Description of the Optimization Problem

Two different optimization formulations are considered in the present analysis. They are the minimization of total annual cost (TAC) and the minimization of total investment cost (IC) of the wastewater treatment strategy which is required to satis@ various water quality standards (ambient andfor effluent). The TAC comprises the operation, maintenance, and replacement cost (OMRC) and the annual component of IC. The ambient water quality standards are imposed at selected points along the river. They spec@ the minimum limits on DO, and maximum limits on BOD and NH4-N at those specific locations.

The above formulations can be mathematically expressed as:

Minimize: C Ci (qi) qi E Ti i

Subject to: Qj,k 2 Sj,k

v

^j,k

Ei,k ² Estandard, k

where Ci (qi) is the treatment cost (TAC or IC) required to achieve a treatment efficiency at the ith wastewater discharge. Ti denotes the feasible set of treatment alternatives for the i

I

discharge. y , k is the water quality at the standard location j, expressed by the quality indicator k he corresponding quality standard is denoted by Sj,k. Ei,k and Estandard,k stand for the effluent quality (expressed by indicator k) at the ifh discharge, and the effluent standard for kfh indicator respectively.

10.4 DP Formulation

The river system was subdivided into a number of reaches, which were hrther divided into

"stages". The river network has been defined by the interconnection of different reaches. A

"stage" was considered as part of the river from a point immediately upstream of a "point of action" (Pl) to a point immediately upstream of the next "point of action" (P2) which is located downstream. A "point of action" can be: a wastewater discharge, an abstraction point, a measurement point, a point with a pre-specified water quality standard, a weir or an artificial point which is introduced in order to maintain the computational procedure.

10.4.1 Discretization of quality states

Application of dynamic programming in the present problem requires a sub-division of the feasible water quality range into a number of discrete intervals. The feasible quality range is defined by the various water quality indicators that are considered. In the case of single quality indicator, the term "interval" is self-explanatory. If two indicators are to be considered (e.g.

DO and BOD), the "interval" would be an "area" defined by the two indicators. Three indicators form quality "intervals" which can be visualized as three-dimensional blocks. Three water quality indicators were explicitly included in the present analysis. They are the ambient concentrations of DO, BOD and NH4-N. Depending on the number of intervals considered for each quality indicator, the total number of feasible "joint" quality intervals can be quite

large. The discretization used in the present study considered 40 levels for DO, 120 levels for BOD, and 80 levels for NH4-N. The practical ranges considered for DO, BOD and NH4-N were 0-10,O-30, and 0-20 (mg/l) respectively. In fact, the last interval of each indicator was of variable size, in order to accommodate concentrations which may be larger than the predefined ranges. All other intervals were of equal size for a particular quality indicator.

10.4.2 Stage-by-stage computations o f

DP

DP computations are started from the most upstream point of the river system. The water quality at this point is known, which implies only one quality state at that point. Using Bellman's (1957) principle of optimality, the DP calculations are performed stage by stage, proceeding towards the most downstream point of the system. As it proceeds, it makes use of river water quality models which estimate the feasible water quality states at the beginning of the subsequent stage. There can be an increase in the number of possible water quality "states"

at a "subsequent" stage, depending on the number of management alternatives available at the current stage. As described above, the allowable water quality range at each stage is divided into a discrete number of "quality intervals". If two or more quality states fall within an interval, the quality state that corresponds to the best value of the objective function within the interval is retained for further computations downstream. This indicates that the maximum number of quality states at any stage is equal to the number of discrete quality intervals.

For each of the quality states that are retained for further computation, the objective function values (cumulative costs up to the particular stage) and the current decision (if any) are recorded. It is also necessary to record, for each of the quality states, the previous stages' quality state which produced the current state. The values of the three quality indicators for each quality state are also kept on record. At a water quality standard (constraint) point, it is possible to eliminate some of the discrete states from further consideration, if they do not satisfy the standards. Therefore this reduces the computational load of the DP problem.

However, in the present study, the optimization is performed repeatedly with different quality constraints; so that the quality constraints are not considered as a predetermined set of limiting factors.

After performing the DP computations to the most downstream stage of the system, the feasible water quality states at the downstream stage will produce various values for the cumulative objective function. Each value indicates an alternative solution which satisfies the water quality standards imposed. It is necessary to select the state which has the best value for the objective function, so that the optimal solution can be traced upstream from this starting point. The non-optimal solutions (nonoptimal in terms of the single objective considered) that are generated by the DP computations, can be analyzed in a multiobjective decision framework if necessary. This is another advantage of the DP approach, because it generates a set of solutions which cannot be obtained by other optimization techniques. Figure 10.1 displays the main elements of the DP-based computational procedure (see Appendix 10.1 for a detailed flow diagram). A graphical representation of the computations is presented in Figure

10.2.

I

Headwater location (PI)

I

Use water quality models to estimate the quality state at the downstream point of action P2, for the decision J (if any) at P2. Determine the discrete quality interval (K) to which this quality state belongs. Record, at the cell Kof

a quality state matrix for location P2, the quality state with its costs, the decision, and quality; (only if previously recorded state of cell

K

is more

expensive, or, if no values are recorded in cell K previously)

All feasible decisions

Next J at P2 evaluated ?

+

^Yes

All quality states at Next I P1 evaluated ?

t

^Yes

I

Consider P2 as the

/ /

^{Is the}

\

new starting point (P 1) downstream boundary ?

, ^\--$xJ

Trace the optimal

,

solution upstream

ri\

Fig. 10.1 Flow diagram for the computational procedure of DP-based optimization

Pollutant

Pollutant Pollutant Water quality discharge

discharge discharge constraint point

I

/

I

t

I

-

.

River flow ^I

I

I

I

1

Upttrearn 1 Bo ndary

P i

' Downstream

!

^Boundary

I

range of river water quality

Legend:

Feasible water quality states Representative water quality states

t

Boundaries of quality intervals

constraint violated

Notes: For the sake of simplicity, this illustration assumes the following 1. Only one quality indicator to define the river water quality

2. Only three feasible treatment alternatives for each wastewater discharge (which can be different from one discharge to the other)

Figure 10.2 Graphical representation of DP computations

As indicated above, only one quality state is retained fiom each discrete quality "interval" at each stage, for fbrther computations. This particular state will subsequently represent the quality interval, hence it can be termed the "representative quality state" for the quality interval at that stage. The elimination of all but one state fiom each interval considerably reduces the amount of computations. However, it can easily lead to a non-optimal solution, especially if the discretization is not fine enough.

10.4.3 Mathematical formulation of the DP computations

The mathematical formulation of the DP solution procedure is given below, for a simplified case of a river without tributaries. An extended form of this procedure was employed to analyze the Nitra system comprising several tributaries. The optimization problem for the most upstream stage can be expressed as: treatment cost required (which is zero if there is no wastewater discharge or if no treatment is to be made at stage n) to achieve an efficiency of qn at stage n. This efficiency qn (the removal rate) is the decision variable at stage n of the problem. For the most upstream stage, QO indicates the headwater quality. Tn( ) represents the transfer function at stage n, and is a water quality simulation model.

The efficiency of a particular treatment alternative is characterized by the influent and effluent quality. For each WWTP, a set of feasible treatment alternatives is developed. Each alternative is represented by its IC, OMRC, and the effluent concentrations (see Chapter 11). The optimization task is to make a (0,l) decision with regard to all feasible treatment alternatives at each of the WWTPs. These decisions should sum-up to one at each WWTP location. For the subsequent stages (fiom n=2 up to the most downstream stage), the recursive relation space and represent each discrete quality state by a representative point. This representative point is usually fixed, the center of the discrete state, for example. Some discontinuity occurs at this point, because the simulated water quality may not coincide with the central point of the quality state. However, the representative point used in the current approach is the simulated quality state which corresponds to the best objective fbnction value within the current state. This allows a correct representation of the river water quality profile (for the selected control actions), as the rounding off of the quality values is not involved.

As indicated above, the current decision (treatment alternative), the previous state (at the previous stage) and the cumulative cost are recorded for each allowable water quality state at each stage of the computation. Having reached the most downstream point, the optimal solution can be traced-back upstream.

10.4.4 Possible improvements

In the present study, uniform ranges of water quality indicators were considered for the DP computations. However the feasible quality range at different locations along the river can be significantly different. An improvement in the accuracy of the results may be obtained by considering location-specific ranges instead of uniform ones. A narrower quality range would lead to a finer discretization, assuming the same number of quality intervals as considered in the uniform case.

An equal number of feasible "quality intervals" was considered at each stage of computations in the present study. Nonetheless, it is advantageous to discretize the quality range into a different number of intervals at different places. This is important due to the fact that it is not practical to use a large number of discrete intervals at each stage. Such a discretization would lead to an impractical computational requirement. However, certain stages need fine discretization, while some other stages need a smaller number of intervals because there are only a few feasible quality states. Significant reductions in computational load can be realized if it is possible to have a non-uniform discretization, combined with non-uniform quality ranges.

For each quality interval, only the "best" quality state falling within that interval was taken as the representative quality state. All the others within the interval were eliminated from hrther computation. However, this introduces an error in the optimization procedure, eliminating some non-identical quality states (and consequently certain feasible decisions) on the basis of a suboptimal objective hnction value. This error might be reduced if the objective hnction is not the sole criteria used to eliminate feasible states which are different from each other in terms of water quality. It may be worthwhile to test other heuristic approaches for selecting the representative quality states. Such an approach can be based on the comparison of the objective hnction value as well as one or more quality indicators.

10.5 Alternative Formulations of the Problem

Im Dokument Water Quality Management in the Nitra River Basin (Seite 123-128)