Modelling and Management of Sustainable Basin-scale Water Resource Systems (Proceedings of a Boulder Symposium, July 1995). IAHS Pub!, no. 231, 1995. 311

Planning and management of water resources systems under uncertain future information using fuzzy optimization

TOSHfflARU KOJIRI Department of Civil Engineering, Gifu University, 1-1 Yanagido, 501-11 Gifu, Japan

Abstract To plan a water resources system, fuzzy theory is introduced to handle uncertain future information. Representing runoff hydrograph and future water demand with fuzzy membership functions, the optimal site and scale of the system are decided through fuzzy linear program­ming. Then the construction schedule is calculated through fuzzy dynamic programming, under the conditions of limited finance and uncertain water utilization. The management plan of the reservoir system is also evaluated through fuzzy theory.

INTRODUCTION

Water resources systems have to be planned by considering uncertain input such as water quality and quantity and future water demand situations, because long construction periods are needed which can include changeable development strategies. Traditional prediction methodologies for water demand analyse averaged values with stochastic meaning, derived from statistical analysis of past known data. However, the runoff hydrograph and water demand will fluctuate, depending on basin development, popula­tion increase, water usage, human activities and other technological developments. To balance sustainable development between human society and the natural environment, unnecessary development should be curtailed. To avoid an over-estimation of planning, prediction factors are shifted to lower or upper values according to the preference and perspective levels of the decision makers.

So, planning and scheduling procedures for water resources systems using fuzzy optimization are proposed. The first stage is to decide on the construction site and scale, and the second is to order the construction schedule. The whole water resources system is decided through fuzzy linear optimization under the constraints of fuzzy water demand. The storage capacities of reservoirs and distribution systems are obtained to maximize the fuzzy membership grade. The total horizon of construction is divided into several periods, depending on limited financial conditions and the construction periods for some facilities. Then fuzzy dynamic programming is applied to decide the construc­tion schedule in order to obtain the highest feasibility.

FUZZY OPTIMIZATION

The fuzzy set was proposed by Zadeh (1968), and Mamdani (1974) presented the fuzzy

312 Toshiharu Kojiri

inference commonly applied in the expert system. The fuzzy set can represent linguistic uncertainty with membership functions different from the crisp concept. The fuzzy theory has the same algorithms as in the crisp set as follows:

Union: /J-in2(x) = MiMV/^M ^

Intersection: /x,,-,2(x) = /x1(x)Ajic2(x) (2)

Complement: /n,(x) = 1 -/^(x) (3)

where x represents the elements of the fuzzy set and /x is the fuzzy membership function represented in exponential or triangular form.

Under the condition of fuzzy goal and fuzzy constraints, the fuzzy optimization is formulated as follows:

V nD(X) subject to ixD(X) = ^OQA^iA^A x>o (4)

../xgm (AmX)..AixBM(AMX)

where D is the comprehensive evaluation function; G is the fuzzy goal; B are the fuzzy constraints with M vectors; and C is the coefficient vector. The parameters Am and Bm are the coefficient vectors of matrix A and B represented by basic linear programming as follows:

Z0 < CX (5)

AX < B and X > 0 (6)

Equation (5) means the value of the objective function is expected to be nearly Z0 and equation (6) denotes the allowable ranges of constraints. By introducing the parameter X as fuzzy grade, the fuzzy optimization is converted into ordinary linear programming.

On the other hand, the goal in a fuzzy multi-stage decision process is formulated as follows:

V-D = MGA • • A MGAA HA •••^VBM W

where N is the number of stages, and the goal is maximized for x. If the system state xl+, is formed as the following transition function of input ut and xt:

*,•, = / (*„« , ) (g)

the fuzzy decision process is represented as follows:

^D(H0,...,«„_,) = n0(uQ)A...AnN_1(uN_1)A(iG(N)(xN) (9)

Therefore, the recursive equation of fuzzy dynamic programming can be formulated as follows:

MaW-oCW = max{/W"AM)AMtf-*+iC%-/+l)} (10) "N-I

The optimum solution provides the maximum membership grade of the goal with the decision variable of n0, /JL1 ,.., fxNA.

Planning and management of water resources systems using fuzzy optimization 313

PLANNING OF SITE AND SCALE

Future circumstances in water resources development are assumed as fuzzy state as follows: (a) inflow hydrographs are known with fuzziness through historical data; (b) water demand is predicted by fuzzy theory. The first condition is recognized as the basin change through future development and the second one denotes the uncertain water demands through advanced technologies for water usage or unexpected events in human activities.

Basically, the water resources system should be designed with a minimum amount of construction and operation cost as follows:

j = i cCjV + oci {op) + E Qkj(t)} 7=1 t=\ k=l

(11)

where CC and OC are the parameters of construction and operation cost, respectively; V is the reservoir capacity, and K and J are the number of demand points and reservoirs. The objective in the fuzzy set, as represented in Fig. 1, sets the preferable range as less than ZiL from a viewpoint of normal national budget for water resources development. Moreover, the objectives on release and storage are defined in Figs 2 and 3. The water supply is expected to be larger than Qd{t) and not less than Q^. By introducing the parameter X, denoting the fuzzy grade for the whole system, it is formulated as follows:

Qp)-{Odft)-OdjL{t))\ > QdjL{t) (12)

W)' **/ (Qdk(t)-QdkL(t))X > QdkL(t) (13)

We assume that the discharge from reservoir Od{t) and O^ are equal to the discharge of river Qd{t) and QdL. As the storage volume has fuzziness according to the inflow, the continuous equations are represented as follows:

Sft)-Sft-\)+Ofi) + E Qkj(t) +dj(t)\ < 7,(0 +dj(t) k=\

(14)

Sj(t)-Sj(t-l) + Oj(t)+lQ 1 J J *=i J

(t)-dj(t)X > Ij(t)-dj(t) (15)

Mzi) A

1.0

o - * - Z i ZiL ZiU

Fig. 1 Membership function of construction cost.

314 Toshiharu Kojiri

j"(Q)

A 1.0-

Qdifo Q*) OdifO OdW

Q(0

O(0

Fig. 2 Membership function of water demand and discharge.

Vrdi(t) Vi

Fig. 3 Membership function of storage volume.

where dt(t) is the considerable range of fuzzy inflow. Through linearization with the stages for a cost function, the whole optimization can be formulated to maximize X and can be solved following normal linear programming, by combining described constraints Objective: max X, subjected to: constraints of cost (Fig. 1); constraints of desirable water supply (Fig. 2) ; constraints of minimum water supply (same as Fig. 2) ; constraints of storage volume (Fig. 3); constraints of continuity (equations (14) and (15)), and constraints of linearization.

The mathematical formulation is omitted here because the description is too compli­cated to be understood. The reservoir operation is also extracted to minimize the operation cost at the same time.

SCHEDULING OF CONSTRUCTION WORKS

Under the condition of limited finances, water resources facilities should be constructed in the proper order. As the necessary facilities to be constructed are decided in advance, the construction work should satisfy the water demand at each stage, as much as possible (Ikebuchi et al., 1985). This objective is formulated as the following minimization:

mmOdij = rmx(DlF) (16)

DIF QdtM)-QJt)\ (17)

Planning and management of water resources systems using fuzzy optimization 315

If the fuzzy membership function of the objective is set as shown in Fig. 4, and the number of construction time stages assumed to be three, the recursive equation for fuzzy dynamic programming is formulated as follows (Kojiri et al., 1993);

^G(N-i) r^GiN-i + iy-^N-i + U (18)

Let us define the construction state as W, where each reservoir gets three conveyances for water supply to cities; 1 = construction and 0 = non-construction. The first three columns represent reservoirs and the last nine columns represent conveyances, where every three items show the construction situation of each reservoir. For instance, W(l) = [100 111 000 000] denotes that the first reservoir and three conveyances from it are constructed at the first stage. The feasible state at each calculation stage in Dynamic Programming (DP) is extracted from among all combinations of reservoirs and conveyances, and compared with the previous and the current construction state. The optimal path in DP is decided to minimize the fuzzy grade of equation (16), as shown in Fig. 5.

^ D I F

Fig. 4 Membership function of deficit water supply.

stage 0 stage 1 stage 2 stage 3

000 000000000

100 111000000

100 110100000

100 110010000

001 111000000

001 000000111

110 111111000

111 111111111

101 111111000

011 000111111

Fig. 5 Conceptual illustration of optimal path in dynamic programming.

316 Toshiharu Kojiri

APPLICATION OF METHODOLOGIES

The proposed methodologies are applied in the assumed river basin. The parameters of construction are listed below:

ZU: CO. OCx:

3 000 000 1000 1000

ZL:

C2:

10 OC

2000 C3: 500

The fuzziness of hydrograph and water demand is set as 10% of the average. The final planning of the reservoir capacities to be constructed were Vl = 900 000 m3, V2 = 1 300 000 m3, V3 = 900 000 m3, respectively. The fuzzy grade was 0.26 because the uncertain discharge was not sufficient when compared with the uncertain water demand. Figures 6 and 7 show the operation strategies of both the reservoir and conveyance. As

o d

-e- iNaow - B - RELEASE

STORAGE VOLUME

°--e

5 6 . -TIME STAGES(MONTH)

12

Fig. 6 Calculated results of release sequence (reservoir 1).

o o

o in

o d

- © - CONVEYANCE WATER(CnY1 - a - CONVEYANCE WATERICITY2 - A - CONVEYANCE WATER(CITY3'

TIME STAGES(MONTH)

Fig. 7 Calculated results of conveyance sequence (reservoir 1).

Planning and management of water resources systems using fuzzy optimization 317

the fluctuation was not restricted to the limited ranges, at some stages the storage volume became empty from its previous full stage. Moreover, as there is no priority among conveyances, the intake volume shows a large fluctuation in spite of a steady water supply. The water demands were almost satisfied with the same fuzzy grade at all the reference points. The least amount of fluctuation in storage and conveyance sequences is possible if the other constraints are introduced. For the scheduling problem, water demands were set at 75% for the first stage and 50% for the second stage against the final ones. Figure 8 shows the construction sequences. Reservoir 1 was constructed with a fuzzy grade of 0.75, and reservoir 2 was constructed to keep the same grade, but using a different water demand. The final grade of 0.75 is different from the planning problem because of different fuzzy membership functions and optimization.

stlsige S I

reservoir 1 /

/ /

/ /

/ / ..--'

^Xi£D/

p \. \ reservoir 2

/ ~XcJtyT)

Fig. 8 Calculated results of construction schedule.

318 Toshiharu Kojiri

CONCLUSIONS

We discussed the planning and scheduling of a water resources system by taking into account future uncertainty. The following results were obtained: (a) the multi-objective plan was formulated using fuzzy membership functions; (b) the optimal plan for a water resources system, where there is future uncertainty due

to natural environment, technological development and human activities, can be formulated through fuzzy linear programming;

(c) reservoir operation was gained through fuzzy linear programming; (d) scheduling for obtaining a water resources system was optimized with the use of

fuzzy dynamic programming and a proposed construction matrix.

REFERENCES

Ikcbuchi, S., Kojiri, T. & Hori, T. (1985) A study on optimal scheduling of construction works for flood control system. Annual Report of DPRI, Kyoto Univ. 28(B-2), 237-252.

Kojiri, T., Sugiyama, Y. & Paudyal, G.N. (1993) Fuzzy reservoir operation with multiobjectiveand few monitoring data. Int. Conf. on Environmentally Sound Water Resources Utilization, 11-280 to 11-287.

Mamdani, E. H. (1974) Application of fuzzy algorithm for control of simple dynamic plant. Proc. IEEE 121(12), 1585-1588.

Zadch, L. A. (1968) Fuzzy Algorithm. J. Inform. <Ê Control. 12, 94-102.