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

نویسندگان

  • TOSHfflARU KOJIRI
  • Toshiharu Kojiri
چکیده

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( ) = 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 () 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: *,•, = / (*„« , ) () 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 = {/W"AM)Mtf-*+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)

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تاریخ انتشار 2008