**2.2. Constraints**

The main constraint of the model is the hydraulic balance verification for the network. To establish such balance, the equations of the conservation of mass at each junction node and the conservation of energy around each loop in the network are satisfied. In order to these conditions be attended it is necessary to accomplish the hydraulic verifications to each sys‐ tem configuration. The hydraulic simulator EPANET (ROSSMAN, 2000) was used to per‐ form this purpose.

The constraints are implicit in the calculation of the objective function. These are equations that need to be solved in order to obtain the total energy cost of the solution to be analyzed. After accomplishing this stage, some variables are verified, from the hydraulic simulation, aiming for obtaining the hydraulic performance of the system that it is evaluated by means of explicit constraints, showed as follows:

Pressure: for each time-step of operational time, the pressures in all the junction nodes must be between the minimum and maximum limits.

This work aims to present an artificial neural network model by the optimization of the best economical hybrid solution configuration applied to a typical water distribution system.

The search for the optimal control settings of pumps in a real drinking network system is seen as a problem of high complexity, due to the fact that it involves a high number of deci‐ sion variables and several constraints, particular to each system. The decision variables are the operational states of the pumps xt (x 1t, x 2t, …, x Nt), where N represents the number of

To represent the states of the decision variables in each time-step, the binary notation was used. The configuration of each pump is represented by a bit where 0 and 1 stated switch‐ ed on and off, respectively. The main goal of the model is to find the configuration of the pumps' status which proceeds to the lowest energy cost scenario for the operational time duration. To calculate this cost, several variables must be considered, in each time-step, such as the variation of consumption, energy tariff pattern and the operational status of

The objective function is the sum of energy consumed by the pumps, in every operational time, due to the water consumption and tanks' storage capacity. It can be expressed accord‐

where E and C stated the consumed energy (kWh) and the energy costs by pumps' opera‐

The main constraint of the model is the hydraulic balance verification for the network. To establish such balance, the equations of the conservation of mass at each junction node and the conservation of energy around each loop in the network are satisfied. In order to these conditions be attended it is necessary to accomplish the hydraulic verifications to each sys‐ tem configuration. The hydraulic simulator EPANET (ROSSMAN, 2000) was used to per‐

The constraints are implicit in the calculation of the objective function. These are equations that need to be solved in order to obtain the total energy cost of the solution to be analyzed. After accomplishing this stage, some variables are verified, from the hydraulic simulation, aiming for obtaining the hydraulic performance of the system that it is evaluated by means

*CntEnt*(*Xnt*) 1 (1)

pumps and t is the time-step throughout the operational time.

*Minimize* Σ

n=1 N Σ t=1 24

**2. Models formulation**

78 Water Supply System Analysis - Selected Topics

**2.1. Objective function**

each pump.

ing to the following equation:

tion in the time-step t.

**2.2. Constraints**

form this purpose.

of explicit constraints, showed as follows:

$$P\mathbf{min}\_i \le P\_{it} \le P\mathbf{max}\_i \qquad \forall\_{i'} \quad \forall\_{t} \tag{2}$$

where Pit represents the pressure on node i in time-step t, Pmin <sup>i</sup> and Pmax <sup>i</sup> are the mini‐ mum and maximum pressures required for node i.

Levels of storage tanks: The levels of storage tanks must be between the minimum and max‐ imum limits for each time-step. Besides at the end of the operational time duration, they must be superior to the levels at the beginning of the time duration. This last constraint as‐ sures the levels of the tanks do not lessen with the repetitions of the operational cycles.

$$\mathsf{Smin}\_{j} \leq \mathsf{S}\_{jt} \leq \mathsf{Smax}\_{j} \forall \ \_{j} \forall \ \_{t} \tag{3}$$

where Sjt: level of tank j in time-step t; Smin <sup>j</sup> e Smax <sup>j</sup> : minimum and maximum levels of storage tank j.

$$S\_j(24h) \triangleleft S\_j(0h) \forall \\_j \tag{4}$$

Pumping power capacity: the power used by each pump during the operational time must be inferior to its maximum capacity.

$$PP\_{kt} \leq PP \max\_{k} \forall\_{k} \tag{5}$$

where PPkt : used power by pump k in time-step t; PPmaxk: maximum capacity of the pump k.

Actuation of the pumps: The number of pumping start-ups in the operational strategy must be inferior to a pre-established limit. This constraint, presented by Lansey and Awumah (1994), influences in the maintenance of each pump, since the more it is put into action in a same operational cycle, the bigger will be its wear. Lansey and Awumah (1994) suggest the maximum pump start-ups 3 in 24 hours. A greater value can cause problems on the pumps inducing the need of maintenance and repair and consequently the interruption of the sys‐ tem operation.

$$NA\_k \le NA\max\_k\tag{6}$$

where: NAk represents the number of start-ups for pump k and NAmaxk the maximum al‐ lowable pump start-ups for the pump k.
