Using a GAMS Modelling Environment to Solve Network Scheduling Problems

ing and treating water. The formula for pump energy cost includes a pump efficiency factor. The pumping cost also depends on the electricity tariff. There are different pricing options offered by the power utilities, the simplest being a unit electricity tariff where a price is paid for a unit of electrical energy. The tariff is usually a function of time with cheaper and more expensive periods. Sometimes maximum demand tariffs are also applied where a cost is incurred for the peak power consumption within a period (e.g. one month). Clearly, the type of tariff affects the time horizon of the scheduling task. The time horizon is typically 24 hours for unit tariffs, and I month for maximum demand tariffs. Objective functions may also include other cost terms for pump switching, penalties for deviation from the final target reservoir levels, etc. The optimisation problem will be considered over a given time horizon. The operational cost is represented by a unit electricity charge and a unit water treatment cost, but other costs can be added as required: (6) 1. 1 r/J = ~ fy~(t)f/qi(t), d(t))dt + ~ Hit) x rlit)dt jEfp to jEf,. to (I) where [ ~~ ] is a vector of node heads, hf is a vector of heads at reservoir nodes (fixed grade nodes) and he is a vector of heads at connection nodes; q is a vector of branch flows; qf and qs are vectors of reservoir flows and source flows respectively (both occurring at network nodes); d is a vector of nodal demands; and Ae,!''J are node branch incidence matrices for connection nodes and reservoir nodes respectively. The system of equations (2) to (5) can be classified as a system of differential-algebraic equations where equation (2) represents the differential part and equations (3) to (5) are the algebraic part. The vector variable e(t}, which appears in the component equation (3) is a decision variable and represents pump and valve control. The vector of source flows qs in (5) is another decision variable. Most of the components do not include decision variables (eg, equation 6 for pipe sections): Ri I qi(t} 10.852 x qi(l} =hLiit} hjeslt) Vector function hJt}, hf.t}, q(t}, q,(t} are internal variables of the model, vector hf is a differential variable and he' q, qf are algebraic variables. Operational Constraints Transformation of the network scheduling problem into a non-linear programming problem Operational constraints are applied in operational scheduling problems to keep the system-state within its feasible range. Practical requirements are translated from the linguistic statements into mathematical inequalities. The typical requirements of network scheduling are concerned with reservoir levels (water network state variables) in order to prevent emptying or overflowing, and to maintain adequate storage for emergency purposes. Similar constraints must be applied to the heads at critical connection nodes in order to maintain required pressures throughout the water network. The control variables, such as the number of pumps switched on, pump speeds or valve positions, are also constrained by lower and upper constraints determined by the features of the control components. The constraints on the control variable corresponding to water production reflect the properties of the water treatment processes. The constraints may be instantaneous or functional constraints covering a period of time. The functional constraints can represent, for example, total water production, or a limitation on the rate of change in water production. (7) min /11ax hf s hjt} S hf for lE [ to, If 1 Network Model where i p is the set of indices for pump stations and is is the set of indices for treatment works. The term Jj(qJ(t) , cl(t)) represents the electrical power consumed by pump station j. The potential energy of the water is obtained by multiplying the flow (q) and the head increase across the pump station. The consumed electrical power can then be calculated from the pump efficiency. The head increase variable flhi(l) can be expressed in terms of flow in the pump hydraulic equation, so that the cost term depends only on the pump station flow qi(t) and the control variable d(t) as illustrated in equation (1). The d(t) vector represents the nl,lmber of pumps on and/or pump speed. The function yt(t) represents the electrical tariff. The treatment cost for each treatment works is proportional to the flow output with the unit price of yf(t). The ontology of the hydraulic model is formulated in the paper by Ulanicka et a14 . The fundamental requirement in an optimal scheduling problem is that all calculated variables satisfy the hydraulic model equation.s. The network equations are non-linear and play the role of equality constraints in the optimisation problem. It is convenient to use a compact vector-matrix notation for writing down the network equations (see references4&5) as follows: dhf d1 = -S-1 qjt) reservoir dynamics (2) R(q(t), e)q = f,T h(l) component equations (3) Ae q(l) = -d(t) mass balance at connection nodes (4) Afq(l} =qjt} + qit) mass balance at the reservoir nodes (5) The network scheduling problem is represented in Figure I. It consists of searching the space of decision variables to minimise the objective function over a given time horizon, whilst satisfying the constraints. The decision variables are operational schedules e(t) for the control elements (e.g. pumps and valves), and water production schedules qit). For given control schedules (e(t), qit)) the network model equations (2) to (5) can be solved and the values of all internal variables heel), hf.t), Measurement + Control, Volume 32. May 1999 111 Using a GAMS modelling environment Ulanicki et al