To determine optimal operations of Old Hickory reservoir, problems are formulated to determine turbine operations that generate maximum power value, subject to operational constraints. The problems are nonlinear with integer decision variables{x1,x2, ...,xn}, representing the number of active turbines at each houri=1 :n. Optimization is performed for a defined planning period, in this case 10 days, a typical river system scheduling operational period (U.S. Army Corps of Engineers, 1998). Computational expense increases substantially as the number of decision variables grows;
therefore, the planning period is divided into daily sub-problems which are solved consecutively.
Old Hickory reservoir must fulfill many requirements, which are formulated as a set of hard and soft constraints. The algorithm seeks to meet soft constraints, but if they are not fulfilled completely the algorithm still proceeds. Soft constraints are integrated into the objective function by use of a penalty parameter. Several hard constraints and a single soft constraint applied in the experiments are described below and in Table III.4. The optimization problem objective and constraints can be written as follows and explained below. Equations III.2-III.8 are firm constraints on the problem that must be satisfied.
minimize
X −h n
i=1
∑
C(i)·xi·r i
+ h
d·(ef−et)2 i
(III.1)
s.t. pl≤E(x1,x2, ...,xi)i≤pu, ∀ i=1 :n (III.2)
i+z
∑
i
xi≥1, ∀ i=1 :(n−z) (III.3)
|xi+1−xi|≤c, ∀ i=1 :(n−1) (III.4) (xi≤xi+1≤xi+2≤xi+3)∨(xi≥xi+1≥xi+2≥xi+3),
∀ i=1 :(n−3) (III.5)
{xi∈Z|0≤xi≤a}, ∀ i=1 :n (III.6)
∑|S|i=1max(0,ol−oi)
|S| ≤0 (III.7)
∑|S|i=1max(0,tl−ti)
|S| ≤0 (III.8)
III.3.1 Objective Function and Soft Constraint
The objective (Equation III.1) is to maximize (formulated as a minimization as is convention) the value of hydropower produced over a set planning period. n is the number of hours in the planning period,C(i)is the power value at timei, andris the turbine power rating in MW. A cost curve defines the relationship between the value of power production and the time of day, which is important due to changes in electricity demand and the use of hydropower traditionally as peaking power to supplement thermal power production. If no cost curve is provided, i.e.,C(i) =1 for all values of i, the problem is equivalent to maximizing the total power generated over the planning period. The employed cost curve (Figure III.2) was created using Old Hickory reservoir historical operating patterns to estimate a relationship between time of day and generation. This approach is intended to be used for planning, not for real-time grid balancing, so a historically-based cost curve is appropriate.
The second term in Equation III.1 is a penalty term representing a soft constraint, penalizing deviations of final water levelef from the final target elevationet. This restricts the solution from draining to the bottom of the power pool at the end of each daily optimized sub-problem. Briefly,
0 2 4 6 8 10 12 14 16 18 20 22 24 Time of day (hours)
30 40 50 60 70 80 90 100 110
Generation value ($/MWh)
Figure III.2: Cost curve used in optimization applications.
for each daily sub-problem potential solution the final water level elevation is found, the penalty is computed, and a deduction to the objective function value is made for water level elevations below target levels. Prior to the start of the GA solver, a penalty coefficient is computed using linear interpolation:
d=ypro jected·
vu+ (vl−vu)pT−pl pu−pl
(III.9) whered is the penalty coefficient in dollars per meter below target (or megawatt-hour, MWh, per meter below target if no cost curve is provided),ypro jected is the estimated power value under pro- jected operations for the sub-problem optimization time period (in dollars if a cost curve is provided, otherwise in MWh),pTis the target water level elevation at the end of the time period,plandpuare lower and upper bounds of the power pool, respectively, andvl andvuare scaling coefficients with vl ≤vu. The penalty coefficient is greater the closer the target water level elevation is to the bottom of the power pool. Scaling coefficients are a function of the value of power and reservoir generation capacity, with larger coefficients aligned with increased penalty. For reservoirs with total capacities of 100 MW, like the one used in this study, and a cost curve with value magnitudes in the range of
$40-$100/MWh as assumed here, values ofvl=500 andvu=1000 perform well.
III.3.2 Hard Constraints
Equation III.2 sets lower and upper bounds (plandpu, respectively) on water levels.E(x1,x2, ...,xi) is an elevation model that predicts water level elevations for all timesteps 1 :i. For reservoirs op- erated on a seasonal guide curve, pl andpu are typically set to the lower and upper bounds of the power pool. The simplified water level elevation model assumes the water level to be consistent along the entire reservoir and is a function of all inflows and outflows. Spill flow is often engaged to improve downstream water quality. An average spill flowrate for each daily sub-problem is com- puted during elevation calculations based on turbine releases, inflows, and user-provided midnight target elevation values. First, water level elevation is computed based on the hourly turbine settings assuming no spill release. If the final elevation for the sub-problem is less than the target elevation, spill remains zero. If the final elevation is greater than the target elevation, an average spill flowrate for the sub-problem is assigned which results in a final water level elevation equal to the target value. This incorporates spill without requiring additional decision variables, which is important since spill flow is often engaged to improve downstream water quality.
In an effort to maintain minimum flows along the river, the maximum number of consecutive hours z allowed without power generation is defined by Equation III.3. The USACE Nashville District implements this rule for water quality purposes as well.
Equation III.4 limits the hourly rate of change in the number of active turbines, with c being is the maximum number of turbine units that can become active or go inactive each hour. Since Old Hickory reservoir exists on a navigable waterway with lock systems, this constraint assists in minimizing fluctuations in the surface elevation and adverse impacts on water level stability.
Equation III.5 attempts to reduce oscillations in the turbine operations over time. This constraint is formulated with logic that states that, except in cases of ramping turbines up or down, the number of active turbines must be fixed for at least three hours consecutively before changing. Reducing oscillations is desired to minimize equipment wear.
Equation III.6 defines the maximum number of turbines at the hydropower facility, a. It is assumed that all turbines operate at the same turbine power rating,r, and that the number of active turbines is selected from a set of integer options.
The Nashville District monitors DO levels in the Old Hickory dam, which is directly upstream of
the metropolitan Nashville area and has historically proven to be a strong indicator of water quality system-wide (U.S. Army Corps of Engineers, 1998). Maintaining cool discharge temperatures is also important as the Cumberland River serves as a source of cooling water for TVA’s thermal power plants both upstream and downstream of Old Hickory dam. Equations III.7 and III.8 define lower constraints on discharge DO and temperature, respectively, whereol andtl are lower limits andoiandtiare DO and temperature estimates at timei. These equations can be modified to account for maximum constraints as well. Discharge water quality over the operating period is computed by:
O(~x) = (o1 o2 · · · on) (III.10) T(~x) = (t1 t2 · · · tn) (III.11) whereO(~x)is a function estimating discharge DO concentration andT(~x)is a function estimating discharge temperature. In this application, O(~x) andT(~x) are ANN models predicting the water quality estimations of a simulation model.S, the set of timesteps with total dam discharge flow not equal to zero, is defined by:
S=
i|(QTi +QSi)6=0 (III.12)
where QTi is the turbine discharge and QSi is the spill discharge at time i. |S| is the size of set S. Dividing by |S|accounts for the fact that at times when there is no release from the turbines or spillway, discharge water quality is undefined. This approach also makes it easier to compare population members which are not fully-feasible with respect to water quality by having a single metric for comparison. Equations III.7 and III.8 require the average hourly constraint violation to be less than or equal to zero; since the constraint violation can never be negative, the average hourly constraint violation is equal to zero.