pd) is unknown. However, a good estimate can be synthesized from Equation 2 using '_d=47 hours. As new values for the DV's are input with each time step, the optimization program's is to compute values for the MV that causes the model to predict a CV that is at or below the 1,500 µs/cm threshold at s50.

Discussion and Conclusions

Possible improvements could come from several directions. One would be to use actual flow measurements from the dam instead of the surrogate WL at s011. Including additional information from the plethora of variables available from throughout the gauging network might lead to dramatic improvements in prediction accuracy. The investigation of alternative non-linear calculation methods for _z, such as average mutual information (Abarbanel, 1996) or single-input-single-output ANN's, may also be beneficial.

Leaving ample room for refinement, the above approach has revealed relationships between measured variables that match a qualitative understanding of this very complex estuarial system. The straightforward mechanics of constructing the various components of the application point to a reliable solution to controlling the location of the salt front. Earlier work in applying these methods to water quality, which is also affected by saltwater migration, suggest combining these problems because of their commonality in serving the public interest.

References

Conrads, P.A., and Roehl, E.A. (1999), "Comparing Physics-Based and Neural Network Models for Predicting Salinity, Water Temperature, and Dissolved-Oxygen Concentration in a Complex Tidally Affected River Basin," South Carolina Environmental Conference, Myrtle Beach, March 15-16.

Press, William H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. (1993), Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press.

Roehl, E.A., and Conrads, P.A. (1999), "Real-Time Control for Matching Wastewater Discharges to the Assimilative Capacity of a Complex, Tidally Affected River Basin," South Carolina Environmental Conference, Myrtle Beach, March 15-16.

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