Hydro-dominated short-term hydrothermal scheduling via a hybrid simulation-optimisation approach: a case study

Hydro-dominated short-term hydrothermal scheduling via a hybrid simulation-optimisation approach: a case study

S. Soares T. 0 his h i

Indexing terms: Hydro power systems, Short-term hydrothermal scheduling, Operational planning, Optimisation, Simulation

Abstract: The paper is concerned with the problem of short-term hydrothermal scheduling for hydro-dominated power systems. The problem is to determine hydro and thermal generation scheduling for a day or a week ahead, usually on an hourly basis, simultaneously satisfying hydraulic, thermal and electrical operational constraints while optimizing a criterion of performance. Reservoir release targets suggested by midilong-term operational planning models are also considered. The paper presents a case study based on a hybrid optimisation- simulation approach. The procedure consists of hydraulic system simulation on an hourly basis, with release decisions based on optimal DC power flow. Reservoir release targets are enforced through a dual-penalty approach which assigns shadow costs to hydro generation. The case study was conducted using part of the Brazilian interconnected power system and progressively developed through the use of various scripts. The study provides an insightful analysis of the nature of the problem and reveals a practical and efficient technique for its solution.

Notation

F = total objective function z = partial objective function t = time index T = time index set i = bus index Z = bus index set c = bus generation cost function vector g = bus active power injection vector at = Weight factor at time t k = branch index K = branch index set r = branch resistance vector

0 IEE, 1995 IEE Proceedings online no. 19952156 Paper frst received 9th January 1995 and in revised form 9th June 1995 The authors are with the University of Campinas, Department of System Engineering, Electrical Faculty, Caixa Postal 6101, 13081-970 Camphas SP. Brazil

f = branch active power flow vector h = Lagrange multiplier vector A = power network incidence matrix x = branch reactance vector X = power network basic loop reactance matrix 1 = bus load vector j = hydroelectric index J = hydroelectric index set v = reservoir storage vector U = reservoir release vector q = maximum discharge vector y = reservoir inflow vector HI = index set of immediate upstream reservoirs 8, = water time delay from reservoir m to its neigh- bour I$ = hydro production function vector p = release target vector hf = forebay elevation function h, = tailrace elevation function T = coordinator iteration index Upper and lower limits on variables are indicated by subscript u and 1, respectively.

1 Introduction

In the operational planning of hydro-dominated power systems, hydrothermal scheduling is usually achieved by hierarchical chains of long, mid and short-term models. MidAong-term models are concerned with opti- mal hydrothermal coordination for one or more years (hydrological cycles) on a weekly or monthly basis. They minimise expected operational costs by optimis- ing hydro energetic resources. The final output is the amount of water to be discharged at each hydroelectric plant throughout the coming week. Short-term opera- tional planning, on the other hand, is concerned with distributing generation among units for a single day or week, usually on an hourly basis, satisfying hydraulic and electrical operational constraints, as well as reser- voir release targets determined by mid/long-term plan- ning models.

For power systems with predominantly hydro gener- ation, load tracking can be performed exclusively by hydro units in order to minimise operational costs. In fact, since thermal units are dispatched in an increasing

569 IEE Proc-Gener. Transm. Distrib., Vol. 142, No. 6, November 1995

marginal cost order the optimal hydrothermal coordi- nation tends to keep thermal generation in the base in order to avoid noneconomical marginal cost oscilla- tions. This means that thermal unit commitment is pre- determined on a daily or weekly basis and, therefore, no decision-making is involved regarding it. The main Brazilian interconnected power system, with 93% hydro generation in a total of 48GW of installed capacity, is typical of such systems. For such a system, the repre- sentation of thermal operation can be greatly simpli- fied, since it is not necessary to consider thermal startuplshutdown operations (and costs) nor power rate restrictions.

The problem of short-term hydrothermal scheduling has been the subject of many research papers during the past few decades; the result has been a vast and diversified literature. The first papers in the 1940s and 1950s represented power system operation in a simpli- fied manner, ignoring hydraulic coupling of cascaded plants and thermal dynamic operating constraints. The variables were considered to be continuous as to time, and the usual techniques utilised were variational cal- culus and classical coordination equations [ l , 21. Since the 1970s, mathematical programming models have been applied to the problem, and techniques such as dynamic programming [3, 41 and the progressive opti- mality principle [5, 61 were suggested. Since the 1980s, network flow algorithms [7-91 have become the main technique. These models, however, have taken only thermal and hydraulic constraints into consideration. Other papers have proposed more complete formula- tions, with transmission constraints being represented through AC [IO-121 or DC load flow equations [13, 141.

None of these papers, however, considers the com- plete hydraulic operation in sufficient detail to repre- sent actual hydro-dominated power systems. Hydro plants have, in general, various operational constraints that must be taken into account in short-term hydro- thermal scheduling. Some common constraints are the effect of cavitation in turbines, limits on power varia- tion rates, and limitations on machine startup in order to minimise equipment deterioration. Two other aspects are becoming more and more important nowa- days: environmental requirements and the multiple use of water resources. These aspects lead to new restric- tions on the operation of hydroelectric systems, making their operation much more complex and restricted.

In a recent paper [ 151, the authors proposed a hybrid approach for the problem of hydro-dominated short- term hydrothermal scheduling. This approach com- bines the simulation of the hydraulic system on an hourly basis with the optimisation of the electrical sys- tem for each hour. The main advantage of the hybrid approach is that it allows the representation of the operational constraints of a hydroelectric system in great detail through its simulation. At the same time, the electric transmission system is represented by an optimal DC power flow model.

The present paper presents a case study using this hybrid approach. The test system is the 440kV CESP (S%o Paulo Energetic Company) power system in Bra- zil. Distinct constraints such as reservoir release targets, power flow transmission limits, reservoir storage limits, and discharge variation limits have been considered by means of progressively more complete scripts. The results provide an elucidative analysis of the problem's

570

nature and show that the hybrid approach is an effi- cient and useful tool for solving the problems of short- term hydrothermal scheduling for hydro-dominated power systems.

2 Problem formulation

2. I Mathematical formula tion The problem can be stated as the determination of the optimal power output of each unit for a single day or week at hourly intervals that satisfies hydraulic, ther- mal and electrical operational constraints. The problem must also consider release targets in accordance with midilong-term operation planning studies. In mathe- matical terms, the problem is stated as follows:

2.2 Objective function The objective function to be minimised comprises sys- tem generation costs and a weighted index representing power losses in the transmission network,

min F = ~ ( t ) t E T

where

%€I k € K The generation costs for nonhydro power injection include fuel costs for thermal plants and interchange costs with neighbouring systems (positive for importa- tion and negative for exportation). Note that the group of thermal plants to be operated during the day or week is specified so that only their power dispatch is a matter of decision.

The second term in eqn. 2 represents power losses in the transmission system according to the DC load flow model. In order to account for the costs of transmis- sion losses, the weights cot should be the marginal gen- eration cost of the system at each time t. They can be calculated iteratively. A possible inital estimate could be the average marginal cost of the thermal system configuration. In practice, mu h better initial estimates can be obtained using the wei 1 hts obtained from solu- tions for the previous day or week. After solving the problem for these weights, the resulting marginal costs at each time interval are assigned as new weights, and a new optimisation is performed. The process is repeated until the marginal costs no longer vary from one itera- tion to the next. If desired, however, cot can also be used to perform a bi-objective trade-off analysis between generation and transmission costs.

2.3 Electricalconstraints Electrical operational constraints are established through a DC load flow model with limits on active power injections and flow. Thus, for each time interval t ,

A ( t ) f ( t ) = 9 ( t ) - Ut) ( 3 )

X ( t ) f ( t ) = 0 (4)

g e ( t ) 5 g ( t ) I s u ( t ) (5)

If(t)l I "fu (6) Eqns. 3 and 4 represent the electric network transmis- sion system via a DC load flow model. Eqn. 3 corre- sponds to Kirchhoff s current law (KCL) at each node whereas eqn. 4 corresponds to Kirchhoffs voltage law (KVL) at each basic loop. It is important to note that

IEE Proc -Gener Transm Dlstrib , Vol 142, No 6, November 1995

matrices A(t) and X(t) are considered to vary in time due to the manoeuvering of the electrical transmission configuration during the planning period. Eqns. 5 and 6 represent limits on active power injections and flow. The representation of DC load flow through a capaci- tated network flow problem with linear side constraints allows the explicit representation of power injections and flow in the model, permitting the inclusion of transmission losses as an additional performance crite- rion [16].

2.4 Hydraulic constraints Hydraulic operational constraints comprise the water balance equations for each hydro plant as well as the bounds on storage and release,

m E H ,

ue 5 u(t) i U , (9) The initial conditions are given by the initial reservoir storage v(0) and the releases u(t), t = -1, -2, -3, . . . prior to the planning period.

Eqns. 7-9 specify the usual constraints considered in most short-term hydrothermal scheduling models. For many actual power systems, however, these constraints are not sufficient to represent all hydraulic operational conditions.

Other constraints such as operational ranges due to the effect of turbine cavitation, bounds on temporal variation in release and storage, and constraints due to multipurpose use of water must also be taken into account. These are represented generically by

One major advantage of the hybrid approach is the fact that it can deal with such constraints. Since the hydrau- lic operation is simulated instead of optimised, it can be represented in great detail.

2.5 Hydroelectric coupling The coupling of hydraulic and electrical variables is established by the following equations which represent the output of hydro plants,

Due to variations in turbine-generator efficiency, water head and penstock losses, 4 is generally considered to be a nonlinear and nondifferentiable function, in some cases available only in tabular form. If so, the represen- tation of hydro generation can only be handled through simulation.

2.6 Reservoir release targets Reservoir release targets suggested by midflong-term operational planning models are established as total release recommended for each hydro plant for the fol- lowing week or day,

t€T

Since it is a recommendation, constraint eqn. 12 is considered to be a soft constraint in the hybrid approach. In other words, this constraint is enforced by

a dual-penalty approach that assigns generation shadow costs to hydro plants.

3 The hybrid approach

Dualising the water release target constraints (eqn. 12), results in the following Lagrangian function:

Express eqn. 13 as an additively separated function results in,

tET 3 E J

where

The solution of the Lagrangian problem involves mini- mising eqn. 13 or eqn. 14, subject to eqns. 3-11, for a given Lagrange multiplier vector A. This will corre- spond to the solution of the original problem (eqns. 1- 12) if the release target constraints (eqn. 12) are satis- fied. If not, h should be updated according to release target mismatches by

The Lagrange multiplier AI can be interpreted as the ‘water value’ at hydro plant j . If the mismatch of plant j in eqn. 12 is negative, meaning that the total release is below the specified target, the ‘water value’ AJ should be proportionally reduced in order to stimulate more generation at this plant. If the mismatch is positive, however, the ‘water value’ AJ should be proportionally increased to discourage generation. This simple first- order dual method is quite efficient due to the highly interconnected nature of the problem. In fact, since the load is met for each time interval during the simula- tion-optimisation procedure, the final mismatches are highly interdependent. This means that adequate changes in the ‘water value’ of a single plant will simul- taneously reduce the mismatch for all other plants.

The performance of the dual approach depends on efficiency in solving the Lagrangian problem. This is, however, a very difficult task due to the hydraulic con- straints that establish interdependence between hydrau- lic variables in time.

To overcome this difficulty, the global optimisation of the Lagrangian problem is replaced by an approxi- mate solution obtained by a hybrid simulation-optimi- sation procedure.

The block diagram of Fig. 1 depicts the proposed approach. The external loop (A) conducts the dual solu- tion with respect to the reservoir release targets. The internal loop ( t ) sequentially performs the simulation- optimisation of the Lagrangian problem. For each time interval, two steps are performed: 1. The hydraulic simulation step determines operational conditions for the hydro system for the current time interval. Based on release information from previous intervals, and taking into account all the hydraulic aspects of the problem, such as water conservation, water time delay, limits on release variation, and other

IEE Proc.-Gener Transm. Distrib., Vol. 142, No. 6, November 1995 571

environmental and multiple use restrictions, this step establishes limits for power generation at each plant for the current time interval. 2. The electrical optimisation step is responsible for minimising eqn. 15 subject to eqns. 3-6, and also the limits for power generation at each plant as a result of the simulation step. Consequently, this step determines the power generation as well as the corresponding water release for each plant during the current time interval.

I initialisation I 'shadow costs'

Fig. 1 Flow chart of hybrid approach

In the hybrid approach, electrical optimisation must be effected for each time interval in order to provide the hydraulic simulation algorithm with reservoir release decisions. The efficiency of the global approach is therefore highly dependent on the efficiency of the electrical optimisation algorithm. In this paper, a spe- cial nonlinear capacitated network flow model with lin- ear side constraints has been used [16]. It is based on the generalised upper bounding technique that reduces the number of constraints to a working basis with the dimension of the number of basic loops in the network. Some additional advantages are obtained from the sparsity of constraints eqn. 4.

r- _ - _ - - * ,'*h, ,-----i<--- Grande river

- _ _ - - ' river -. Fig.? 440kV CESPpower system

1 bus bar; A hydro plant river; ___ transmission line ~ _ _

SI2

4 Casestudy

The hybrid approach has been applied to the 440kV CESP (Companhia Energetica do Estado de S%o Paulo) power system, depicted in Fig. 2. The system has 15 buses and 21 branches, four hydroelectric plants with three in the same cascade, and 11 load buses. The transmission network data is presented in Table 1.

Table 1: Transmission network data

k from to r(pu) x(pu) fu(MW)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

1

1

1

1

1

2

2

3

3

3

4

4

4

4

5

5

6

7

8

9

10

3

4

8

11

12

4

6

8

10

13

5

7

8

9

7

12

15

12

9

14

11

0.0042

0.0040

0.0018

0.0013

0.0018

0.0015

0.0014

0.0019

0.0009

0.0024

0.0018

0.0021

0.0020

0.0007

0.0023

0.0018

0.0014

0.0009

0.0256

0.0010

0.0320

0.0568

0.0639

0.0236

0.0172

0.0236

0.0188

0.0182

0.0243

0.0121

0.0304

0.0241

0.0258

0.0248

0.0096

0.0306

0.0233

0.0185

0.0124

0.0675

0.0833

0.1070

1530

1200

1500

1050

1300

1500

1600

1250

1450

1450

1450

1500

1450

1500

1400

1300

800

1200

650

500

1100

Table 2: Hydroelectric system data

Plant H, H* H3 H, v,(hm3) 10000 19500 2800 10000

ve (hm3) 4400 12740 2450 3730

v,(hm3) 11000 21160 3680 10570

U, (m3/s) 482 1400 1105 244

u(m3/s ) 3216 8808 9365 1750

y(m3/s) 1950 4950 5750 900

p(m3/s) 20400 118800 138000 21600

0 (hours) 6 3 0 0

k 0.87-02 0.88-02 0.89-02 0.87-02

{CJ .31+03 .27+03 .25+03 .28+03

.82-02 .15-03 .87-03 .16-02

-.35-05 ,1407 -.33-07 -.26-06

.79-09 -.12-11 .18-11 .OO-00

-.64-13 .26-16 -.38-16 .OO-00

Id,) .34+03 .29+03 .27+03 .29+03

.82-02 .32-02 .OO-00 .65-02

-.92-06 -.15-06 .OO-00 .45-06

-.62-10 ,4411 .OO-00 .15-10

-.16-14 -.53-16 .OO-00 .OO-00

IEE Proc.-Gener. Transm. Distrib., Vol. 142, No. 6, November I995

Data for the hydroelectric system are given in Table 2. The tailrace (h,) and forebay (hf) elevations are fitted by fourth-degree polynomial functions of the release and storage variables, respectively

h,(u) = + C ~ U + c2u2 + ~ 3 2 1 ~ + c4u4 (17)

hf(v) = do + d i u + d2v2 + d3v3 + d4v4 (18)

with {ci} and {di}, for i = 0, 1, 2, 3, 4, given in Table 2. The hydro production function of each hydro- electric plant is given by a nonlinear and nondifferenti- able function of the reservoir release and storage variables

$(U, U ) = k [ h f ( V ) - M u ) ] min(u, U) (19) where k is a constant and q is the maximum discharge through the turbines. Note that at U = q the function is nondifferentiable with respect to U. The constant k includes average turbine-generator efficiency, gravity acceleration, and water density, and is given in kg/m2s2.

The case study presented in this paper has considered a planning period of 24h. The load distribution for the buses in the first hour of the day is given in Table 3 .

Table 3: Load distribution at tl

B u s Load B u s Load B u s Load (MW) ( M W ( M W

1 140 6 0 11 710

2 29 7 1200 12 600

3 0 8 0 13 100

4 0 9 0 14 61

5 700 10 105 15 56

The load at each bus is assumed to vary throughout the day according to the normalised load curve shown in Fig. 3.

0.95

0.85

0.7-

0.65 - 0.6 -

0.55 -

A r 0.51

0 5 10 15 20 25 time,h

Fig.3 Normalised load curve

This case study has been conducted progressively to highlight the main characteristics of the problem.

4.1 Script I The first script ignores release targets and focuses on the minimisation of power losses in the transmission network. It corresponds to a single global simulation- optimisation application of the hybrid algorithm with hydro generation shadow costs h equal to zero. The solution can be seen as the best hydro schedule from an

IEE Proc-Gener. Transm. Distrib., Vol. 142, No. 6, November 1995

electrical point of view. The results are shown in Fig. 4.

2400-

2200 - 2000 -

3 1800- I:

.= 1400- =- 1600-

5 1200-

0

-1 - 2 i

L.. L..

. . .-.

.i ..,..,... .... ....... \._I' ... I... ...... 2001 I

0 5 10 15 20 25 time,h

Fig.4 Minimum loss scheduling ~ HI; - - - HZ' ... Hj' ...... H 3 4

Note the parallelism in the scheduled production of the four hydro plants. This can also be verified by the power flow through the transmission system, as is shown in Fig. 5 for the two most heavily loaded branches in the network. This behaviour is due to the minimum loss dispatch which maintains a given ratio between power flow in all parts of the network. Thus, if the load oscillates proportionally according to Fig. 3 , power generation and flow will oscillate simi- larly to maintain operation at the point of minimum loss.

2200r

2ooo 1800 t 1600-

I: ; 1400-

= 1200- 0

10oot

n J I

I L-

6001 I

0 5 10 15 20 25 time,h

Fig.5 Power flow in branches I2 and 14 12 ~ ; 14 - - -

In this solution, the generation dispatch tends to allocate more generation for plants that are close to the load buses. Therefore, the corresponding release deci- sions may be considerably different from the targets established by mid/long-term planning models. In this script, the errors in reservoir release targets (in m3/s) are -14 400, -14 400, +84 000 and +6240, respectively.

4.2 Script2 In the second script, the release targets suggested by mid/long-term planning models are enforced through the dual-penalty approach. As can be seen, the new optimal solution corresponds to a trade-off between transmission losses and reservoir release targets. This can be observed in Fig. 6, where the scheduling pro- posed has been shifted vertically in relation to that of the first script. Hydro plants 1 and 2 have increased production whereas hydro plants 3 and 4 have decreased it. This kind of variation maintains the

S I 3

power injection ratio, providing the minimum loss solu- tion.

3000r 3 ~ . 2 0 0 0

0 5 1500-

1000-

c

a,

increased their generation. in comparison with the sec- ond script, these results show a significant change in the power dispatch for peak hours, with a similar pro- file for the remaining day. A slight compensation in generation at non-peak hours is necessary to preserve reservoir release targets. Fig. 8 shows the power flow in branch 14 for the second and third scripts. -

25001

3 I 2000- c- 0 5 1500-

m 1000

500

0

a, -

-

z 2000- C'

e 1500- e 0 .-

a,

1 OOOr

I

4.4 Script4 , This script was designed to analyse the influence of ' run-off-river plants in cascaded hydro systems. As this kind of hydro plant does not have any storage capac- ity, its scheduling should follow that of its upstream i neighbour, although allowing for the time delay 1 involved in water displacement. To simulate the pres- ence of a run-off-river plant, the storage capacity of plant H3 was considered to be zero. The resulting schedule is shown in Fig. 9.

0 5 10 15 20 25 t ime,h

Fig. 9 Run-off-river plant ~ HI; - ~ H2. ... H ....... 3, H4

'.. - L.3..d-a 1 1 . 1 ........................... ... #.... ...... .,,.. .... ....... .... ...,..(, .#

so,} ......

0 5 10 15 20 25 time,h

Fig, 6 Release targets attainment ---HI, - - -H2, - - - H 3. H4

4.3 Script3 The two previous scripts do not take into account transmission limits of the electric network. The solution for the second script reveals overloads in branches 12 and 14 during peak periods. The third script includes transmission limits; the effects can be seen in Fig. 7.

3000r r-l

2500} . . I I

- 1 r-(

Note the 3 h delay between the scheduling of plants H2 and H3. The first hours do not reveal the same conection because of the initial releases adopted. The same effect can be observed in hydro plants with a small storage capacity when the reservoir reaches one of its limits. At this moment, the hydro plant begins functioning as a run-off-river plant and its scheduling becomes linked to that of its upstream neighbour.

3000Y I ,

:-,: .... ..,.. :'" ..... : ..... ...... ...,...... .... . . . . . ....... ....... .... ............... 8 ... d

25 0 5 10 15 20 t i m e , h

Fig. 7 Transmission limit attainment ...... ----HH1; ---H2....H3. H 4

laoor 25001

1600 1400 t h 1 0 0 0 ~

5001 8001

01 I

0 5 10 15 20 25 600L I

0 5 10 15 20 25 tlme. h

F19.8 Powerjlow in branch 14 __ scenario 2 , - - - scenario 3

time,h

Note that there is a reduction in the peak generation of plants H3 and H,, which provide most of the power for branches 12 and 14. In order to maintain the power balance during peak hours, the other two plants have

574

4.5 Script 5 This final script reveals the effects of imposing limits on water release variation in time, characteristic of

, IEE Proc -Gene? Transm Dlstrib , Vol 142, No 6, November 1995

some hydro plants. This constraint is usually due to navigational and environmental concerns. To simulate this constraint, a maximum variation of 800 m3/s per hour at hydro plant H2 was considered. Fig. 10 shows the new scheduling. Note that the major difference with respect to script 2 is found in the final 3h because the constraints are only violated between 10 and 11 pm. To maintain the release targets, the entire day’s scheduling has been slightly modified again.

This case study highlights the main features of the hybrid solution for hydro-dominant short-term hydro- thermal scheduling problems. The computational effi- ciency of the approach and its suboptimality are referred to in the following.

4.6 Computational performance As has already been pointed out, the computational efficiency of the hybrid approach depends largely on the resolution of the electrical optimisation problem. The code has been implemented using Fortran 77 in a SUN SparcStation IPX. The average CPU time for script 2, starting with h = 0, is -20s. In the normal sit- uation, however, the performance is improved because the optimal multiplier obtained in the solution for the previous day or week is used as the initial parameter. This is quite a good initialisation procedure because the load and water inflow usually present minimal varia- tions from one day or week to the next. Considering this initialisation and a load variation of 5%, the aver- age CPU time is reduced to 7s.

4.7 Suboptimality The hybrid approach provides a suboptimal solution because instead of optimising the Lagrangian problem a simulation-optimisation procedure is implemented. In other words, once the hydraulic simulation is driven only by an OPF dispatch the resulting reservoir storage trajectories do not take into account the hydro genera- tion efficiency optimisation. Thus, the degree of subop- timality in the proposed approach will depend on how much the solution could be improved if the hydraulic simulation were replaced by a multiperiod hydraulic optimisation which considered hydro efficiency in the cost trade-off. Nevertheless, hydro generation efficiency is achieved by maximising net water head. Therefore, when water head shows small variations with respect to different operational policies the possible hydraulic effi- ciency gain is neglectable. In the case study performed, the maximum water head variations with respect to the average heads were 3.2%, 1.9%, 1.2% and 3.0% for plants 1, 2, 3 and 4, respectively. As variations in fore- bay elevation are small due to the short period of anal- ysis, these water head variations are mostly explained by unavoidable variations in the tailrace elevation due to hydro plants participation on load tracking. There- fore, since any operational policy will have to track the load, there is not much room for hydro efficiency opti- misation in the short-term operational planning and, as a consequence, the approach’s suboptimality is not sig- nificant.

5 Conclusions

This paper has presented a case study concerning the short-term hydrothermal scheduling for dominantly hydro system. The study was performed using a hybrid approach consisting of the simulation of the hydraulic

system with release decisions provided by an optimal DC power flow algorithm. The release targets sug- gested by mid/long-term planning models have been enforced by a dual-penalty method that assigns shadow costs to hydro generation.

The case study was performed with the 440kV CESP power system in the Southeast Interconnected Brazilian Power System. The tests showed the influence of spe- cific constraints such as reservoir release targets, power flow transrnission limits, reservoir storage limits, and limits on discharge variation. The result is an insightful analysis of the nature of the problem that simultaneous reveals the procedure itself to be both efficient and practical.

6 Acknowledgments

The authors would like to acknowledge the support of the Fundagio de Amparo a Pesquisa do Estado de S2o Paulo (FAPESP) and the Conselho Nacional de Desen- volvimento Cientifico e Tecnologico (CNPq), govern- ment agencies for research support in Brazil.

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1

2

3

4

5

6

7

8

9

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