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Optimizing Plant Operations: What Can We Learn from Drunken Kangaroos?

Hydroelectric facility managers often struggle to develop flow release schedules that provide low-cost electricity while also preventing long-term damage to the downstream river. Use of a simulated annealing optimization process at Glen Canyon Dam has the potential to restore the local environment while actually increasing revenue from generation by 2%.

By Brent Travis

Operation of dams that impound water for hydroelectric facilities must take into account the potential effects of daily flow fluctuations on the downstream environment. Worldwide, dams equipped with generating facilities produce enough electricity to supply nearly one-fifth of the world's energy. Operation of these dams has the possibility to affect more than half of the world's large water systems in the process.1 Potential adverse effects can include ecological damage, environmental changes and water quality reduction.2

The Colorado River, downstream of Glen Canyon Dam in the Grand Canyon, provides a case study of the effects of river regulation on the environment. It is well-documented that eddy sandbars, which are formed by fine sands transported into eddies adjacent to the riverbanks, have been disappearing in this stretch of the river for more than 20 years.

The author presents an approach, called simulated annealing optimization, that can be used to develop an operational schedule for the Glen Canyon Dam powerhouse to offer the best outcome for the downstream environment while having the least negative effect on generating revenue. Discover how this process can be likened to a drunken kangaroo hopping around looking for the best way to climb Mount Everest.

Glen Canyon Dam

Owned and managed by the U.S. Department of the Interior's Bureau of Reclamation, the 1,296-MW Glen Canyon Dam project powers more than 600,000 homes. Water is run through the turbines to meet peak demand during the day, introducing high flows and an elevated river stage. As energy needs decrease at night, the flow from the powerhouse is reduced. As a result, the flow release schedule has created a sort of manmade daily tide.

During high flows, the water table rises and the corresponding river stage pressure stabilizes the sandbars. But when the water level begins to drop, the water table is still high, and seepage pressures build. With nothing to restrain this internal force, the sandbars break apart, often shrinking by 30% or more within just a few hours.

To minimize the risk of further damaging the area, managers made significant changes to the flow releases; most importantly, downramp and upramp rates were limited to about 40 m3/sec/hr and 70 m3/sec/hr respectively.3 Under these restrictions, it takes four hours to increase dam outflow from 200 m3/sec to 480 m3/sec and five hours to reduce back down to 200 m3/sec.

Electricity pricing follows demand and can fluctuate by as much as 30% in a single day. From an economic standpoint, flows should be maximized during peak hours and minimized during non-peak hours. Long ramp times equal wasted revenue, a problem that begs the question: Is there a most cost-effective way to protect the beaches?

The use of controlled flooding

Since 1996, the method of controlled flooding, also known as "beach building floods," has been used to release sediment trapped behind Glen Canyon Dam back into the river system, where it forms sandbars. Four beach building floods have been conducted, the most recent one in 2012.

Although effective, this mitigation method carries a significant cost. To meet the high flow requirements for a beach building flood, water must be let out of the bypass tubes as well as through the turbines. All flow through the bypass tubes is water that could have been used to generate power, and so there is a significant cost associated with this mitigation method. The 2012 flood is estimated to have cost $1.5 million. Despite the cost, the technique has been successful.

Of particular importance here are the results of a 1999 study.4 The scientists who conducted this study estimated that 84 high-elevation sandbars had been formed by the first beach building flood, but about 44% of these failed over the following six months, leaving 262 high-elevation sandbars of sufficient size to be used as campsites.4 If taken as representative data, the results of the 1996 flood suggest that each controlled flood will build about 47 new sandbars, constituting about 20% of the total stable sandbars after the flood.

Seeking a solution

Because research shows that controlled flooding replaces at least some of the lost sandbars, using this method more often as a mitigation measure should prevent further loss. However, questions arise as to how often controlled floods should be used and how the revenue loss plays into the overall goal of the floods.

Looking at these issues together results in one overarching question: What is the best way to combine ramping reductions and controlled flooding to minimize the cost of mitigating sandbar loss?

Formally speaking, the challenge is to optimize dam operations so as to minimize the cost of mitigating downstream slope failures subject to the constraints specific to physical limitations, water balance targets and environmental concerns.

Unfortunately, the extreme complexity of dam operations prevents normal optimization techniques from being successful. Even simple models of the Grand Canyon river system are too nonlinear to use traditional optimization methods. The optimal costs determined by such methods can vary from 18% to 52%, which is not particularly helpful.

When standard optimization techniques fail, metaheuristics may be the best alternative. Metaheuristics attempt to optimize a problem by iteratively improving candidate solutions, based on a predetermined success score. Metaheuristic solvers are typically simple to apply and easy to adjust for new constraints. The major drawback is that global optimization is not assured.

Simulated annealing (SA), one of the first published of the metaheuristic techniques, has been applied in many areas of water resource engineering, including water distribution networks,5 irrigation,6 reservoir systems,7 and storm water management.8 Based on the available knowledge and history of water resource applications, SA is a viable optimization tool.

Simulated annealing explained

SA optimizes by making an analogy between the problem parameters with the internal energy of the atoms in physical annealing. Physical annealing is a metallurgical process wherein a material is repeatedly heated and cooled.9 The success of physical annealing comes from exciting the energy states of the atoms within the material, which allows them to settle in different positions that reduce the material internal energy. The lower internal energy results in a material with greater ductility, fewer defects and other desirable qualities. So just like physical annealing seeks to find the "best" (minimum) energy states of the atoms, SA seeks to find the problem parameters that give the best solution.

A less technical but much more entertaining description of SA comes from Warren S. Sarle, an eminent statistician with the SAS Institute, who said:10

"[Numerical] optimization … can be likened to a kangaroo searching for the top of Mount Everest. … In simulated annealing, the kangaroo is drunk and hops around randomly for a long time. However, she gradually sobers up and the more sober she is, the more likely she is to hop up hill."

Figure 1 shows an example of Sarle's kangaroo. Each of the sketches (A through E) shows the progression of the kangaroo to the summit of the mountain, attained in 10 easy hops:

A. The drunken kangaroo initially hops to the top of a false summit (hop 1)

B. Having reached the local maximum, a sober kangaroo would stop, but the drunken kangaroo still jumps randomly about (hops 2-5), trying different spots to see if they are higher.

C. Eventually the kangaroo leaps over the nearby valley and lands on the primary incline (hop 6).

D. Now near the real maximum, the kangaroo will keep doing random hops, but at smaller and smaller distances (hops 7-10)

E. After 10 hops, the kangaroo, now mostly sober and at the global maximum elevation, gives up trying to find higher ground and enjoys the view.

Applying the method

Unfortunately, no analogy is perfect, and the kangaroo analogy suffers several flaws for the present application. The first flaw is that kangaroos almost never drink while climbing mountains. The second, more serious, flaw is that the goal in SA is to reach the equivalent of a minimum energy state, rather than a maximum. Thus, both analogies are needed. Table 1 shows the particulars.

An inspection of the table shows that the biggest challenge in applying SA is determining how to define the constraints. Glen Canyon Dam operations are constrained by dozens of factors, including turbine flow capacities, cavitation risk, vortex prevention, sandbar protection, operational/manpower limi-tations, storm flow management and environmental concerns.

Generally speaking, constraints due to turbine capacities, cavitation risk and vortex prevention tend to be minor. As discussed, ramping rates and controlled flooding must be used to ensure the same number of sandbars in the canyon is maintained, at least on a periodic basis. Operational constraints require that no more than one flow change be made per hour. Storm flow management requires that the dam release all the flows conveyed to it (e.g. run of river). This is usually implemented by setting a target volume of water that the dam is to release each month to prevent overtopping.

Of all of the constraints, the environmental constraints are the most complex. These constraints limit the maximum outflow to 566 m3/sec, require specific minimum flows at specific times of the day, and even restrict the amount of flow fluctuation over a 24-hour period.

In general, constraints in SA may be applied through one of two methods. The first, most desirable method is to limit search space to only those states that meet constraints. This would be like limiting the kangaroo hops to no more than the nearest edge of the mountain range. But most problems that utilize SA are of sufficient complexity that this method is not feasible. The alternative is to apply a penalty to the energy function. This would be like stopping the kangaroo from jumping out of the mountain range by building an electrified fence. For the Glen Canyon Dam optimization, penalties were implemented by imposing large cost increases when constraints were violated.

A detailed description of how the SA methodology is implemented in computer code is beyond the scope of this article. Implementation of SA is shown by way of the pseudocode in the full study.11

Putting it together

The final step is to be able to predict how the flow scheduling affects sandbar failure risk. This is a nontrivial task. I tackled the challenge by developing an analytical solution to the river stage/groundwater response relationship, ex-tended the solution by a Monte Carlo simulation throughout the entire canyon, then fit the results using non-linear multiple regression techniques. The result was a very complicated, very long (37 separate terms) but entirely usable equation.

With the optimization method selected, the constraints established, a physical model developed, and controlled flooding costs tabulated, the best operations schedule could be determined.

Results

A stopping criteria of 10,000,000 iterations was chosen (i.e., 10,000,000 "hops") for the initial test run. The initial state for the optimization was arbitrarily chosen to be a constant flow 24 hours a day at 450 m3/sec. The model predicted doing so would cause nearly 50% of the sandbars in the canyon to fail and would require so many controlled floods that it would cost millions of more dollars then present day operations.

Luckily, the optimization program quickly improved upon this initial state. After improving five times after 37,586 trials (e.g., the kangaroo managed to go up in elevation five times after 37,586 hops), the optimization schedule was compared to the initial run. The new solution improved upon the constant flow solution by releasing higher flows during the day and less at night (thus taking advantage of the daily electricity rates), as well as by limiting flows on Sunday when the electricity rates are low all day.

After six more changes and 3,579,985 iterations later, the best operations schedule was found to be much more mercurial. The maximum allowable flow is now being scheduled two days a week, followed by a 300+ m3/sec drop two days later. With this schedule, sandbar failure is high but revenue for frequent building floods is being generated.

At trial 7,697,425, the final change of state occurred (see Figure 2). This is the best solution found by the optimization technique. There are several aspects of the optimal returned schedule that are non-intuitive:

- Neither the maximum allowable flow (570 m3/sec) or the minimum allowable flows (140 m3/sec and 230 m3/sec ) were scheduled for any day;

- Peak flow was not maintained over all the peak hours, and instead slightly lowered from 7 p.m. to midnight;

- Sunday's schedule was reversed from normal. That is, the daily flow was scheduled as lower than the nightly flow;

- The Tuesday and Friday peak flows were lower than the Wednesday and Thursday peak flows.

More surprising than the peculiarities of the optimal schedule were the associated costs. The optimal schedule would actually increase revenue by more than 2%; the biggest change would be the building flood budget. The optimal solution would require more frequent building floods then previously scheduled.

Although controversial in 2010 when first proposed, this general strategy of sandbar depletion mitigation by utilizing frequent building floods has now been adopted at the highest levels of government. U.S. Department of the Interior Secretary Ken Salazar recently announced that controlled floods will be implemented in the canyon whenever conditions allow, and he proved his intent in November 2012, when he personally opened the bypass tubes to start the first controlled flood in four years.

Conclusion

Using SA and some physical modeling, the best solution for Glen Canyon Dam is a radical change from previous strategies, the recommendation being that frequent building floods be scheduled instead of losing revenue by attempting to gently downramp and upramp daily flows. The daily oscillations will be high, but so will the minimum flows, which should prove to be a boon to the environment and wildlife. The cost savings to the taxpayer are substantial and may even result in greater revenue then currently generated.

These results are significant, but the method used to get there is ultimately more important. Every step in the method can be executed using standard software that comes with a personal computer.

The optimization tools given here are simple, the implementation straightforward, and yet the results can be profound. The lesson may be to act like a drunken kangaroo once in a while. We might end up somewhere we didn't expect, but odds are the view will be magnificent.

Notes

1Jacquot, Jeremy, "Dams, From Hoover to Three Gorges to the Crumbling Ones," Discover Magazine, March 2009, http://discovermagazine.com/2009/mar.

2Goodwin, P., M. Falte, and A.D.K. Betss, "Managing for Unforeseen Consequences of Large Dam Operations," Operations, Monitoring, and Decommissioning of Dams, prepared as input to the World Commission on Dams, Cape Town, South Africa, 2000.

3Budhu, M., and R. Gobin,"Instability of Sandbars in Grand Canyon," Journal of Hydraulic Engineering, Volume 120, No. 8, August 1994, pages 919-932.

4Kearsley, L.H., R.D. Quartaroli, and M.J.C. Kearsley, "Changes in the Number and Size of Campsites as Determined by Inventories and Measurement," The Controlled Flood in Grand Canyon, Geophysical Monograph 110, American Geophysical Union, Washington, D.C., 1999.

5Geem, Z. "Particle-swarm Harmony Search for Water Network Design," Engineering Optimization, Volume 41, No. 4, 2009, pages 297-311.

6Georgiou, P. E., D.M. Papamichail, and S.G. Vougioukas, "Optimal Irrigation Reservoir Operation and Simultaneous Multi-crop Cultivation Area Selection using Simulated Annealing," Irrigation and Drainage, Volume 55, No. 2, 2006, pages 129-144.

7Vasan, A., and K.S. Raju,"Comparative Analysis of Simulated Annealing, Simulated Quenching and Genetic Algorithms for Optimal Reservoir Operation," Applied Soft Computing, Volume 9, No. 1, 2009, pages 274-281.

8Avellaneda, P., et al, "On Parameter Estimation of an Urban Stormwater Runoff Model," Journal of Environmental Engineering, Volume 135, No. 8, 2009, pages 595-608.

9Verhoeven, J.D., Fundamentals of Physical Metallurgy, Wiley, New York, 1975.

10Sarle, W., "Neural Network Implementation in SAS Software," Proceedings of the 19th Annual SAS Users Group International Conference, SAS, Cary, N.C., 1994.

11Travis, Q.B., "Ebb and Flow: Preserving Regulated Rivers through Strategic Dam Operations," Doctoral Dissertation, Arizona State University, Tempe, Ariz., 2010.


Brent Travis is a senior hydraulic engineer with WEST Consultants Inc.


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