Following each iteration, two CE-QUAL-W2 simulations (confirmation and confirmation with re- leases swapped) are added to the training data set; therefore, following the first iteration (which uses a robust training data set described below) each training data set consists of 2·(iteration−1) CE-QUAL-W2 simulations.
The algorithm’s stopping condition is based upon the “best iteration” index at the end of each iteration. If any water quality constraint is “active”, meaning not fully satisfied, the absolute mean error (AME) between the NARX and CE-QUAL-W2 water quality predictions is checked. If the AME is greater than 0.5 °C for temperature or 0.5 mg/L for DO (AME thresholds lower than ac- ceptable levels given byCole and Wells(2007)), the iterations solution is not acceptable and the best iteration is set to the previously found best iteration, or null in the case of no acceptable solution found thus far. If the AME is acceptable and the best iteration is null thus far, the current iteration is the best iteration. If a best iteration has been determined already, the water quality violation from the constraint limit is compared between the current iteration and the previously found best iteration.
If the current iteration achieves a smaller water quality violation, it becomes the new best iteration.
If there are no “active” water quality constraints (i.e., these constraints are fully satisfied), then the best iteration is based on the objective function valuation, which represents the power value. The power value of each iteration is compared to the power value of the best iteration found thus far, and if the new solution results in greater power value, it then becomes the new best iteration.
AME values are less than 1 °C for temperature and 1.5 mg/L for DO (Cole and Wells, 2007). Error metrics are within this threshold with the exception of Old Hickory in-stream temperature profiles for both years, likely due to only having daily temperature values available for the mainstem inflow.
Old Hickory reservoir’s main inflows consist of releases from two upstream dams, both of which are stratified in the summer and have outlet structures at multiple depths. The temperature and other water quality characteristics of the upstream dams’ discharges are strongly impacted by release decisions, which much like Old Hickory reservoir are adjusted by operators on a short timescale.
Consequently, the water quality of upstream releases is not adequately captured by a single mea- surement each day, thereby resulting in larger water quality prediction errors at profile locations in the upstream half of the reservoir.
Additionally, the original developers of the Old Hickory model separated side bank storage volume from mainstem conveyance volume by use of a separate branch and a series of weirs con- necting the storage branch to the mainstem. While this may improve hydrodynamics modeling, this methodology does not properly represent the water quality phenomenon of the system. This makes a particular impact in the forebay of the reservoir, where the additional storage branch (Branch 10) enters the mainstem (Branch 1) as shown in Figure III.4. While the model is not constructed as de- sired, CE-QUAL-W2 emulation by surrogate model and integration within an optimization scheme is demonstrated using the Old Hickory model regardless of model structure and accuracy. The focus here is transition from high-fidelity simulation to reduced surrogate model, not transition from the true system to high-fidelity simulation model.
(a) (b)
Figure III.4: Bathymetry of Old Hickory reservoir CE-QUAL-W2 model, showing (a) plan view of all branches and (b) elevation view of the mainstem, Branch 1 (created using AGPM-2D v3.5 post-processor for CE-QUAL-W2 by Loginetics, Inc.).
Table III.1: Summary of Old Hickory CE-QUAL-W2 model calibration and validation results.
Calibration Validation
Year 1988 2005
Computational Time (minutes) 9 9
Elevation AMEa(meters) 0.025 0.053
Dam Releases:
Temperature AMEa(°C) 0.963 0.617
DO AMEa(mg/L) 1.010 1.196
In-stream Profiles:
Temperature AMEa(°C) 2.076 1.350
DO AMEa(mg/L) 0.943 0.716
a Errors are presented as absolute mean error (AME). In-stream profile measurements of temperature and DO were collected at 8 locations on 7 dates in the calibration year (1988) and at 7 locations on 2 dates in the validation year (2005).
The Old Hickory tailwater is considered the point of compliance and monitoring for water qual- ity by dam operators; therefore, ANN models were trained to emulate the hourly discharge tem- perature and DO predictions of the CE-QUAL-W2 model. Based on observations made during CE-QUAL-W2 model calibration and validation, the discharge temperature and DO at Old Hick- ory are sensitive to only the two most dominant upstream inflows: Branch 1 (the mainstem) and Tributary 2 (Caney Fork, and the Center Hill dam discharge). Flowrates, temperatures, and DO concentrations for these two inflows were included in an initial exogenous input set. Additionally, meteorological data and operational data (spill and turbine flowrates) were included. Using the 2005 Old Hickory CE-QUAL-W2 model inputs and outputs, correlation tests were performed to narrow the set of exogenous inputs to the main driving factors for discharge temperature and to esti- mate the appropriate sets of input and feedback delays. Examples of correlation plots for discharge temperature are shown in Figure III.5 for demonstration. Exogenous inputs with low correlations were removed from the set. For the narrowed exogenous variable set, correlations with discharge temperature and DO were maximized in the vicinities of 0, 1, and 12 hour delays; hence, the input delay set was assigned to these values. Lagged autocorrelation testing of the discharge temperature and DO output time series show decreasing correlation over time, meaning a single feedback of 1 is appropriate. The resulting sets of exogenous variables for temperature and DO NARX models are given in Table III.2. The number of hidden layers and neurons in each layer were assigned to
the default values of 1 and 10, respectively, following sensitivity testing that revealed an increase in these values yielded little to no improvement in prediction ability at considerable computational expense.
Training data for Old Hickory NARX WQMs was generated by combining dominant inflows, outflows, and meteorological data time series. For each input type, three variations were considered.
Meteorological conditions consisted of the 2005 (average year), 2006 (wet year), and 2007 (dry year) values. Inflow temperatures and DO concentrations consisted of the values from 2005 and the 2005 values were increased and decreased by 5%. Inflows were not varied, but outflows were varied to create heavy spill and heavy turbine scenarios. The heavy spill scenario was created by allocating 20% of the 2005 turbine outflow to the spill gates, and the heavy turbine scenario was created by allocating 20% of the 2005 spill outflow to the turbine structure outflow. Spill and turbine scenarios were not combined exhaustively, but instead were paired to maintain an equivalent total outflow to maintain water balance stability in the CE-QUAL-W2 simulations. This process creates a surrogate model which can be used to explore the trade-off between releases through the turbines and spill gates. An exhaustive combination of all variables, with the exception of the paired spill and turbine inputs as explained, resulted in a total of 729 CE-QUAL-W2 model simulations.
Seventy percent of the simulations were provided to the training algorithm and the remaining thirty percent saved for final validation. To minimize the impact of substantial oscillatory noise found in some CE-QUAL-W2 simulation results, the water quality predictions were smoothed us- ing a 24-hour moving average process prior to training. A smoothing approach was selected in order to avoid removing runs from the design of experiments set; with the understanding that the
Table III.2: Exogenous variables lists for Old Hickory discharge NARX models.
Discharge Temperature Discharge DO Branch 1 Inflow Branch 1 Inflow Branch 1 Temperature Branch 1 Temperature Tributary 2 Temperature Branch 1 DO
Air Temperature Tributary 2 Temperature
Dew Point Tributary 2 DO
Turbine Flow Air Temperature
Spill Flow Dew Point
Turbine Flow Spill Flow
-30 -20 -10 0 10 20 30 Lag (hours)
-1 -0.5 0
Cross Correlation
Discharge Temperature & Turbine Outflow Cross Correlation Function
(a)
-30 -20 -10 0 10 20 30
Lag (hours) -1
-0.5 0
Cross Correlation
Discharge Temperature & Branch 1 Inflow Cross Correlation Function
(b)
-30 -20 -10 0 10 20 30
Lag (hours) 0
0.5 1
Cross Correlation
Discharge Temperature & Air Temperature Cross Correlation Function
(c)
-30 -20 -10 0 10 20 30
Lag (hours) -1
-0.5 0
Cross Correlation
Discharge Temperature & Tributary 2 Inflow Cross Correlation Function
(d)
Figure III.5: Old Hickory discharge temperature lagged cross correlation test examples for (a) tur- bine outflow, (b) branch 1 inflow, (c) air temperature, and (d) tributary 2 inflow with 95% confidence bounds. Inputs shown in (a), (b), and (c) are considered correlated with discharge temperature and are included in the NARX model exogenous variables, while input (d) is not.
initial set of NARX models provides somewhat “smoothed” predictions due to the wide range of conditions in the training data set; and due to the fact that the NARX models are later updated in a retraining step within the optimization process, which is based upon non-smoothed CE-QUAL-W2 outputs. The training algorithm randomly divides its portion of data between training (70%), vali- dation (15%), and test (15%) subsets. The training subset is used to compute gradients and update network weights and biases, the validation subset for computing errors and determining when to halt the training routine, and the test subset for confirming an appropriate division of data by com- paring when the test subset and validation subset errors reach their minimums. Figure III.6 provides a visual demonstration of the random data division, with each box representing a CE-QUAL-W2 simulation.
Because the models are trained using an optimization algorithm that incorporates a random process, temperature and DO networks were each trained five times. After five networks were constructed and bias correction performed, an interior point constrained nonlinear optimization al- gorithm was employed to compute network weights (which sum to 1) that minimize the validation set error. After the first weight set was computed, any networks with a weight less than 25% of the maximum weight were removed and the weights recomputed for the smaller set of NARX models.
This removes inferior networks from the set while still maintaining a “family” of networks that may provide better global predictions than a single trained network. In this application, the temperature surrogate model consists of 4 weighted NARX models and the DO surrogate model consists of 4 weighted NARX models.
Final Validation Set Set Provided to train()Function
Set aside 30%
for final validation NARX #1 NARX #2 NARX #3 NARX #4 NARX #5
Final Validation Train (70%) Validate (15%) Test (15%)
Figure III.6: Data division demonstration for NARX model training. Each box represents 1% of the total set of CE-QUAL-W2 simulations resulting from design of experiments.